[studies in fuzziness and soft computing] formal languages and applications volume 148 ||

2

Formal Languages: Foundations, Prehistory, Sources, and Applications

Solomon Marcus

Romanian Academy, Mathematics Calea Victoriei 125" Bucure§ti, Romania E-mail: solomon.marcusClimar.ro

2.1 Language as a Universal Paradigm

Arguments are brought in favor of the universality of the language paradigm. In this respect, it is essential to take in consideration the fact that language is at the crossroad of linguistics, semiotics, biology, psychology, psychiatry, logic, art, literature, information and communication theory, computer science, artificial intelligence etc. The history of the slogan "linguistics as a pilot science" is analyzed and arguments are brought in favor of its replacement with the slogan in the title above.

Language is recognized now as a universal paradigm, like "space", "time" or "logic". However, when we try to detail this fact, we are faced with some difficulties. This happens because we are far from a generally accepted definition of language, separating clearly its denotative from its metaphorical use. For instance, Pescador (1986-1989: 36) considers that "in generic terms, one speaks of a language any time when there is a plurality of signs of similar nature whose primary function is the communication between organisms". So, Pescador limits language to the organic world, but his examples include also "the language of colors". The same author seems to make no difference between the proper and the figurative way to consider a language, because his list includes "the languages of different species of animals", "the language of music", "the language of gestures" , "the language of painting" , and "the language of flowers". Then, what kinds of sign systems don't have the status of a language? Otherwise, many authors (among them, Roman Jakobson and Noam Chomsky) accept, more or less explicitly, the existence of non-human languages. According to another line of thinking, language is "uniquely human" (see Miller 1991: 260). Similarly, Sebeok (1995) considers language as belonging exclusively to the genus homo: "a distinction is maintained among three commonly confounded notions: (1) communication (or, technically, semiosis), which is a criterial attribute of all living creatures [ ... J; (2) language, a criterial

C. Martín-Vide et al. (eds.), Formal Languages and Applications© Springer-Verlag Berlin Heidelberg 2004

12 Solomon Marcus

attribute of all 'normal' members solely of all species in the genus homo; and (3) speech, one of the linear manifestations of language, occurring in most but not all members of our species homo sapiens."

Other ways to use "language" cannot be ignored. In some parts of his work, Lyons (1977) uses this term with its meaning in formal logic: a set of finite strings over a given finite non-empty alphabet; in other parts of the same work, "language" means "ordinary language" or "natural human language" , in contrast with "constructed (artificial) languages". Another attitude belongs to an author interested in programming languages. His name is Higman (1967: 7) and he proposes a definition of language stressing its discrete-linear structure and its communicational function: "A precise definition of language is an elusive thing, but we come fairly close to one if we say that a language consists of a set of objects called its vocabulary, which can be combined into linear strings in accordance with certain rules known as its grammar, for communication to a recipient with the intention of inducing activity in the recipient relative to certain specific features abstracted from a general situation" .

Language is recognized as a pattern under the control of the left hemisphere of the brain, where already in 1861 Pierre Paul Broca localized speech (see, for instance, Miller 1991: 263). In a more recent variant, left hemisphere is specialised in coping with sequential processes, to which the definition of a formal language makes explicit reference. "Sequential" is the discrete variant of "linear", to which Sebeok makes reference when discussing on speech. Strings are representations of sequential structures and occur in various fields. DNAs and RNAs in molecular genetics are strings over the alphabet of nucleotide bases, proteins are strings on the alphabet of amino acids, English words are strings on the alphabet of English, English statements are strings on the vocabulary of English, any input and any output in a computational process are strings on the alphabet of a programming language, statements in propositional calculus or in the predicate calculus are strings on their respective alphabets. In Marcus (1969, 1974) we tried to argue in favor of the old slogan "linguistics as a pilot science" proposed by Claude Levi-Strauss, but, as a matter of fact, the arguments we brought were in favor of the slogan "formal linguistics as a pilot science" . The pattern called formal language has all features of a universal paradigm: it expresses a biological reality, the sequential structure of processes under the privilege of the left hemisphere of the brain; equally sequential are the basic life processes related to DNA, RNA and proteins; moreover, as it was proved in a lot of papers by various authors, strings on a given alphabet occur in the mathematical modeling of some basic operations in logic, combinatorics, quantum physics, organic chemistry, cellular biology, computer programming languages, linguistics (mainly computational linguistics), anthropology of kinship relations, medical diagnosis, tennis game, international relations, musical composition, painting, architecture, poetics, theatrical plays, narrativity, etc.

If "formal" is a necessary condition of universality, can we still claim for "language" understood as "ordinary language", Le., as "natural human lan-

2 Formal Languages: Foundations, Prehistory, Applications 13

guage" , the status of a universal paradigm? The answer is affirmative, because the language of the everyday life is the term of reference, the intuitive basis and the main source of problems and concepts in the study of any other language considered by scientists or by artists. Natural language is, in respect to any other language, what the Euclidean space is in respect to any other space imagined by human beings in their creative work. It is not by chance that the generative grammars by means of which we investigate programming languages are just the same devices introduced initially by Noam Chomsky in respect to the needs of linguistics. It is not by chance that the same semiotic dimensions proved to be relevant in the study of natural languages, the syntactic, the semantic, and the pragmatic ones, show their relevance in the study of any sign system, be it natural or cultural, scientific or artistic. Particularly, the study of computer programming languages is essentially based on this triad. We are so deeply motivated to consider natural human language as a universal paradigm.

Now, accepting language as a universal paradigm, are we motivated to infer for linguistics the status of a pilot science? It is the moment to recall that many authors argued, beginning with the sixties, against this special status of linguistics. Let us refer, for instance, to Greenberg (1970), whose article has the same title as our articles (Marcus 1969, 1974): Linguistics as a pilot science. Grenberg recalls that already in the XIXth century, long time before the rise of structuralism, "linguistics furnished the basic model of a science which by the use of a comparative method obtained significant historical results" and he observes that in historical linguistics, "the acceptance of evolutionary explanations preceded that in biology by about half a century". In the XXth century, when linguistics, directed towards the internal structure of languages, proved a level of rigor higher than in any other human science, continued to have a special methodological role. However, the first failures appeared in the fifth and the sixth decades of the XXth century, when some human and social fields "might be advanced by the analysis of the language employed in the science itself'. Later, the strategy changed. Many authors looked for the cultural equivalent of the phoneme or of the morpheme, checking various procedures equivalent to phonemic or to morphemic segmentation. Some of these attempts succeeded, other attempts failed. Such failures, mainly related to poetics, anthropology, and psychoanalysis, were analysed in details by Pavel (1988). Greenberg proposes another explanation, of an apriori nature, of the unavoidable failure of linguistics in its capacity to become a model for other fields: "[ ... ] language as a subject matter possesses certain peculiarities such as the arbitrariness of the relation between form and meaning [ ... ]", while in certain other fields "[ ... ] the relationship between form and meaning is far less arbitrary than in linguistics." But, on the other hand, Greenberg accepts the methodological power of a distinction such as "emic-etic" , suggested by "phonemics-phonetics", and of a device such as the Chomskian generative grammar, which is a new way to consider language as a human competence and so, by extension, to learn to question, for any other

14 Solomon Marcus

human activity, what kind of "machine" is behind it. Greenberg (as well as Pavel later and as well as some other authors) had in view some possible direct transfers from linguistics into other fields. In contrast with them, the transfers we have considered in Marcus (1969, 1974) were operated via some formal languages. The last decades brought new arguments in favor of the capacity of language to be a universal paradigm. Let us refer to the idea of a text, which was initially related to natural language, but which became much more comprehensive, partly by its metaphorical virtue ("the world as a text"), partly by its extrapolation in apparent completely heterogeneous fields (for instance, in theoretical computer science).

Moreover, by means of various prefixes, a lot of new concepts were introduced, such as "context", "cotext", "subtext", "intertext", "paratext" and mainly "hypertext", which became the start of a new chapter in the field of artificial intelligence and of literary analysis, transgressing the linearity of most linguistic and literary representations. Another step forward came from the new field of DNA computing, where the language representation of DNAs is bridging two apparently very different activities, such as computation and heredity. A third example is that of metaphorical processes. Born in direct link with figurative expressions in natural languages, metaphor became a universal procedure to generate meaning, not only in natural languages but in any sign system. Cognitive and creative metaphors are considered now a general procedure to create meaning and a fundamental aspect of human behavior. These metaphors belong now to the common denominator of science, art, religion, and philosophy. A fourth example is related to the expansion of the grammatical paradigm, under the aspect promoted by the generative theory in the study of natural languages, a theory showing its relevance in a lot of domains of science or of art. The list could continue. What is shown by all these examples? Despite some failures (only one example: some trends, in the years 1960 and 1970, to reduce semiotics to linguistics), the universality of the language paradigm is more and more beyond any doubt. But in order to understand and explain this phenomenon it is necessary to adopt a transdisciplinary perspective, taking into account that language is at the crossroad of linguistics, logic, psychology, art, literature, computer science, information and communication theory, biology, anthropology, sociology, psychiatry, philosophy, semiotics etc. Far from being equivalent to "linguistics as a pilot science", "language as a universal paradigm" expresses an important aspect of today culture. Linguistics alone cannot account for the universality of the language paradigm; it needs the help of some other fields where language aspects are essential.

References

1. J.H. Greenberg (1970) Linguistics as a pilot science. In J.H. Greenberg, N.A. Mc Quown, M. Halle, W. Labov, Linguistics in the 1970's. Center for Applied Linguistics, Washington D.C.

2. B. Higman (1967) A Comparative Study of Programming Languages. New York: American Elsevier.


3. J. Lyons (1977) Semantics, vol. I. Cambridge: Cambridge University Press. 4. S. Marcus (1969) "Lingvistica, stiin~a pilot". Studii §i Cercetif.ri Lingvis

tice, 3, 235-245. 5. S. Marcus (1974) "Linguistics as a pilot science". Current Trends in Lin

guistics (ed. T.A. Sebeok), 12, The Hague: Mouton, 2871-2887. 6. G.A. Miller (1991) The Science 0/ Words. New York: The Scientific Amer

ican Library. 7. T.A. Pavel (1988) Le mimge linguistique. Paris: Editions de Minuit. 8. J.H.S. Pescador (1986-1989) Principios de filosofia dellenguaje. Madrid:

Alianza. 9. T.A. Sebeok (1995) Brief remark about the universe. The evolution of

communication and the origin of language. Seminar. Budapest: Collegium Budapest.

2.2 Formal Languages and the Artificial-Natural Controversy

Any claim for a sharp opposition between artificial and natural languages has to face some counterexamples. There is a general trend of attenuation of these oppositions. Within artificial languages, 'formal' and 'non-formal' are in a similar situation. If the propositional calculus and the predicate calculus are purely formal languages, we cannot make a similar claim for the language of mathematics (which has a mixed structure, with a natural and an artificial component, the latter being only partially formalized) or for the computer programming languages, sharing features with both natural and artificial languages and with both formal and non-formal languages.

Our attention will focus on the elementary structure called "formal language", defined, as we have seen, as a set of finite strings over a finite nonempty alphabet. Its relations with the general, but controversial structure called "language" was explained in the preceding section. To be formal is both an advantage and a shortcoming. It is an advantage because, as a mathematical object, it can be approached in a rigorous way and it can take profit of a variety of mathematical methods, some of them already elaborated in logic, set theory, combinatorics, algebra, number theory, topology, probability theory etc., some of them deliberately introduced in order to approach the problems raised by formal language theory. For instance, a large part of automata theory was developed in order to solve problems raised by formal grammars and languages.

Formal languages are a special type of artificial languages, which can be defined as languages which are deliberately created in order to approach a definite area of problems. For instance, espemnto was created by L.L. Zamenhof

16 Solomon Marcus

more than one hundred years ago, in order to facilitate international communication. Lincos was created in the sixties (XXth century) by Hans Freudenthal (1960) in order to make possible cosmic communication, i.e., communication with hypothetical intelligent beings from other celestial bodies. These examples show that only some artificial languages are formal languages. Esperanto, made of pieces of natural languages, is not a formal language, while Lincos, as it was described by its author, is a formal language, where we can identify a basic set of primitive signs (the alphabet) and some rules to form well formed strings. Artificial languages have, generally, a precise date of birth and a (some) precise author (authors). In opposition with them, natural languages are the result of a long evolution, with no precise date of birth; only an approximate period can be indicated, to not speak about the usual controversies related to this evolution. If natural and artificial languages clearly differ in respect to their history (the oldest known artificial languages belong to the XVIth century), a more delicate problem is to what extent do they differ in respect to their structural properties.

So far, we had to distinguish between human and non-human languages. The latter may belong to the living or (at least for some authors) to the inert universe, while the former may be natural or artificial. In their turn, artificial languages may be formal or non-formal. Before going further, we may question the binary nature of the considered distinctions. We are now especially concerned with the formal-non-formal distinction and it is important to warn the reader about the existence of some important languages which are neither purely formal, nor purely non-formal: the clearest example in this respect are the computer programming languages.

Various structural properties were proposed as specific distinctive features of natural languages. Greenberg's above discussed proposal is one of them. We discussed in some details this problem in Marcus (1996: 114-131). Here we give only an abstract of this topic.

We call attention on the fact that the meaning of the word 'artificial' in the syntagm 'artificial language' is different from its meaning in the syntagm 'artificial intelligence'; in the former, the reference is to a language directly produced by humans, while in the latter we mean the intelligence directly produced by machine (and only indirectly by humans).

A first claim asserts that infinity is a specific property of natural languages. There are many ways to argue that natural languages are potentially infinite (for instance, constructions such as coordination, repetition, insertion of relative clauses or if-then constructions can be iterated as many times as we want). However, the formal language of statements in predicate logic is also infinite and the same is true for the programming languages such as Algol, Fortran, Lisp, Prolog which are infinite in their general competence. As a matter of fact, any human language of enough high complexity is infinite. A generally accepted conjecture asserts that infinity is a privilege of human languages.


A second claim asserts that the fuzziness of the well-formedness property is a specific feature of natural languages. There is no necessary and sufficient condition for a string in a natural language to be well-formed. But this situation occurs also in programming languages, because there is no necessary and sufficient condition for a finite sequence of instructions to be a computer program; see, for instance, Moreau (1968).

A third claim asserts that the integrative nature of semantics is specific to natural languages, in contrast with the additive nature of semantics in artificial languages. This means that in artificial languages the meaning of a well-formed string is obtained by concatenation of the meanings of its terms, while this is generally no longer valid in a natural language. Indeed, natural languages are rich in idiomatic expressions such as 'it rains with cats and dogs' in English, or 'sin embargo', in Spanish. However, contrary to expectations, advanced artificial languages are also rich in idiomatic expressions. For instance, in the artificial component of the mathematical language the sign of the integral of a function on a real interval has a meaning which is very far from the concatenation of the meanings of the components of this sign. Similar 'idiomatic expressions' occur also in some programming languages; see Calude (1976). The motivation of such sign is usually of iconic or indexical nature.

A fourth claim asserts the existence of morphology as a specific feature of natural languages (although some natural languages, the so-called isolating languages, such as Chinese, have a very poor morphology), in contrast with formal artificial languages, where only syntax is considered. As a counter example to this claim, see the situation in Fortran IV (Steriadi, 1974) and in Assembler-360 (Calude, 1973).

A fifth claim asserts the presence of ambiguities as a specific feature of natural languages. There are however ambiguities in both the mathematical language and the programming languages (see S. Marcus (ed.), 1981, 1983 for contextual ambiguities and Culik II (1973) for the inherent ambiguity of programming languages.

A sixth claim refers to the duality of patterning as a possible specific feature of natural languages (Martinet, 1948). However, the language of DNAs is endowed with the same duality, although the way we should interpret it is controversial (Marcus, 2004). We take the level of codons as the first articulation and the level of nucleotide bases as the second articulation. 'Meaningful' means 'with genetic meaning', while 'meaningless' means 'with chemical meaning only'.

A seventh claim asserts the specific of natural languages to can be their own metalanguage. However, Smullyan (1957) has indicated a class of languages where self reference is possible and Freudenthal's Lincos has the same capacity.

Other claims refer to: the use of some artificial languages beyond their initial purpose, their poly-functionality and, even, quasi-universality; the presence of phenomena of imprecision related to randomness, fuzziness, roughness, genericity, indecidability, indetermination etc. in many artificial languages;

18 Solomon Marcus

the impossibility to use (learn) a language before learning (using) it; the primordiality of the spoken form in any natural language, in contrast with the primordiality of the written form in most artificial languages; the tendency to develop rhetorical figures and connotative meanings in both natural and artificial languages.

A history of artificial languages was proposed by Pei (1974). A possible relation between Esperanto and some programming languages was discussed by Naur (1975).

References

1. C. Calude (1973) Sur quelques aspects morphologiques dans Ie langage Assembler-360. Cahiers de Linguistique Theorique et Appliquees, 10, 2, 153-162.

2. C. Calude (1976) Quelques arguments pour Ie caractere non-formel des langages de programmation. Revue Roumaine de Linguistique, Cahiers de Linguistique Theorique et Appliquees, 13, 1, 257-264.

3. K. Culik II (1973) A model for the formal definition of programming languages. International Journal of Computer Mathematics, 3, 315-345.

4. H. Freudenthal (1960) Lincos. Design of a Language for Cosmic Intercourse. Amsterdam: North Holland.

5. S. Marcus (1996) Language, Logic, Cognition and Communication. Universitat Rovira i Virgilio Research Group on Mathematical Linguistics, Report 9/96, Tarragona, Spain.

6. S. Marcus (2003) The duality of patterning in molecular biology. In Eds. N. Jonoska, Gh. Paun, G. Rozenberg, Aspects of Molecular Computing. Essays in Honor of the 70th Birthday of Tom Head, LNCS 2933, Berlin: Springer, 318-321.

7. S. Marcus, ed. (1981,1983) Contextual Ambiguities in Natuml and in Artificial Languages, 2 volumes. Ghent, Belgium: Communication and Cognition.

8. A. Martinet (1968) Le lang age. Paris: Gallimard. 9. R. Moreau (1968) Langages naturels et langages de programmation. In

Linguaggi nella societa e nella tecnica, Milano: Comunita, 303-324. 10. P. Naur (1975) Programming languages, natural languages, and mathe

matics, Communications of the ACM, 18, 12, 676-683. 11. M. Pei (1974) Artificial languages: international, in Th.A. Sebeok, ed.

(1974), 999-1017. 12. Th.A. Sebeok, ed. (1974) Current Trends in Linguistics 12, The Hague:

Mouton. 13. R.M. Smullyan (1957) Languages in which selfreference is possible. J. of

Symbolic Logic, 22,55-67. 14. M. Steriadi (1974) Morphological aspects in Fortran IV (in Romanian).

Studii §i CerceUiri Matematice, 26, 5, 755-761.


2.3 Explicit and Implicit Formal Languages

The theory of formal grammars and languages is born in three steps: Chomsky (1956) proposes a hierarchy of generative devices claiming a new way to look at the syntax of natural languages, according to which linguistics becomes a chapter of cognitive psychology; Ginsburg and Rice (1961) show that Chomsky hierarchy of grammars is able to cope with the syntactic and the semantic problems of programming languages; Salomaa (1973) gives the first presentation, both accurate and complete, of formal languages as a chapter of theoretical computer science.

The explicit beginning of formal language theory had three steps: the year 1956, when Noam Chomsky published his pioneering paper "Three models for the description of language" (IRE Transactions on Information Theory, IT-2, 1956,3, 113-124); the year 1961, when S. Ginsburg and H.G. Rice publish the paper "Two families of languages related to Algol" (Technical Memo 578/000/1, SDC, Santa Monica, Calif. July 1991), and the year 1973, when Arto Salomaa publishes his book Formal Languages (New York: Academic Press, 1973). Chomsky is the author of the famous generative hierarchy of languages, changing radically the way to look at natural languages. Ginsburg and Rice called attention on the fact that Chomsky hierarchy is relevant not only for natural languages, but for programming languages too. With them, the theory of programming languages becomes almost identical to the theory of generative formal grammars and languages. Salomaa is the first author of an accurate mathematical presentation and clarification of the whole theory, after successive presentations either incomplete or with many obscurities or mistakes. As a matter of fact, Chomsky's 1956 paper, as most pioneering papers, is full of local mistakes, although revolutionary in its main ideas.

If the explicit status of the theory of formal languages was obtained in a fight covering 17 years, the implicit involvement of formal languages in a large variety of situations can be observed beginning with the end of the XVIIIth century. A period of more than 150 years is full of objects whose status, in a today reading, is that of a formal language. Like Monsieur Jourdain, the famous character of Moliere, who was not aware of the nature of his action, many scholars in various fields were using a formal languages it la Jourdain, which are waiting to be discovered and made explicit. Some attempts in this respect were already successful and we will point out some of them. They belong to the prehistory of the theory of formal languages. A good knowledge of this prehistory is a condition to understand the history of this field and its today situation.

20 Solomon Marcus

2.4 Gauss Languages

This is perhaps the chronologically first example of implicit involvement of a formal language structure into a combinatorial problem. The itinerary from syntactic combinatorics to formal languages transforms an individual problem into a global one, a finite entity into a potentially infinite structure and a descriptive analytic approach into a generative one.

One of the oldest problem leading to a formal language belongs to Gauss (1777-1855). Consider a planar closed curve with simple crossing points. (A point is simple if it is not a tangent point and the curve crosses itself only once at that point.) Assign the numbers 1,2, ... , n to the n crossing points of a given curve c. A sequence x(c) containing exactly two occurrences of each i (i between 1 and n) and describing the passing of the curve through the crossing points is called a Gauss code (of the curve c). Gauss (see pp. 272 and 282-286 of Werke, Teubner, Leipzig, 1900) proposed to characterize Gauss codes in terms of interlacement properties of their symbols. He gave a necessary syntactic-like condition for a string to be a Gauss code. Besides a topological and a graph theoretical solution, there is a syntactic solution too, of the type: a word x is a Gauss code iff it contains no subwords of the form ....

For the bibliography related to these results, see Kari et al (1992), where the problem is further investigated by transforming Gauss combinatorial problem into a generative one. The itinerary from combinatorics to formal languages is at the same time an itinerary from individual to global, from descriptive to generative, and from finite to infinite. This itinerary can be applied to any combinatorial problem which is of a syntactic nature.

Let us consider paths of arbitrary finite lengths along a sccc (closed curve with simple crossing points). Describe such a path by the sequence of visited pointsj call such a sequence a weak Gauss code. Denote by WG(c) the set of all weak Gauss codes associated to the curve Cj WG(c) is an infinite language and WG(c) = mi(WG(c)) for any sccc c, where mi denotes the mirror image operationj for any sccc c, the language WG(c) is regular. For any sccc, another language can be considered too, taking paths along c, but permitting returnings along segments, not on intersection points. We can go freely forward and backward on c. Call such strings double weak Gauss codes and denote by DWG(c) the language associated in this way to c. DWG(c) includes WG(c), hence DWG(c) is infinite, and DWG(c) = mi(DWG(c)) for any sccc c. If cis a sccc with at least two intersection points, then WG(c) is strictly included in DWG(c). DWG(c) is regular for any sccc c. The Gauss criterion is a necessary condition for a string to be in DWG(c). Given n points, consider all planar closed curves which cross arbitrarily many times in these points, in the sense that each passing through a point intersects all other passings of the curve through that point (no two curve branches are tangent in a crossing point). Denote by SG(n) the set sub{x I x is a Gauss code of a curve passing arbitrarily many times through 1,2, ... , n}. We call such strings semi-Gauss codes.


All languages SG(n) are infinite, while for n = 1 we get a regular language. The further investigation of these notions is done in L. Kari, S. Marcus, Gh. Paun, and A. Salomaa, In the prehistory of formal language theory: Gauss languages. Bull. EATCS, 46, 1992, 124-139

2.5 Langford Languages

Three hyposthases of mathematics meet here, although each of them developed on its own: the purely combinatorial-mathematical investigation, the game aspect, the practical problems. This is one more example of how a syntactic combinatorial problem Can be transformed into a generative one, replacing the finite by the infinite, the individual by the collective, and the analytic-taxinomic approach by a generative one.

Implicitly introduced by Netto (1901), explicitly by Langford (1958), as a game about colored bricks: given the symbols 1,1,2,2, ... , n, n, can we arrange them in a sequence such that the two 1 's be separated by one symbol, the two 2's by two symbols and so on, the two i's by i symbols?

Example for n = 4: 41312432. Generalization by considering 3,4, ... , m copies of each symbol. This problem proved to be very hard in the general case; see Davies (1959, 1982). The number of known Langford strings for given n is finite. Langford claims that this problem was suggested to him by his child, who, playing with colored bricks, showed some clear preference for some arrangement. Trying to discover its secret, Langford replaced red by one, blue by two, yellow by three, and green by four and so he introduced his strings.

Started in the framework of combinatorics, continued as a game, Langford strings were later identified as a very useful tool in the construction of noise resistant codes (Eckler 1960) and of Steiner systems (Alekseev 1967). Denote by L( m, n) the language of all Langford strings over an ordered alphabet of n elements, where each element of the alphabet occurs exactly m times. Weakning some conditions, we get weak Langford strings, semi-Langford strings or their combinations. See, for details, Marcus, Paun (1989), where some open problems are also proposed.

Besides their interest as a purely mathematical problem, Langford strings develop a capacity of modeling the real world. For instance, there is a need to create a distance between iterative uses of the same word. But the requirements to avoid repetitions are weaker for short words like a, an, and, at, in, on, to, than for nouns, adjectives, and verbs. Rhyme and other prosodic restrictions are of Langford type. Pieces in a chess game, in their initial position, are near to Langford's rule: pawns are at distance zero, bishops at distance 2, knights at distance 4, and rooks at distance 6. Given a palindromic string s where no element from the alphabet appears twice, the concatenation of s

22 Solomon Marcus

with its reverse leads to a string u where each element in s appears exactly twice and, if we denote by 2n the length of s, between the element of rank i and its second occurrence in u there are 2(n - i) intermediate elements. If the length of u is odd, then there are between the same elements 2(n - i) + 1 intermediate elements. No (m, n)-Langford string exists if m is strictly larger than n. It follows that L(n), the union of all L(m, n), for m = 1,2,3, ... is a finite language for any nj see for more details Gilespie and Utz (1966).

References

1. V.E. Alekseev (1967) On the Skolem method of construction of Steiner systems of triples. Mat. Zametki, 2, 145-156.

2. RO. Davies (1959) Problem. Math. Gazette, 43, 253-255. 3. RO. Davies (1982) Hayasaka and Saito build tall towers. Univ. of Leices

ter, Depth. of Math. 4. A.R Eckler (1960) The construction of missile guidance codes resistant

to random interference. Bell Syst Techn. J., 39, 973-974. 5. F.S. Gilespie, W.R Utz (1966) Problem. Fibonacci Quarterly, 4,184-186. 6. S. Marcus, Gh. Paun (1989) Langford strings, formal languages and con

textual ambiguities. Intern. J. of Computer Math., 26,3-4, 179-191.

2.6 Unavoidable Patterns in Infinite Words

Most presentations of formal languages indicate as a first author in this field the name of Axel Thue (1906, 1912). As a matter of fact, the object of his interest was the identification of some unavoidable patterns in infinite words. For instance, there exists no square free infinite binary word, but there exists a cube free infinite ternary word.

The study of infinite words is older than the study of formal languages and these two lines of research developed until the nineth decade of the XXth century rather independently each other. Thue (1906, 1912) is considered the initiator of the theory of avoidable and unavoidable patterns in infinite words. Does there exist an infinite word over a given finite alphabet A which avoids a certain pattern, i.e., does not contain as a factor any word of the form of the pattern? The results in this respect depend on the size of A. From Thue we learn the first results in this respect: there exists an infinite square free word if A has three elements, but there exists no such infinite word if A has only two elements. More generally, for each i = 1,2,3 there exists a pattern P( i) which is unavoidable if A has exactly i elements, but avoidable if A has exactly i+ 1 elements. The situation for i larger than 3 is still a challenge. Here are some bibliographic indications about infinite words, in chronological order: M.E. Prouhet, C. R. Acad. Sci. Paris, 33, 1851j A. Thue, Uber unendliche Zeichenreihen, Norske Vid. Selsk. Skr. I Mat. Nat. Kl., Kristiania, 7, 1906, 1-22j A. Thue, Uber die Gegenseitige Lage gleicher Teile gewisser Zeichenreihen,


Norske Vid. Selsk. Skr., I Mat. Nat. Kl., Kristiania, 1, 1912, 1; M. Morse, Recurrent geodesics on a surface of negative curvature. Trans. AMS, 22, 1921, 84-100; S.E. Arson, Proof of the existence of n valued infinite asymmetric sequences, Math. Sbornik, 2 (44), 1937, 769-779; M. Morse, A solution of the problem of infinite play in chess, Bull. AMS, 44, 1938, 832; M. Morse, G. Hedlund, Symbolic dynamics, Amer. J. of Math., 60, 1938, 815-865. The study of words in relation with the theory of groups or semigroups emerged in the second half of the XX-th century: R.C. Lyndon, M.P. Schiitzenberger, The equation am = bncp is a free group. Michigan Math. J., 9, 1962, 289-298; W. Magnus, A. Warrass, D. Solitar, Combinatorial Group Theory, New York: Wiley, 1966; R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory, Berlin: Springer, 1977; M.P. Schiitzenberger, P.S. Novikov, S.l. Adian, The Burnside Problem and Identities in Groups, Berlin: Springer, 1979. A basic source for the whole problem of infinite words is M. Lothaire, Combinatorics of Words, New York: Addison-Wesley, 1983. More recent presentations are those by C. Choffrut and J. Karhumaki in voU of Handbook of Formal Languages (eds. G. Rozenberg, A. Salomaa), Berlin: Springer, 1997, and by J. Berstel and J. Karhumaki in Bull. EATCS, 79, 2003, 178-228.

2.7 Linguistic Roots of Formal Languages

Beginning with the XIX-th century, the idea to bridge linguistics with mathematics emerged step by step, mainly in two ways: the quantitative approach, leading to quantitative and statistical linguistics, and the qualitative, structural, formal approach, leading to structural linguistics and to algebraic linguistics. This trend was a reply to the previous period, when comparative-historical aspects prevailed in the study of languages, at the expense of analytic, structural methods permitting to deepen the internal, intrinsic properties of language and to introduce some rigor in linguistic conceptualization and taxinomy.

Linguistics showed always two apparently contradictory faces: one of them oriented towards physics, logic, computation, mathematics, and engineering; the other directed towards life sciences, social sciences, and the humanities. At various moments, one or the other face was brought in the main attention, but a careful examination shows that both of them are very important and any attempt to make of them a conjugate pair, where each of them can develop only at the expense of the other, fails. As a matter of fact, the two faces need each other.

The idea to bridge probability and language was argued already in 1847, by V.l. Buneakovski and developed later by A.A. Markov, who introduced his chains with motivation coming from linguistic phenomena (alternation of vowels and consonants in poetry). On the other hand, already Newton and Leibniz developed and transformed traditional approach to the linguistic sign.

24 Solomon Marcus

Newton focused on the phonetic side of language, paying attention to the continuous aspect of information; the meaning of words, in the framework of a semantic field, changes continuously. Leibniz is mainly interested in what we call today the discrete approach. The continuous-discrete distinction is fundamental in Ferdinand de Saussure's conception on language, claiming for a discrete approach as the only one able to transform linguistics in a science. According to this idea, the itinerary of linguistics in the XX-th century was one from phonetics to phonemics. This became one of the main aim of structurallinguistics, started by F. de Saussure, who already in 1894 wrote: "Les quantites du langage et leurs rapports sont rt1gulierement exprimables dans leur nature fondamentale par des formules mathematiques" and: "L'expression simple sera algebrique ou elle ne sera pas". Later, in 1911, always in respect to linguistics, he wrote: "On aboutit a des theoremes, qu'il faut demontrer". Concomitantly, Baudoin de Courtenay states, in 1909, that linguistics needs to move from continuity to discreteness in a way similar to that in which mathematics moves from infinite sets to countable sets. The need of collaboration between mathematics and linguistics is stressed by Emile Borel, at the 4th International Congress of Mathematicians (1909), while later, in 1943, Jacques Hadamard sees linguistics as a bridge between mathematics and the humanities. For all these references, see Roman Jakobson's Preface to "Structure of language and its mathematical aspects" (American Math. Soc., 1961), where the theory of recursive functions, automata theory, algebra, topology, communication theory, and probability theory are indicated as the main branches of mathematics relevant for linguistics. (For the continuous-discrete distinction, see also S. Marcus, Discreteness versus continuity; competition or cooperation? in Essays in poetics, literary history and linguistics (eds. L. Fleishman et al.), OGI Moscow, 1999, 659-665). The Prague Linguistic Circle, in the third and the fourth decades of the XX-th century, the Danish glosematics, with Louis Hjelmslev (who proposed algebra as a model for linguistic description), in the fifth and the sixth decades of the XX-th century, and the American structuralism, with the axiomatic approach by L. Bloomfield and B. Bloch, continued by the school of distributional descriptive linguistics of Z Harris and Ch. Hockett bringing linguistics very near to algebra of free semigroups, all of them prepared the way for the development of mathematical linguistics.

2.8 From Formal Systems to Formal Grammars

We follow the itinerary from Thue to Hilbert, from Hilbert to Post, and from Post to Chomsky.

A semi-Thue system T = (A, R) consists of a finite alphabet A and a part R of the cartesian product A * x A *, where elements in R are said to be rules in T. Given two finite words p and q over A, we say that q directly derives


from p in T if there exists words u, v, y, and z such that (u, v) is a rule in T and p = yuz, q = yvz. Define a derivation as the reflexive and transitive closure of the relation of direct derivation. The word problem for T is: Given any two words p and q over A, decide whether q derives from p in T.

The semi-Thue system is a Thue system if its relation R is symmetric, i.e., if (u, v) is a rule in T, then so is (v, u). In this case, the derivation relation is a congruence of the word monoid A * , hence each Thue system corresponds to a finite presentation of a semigroup S(T) = Aj u = v for (u, v) in R, where x = y in S(T) iff y derives from x and x derives from y in T. From this it follows that the word problem for Thue systems as well as for semi-Thue systems is undecidable (T. Harju, J. Karhumaki, Morphisms, p. 445-446, vol. 1 of Handbook of Formal Languages (eds. G. Rozenberg, A. Salomaa), Berlin: Springer, 1997).

Hilbert introduces his formal systems in the framework of a theory of proof, requested by the difficulties which appeared in connection with the crisis of the principle of non-contradiction. A formal system S is given by a finite alphabet A, a language L over A, where L is the union of two disjoint languages M and N over A, where M is the set of terms and N is the set of relations, and a subset T of N, where the elements ofT are said to be theorems. As an example, let us consider A = {I, <, >, =}, M = {n repetitions of Ij n = 0,1,2, ... }, N = {u < v, u = v, u > Vj u is equal to m repetitions of I, v is equal to n repetitions of I,m,n = 0,1,2, ... }j T is that part of N where m is strictly smaller than n in the relation <, m is equal to n in the relation =, and m is strictly larger than n in the relation >.

In a scientific theory F we distinguish some concepts, some statements and, among statements, some true statements. An interpretation of a formal system S into a scientific theory F is a mapping from S into F, where to any term in S corresponds a concept in F, to any relation in S corresponds a statement in F and to any theorem in S corresponds a true statements in F. For instance, if F is the arithmetics of positive integers, then F is an interpretation of the above formal system (A, M, N, T), where terms correspond to positive integers, relations correspond to all possible inequality or equality relations between positive integers, irrespective their truth value, while theorems correspond to true inequalities and equalities between positive integers.

The above presentation introduces the theorems in a descriptive way that does not allow to see how theorems are obtained. In order to obtain the theorems, Hilbert considers a subset B of N whose elements are called axioms and a finite set R of rules transforming in a mechanical, algorithmic way, some relations into some other relations. A demonstrative text in the formal system S is a finite sequence of relations such that any relation in the sequence is either an axiom or it is obtained from the preceding relations by applying some rules in R. Any relation that appears in a demonstrative text is said to be a theorem. It follows that axioms are a particular case of theorems.

When a formal system S is interpreted into a scientific theory F, we say that F is a model of the system S and that the theory F has been formalized

26 Solomon Marcus

by means of the system S. Such interpretative models should not be confused with the cognitive models. The example we gave of interpretation of a formal system S into the arithmetics of positive integers shows that the two types of models may be in a reciprocal relation: if F is an interpretative model of S, then S is a cognitive model of F. But we should be cautious in respect to a possible generalization of this statement and anyway its converse is not true.

For both semi-Thue systems and Hilbert formal systems their language structure is obvious. In both cases we have to cope with words over a finite non-empty alphabet. In both of them we define a generative device having the status of a grammar, as a finite device able to generate infinitely many words. Post combinatorial systems follow the same pattern, leading to Post correspondence problem; see, for instance, pp. 446-456 in the already quoted Harju-Karhumaki (1997). Wellformedness is essential in all these cases, although differently interpreted. Comparing Chomsky's generative grammars with Thue's, Hilbert's and Post's proposals, we realize how strong was Chomsky inspired by his predecessors in the field of mathematical logic.

2.9 Roots in the Algebraic Theory of Semigroups

The theory of free semigroups, as a chapter of pure algebra, developed and anticipated some basic aspects of formal languages, in both their analytic and generative aspects. This line of research is far from being exploited as much as it deserves.

The concept of a formal language, where both semantics and pragmatics are ignored, was an object of research, in a different framework and terminology, in the algebraic theory of free semigroups, where some important notions in the study of languages were introduced with purely algebraic motivations. It is natural to expect that some ideas and theorems coming from algebra may show linguistic relevance too. This line of thought was followed by Miroslav Novotny and his school, as it can be seen in their papers in the journals Prague Studies in Mathematical Linguistics and, more recently, in F'undamenta Informaticae. Let A" be the free monoid generated by the finite non-empty alphabet A and let r be a congruence on A". What property must r have to assure the existence of a language L over A, such that r is just the equivalence relation c(L) defined as follows: the words u and v on A are equivalent iff, for any words x and y on A, xuy is in L iff uyv is in L.

Some basic sources in this respect are: A.H. Clifford, G.R. Preston, The algebraic theory of semiroups, Amer. Math. Soc., 1964; a Russian book on semigroups, by A. Liapin; M. Teissier, C. R. Acad. Sci. Paris, 232, 1951, 1897-1899; R. Pierce, Ann. of Math., 59, 1954,287-291; M.P. Schiitzenberger, C. R. Acad. Sci. Paris, 242, 1956, 862-864; B.M. Schein, Pacific J. Math., 17, 1966, 3, 529-547; J. Zapletal, Spisy Prirodov Fak. Univ. Purkyne v Brno, 4, 1968, 241-252. One can dress a dictionary translating the algebraic terminology into


the linguistic one. For instance, if the considered semigroup in Schein (1966) is free, then congruence classes correspond to distributional classes (in the sense of American descriptive linguistics). An indivisible set corresponds to a set contained in a distributional class. Schein's saturated sets correspond to unions of distributional classes. Schein's bilateral residue corresponds to parasitic strings in algebraic linguistics. Schein's neat corresponds to a language in respect to which no parasitic string exists.

2.10 Computational Sources of Formal Languages

Recursive functions, Turing machine, Markov's normal algorithms, automata and programming languages lead to sequential structures pointing out a formal language pattern.

All variants of computation, from recursiveness and Turing machine to normal algorithms and lambda calculus, share a sequential structure leading to words on a finite alphabet, i.e., to a formal language pattern. Finite automata, for instance, in their Rabin-Scott variant, correspond to Chomsky's finite state grammars or type 3 grammars, later called regular grammars. Pushdown automata correspond to context-free grammars. Linear bounded automata correspond to context-sensitive grammars. So, all types of classical computation are associated with a formal language structure.

In 1960, P. Naur et al. publish the Report on the programming language Algol 60 (Comm. ACM, 3, 299-314) and in 1963 its revised form (Comm. A CM, 6, 1-17). The syntax of Algol, defined by means of Backus normal forms, is shown to be of a context-free grammar type (S. Ginsburg, H.G. Rice, Two families of languages related to Algol, J. of ACM, 9, 1962, 350-371). This discovery changed the direction of development in the study of Chomsky's generative grammars. On the one hand, linguists were disappointed by the inadequacy of context-free grammars to the modeling of natural languages and left them in favor of transformational grammars; on the other hand, computer scientists became enthusiastic with the perspective to develop the study of Chomsky grammars as a tool to build a theory of programming languages. Not only context-free grammars were involved in this perspective. In 1962, R.W. Floyd (Comm. ACM, 5, 483-484) calls attention on the fact that as soon as syntactic definitions are supplemented with semantic conditions, we need also some context-sensitive rules which are no longer context-free. The new problem was: How to increase the generative power of context-free rules, by various types of restrictions in derivation. In other words, taking advantage from the simplicity of context-free rules, how can we increase their generative capacity by keeping the form of the rules, but changing adequately the derivation process? A lot of new types of grammars were introduced, situated between context-free and context-sensitive grammars, according to the fact that both natural and programming languages are situated somewhere

28 Solomon Marcus

between context-free and context-sensitive. The monograph by J. Dassow and Gh. Paun (Regulated Rewriting in Formal Language Theory, Berlin: Springer, 1989) gives a systematic picture of this line of research. Various parts of Algol, Fortran, Cobol which are not context-free are regular simple matrix languages (Floyd 1962, Brainerd 1968, Ibarra 1970, Aho-Ullman 1972; for all these references see Gh. Paun (J. Computer System Sci., 18, 1979,267-280)). See also S. Marcus, Linguistics for programming languages, Revue Roumaine de Linguistique, Cahiers de Linguistique Theorique et Appliquee, 16, 1979, 1, 29-39.

2.11 The Relation Between 'Formal' and 'Syntactic'

Is the expression 'formal syntactics' a pleonasm? We try to answer this question.

Witold Marciszewski (Dictionary of Logic as Applied to the Study of Language, The Hague: M. Nijhoff Publ., 1981) proposes the following explanation: "The adjective 'formal' when prefixing the word 'syntax' might be deemed pleonastic as syntax is formal in virtue of a definition, In fact, it is pleonastic only in some context. There is a well-known controversy as to whether syntax depends on the study of meaning or not. Insofar as it does depend of it, it is not formal, and then just those parts of syntax which are free of semantic problems coud be called formal" .

In a semiotic perspective, syntax is that part of a general theory of signs which is related to form. We distinguish between formation rules, defining the logical syntax, and transformation rules, defining the linguistic syntax. In both types of syntax, we distinguish between the formal aspect (where no reference to the semantic aspect is made) and the non-formal aspect (using reference to the semantic aspect).

In a formal system S (in Hilbert's sense), the syntactic dimension is given by what happens in the interior of S. The semantic dimension of S is given by all its possible interpretations in various scientific theories. The pragmatic dimension of S is given by its relations with the users of S.

In the second half of the XX-th century, a new field emerged: formal semantics. In the light of some of the considerations above, this expression seems to be oxymoronic. It concerns the syntactic component of semantics, i.e., that part which is related to form, not to substance. Similarly, formal pragmatics refers to the syntactic component of pragmatics. So, the itinerary from syntax to semantics to pragmatics closes the circle, going back to syntax.

2.12 Roots in Psychology

Formal grammars are associated with the move of learning processes from stimulus-response interaction to innate-acquired interaction. Linguistics becomes a chapter of cognitive psychology.


Traditional theories of learning processes see learning as an interaction between external stimuli and our responses to them. The corresponding mathematical models are based on probability theory, Markov chains and OnicescuMihoc's chains with complete connections. Some basic references for this approach are: M. Iosifescu, R. Theodorescu, Random Processes of Learning, Berlin: Springer, 1969; M.F. Norman, Marko'/} Processes of Learning, New York: Academic Press, 1972. One of Chomsky's motivations to start his generative approach is concerned with the language learning process, seen as the fundamental human learning process; any human learning has as a term of reference the language learning process. In this respect, Chomsky brings in the main attention of the researcher the interaction between the innate and the acquired factors. Our brain is seen as a generative device, a machine defining the basic human competence, whose various sub-devices define various particular human competences. Human competence is opposed to human performance, seen as what we actually can do effectively. Only a part of human competence becomes human performance. The former is infinite, the latter is finite. This distinction should not be confused with that between language and speech, proposed already by F. de Saussure, where the basic difference is that between the discreteness of language, as opposed to the continuous nature of speech.

Chomsky exposes his main ideas in this respect in a famous polemical paper entitled modestly "Review of 'Verbal Behavior' by B.F. Skinner", in Language, 35, 1959, 26-58. One of the main slogans proposed by Chomsky is "Linguistics is a chapter of cognitive psychology". Obviously, we have in view the new, generative linguistics, where the main stress is not on description and taxinomy (as in traditional structuralism including also the distributional structuralism of his former teacher Zellig S. Harris), but on the machine aspect, defining human linguistic competence. Is this approach (for which Chomsky finds roots in some old Indian grammars and later in what he calls Cartesian linguistics) excluding the traditional one, based on analytic investigation? Chomsky brings in front of attention one more dichotomy, that between surface structure and deep structure. Traditional linguistics focused on surface structures, generative linguistics focuses on deep structure, i.e., on the grammatical mechanism conceived as a generative device. In the late seventies of the past century, a great debate took place between Chomsky's supporters and Jean Piaget's supporters. It was one of the most important scientific debates of the XX-th century; see their proceedings in M. Piattelli-Palmarini (ed.), Theorie du langage, theories de l'apprentissage, Paris: SeuiI, 1979. The conclusion? Chomsky and Piaget are complementary, they need each other.

In order to make clearer where are they situated, we will propose a cognitive map, where the North is occupied by the reflexive (theoretical) approach, the South by the empirical approach, the East by the sequential approach (under the control of the left hemisphere of the brain), and the West by the non-sequential (including also the polydimensional) approach, under the control of the right hemisphere of the brain. At the interference of the reflexive and the sequential approach is the analytic-synthetic approach, traditionally

30 Solomon Marcus

associated with mathematics, theoretical physics, theoretical linguistics etc., and with names such as Democrit, Descartes, Hilbert, Russell, Einstein, and Bourbaki. At the interference of the sequential and the empirical approach we locate the experimental approach, traditionally associated with experimental physics and experimental sciences in general and with names such as Archimede, Galilei, Faraday, Lebesgue, Brouwer, John von Neumann, Benoit Mandelbrot. At the interference of the empirical and the non-sequential approach is the experiential approach, related to the natural experiences of human life and associated traditionally with human and social sciences and with names such as Hegel, Whitehead, Bergson. At the interference of the nonsequential and the reflexive approach it is the holistic approach, traditionally associated with arts and literature and with names such as Parmenide, Zenon, and Plato. The intuitive approach is a part of the non-sequential and it is predominant to Riemann and Poincare.

Obviously, we should be aware of the relativity of these distinctions. In each personality there is a specific interaction of all parts of the cognitive map. The problem is one of the way and proportions in which various factors are involved. No part of the cognitive map can be totally absent in a creative process. The typology of approaches and personalities concerns just this variety of ways and proportions.

Now, where is language located in the cognitive map? It is genuinely related to the Western part (the left hemisphere of the brain), as we have already pointed out. But this fact is much more visible in the variant of a formal language than in the case of languages in general. But the ChomskyPiaget controversy was concerned mainly with the North-South interaction (to use the terminology of the cognitive map), i.e., with the two basic ways to model learning processes and mainly language learning processes, one via stimulus-response representation, associated essentially with the empirical aspect of learning, the other via innate-acquired representation, accounting of the interaction between the empirical side and the hereditary side of learning.

A lot of experimental research was done, in order to test the explanatory capacity of these models. The results were very controversial. What are we telling here is only the start of a long series of learning models, constituting a basic part of the field of Artificial Intelligence. But we try to give an idea of the link of this problem with formal language theory. Here are some bibliographic references about this experimental research: J. Greene, Psycholinguistics. Chomsky and Psychology, Penguin Books, New York, 1972, where it is shown that only some experiments confirm the use by children of Chomskian rules, but these rules are sometimes relevant not by their form but by the result of the generative process in which they are involved. Another significant source is R. Moreau. M. Richelle, L'acquisition du langage, Mardaga, Bruxelles, 1981. A totally negative appreciation of Chomsky's learning model belongs to M. Labelle, Failure of Chomsky's approach, Cahiers Linguistiques d'Ottawa, 13, 1985, 1-36, where however the importance of the competence-performance distinction, of the role of generative rules and of the linguistic universals are


fully recognized. For the role of universal grammar as a hypothetical brain managing the various specific human competences, see C. Calude, S. Marcus, Gh. Paun, The universal grammar as a hypothetical brain, Revue Roumaine de Linguistique, 24, 1979, 5, 479-489.

2.13 Biological Sources

After a long period, when Darwinian and post Darwinian biology were the main model to approach historical linguistics, the linguisticsbiology relation changed the direction: from biological models of historical linguistics to linguistic models for nervous activity and for molecular genetics. Now, the triangle language-biology-computation is at the heart of contemporary research, due to its involvement in genomics and in DNA and membrane computing.

Chronologically the first involvement of biology in a language-Iogiccomputation context is the paper by W.S. McCulloch and E. Pitts, A logical calculus of the ideas imminent in nervous activity, Bull. of Math. Biophysics, 5, 1943, 115-133. We will however stop on another pioneering paper, by S.C. Kleene, Representation of events in nerve nets and finite automata, from A utomata studies (C.E. Shannon & J. McCarthy, eds.), Princeton Univ. Press, 1956. An experience with k neurons, of duration h, is described by a binary matrix with k columns and h lines, where the digit 1 at the intersection of the ith column and jth line shows that the ith neuron is active at the moment j; otherwise, we put the digit o. An event is defined as a partition of the set of experiences in two subsets: one of them includes the experiences where the considered event appears, while the other set includes the experiences where the respective event does not appear. When h runs over all positive integers, we get all experiences with k neurons, of finite duration. Adequate operations are defined with these events (union, product, star). Let R be the smallest class of sets of matrices including the sets containing only one matrix and the empty set and which is closed in respect to the considered operations. Any set of matrices belonging to R is by definition a regular set; the corresponding event is said to be a regular event. There is an isomorphism between regular events and events that can be represented in finite automata. A whole chapter is devoted to this topic in S. Marcus, Grammars and Finite Automata (in Romanian), Romanian Academy Publ. House, Bucharest, 1964.

Another important direction is related to molecular biology. Z. Pawlak was the first author calling attention on the relation between the heredity mechanism and the formal language structure, by pointing out a way to model molecular genetics by dependency grammars (Z. Pawlak, Gramatyka i matematika, Panstwowe Zakady Wydawnietw Szkolnych, Warsaw, 1965). Then, we developed farther his model (S. Marcus, Linguistic structures and generative devices in molecular genetics, Cahiers de Linguistique Theorique et Appl.,

32 Solomon Marcus

11, 1974, 1, 77-104). For the whole development of the triangle languagecomputation-biology, see S. Marcus, Language at the crossroad of computation and biology, in Gh. Paun (ed.), Computing with Biomolecules. Theory and Experiments, Singapore: Springer, 1998, 1-35.

A very productive line of research in the field of language-biology interaction was the modeling of growth processes in cellular biology, mainly concerned with filamentous organisms. It was started by Aristid Lindenmayer in 1967 and became in short time one of the most active direction in formal language theory. Lindenmayer systems became an object of intrinsic mathematical interest, irrespective its initial biological motivation. But for those who are interested in this motivation we propose the paper presented by Aristid Lindenmayer to an internatioal congress of philosophy: Cellular automata, formal languages and developmental systems, Proceedings of the IV-th International Congress of Logic, Methodology and Philosophy of Science, Bucharest, 1971, North Holland Publ. Comp., Amsterdam, 1973. Surpringly, but according to a well known rule, Lindenmayer systems showed applications in a field very far from cellular biology: computer graphics.

2.14 At the Crossroad of Logic, Mathematics, Linguistics, and Semiotics

Messages from Carnap, Ouine, Morris, Tarski, Vasiliu, Barbara Hall Partee et al. about formalization, logical semantics and formal semantics.

According to Carnap, syntax is linguistic when it refers to natural languages and it is logical when it is applied to a language of science; but if the language of science is mainly logical and mathematical, then the science we are concerned with may be linguistics too, so syntax may be both linguistic and logical. Rudolf Carnap (Logische Syntax der Sprache, Springer, Vienna, 1934) defined the logical syntax of a language as "the formal theory of the linguistic form of that language - the systematic statement of formal rules which govern it together with the development of the consequences which follow from these rules. A theory, a definition or the like is to be called formal when no reference is made in it either to the meaning of the symbols or to the sense of the expressions but simply and solely to the kinds and order of the symbols from which the expressions are constructed" . This view is consistent, although different, with that of Hilbert's formal systems, with R. Feys, F. Fitch (Dictionary of Symbols of Mathematical Logic, North Holland, Amsterdam, 1969), with W.V.O. Quine (Mathematical Logic, Harvard Univ. Press, 1955), with N. Chomsky (Syntactic Structures, Mouton, The Hague, 1957) and with Ch.S. Morris (Foundations of the Theory of Signs, Univ. of Chicago Press, 1938), according to whom the syntactic dimension of semiosis is given by the relation of sign-vehicles to sign-vehicles within a complex sign or a


sign-system. However, today it is no longer possible to identify "logical" and "formal" or logical and syntactic, because we cannot ignore the field of logical semantics (the study of denotations) and of formal semantics, so important in contemporary linguistics (see, for instance, B. Hall Partee, A. ter Meulen, R. E. Wall, Mathematical Methods in Linguistics, Dordrecht et al, Kluwer, 1990) and in contemporary formal semantics of programming languages. For the status of a scientific semantics, seen as a theory of the relations between signs and what they refer to, one should consider also the work of Alfred Tarski and Charles Morris in the thirties of the XX-th century. Logical semantics is for Tarski the study of the connections between the expressions of a language and the objects or state of affairs referred to by these expressions. This study requires the use of a formal metalanguage.

In respect to the logic-linguistics interaCtion, very significant are the works of Emanuel Vasiliu. He observes, in the Preface of his Logical Preliminaries to the Semantics of Phrase (in Romanian), Scientific and Encyclopedic Publ. House, Bucharest, 1978, that words such as 'sense', 'denotation' (or 'reference'), 'sense identity', 'synonymy' etc., initially used mainly in linguistics, have been clarified in the field of logic, not in that of linguistics. The semantics of natural languages and the semantics of logical languages are, first of all, semantics, i.e., the study of some sign systems in respect to their denotations. Continuing the line of Carnap's thinking (mainly that from Introduction to Symbolic Logic and Its Applications, New York, Dover, 1958, The Logical Syntax of Language, Paterson, New Jersey, Little Field, 1959, Meaning and Necessity, Chicago Univ. Press 1960), Vasiliu is arguing that logical semantics offers an unavoidable framework for semantics in general and especially for the semantics of natural language. In the meantime, this framework has been enlarged with a semiotic and a computational perspective.

2.15 Syntactics as Calculus

We sketch the Ajdukiewicz - Bar Hillel - Lambek line of research, leading to categorial grammars, so important today in logic and in computer science.

Since any formal system is a calculus (see the derivation of theorems by means of some rules, starting from axioms), logical syntax is closely related to a calculus. A generative grammar is also a calculus (derivations in a grammar have the same structure as proofs in a Hilbert formal system). 'lUring machine develops a symbolic calculus. Calculus aspects of syntax are clear in K. Ajdukiewicz (Syntaktische Konnexitaet, Studia Philosophica, 1, 1935, 1-27) and in Emil L. Post's combinatorial systems (1936). One step forward in this direction is made by Y. Bar-Hillel (A quasi-arithmetical notation for syntactic description, Language, 29, 1953, 47-58) and by Joachim Lambek {The calculus of syntactic types, in "The structure of language and its mathematical aspects" (R. Jakobson, ed.), American Math. Soc., Providence, R.I.,

34 Solomon Marcus

1961,166-178). Lambek's starting point is the following analogy: In classical physics, it was possible to decide whether an equation was "grammatically correct" by comparing the dimensions of the two sides of the equation. One may ask whether it is similarly possible to assign grammatical types to the words of a natural language in such a way that the grammatical correctness of a sentence can be determined by computation with these types. Such possibilities already exist in certain artificial languages, for example, propositional calculus, where there are rules which distinguish between well-formed and non well-formed formulas.

Let us now sketch Lambek's method. Starting with a vocabulary V, we begin by assigning certain primitive types to some words and some strings on V. From these primitive types, compound types are built by three formal operations: multiplication, left division, right division, denoted by juxtaposition, by \ and by / respectively. We write X -+ x to indicate that the string X has the type x. The defined compound types have the following significance: If X -+ x and Y -+ y, then XY -+ xy. If XY -+ z and Y -+ y, then X -+ z/y (read "z over y"); if XY -+ Z and X -+ x, then Y -+ x\z (read "x under z"). In other words, an expression of type x/y, when followed by an expression of type y, produces an expression of type x, as does an expression of type y\x when preceded by an expression of type y. Among the primitive types there always exists the type s ascribed to all sentences and only to sentences. If we could say whether a given string u is well-formed, then we could compute the types ascribed to the terms of u and we verify whether the compound type is precisely s.

Another type of syntactic calculus has been investigated by Bar-Hillel (Four Lectures on Algebraic Linguistics and Machine 7ranslation, Venice, Italy, 1962) and by Y. Bar-Hillel, C. Gaifman and E. Shamir, On categorial and phrase structure grammars, Bull. Res. Council Israel, Sec. F, 9, 1960, 1-16. The aim of a categorial grammar is to determine the syntactic structure of a given string, by using only the shape and the order of elements in the string. This is achieved by assuming that each of the finitely many elements of the considered natural language has finitely many syntactic functions, to which a suitable notation is associated and an algorithm operating on this notation is designed. For instance, a bidirectional categorial grammar is a quintuple (V, C, W, R, j), where V is a finite set of elements (the vocabulary), C is the closure of a finite set of fundamental categories, say Cl, ••. ,en, under the operation of right and left diagonalization (that is, whenever a and b are categories, (a/b) and (a\b) are categories), W is a distinguished category of C (the category of sentences), R is the set of the two cancellation rules (Ci/Cj)Ci -+ Cj and similarly for \, and f is a function from V to finite subsets of C (the assignment function). We say that a category sequence a directly cancels to b, if b results from a, by one application of one of the cancellation rules. We say that a cancels to b if b results from a by finitely many applications of these rules. A string x over V is defined as a sentence iff at least one of the category sequences assigned to x by f cancels to W. The set of


all sentences is then the language determined by the given bidirectional categorial grammar. Such a language is called a bidirectional categorial language. IT instead of C one considers the closure of a finite set of fundamental categories under the operation of right (left) diagonalization, the other objects, V, W, R, and f remaining unchanged, we obtain the definition of a right (left) categorial grammar, which determine right (left) categoriallanguages. They are unidirectional categorial grammars and languages, respectively. It is this last variant that was considered by Ajdukiewicz. Bar-Hillel proved that every bidirectional categorial language is a unidirectional categorial language. Moreover, for any bidirectional categorial grammar H one can construct a context-free grammar G such that the language generated by G is just the language determined by H.

A rich literature was devoted to Lambek calculus and to categorial grammars, which deserves the attention of formal language theorists.

2.16 Dependency Grammars and Syntactic Projectivity

The itinerary from Tesniere to Hays and from Hays to syntactic projectivity is sketched.

The idea of calculus is subjacent in L. Hjelmslev's Prolegomena to a Theory of Language (Baltimore, 1953), who refers to the need to build an algebra of language. Dependency grammars, introduced by David G. Hays (Dependency theory; a formalism and some observations, Language, 40 (4), 1964, 511-525) and having as source of inspiration Lucien Tesniere's Elements de syntaxe structumle (Klincksieck, Paris, 1959), where a geometric approach, via rooted-trees (whose mathematical model is borrowed from graph theory), is for the first time developed, had great impact in linguistics. Dependency relations and subordination relations find here an adequate framework, which leads to a rigorous approach to many syntactic problems. Two of them are particularly relevant. One is related to the modeling of dependencies of various degrees, like in "very nice pictures"; we distinguish here between a first degree dependency in "very nice" and a second degree dependency in "nice pictures" . A whole theory of syntactic configurations emerges (see, for instance, Chapter 5 in S. Marcus, Algebmic Linguistics; Analytical Models, New York, Academic Press, 1967). The other interesting problem successfully approached by dependency relations concerns syntactic projectivity. Dependency is a specific binary relation d between some terms of a string over a finite vocabulary. Subordination is the reflexive and transitive closure of d. Grammatical propositions are modeled by rooted trees, where an oriented arc between two vertices indicates a dependency relation of the second vertex in respect to the first. The typical projectivity property, introduced by Y. Lecerf and P.!hm (E:lements pour une grammaire generale des langues projectives, Rapport CETIS no.l, Euratom, 1-19) is defined as follows; the string x endowed with a dependency relation

36 Solomon Marcus

is projective if for any two terms u and v of x such that u is subordinated to v, all intermediate terms between u and v are also subordinated to b. For a typology of projective relations and their relationships see chapter VI in S. Marcus's 1967 book quoted in this section. Projectivity property had a great impact in the study of natural language syntax and surprisingly it opened a new chapter in graph theory; see a book by Ladislav Nebesky on this topic. The accurate presentation of dependency grammars is due to H. Gaifman (Dependency systems and phrase structure systems, Information and Control, 8 (3), 1965, 304-337.

2.17 Formal Grammars in Number Theory and in Combinatorial Geometry

Friant was the first to bridge formal grammars and number theory, while Gross and Lentin did a similar job for combinatorial geometry.

Fermat great theorem states that the equations xn + yn = zn (n strictly larger than 2) admit no entire non-null solutions. The theorem has been proved only recently. Jean Friant (Les langages context-sensitives, Annales de l'Institut Henri Poincare, Section B: Calcul des probabilites et statistique 3/1, 1967, 35-120) proposed a way to state this theorem in formal language terms. Let us consider an alphabet A = {a}. Define, for every integer number i strictly larger than 2, the language L(i) including exactly the powers of the form ni of a, when n runs over all non-null positive integers. Friant has shown that this language is context-sensitive, but not context-free, for every positive integer i strictly larger than 2. Since the product and the intersection of two context-sensitive languages are context-sensitive languages themselves, it follows that the language obtained as the intersection (1) between L(i) L(i)

and L(i) is context-sensitive for every positive integer i strictly larger than 2. This fact leads to the following way to formulate Fermat great theorem: For every positive integer i strictly larger than 2, the language represented by the intersection (1) is empty. Indeed, in the contrary case an ordered triple (k, m, n) of non-null positive integers would exist, such that the power of exponent ni of a would be equal to the product between the power of exponent k i and the power of exponent mi of a. This situation would imply the equality k i + mi = ni, in contradiction with the Fermat great theorem. Now let us recall that the question whether the language generated by an arbitrary given context-sensitive grammar is or is not empty is algorithmically nondecidable. Fermat theorem removes this undecidability in the particular case of the languages of the form (1).

Now there are many results bridging number theory and formal languages, for instance, in respect to the development of real numbers in their decimal expansion or in a continuous fraction. The French habit to call regular languages rational languages and context-free languages algebraic languages can be motivated by some theorems pointing out just this association.


Given a Jordan polygon (i.e., a simply connected topological polygon with an inside and an outside) whose n vertices have been labelled, in how many ways can it be decomposed into triangles by non-intersecting diagonals (except for intersections at the vertices)? The basic idea of a solution to this problem using a grammar consists in generating the triangulated polygons in the same way that the words of a language are generated. If the grammar is unambiguous, it will provide us with a count of the triangles as a by-product. The method that follows is a result of the application of simple ideas, in order to make the triangles 'sprout'. Let us consider the grammar S ~ aSS, S ~ b, which is known to be unambiguous, and attribute to the symbols and the productions the following semantic values. S is taken to be a directed topological segment which we shall call 'virtual'; aSS is taken to be a directed topological triangle having a side a, which is called 'real', and two virtual sides S and S. A production that consists of replacing some S by aSS means: construct a triangle aSS whose real side a coincides, direction included, with the S in question, and which is located outside the polygon thus far constructed. It is easy to show by recursion that this last condition can always be satisfied: the application of the rule is not subject to any restrictions of context. For the whole approach, including furter details and examples, see M. Gross and A. Lentin, Introduction to Formal Grammars, Springer, Berlin, 1970,210-215.

2.18 Formal Grammars in Chemistry and in Medical Diagnosis

Isoprenoid structures in organic chemistry can be approached both by Chomsky grammars and by picture grammars. Medical diagnosis raises problems of contextual ambiguity similar to those occurring in natural languages

Starting from an idea of R.C. Read (1969) for coding tree-like graphs, acyclic isoprenoid structures are coded using a binary notation system. Context-free grammars are used for generating either regular (head-to-tail) or standard acyclic polyisoprenoid structures (head-to-tail, tail-to-tail, tailto-head, head-to-head). It is shown that the characteristics of these codes are the following: for regular acyclic polyisoprenoid structures the code detects nine errors and may correct four errors; for standard acyclic polyisoprenoid structures the code detects a single error. By using deterministic pushdown automata, one may check the presence or absence of errors and in the former case one may correct them in part. (M. Barasch, S. Marcus, A.T. Balaban, Codification of acyclic isoprenoid structures using context-free grammars and pushdown automata, Match (Mathematical Chemistry), 12, 1981,25-64.)

Given a molecular formula of a substance and knowing that the substance has a isoprenic skeleton, how can we find all the isoprenoid structural formulas corresponding to the molecular formula? In other words, how can we

38 Solomon Marcus

build the isoprenoid graphs which can be decomposed into n isoprene units? We recognize here a typical generative problem, because we have to generate a potentially infinite set of isoprenoid structures. The tool to be used is the theory of automata and formal grammars, which has already been used, in the seventies, in problems of information retrieval in the field of chemistry. But in that period the only applications of formal grammars in chemistry were concerned with their classical Chomskian form, related to concatenative, linear sequential structures. Typical in this respect are works like those of E.I. Burkart and W.E. Underwood, A grammar for the line formulas of organic compounds, Univ. of Maryland, Computer Science Center, Report TR-530, April 1977, and of G.S. Mancill and W.E. Underwood, A grammar for the nomenclature of acyclic compounds, Univ. of Maryland, Computer Science Center, Report TR-529, April 1977. Structural formulas in chemistry are no longer built in a linear way. How could they take profit of the grammar-like devices? This problem, which occurs in many fields of research, was raised already in the sixties: Can one build some generative devices which, on the one hand, are similar to Chomskian grammars, and, on the other hand, are suitable to generate polydimensional structures (i.e., structures represented in Euclidean spaces with several dimensions)? This question was the starting point of picture grammars, where the elements of the terminal alphabet are geometric entities, while the concatenation operation is replaced with some geometric operations. The 40 years long history of picture grammars is strongly related to the field of syntactic pattern recognition, with many applications in physics, biology, genetics, architecture, visual arts etc. Two types of picture grammars proved to be useful in organic chemistry (A.T. Balaban, M. Barasch, and S. Marcus, Picture grammars in chemistry: Generation of acyclic isoprenoid structures, Match (Mathematical Chemistry), 8, 1980, 193-213): web grammars and array grammars. The terminal alphabet is the set of isoprenoid units. Head-to-tail isoprenoid acyclic structures are obtained by means of head-to-taillinkings of isoprenoid units. The strings which are generated in this way correspond to all head-to-tail isoprenoid acyclc structures. The advantage of web grammars is the possibility they offer to analyze the given structures by using reduction rules, obtained by inversion of rewriting rules of the grammar. This corresponds to testing whether a given structure is or is not of a given type. In this way, the same grammar is used to solve both the problem of generation and that of checking.

Picture grammars allow to build a compiler and to achieve, by means of a computer, both the construction and the verification of structural formulas. Thus, the problem of isomeric structures (to each molecular formula correspond several structural formulas) can be easily solved. Array grammars show their advantage when dealing with generation of isoprenoid acyclic chemical structures with an odd number of isoprenoid units, with head-to-taillinkage, which admit a symmetry axis.

Web grammars were introduced by Pfaltz and Rosenfeld in 1969. The associated languages are sets of labelled graphs ('webs'). The productions in


these grammars replace some subwebs by other subwebs. Let V be a finite nonempty set whose elements are called 'labels'. A web on V is a triple W = (Nw, Ew, Iw) where Nw is a finite nonempty set whose elements are called the vertices of Wj Ew is a set of unordered pairs of distinct elements of Nw, pairs which are called 'arcs' of Wj Iw is a function from Nw into V. The pair (Nw, Ew) = Gw is a graph, called the underlying graph of w. The distinctness of the terms of the pairs in Ew implies that Gw is a loop-free graph. The vertices m and n are called neighbours if (m, n) is in Ew. We call m and n connected if there exists a path from m to n, such that each consecutive terms in the path are neighbours. If any two nodes of W are connected, W is said to be connected. Given two webs u and v on V, we say that u is a subweb of v if N.u is a subset of NtJj Ell. is the restriction of EtJ to Nuj III. is the restriction of Iw to Nil.. A web grammar is a 4-tuple G = (V, VT'S,P), where V is a finite non-empty set (the vocabulary), VT is a part of V, the terminal vocabulary, S belongs to V, but not to VT and it is the initial symbol, while P is a finite, non-empty set of triplets p = (c,d, e) called 'productions', where c and d are connected webs on V and e is a function from Nd x Nc into the set of subsets of V. Adequate direct derivability and derivablity in G are defined and then a sentential form is defined as a web which is derivable in G from the initial web. A web grammar is constructed that generates a language which is the set of all acyclic regular isoprenoid structures (see also A.T. Balaban, M. Barasch, and S. Marcus, the paper already quoted in Match, 5, 1979,239-261).

Medical diagnosis becomes a formal language problem as soon as we observe that a syndrom is a finite sequence of symptoms. The clinical phenomenon is regarded as a language over the alphabet of the possible symptoms of a considered illness. So, to any illness M we associate the set L(M) of all possible syndroms of M. Obviously, any subword of a word in L(M) is still in L(M). This situation leads to the idea to consider the sub language Lo(M) of L(M) formed by the saturated substrings of the strings in L(M), i.e., of those strings in L(M) which are no longer substrings of other strings in L(M). We have Lo = the intersection of L-(M) (the sublanguage of L(M) including the strings in L(M) saturated at left) and L+(M) (the sublanguage of L(M) incuding the strings saturated at right). Usually, the clinical examination concerns strings in L( M) - L( Mo), because rarely happens to have a person under clinical examination from the very beginning of his illness till to its very end. Longer is the sequence of observed symptoms, better is the chance to identify the illness M. So, practically things happen in the opposite order in respect to theory. Theoretically, we start with an ilness and we describe its possible sequences of symptoms. Practically, we ignore the illness and we try to infer it starting from the observed sequences of symptoms. But there are many illnesses whose cause is not yet known and we can describe only the effects, so in these cases the only available data are the respective sequences of symptoms. We accept the compromise of approximation by syndroms, but the price we have to pay is to cope with the ambiguity of this approximation. Rarely happens to have syndroms determining with no ambiguity the illness. When this

40 Solomon Marcus

happens, we call the respective syndrom pathognomonic. Otherwise, we associate to each syndrom an index of ambiguity, equal to the number of illnesses that could be associated to the respective syndrom. Given two syndroms x and y over the alphabet A of all possible symptoms, we consider that x is less ambiguous than y, in respect to a language L over A, if for any context (u, v) over A for which uxv is in L, the string uyv is also in L. Since L is a set of strings on A, it may represent a hypothetical illness, in respect to which the ambiguity of the syndrom x is not larger than the ambiguity of y. There is a similarity between this situation and the contextual ambiguity leading in natural languages to what is called morphological homonymy. For more details see E. Celan and S. Marcus, Le diagnostic comme langage, Cahiers de linguistique tMorique et appliquee, 10, 1973, 2, 163-173.

2.19 Anthropology and Formal Languages

Marriages in some primitive societies lead to some formal language problems.

The starting point in this surprising itinerary is a note published by Andre Weil as an annex to a work by the famous anthropologist Claude Levi-Strauss. It was Jean Friant (1967, op. cit.) who observed that the problem raised by Levi-Strauss has an interesting formal language face.

In some primitive societies, marriages are subject to very restrictive rules. A significant case (encountered to some populations in Australia) is that in which each person belongs to a 'marriage class', such that two persons may contract a marriage only if they belong to the same class. The marriage class of each person is determined by his sex and by the class of his parents, such that two boys (or two girls) whose parents have distinct types are themselves of distinct types. Let us suppose that we have n distinct classes of marriage, denoted by C l , ... , Cn. The mentioned hypotheses imply the existence of two permutations, g and J, of the set {l, ... ,n}, such that Cg(i) (Cf(i), respectively) denotes the class of the parents of a man Mi (of a woman Wi, respectively) of class Ci (i between 1 and n). The parents of a man of type Ci are thus well determined as to what concerns their type; they will be denoted by Mg(i)Fg(i). The grandparents will thus be denoted by the symbolism:

Mg(g(i» Wg(g(i»Mf(g(i» Wf(g(i»·

In fact, the class of the parents of M i , at the second ascending generation, is shown as Cg(g(i»Cf(g(i». More precisely, given a person of the considered society, one introduces the signature of order k, defined by the sequence of the 2k- l types of the 2k ancestors from the generation of order k. Hence, this signature is a string of length 2k-l on the alphabet {C1, ... , Cn }. The set of strings thus obtained form an infinite language (since with each person one associates infinitely many signatures, one for each order k). As it was


shown by Friant (1967), this language can be generated by a context-sensitive grammar, but it cannot be generated by a context-free grammar.

The mathematics of kinship relations, a chapter initiated by Andre Weil, may have many other interesting problems for formal language theory; they deserve our attention; for instance, where, between context-free and contextsensitive, is located the language investigated by Friant?

2.20 The Formal-Linguistic Space of Learning

The set of languages over the alphabet A is organized as a normed vector space on {O, 1} in the following way. For two given lallguages E and F we define E + F = (E - F U (F - E) (the symmetric difference of E and F); 0 . E = 0; 1· E = E. Denote by s(E) the smallest length of a word in E. Define the norm IIEII of E as being equal to 2-s(E) and the distance d(E, F) = liE - FII. The metric space so obtained is called the Bodnarciuk metric space of languages and it is complete, separable, and compact. It is homeomorphic to a part of the learning space (G, t) (to be defined), while the latter is homeomorphic to a part of the former. (For the metric space we have considered, see V. G. Bodnarciuk, The metric space of events (in Russian), Kibernetika (Kiev), 1, 1965, 1, 24-27.)

An object to be learned is a mapping f from N into N, where N is the set of positive integers; f(i) is the response to the stimulus i in N. Here, N is the symbolic representation of an infinite sequence of stimuli. This definition belongs to Y. Uesaka, T. Aizawa, T. Ebara, and K. Ozeki (A theory of learnability, Kybernetik, 3, 1973, 123-131) and is the starting point of a topological model of learning processes; its continuation was done by T. Aizawa, T. Ebara, K. Ozeki, and Y. Uesaka (Sur l'espace topologique lie a une nouvelle theorie de l'apprentissage, Kybernetik, 4, 1974, 144-149). At a first glance, this definition may seem to be anti-Chomskian; it refers to the traditional, empirical representation of learning processes as stimulus-response processes. The empirical approach is necessary, but not sufficient; it needs to be correlated with its interaction with the reflexive approach. However, if we look more carefully at the definition above, we observe that it includes an element making it to be able to capture the reflexive side too. Indeed, in contrast with the empirical approach, prisoner of a finite number of stimuli, the considered definition refers to infinitely many stimuli (the infinity of N). What does it mean? Suppose, for instance, that the object to be learned is the notion of an odd number. Empirically, we can learn a finite number of such numbers, Learning, for instance, that 1,3, and 5 cannot be divided exactly by 2 means to learn the first three ordered pairs (1,1), (2,3), (3,5) from the infinite sequence of ordered pairs having as a general term the pair (n, 2n + 1). But this second step of the learning process, the move from an initial, finite segment of the sequence to its general term can be achieved only by a theoretical effort, extrapolating the finite number of steps that can be empirically realized.

42 Solomon Marcus

Things become more convincing when we replace the learning of odd numbers with the learning of prime numbers, where the theoretical step accounting for the generic form of the n-th prime number is still an open problem.

In the light of the considerations above, the next step will appear very natural. To each finite initial section d of the sequence (i, /(i)) (i = 1,2,3, ... ) we associate the set p(d) of all objects to be learned which are compatible with d, i.e., which have d as their initial section. Obviously, p(d) is infinite. For instance, if d = {(I, 1), (2,3), (3, 5)}, then p(d) includes objects such as 'odd number', 'prime number', 'Fibonacci sequence' and infinitely many other possible objects to be learned. If theoretically we associate to each object / to be learned its initial sections d(f), practically things happen in the opposite way. Children develop a rich empirical learning activity, by acquiring a lot of stimulus response pairs, i.e., of potentially sectioned d of some objects to be learned. But the itinerary from d to some objecte in p(d) is very long, very often never completely accomplished. This fact legitimates the following procedure. Denote by F the set of all objects to be learned and let Sand T the subsets of F. We say that the object / situated in F can be learned by knowing that "/ is in T" and under the innate information "is in S" as soon as it exists a finite section d of /, such that / is in both Sand p(d), and the common part of Sand p( d) is contained in T.

We consider a topology t in F, whose base is the family of all sets of the form p( d), when d runs over all finite sequences of ordered pairs of positive integers, the first terms in the sequences being always an initial section in the sequence of positive integers. One can acquire the knowledge "/ is in T" under the presupposition "/ is in S" iff the intersection of T and S is a neighbourhood of / in the topology of (F, t) relative to S. On the other hand, the distance between two objects / and 9 to be learned is given by the sum of the series whose general term is the product between 1/ qn and u(f (n), g( n)), where u(x, y) = 0 if x = y, u(x, y) = 1 if x is different from y, q is strictly larger than 1 and x and y run over the sequence 1,2,3, ... ,n, ... The metric space so obtained is complete and homeomorphic to a part of the topological space (F, t), which, as we have already pointed out, is homeomorphic to a part of the Bodnarciuk metric space. For further developments in this direction see V. Vianu, The structure of formal languages from the viewpoint of measure theory (in Romanian), Studii §i Cercetiiri Matematice, 20, 1978,2,227-235; S. Marcus, Learning as a generative process, Revue Roumaine de Linguistique, 24, 1979, Cahiers de Linguistique The6rique et Appl. 16, 1979, 2, 117-130; S. Marcus, Tolerance rough sets, Cech topologies, learning processes, Bull. Polish Acad. 0/ Sciences, 42, 1994,3,471-487.


2.21 Formal Languages in Praxiology

Tadeusz Kotarbinski's praxiology, defined as a theory of human action, was brought within the framework of formal languages by the work of Marie N owakowska.

Marie Nowakowska (Language of Actions, Language of Motivations, Mouton, The Hague 1973) defines the language of actions as a system S = (D, L, f, R, r), where D is a finite set of elementary actions, L is a language over D whose elements are caled actions, f (d) is a positive integer indicating the duration of an action d in L, R is the set of all possible results of the actions in L, r is a mapping from L x N into R, where r(d, n) indicates the result obtained at the moment n by performing the action d. Nowakowska's book provides a comprehensive theory of the language of actions, where many fundamental notions and results from the analytical approach to languages are used, mainly in the form they were developed by S. Marcus, Algebraic Linguistics; Analytical Models, Academic Press, New York, 1967; S. Marcus, Introduction mathematique a la linguistique structurale, Dunod, Paris, 1967. The philosophy of this approach is to trust the possibility to represent human and social actions as concatenations of some elementary actions, forming a basic alphabet. In other words, it is assumed that in most cases we can quantify human and social actions, in the way alphabetic languages are represented as discrete sequential structures. Obviously there are many situations where this representation is successful and this fact explains the success of computer science, where algorithmic models show their relevance just in view of the possibility to quantify some processes, by decomposing them in elementary steps having behind them an elementary alphabet.

One of the simplest human action is that of getting a telephone connection from a public telephone, with a person having a telephone at home. According to the procedure that was valid some years ago, one can decompose this action in the following elementary steps: a = pick up the receiver, b = introduce the coin, c = wait for at most s seconds, d = getting the signal, e = dial the wanted number, f = wait for at most t seconds, g = getting the answer and carrying out the conversation, h = hanging up the receiver. Let us simplify things a little: just suppose that we dial the right number from the very beginning and the very person we are wishing to talk to is answering the phone. There are more possibilities. If after waiting s seconds we get no signal, we go over the sequence abch, that is, we put down the receiver and try again. If we hear the signal, but there is no answer, then we cover the sequence abcdefh. Yet, we may fail, twice or more times in getting either the signal or the answer. After each failure, we may try again. For expressing more exactly what is happening, let us consider x = abch and y = abcdefh. We get the general form of a successful attempt by composing an arbitrary finite sequence made up of x and the sequence z = abcdefgh. Such a sequence is, for example, x3y2x2y4z.

It shows that the telephonic connection was established after three failures in

44 Solomon Marcus

getting the signal, two in getting the answer, followed by other two failures in getting the signal and four in getting the answer, that is, eleven failures in all. Generally, the exponents of x and y are arbitrary natural numbers, with an arbitrary finite occurrence.

Other times, things are more complicated. Consider car-driving. Even if we refer only to the basic manoeuvres: stopping, overtaking, backing, turning, cornering to the right or to the left, we need an alphabet of 25 elementary actions: contact, breaking of the contact, releasing of the hand brake, lifting of the hand brake, assuring by looking forwards, backwards, to the right and to the left, or in the observation mirror, bringing the change gear to the dead point, then to the first speed, respectively the second, the third, the fourth or on the reverse, pushing the clutch pedal, respectively the throttle pedal or the brake pedal, turning the wheel hand 90 degrees to the right or to the left respectively, signalling to the right, respectively to the left, stopping the signalling, waiting without moving, moving at an almost constant speed; so, the corresponding infinite languages are much more complicated than in the case of the telephone. See M. Scortaru, Generative devices of the driver manoeuvres, Foundations of Control Engineering, 2, 1977, 2, 107-115. For a real, flesh and blood driver, however, only a finite part of these languages is useful. Scortaru pointed out that the language describing the overtaking or the turning manoeuvres cannot be generated by a finite automaton, in contrast with what happens with the telephone calls. There are set into motion more complex generative mechanisms than those related to the cornering to the left or to the right, for which finite automata are enough.

Some general remarks could be made about the above actions.

1. The actions proved to be quantifiable, that is, any of their variants can be represented by an adequate combination of a relatively small number of elementary actions, the same for each variant.

2. The actions have a language structure, any of their realizations being a word on the same finite alphabet.

3. The language we obtain includes infinitely many words, because there is no length limit for them.

4. The examples discussed show that human actions may be considered either from the point of view of its general structure or as related to the particular circumstances, different in each case. The former aspect defines the competence, the latter the performance aspect of the considered action. The formal language viewpoint is referring mainly to the competence aspect.

2.22 Generative Mechanisms of Economic Processes

This is the title of a book by Paun (1980), where Nowakowska's action systems are used in their syntactic dimension, ignoring, in a first step,


their semantic components R and r. Economic processes are the object of investigation.

The title of this section is just that of a book by Gh. Paun (in Romanian) published at the Technical Publ. House, Bucharest, 1980. Links between economics and linguistics were already studied, the main contribution in this respect being that of Ferrucio Rossi Landi (Linguistics and Economics, Janua Linguarum, series Minor 81, Mouton, The Hague, 1975), where the empirical analogy between the circulation of goods within an economic system and the circulation of words in language is exploited. Language is reduced to its pragmatic dimension, in contrast with Paun's approach, where language is conceived as a theoretical construct and the analogy is between the language competence and the economic competence.

The idea to associate to an economic process a grammar or an automaton was already expressed by J. Kornai in his book Anti-equilibrium, Romanian translation from Hungarian, Scientific Publ. House, Bucharest, 1975. Paun is effectively developing this program, in a large variety of situations. Various economic actions are represented as strings on a finite non-empty alphabet, but in contrast with Nowakowska, who directs her attention towards descriptive aspects investigated with tools of analytical models in algebraic linguistics, Paun is interested in generative aspects, looking for the grammar of various economic processes. An important class of such processes is that related to paths in graphs, including the salesman problem (with or without time restrictions), variants of the delivery problem, the inspector's problem and the courier's problem (a courier moves in a graph, picking-up letters from certain points and delivering them to other points).

Previously, such problems were studied mainly by means of graph theory (see, for instance, M. Mercatanti and L. Spanedda, La recherche de l'ensemble optimal des itineraires des vehicules dans une entreprise de transports automobiles extraurbaines, R.A.I.R.O., 9, V-I, 1975, 59-75). Paun transforms such finite combinatorial optimisation problems in problems of competence leading to potentially infinite possibilities. In a next step, he approaches what he defines as simple systems of action and then he proposes a formal language model of the production process. The next step is that of compound systems of action, including the scheduling problem, and parallel systems of action, including Zilinskaya problem, planning the rhythmic passing through a marshalling yard. Queueing processes are then studied, with their associated decision problems. Finally, the importance of time unit for the formallinguistic modeling is pointed out, and it is shown how the increase of the working time interval simplifies the model. Aggregation and defalcation in hierarchical systems are investigated and the necessity of hierarchically structured management is explained.

Paun's general strategy is to associate to each indicator imposed to an economic process a specific formal language (including the strings fitting the respective indicator) and then to consider the intersection of all these lan-

46 Solomon Marcus

guages. This strategy leads to the need to introduce many new operations with languages (their number is 12), having an interest going beyond their initial motivation. New languages account for those evolutions which fulfil a given indicator with a high quality degree.

Here are some results obtained by Paun: 1) The solutions of a variant of the salesman problem with time restrictions and of some variants of the delivery problem lead to regular languages. 2) The production processes lead to context-sensitive languages which are not context-free; however, if the semifinished products stock is bounded, then we get a regular language. A similar situation occurs in the case of scheduling problem. 3) A specific type of marshalling yard, related to Zilinskaya problem, leads to a regular language. 4) The inspector's problem and the courier's problem lead to non-context-free languages. 5) Under some conditions concerning the languages that describe the input and the serving laws, a queuing process can be modeled by a generalized sequential transducer, permitting to solve algorithmically problems such as the boundedness of the queue or the existence of some unworking moments in the process. 6) Many signalling devices for continuous or discrete processes can be designed as generalized sequential transducers. 7) The aggregation and the parcelling of information in hierarchical systems can be successfully formulated and investigated as operations with formal languages. 8) The greater the time unit in a system is the simpler the associated language in Chomsky's hierarchy or according to certain measures of syntactic complexity.

In approaching the generative typology of the languages occurring in various economic processes, Paun uses, besides the Chomskian well-known hierarchy, some other types, mainly situated between context-free and contextsensitive: simple matrix grammars, parallel matrix grammars, matrix grammars with appearance checking, scattered context grammars. Among the types of automata, Paun uses besides the classical types of finite automata, pushdown automata and linear bounded automata, the contraction automata, the generalized sequential transducers, the a-transducers etc. Here are some practical consequences, as they are stated by Paun: the effective generation of the paths fulfilling the restrictions in the variants of the salesman and the delivery problems (a computer program, called REPIGAL, has been written on this basis); the formal proof of the necessity of the top-down strategy of designing and implementing the management (information) systems; the formal proof of the hierarchically structuring the socio-economic systems; ways to reduce the complexity of management (bounding the semifinished products stocks, aggregation, etc.), and so on.

Further research is needed in order to check the applicability of the proposed solutions to other processes and the possibility to use some other tools and results from formal language theory in the modeling of economic processes. But what it is already obtained proves beyond any doubt that both economics and formal language theory take profit from their interaction. Some bibliographic references in English are available: Gh. Paun, A formal linguistic approach to the production process, Foundations of Control Engineering, 1,


1977, 3, 169-178; Generative grammars for some economic activities (scheduling and marshaling problems), Foundations of Control Engineering, 2, 1977, 1, 15-25; The influence of the degree of information aggregation on the management efficiency, CEPECA Management Training Review, 1, 1977,29-33; A formal linguistic model of action systems, Ars Semiotica, 2, 1979, 1,33-47.

2.23 Formal Grammars for Fairy-tales

Fairy-tales have a relatively simple, canonical, standard typology, pointing out repetition structures that lead, in the reading process, to recursive rules that can be simulated by generative grammars.

Repetition is an unavoidable phenomenon in any text in a natural lan-guage and it occurs at various levels: phonemes, letters, morphemes, words, morphological, syntactic or semantic categories etc. In a narrative text, some narrative entities such as events, motives, markers, narrative segments, stories become interesting and the way they are repeated becomes relevant. But the presence of all these types of units is more or less explicit, to not speak of the controversies related to their definition. It was however observed for long time that folk literature represents a favored field in this respect, the types of units which are used and their repetition being much more simple and more explicit than in other fields. The first author who proposed a method to quantify and segment a folktale was Vladimir Propp (M orfologia skazki, Moscow, N auk, 1928; Morphology of the Fairy tale, Indiana Univ, Bloomington 1958, 1968), then a lot of other authors developed further this idea, remaining however within the framework of morphological, taxinomic approach, which was, for S. Fotino, S. Marcus (see their contribution in S. Marcus (ed.), La semiotique formelle du folklore; Approche linguistique-mathematique, Klincksieck, Paris, 1978,105-141) the starting point of a formal language model of fairy tales. For Propp and his continuators such as A. Dundes, A.J. Greimas, E.M. Meletinski, C. Bremond, T. Todorov, L. Dolezel and others the investigation of narrative structures had especially in view the reduction of some variants to a small number of invariants (Propp's functions, Greimas's actants, Dundes's motifemes, Dolezel's motifemes and motifs, Bremond and Todorov's 'micronarration a structure stable'), in order to discern the narrative architecture and the link between various levels of organization of a narrative structure.

Placing in the center of attention the repetition phenomenon as the key to the understanding of the dynamics of a fairy tale, the following five steps are essential: 1) segmentation in events (e.-s); 2) grouping of events in narrative segments (n.s.-s); 3) introduction of an equivalence relation in the set of n.s.-s; each of the corresponding equivalence classes will be associated to a semantic marker (s.m.); 4) construction of some infinite languages on the terminal alphabet of s.m.-s (these languages are selected in order to simulate the recursive tendencies of the s.m.-s in the narrative development, but they

48 Solomon Marcus

essentially depend on the various possible readings); 5) construction of some generative grammars which generate the previously obtained languages.

The events of a fairy tale are obtained by segmenting its text in the most primitive units having a narrative meaning. The influence of the reader is here unimportant. These events have an essential predicative character; otherwise, they are null-events. Each n.s. is the result of concatenation of one or several events. This operation is more influenced than the preceding one by the reading we adopt. The meaning of a n.s. is, in most cases, more general, more abstract than the meaning of an event. The polysemy of a fairy tale begins here to act. Why do we need to pass from 1 to 27 Because at the level of events there is almost no repetition; at the level of n.s.-s, we begin to observe some semantic similarities which suggest the choice of s.m.-s. Events have, usually, no structural function, they are not able to reveal the essence of a fairy tale: the symmetry and the recurrence. Both of them are revealed by the syntax of the s.m.-s.

As it can be seen, the first three steps presented above belong to the Proppian tradition, helping to reduce some variants to a small number of invariants. But they are only the starting point permitting a generative reading which transforms the initial finite text into a potentially infinite one. As it was shown in Marcus, Fotino (1978), some germs of this generative attitude alreadyexist to J. Kristeva (Semeotike. Recherche pour une semanalyse, Paris, Seuil, 1962), who refers to 'the productivity called text', to Ju.M. Lotman (Struktum hudojestvennogo teksta, Izd. Iskustva, Moscow, 1970: 360), who considers the literary text as a device, to 1. Ihwe (On the foundations of a general theory of narrative structures, Poetics, 3, 1972, 5-14), who makes distinction between narrative competence and narrative performance etc. But never to such authors the investigation of the recurrences is in the center of attention. The generative typology so obtained gives a uniform and global procedure of a comparative study of fairy tales, enabling to understand the deep relation between the tendencies of symmetry and those of repetition, both essential in fairy tales. Against expectations, symmetry increases the generative complexity and the magic numbers dominating folktales, like number 3 in Romanian fairy tales, have their contribution in this respect. Romanian fairy tales belong to a scale of three generative types: regular, context-free, and matrix grammars. But inside of each of these types there is a large, practically infinite sub-typology, given by the different situations of the parameters involved (involvement of auxiliary symbols, form and number of generative rules etc.). More important is the fact that we have, for the same fairytale, different possible readings, of different generative complexity. So, it is not the fairy tale, but its reading, i.e., a specific relation between the text and the person reading it which decides the generative complexity. We will illustrate these ideas by an example.

We refer to the anecdotic fairy tale "Story from the world", which is com posed of the following events: 1. A boy is engaged as servant by a pope and they agree to an understanding; 2. the first day the pope tries to make the


boy brake the understanding, but the boy resists; 3. the next day the same thing; 4. the third day the boy got punished for braking the mischievous understanding of the master; 5. the boy's brother, Pacala, is engaged in the same working conditions; 6. the master resumes his first attempt, but Pacala resists; 7. he resumes the second one, with the same result; 8. the same thing the third time; 9. now it is the pope that, within the mutual understanding, tries to resist to Pacala; in a new happening, the agreement is respected; 10. in another happening, the same thing, and 11. the third the same thing; 12. now the odds are against the pope, he begins to give in and he decides to leave with his son to get rid of Pacala; 13. but P8.cala cheats them and follows them; 14. they decide to use a new procedure to get rid of him; 15. but Pacala cheats them again, therefore 16. the pope's patience gives in, he brakes the initial understanding, and 17. he gets his punishment. Excepting event 1, which works like a pregeneric (that is, a nonsignificant narrative segment), and event 5, which introduces a new character (so, it is not a significant narrative segment), all events are tests having the meaning of proof-happenings and are different either by the nature of the test or by the person they are aimed at. The first narrative part is composed by the e.-s 2, 3, and 4, meaning the first, the second, respectively the third test. The second part, composed by the e.-s 6, 7, and 8, repeat the meaning of e.-s 2, 3, and 4, so 2 and 6 are the same narrative segment and so are 3 and 7, 4 and 8. The third part is composed by the e.-s 9, 10, and 11 (corresponding again to some tests, so they are the same narrative segments represented by 2, 3, and 4.). The next narrative segment is formed of events 12 and 13 (a new first test), then a n.s. formed of e.-s 14 and 15 (a new second test) and a n.s. formed by 16 and 17 (a new third test).

But the analogies observed so far are rather weak, because, for example, 4 and 8 are different in respect to the addressee and the result of the test (positive or negative); 8 is different of 11 by the test itself, while 11 is different from 16-17 by a reversal position. Indeed, if the sequence 2, 3, 4 brings in opposition the pope and Pacala's brother, the second and the third sequences of length three replace Pacala's brother with Pacala himself, whereas the fourth sequence, maintaining the same heros, builds such a situation that the pope is no longer one who is testing other people, but one who is tested by pacala.

We may now express the situation as follows. 2-3-4 the dominant pope vs the dominated Pacala's brother; 6-7-8 and 9-10-11 balance between the pope and Pacala (the latter is able to face the challenge ofthe former) 12-13, 14-15, 16-17 again, like in the initial sequence 2-3-4, there is a dominant-dominated relation, but the pope is no longer the dominant, he is dominated (by pa.cala). We move in this way from the level of narrative segments to that of semantic markers. The fairy tale takes the shape

ABCabcabcABC,

where A is the semantic marker (s.m.) of a first dominant-dominated relation (2 and 12-13), B is the s.m. of a second dominant-dominated relation (3 and

50 Solomon Marcus

14-15), C is the s.m. of a third dominant-dominated relation (4 and 16-17), a is the s.m. of a first balanced relation (6 and 9), b is the s.m. of a second balanced relation (7 and 10), while c is the s.m. of a third balanced relation (8 and 11).

Now, what does it mean to read this fairy tale? It means to give some meaning to these repetitions and symmetries. The reader may be tempted to interpret the structure (*) not only as the capacity of PacaHi to dominate the pope, after the domination of his brother by Pacala, but in the stronger sense giving to Pacala the capacity to dominate anybody dominating his brother. This means to iterate the sequence ABC n times in both the beginning and the end of (*) (for each failure of Pacala's brother there is a success of Pacala). We get in this way the structure

For n running over the sequence 1, 2, 3, ... we get a formal language which is context-free, but not regular. But another reading is also possible, according to which the number of equilibrium situations is equal to the number of domination situations. We get the structure

which, for n = 1,2,3, ... leads to a matrix language which is not context-free. So, this reading is of higher complexity than that given by (**).

2.24 The Formal Language Associated to a Theatrical Play

The strategy applied to fairy tales succeeds in the field of theater, at least in the case of the classical one, where some regularities are very visible. This fact is only a part of what can be called 'mathematical theatrology', a syntagm that began its life in the XVIIIth century.

A detailed examination of a theatrical play shows the presence of some elements that 'cut' the action, such as vengeance, suicide, murder ofthe opponent and the presence of some elements which recurrently stimulate the conflict, like questions or announcements. A concrete investigation of this type was made for some of Sophocles's tragedies: Ajax, Antigone, Electra, and Oedipus Tyrannus (1. Gorun, On recurrent dramatic structures, Poetics, 6, 1977, 3/4, 287-304). The following general categories are considered: 1 = lyrical monologue, s = story told by a character, d = dialogue, c = comment, f = non-verbal action. There are scenes belonging to several such general categories. Sometimes, a refinement is used: h = happy effusion, u = lamentation, a = story of a past event, p = story of an event to come (a prophecy, a plan, a promise), c = antagonistic dialogue, m = friendly dialogue. In the analysis of


the considered four Sophoclean plays the non-verbal action I is represented by suicide or murder. On the ground of a similar dramatic function, one denotes also by I other scenes, where one of the main characters is defeated by verbal action, scenes that would be otherwise denoted by d.

A careful analysis of the play Ajax (the model described in the preceding section is valid here too) leads to its representation as follows:

auCuuuulhCpuuluCcCcC IC,

or, shortly, auCu4 IhCpu2 IUC(CC)2 IC.

Now the reading process should choose the semantic markers that deserve to be transformed in recurrent markers. In a first step, Gorun argues for the marker u, "because the specificity of the Ajax' drema dwells in the sequence of lamentations which are a lyrical comment of his mischance and this seems to be the only possible reaction of Ajax to the vexation he suffered". A careful literary and mathematical analysis (sorry we cannot reproduce them here) lead to several possible readings, each of them having its motivation. Among these readings, there is one which is not context-free and another one which is context-free, but not regular. Ajax and Antigone are characterized by a sharp distinction between their conflictual and lyrical sub-plays, which no longer appears in Sophocles' later plays. In contrast with the linearity of Ajax and Antigone, Electra and Oedipus make use of a mixture of different dramatic directions.

Daniela Gabrielescu (Syntax, semantics, and pragmatics in a theatrical play, Poetics, 6, 1977, 3/4, 319-338) analyzed the grammar of Moliere's Tartuffe and found among the possible readings one which is not contextfree, but can be generated by a matrix grammars; the explanation of this situation refers to the fact that "the segments which represent the failure of the imposture must be more numerous than those which represent its success, in order to convince all the character (inclusively Orgon and Mme Pernelle of the real nature of Tartuffe)". It is also involved the wish of Moliere to increase the symmetry of the play.

Let us recall (B. Brainerd, On a class of languages occurring in the study of theater, Discrete Mathematics, 21, 1972, 195-198; Gh. Paun, Languages associated to a dramatic work, Cahiers de Linguistique TMorique et Appliquees, 13, 1976, 2, 605-611) that, on the alphabet of the characters of a play, all possible languages describing the syntagmatic strategy of the characters are regular. This property shows a basic limitation of theater. It may be an a posteriori explanation of the fact that the syntagmatic theatrical strategy cannot be too sophisticated.

52 Solomon Marcus

2.25 Generative Mechanisms in Music

Various types of repetitions in music can be simulated by formal grammars.

Bogdan Cazimir (Semiologie musicale et linguistique mathematique, Semiotiea, 15, 1976, 1,48-57) used Lindenmayer systems in order to simulate various types of musical transformations. Cristina Patraulescu (Mecanismes generatifs du rythme palestrinien, Bull. Math. de La Soe. Sei. Math. de Roumanie, 20,1976,3/4,341-351) studied the generative mechanisms of the Palestrinian rhythm. Florentina Simionescu (The grammar of menuet (in Romanian), Studii §i Cereetilri Matematiee, 28, 1976, 2, 243-249) gave a generative typology of the menuet. A generative study of the Palestrinian melodical structures was done by Venera Turcu (The generative grammars of the Palestrinian melodical structures (in Romanian), Studii §i Cercetilri Matematiee, 30, 1978, 2,217-225). The link between generative grammars and musical harmony was investigated by Luana Irina Stoic a (Generative grammars and musical harmony, Revue Roumaine de Linguistique, Cahiers de Linguistique Theorique et Appl., 24, 1987,2,169-179).

Let us detail one of these studies. The ideal of the Palestrinian melodic line can be summed up in the formula: silence - movement - silence.

There are nine rules to which one associates nine rhytmic formulas having respectively the marks 1, 2, ... , 9. In the set of these formulas, we insert the relation R: xRy means that xy occurs in a Palestrinian rhythmic structure.

There is a graph with nine vertices: if xRy, then there is a line from x to y. The grammar of Palestrinian rhythmic structures has the terminal alphabet {a, b, e, d, e, I, g} and the non-terminal alphabet resulting from the following rules: 1) S - eAIIClhdClheA; 2) A - egeB; 3) B - eB'; 4) B' - eC; 5) C - egbDlegeE; 6) D - bE; 7) E - bE'; 8) E' - bF; 9) F - eSleGleh; 10) G - dK; 11) K - aLlbK'; 12) K' - egeL'; 13) L - aL'; 14) L' - eSleM; 15) M - leNlleSlleh; 16) N - eeR; 17) R - eSelh, where h means ee.

This is a context-free grammar. A probabilistic grammar associated to rhythm in the soprano part of Missa Papae Marcelli is also proposed, associating to x(i,j) (i,j = 1,2, ... ,9) the probability that the formula j should follow the formula i; it is a context-free grammar with 9 terminals, 4 auxiliary symbols and 22 rules, each of the rules having an associated probability vector with 22 compounds (the probability that the rule j can be applied after rule i).

A similar method is proposed for the interior musical rhythm. (For all these facts, see Patraulescu's above quoted paper.)


2.26 Formal Grammars and Games

Games are based on rules, which may have the shape of a generative grammar. Words games are particularly interesting in this respect. The case of tennis will be specially analyzed.

Ludwig Wittgenstein observes (Remarks on the Foundations of Mathematics (eds. G.H. von Wright, Rhees and G.E.M. Anscombe), Blackwell, Oxford, 1956-1967-1978, p.401) that "the rules of logical inference are rules of the language game". Calculus and computation have a ludic face. The rules of inference have a potentially infinite combinatorial capacity. There is a natural tendency and an attractive exercise to check the meaningful, interesting theorems of a formal system against the combinatorial capacity of this system. The syntactic game means to try to go beyond those theorems that answer some already existing problems and to obtain theorems whose reason could be their global coherency and their symmetry and harmony aspects. Aesthetic requirements may lead the investigation of a formal system to some developments having a purely internal motivation, with respect to the architecture of the system, but no motivation with respect to possible interpretations of the system. But it happens that some developments which initially were motivated only by aesthetic factors became latter of great practical interest.

The need to check the combinatorial possibilities of phonemes, letters, syllables, morphemes and words in a natural language lead to the development of various linguistic games. Crosswords seem to be the most famous in this respect. Sorin Ciobotaru (Jeux enigmistiques et langages formels, A Semiotic Landscape, Proceedings of the first Congress of the Intern. Assoc. for Semiotic Studies, Milano 2-6 June 1974; The language of anagramming and the generative grammars (in Romanian), Studii §i Cercetari Matematice, 28, 1976,5,521-532) proposed some formal grammars for some words games. The same author studied word puzzles by means of some analytical tools in formal languages, such as the contextual domination relation (S. Ciobotaru, An algebraic treatment of word puzzles, Foundations of Control Engineering, 2, 1977,4, 163-174). A link between word puzzles and Artificial Intelligence was pointed out in Revue Roumaine de Linguistique, Cahiers de Linguistique Theorique et Appl., 15, 1988, 1, 11-16) by Florentin Smarandache.

In the following, we will pay attention to the generative approach proposed by Gabriela Sfarti (The tennis game as a language, Revue Roumaine de Linguistique, 25, 1980, 3, 243-260).

The terminal alphabet includes the semantic marks of the main moments during an exchange of balls: a = serve; b = passing the ball over the net; c = the ball is stoped by the net; d = right service; e = wrong service; f = net right (the serve is to be repeated); g = return; h = right ball (the ball falls within the court delimitation lines); i = wrong ball (the ball falls out of the court); j = non-return (the ball is not returned by one of the players, because he could not intercept it in time); k = one player hits the ball twice at the impact moment.

54 Solomon Marcus

Denote by x one of the following four strings: gbi,gc,j, and k. The winning of a point by the serving player could be achieved in one of the following ways: 1) (af)Pabd(gbht)nx; 2) (af)Pac(af)q(abd)(gbht)nx; 3) (af)Pabe(af)q (abd) (gbh)nx, where n = 2m (m EN); p, q E N represent the number of consecutive net right serves before the first respectively the second (good or wrong) serve; t = 1, or t = 0 (t is 1 for a regular shot and 0 for a volley). With the same conditions for n,p,q, and t, the winning language M of the serving player is composed exactly of the strings of the type: (af)Pabd(gbht)ngc; (af)Pabd(gbht)n and nine more types.

The winning of a point by the receving player can take place in one of the following seven ways: 1) (af)Pac(af)qac and six more types, the last one being: (af)Pabd(bht)nx, where x is one of the following four sequences: gbi,gc,j, and k; (n = 2m + 1, m,p, q E N, t = 0 or t = 1). The language P of the receiving player consists of the following types of strings: (af)Pac( af)q ac and seven more types of strings, the last one being (af)Pabe(af)qabd ... (n = 2m+ 1, m,p, q E N, t = 0 or t = 1).

The tennis game language L is the union of M and P and it is proved that it is regular (M and P are disjoint). Tennis is a non-cooperative twopersons game with a null sum. Subsequent study of tennis involves finite non-deterministic automata, generalized sequential machines (according to the tie-break system) and Mealy automata.

By means of these tools one can rigorously distinguish between an offensive and a defensive play. It is analytically confirmed that the initial positions of the tennis players and the starting shot are determining factors for the subsequent evolution of the game.

* Many other topics, such as formal languages in international relations, in

visual arts, in the study of dreams, in the field of infinite words, in quantum computing, etc. remained outside our approach. We leave them for a next occasion.

[studies in fuzziness and soft computing] formal languages and applications volume 148 ||

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