c-command dependencies as tsl string constraints
TRANSCRIPT
Proceedings of the Society for Computation in Linguistics
Volume 2 Article 22
2019
C-Command Dependencies as TSL StringConstraintsThomas GrafStony Brook University, [email protected]
Nazila ShafieiStony Brook University, [email protected]
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Recommended CitationGraf, Thomas and Shafiei, Nazila (2019) "C-Command Dependencies as TSL String Constraints," Proceedings of the Society forComputation in Linguistics: Vol. 2 , Article 22.DOI: https://doi.org/10.7275/4rrx-x488Available at: https://scholarworks.umass.edu/scil/vol2/iss1/22
C-command dependencies as TSL string constraints
Thomas GrafStony Brook University
Nazila ShafieiStony Brook University
AbstractWe provide a general formal framework foranalyzing c-command based dependencies insyntax, e.g. binding and NPI licensing, froma subregular perspective. C-command rela-tions are represented as strings computed fromMinimalist derivation trees, and syntactic de-pendencies are shown to be input-output tier-based strictly local over such strings. Thecomplexity of many syntactic phenomena thusis comparable to dependencies in phonologyand morphology.
1 Introduction
The last decade has seen tremendous progress inthe subregular analysis of phonology and mor-phology (see Chandlee 2017, Heinz 2018 and ref-erences therein). The subregular program is con-cerned with identifying proper subclasses of theregular languages that are sufficiently powerful fornatural language, and to convert these tighter com-plexity bounds into typological predictions andnew learning algorithms.
Extending the subregular approach to syntaxhas proven more challenging because syntactic de-pendencies are not regular over strings, let alonesubregular (Chomsky, 1957; Huybregts, 1984;Shieber, 1985; Michaelis and Kracht, 1997; Ko-bele, 2006). But syntactic dependencies seem tobe subregular over tree structures (Graf, 2012).More recently, a series of arguments (Graf,2018a,b; Graf et al., 2018) has been presented thatdependencies in phonology, morphology, and syn-tax are all of comparable subregular complexityonce one picks a suitable data structure (string-likevs. trees). More precisely, all dependencies areconjectured to belong to natural extensions of theclass tier-based strictly local (TSL; Heinz et al.,2011; Graf and Mayer, 2018).
While this claim is well-supported for phonol-ogy and morphology, it is on shakier ground for
syntax. It seems to hold for subcategorizationand displacement (Merge and Move in Chom-skyan terms) and some kinds of NPI-licensing(Vu, 2018). But it may fail for constraints based onc-command, which are numerous in syntax. Thispaper argues that these dependencies are also TSLgiven the right kind of representation.
We define a general procedure for converting c-command dependencies to string languages. Intu-itively, our procedure amounts to little more thanmaking c-command a basic relation in syntac-tic representations (cf. Frank and Vijay-Shanker,1999). Formally, we represent Minimalist gram-mar derivations (Stabler, 1997) as dependencytrees (cf. Brody, 2000; Kobele, 2002) and then as-sociate each node with a string of nodes that c-command it. Principle A of binding theory, forexample, then requires that a string that ends ina reflexive must contain a potential binder some-where between the reflexive and the head of itsbinding domain. Based on a preliminary analy-sis of c-command dependencies in the literature,we contend that they all fit into extensions of TSLthat have been proposed for phonology and mor-phology.
The paper proceeds as follows. We first discusssome general preliminaries (§2), including MGs,dependency trees, and subregular syntax. Afterthat, c-strings are introduced as a string-based rep-resentation of c-command relations (§3.1), andvarious syntactic phenomena are characterized inthese terms (§3.2). We then show that the depen-dencies of these phenomena are input-output tier-based strictly local (Graf and Mayer, 2018) over c-strings (§4), and we explain why this likely holdsfor all c-command dependencies in syntax. Somefinal complications introduced by movement arediscussed in §5.
205Proceedings of the Society for Computation in Linguistics (SCiL) 2019, pages 205-215.
New York City, New York, January 3-6, 2019
2 Preliminaries
While the ideas presented in this paper are fairlyintuitive, a sufficiently rigorous description re-quires a broad formal background. We first intro-duce some basic notation (§2.1) and then discussMinimalist grammars as a formal model of syntax(§2.2). We also explain how their derivation can beencoded as dependency trees, which will be usedin §3 to easily compute the c-commanders for anygiven node. After that, §2.3 surveys some recentdevelopments on subregular syntax and why thispaper pursues a different route.
2.1 Formal language theoryWe follow the standard notation for strings: ⌃ bydefault denotes a fixed alphabet, " is the emptystring, S⇤ is the Kleene closure of set S, S+ isS⇤ � {"}, S is ⌃� S, and instead of {a} we maysimply write a.
2.2 Minimalist grammarsIn order to assess the complexity of syntacticdependencies, we need a formal model of syn-tax. We choose Minimalist grammars (MGs; Sta-bler, 1997, 2011). MGs are inspired by Chomsky(1995) but can incorporate a wide range of ideasfrom the syntactic literature (see Graf 2013 andreferences therein). This makes them an ideal test-ing ground for c-command dependencies.
Every MG is a finite set of lexical items thatare annotated with strings of features to drive thestructure-building operations Merge and Move.
Example 1 (Merge). The noun car only carriesthe category feature N�. The determiner the car-ries a matching selector feature N+. The full fea-ture string for the is N+D�. This indicates thatthe first selects a noun and then acts as DP thatcan be selected by anything with the matching se-lector feature D+. The possessive marker ’s, onthe other hand, must select both a noun as its com-plement and a DP as its specifier. Hence its featurestring is N+D+D�.
Example 2 (Move). The determiner which is sim-ilar to the, except that it also undergoes wh-movement after it has been selected. Thereforeits feature string must include a licensee featurewh� at the very end: N+D�wh�. At least insome cases the landing site of the wh-mover isan empty CP-specifier. In order to allow for this,the grammar must contain a lexical item whosephonetic exponent is " and whose feature string
is T+wh+C�. Here wh+ is a licensor feature thatmarks the empty C-head as a landing site for wh-movement.
As every MG builds a phrase structure tree viaa sequence of Merge and Move steps, every syn-tactic tree can be fully specified by its derivation.The derivation, in turn, can be represented as atree, which may take the form of a dependencytree as shown in Fig. 1 (cf. Brody, 2000; Kobele,2002). In an MG dependency tree, a node d isthe daughter of node n iff n selects d. For MGs,this means that X� on d and a matching X+ ofn were checked as part of the same Merge opera-tion. We assume that the linear order of siblingsis the reverse of the order of selection, so that thefirst argument of a head is its rightmost daughter.Movement is not indicated in dependency trees asit can be inferred from the distribution of licensorand licensee features.
2.3 MGs as subregular tree languages
It has been known for a long time that everyMG’s set of well-formed derivation trees formsa regular tree language (Michaelis, 2001; Kobeleet al., 2007). While these results build on a dif-ferent derivation tree format, they also hold forthe dependency tree format in this paper becausethe latter can be obtained from the former by aregularity-preserving tree transduction.
It has also become clear, though, that MGderivation tree languages are not just regular, butsubregular (Graf, 2012). Most recently, Graf(2018b) showed that they belong to the formalclass TSL. This class was first defined for stringsto handle some phenomena in phonology (Heinzet al., 2011). Graf lifts TSL from strings to trees.Intuitively, TSL masks out parts of the tree thatare irrelevant for a given dependency and thenputs locally bounded constraints on the remain-der. Graf limits his attention to Merge and Move,but follow-up work has since argued that NPI-licensing and morphological case also fit into thisclass (Vu, 2018; Vu et al., 2019). If correct, thiswould not only suggest a new kind of computa-tional parallelism between language modules, butmight also prove useful for parsing and learningalgorithms.
Although we believe this work to be verypromising, it is still based on detailed case stud-ies and has not yielded any general properties thatguarantee that TSL is a safe upper complexity
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" :: T+top+C�
" :: V+nom+T�
likes :: D+D+V�
’s :: N+D+D�nom�
John :: D� mother :: N�
this :: N+D�top�
car :: N�
CP
DP
this car
C0
C TP
DP
John D0
’s mother
T0
T VP
t V0
likes t
Figure 1: A dependency tree representation of an MG derivation, with its corresponding phrase structure tree to theright. Movement is indicated by arrows for the reader’s convenience, it is not part of the tree. Crucially, movingphrases remain in their base position in the dependency tree.
bound for all attested syntactic dependencies. Inaddition, the step from strings to trees compli-cates the comparison to phonology and morphol-ogy. Several subregular classes besides TSL havebeen argued to play a central role in these domains,e.g. SP (Rogers et al., 2010) and IBSP (Graf, 2017,2018a). The latter is difficult to lift from stringsto trees, whereas the former seems ill-suited forsyntactic dependencies based on preliminary workof ours. For these reasons, a string-based viewof syntactic dependencies is a welcome alterna-tive that makes it easier to compare phonology andsyntax. As we argue in this paper (§4.2), it alsoprovides some general insights that make it verylikely that all c-command dependencies fall withina subregular region that is also occupied by depen-dencies in phonology and morphology.
3 C-command relations as strings
We are finally in a position to demonstrate howc-command dependencies reduce to constraints onstrings. We first present the general idea in §3.1and then apply it to several examples in §3.2. Weonly describe the relevant string languages andwait until §4 to show that they fit into (extensionsof) TSL.
3.1 Idea and definitionThe dependency graphs make it very easy to en-force c-command requirements for any given node— at least if one puts aside movement, as we willdo until §5. Until then, “c-command” only means“c-command between base positions before move-ment”.
If u is a left sibling of v in an MG dependencytree t, then (the phrase headed by) u c-commands
(the phrase headed by) v. This is illustrated byJohn and mother in Fig. 1. In addition, v is c-commanded by almost every node that dominatesit. Looking again at Fig. 1, we see that the T-head and the C-head dominate John in the de-pendency tree and c-command John in the phrasestructure tree (at the position marked by the tracein Spec,VP).
One minor wrinkle is the relation betweenheads and their specifiers. The former do not c-command the latter, but they do so according toour definition. Hence our notion of command isactually a hybrid of c-command and m-command(Aoun and Sportiche, 1983). One may call itd[erivational]-command to emphasize this differ-ence: a node d-commands all its right siblings, allits daughters, and everything that is d-commandedby its siblings or daughters. As far as we cantell, the minor differences between c-commandand d-command are immaterial for all syntacticdependencies that have been argued to involve c-command, and we will continue to use c-commandas the general term instead of d-command when-ever this does not cause any confusion.
Note that d-command is a highly natural rela-tion over derivation trees as it encodes prominencewith respect to feature checking. Based on thedefinition above, x d-commands y iff x had oneof its Merge features checked more recently thany. A specifier s of head h d-commands a com-plement c of h because s has been selected morerecently than c. And h d-commands s because his selected by some other head after s has been se-lected. C-command as a basic relation of syntaxis difficult to motivate (but see Epstein et al. 1998and Chametzky 2000, 2011). D-command, on the
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other hand, falls out naturally from the MG fea-ture calculus and thus is a plausible primitive forsyntactic dependencies.1
One can automatically calculate for every nodea string representation of the relevant command re-lations. Let T be a tree such that node m has thedaughters d1, . . . , di, d, di+1, . . . , dn, with n � 0.The immediate c[ommand]-string ics(d) of d isthe string d di · · · d1. For every node n of T , itsc[ommand]-string cs(n) is recursively defined asshown below, where · indicates string concatena-tion:2
cs(n) :=
(ics(n) if n is the root of T
ics(n) · cs(m) if m is n’s mother
Note that c-strings sort d-commanders in abottom-up fashion, which usually yields the re-verse of their linear order in the string. For im-proved readability, we omit all features and re-place empty heads by their category.Example 3. Consider once more the dependencygraph in Fig. 1. While the immediate commandstring of car is just car, its c-string is car this ’slikes T C.
The next subsection illustrates how a number ofsyntactic phenomena can be reinterpreted as con-straints on c-strings. Even more strikingly, theseconstraints are fairly intuitive, and as we will seein §4, their subregular complexity is largely in linewith what has been found in phonology and mor-phology.
3.2 Examples of syntactic constraintsIt is beyond the purview of this paper to exhaus-tively analyze every syntactic dependency thathas been claimed to involve c-command. Con-sequently, we limit ourselves to a representativesample of NPI-licensing (§3.2.1) and binding ef-fects (§3.2.2–§3.2.4). Due to space restrictions,we are also limited to high-level discussions thatillustrate the general ideas but do not account forall details and potential complications (cf. §3.2.5).
1If one wants to include c-command relations created bymovement, it suffices to broaden the definition such that anykind of feature checking induces d-command relations, notjust those affecting category and selector features.
2C-strings over dependency trees are closely related to theconcept of spine languages (cf. Martens et al., 2008; Graf,2012). In spine languages, ics(n) and cs(m) would be sep-arated by a distinguished symbol, e.g. |. We omit this herefor simplicity, but point out that preliminary work of ourssuggests that | is needed in some cases to correctly definelocality domains.
3.2.1 NPI licensingNPIs can occur in a variety of contexts, but thecentral syntactic licensing condition is c-commandby a downward-entailing operator. We can capturethis in terms of conditions on string languages. Ifa c-string ends in an NPI, then it must also containa licensor such as no or nobody. So the requiredstring language is NPI · · · {no, nobody} · · · . Thisensures that no NPI can ever occur without a c-commanding licensor.
Example 4. Consider the ill-formed ⇤Every stu-dent said that the train ever arrives on time. As-suming that ever is a functional head between Vand T, its c-string is ever T that every said T C.Since this string is not a member of the NPI-licensing language, ever is not licensed and thesentence is illicit.
As is witnessed by Have you ever been to Rus-sia, NPIs are also licensed in questions. We canreadily accommodate this by adding interrogativeC-heads to the set of licensors. Overall, then, thec-string pattern for NPI-licensing is NPI · · · Lic· · · , where Lic is a downward entailing quantifieror an interrogative C-head.
3.2.2 Locally bound reflexivesPrinciple A requires that reflexives must be boundwithin their binding domain, usually a TP contain-ing an abstract subject. When construed as a dis-tributional constraint, this reduces to the presenceof a DP in the binding domain that c-commandsthe reflexive and matches its �-features (person,number, and gender).
In MGs, every TP with an abstract subject isheaded by a lexical item with category feature T�
and licensor feature nom+. Let T be the short-hand for any such LI, R[�] a reflexive, and D[�] amatching determiner. Since MGs have finite lex-icons, one can list for each reflexive which deter-miners match its �-features. Then Principle A cor-responds to the string language R[�] T
⇤D[�] · · · ,
where T⇤ is a (possibly empty) string that contains
no instances of T .
Example 5. Contrast the well-formed John saidthat Mary likes herself with the ill-formed ⇤Johnsaid that Mary likes himself. The former has thec-string herself Mary likes T that John said T C.Since Mary is �-feature compatible with herselfand occurs between the reflexive and the T-head,Principle A is satisfied. In the ill-formed sentence,herself is replaced by himself. The closest com-
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patible D-head is John, but since a finite T-headoccurs between John and the reflexive, Principle Ais violated.
3.2.3 Non-locally bound reflexivesThe reflexive sig in Swedish only partially obeysPrinciple A. It still requires a syntactic binder, butsaid binder must not be part of the same bindingdomain (Kiparsky, 2002). The corresponding c-string language is R[�] · · · T · · · D[�] · · · .
Example 6. Suppose for the sake of argumentthat English himself behaves like Swedish sig, andconsider again John said that Mary likes himself/herself. Then the sentence with herself would beill-formed, but not the one with himself. And asthe reader may verify for themselves, only the c-string of himself fits the pattern above.
3.2.4 Principle BPrinciple B regulates the distribution of syntacti-cally bound pronouns. Pronouns in English canalways be discourse-bound, so their distributionis unproblematic. Marathi, on the other hand,has pronouns that must be syntactically bound(Kiparsky, 2002). Like standard pronouns, theyI) must be non-locally bound, and II) cannot havethe same referent as a local c-commander. Condi-tion II) makes the distribution of Marathi pronounsmuch harder than that of Swedish sig-reflexives.
From a distributional perspective, Principle Benforces a counting condition: if a pronoun isc-commanded by n pronouns in the same bind-ing domain, then at least n potential antecedentsmust c-command the binding domain in order tofurnish at least one grammatical reading. Tem-porarily putting aside �-features, this yields thestring template ↵T� such that ↵,� 2 ⌃⇤, ↵starts with a pronoun, and the total number of R-expressions (i.e. D-heads) in � exceeds the num-ber of pronouns in ↵; pronouns are counted as R-expressions in � because they can bind pronounsin ↵. Without further assumptions, this pattern iscontext-free and thus much more complicated thanany of the previous c-string languages.
But Graf and Abner (2012) show that eventhough a binding domain may contain an un-bounded number of pronouns, most of them canshare the same referent. Pronouns within distinctadjuncts can pick out the same referent, and otherconstructions such as coordination are very lim-ited in how they interact with bound pronouns. Asa result, there is an upper bound n on the number
of distinct antecedents that are needed to licenseall pronouns in a binding domain. This guaranteesregularity and, as we will see in §4.3, subregularityby virtue of being definable in IO-TSL.
Example 7. Suppose for the sake of argumentthat English pronouns always require syntactic an-tecedents. Then the sentence John thinks that helikes him is ungrammatical, whereas John told Billthat he likes him is well-formed. This immediatelyfollows from the c-string of the second pronoun ineach sentence. One is him he likes T that Johnthinks T C and thus does not furnish enough an-tecedents for him and he. The other c-string pro-vides the required minimum of two antecedents,namely John and Bill: him he likes T that Bill Johntold T C.
3.2.5 CaveatsEach one of the phenomena above has severalcomplications. What counts as an NPI licensordepends on the position of the NPI. For exam-ple, every can license an NPI if it occurs in a rela-tive clause modifying the argument of every. Simi-larly, many reflexives are exempt from Principle Awhen they occur inside certain adjuncts, and somelanguages like Icelandic allow for long-distancebinding only under specific circumstances. Theseare serious complications. A careful analysis ofany one of these phenomena arguably requires afull-length journal paper. Nonetheless we believethat these complications would render the stringpatterns more complicated, but not more complex.The reason for this is explained at the end of thenext section, which analyzes the subregular com-plexity of syntactic dependencies over c-strings.
4 Subregular complexity
We have successfully reduced c-command depen-dencies to regular string languages. But regularityis a loose upper bound. As we explain next, thec-string languages from the previous section all fitinto IO-TSL, the class of input-output tier-basedstrictly local string languages. This class has re-cently been proposed as an upper bound on phono-tactic complexity (Graf and Mayer, 2018). We firstpresent the defintion IO-TSL (§4.1) and provide ageneral IO-TSL strategy for capturing dependen-cies over c-strings (§4.2). This strategy is thenexemplified with IO-TSL grammars for each oneof the templates from the previous section (§4.3).In §4.4, we explain why IO-TSL is a meaningful
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upper bound that makes testable predictions aboutthe shape of syntactic dependencies.
4.1 Defining IO-TSLIO-TSL is an extension of the strictly local (SL)languages. A language is SL iff it can be de-scribed by a list of finitely many forbidden sub-strings. IO-TSL enhances this with a local tierprojection mechanism: whether a symbol is pro-jected on a tier depends on the symbol itself, itslocal context in the string (i.e. up to m symbolsbefore and/or after it), and up to n previous sym-bols on the tier.
Example 8. Unbounded tone plateauing forbidsany instances of the low tone L to occur betweentwo high tones H. So HLLL and LLLH are well-formed, but HLLLH is not. One can capture thisby projecting L iff it occurs immediately to theright of H, and projecting H iff the previous sym-bol on the tier is L. The strings above would getthe tiers L, empty, and LH, respectively. If one for-bids the substring LH on tiers, HLLLH is ruled outas desired while HLLL and LLLH are still gener-ated.
The rest of this subsection formalizes this in-tuitive idea. Let ⌃ be some fixed alphabet ands 2 ⌃⇤. The set fk(s) of k-factors of s consists ofall the length-k substrings of ok�1snk�1, whereo,n /2 ⌃ and k � 1.
Definition 1. A stringset L ✓ ⌃⇤ is strictly k-local (SL-k) iff there is some G ✓ (⌃[ {o,n})k
such that L = {s 2 ⌃⇤ | fk(s) \ G = ;}.
Intuitively, G defines a grammar of forbiddensubstrings that no well-formed string may con-tain. The class SL of strictly local stringsets isS
k�0 SL-k.
Example 9. The string language (ab)+ is gener-ated by the grammar G := {on,ob, aa, bb, an}and thus is SL-2. For instance, aba is illicitbecause f2(aba) \ G = {an} 6= ;, whereasf2(abab) \ G = ;.
Example 10. The finite language of allstrings of length 3 is SL-4. It is generatedby the grammar {abcd | a, b, c, d 2 ⌃} [{abcd | a = o, d = n, b, c 2 ⌃ [ {o,n}}.
The tier projection mechanism of IO-TSL isdefined in terms of contexts that specify when agiven symbol should be added to the tier. An(i, j)-context c is a 4-tuple h�, b, a, ti with � 2 ⌃,
t a string over ⌃ [ {o} of length j � 1, anda and b strings over ⌃ [ {o,n} of combinedlength i � 1. The basic idea is that c specifiesthat � should be projected whenever both of thefollowing hold: it occurs between the substrings b(look-back) and a (look-ahead), and the tier con-structed so far ends in t. Given a set of contextsc1, c2, . . . , cn, we call it an (i, j)-context set iff forevery cm (1 m n) there are im i andjm j such that cm is an (im, jm)-context.
Definition 2. Let C be an (i, j)-context set.Then the input-output strictly (i, j)-local (IOSL-(i, j)) tier projection ⇡C maps every s 2 ⌃⇤
to ⇡0C(oi, sni,oj), where for � 2 ⌃ and
a, b, t, u, v, w 2 (⌃ [ {o,n})⇤, it holds that⇡0
C(ub,�av, wt) is
" if �av = ",�⇡0
C(ub�, av, wt�) if h�, b, a, ti 2 C,⇡0
C(ub�, av, wt) otherwise.
Example 11. Let ⌃ := {a, b, c} and consider thetier projection that always projects the first andlast symbol of the string, always projects a, neverprojects c, and projects b only if the previous sym-bol on the tier is a. This projection is IOSL-(2,2).The context set contains all the contexts below,and only those:
• h�,o, ", "i for all � 2 ⌃,
• h�, ",n, "i for all � 2 ⌃,
• ha, ", ", "i,
• hb, ", ", ai.
Definition 3. A stringset L ✓ ⌃⇤ is input-output tier-based strictly (i, j, k)-local (IO-TSL-(i, j, k)) iff there exists an IOSL-(i, j) tier pro-jection ⇡C and an SL-k language K such thatL := {s 2 ⌃⇤ | ⇡C(s) 2 K}. It is IO-TSL iff itis IO-TSL-(i, j, k) for some i, j, and k.
The class TSL-k defined in Heinz et al. (2011) isidentical to IO-TSL-(1, 1, k). This shows that IO-TSL is indeed a generalization of TSL. In fact,I-TSL and O-TSL have been independently pro-posed in computational phonology (Baek, 2017;De Santo and Graf, 2017; Mayer and Major, 2018;Yang, 2018), and Graf and Mayer (2018) showthat the combination of the two into IO-TSL fur-nishes the additional power that is required forSanskrit n-retroflexion. Under the assumption that
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dependencies in phonology and syntax are of com-parable complexity, IO-TSL is a natural candidatefor a tighter upper bound on the complexity ofconstraints on c-strings.
4.2 General IO-TSL strategyThe dependencies in §3.2 — and syntactic depen-dencies in general — all share the common prop-erty that even though there is no upper bound onthe length of c-strings, only a finitely boundednumber of elements actually matter. Which ele-ments matter depends only on the local contextand which symbols are already on the tier. Thisallows for a very general IO-TSL strategy: con-struct tiers such that they only contain the rele-vant elements. Then there are only finitely manydistinct tier configurations, and hence the set ofwell-formed tiers is finite. As every finite stringlanguage is SL-k for some k (cf. Ex. 10), sep-arating the well-formed tiers from the ill-formedones becomes trivial. So the central challenge foran IO-TSL treatment of c-command dependencieslies in the construction of tiers, not the constraintson those tiers.
The tier construction process also follows a gen-eral template. We project tiers from left to right,and the very first element is always projected —this is an input-sensitive projection step. We thenuse an output-sensitive projection strategy to onlyproject relevant elements.
Example 12. Suppose the first symbol to be pro-jected is an NPI. Then only an NPI-licensor canbe projected next. After the first NPI-licensor hasbeen projected, no more symbols are put on thetier. Since a single NPI-licensor is enough, wenever project more than one. So the tier either hasthe form NPI NPI-licensor or just NPI. The formeris well-formed, the latter is not.
Example 13. If the first element is a reflexive thatis subject to Principle A, only T-heads and match-ing D-heads need to be projected. Again nothing isprojected after this second projection step as Prin-ciple A is already satisfied or violated dependingon whether the second symbol is T or D[�]. Thetier then has one of the following three forms: re-flexive D, reflexive T, or just reflexive. The first oneis well-formed, the other two ill-formed.
If the first symbol on the tier is not subjectto any constraints, then nothing else is projected.Proceeding in this fashion, we can define a sin-gle IO-TSL grammar that generates every well-
formed c-string while forbidding those that violateone of our string patterns from §3. This shows thatthe whole system of syntactic dependencies is IO-TSL, not just each individual dependency.
4.3 Example grammars for dependenciesLet us now apply this idea to the c-string tem-plates from §3, repeated here for the readers con-venience:
NPI · · · Lic · · ·R[�] T
⇤D[�] · · ·
R[�] · · · T · · · D[�] · · ·↵T� (↵ starts with pronoun,� has more R-expressions than ↵ has pronouns)
Now consider the corresponding projectioncontexts for any arbitrary MG G. We first includeh�, ",n, "i for all � 2 G so that the first symbolis always projected. Next we add hl, ", ", ti for alll 2 G and t 2 G⇤ such that:
NPI t is an NPI and l an NPI licensor, or
Reflexive t is a reflexive, and l is either a T-headcarrying nom+ or a D-head with matching �-features, or
sig either t is a sig-reflexive and l is a T-head car-rying nom+, or t = uv with u a sig-reflexiveand v a T-head carrying nom+ and l is a D-head with matching �-features.
The definition for sig is convoluted because it ac-counts for both projection of T after sig and pro-jection of D[�] after T and sig.
Next we forbid all bigrams of the form lp suchthat either l is an NPI and p is not an NPI-licensor,or l is a reflexive and p is not a matching D-head.Note that this includes cases where p is no LI at all,but rather the start/end of a tier. We also forbid alltrigrams that start with a sig-reflexive but do notcontinue with a suitable T-head and a matching D-head.
Example 14. Consider once more the illicit NPIin example 4. Given the c-string ever T that everysaid T C, the constructed tier is just ever, whichis forbidden. If every is replaced with no, the tierbecomes ever no and thus is well-formed. If everwere always instead, the tier would just be always,which is never illicit because no special licensingis needed. Note that the tier never grows past everno, even for sentences like No student told no pro-fessor that no train ever arrives on time.
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Example 15. Suppose as in example 6 that En-glish reflexives behaved like Swedish sig and thusrequire a non-local binder. The c-string herselfMary likes T that John said T C results in the illicittier herself T. But if herself is replaced by himself,one obtains the well-formed tier himself T John in-stead. This is also the tier for much longer sen-tences such as Bill thinks Peter doubts that Johnsaid that Mary likes him.
For Principle B, the projection strategy is verysimilar.
• Project the first element, which is a pronounthat must be syntactically bound.
• Also project such pronouns if the tier so farcontains i such pronouns and nothing else,where i is less than some fixed bound n.
• Project a T-head carrying nom+ if the tieronly contains pronouns so far.
• Project an R-expression if the tier alreadycontains a T-head and at most n � 1 R-expressions.
With this projection strategy, the longest possibletier is of the form pron T Rn, and all illicit tierscan be filtered out by (2n + 1)-grams.
As pointed out earlier, this analysis ignorescomplicating factors such as additional licensingconfigurations for NPIs, or �-feature matching inPrinciple B. But this does not affect the coreproperty that guarantees the IO-TSL nature ofc-command dependencies: Each dependency re-quires only a finite number of elements on thetier, and by considering the entire tier built so farone can ensure that no extraneous material is pro-jected. In addition, each LI can only be subjectto finitely many licensing conditions. These twofacts jointly entail that the length of tiers can al-ways be finitely bounded, which makes it trivialto provide an SL grammar that rules out all il-licit tiers. It is because of this fixed bound on thenumber of elements that matter for any given c-command dependency that IO-TSL is a safe upperbound on the complexity of syntactic dependen-cies over c-strings. This is also the reason why wecontend that IO-TSL could accommodate empiri-cally more adequate characterizations of the phe-nomena in §3 — the number of relevant lexicalitems would still be finitely bounded within anygiven c-string.
Admittedly, this strategy comes at the poten-tial cost of large contexts. Fairly small contextsseem to suffice for realistic examples, though, anda smarter, less brute-force strategy may be able toreduce their size even further. We would not besurprised if most c-command dependencies turnout to belong to IO-TSL-(3, 3, 3) or perhaps evenIO-TSL-(2, 2, 2).
4.4 Limits of IO-TSL dependencies
IO-TSL makes some strong predictions aboutwhat shape linguistic dependencies can take andhow multiple c-command conditions may be in-terwoven.
Example 16. Consider an unattested variant of thelong-distance Principle A that applies to Swedishsig. In this variant, the reflexive and the antecedentnot only have to be separated by a T-head, buteach intervening T-head must be c-commanded bysome functional head F that is lower than the nexthigher T-head. So the c-string language is not just· · · D[�] · · · T · · · R[�], but rather ↵R[�] such thatI) ↵ contains at least one D[�] and, II) F occursbetween every two instances of T.
This string language is not IO-TSL because wecan no longer omit projecting all T-heads to thetier. To see this, contrast the well-formed pat-tern D[�](F T )⇤ F T (F T )⇤R[�] against the ill-formed D[�](F T )⇤ T (F T )⇤R[�]. The only dif-ference is the unlicensed T-head in the middle, sothis T-head must end up on the tier in order to dis-tinguish well-formed from ill-formed strings. Butthere is no context that can uniquely identify justthis T-head without projecting other T-heads or F-heads. The set of well-formed tiers then would bean infinite subset of {T, F}⇤ D[�] {T, F}⇤ R[�],which is not SL unless the distance between D[�]and R[�] is finitely bounded. Since there are noadditional factors that guarantee such a bound onthe distance between reflexive and licensor, ourunattested variant of sig-licensing is not IO-TSL.
This example highlights a crucial limitation ofIO-TSL: a long-distance dependency cannot applyacross an unbounded number of long-distance de-pendencies that interact with it. The F -licensingof T -heads would be unproblematic if it were anindependent constraint that must always be satis-fied, rather than just being an extra condition onsig-licensing. For then it would be a condition onall c-strings that start with T-heads and could beomitted in c-strings that start with reflexives. Al-
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ternatively, F -licensing of T -heads could be cap-tured if the distance between F and T were lo-cally bounded. In this case, one could project Fand T only once and skip all other locally licensedT -heads. Similarly, it would suffice to project atmost one unlicensed T -head. The result would betiers of the form D[�]↵R[�] where ↵ is T F T orF T T . Then the tier language would once againbe finite. So a long-distance dependency can inter-act with other dependencies, but either the numberof those dependencies or their locus of applicationmust be finitely bounded.
It is worth mentioning that this unattested ex-ample pattern is easily defined in first-order logic,which so far has been the only safe upper boundon syntactic dependencies (Graf, 2012). The IO-TSL perspective of c-command dependencies thusimproves on previous work by ruling out some lin-guistically undesirable patterns.
5 Adding movement
The discussion so far has largely ignored the ef-fects of movement on c-command relations. Un-der the standard raising analysis, no does notc-command ever from its base position in [Notrain]i has ever seemed to me ti to ti arrive ontime. The c-command relation is derived by move-ment. Our current notion of c-strings fails to cap-ture this.
For most dependencies discussed in this paper,it only matters that a dependent element has atleast one licensor in its c-string. In this case,movement effects can be accommodated by mod-ifying the construction of c-strings such that amover is also copied into its landing sites. Thec-string for ever in the previous example would beever no has C after the mover no is copied into theposition corresponding to its landing site.
But this covers only movement of a licensor.Sometimes, a dependent element is licensed onlybecause it has moved out of a locality domain intoa higher position where it is accessible to its licen-sor. In such cases, the c-string would be truncatedby deleting all material between the licensor andthe relevant occurrence, effectively pushing the li-censee into a higher position.
This strategy shifts a lot of the burden to the cor-rect construction of c-strings, which may be par-ticularly complicated when both the licensor andthe licensee move. The construction of the appro-priate c-string from a dependency tree is still de-
finable in first-order logic and hence subregular,but this is a very generous upper bound. An in-depth exploration of movement from a subregularperspective has to be left for future work.
Conclusion
We have defined a string-based representationformat over dependency trees that allows for c-command dependencies to be easily evaluated forany given node. The dependencies over thesestrings all fall within the class IO-TSL, whichwas first defined for phonology (Graf and Mayer,2018). This paper marks but a first step towards asubregular theory of syntactic dependencies, and alot remains to be done.
The current approach only measures the com-plexity of a syntactic dependency with respect toa specific node. To check the whole dependencytree, one has to evaluate the c-string of each node.We do not know whether the TSL-approach ofGraf (2018b) provides a method for doing so. Aslong as all dependencies are IO-TSL, though, thewell-formedness of the whole dependency tree canbe verified by a deterministic top-down tree au-tomaton with a look-ahead of 1. This implies sub-regular complexity and may even allow for highlyefficient parsing algorithms.
Dependencies that go beyond c-command can-not be handled with our approach. This includesbinding via sub-command, parasitic gaps, andacross-the-board movement as an exception to thecoordinate structure constraint. It remains to beseen whether they can be accommodated with treetiers as proposed in Graf (2018b), or whether acompletely new perspective is needed for thesephenomena.
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