consistency and completeness of the theory of combinators

Consistency and Completeness of the Theory of CombinatorsAuthor(s): Haskell B. CurrySource: The Journal of Symbolic Logic, Vol. 6, No. 2 (Jun., 1941), pp. 54-61Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266656 .

Accessed: 16/06/2014 09:27

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

This content downloaded from 185.2.32.28 on Mon, 16 Jun 2014 09:27:33 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=asl

http://www.jstor.org/stable/2266656?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


THa JouVEAL or SBouc LoGC Volume 6, Number 2, June 1941

CONSISTENCY AND COMPLETENESS OF THE THEORY OF COMBINATORS

HASKELL B. CURRY

1. Introduction. The consistency of the theory of combinators, when based on the combinatory axioms, the rules of equality, and the equations

Bxyz = x(yz),

Cxyz = XZy,

Wzy = xyy,

Kxy = x,

was proved in my thesis;' likewise some theorems expressing certain completeness properties of this theory of combinatory equivalence formed the principal subject of that thesis and a succeeding paper. Further theorems of this sort were later demonstrated by Rosser;3 his theorems are, in some cases, more powerful than those just mentioned, but they apply to a weakened system. The purpose of this note is to discuss the extension of these theorems to the case where the whole of the theory of combinators is taken into account, and the rules for equality are derived from more fundamental considerations.

The paper is based on a formulation of the theory of combinators contained in a previous paper,4 acquaintance with which is presupposed.

2. The Church-Rosser theorem. A key theorem in investigations of this character is a theorem proved in 1936 by Church and Rosser." It is necessary to give a brief account of this theorem together with a generalization of it.

The theorem concerns a certain formalism, which I shall here call the X-formalism. The terms of this formalism, which I shall call X-terms, are constituted as

Received April 9, 1941. Presented to the American Mathematical Society September 10, 1940.

1 Grundlagen der kombinatorischen Logik, American journal of mathematics, vol. 52 (1930), pp. 509-536, 789-834. This paper will be referred to hereafter as GKL, and the various subdivisions of the paper will be referred to according to the system used in the paper itself. The consistency was proved in II C 1 Satz 13, on p. 799.

2 Some additions to the theory of combinators, American journal of mathematics, vol. 54 (1932), pp. 551-558. This paper will be referred to as ATC.

3A mathematical logic without variables, Annals of mathematics, vol. 36 (1935), pp. 127-150, and Duke mathematical journal, vol. 1 (1935), pp. 328-355. References to Rosser refer, when no further explanation is given, to this paper. See also the paper by Church and Rosser cited below.

4A revision of the fundamental rules of combinatory logic, this JOURNAL, vol. 6 (1941), pp. 41-53. This paper will be cited as RFR.

6 Some properties of conversion, Transactions of the American Mathematical Society, vol. 39 (1936), pp. 472-482.

54



CONSISTENCY OF THE THEORY OF COMBINATORS 55

follows. For primitive terms we have an infinite set of variables, together with certain constants,6 the nature of which is arbitrary. Further terms are formed by two operations: a binary operation (analogous to application in the- theory of combinators) which combines two X-terms 9) and 9 to form a new X-term here symbolized by '91' 7 and another binary operation combining a variable x and a term 9) containing x as free variables to form a term Xx9)1.9

Between the X-terms Church defines a relation, which he calls conversion, as follows. In the first place it has the properties of equality-i.e. it is symmetric and transitive and has the usual properties in relation to replacement. Again it holds between 9)1 and W whenever 9) is carried into 91 by a change of bound variables-i.e., a replacement of a term Xx$ by Xy[y/x]$3, where y is a variable not occurring in $ and [y/x]3 is the result of substituting y for x in $3. Finally the relation holds between 9) and 91 when 9) is carried into 9 by a reduction process (a) viz.:

(a) Replacement of (Xxj3)? by [Z2/x] J3, where [O/x] $ is the result of substituting eC for x in e (with necessary changes in bound variables to avoid collision).

Two further conventions will be expedient before we proceed. In the first place an additional relation, called reduction, shall be defined by replacing the word 'symmetric' in the above definition by 'reflexive.' Thus 9) reduces to 91 if and only if 9) is identical with W or can be transformed into 91 by a sequence of reduction processes, changes in bound variables, or both. This relation is what Church and Rosser call I-I1 conversion. In the second place, although changes in bound variables are often necessary to prevent confusion, yet throughout the rest of this paper, as in the paper by Church and Rosser, they will not be explicitly mentioned.

The theorem of Church and Rosser with which we are here concerned is the following: If 91 is convertible to 9, then there can be found (constructively) a X-term $, such that 9) and 9 reduce to a.

This theorem is capable of generalization. In the first place we can drop the requirement that 9) in Xx) shall actually contain x; i.e., we can allow that Xx9) is a term whenever x is a variable and )1 is a term. This kind of generalization is considered by Church and Rosser at the end of their paper. I shall as-

6 For the applications it is convenient to suppose that there are primitive X-terms which cannot function as variables-i.e., as the x in Xx9J.

Church writes this '19)11(9)'. It is interpreted as meaning the value obtained by applying y, considered as a function, to the argument 91. Church calls what is here designated 'X-term' a 'well-formed formula.'

8 This notion is here taken intuitively. For precise definitions of this and related notions in the X-formalism see S. C. Kleene, Proof by cases informal logic, Annals of mathematics, vol. 35 (1933), see p. 530.

9 Since the X is a prefixed binary operation, parentheses are unnecessary. Thus Xx9xO means automatically Xx(9)1), not (Xx9M)91. However parentheses will sometimes be in- serted in such cases for the sake of perspicuity, and of course they are necessary if the second of the above terms is meant rather than the first. If the other operation were indicated by a distinctive prefixed symbol, say 'a' or Chwistek's '*", no parentheses at all would be necessary.



56 HASKELL B. CURRY

sume henceforth that this generalization is adopted. Again, we may admit further kinds of reduction, viz.:

(fl) Replacement of Xx(9x), where 9)1 does not contain x as free variable, by 931.

(my) Replacement of certain sorts of terms 9), which are not replaceable by processes (a) and (p3) and do not contain any free variables nor any parts which are replaceable, by a term % not containing any free variables.

The proof of the theorem under these more general conditions is essentially the same as that given by Church and Rosser. Indeed they give themselves a special case of a reduction process (y).

The reduction processes ('y) will not be needed in this paper. But we do need the generalization where 9)1 need not contain x, and for investigation the theory of combinators with BI=I we shall need (p3).

3. Some definitions. The system ,, which forms the subject matter of this paper, is to be constituted as in Theorem 25 of RFR except as follows:"

(i) the two axioms for E are replaced by any set of axioms, called E-axioms, of the form [EX;

(ii) the rule for E is replaced by any rule or set of rules, called E-rules, such that the conclusion of each is of the form FEZ;

(iii) if the combinatory axioms are

FQWWA., i - 1, 2, *.., then FE21i

holds in addition. (iv) the primitive terms may include others besides S, K, E, S. Let 2 be a term of R. If S and K are replaced throughout 2 by S>, KA, viz.,

S>--XXXYXXZ(xz), (2) K> _ XxXyx,

then the result, which I shall write !, shall be called the X-repre8entative of 2." It is evident that 2h. is a X-term.

10?The motivation of these changes is that it may be easier to prove the consistency of stronger systems based on the theory of combinators if a normality condition ouch as is here expressed by FEX is introduced. Such conditions were introduced by Rosser. If his R 2 is replaced by his M 13 his conditions are a special case of those here formulated. For the case where R 2 is used, we may note that the theorems below hold if we interpret the E in (i) and (ii) and Type I as Rosser's 2; while in (iii) and Types II and III it is Rosser's E.

11 In case the system is formulated with B, C, W, K as primitives instead of S and K, then the former are to be replaced respectively by B),, Co, Wx, Kx, where

Bx XxXyXzx(yz),

CX XxXyXzxzy,

WX XxXyxyy.




A term 2[ of R shall be said to be convertible to a term e3 when WA is convertible to A.

The following discussion is intended to apply either to the case that Ax. I2, viz.Y

FBI = I,12

is assumed along with the other axioms of GKL, or else Ax. I2 is omitted and its place taken by the following (in Rosser's notation)

I B2I X B = BY

F B2I X C = C,

j-BI X W = WY (3)

FBI X K = K.

The latter (weaker) system will be known as M1, the former as R2. The sense of 'convertible' in the above definition is to be such that in the case of R, only reduction processes of type (a) are allowed, while in the case of S2 those of type (I) are allowed also. Presumably then the argument would apply to other modifications of R2, with corresponding definitions of conversion;"3 but I have not had such specifically in mind.

4. The consistency theorem. Let us first make the following definition: a term 2f shall be said to be of type I, II or III respectively, if and only if it is convertible into a term of the corresponding form in the following list of standard forms:

I. EX, II. Qua for I a term such that HEX,

III. PE(WQ).4 Then we have

THEOREM 1. Every term 21 asserted in 9 is of one of the types I, II, or III.15 Proof. We note first a lemma. Lemma 1. If 21 is convertible to e3 and e3 is of type I, II, or III, then St is of

the same type. For convertibility is a transitive relation; and if e is convertible to a E in

standard form, 21 is convertible to the same (. The proof of the theorem will proceed by induction on the derivation of Fif. Case 1. Suppose 21 is asserted in an axiom. If the axiom is an E axiom, then 91 is of type I by (i). If the axiom is a combinatory axiom, then 2f is of the form QXua where 2) is

12 The sign '=' is to be regarded as defined by the equivalence X= 2) 2 FQ2). This is not to be confused with the equality of ?5 below.

13 E.g., if we require that x shall not appear more than once in 9R (in order that XxSR should be a X-term), we have a form of conversion which goes with the system R with omission of the rules and axioms for W.

14 There is only one term of this form. 11 Cf. Rosser's T 18.



58 HASKELL B. CURRY

convertible to ?. Hence 2[ is convertible to QXX. By (iii) it then follows that 21 is of type II.

If the axiom is Qo, then 2f is of type III. Case 2. Suppose 2E is asserted in the conclusion of a rule, and that the theorem

holds for the premises. If the rule is an E rule, then 2f is of Type I by (ii). If the rule is one of the combinatory rules - I I[SKI I then the conclusion is con-

vertible to the premise, and the theorem follows by the lemma. Suppose now the rule is Rule :. Let the particular instance in question be

i-X2 & X3 -)3 Then W 2)8. Let e --12). Then by the hypothesis of the induction there is a V' of one of the standard forms such that e is convertible to V'. By the Church-Rosser Theorem there is a X-term i such that Zx and Q'x are both carried to $3 by reductions only. Then 3 must be of the form :-9)1, where Xx reduces to WZ and 2x to 9. Then 58' cannot be of form I, i.e., it cannot be EYE (otherwise E would be of form E9, where S' reduces to 9I). Hence d' must be of form II, i.e., Q3' is QX'T' for some term E', or V' is of form III, i.e., V' is AE(WQ).

If V' is of form II, then, using the definition of Q from Theorem 15 of RFR, we find Q' is convertible into '(TX')(TX'). Hence X and TX' are interconvertible (since X and (TV') both reduce to 9)?); and also 2) and TX'. Hence X is convertible to 2J. Then the conclusion is convertible to the second premise and the theorem follows by the lemma.

If V' is of form III, then X and E are interconvertible; and also 2J and WQ. Hence 2I is convertible to Q88, where fE3 follows from the second premise. Thus W is of type II-which completes the proof.

Remark 1. This theorem would also be valid, with suitable modifications as to type III, if the axiom Qo were replaced by any axiom forming a distinct standard form Add such that the conclusions deducible by Rule ' from that axiom as first premise are always of Type II. Likewise if the axiom Qo were replaced by a rule whose conclusion is always of Type II. Moreover if we introduced an axiom or rule leading to conclusions of the form HO2), where HEX and X is convertible to 2), the case could be handled by considering an

additional type form ZXt with HEN. Remark 2. In the above I have assumed that E is a primitive term (in the

proof that Q' is not of form I). But if E -WQ a modified proof is valid. In that case Axiom Qo is superfluous.

Remark 3. The theorem also applies to other formulations, such as that of GKL in terms of Q and HI, or that of Remark 1 to Theorem 17 of RFR in terms of P and II. For the primitives for those formulations could be defined from the primitives of the present formulation in such a way that their axioms and

16 rules are deducible from the present ones as theorems. For the case of GKL, Axiom Qo has to be replaced by Axiom Q of that paper;

for the formulation in Remark 1 to Theorem 17 of RFR, another axiom as

16 A modification of the axiom Qo is necessary in the case of GKL. Cf. Remark 1.




indicated there is appropriate. For the consistency proof then cf. Remark 1 above.

5. Completeness theorems. Let us define an equality relation between St-terms as in RFR ?3; i.e. for St-terms W and A,

is true when and only when it follows from the first system of the section cited, which system I shall call the system t.17 Then the completeness proof depends on the following theorem:

THEOREM 2. If 2I and e are St-terms ah that 2x is convertible to Z; then 9-_z

To prove this theorem let us suppose first that, in the system R' obtained by adjoining variables to S, we have Rosser's M 65 and M 69 with 'conv' replaced by '-'. Next define, for each variable x and 5V-term X, a term [x]' as in my paper on apparent variables,'8 so that

(a) [x]% does not contain x, (b) ([x]3E)x - , (e) BI([x]3)-- []x1. Then it will follow, if X is a term not involving z, that (d) Xx -X BIX [x]l, (e) X11 -x2 []X11 [X]X2, Mf ([]X)v - [2)/x]x. Next for every X-term WZ define inductively a M'-term 91W* thus: If WD is primitive, 9* 9,

(M1042)* -= 1192*

(XEn)* - [x]9o*.

Then the proof of Theorem 2 depends on the following sequence of lemmas. (Here W, Q are S'-terms and 9JZ, 9, X, are X-terms).

Lemma 1. - 2 [x*. Lemma 2. ([9/x]9)* _ [%*/x]n*. Lemma S. If 9) is carried into W by a single reduction process, then 9* _ 9** Lemma 4. If 9) is convertible to 9, then 9)* 91*. Of these lemmas Lemma 4 follows from Lemma 3. The proofs of the others

are so similar to those of Rosser's M 75, M 73, and M 74 respectively that it is not worth while to publish them separately."9 The theorem follows immediately from Lemmas 1 and 4.

17 Note that -here designates the primitive relation of the system A~; whereas in ?3 '= was a defined relation in regard to R. Of course ! can be interpreted so that the two relations are the same; but this is a different interpretation from that used in Theorem 3.

18 Apparent variables from the standpoint of combinatory logic, Annals of mathematics, vol. 34 (1934), pp. 381-404. The definition referred to is stated on page 390. Only obvious modifications are necessary to adapt it so as to be suitable for the system M1.

19 Instead of assigning a unique 9J* to every XN, Rosser defines a one-many relation "9 denotes 9N*", and then proves (M 72) that if 9N denotes 9N* and 9*, then * 94'. This procedure can, if desired, be followed here.



60 HASKELL B. CURRY

Let us now turn to the consideration of Rosser's lv 65 and M 69, which were presupposed in the above proof. For the system R2 M 65 is obvious and M 69 was proved in A TC. For the system S1 it suffices to note that Rosser's proofs will go through with only trivial modifications.20

THEOREM 3. If 2 and Q3 are M-terms such that 24 is convertible to Qf and FE I; then FQ IQW is valid ink .

Proof. Consider first a combinatory axiom, say

FQ21it3. (4)

Then (by (iii)) FE21j.

.,. (Ax. Qo, Rule F)Q2%.

Hence by the proof of RFR Theorem 13c,

FQ03iS,. (5)

From (4), (5), and RFR Theorem 15 it follows that

241[F~tIA. (6)

This holds for all the combinatory axioms. Hence by RFR Theorem 9 all the axioms and rules of the system ! hold if _ is interpreted as j[FI1.

20 For the reader who is familiar with GKL the argument of Rosser, up to M62, can be simplified as follows. In GKL, through II D, Ax. I2 has been used only in II B 2, II B 4 Satz 4, and in II D 1, 3 ff. to drop out factors of the form BmL. Without Ax. I2, however, we can replaced II B 4 Satz 4 by XXI = IXX. Then we can derive Rosser's M 53 directly; this requires first his F 12, F 16, then a proof by induction and II B 4 Satz 6 that

BmI X Bm+nI = Bm+nI X BnI = Bm+nI,

whence his M 53 follows as he indicates. (Here I write Bm instead of Bm because Rosser has shown it is the mth power of B). The proof of II D 2 Satz 1 then goes through with obvious modifications (cf. Rosser's F 26). Next by the method of The universal quantifier in combinatory logic ?3 Theorem 8 (Annals of mathematics vol. 32 (1931), p. 165), we can show without using variables or the theorems of II E, that if X is a normal combinator (in the sense of GKL, or better ATC convention 3) of minimum order m and degree n, then

B'I X X - X X BVI - X

for pam and q5n, and also

Bm+kI X X = X X Bn+kI

(or, if preferred

CBm+k+lX = BX X Bn+k).

(The first of the properties to be established does not satisfy hypothesis (a) of Theorem 8 (l.c.); but if we use Theorem 7 and replace "order m and degree n" by "minimum order m and degree n," then Theorem 8 can be established for normal combinators; cf. remark on p. 164, (l.c.).) From these properties it follows that where a factor of the form B-I occurs in a regular combinator, it can be absorbed in the factor to the left of it or passed across it, and this process can be continued until all these factors are at the beginning, where they may be combined by M 53. With this modification in II D 1 Satz 1 the trans- formation to a normal form of Rosser type goes through as in II D 3 ff.



CON SISTENCY OF THE THEORY OF COMBINATORS 61

Now suppose that 21 and e3 satisfy the hypothesis of the theorem. Then by Theorem 2

911[F1I1. Hence, W[ F]Q .., by the definition of [f],

FQW!K -+ FQ2W. But the premise of this implication is true by hypothesis, Axiom Qo and Rule E; hence the conclusion follows, q.e.d.

THE PENNSYLVANIA STATE COLLEGE



consistency and completeness of the theory of combinators

Documents