Alonzo Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 36(3):472--482, May 1936.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Lemma 6.8. That projection equivalence is a congruence follows from the defining identities for a term residual system, and from Theorem 6.12 showing that residuals can be computed by orienting these identities into a TRS. The third item is trivial. # Remark 6. 14 (i) Church and Rosser showed in [6] that # calculus with # reduction is confluent. Their proof is based on a construction, projection, for finding the common reduct of two diverging reductions. The construction is based on the observation that a common reduct of two diverging steps can be found by contracting the set of residuals ....

A. Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, January to June 1936.

....5.3(i) ut 5.9. Proposition. Let S be generalized non dependent, weakly normalizing, and clean, and s 2 S be negatable. For all M 2 Term s : M ] s Gamma 2 SN fi ) M 2 SN fi Proof. By Lemma 5.7 and Lemma 5.8. ut 5.2. A conservation result In this subsection we prove a version of Church s [7] conservation theorem see [1] for expressions. 5.10. Definition. Let K L mean that K fi L by a left most reduction. 5.11. Definition. Let S be generalized non dependent. An s 2 S is secure if, for all N 2 Neu s , N 2 SN fi . 5.12. Lemma. Let S be generalized non dependent, s 2 S be ....

A. Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:11--21, 1936.

....the techniques needed to 186 HENK BARENDREGT prove G odel s incompleteness theorem 2 . Then [28] isolated the (untyped) lambda calculus from the system T by deleting the part dealing with logic and keeping the essence of the part dealing with functions. This system was proved consistent by [31], who showed the confluence of # reduction. Curry, who also wanted to build a foundation for mathematics based on functions (in his case in the form of combinators that do not mention free or bound variables) found a paradox for a system with a similar aim as T , that is very easy to derive, ....

....Let us first give the promised argument that eager functional languages are computationally complete. Every computable (recursive) function is lambda definable in the #I calculus (see [30] or [6, Theorem 9.2. 16] In the #I calculus a term having a normal form is strongly normalizing (see [31] or [6, Theorem 9.1.5] Therefore an eager evaluation strategy will find the required normal form. The first functional language, LISP, was designed and implemented by [83] The evaluation of expressions in this language is eager. LISP had (and still has) considerable impact on the art of ....

A. Church and J. B. Rosser, Some properties of conversion, Transactions of the American Mathematical Society, vol. 39 (1936), pp. 472--482.

....executing first a and then b, then we can reach the same state by first executing b and then a, and vice versa. a b b a b b a a s1 s2 s2 Figure 2.1: Commutativity of actions a and b. 2 Remarks about Definition 2.13. ffl a and b commute exactly when a and b have the diamond property [25, 53]. 2 We now define the desired condition. 20 Definition 2.14 (arb compatible) Programs P 1 ; PN are arb compatible exactly when they can be composed (Definition 2.10) and any action in one program commutes (Definition 2.13) with any action in another program. 2 2.2.2 Equivalence of ....

A. Church and J. B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, 1936.

....using such rules. Finally, the , system of Principia was generalized to arbitrary systems of truth functions that are complete in the sense that every truth function can be obtained by their suitable iteration; in this last case, only the main result is given. 5 In Alonzo Church s dissertation [2] of 1926, Church proposed to do for set theory (including Cantor s theory of transfinite numbers) what the creators of non Euclidean geometry had done for geometry. Lobachevsky had explicitly considered the parallel postulate of Euclid as a mere assumption the consequences of whose denial could be ....

Alonzo Church, Alternatives to Zermelo's assumption, Transactions of the American Mathematical Society, vol. 29 (1926), pp. 178--208.

....combinatory reduction systems (CRSs) Khasidashvili s expression reduction systems (ERSs) and Nipkow s higher order pattern rewriting systems (PRSs) 1 Introduction This paper is concerned with a method to prove confluence of rewriting systems. It s an extension of some confluence results in [CR36, Hue80, Toy88, Klo80, Kha92, Raa93, Tak, MN94, Oos94, ORb] and we refer the reader to these papers and to the handbook chapters [DJ, Klo] for motivation and for standard definitions as well. Here we will mainly be concerned with proving our result: Left linear development closed PRSs are confluent. Let s explain the terminology used. A rewrite system ....

....It took two years before looking at it again and realising that a straightforward adaptation of the measure function did the trick. The proof is modular in the following sense. The basis of the result is formed by a more or less abstract theory of independence of redexes as found at many places ([CR36, Klo80, HL, Kha92, Hue93, Oos94, Mel95]) and briefly recapitulated here. For orthogonal systems this immediately yields confluence. The development closed condition requires on top of that also a term structure of the objects of the rewriting system. The known relaxation of orthogonality to weak orthogonality (having only trivial ....

Alonzo Church and J. B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, January to June 1936.

....is presented in Section 3. The proof is based on a labelled calculus. The alternation of labels mirrors the alternation of abstractions mentioned above. The proof establishes a relationship between normalization and superdevelopments [49, 36] an extension of the classical notion of development [14]. We show that the number of superdevelopments needed to reduce a hyperbalanced term M to its normal form can be determined at the beginning of reduction just by analyzing the structure of M . Due to the parallel nature of superdevelopments, this 4 also gives a lower bound on the steps taken to ....

A. Church, J.B. Rosser. Some properties of conversion. Transactions of American Mathematical Society 39:11-21, 1936.

....is not a development. In contrast, in Finite Family Developments in Section 4 distinct residuals can be contracted by distinct rules. Finiteness of developments (and strengthened versions of it) have been studied extensively in the literature for various classses of rewriting systems (see e.g. CR36, Sch65, Hin78, Klo80, Kha92, Raa, Oos94, Melon] In Subsection 3.1 a simple proof of FD is presented for the class of PRSs. In Subsection 3.2 upperbound information is added to the termination proof of Subsection 3.1, yielding (exact) upperbounds on the lengths of marked rewrite sequences. As a ....

Alonzo Church and J. B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, January to June 1936.

....on descendants This paper does not intend to give a complete historical account of the origins of the residual notion in lambda calculus and term rewriting, but we will shortly remember some of the prominent early contributions. The notion of residual seems to originate with Church Rosser [CR36], where it as used in the proof of the Church Rosser theorem. There, and in Church [Chu41] one finds a lengthy verbal description of the notion of residual of a fi redex (after a sequence of ff and fi reductions) A detailed definition of residual for fi calculus in the same style as that of ....

A. Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, 1936.

....P; Q 2 SN fi . By Lemma 6.9 also Pfx : Qg 2 SN fi . By the fundamental lemma of perpetuality for developments, M 2 SN fi . ut There are many proofs of the finite developments theorem in the literature; the following is an incomplete list. The theorem was first proved by Church and Rosser [12, 13] for I ; they also sketch a proof for K . 6 Curry and Feys [15] and Schroer [61] give full proofs of the theorem for K . Other proofs were later given independently by Hyland [27] and Hindley [21] Barendregt et al. 4] subsequently simplified Hyland s proof see also [2] Xi [82] gives a ....

....has also been proved in several ways for various notions of higher order rewrite systems. Klop [38] proves it for orthogonal combinatory reduction systems by means of his technique to reduce weak normalization to strong normalization. Van Oostrom [49, 51] proves finiteness of 6 See the end of [13], or the beginning of Chapter V of [12] 36 developments for orthogonal higher order rewriting systems and for pattern rewriting systems. Each of these two results implies finite developments for orthogonal combinatory reduction systems. Melli es [45] gives an axiomatic formulation of ....

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A. Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:11--21, 1936.

.... we often use orthogonal instead of non overlapping (see Figure 1) One can give an inductive definition of performing a non overlapping set of rewrite steps in one go in the style of Tait Martin Lof [Bar84] For our purposes it is more convenient to do this via the possible developments [CR36] of the set. Definition 8 (simultaneous) 1. A development of a non overlapping set of rewrite steps is (a) the empty rewrite sequence if the set is empty, and (b) an arbitrary rewrite step from the set, followed by a development of the residual of the set after that rewrite step otherwise. 2. ....

Alonzo Church and J. B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, January to June 1936.

....solve the problem by changing or eliminating the notion of named variable . These seem too radical for the task at hand. Instead, we will circumvent the problem by giving a different meaning to (4.1) and (4. 2) so they are actually the same mathematical object, as was done in [Bar80, page 26] and [CR36] In this approach, we identify two expressions if we can transform one into the other by renaming bound variables, and whenever we write an expression, we really mean the equivalence class containing that expression. In this case the proper definition of substitution still forbids variable ....

Alonzo Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 36(3):472--482, May 1936. BIBLIOGRAPHY 307

....of calculus obtained by requiring that for all abstractions x:M , x occur free in M exactly once, see e.g. 12] Proof. Obvious: all abstractions in I have form x:M where x 2 FV(M ) ut Proposition31 (Curry and Feys [7] Klop [15] In Ifi and Ifij, F l is normalizing. Corollary32 (Church [6]) i) In Ifi and Ifij: WN(M ) SN(M ) ii) In Ifi and Ifij all strategies are perpetual. Proof. i) follows by the following equivalences that hold both in Ifi and Ifij due to Corollary 12, Lemma 30, and Proposition 31: WN(M ) 9n : F n l (M ) 2 NF , 9n : F n 1 (M ) 2 NF , SN(M ) ii) ....

A. Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:11--21, 1936.

....functions. There were many axioms to deal with logical notions. The system T turned out to be inconsistent, as was shown by Church s students Kleene and Rosser [1935] using a tour de force argument involving all the techniques needed to prove Godel s incompleteness theorem 2 . Then Church [1936] isolated the (untyped) lambda calculus from the system T by deleting the part dealing with logic and keeping the essence of the part dealing with functions. This system was proved consistent by Church and Rosser [1936] who showed the confluence of fi reduction. Curry, who also wanted to build a ....

....involving all the techniques needed to prove Godel s incompleteness theorem 2 . Then Church [1936] isolated the (untyped) lambda calculus from the system T by deleting the part dealing with logic and keeping the essence of the part dealing with functions. This system was proved consistent by Church and Rosser [1936], who showed the confluence of fi reduction. Curry, who also wanted to build a foundation for mathematics based on functions (in his case in the form of combinators that do not mention free or bound variables) found a paradox for a system with a similar aim as T , that is very easy to derive, see ....

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Church, Alonzo and J.B. Rosser [1936] Some properties of conversion, Transactions of the American Mathematical Society 39, 472--482.

....FOUNDATION 16 is considered as an optimal reduction strategy although it may be less efficient in some cases. If the sequence of reductions for a expression is known to terminate with an answer, the choice of reduction order for its evaluation is immaterial and the Church Rosser theorem [ChRo36] guarantees the uniqueness of the result using either strategy. The referentially transparent nature of functional languages, which is so beneficial for simultaneous evaluation of expressions and which we alluded to above, is a consequence of this important result. Some implementations of ....

A Church and JB Rosser, Some Properties of Conversion. Transactions of the American Mathematical Society, 39, pp. 472-482.

....s. Then it is easy to see that a rewriting system is UN iff all of its redexes are perpetual. Studying uniform normalization therefore reduces to studying the perpetuality of redexes, and this has already been studied quite extensively. The classical results in this direction are Church s Theorem [CR36], stating that the I calculus is UN, and the Conservation Theorem of Barendregt, Bergstra, Klop and Volken [BBKV76, Bar84] stating that fi I redexes are perpetual in the calculus. Bergstra and Klop [BK82] gave a necessary and sufficient criterion for the perpetuality of fi K redexes in every ....

....ordering on reductions in orthogonal rewriting systems [L ev80, HL91] Even though our criteria are simple and intuitive, they are strong tools in proving strong normalization from weak normalization in orthogonal (typed or type free) rewrite systems. We will show that all known related criteria [CR36, BBKV76, BK82, Klo80, Klo92, Kha94c] can be obtained as special cases. We will also demonstrate that uniform normalization for a number of variations of fi reduction (which cannot be derived from previously known perpetuality criteria) Plo75, dGr93, BI94, HL93] is an immediate consequence of our criteria. The format of ERSs is ....

Church A. and Rosser, J.B., Some properties of conversion. Transactions of the American Mathematical Society, 39:472-482, 1936.

....the diamond property if whenever a Gamma b and a Gamma c, then there exists d such that b Gamma d and c Gamma d. In diagrams: a b c ) a b c d Also, a relation Gamma is said to be Church Rosser if the transitive closure Gamma has the diamond property. Theorem 2. 1 (Church, Rosser [12]) The relations fi Gamma and fij Gamma are Church Rosser. 2 This theorem was first proved by Church and Rosser in 1936 [12] Since then, the proof has been adapted and streamlined in various ways by Tait, Martin Lof, Girard and others. One can find a proof in Barendregt s book [5] The ....

....a b c ) a b c d Also, a relation Gamma is said to be Church Rosser if the transitive closure Gamma has the diamond property. Theorem 2.1 (Church, Rosser [12] The relations fi Gamma and fij Gamma are Church Rosser. 2 This theorem was first proved by Church and Rosser in 1936 [12]. Since then, the proof has been adapted and streamlined in various ways by Tait, Martin Lof, Girard and others. One can find a proof in Barendregt s book [5] The Church Rosser Theorem has several important consequences. As a first consequence, one proves that for each pair of fi convertible ....

A. Church and J. B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, 1936.

....Of course, there is in general no guarantee that such a direct proof exists for a particular consequence of the equations represented by R. When that is the case, i.e. when the relation R is contained in R ffi R , the system is called Church Rosser , after a property in [ Church Rosser, 1936 ] See Figure 1(a) Equivalent properties are defined in Section 4 and methods of establishing them are described in Section 7. 2.4 Decision Procedures One of our main concerns is in decision procedures for equational theories; an early example of such a procedure for groups is [ Dehn, 1911 ] ....

A. Church and J. B. Rosser, Some properties of conversion, Transactions of the American Mathematical Society 39, pp. 472-482 (1936).

....that reduce terms having an infinite reduction, which we call 1 terms, to 1 terms. Therefore, studying the UN property reduces to studying perpetuality of redexes. The latter has already been studied quite extensively in the literature. The classical results in this direction are Church s Theorem [CR36], stating that the I calculus is uniformly normalizing, and the Conservation Theorem of Barendregt et al. [BBKV76, Bar84] stating that fi I redexes are perpetual in the calculus. Bergstra and Klop [BK82] give a sufficient and necessary criterion for perpetuality of fi K redexes in every ....

....orthogonal rewriting systems [L ev80, HL91] Despite the fact that our criteria are simple and intuitive, they appear to be strong tools in proving strong normalization from weak normalization in orthogonal (typed or type free) rewrite systems. We will show that previously known related criteria [CR36, BBKV76, BK82, Klo80, Klo92, Kha94c] can be obtained as special cases. We will also derive the UN property for a number of variations of fi reductions [Plo75, dGr93, BI94, HL93] which cannot be derived from previously known perpetuality criteria, as immediate consequences of our criteria. 2 Conditional Expression Reduction ....

Church A., Rosser, J. B. Some Properties of Conversion. Transactions of the American Mathematical Society, 39:472-482, 1936.

....than orthogonality, because the rules may operate on the same part of a term, but in that case, both applications should result in exactly the same term. weak orthogonality implies confluence. It is well known that orthogonality implies confluence for many classes of term rewriting systems ([CR36, Ros73, Klo80, Raa93, Nip93]) Basically, two methods are used to prove this. The first method known as confluence via developments , is due to Church and Rosser [CR36] and employed in the first three papers above. The second method, due to Tait and Martin Lof (see [Bar84] is known as confluence via parallel reductions , ....

....implies confluence. It is well known that orthogonality implies confluence for many classes of term rewriting systems ( CR36, Ros73, Klo80, Raa93, Nip93] Basically, two methods are used to prove this. The first method known as confluence via developments , is due to Church and Rosser [CR36], and employed in the first three papers above. The second method, due to Tait and Martin Lof (see [Bar84] is known as confluence via parallel reductions , and employed in the last two papers. In the case of weak orthogonality (and generalisations thereof) confluence has been proved only for ....

Alonzo Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, 1936.

....ff conversions can be performed implicitly, or, as Barendregt [Bar80] puts it: Terms that are ff congruent are identified. Conventions of this kind are common right from the beginning of the study of the calculus (see, for example, the original paper with a proof of the Church Rosser theorem [CR36]) In order to avoid any possible problems which arise from this convention, a common route is to go to combinatory calculi [CF58] or to use de Bruijn indices [dB72] It is interesting to note that de Bruijn s motivation for his notation for terms came from a proof of the Church Rosser theorem, ....

.... M 00 M Gamma M 0 red M M 0 The representation in Elf is a direct transcription. term term type. infix none 10 name C refl : M M. sym : M M M M. trans: M M M M M M . red : M M M M . The Church Rosser theorem [CR36] now states that if M M 0 then there exists some N such that M Gamma N and M 0 Gamma N . We are taking the liberty of simply using a judgment J to stand for the meta language proposition J is derivable or J is evident . We hope that this will not lead to any confusion on the ....

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Alonzo Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 36(3):472--482, May 1936.

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Alonzo Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 36(3):472--482, May 1936.

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Alonzo Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 36(3):472--482, May 1936.

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Alonzo Church, Alternatives to Zermelo's assumption, Transactions of the American Mathematical Society, vol. 29 (1927), pp. 178--208, Ph.D. thesis, Princeton.

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Alonzo Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 36(3):472--482, May 1936. 306

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A. Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:472--482, 1936.

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A. Church, J. Rosser, Some properties of conversion, Transactions of the American Mathematical Society 39 (1936) 472482.

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A. Church and J. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 36(3):472-482, May 1936.

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A. Church and J.B. Rosser, Some properties of conversion, Transactions of the American Mathematical Society 39 (1936) 472-482.

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