D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101(2):251267, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....notion of unification, a notion of principal typing which is more general than ML s principal type property, since also the types for the free variables of terms are inferred. Introduction Since the first investigations on combinations of Lambda Calculus (LC) and Term Rewriting Systems (TRS) [13, 19, 14, 26], this topic has drawn attention from the theoretical computer science community. At first, the area of programming language design was consider to be the typical field on which the theoretical results for combinations of the two computational paradigms could better exploit their potentialities. ....

D. J. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101:251--267, 1992.

....2 fv(l) F ) 0 (F ) l 0 r . Note that for OHRS the two versions of (R) coincide because left hand sides do not overlap. Nevertheless it seems that by using the stronger form above, the confluence proof is simplified. Further variations of this technique appear in the literature [29, 6]. In the sequel R is fixed and we simply write and . Parallel reduction has a number of interesting properties. Lemma 6.1 s s holds for all terms s in long fij normal form. Proof by induction on the structure of s. 2 Because all rules are of base type we again have Lemma 6.2 If x :s t ....

Daniel J. Dougherty. Adding algebraic rewriting to the untyped lambda-calculus. In Ronald V. Book, editor, Proc. 4th Int. Conf. Rewriting Techniques and Applications, volume 488 of Lect. Notes in Comp. Sci., pages 37--48. Springer-Verlag, 1991.

....work has been carried out while the second author was at INRIA Sophia Antipolis, France, on a grant of the HCM Cooperation Network EXPRESS. The second method, which we call termination by stability, has not only useful applications but is also interesting in itself. The method is inspired from [16] where Dougherty considers untyped calculus with fi reduction in combination with a first order, single sorted, term rewriting system R. In particular, Dougherty shows that the union of fi and R is terminating on a suitably defined subset Stable(R) of the set of fi strongly normalising terms. ....

....course well known, but to show that Proposition 14 has useful applications. 3.2 Termination by Stability In this subsection we present a second technique to infer termination of an algebraic type system: termination by stability. The principle of this technique is due to Dougherty. He shows in [16] that termination of fi [ R follows from termination of fi and termination of R , provided that we restrict attention to a set of stable terms. Stability is in fact an abstract form of typing, and Dougherty s result is obtained for untyped calculus. In this subsection we adapt Dougherty s ....

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D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101:251--267, 1992.

....recent years, both in typed and untyped contexts. In the absence of types, the two systems do not interact in a very smooth manner. For instance, in [21] Klop showed that confluence, a highly desirable property in practice, is lost if a surjective pairing operation is added to the untyped LC. In [16], Dougherty provided some restrictions on terms, thus ensuring that properties that LC and TRS both possess can be preserved when these systems are combined. Instead, in the presence of types the combination proved to be much safer. Type disciplines provide an environment in which rewrite rules ....

D. J. Dougherty. Adding Algebraic Rewriting to the Untyped Lambda Calculus. In Proceedings of RTA '91, volume 488 of LNCS, pages 37--48. 1991.

.... the termination or strong normalisation of such systems and is as follows: given a terminating type system T and a terminating rewriting system R, is the combination of T and R terminating It is not surprising that this question has already received considerable attention, see for example [2, 3, 5, 9, 13, 14, 17, 18, 19, 23, 28]. However, the situation is in our opinion not yet satisfactory, since most of the proofs of termination of a combination of a type theory and a rewriting system consist basically in redoing the proof of termination of the type theory. Ideally, one would like to have a modular proof of these ....

....rewriting. Despite its extreme simplicity, it permits to obtain, in a very easy way, useful termination results for algebraic type systems. The second method, which we call normalisation by stability, has not only useful applications but is also interesting in itself. The method is inspired from [18] where Dougherty considers untyped calculus with fi reduction in combination with a first order, single sorted, term rewriting system R. In particular, Dougherty shows that the union of fi and R is strongly normalising on a suitably defined subset Stable(R) of the set SN(fi) of fi strongly ....

[Article contains additional citation context not shown here]

D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101:251--267, 1992.

....know that an algebraic pure type system is strongly normalising if its underlying pure type system is. Note that such a result would require a purely syntactic proof as no assumption is made on the algebraic pure type system. One idea would be to try to use a generalisation of Dougherty s results ([9]) However, it requires to prove subject reduction and also that the algebraic pure type system is strongly normalising with fi reduction. One approach would be to try to define a fi reduction preserving mapping from the algebraic pure type system to its underlying pure type system. ....

D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101:251--267, 1992.

....the author was at LRI, Universit e de Paris Sud. The study of systems based on calculus and algebraic rewriting has been carried out both in untyped and typed contexts. If no type discipline is imposed on the languages, the interactions between these computational models raise several problems [Klo87, Dou91]. For typed languages things work out nicely. In [BTG90] and [Oka89] it is shown that the system obtained by combining a terminating first order many sorted term rewrite system with the second order typed calculus is again terminating with respect to fi reduction and the algebraic reductions ....

D. J. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. In Proc. 4th Rewriting Techniques and Applications, Como, LNCS 488, Springer-Verlag, 1991.

....is by induction on the structure of derivations. See [1] for a (non modular) model theoretic proof of a similar result. 4.2 Algebraic rewriting Combinations of type theory and rewriting are of interest for higher order programming and proof checking. A central question in this field see e.g. [3, 7, 8, 14, 20] is whether the combination of CC and of a terminating rewriting system R is itself terminating. Most proofs of termination of a combination of a type theory and a rewriting system consist in redoing the proof of termination of the type theory. This is unsatisfactory and one would like to have a ....

....and one would like to have a modular proof of these modularity results, i.e. a proof that uses but does not re prove the facts that the type system and the term rewriting system are terminating. In [7] F. van Raamsdonk and the author exploit Dougherty s technique of normalisation by stability [14] to provide a modular proof of the following criterion. 2 Proposition 4 ( 7] Let Sigma = F ; decl) be a first order many sorted signature and let R be a terminating term rewriting system over Sigma . Let cnv be an equivalence relation and set S = DS Sigma ; cnv) Suppose that 2 The ....

D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101(2):251--267, December 1992.

....(but otherwise not changing the strategy) also produces a non terminating completion sequence Problem 50. Combinations of typed calculi with term rewriting systems have been studied extensively in the past few years [ Barbanera, 1990; Breazu Tannen and Gallier, 1989; Dershowitz and Okada, 1990; Dougherty, 1991 ] The strongest termination result allows first order rules as well as higher order rules defined by a generalization of primitive recursion. Suppose all rules for functional constant F follow the schema: F ( l[ X] Y ) v[F ( r 1 [ X] Y ) F ( r m [ X] Y ) Y ) where ....

Daniel Dougherty. Adding algebraic rewriting to the untyped lambda calculus (extended abstract). In Ron Book, editor, Proceedings of the Fourth International Conference on Rewriting Techniques and Applications (Como, Italy), volume 488 of Lecture Notes in Computer Science, pages 37--48, Berlin, April 1991. Springer-Verlag.

.... #50 1 Problem #50 Originator: Jean Pierre Jouannaud Date: December 1991 Combinations of typed calculi with term rewriting systems have been studied extensively in the past few years [Bar90] BTG89] DO90][Dou91]. The strongest termination result allows rst order rules as well as higher order rules de ned by a generalization of primitive recursion. Suppose all rules for functional constant F follow the schema: F ( l[ X ] Y ) v[F ( r 1 [ X] Y ) F ( r m [ X] Y ) Y ) where the ....

Daniel Dougherty. Adding algebraic rewriting to the untyped lambda calculus (extended abstract). In Ronald. V. Book, editor, 4th International Conference on Rewriting Techniques and Applications, volume 488 of Lecture Notes in Computer Science, pages 37-48, Como, Italy, April 1991. Springer-Verlag.

....2 fv(l) F ) 0 (F ) l 0 r. Note that for OPRSs the two versions of (R) coincide because left hand sides do not overlap. Nevertheless it seems that by using the stronger form above, the confluence proof is simplified. Further variations of this technique appear in the literature [39, 6]. Parallel reduction has a number of interesting properties. Lemma 6.1 s s holds for all terms s in long fij normal form. Proof by induction on the structure of s. 2 Because all rules are of base type we again have Lemma 6.2 If x:s t 0 then t 0 = x:t and s t for some t. More ....

Daniel J. Dougherty. Adding algebraic rewriting to the untyped lambda-calculus. In Ronald V. Book, editor, Proc. 4th Int. Conf. Rewriting Techniques and Applications, volume 488 of Lect. Notes in Comp. Sci., pages 37--48. Springer-Verlag, 1991.

....ML) in a unified framework. The study of systems based on calculi and algebraic rewriting has been carried out both in untyped and typed contexts. If no type discipline is imposed on the languages the interactions between these computational models raise several problems, as shown in [21] and [14]. For typed languages (typed versions of calculus and typed term rewriting systems) things work out nicely. In [8] and [24] it is shown that the system obtained by combining a terminating first order many sorted term rewrite system with the second order typed calculus is again terminating with ....

D. J. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. In Proc. 4th Rewriting Techniques and Applications, Como, LNCS 488, 1991.

....properties of its components, for example, combinations of manysorted rst order term rewriting systems and simply typed calculus are modular with respect to conAEuence and termination. These properties have been studied for a variety of combinations of term rewriting systems and calculi [6, 7, 8, 31, 1, 19, 10, 3, 2, 5]. Our aim is to show that a programming language based on a combination of calculus and term rewriting can be modeled in a uniform way using interaction nets. For this we will de ne a translation function that transforms an algebraic functional program into an interaction net program that ....

D. J. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. In Proc. 4th Rewriting Techniques and Applications, LNCS 488, Como, Italy.

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D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101(2):251267, 1992.

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D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101(2):251-267, 1992.

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D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. In 488, 1991. Extended version in [48].

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Daniel J. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. Information and Computation, 101(2):251--267, December 1992.

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D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. In Proc. of the 4th Int. Conf. on Rewriting Techniques and Applications, LNCS 488, 1991.

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Daniel Dougherty. Adding algebraic rewriting to the untyped lambda calculus (extended abstract). In Book [Boo91], pages 37-48.

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D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus #extended abstract#. In R. Book, editor, Proceedings of the Fourth International Conferenceon Rewriting Techniques and Applications #Como, Italy#,volume 488 of Lecture Notes in Computer Science, pages 37#48, Berlin, April 1991. Springer-Verlag.

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D. Dougherty. Adding algebraic rewriting to the untyped lambda calculus (extended abstract). In R. Book, editor, Proceedings of the Fourth International Conference on Rewriting Techniques and Applications (Como, Italy), volume 488 of Lecture Notes in Computer Science, pages 37--48, Berlin, April 1991. Springer-Verlag.

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