Congruence of Bisimulation in a Non-Deterministic Call-By-Need Lambda Calculus

Lecture Notes in Computer Science

Sabel

2014

Abstract. Motivated by the question whether sound and expressive applicative similarities for program calculi with should-convergence exist, this paper investigates expressive applicative similarities for the untyped call-by-value lambda-calculus extended with McCarthy's ambiguous choice operator amb. Soundness of the applicative similarities w.r.t. contextual equivalence based on mayand should-convergence is proved by adapting Howe's method to should-convergence. As usual for nondeterministic calculi, similarity is not complete w.r.t. contextual equivalence which requires a rather complex counter example as a witness. Also the call-by-value lambda-calculus with the weaker nondeterministic construct erratic choice is analyzed and sound applicative similarities are provided. This justifies the expectation that also for more expressive and call-by-need higher-order calculi there are sound and powerful similarities for should-convergence.

Section: Resultsmentioning

confidence: 99%

“…Proof. Reflexivity holds since η : But note that ≈ ↓ is not complete using a similar example as in [15]:…”

Section: Preliminaries On Howe's Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Applicative May- and Should-Simulation in the Call-by-Value Lambda Calculus with AMB

Lecture Notes in Computer Science

Sabel

2014

“…Several methods and theoretical tools have been developed to ease the proofs, however, depending on properties of the program calculus. In this paper we concentrate on the so-called diagram-based method to prove correctness of program transformations, which was successfully used for several calculi, e.g., [7,9,11,21,18,19]. Diagram uses that are similar to ours also appear in [1].…”

Section: Introductionmentioning

confidence: 99%

Correctness of Program Transformations as a Termination Problem

Rau

Sabel

2012

Automated Reasoning

Abstract. The diagram-based method to prove correctness of program transformations includes the computation of (critical) overlappings between the analyzed program transformation and the (standard) reduction rules which result in so-called forking diagrams. Such diagrams can be seen as rewrite rules on reduction sequences which abstract away the expressions and allow additional expressive power, like transitive closures of reductions. In this paper we clarify the meaning of forking diagrams using interpretations as infinite term rewriting systems. We then show that the termination problem of forking diagrams as rewrite rules can be encoded into the termination problem for conditional integer term rewriting systems, which can be solved by automated termination provers. Since the forking diagrams can be computed automatically, the results of this paper are a big step towards a fully automatic prover for the correctness of program transformations.

“…This method was also used for other lambda calculi, see e.g [Abr90,How89,Gor99]. Application of simulation to a non-deterministic callby-need calculus was done in [Man04], where Howe's [How89,How96] proof technique is extended to call-by-need by using an intermediate approximation calculus. This was generalized to a calculus also including constructors in [SSM07].…”

Section: Introductionmentioning

confidence: 99%

A Finite Simulation Method in a Non-deterministic Call-by-Need Lambda-Calculus with Letrec, Constructors, and Case

Rewriting Techniques and Applications

Machkasova

Abstract. The paper proposes a variation of simulation for checking and proving contextual equivalence in a non-deterministic call-by-need lambda-calculus with constructors, case, seq, and a letrec with cyclic dependencies. It also proposes a novel method to prove its correctness. The calculus' semantics is based on a small-step rewrite semantics and on may-convergence. The cyclic nature of letrec bindings, as well as nondeterminism, makes known approaches to prove that simulation implies contextual equivalence, such as Howe's proof technique, inapplicable in this setting. The basic technique for the simulation as well as the correctness proof is called pre-evaluation, which computes a set of answers for every closed expression. If simulation succeeds in finite computation depth, then it is guaranteed to show contextual preorder of expressions.