On the confluence of lambda-calculus with conditional rewriting

The confluence of untyped \lambda-calculus with unconditional rewriting is now well un- derstood. In this paper, we investigate the confluence of \lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of M\"uller and Dougherty for unconditional rewriting. Two cases are considered, whether \beta-reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. We also provide examples showing that outside these conditions, modularity of confluence is difficult to achieve. Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules

Narrowing Trees for Syntactically Deterministic Conditional Term Rewriting Systems

A narrowing tree for a constructor term rewriting system and a pair of terms is a finite representation for the space of all possible innermost-narrowing derivations that start with the pair and end with non-narrowable terms. Narrowing trees have grammar representations that can be considered regular tree grammars. Innermost narrowing is a counterpart of constructor-based rewriting, and thus, narrowing trees can be used in analyzing constructor-based rewriting to normal forms. In this paper, using grammar representations, we extend narrowing trees to syntactically deterministic conditional term rewriting systems that are constructor systems. We show that narrowing trees are useful to prove two properties of a normal conditional term rewriting system: one is infeasibility of conditional critical pairs and the other is quasi-reducibility

Confluence of an Extension of Combinatory Logic by Boolean Constants

We show confluence of a conditional term rewriting system CL-pc^1, which is an extension of Combinatory Logic by Boolean constants. This solves problem 15 from the RTA list of open problems. The proof has been fully formalized in the Coq proof assistant

Faithful (meta-)encodings of programmable strategies into term rewriting systems

Rewriting is a formalism widely used in computer science and mathematical logic. When using rewriting as a programming or modeling paradigm, the rewrite rules describe the transformations one wants to operate and rewriting strategies are used to con- trol their application. The operational semantics of these strategies are generally accepted and approaches for analyzing the termination of specific strategies have been studied. We propose in this paper a generic encoding of classic control and traversal strategies used in rewrite based languages such as Maude, Stratego and Tom into a plain term rewriting system. The encoding is proven sound and complete and, as a direct consequence, estab- lished termination methods used for term rewriting systems can be applied to analyze the termination of strategy controlled term rewriting systems. We show that the encoding of strategies into term rewriting systems can be easily adapted to handle many-sorted signa- tures and we use a meta-level representation of terms to reduce the size of the encodings. The corresponding implementation in Tom generates term rewriting systems compatible with the syntax of termination tools such as AProVE and TTT2, tools which turned out to be very effective in (dis)proving the termination of the generated term rewriting systems. The approach can also be seen as a generic strategy compiler which can be integrated into languages providing pattern matching primitives; experiments in Tom show that applying our encoding leads to performances comparable to the native Tom strategies

A Fast Decision Procedure For Uniqueness of Normal Forms w.r.t. Conversion of Shallow Term Rewriting Systems

Uniqueness of normal forms w.r.t. conversion (UNC) of term rewriting systems (TRSs) guarantees that there are no distinct convertible normal forms. It was recently shown that the UNC property of TRSs is decidable for shallow TRSs (Radcliffe et al., 2010). The existing procedure mainly consists of testing whether there exists a counterexample in a finite set of candidates; however, the procedure suffers a bottleneck of having a sheer number of such candidates. In this paper, we propose a new procedure which consists of checking a smaller number of such candidates and enumerating such candidates more efficiently. Correctness of the proposed procedure is proved and its complexity is analyzed. Furthermore, these two procedures have been implemented and it is experimentally confirmed that the proposed procedure runs much faster than the existing procedure

Decidability for Non-Standard Conversions in Typed Lambda-Calculi

This thesis studies the decidability of conversions in typed lambda-calculi, along with the algorithms allowing for this decidability. Our study takes in consideration conversions going beyond the traditional beta, eta, or permutative conversions (also called commutative conversions). To decide these conversions, two classes of algorithms compete, the algorithms based on rewriting, here the goal is to decompose and orient the conversion so as to obtain a convergent system, these algorithms then boil down to rewrite the terms until they reach an irreducible forms; and the "reduction free" algorithms where the conversion is decided recursively by a detour via a meta-language. Throughout this thesis, we strive to explain the latter thanks to the former

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions