Combining first order algebraic rewriting systems, recursion and extensional lambda calculi (original) (raw)
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Combining algebraic rewriting, extensional lambda calculi, and fixpoints
1996
It is well known that confluence and strong normalization are preserved when combining algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual contraction rule for η, or recursion together with the usual contraction rule for surjective pairing.
Confluence via strong normalisation in an algebraic λ-calculus with rewriting
Electronic Proceedings in Theoretical Computer Science, 2012
The linear-algebraic λ-calculus and the algebraic λ-calculus are untyped λ-calculi extended with arbitrary linear combinations of terms. The former presents the axioms of linear algebra in the form of a rewrite system, while the latter uses equalities. When given by rewrites, algebraic λ-calculi are not confluent unless further restrictions are added. We provide a type system for the linear-algebraic λ-calculus enforcing strong normalisation, which gives back confluence. The type system allows an abstract interpretation in System F.
Rewriting with extensional polymorphic λ-calculus
1996
We provide a confluent and strongly normalizing rewriting system, based on expansion rules, for the extensional second order typed lambda calculus with product and unit types: this system corresponds to the Intuitionistic Positive Calculus with implication, conjunction, quantification over proposition and the constant True.
On constructor rewrite systems and the lambda-calculus
Automata, Languages and Programming, 2009
We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate each other with a linear overhead. In particular, weak call-by-value betareduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in an term rewrite system can be simulated by a constant number of beta-reduction steps. This is relevant to implicit computational complexity, because the number of beta steps to normal form is polynomially related to the actual cost (that is, as performed on a Turing machine) of normalization, under weak call-by-value reduction. Orthogonal constructor term rewrite systems and lambda-calculus are thus both polynomially related to Turing machines, taking as notion of cost their natural parameters.
1993
We add extensional equalities for the functional and product types to the typed λ-calculus with not only products and terminal object, but also sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing (thus decidable) rewriting system for the calculus, that stays confluent when allowing unbounded recursion. For that, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We first prove the calculus to be weakly confluent, which is a more complex and interesting task than for the usual λ-calculus. Then we provide an effective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non extensional system. These results give us the confluence of the full calculus, but we also show how to deduce confluence without the weak confluence property, using only our technique of simulating expansions.
Polymorphic Rewriting Conserves Algebraic Strong Normalization and Confluence
1989
We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants. We show that if a many-sorted algebraic rewrite system R is strongly normalizing (terminating, noetherian), then R + β + η + type-β + type-η rewriting of mixed terms is also strongly normalizing. We obtain this results using a technique which generalizes Girard's “candidats de reductibilité”, introduced in the original proof of strong normalization for the polymorphic lambda calculus. We also show that if a many-sorted algebraic rewrite system R has the Church-Rosser property (is confluent), then R + β + type-β + type-η rewriting of mixed terms has the Church-Rosser property too. Combining the two results, we conclude that if R is canonical (complete) on algebraic terms, then R + β + type-β + type-η is canonical on mixed terms. η reduction does not commute with algebraic reduction, in general. However, using long η-normal forms, we show that if R is canonical then R + β + η + type-β + type-η convertibility is still decidable.
Polymorphic rewriting conserves algebraic confluence
1992
Abstract We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants.
Confluence of algebraic rewriting systems
Mathematical Structures in Computer Science, 2021
Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that proves local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce a critical branching lemma for rewriting systems on algebraic models whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.
Separability and translatability of sequential term rewrite systems into the lambda calculus
2001
Orthogonal term rewrite systems do not currently have any semantics other than syntactically-based ones such as term models and event structures. For a functional language which combines lambda calculus with term rewriting, a semantics is most easily given by translating the rewrite rules into lambda calculus and then using well-understood semantics for the lambda calculus. We therefore study in this paper the question of which classes of TRS do or do not have such translations.
Natural Rewriting for General Term Rewriting Systems
2004
We address the problem of an efficient rewriting strategy for general term rewriting systems. Several strategies have been proposed over the last two decades for rewriting, the most efficient of all being the natural rewriting strategy . All the strategies so far, including natural rewriting, assume that the given term rewriting system is a left-linear constructor system. Although these restrictions are reasonable for some functional programming languages, they limit the expressive power of equational languages, and they preclude certain applications of rewriting to equational theorem proving and to languages combining equational and logic programming. In this paper, we propose a conservative generalization of natural rewriting that does not require the rules to be left-linear and constructor-based. We also establish the soundness and completeness of this generalization.