Rewriting systems with limited distance forbidding context

A mechanical proof of the Knuth---Bendix Critical Pair Theorem in the higher-order language of the theorem prover PVS is described. This well-known theorem states that a Term Rewriting System is locally confluent if and only if all its critical pairs are joinable. The formalization of this theorem follows Huet's well-known structure of proof in which the restriction on strong normalization or Noetherian was dropped and the result presented as a lemma. In order to formalize the Knuth---Bendix Critical Pair Theorem we rely on previously developed PVS theories for abstract reduction systems, named ars, and term rewriting systems, named trs, which were built upon the PVS libraries for finite sequences and sets. On the one hand, the theory trs is composed of subtheories for dealing with the structure of terms, for replacements of subterms and substitutions and jointly with the theory ars it allows for adequate specifications of elaborate notions of term rewriting systems such as the one of critical pairs. On the other hand, ars specifies basic definitions and notions of abstract reduction systems such as reduction, termination, normal forms, and confluence as well as non basic concepts such as strong normalization.

Given a term rewriting system R, a term t is inductive reducible if every ground instance ts of it is reducible. A pair (t, s) of terms is inductive co-reducible if every ground instance (ts, ss) of it for which ts and ss are distinct, is reducible. Inductive (co)-reducibility has been proved to be the fundamental tool for mechanizing inductive proofs, together with Knuth-Bendix completion (Jouannaud and Kounalis [JK, 86 and 89]).In [JK, 86 and 89] an algorithm for testing inductive reducibility is also presented which is tractable in pratical cases, but restricted to left-linear term rewriting systems. The solution of the inductive-(co)-reducibility problem, for the general case, turned out to be surprisingly complicated. Decidability of inductive reducibility for arbitrary term rewriting systems has been first proved by Plaisted [PLA, 85] and next by Kapur, Narendran, and Zhang [KNZ, 87]. However Plaisted's and Kapur, Narendran, and Zhang's algorithms amount to intractable computation, even in very simple cases.We present here a new algorithm for the general case which outperforms Plaisted's and Kapur, Narendran and Zhang's algorithms and even our previous algorithm in case of a left-linear term rewriting system. We then show how to adapt it to check for inductive co-reducibility.KeywordsEquationsTerm rewriting systemsProof by consistencyTest SetsGround terms in normal form

On abstract normalisation beyond neededness

This paper generalizes the Huet and Lévy theory of normalization by neededness to an abstract setting. We define Stable Deterministic Residual Structures (SDRS) and Deterministic Family Structures (DFS) by axiomatizing some properties of the residual relation and the family relation on redexes in an Abstract Rewriting System. We present two proofs of the Relative Normalization Theorem, one for SDRSs for regular stable sets, and another for DFSs for all stable sets of desirable ‘normal forms’. We further prove the Relative Optimality Theorem for DFSs. We extend this result to deterministic Computation Structures which are deterministic Event Structures with an extra relation expressing self-essentiality.

A PVS Theory for Term Rewriting Systems

Polymorphic Rewriting Conserves Algebraic Confluence

This paper introduces positive/negative conditional term rewriting systems, with rules of the generic form: $$u = v \Lambda u' \ne v' \Rightarrow \lambda \to \rho ,$$ as they often appear in algebraic specifications. We consider the algebraic semantics of such systems (viewed as sets of axioms). They do not in general have initial models; however, we show that they admit quasi-initial models, that are in some sense extremal within the class of all models. We then introduce the subclass of reducing rewrite systems, constrained by the condition: λ>ρ,u,v,u',v' (for some reduction ordering >). For such systems, we show that an optimal rewrite relation → may be defined, and constructed as a "limit". We prove the total validity of an interpreter that computes the normal forms of terms for →. It is then shown that when → is confluent, the algebra of normal forms is a quasi-initial model. We state a general result about the converse. Lastly, we present a complete critical-pair criterion a la Knuth-Bendix to check for the confluence of reducing systems.

Unique normal forms for lambda calculus with surjective pairing

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. This result is an important step towards a new theory of reduction based on expansion rules, and gives a natural interpretation to the notion of second order η-long normal forms used in higher order resolution and unification, that are here just the normal forms of our reduction system.

On declarative rewriting for sound and complete union, intersection and negation types

We propose a new semantics for rewrite systems based on interpreting rewrite rules as inequations between terms in an ordered algebra. In particular, we show that the algebra of normal forms in a terminating system is a uniquely minimal covering of the term algebra. In the non-terminating case, the existence of this minimal covering is established in the completion of an ordered algebra formed by rewriting sequences. We thus generalize the properties of normal forms for non-terminating systems to this minimal covering. These include the existence of normal forms for arbitrary rewrite systems, and their uniqueness for confluent systems, in which case the algebra of normal forms is isomorphic to the canonical quotient algebra associated with the rules when seen as equations. This extends the benefits of algebraic semantics to systems with non-deterministic and non-terminating computations. We first study properties of abstract orders, and then instantiate these to term rewriting systems.KeywordsNormal FormMinimal CoveringConfluent SystemAlgebraic SemanticGround SignatureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Convergence of an abstract reduction system (ARS) is the property that any derivation from an initial state will end in the same final state, a.k.a. normal form. We generalize this for probabilistic ARS as almost-sure convergence, meaning that the normal form is reached with probability one, even if diverging derivations may exist. We show and exemplify properties that can be used for proving almost-sure convergence of probabilistic ARS, generalizing known results from ARS.

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 beta-reduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in a 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.

In this paper we describe a teaching tool for proving equivalences between propositional logic formulae, using rewrite rules such as De Morgan’s laws and double negation. This tool is based on an earlier tool for rewriting logical formulae into disjunctive normal form (DNF). Both tools make use of a rewrite strategy, which specifies how an exercise can be solved stepwise. Different types of feedback can be calculated automatically from such a strategy specification. We describe a strategy for constructing expert-like equivalence proofs, and present two techniques for improving the proofs that are generated by the strategy.Keywordspropositional logicequivalencese-learningfeedback

Term rewriting is generally implemented using graph rewriting for efficiency reasons. Graph rewriting allows sharing of common structures thereby saving both time and space. This implementation is sound in the sense that computation of a normal form of a graph yields a normal form of the corresponding term. In this paper, we study modularity of termination of the graph reduction. Unlike in the case of term rewriting, termination is modular in graph rewriting for a large class of systems. Our results generalize the results of Plump [14] and Kurihara and Ohuchi [10].