Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1

The simultaneous rigid E-unification problem arises naturally in theorem proving with equality. This problem has recently been shown to be undecidable. This raises the question whether simultaneous rigid E-unification can usefully be applied to equality theorem proving. We give some evidence in the affirmative, by presenting a number of common special cases in which a decidable version of this problem suffices for theorem proving with equality. We also present some general decidable methods of a rigid nature that can be used for equality theorem proving and discuss their complexity. Finally, we give a new proof of undecidability of simultaneous rigid E-unification which is based on Post's Correspondence Problem, and has the interesting feature that all the positive equations used are ground equations (that is, contain no variables). Contents 1 Introduction 2 2 Paths and Spanning Sets 2 3 Critical Pairs and Rigid E-Unification 4 3.1 NP-Completeness of Rigid E-Unification : : :...

The confluence property of ground (i.e., variable-free) term rewrite systems (GTRS) is well-known to be decidable. This was proved independently in [4, 3] and in [13] using tree automata techniques and ground tree transducer techniques (originated from this problem), yielding EXPTIME decision procedures (PSPACE for strings). Since then, it has been a well-known longstanding open question whether this bound is optimal (see, e.g., [15]). Here we give

This paper surveys a part of the theory of fi-reduction in -calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from -terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in -terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in -calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in type-free -calculus [4, 6, 7, 15, 38, 44, 81]---see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...

We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the -calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Context-sensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...

: In this paper we initiate a systematic study of polynomial-time reductions for some basic decision problems of rewrite systems. We then give a polynomial-time algorithm for Unique-normal-form property of ground systems for the first time. Next we prove undecidability of these problems for string rewriting using our reductions. Finally, we prove partial decidability results for Confluence of commutative semi-thue systems. The Confluence and Unique-normal-form property are also shown Expspace-hard for commutative semi-thue systems. Key-words: Rewriting, Complexity, Reduction (R'esum'e : tsvp) Unite de recherche INRIA Lorraine Technopole de Nancy-Brabois, Campus scientifique, 615 rue de Jardin Botanique, BP 101, 54600 VILLERS L ES NANCY (France) Telephone : (33) 83 59 30 30 -- Telecopie : (33) 83 27 83 19 Antenne de Metz, technopole de Metz 2000, 4 rue Marconi, 55070 METZ Telephone : (33) 87 20 35 00 -- Telecopie : (33) 87 76 39 77 Algorithmes et R'eductions en R'e'ecriture R'e...

. We present here an algorithm for proving termination of term rewriting systems by gpo ordering constraint solving. The algorithm gives, as automatically as possible, an appropriate instance of the gpo generic ordering proving termination of a given system. Constraint solving is done eOEciently thanks to a DAG shared term data structure. 1 Introduction To prove termination of a Term Rewrite System (TRS for short), the most commonly used method is to deøne a well-founded ordering between terms and show that each rewrite step is a strictly decreasing step. In general, the proof is made by veriøcation: orderings are proposed by the user and tested until an appropriate one is found. Our goal here is to reduce human expertise by working in a constructive way: starting from constraints on a generic ordering, we help the user to build an appropriate speciøc instance of this ordering by using semi-automatic constraint solving methods. The generic ordering, we start from, is the general path...

This paper surveys a part of the theory of fi-reduction in λ-calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λ-terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λ-terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in λ-calculus and type theory.

. We define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redexes whose contractions retain the existence of infinite reductions. These characterizations generalize existing related criteria for perpetuality of redexes. We give a number of applications of our results, demonstrating their usefulness. In particular, we prove equivalence of weak and strong normalization (the uniform normalization property) for various restricted -calculi, which cannot be derived from previously known perpetuality criteria. 1 Introduction The objective of this paper is to study sufficient conditions for uniform normalization, UN, of a term in an orthogonal (first or higher-order) rewrite system, and for the UN property of the rewrite system itself. Here a term is UN if ei...

Programming language interpreters, proving equations (e.g. x3 = x implies the ring is Abelian), abstract data types, program transformation and optimization, and even computation itself (e.g., turing machine) can all be specified by a set of rules, called a rewrite system. Two fundamental properties of a rewrite system are the confluence or Church–Rosser property and the unique normalization property. In this article, we develop a standard form for ground rewrite systems and the concept of standard rewriting. These concepts are then used to: prove a pumping lemma for them, and to derive a new and direct decidability technique for decision problems of ground rewrite systems. To illustrate the usefulness of these concepts, we apply them to prove: (i) polynomial size bounds for witnesses to violations of unique normalization and confluence for ground rewrite systems containing unary symbols and constants, and (ii) polynomial height bounds for witnesses to violations of unique normalization and confluence for arbitrary ground systems. Apart from the fact that our technique is direct in contrast to previous decidability results for both problems, which were indirectly obtained using tree automata techniques, this approach also yields tighter bounds for rewrite systems with unary symbols than the ones that can be derived with the indirect approach. Finally, as part of our results, we give a polynomial-time algorithm for checking whether a rewrite system has the unique normalization property for all subterms in the rules of the system.