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Abstract. We describe the concept of an abstract congruence closure and provide equational inference rules for its construction. The length of any maximal derivation using these inference rules for constructing an abstract congruence closure is at most quadratic in the input size. The framework is used to describe the logical aspects of some well-known algorithms for congruence closure. It is also used to obtain an efficient implementation of congruence closure. We present experimental results that illustrate the relative differences in performance of the different algorithms. The notion is extended to handle associative and commutative function symbols, thus providing the concept of an associative-commutative congruence closure. Congruence closure (modulo associativity and commutativity) can be used to construct ground convergent rewrite systems corresponding to a set of ground equations (containing AC symbols). Key words: term rewriting, congruence closure, associative-commutative theories. 1.

We use the uniform framework of abstract congruence closure to study the congruence closure algorithms described by Nelson and Oppen [9], Downey, Sethi and Tarjan [7] and Shostak [11]. The descriptions thus obtained abstracts from certain implementation details while still allowing for comparison between these different algorithms. Experimental results are presented to illustrate the relative efficiency and explain differences in performance of these three algorithms. The transition rules for computation of abstract congruence closure are obtained from rules for standard completion enhanced with an extension rule that enlarges a given signature by new constants.

A term rewrite system (TRS) terminates iff its rules are contained in a reduction ordering ?. In order to deal with any set of equations, including inherently non-terminating ones (like commutativity), TRS have been generalised to ordered TRS (E; ?), where equations of E are applied in whatever direction agrees with ?. The confluence of terminating TRS is well-known to be decidable, but for ordered TRS the decidability of confluence has been open. Here we show that the confluence of ordered TRS is decidable if ordering constraints for ? can be solved in an adequate way, which holds in particular for the class of LPO orderings. For sets E of constrained equations, confluence is shown to be undecidable. Finally, ground reducibility is proved undecidable for ordered TRS. 1 Introduction Term rewrite systems (TRS) have been applied to many problems in symbolic computation, automated theorem proving, program synthesis and verification, and logic programming among others. Two fundamental pr...

We present an extended completion procedure with builtin theories defined by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm for constructing Gröbner bases for polynomial ideals in this formalism. The presentation of completion is at an abstract level, by transition rules, with a suitable notion of fairness used to characterize a wide class of correct completion procedures, among them Buchberger's original algorithm for polynomial rings over a field.

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An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation, saturation, redundancy elimination, and rewrite-system reduction are connected to proof orderings. Fairness of deductive mechanisms is defined in terms of proof orderings, distinguishing between (ordinary) “fairness, ” which yields completeness, and “uniform fairness, ” which yields saturation.

Solving goals—like deciding word problems or resolving constraints—is much easier in some theory presentations than in others. What have been called “completion processes”, in particular in the study of equational logic, involve finding appropriate presentations of a given theory to solve easily a given class of problems. We provide a general proof-theoretic setting within which completion-like processes can be modelled and studied. This framework centers around well-founded orderings of proofs. It allows for abstract definitions and very general characterizations of saturation processes and redundancy criteria.

All current completeness results for ordered paramodulation require the term ordering Ø to be well-founded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for Ø are well-foundedness and the subterm property 1 . The technique is a relatively simple and elegant application of some fundamental results on the termination and confluence of ground term rewrite systems (TRS). By a careful further analysis of our technique, we obtain the first Knuth-Bendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly non-totalizable) reduction ordering Ø whenever it exists 2 . Note that being a reduction ordering is the minimal possible requirement on Ø, since a TRS terminates if, and only if, it is contained in a reduction ordering. Keywords: term rewriting, automated deduction. 1 Introduction Deduction with equality is fundamental in mathematics, logics and many applications of ...