Paliath Narendran

Church-Rosser Thue systems and formal languages

Since about 1971, much research has been done on Thue systems that have properties that ensure viable and efficient computation. The strongest of these is the Church-Rosser property, which states that two equivalent strings can each be brought to a unique canonical form by a sequence of length-reducing rules. In this paper three ways in which formal languages can be defined by Thue systems with this property are studied, and some general results about the three families of languages so determined are studied.

On sufficient-completeness and related properties of term rewriting systems

SummaryThe decidability of the sufficient completeness property of equational specifications satisfying certain conditions is shown. In addition, the decidability of the related concept of quasi-reducibility of a term with respect to a set of rules is proved. Other results about irreducible ground terms of a term rewriting system also follow from a key technical lemma used in these decidability proofs; this technical lemma states that there is a finite bound on the substitutions of ground terms that need to be considered in order to check for a given term, whether the result obtained by any substitution of ground terms into the term is irreducible. These results are first shown for untyped systems and are subsequently extended to typed systems.

A finite thue system with decidable word problem and without equivalent finite canonical system

Abstract We present a single-axiom. Thue system with a decidable word problem for which there does not exist any finite equivalent canonical system. However, an equivalent finite canonical system for this Thue system can be obtained if new symbols are introduced in the presentation. This result settles an open question by Jantzen (1982) who asked whether every Thue system with a decidable word problem has an equivalent finite canonical system. We also discuss relationships between Thue systems and term rewriting systems.

Proof by induction using test sets

A new method for proving an equational formula by induction is presented. This method is based on the use of the Knuth-Bendix completion procedure for equational theories, and it does not suffer from limitations imposed by the inductionless induction methods proposed by Musser and Huet and Hullot. The method has been implemented in RRL, a Rewrite Rule Laboratory. Based on extensive experiments, the method appears to be more practical and efficient than a recently proposed method by Jouannaud and Kounalis. Using ideas developed for this method, it is also possible to check for sufficient completeness of equational axiomatizations.

Only prime superpositions need be considered in the Knuth-Bendix completion procedure

The Knuth and Bendix test for local confluence of a term rewriting system involves generating superpositions of the left-hand sides, and for each superposition deriving a critical pair of terms and checking whether these terms reduce to the same term. We prove that certain superpositions, which are called composite because they can be split into other superpositions, do not have to be subjected to the critical-pair-joinability test; it suffices to consider only prime superpositions. As a corollary, this result settles a conjecture of Lankford that unblocked superpositions can be omitted. To prove the result, we introduce new concepts and proof techniques which appear useful for other proofs relating to the Church-Rosser property. This test has been implemented in the completion procedures for ordinary term rewriting systems as well as term rewriting systems with associative-commutative operators. Performance of the completion procedures with this test and without the test are compared on a number of examples in the Rewrite Rule Laboratory (RRL) being developed at General Electric Research and Development Center.

Complexity of matching problems

We show that the associative-commutative matching problem is NP-complete; more precisely, the matching problem for terms in which some function symbols are uninterpreted and others are both associative and commutative, is NP-complete. It turns out that the similar problems of associative-matching and commutative-matching are also NP-complete. However, if every variable appears at most once in a term being matched, then the associative-commutative matching problem is shown to have an upper-bound of O (|s| * |t|3), where |s| and |t| are respectively the sizes of the pattern s and the subject t.

Rigid E-unification is NP-complete

Rigid E-unification is a restricted kind of unification modulo equational theories, or E-unification, that arises naturally in extending P. Andrews (1981) theorem-proving method of mating to first-order languages with equality. It is shown that rigid E-unification is NP-complete and that finite complete sets of rigid E-unifiers always exist. As a consequence, deciding whether a family of mated sets is an equational mating is an NP-complete problem. Some implications of this result regarding the complexity of theorem proving in first-order logic with equality are discussed.<<ETX>>

Rigid E -unification: NP-completeness and applications to equational matings

Abstract Rigid E -unification is a restricted kind of unification modulo equational theories, or E -unification, that arises naturally in extending Andrews theorem proving method of matings to first-order languages with equality. This extension was first presented by J. H. Gallier, S. Raatz, and W. Snyder, who conjectured that rigid E -unification is decidable. In this paper, it is shown that rigid E -unification is NP-complete and that finite complete sets of rigid E -unifiers always exist. As a consequence, deciding whether a family of mated sets is an equational mating is an NP-complete problem. Some implications of this result regarding the complexity of theorem proving in first-order logic with equality are also discussed.

A Path Ordering for Proving Termination of Term Rewriting Systems

A new partial ordering scheme for proving uniform termination of term rewriting systems is presented. The basic idea is that two terms are compared by comparing the paths through them. It is shown that the ordering is a well-founded simplification ordering and also a strict extension of the recursive path ordering scheme of Dershowitz. Terms can be compared under this path ordering in polynomial time.

Finding Canonical Rewriting Systems Equivalent to a Finite Set of Ground Equations in Polynomial Time

In this paper, it is shown that there is an algorithm which, given any finite set E of ground equations, produces a reduced canonical rewriting system R equivalent to E in polynomial time. This algorithm based on congruence closure performs simplification steps guided by a total simplification ordering on ground terms, and it runs in time O(n3).