Knuth-Bendix Completion Procedure Research Articles

A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems

We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating them as equational term rewriting systems and considering E-critical pairs and/or termination modulo E. In contrast, our method is based solely on usual critical pairs and it also (partially) works even if the system is not terminating modulo E. We first present confluence criteria for term rewriting systems whose rewrite rules can be partitioned into a terminating part and a possibly non-terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system.

Paramodulation with Non-Monotonic Orderings and Simplification

Ordered paramodulation and Knuth-Bendix completion are known to remain complete when using non-monotonic orderings. However, these results do not imply the compatibility of the calculus with essential redundancy elimination techniques such as demodulation, i.e., simplification by rewriting, which constitute the primary mode of computation in most successful automated theorem provers. In this paper we present a complete ordered paramodulation calculus for non-monotonic orderings which is compatible with powerful redundancy notions including demodulation, hence strictly improving the previous results and making the calculus more likely to be used in practice. As a side effect, we obtain a Knuth-Bendix completion procedure compatible with simplification techniques, which can be used for finding, whenever it exists, a convergent term rewrite system for a given set of equations and a (possibly non-totalizable) reduction ordering.

Completion after Program Inversion of Injective Functions

Given a constructor term rewriting system that defines injective functions, the inversion compiler proposed by Nishida, Sakai and Sakabe generates a conditional term rewriting system that defines the inverse relations of the injective functions, and then the compiler unravels the conditional system into an unconditional term rewriting system. In general, the resulting unconditional system is not (innermost-)confluent even if the conditional system is (innermost-)confluent. In this paper, we propose a modification of the Knuth-Bendix completion procedure, which is used as a post-processor of the inversion compiler. Given a confluent and operationally terminating conditional system, the procedure takes the resulting unconditional systems as input. When the procedure halts successfully, it returns convergent systems that are computationally equivalent to the conditional systems. To adapt the modified procedure to the conditional systems that are not confluent but innermost-confluent, we propose a simplified variant of the modified procedure. We report that the implementations of the procedures succeed in generating innermost-convergent inverse systems for all the examples we tried.

Paramodulation and Knuth–Bendix Completion with Nontotal and Nonmonotonic Orderings

Up to now, all existing completeness results for ordered paramodulation and Knuth–Bendix completion have required term ordering s to be well founded, monotonic, and total(izable) on ground terms. For several applications, these requirements are too strong, and hence weakening them has been a well-known research challenge. Here we introduce a new completeness proof technique for ordered paramodulation where the only properties required on s are well-foundedness and the subterm property. 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 s whenever it exists. Note that being a reduction ordering is the minimal possible requirement on s, since a TRS terminates if, and only if, it is contained in a reduction ordering.

Combining word problems through rewriting in categories with products

We give an algorithm solving combined word problems (over non-necessarily disjoint signatures) based on rewriting of equivalence classes of terms. The canonical rewriting system we introduce consists of few transparent rules and is obtained by applying Knuth–Bendix completion procedure to presentations of pushouts among categories with products. It applies to pairs of theories which are both constructible over their common reduct (on which we do not make any special assumption).

Knuth-Bendix Completion for Non-Symmetric Transitive Relations

We extend the Knuth-Bendix completion procedure from equational rewriting to rewriting with non-symmetric transitive relations and quasi-orderings. The main differences are the following: Specification of the general non-ground case seems beyond first-order logic. It is within that realm when terms are linear or functions non-monotonic. The procedure requires critical-pair computations and need not terminate even in the ground case. Simplification is not don't care non-deterministic, but search based. Applications include ordered resolution and ordered chaining calculi, development of rule-based declarative procedures and algorithms, program and reachability analysis (in rewriting logic) and propagation of inequality constraints.

SOUR graphs for efficient completion

We introduce a data structure called \emphSOUR graphs and present an efficient Knuth-Bendix completion procedure based on it. \emphSOUR graphs allow for a maximal structure sharing of terms in rewriting systems. The term representation is a dag representation, except that edges are labelled with equational constraints and variable renamings. The rewrite rules correspond to rewrite edges, the unification problems to unification edges. The Critical Pair and Simplification inferences are recognized as patterns in the graph and are performed as local graph transformations. Our algorithm avoids duplicating term structure while performing inferences, which causes exponential behavior in the standard procedure. This approach gives a basis to design other completion algorithms, such as goal-oriented completion, concurrent completion and group completion procedures.

Equational inference, canonical proofs, and proof orderings

We describe the application of proof orderings—a technique for reasoning about inference systems-to various rewrite-based theorem-proving methods, including refinements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion (a refutationally complete extension of standard completion); and a proof by consistency procedure for proving inductive theorems.

On The Problem Of Generating Small Convergent Systems

A sequence (Rn,m)n,mϵN of normalized string-rewriting systems on a fixed finite alphabet Σ is constructed such that, for all n,m ϵ N,1. Rn,m contains 44 rules, it is of size O(n + m ), and it is compatible with a length-lexicographical ordering > on Σ*, but2. given the system Rn,m and the ordering > as input, the Knuth-Bendix completion procedure will generate more than A(n, m) intermediate rules before a finite convergent system Sn,m of size O(n + m) is obtained. Here A denotes Ackermann's function.

Testing for the ground (co-)reducibility property in term-rewriting systems

Given a term-rewriting system R, a term t is ground-reducible by R if every ground instance tσ of it is R-reducible. A pair ( t, s) of terms is ground-co-reducible by R if every ground instance ( tσ, sσ] of it for which tσ and sσ are distinct is R-reducible. Ground (co-)reducibility has been proved to be the fundamental tool for mechanizing inductive proofs, together with the Knuth-Bendix completion procedure presented by Jouannaud and Kounalis (1986, 1989). Jouannaud and Kounalis (1986, 1989) also presented an algorithm for testing ground reducibility which is tractable in practical cases but restricted to left-linear term-rewriting systems. The solution of the ground (co-)reducibility problem, for the general case, turned out to be surprisingly complicated. Decidability of ground reducibility for arbitrary term-rewriting systems has been first proved by Plaisted (1985) and independently by Kapur (1987). However, the algorithms of Plaisted and Kapur amount to intractable computation, even in very simple cases. We present here a new algorithm for the general case which outperforms the algorithms of Plaisted and Kapur and even our previous algorithm in case of left-linear term-rewriting systems. We then show how to adapt it to check for ground co-reducibility.

The use of Knuth-Bendix methods to solve the wordproblem in automatic groups

Certain classes of infinite groups arising from geometry and topology are known to have solvable word problem. We describe the development of practical methods for the solution of the word problem based on the reduction of words in the generators to a normal form. The Knuth-Bendix completion procedure is the principal tool used but, in the case that this process does not halt, we use alternative methods involving the construction of finite-state automata. A computer implementation of these procedures together with some performance statistics on some simple examples are also described.

What's so special about Kruskal's theorem and the ordinal Γ o? A survey of some results in proof theory

This paper consists primarily of a survey of results of Harvey Friedman about some proof-theoretic aspects of various forms of Kruskal's tree theorem, and in particular the connection with the ordinal Γ 0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Verlen hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard's result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the ‘tree theorem’, as well as a ‘finite miniaturization’ of Kruskal's theorem due to Harvey Friedman. These versions of Kruskal's theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called ‘natural independence phenomena’, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal's tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ o and Kruskal's theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence.

Semi-unification

Semi-unification is a generalization fo both matching and ordinary unification: for a given pair of terms s and t, two substitution ρ and σ are sought such that ρ( σ( s))= σ( t). Semi-unifiability can be used as a check for non-termination of a rewrite rule, but constructing a correct semi- unification algorithm has been an elusive goal; for example, an algorithm given by Purdom in his RTA-87 paper was incorrect. This paper presents a decision procedure for semi-unification based on techniques semilar to those used in the Knuth-Bendix completion procedure. When its substitutions ρ and σ are easily extracted. Though exponential in its computing time, the decision procedure can be improved to a polynomial-time algorithm, as will be shown.

Automating inductionless induction using test sets

The inductionless induction (also called proof by consistency) approach for proving equations by induction from an equational theory, requires a consistency check for equational theories. A new method using test sets for checking consistency of an equational theory is proposed. Using this method, a variation of the Knuth-Bendix completion procedure can be used for automatically proving equations by induction. The method does not suffer from limitations imposed by the methods proposed by Musser as well as by Huet and Hullot, and is as powerful as Jouannaud and Kounalis’ method based on ground-reducibility. A theoretical comparison of the test set method with Jouannaud and Kounalis’ method is given showing that the test set method is generally much better. Both the methods have been implemented in RRL, Rewrite Rule Laboratory, a theorem proving environment based on rewriting techniques and completion. In practice also, the test set method is faster than Jouannaud and Kounalis’ method. The test set construction can also be used to check for the sufficient-completeness property of equational axiomatizations including algebraic specifications of abstract data types as well as for identifying constructors in an algebraic specification.

Chain properties of rule closures

Abstract This article introduces a generalisation of the crossed rule approach to the detection of Knuth-Bendix completion procedure divergence. It introduces closure chains, which are special rule closures constructed by means of particular substitution operations and operators, as a suitable formalism for progress in this direction. Supporting substitution algebra is developed first, followed by considerations concerning rule closures in general, concluding with an investigation of closure chain properties. Issues concerning the narrowing process are not discussed here.

On the Knuth-Bendix completion for concurrent processes

We consider critical pairs for replacement systems over free partially commutative monoids. This is done in order to apply the Knuth-Bendix completion procedure to concurrent processes. We will see that there are one-rule systems which have no finite set of critical pairs. Therefore we develop a sufficient (and computable) condition such that finite trace replacement systems have a finite set of critical pairs. This condition is always satisfied in the purely free case of semi-Thue systems or purely commutative case of vector replacement systems. In fact, it generalizes (and unifies) both cases. We will give examples of how one can use this generalization.

Existence, Uniqueness, and Construction of Rewrite Systems

The construction of term-rewriting systems, specifically by the Knuth–Bendix completion procedure, is considered. We look for conditions that might ensure the existence of a finite canonical rewriting system for a given equational theory and that might guarantee that the completion procedure will find it. We define several notions of equivalence between rewriting systems in the ordinary and modulo case, and examine uniqueness of systems and the need for backtracking in implementing completion.

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.

The Knuth-Bendix Completion Procedure and Thue Systems

The Knuth-Bendix completion procedure for term rewriting systems in many cases provides a decision procedure for equational theories and has been found to have many applications in various areas. We discuss the application of the Knuth-Bendix procedure to Thue systems. We use the notion of a reduced Thue system and show that for every Church-Rosser Thue system, there is a unique reduced Church-Rosser Thue system equivalent to it. Furthermore, the Knuth-Bendix completion procedure, when applied to a Thue system T, always produces the finite reduced Church-Rosser Thue system equivalent to T whenever such a system exists. Similar results can also be proved for almost-confluent Thue systems. Using properties of reduced Church-Rosser systems, we develop conditions under which a class of special Thue systems have equivalent finite Church-Rosser systems. In addition, we show that the completion procedure always terminates on finite parenthesized Thue systems, from which the termination of the completion procedur...

Refutational theorem proving using term-rewriting systems

In this paper we propose a new approach to theorem proving in first-order logic based on the term-rewriting method. First for propositional calculus, we introduce a canonical term-rewriting system for Boolean algebra. This system enables us to transform the first-order predicate calculus into a form of equational logic, and to develop several complete strategies (both clausal and nonclausal) for first-order theories based on the Knuth-Bendix Completion Procedure. More importantly, our strategies can deal with predicate logic and built-in (equational) theories in a uniform and effective way. We also describe an implementation and comparisons with some other first-order theorem-proving methods.