Hans Zantema

Termination of term rewriting

We investigate proving termination of term rewriting systems by interpretation of terms in a well-founded monotone algebra. The well-known polynomial interpretations can be considered as a particular case in this framework. A classification of types of termination, including simple termination, is proposed based on properties in the semantic level. A transformation on term rewriting systems eliminating distributive rules is introduced. Using this distribution elimination a new termination proof of the system SUBST of Hardin and Laville (1986) is given. This system describes explicit substitution in ?-calculus.Another tool for proving termination is based on introduction and removal of type restrictions. A property of many-sorted term rewriting systems is called persistent if it is not affected by removing the corresponding typing restriction. Persistence turns out to be a generalization of direct sum modularity, but is more powerful for both proving confluence and termination. Termination is proved to be persistent for the class of term rewriting systems for which not both duplicating rules and collapsing rules occur, generalizing a similar result of Rusinowitch for modularity. This result has nice applications, in particular in undecidability proofs.

Termination Of Term Rewriting By Semantic Labelling

A new kind of transformation of term rewriting systems (TRS) is proposed, depending on a choice for a model for the TRS. The labelled TRS is obtained from the original one by labelling operation symbols, possibly creating extra copies of some rules. This construction has the remarkable property that the labelled TRS is terminating if and only if the original TRS is terminating. Although the labelled version has more operation symbols and may have more rules (sometimes infinitely many), termination is often easier to prove for the labelled TRS than for the original one. This provides a new technique for proving termination, making classical techniques like path orders and polynomial interpretations applicable even for non-simplifying TRS’s. The requirement of having a model can slightly be weakened, yielding a remarkably simple termination proof of the system SUBST of [11] describing explicit substitution in λ-calculus.

Matrix Interpretations for Proving Termination of Term Rewriting

We present a new method for automatically proving termination of term rewriting. It is based on the well-known idea of interpretation of terms where every rewrite step causes a decrease, but instead of the usual natural numbers we use vectors of natural numbers, ordered by a particular nontotal well-founded ordering. Function symbols are interpreted by linear mappings represented by matrices. This method allows us to prove termination and relative termination. A modification of the latter, in which strict steps are only allowed at the top, turns out to be helpful in combination with the dependency pair transformation. By bounding the dimension and the matrix coefficients, the search problem becomes finite. Our implementation transforms it to a Boolean satisfiability problem (SAT), to be solved by a state-of-the-art SAT solver.

Basic process algebra with iteration: completeness of its equational axioms.

Bergstra, Bethke and Ponse proposed an axiomatization for Basic Process Algebra extended with (binary) iteration. In this paper, we prove that this axiomatization is complete with respect to strong bisimulation equivalence. To obtain this result, we will set up a term rewriting system, based on the axioms, and prove that this term rewriting system is terminating, and that bisimilar normal forms are syntactically equal modulo commutativity and associativity of the +.

The termination competition

Since 2004, a Termination Competition is organized every year. This competition boosted a lot the development of automatic termination tools, but also the design of new techniques for proving termination. We present the background, results, and conclusions of the three first editions, and discuss perspectives and challenges for the future.

Termination of Context-Sensitive Rewriting

Context-sensitive term rewriting is a kind of term rewriting in which reduction is not allowed inside some fixed arguments of some function symbols. We introduce two new techniques for proving termination of context-sensitive rewriting. The first one is a modification of the technique of interpretation in a well-founded order, the second one is implied by a transformation in which context-sensitive termination of the original system can be concluded from termination of the transformed one. In combination with purely automatic techniques for proving ordinary termination, the latter technique is purely automatic too.

Finding small equivalent decision trees is hard

Two decision trees are called decision equivalent if they represent the same function, i.e., they yield the same result for every possible input. We prove that given a decision tree and a number, to decide if there is a decision equivalent decision tree of size at most that number is NP-complete. As a consequence, finding a decision tree of minimal size that is decision equivalent to a given decision tree is an NP-hard problem. This result differs from the well-known result of NP-hardness of finding a decision tree of minimal size that is consistent with a given training set. Instead our result is a basic result for decision trees, apart from the setting of inductive inference. On the other hand, this result differs from similar results for BDDs and OBDDs: since in decision trees no sharing is allowed, the notion of decision tree size is essentially different from BDD size.

Simple termination of rewrite systems

Abstract In this paper we investigate the concept of simple termination. A term rewriting system is called simply terminating if its termination can be proved by means of a simplification order. The basic ingredient of a simplification order is the subterm property, but in the literature two different definitions are given: one based on (strict) partial orders and another one based on preorders (or quasi-orders). We argue that there is no reason to choose the second one as the first one has certain advantages. Simplification orders are known to be well-founded orders on terms over a finite signature. This important result no longer holds if we consider infinite signatures. Nevertheless, well-known simplification orders like the recursive path order are also well-founded on terms over infinite signatures, provided the underlying precedence is well-founded. We propose a new definition of simplification order, which coincides with the old one (based on partial orders) in case of finite signatures, but which is also well-founded over infinite signatures and covers orders like the recursive path order. We investigate the properties of the ensuing class of simply terminating systems.

Termination of String Rewriting Proved Automatically

This paper describes how a combination of polynomial interpretations, recursive path order, RFC match-bounds, the dependency pair method, and semantic labelling can be used for automatically proving termination of an extensive class of string rewriting systems (SRSs). The tool implementing this combination of techniques is called TORPA: Termination of Rewriting Proved Automatically. All termination proofs generated by TORPA are easy to read and check; but for many of the SRSs involved, finding a termination proof would be a hard job for a human. This paper contains all underlying theory, describes how the search for a termination proof is implemented, and includes many examples.

Termination of Logic Programs Using Semantic Unification

We introduce a transformation of well-moded logic programs into constructor systems, a subclass of term rewrite systems, such that left-termination of the logic program follows from termination of the derived constructor system. Thereafter, we present a new technique to prove termination of constructor systems. In the technique semantic unification is used. Thus, surprisingly, semantic unification can be used For giving termination proofs for logic programs. Parts of the technique can be automated very easily. Other parts can be automated for subclasses of constructor systems. The technique is powerful enough to prove termination of some constructor systems that are not simply terminating, and therefore, the technique is suitable to prove termination of some difficult logic programs.