Proving Termination of Higher-Order Rewrite Systems

Mathematical Foundations of …, 1985

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 ...

In this paper we present some original variations of the recursive path ordering. Additionally we de ne a restricted semantic path ordering which, in general, does not include the subterm relation, but is shown to be monotonic. By combining both kind of orderings we can prove (automatically) the termination of several (non-simply terminating) examples.

IEICE Transactions on Information and Systems, 2005

This paper explores how to extend the dependency pair technique for proving termination of higher-order rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the non-existence of an infinite R-chain of the dependency pairs. However, the termination and the non-existence of an infinite R-chain do not coincide in the higher-order case. We introduce a new notion of dependency forest that characterize infinite reductions and infinite R-chains, and show that the termination property of higher-order rewrite systems R can be checked by showing the non-existence of an infinite R-chain, if R is strongly linear or non-nested.

Lecture Notes in Computer Science, 2014

This paper presents several new results on conditional term rewriting within the general framework of order-sorted rewrite theories (OSRTs) which contains the more restricted framework of conditional term rewriting systems (CTRSs) as a special case. The results uncover some subtle issues about conditional termination. We first of all generalize a previous known result characterizing the operational termination of a CTRS by the quasi-decreasing ordering notion to a similar result for OSRTs. Second, we point out that the notions of irreducible term and of normal form, which coincide for unsorted rewriting are totally different for conditional rewriting and formally characterize that difference. We then define the notion of a weakly operationally terminating (or weakly normalizing) OSRT, give several evaluation mechanisms to compute normal forms in such theories, and investigate general conditions under which the rewriting-based operational semantics and the initial algebra semantics of a confluent OSRT coincide thanks to a notion of canonical term algebra. Finally, we investigate appropriate conditions and proof methods to ensure good executability properties of an OSRT for computing normal forms.

Applicable Algebra in Engineering, Communication and Computing, 2005

, a new size-change principle was proposed to verify termination of functional programs automatically. We extend this principle in order to prove termination and innermost termination of arbitrary term rewrite systems (TRSs). Moreover, we compare this approach with existing techniques for termination analysis of TRSs (such as recursive path orders or dependency pairs). It turns out that the size-change principle on its own fails for many examples that can be handled by standard techniques for rewriting, but there are also TRSs where it succeeds whereas existing rewriting techniques fail. Moreover, we also compare the complexity of the respective methods. To this end, we develop the first complexity analysis for the dependency pair approach. While the size-change principle is PSPACE-complete, we prove that the dependency pair approach (in combination with classical path orders) is only NP-complete. To benefit from their respective advantages, we show how to combine the size-change principle with classical orders and with dependency pairs. In this way, we obtain a new approach for automated termination proofs of TRSs which is more powerful than previous approaches. We also show that the combination with dependency pairs does not increase the complexity of the size-change principle, i.e., the combined approach is still PSPACE-complete.

This workshop delves into all aspects of termination of processes. Though the halting of computer programs is undecidable, methods of establishing termination play a fundamental role in many applications and the challenges are both practical and theoretical. From a practical point of view, proving termination is a central problem in software development and formal methods for termination analysis are essential for program verification. From a theoretical point of view, termination is central in mathematical logic and ordinal theory.

1995

We propose a new semantics for rewrite systems ba.~ed on interpreting rewrite rules as in equatioIlB between terms in an ordered algebra. In part.icular, we show thai the algebra. of normal forms in a terminating system is a uniqnely minimal covering of the term algebra. In the non-terminating ca..~e, the existence of this minimal covering is established in the comple tion of an ordered algebra formed by rewrit.ing sequences. We thus generalize the properties of normal forms far: non-terminating systelil~ to this minimal covering. ThesE' include the exi~tence of normal forms for arbitrary rewrite ~ystems, and their uniqueness for conBue-nt ~ystems, in which Ca<le the algebra of normal forms i~ isomorphic to the canonical quotient. algebra associated with the rule~ when seen as eqnations. This extend!> the benefits of alge braic semantics to systems with non-determinist.ic and non-t.erminating computations. V•le first study properties of abstract. order~, and then instantiat.e the~e to term rewriting sy~tems.

Lecture Notes in Computer Science, 2001

We present a new approach to termination analysis of logic programs. The essence of the approach is that we make use of general term-orderings (instead of level mappings), like it is done in transformational approaches to logic program termination analysis, but that we apply these orderings directly to the logic program and not to the term-rewrite system obtained through some transformation. We de ne a variant of acceptability, based on general term-orderings, and show how it is equivalent to LD-termination for well-moded, simply moded programs. We develop a demand driven, constraintbased approach to verify this acceptability-variant.

Journal of Symbolic Computation - JSC, 1995

Usually termination of term rewriting systems (TRS&#x27;s) is proved bymeans of a monotonic well-founded order. If this order is total on groundterms, the TRS is called totally terminating. In this paper we prove thattotal termination is an undecidable property of finite term rewriting systems.The proof is given by means of Post&#x27;s Correspondence Problem.1 IntroductionTermination of term rewriting systems (TRS&#x27;s) is an important property. Oftentermination proofs are given by defining an order ...

IPSJ Online Transactions, 2009

The present paper discusses a head-needed strategy and its decidable classes of higher-order rewrite systems (HRSs), which is an extension of the headneeded strategy of term rewriting systems (TRSs). We discuss strong sequential and NV-sequential classes having the following three properties, which are mandatory for practical use: (1) the strategy reducing a head-needed redex is head normalizing (2) whether a redex is head-needed is decidable, and (3) whether an HRS belongs to the class is decidable. The main difficulty in realizing (1) is caused by the β-reductions induced from the higher-order reductions. Since β-reduction changes the structure of higher-order terms, the definition of descendants for HRSs becomes complicated. In order to overcome this difficulty, we introduce a function, PV, to follow occurrences moved by βreductions. We present a concrete definition of descendants for HRSs by using PV and then show property (1) for orthogonal systems. We also show properties (2) and (3) using tree automata techniques, a ground tree transducer (GTT), and recognizability of redexes.