Proving Termination of Rewrite Systems Using Bounds

Abstract

Abstract. The use of automata techniques to prove the termination of string rewrite systems and left-linear term rewrite systems is advocated by Geser et al. in a recent sequence of papers. We extend their work to non-left-linear rewrite systems. The key to this extension is the introduction of so-called raise rules and the use of tree automata that are not quite deterministic. Furthermore, we present negative solutions to two open problems related to string rewrite systems.

Match-Bounds with Dependency Pairs for Proving Termination of Rewrite Systems

“…In [17] we showed that deterministic tree automata-a common approach to handle non-linearity with automata techniques (cf. [3,18,19])-are unsuitable.…”

“…However as soon as we add those transitions again, they are removed since they cause A to be non-deterministic. In [17] we introduced quasi-deterministic tree automata to solve this problem. This carries over to the present setting without any problems.…”

“…In [8] tree automata were used to cover left-linear term rewrite systems. We extended the latter work to arbitrary term rewrite systems in [17] by considering so-called quasideterministic tree automata. Variations and improvements of the basic method for string rewrite systems are discussed in [6,7].…”

Proving Quadratic Derivational Complexities Using Context Dependent Interpretations

“…(Employing [16] (as in [5]) one sees that any match-raise bounded TRS has linear derivational complexity. Then the claim follows from Lemma 8 in [15].) Note that the restriction to non-duplicating TRS is harmless, as any duplicating TRS induces at least exponential derivational complexity.…”

“…Clearly compatibility with strongly linear interpretations induces linear derivational complexity. Secondly, T T Tbox refers to the implementation of the matchbound technique as in [15]: Linear TRSs are tested for match-boundedness, nonlinear, but non-duplicating TRSs are tested for match-raise-boundedness. This technique again implies linear derivational complexity.…”