Proving Termination by Dependency Pairs and Inductive Theorem Proving

Abstract

Current techniques and tools for automated termination analysis of term rewrite systems (TRSs) are already very powerful. However, they fail for algorithms whose termination is essentially due to an inductive argument. Therefore, we show how to couple the dependency pair method for termination of TRSs with inductive theorem proving. As confirmed by the implementation of our new approach in the tool AProVE, now TRS termination techniques are also successful on this important class of algorithms.

Dynamic Dependency Pairs for Algebraic Functional Systems

“…The formative rules technique is also applicable to first-order rewriting, in particular for many-sorted TRSs (or for innermost rewriting where types may be added by [7]). However, we have not yet investigated whether the technique leads to an improvement in current state-of-the-art termination provers.…”

Analyzing Program Termination and Complexity Automatically with AProVE

“…While QF NIA is undecidable and QF NRA is decidable, decision procedures for QF NRA are extremely inefficient. AProVE applies its bitblasting approach to solve QF NIA also for other extensions of classic polynomial interpretations, such as polynomial interpretations with negative coefficients to prove bounded increase [39], polynomial interpretations with max and min operators [30,31], matrix interpretations [26], and partly strongly monotonic polynomial interpretations suitable for a combination with inductive theorem proving [34]. Moreover, AProVE uses polynomial interpretations not only for termination proving, but also for inferring bounds for the runtime complexity of TRSs [51] and Prolog programs [41].…”

“…A graphical overview of our approach is shown below. Technical details on the techniques for transforming programs to (int-) TRSs and for analyzing rewrite systems can be found in, e.g., [10,11,12,14,17,25,26,27,28,29,30,31,32,33,34,36,37,39,40,41,43,44,51,52,57,58]. Since the current paper is a system description, we focus on the implementation of these techniques in AProVE, which we have made available as a plug-in for the popular Eclipse software development environment [23].…”

Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories

“…In this way, better techniques to prove termination of OS-TRSs become available for proving other termination properties which are persistent and remain unchanged after sort introduction [123]. This is the case of termination of TRSs when some syntactical restrictions are required on their rules [6,70] and of innermost termination of TRS [38,71], which has been recently proved useful to prove termination of programs with pre-defined data structures and operations like integer arithmetic [43,44,102]. In other settings, like higher-order rewriting, type information has also been proved important to prove termination [39] and the techniques developed in this paper could be useful as well.…”