Lazy Abstraction for Size-Change Termination

Abstract

Abstract. Size-change termination is a widely used means of proving termination where source programs are first abstracted to size-change graphs which are then analyzed to determine if they satisfy the sizechange termination property. Here, the choice of the abstraction is crucial to the success of the method, and it is an open problem how to choose an abstraction such that no critical loss of precision occurs. This paper shows how to couple the search for a suitable abstraction and the test for size-change termination via an encoding to a single SAT instance. In this way, the problem of choosing the right abstraction is solved en passant by a SAT solver. We show that for the setting of term rewriting, the integration of this approach into the dependency pair framework works smoothly and gives rise to a new class of size-change reduction pairs. We implemented size-change reduction pairs in the termination prover AProVE and evaluated their usefulness in extensive experiments.

The Certification Problem Format

“…Concerning the individual techniques, currently CPF supports several classes of reduction orders (in alphabetical order): argument filters [2], matrix orders [9], polynomial orders over several carriers [18,20,22], recursive path orders [7], the Knuth-Bendix order [17], and SCNP reduction orders [5]. Moreover, the techniques of dependency graph decomposition [2], dependency pairs [2,13], dependency pair transformations (instantiation, narrowing, rewriting) [2,13], loops, non-looping nontermination [8], matchbounds [11], root-labeling [23], rule removal [15,20], semantic labeling and unlabeling [33], sizechange termination [21,26], string reversal, the subterm criterion [15], switching to innermost termination [14], uncurrying [16,24], and usable rules [2,13,28] are supported.…”

Bounded Termination of Monotonicity-Constraint Transition Systems

“…We mention two of the directions taken. [42,41,7,62] employ the dependency pair method, which like the model we are studying, was originally conceived for termination, and in fact has been effectively combined with size-change termination [67,35,25].…”