Proving termination of logic programs by exploiting term properties

Abstract

In this paper semi-linear norms, a class of functions to weight the terms occurring in a program, are defined and studied. All the functions in this class have the nice property of allowing a syntactical characterization of rigid terms, that is terms whose weight does not change under substitution.Based on these norms, a general proof method for termination of pure Prolog programs can be adapted to deal with a large class of programs in a simple way. The sin~lified method requires pre/post specifications well-behaved with respect to substitutions, quite a general case in practice, and ordering functions not increasing with respect to substitutions, which can be based on semi-linear norms, to be associated to program predicates. A few examples of this simplified proof method are given.

Polytool: Proving Termination Automatically Based on Polynomial Interpretations

“…the polynomial order due to instantiations caused by resolutions steps. In Polytool we use rigidity [3] constraints to impose this. Rigidity means that the interpretation of an atom/term should be invariant for any instance of the atom/term.…”

A declarative approach for first-order built-in's of Prolog

“…Such level mappings were already considered in Bossi, Cocco and Fabris [8] and subsequently in Plumer [16] but they were applied only to prove termination of pure Prolog programs. In our case it is convenient to allow a level mapping yielding values in a well-founded ordering.…”

A Transformational Approach to Resource Analysis with Typed-Norms

“…Size is measured using so-called norms [5] which define how the size of a term is computed. Examples of norms are list-length which counts the number of elements of a list, tree-depth which counts the depth of a tree, term-size which counts the number of constructors, etc.…”

“…The last two decades have witnessed a wealth of research on using norms in termination analysis, especially in the context of logic programming [5,6,9]. Early work pointed out that the choice of norm affects the precision such that the analyzer may only succeed to prove termination if a certain norm is used, while it cannot prove it with others.…”

“…Early work pointed out that the choice of norm affects the precision such that the analyzer may only succeed to prove termination if a certain norm is used, while it cannot prove it with others. Later on, there has been further investigation on applying multiple norms, i.e., using two or more norms by applying them simultaneously [5]. This means that the same data in the original program is replaced by two or more abstract data each one specifying its size information w.r.t.…”