A Model Guided Instantiation Heuristic for the Superposition Calculus with Theories

Abstract

Generalised Model Finding (GMF) is a quantifier instantiation heuristic for the superposition calculus in the presence of interpreted theories with finitely quantified free function symbols ranging into theory sorts. The free function symbols are approximated by finite partial function graphs along with simplifying assumptions which are iteratively refined. Here we present an outline of the GMF approach, give an improvement that addresses some efficiency issues and then present some ideas for extending it with concepts from instantiation based theorem proving. OverviewThe inclusion of interpreted theories in a theorem proving context naturally leads to completeness issues. In the classical first-order logic satisfiability problem we are at least guaranteed refutational completeness, but reasoning modulo theories and including uninterpreted function symbols is often not even semi-decidable. Many applications of theorem proving reduce to the satisfiability problem, from ranking function and loop invariant synthesis in software verification to counter-example finding. In addition to reasoning modulo theories, these applications also require the introduction of uninterpreted function symbols and quantifier reasoning.We aim to recover some completeness in the specific case where there are only finitely many ground instances of (uninterpreted) terms ranging into the interpreted theory. This is guaranteed with the assumption that all variables inside uninterpreted terms of certain sorts are quantified over finite ranges.In the following the interpreted theory will be Linear Integer Arithmetic (LIA), which we view as a particular class of models that satisfy the LIA axioms. Concretely, clauses over the LIA signature (+ 2 , − 1 , 0, s 1 , < 2 ) are evaluated using Cooper's Quantifier Elimination algorithm which decides ∃∀ quantified LIA formulas. Sorts, operators, terms and literals from the LIA theory are called background. New operators which extend the signature of the interpreted theory are called foreground or free.Define background-sorted foreground (BSFG) terms as those which have at their head an uninterpreted function symbol with a background sort.The running example will use the usual array theory with integer indices and elements, this has the operators read : Array × Z → Z, write : Array × Z × Z → Array (these are foreground operators; read is a BSFG operator) and axioms T Array = (1) read(write(A, I, X), I) ≈ X (2) read(write(A, I, X), J) ≈ read(A, J) ∨ I ≈ J * NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.