Quantifier Instantiation Techniques for Finite Model Finding in SMT

Abstract

Abstract. SMT-based applications increasingly rely on SMT solvers being able to deal with quantified formulas. Current work shows that for formulas with quantifiers over uninterpreted sorts counter-models can be obtained by integrating a finite model finding capability into the architecture of a modern SMT solver. We examine various strategies for on-demand quantifier instantiation in this setting. Here, completeness can be achieved by considering all ground instances over the finite domain of each quantifier. However, exhaustive instantiation quickly becomes unfeasible with larger domain sizes. We propose instantiation strategies to identify and consider only a selection of ground instances that suffices to determine the satisfiability of the input formula. We also examine heuristic quantifier instantiation techniques such as E-matching for the purpose of accelerating the search. We give experimental evidence that our approach is practical for use in industrial applications and is competitive with other approaches.

Congruence Closure with Free Variables

“…Model checking is performed with a separate copy of the ground SMT solver searching for a conflicting instance. Alternative methods for model construction and checking were presented by Reynolds et al [25]. Both these model based approaches [19,25] allow integration of theories beyond equality, while CCFV for now only handles equality and uninterpreted functions.…”

“…The successive rounds of instantiation either lead to unsatisfiability or, when no conflicting instance is generated, to satisfiability with a concrete model. Here we follow the model construction guidelines by Reynolds et al [25]. A distinguished term e τ is associated to each sort τ ∈ S. For each f ∈ F with sort τ 1 , .…”

“…Although there are first attempts to unify such techniques with SMT [13], the main approach used in SMT is still instantiation: quantified formulas are reduced to ground ones and refuted with the help of decision procedures for ground formulas. The main instantiation techniques are E -matching based on triggers [12,17,26], finding conflicting instances [24] and model-based quantifier instantiation (MBQI) [19,25]. Each of these techniques contributes to the efficiency of state-of-the-art solvers, yet each one is typically implemented independently.…”

A Decision Procedure for (Co)datatypes in SMT Solvers

“…Third, it must be possible to express cyclic (regular) values as closed terms and to enumerate them. This is necessary both for finite model finding [24] and for theory combinations. The µ-binder notation associates a name with a subterm; it is used to represent cyclic values in the generated models and in the metatheory.…”

Challenges for Fast Synthesis Procedures in SMT

“…This is challenging because it requires the solver to construct internally (a finite representation of) an interpretation of f that is guaranteed to satisfy P [f, x] for every value of x [15,25].…”