Fast Approximations of Quantifier Elimination

TL;DR

This paper introduces QEL, a unified framework leveraging egraphs for fast approximations of quantifier elimination, improving efficiency in SMT solving and related tasks.

Contribution

The paper presents QEL, a novel egraph-based framework for fast quantifier elimination approximations, applicable to theories like Arrays and ADTs, integrated into Z3.

Findings

QEL outperforms existing methods in quantifier approximation tasks.

Integration with Z3 improves performance on SMT problems.

QEL effectively handles theories of Arrays and ADTs.

Abstract

Quantifier elimination (qelim) is used in many automated reasoning tasks including program synthesis, exist-forall solving, quantified SMT, Model Checking, and solving Constrained Horn Clauses (CHCs). Exact qelim is computationally expensive. Hence, it is often approximated. For example, Z3 uses "light" pre-processing to reduce the number of quantified variables. CHC-solver Spacer uses model-based projection (MBP) to under-approximate qelim relative to a given model, and over-approximations of qelim can be used as abstractions. In this paper, we present the QEL framework for fast approximations of qelim. QEL provides a uniform interface for both quantifier reduction and model-based projection. QEL builds on the egraph data structure -- the core of the EUF decision procedure in SMT -- by casting quantifier reduction as a problem of choosing ground (i.e., variable-free) representatives for equivalence classes. We have used QEL to implement MBP for the theories of Arrays and Algebraic Data Types (ADTs). We integrated QEL and our new MBP in Z3 and evaluated it within several tasks that rely on quantifier approximations, outperforming state-of-the-art.