Invariant Inference With Provable Complexity From the Monotone Theory

TL;DR

This paper introduces invariant inference algorithms for propositional transition systems with provable upper bounds on SAT calls, leveraging the monotone theory to improve efficiency and handle complex invariants.

Contribution

It develops invariant inference algorithms with guaranteed complexity bounds using the monotone theory, extending previous results to more expressive invariants.

Findings

Efficient algorithms for invariants with short CNF and DNF representations.

Complexity bounds on the number of SAT calls for invariant inference.

A novel procedure for computing least monotone overapproximations.

Abstract

Invariant inference algorithms such as interpolation-based inference and IC3/PDR show that it is feasible, in practice, to find inductive invariants for many interesting systems, but non-trivial upper bounds on the computational complexity of such algorithms are scarce, and limited to simple syntactic forms of invariants. In this paper we achieve invariant inference algorithms, in the domain of propositional transition systems, with provable upper bounds on the number of SAT calls. We do this by building on the monotone theory, developed by Bshouty for exact learning Boolean formulas. We prove results for two invariant inference frameworks: (i) model-based interpolation, where we show an algorithm that, under certain conditions about reachability, efficiently infers invariants when they have both short CNF and DNF representations (transcending previous results about monotone invariants); and (ii) abstract interpretation in a domain based on the monotone theory that was previously studied in relation to property-directed reachability, where we propose an efficient implementation of the best abstract transformer, leading to overall complexity bounds on the number of SAT calls. These results build on a novel procedure for computing least monotone overapproximations.