Simple Linear Loops: Algebraic Invariants and Applications

TL;DR

This paper presents a polynomial-space algorithm for generating the strongest algebraic invariants in simple linear loops, with applications in verification and synthesis, and analyzes their computational complexity.

Contribution

It introduces a new polynomial-space algorithm for invariant generation in simple linear loops and explores the complexity of related verification and synthesis problems.

Findings

Algorithm computes all polynomial invariants efficiently for fixed variables.

Invariant verification is coNP-complete and in PSPACE with different input representations.

Loop synthesis problems are as hard as Hilbert's Tenth problem, but become decidable under certain constraints.

Abstract

The automatic generation of loop invariants is a fundamental challenge in software verification. While this task is undecidable in general, it is decidable for certain restricted classes of programs. This work focuses on invariant generation for (branching-free) loops with a single linear update. Our primary contribution is a polynomial-space algorithm that computes the strongest algebraic invariant for simple linear loops, generating all polynomial equations that hold among program variables across all reachable states. The key to achieving our complexity bounds lies in mitigating the blowup associated with variable elimination and Gr\"obner basis computation, as seen in prior works. Our procedure runs in polynomial time when the number of program variables is fixed. We examine various applications of our results on invariant generation, focusing on invariant verification and loop synthesis. The invariant verification problem investigates whether a polynomial ideal defining an algebraic set serves as an invariant for a given linear loop. We show that this problem is coNP-complete and lies in PSPACE when the input ideal is given in dense or sparse representations, respectively. In the context of loop synthesis, we aim to construct a loop with an infinite set of reachable states that upholds a specified algebraic property as an invariant. The strong synthesis variant of this problem requires the construction of loops for which the given property is the strongest invariant. In terms of hardness, synthesising loops over integers (or rationals) is as hard as Hilbert's Tenth problem (or its analogue over the rationals). When the constants of the output are constrained to bit-bounded rational numbers, we demonstrate that loop synthesis and its strong variant are both decidable in PSPACE, and in NP when the number of program variables is fixed.