Algebra-based Synthesis of Loops and their Invariants (Invited Paper)

TL;DR

This paper presents algebraic methods for synthesizing loops and their invariants in programs modeled by linear recurrence equations, enabling correctness proofs, optimizations, and sequence generation.

Contribution

It introduces algorithms for synthesizing polynomial invariants and loops satisfying given invariants for systems modeled by C-finite recurrences.

Findings

Algorithms for polynomial invariant synthesis

Automated methods for loop synthesis from invariants

Applications in correctness proofs and compiler optimization

Abstract

Provably correct software is one of the key challenges in our softwaredriven society. While formal verification establishes the correctness of a given program, the result of program synthesis is a program which is correct by construction. In this paper we overview some of our results for both of these scenarios when analysing programs with loops. The class of loops we consider can be modelled by a system of linear recurrence equations with constant coefficients, called C-finite recurrences. We first describe an algorithmic approach for synthesising all polynomial equality invariants of such non-deterministic numeric single-path loops. By reverse engineering invariant synthesis, we then describe an automated method for synthesising program loops satisfying a given set of polynomial loop invariants. Our results have applications towards proving partial correctness of programs, compiler optimisation and generating number sequences from algebraic relations. This is a preprint that was invited for publication at VMCAI 2021.