From Affine to Polynomial: Synthesizing Loops with Branches via Algebraic Geometry

Ensuring software correctness remains a fundamental challenge in formal program verification. One promising approach relies on finding polynomial invariants for loops. Polynomial invariants are properties of a program loop that hold before and after each iteration. Generating such invariants is a crucial task in loop analysis, but it is undecidable in the general case. Recently, an alternative approach to this problem has emerged, focusing on synthesizing loops from invariants. However, existing methods only synthesize affine loops without guard conditions from polynomial invariants. In this paper, we address a more general problem, allowing loops to have polynomial update maps with a given structure, inequations in the guard condition, and polynomial invariants of arbitrary form. We use algebraic geometry tools to design and implement an algorithm that computes a finite set of polynomial equations whose solutions correspond to all nondeterministic branching loops satisfying the given invariants. Furthermore, we introduce a new class of invariants for which we present a significantly more efficient algorithm. In other words, we reduce the problem of synthesizing loops to find solutions of multivariate polynomial systems with rational entries. This final step is handled in our software using an SMT solver.