Quantified Invariants via Syntax-Guided Synthesis

Abstract

Programs with arrays are ubiquitous. Automated reasoning about arrays necessitates discovering properties about ranges of elements at certain program points. Such properties are formally specified by universally quantified formulas, which are difficult to find, and difficult to prove inductive. In this paper, we propose an algorithm based on an enumerative search that discovers quantified invariants in stages. First, by exploiting the program syntax, it identifies ranges of elements accessed in each loop. Second, it identifies potentially useful facts about individual elements and generalizes them to hypotheses about entire ranges. Finally, by applying recent advances of SMT solving, the algorithm filters out wrong hypotheses. The combination of properties is often enough to prove that the program meets a safety specification. The algorithm has been implemented in a solver for Constrained Horn Clauses, Freq-Horn, and extended to deal with multiple (possibly nested) loops. We show that FreqHorn advances state-of-the-art on a wide range of public array-handling programs.