Classifying and Solving Horn Clauses for Verification

Abstract

Abstract. As a promising direction to overcome difficulties of verification, researchers have recently proposed the use of Horn constraints as intermediate representation. Horn constraints are related to Craig interpolation, which is one of the main techniques used to construct and refine abstractions in verification, and to synthesise inductive loop invariants. We give a classification of the different forms of Craig interpolation problems found in literature, and show that all of them correspond to natural fragments of (recursion-free) Horn constraints. For a logic that has the binary interpolation property, all of these problems are solvable, but have different complexity. In addition to presenting the theoretical classification and solvability results, we present a publicly available collection of benchmarks to evaluate solvers for Horn constraints, categorized according to our classification. The benchmarks are derived from real-world verification problems. The behavior with our tools as well as with Z3 prover indicates the importance of Horn clause solving as distinct from the general problem of solving quantified constraints by quantifier instantiation.

On recursion-free Horn clauses and Craig interpolation

“…According to [37], a tree interpolant I T exists for this labelling function. By construction, for non-root positions p ∈ pos\{root} the interpolant labelling is equivalent to I T ( p) ≡ ¬a p ∨ I p , where I p does not contain any further auxiliary Boolean variables.…”

“…Vice versa, for every pair of complementary Horn clauses C 1 → p(t) and C 2 ∧ p(s) → false, we can determine solutions by first normalising the clauses to C 1 ∧t =x → p(x) and C 2 ∧s =x ∧ p(x) → false, and then finding solutions for the binary interpolation problem (C 1 ∧t =x) ∧ (C 2 ∧s =x). Proofs for the most cases of Theorem 1, in particular for inductive sequences of interpolants and DAG interpolants, are given in [38]. On the next pages, we first give a proof for a particularly important form of interpolation, tree interpolation, and then present a generalisation to disjunctive interpolation.…”

“…An essential advance was to use interpolants derived from unsatisfiability proofs to refine the abstraction [22]. In recent years, we have seen significant progress in interpolating methods for different logical constraints [8,9,34], and a wealth of more general forms of interpolation [1,21,34,37].…”

Cell Morphing: From Array Programs to Array-Free Horn Clauses

“…We convert the verification problem to Horn clauses, a common format for program verification problems [35] supported by a number of tools. Usual conversions [18] map variables and operations from the program to variables of the same type and the same operations in the Horn clause problem:…”

Guiding Craig interpolation with domain-specific abstractions

“…Predicate abstraction-based model checkers usually consider more general interpolation problems than just binary Craig interpolation: the interpolation queries might concern sequences of interpolants, tree interpolants, or other interpolation schemata [49]. For instance, given an unsatisfiable conjunction A 1 ∧ .…”