Tree dimension in verification of constrained Horn clauses

Abstract

In this paper, we show how the notion of tree dimension can be used in the verification of constrained Horn clauses (CHCs). The dimension of a tree is a numerical measure of its branching complexity and the concept here applies to Horn clause derivation trees. Derivation trees of dimension zero correspond to derivations using linear CHCs, while trees of higher dimension arise from derivations using non-linear CHCs. We show how to instrument CHCs predicates with an extra argument for the dimension, allowing a CHC verifier to reason about bounds on the dimension of derivations. Given a set of CHCs \(P\), we define a transformation of \(P\) yielding a \emph{dimension bounded} set of CHCs \(P^{\atmost{k}}\). The set of derivations for \(P^{\atmost{k}}\) consists of the derivations for \(P\) that have dimension at most \(k\). We also show how to construct a set of clauses denoted \(P^{\exceeds{k}}\) whose derivations have dimension exceeding \(k\). We then present algorithms using these constructions to decompose a CHC verification problem. One variation of this decomposition considers derivations of successively increasing dimension. The paper includes descriptions of implementations and experimental results. Under consideration for publication in Theory and Practice of Logic Programming (TPLP).