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In this paper, we show how the notion of tree dimension can be used in theverification of constrained Horn clauses (CHCs). The dimension of a tree is anumerical measure of its branching complexity and the concept here applies toHorn clause derivation trees. Derivation trees of dimension zero correspond toderivations using linear CHCs, while trees of higher dimension arise fromderivations using non-linear CHCs. We show how to instrument CHCs predicateswith an extra argument for the dimension, allowing a CHC verifier to reasonabout bounds on the dimension of derivations. Given a set of CHCs $P$, wedefine a transformation of $P$ yielding a dimension bounded set of CHCs$P^{\leq{k}}$. The set of derivations for $P^{\leq{k}}$ consists of thederivations for $P$ that have dimension at most $k$. We also show how toconstruct a set of clauses denoted $P^{>{k}}$ whose derivations have dimensionexceeding $k$. We then present algorithms using these constructions todecompose a CHC verification problem. One variation of this decompositionconsiders derivations of successively increasing dimension. The paper includesdescriptions of implementations and experimental results. Under considerationfor publication in Theory and Practice of Logic Programming (TPLP).

Tree dimension in verification of constrained Horn clauses