Solving non-linear Horn clauses using a linear solver

Developing an efficient non-linear Horn clause solver is a challenging task since the solver has to reason about the tree structures rather than the linear ones as in a linear solver. In this paper we propose an incremental approach to solving a set of non-linear Horn clauses using a linear Horn clause solver. We achieve this by interleaving a program transformation and a linear solver. The program transformation is based on the notion of tree dimension, which we apply to trees corresponding to Horn clause derivations. The dimension of a tree is a measure of its non-linearity – for example a linear tree (whose nodes have at most one child) has dimension zero while a complete binary tree has dimension equal to its height. A given set of Horn clauses P can be transformed into a new set of clauses P≤k (whose derivation trees are the subset of P’s derivation trees with dimension at most k). We start by generating P≤k with k = 0, which is linear by definition, then pass it to a linear solver. If P≤k has a solution M, and is a solution to P then P has a solution M. If M is not a solution of P, we plugged M to P≤k+1 which again becomes linear and pass it to the solver and continue successively for increasing value of k until we find a solution to P or resources are exhausted. Experiment on some Horn clause verification benchmarks indicates that this is a promising approach for solving a set of non-linear Horn clauses using a linear solver. It indicates that many times a solution obtained for some under-approximation P≤k of P becomes a solution for P for a fairly small value of k.