Satisfiability of Constrained Horn Clauses on Algebraic Data Types: A Transformation-based Approach

Abstract: We address the problem of checking the satisfiability of Constrained Horn Clauses (CHCs) defined on Algebraic Data Types (ADTs), such as lists and trees. We propose a new technique for transforming CHCs defined on ADTs into CHCs where the arguments of the predicates have only basic types, such as integers and booleans. Thus, our technique avoids, during satisfiability checking, the explicit use of proof rules based on induction over the ADTs. The main extension over previous techniques for ADT removal is a new transformation rule, called differential replacement, which allows us to introduce auxiliary predicates, whose definitions correspond to lemmas that are used when making inductive proofs. We present an algorithm that performs the automatic removal of ADTs by applying the new rule, together with the traditional folding/unfolding rules. We prove that, under suitable hypotheses, the set of the transformed clauses is satisfiable if and only if so is the set of the original clauses. By an experimental evaluation, we show that the use of the new rule significantly improves the effectiveness of ADT removal. We also show that our approach is competitive with respect to tools that extend CHC solvers with the use of inductive rules.