Removing Algebraic Data Types from Constrained Horn Clauses Using Difference Predicates

Abstract

We address the problem of proving the satisfiability of Constrained Horn Clauses (CHCs) with Algebraic Data Types (ADTs), such as lists and trees. We propose a new technique for transforming CHCs with ADTs into CHCs where predicates are defined over basic types, such as integers and booleans, only. Thus, our technique avoids the explicit use of inductive proof rules during satisfiability proofs. The main extension over previous techniques for ADT removal is a new transformation rule, called differential replacement, which allows us to introduce auxiliary predicates corresponding to the lemmas used when making inductive proofs. We present an algorithm that applies the new rule, together with the traditional folding/unfolding rules, for the automatic removal of ADTs. We prove that if the set of the transformed clauses is satisfiable, then 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, and that our approach is competitive with respect to a state-of-the-art tool that extends the CVC4 solver with induction.

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

“…We have also performed analogous experiments on the set of valid properties by using Z3-SPACER[35], instead of Eldarica, as reported on an earlier version of this paper[18]. The results of those experiments, which we do not report here, are very similar to those shown in Table1.This work has been SUBMITTED (under consideration) to the CILC 2020 special issue of the Journal of Logic and Computation…”

“…This paper is an improved, extended version of a paper that appears in the Proceedings of the 10th International Joint Conference on Automated Reasoning (IJCAR 2020) [18]. The paper was also presented at the 35th Italian Conference on Computational Logic (CILC 2020).…”

An Overview of the HFL Model Checking Project

“…Inspired by the unfold/fold transformation techniques for CHC solving [10,11], we have also studied unfold/fold transformations for the first-order fragment of HFL(Z) [23] to enhance the power of an automated HFL(Z) validity checker. The transformations are useful for reasoning about relations between fixpoint formulas.…”

Symbolic Automatic Relations and Their Applications to SMT and CHC Solving

“…There are other approaches to solving CHCs on recursive data structures. Unno et al [27] proposed a method for automatically applying induction on data structures, and De Angelis et al [9,10] proposed a method based on fold/unfold transformation. An advantage of our approach is that we can generate a symbolic automatic relation as a certificate of the satisfiability of CHCs.…”

“…This makes sense because solving CHCs without data structures is easier in practice (although the problem is undecidable in general, even without data structures). One may wonder why we do not directly translate CHCs with data structures to those without data structures, as in[9,10]. The detour through SARs has the following advantages.…”