Verifying Higher-Order Functional Programs with Pattern-Matching Algebraic Data Types

Type-based model checking algorithms for higher-order recursion schemes have recently emerged as a promising approach to the verification of functional programs.We introduce pattern-matching recursion schemes (PMRS) as an accurate model of computation for functional programs that manipulate algebraic data-types. PMRS are a natural extension of higher-order recursion schemes that incorporate pattern-matching in the defining rules. This paper is concerned with the following (undecidable) verification problem: given a correctness property φ, a functional program P (qua PMRS) and a regular input set I, does every term that is reachable from I under rewriting by P satisfy φ? To solve the PMRS verification problem, we present a sound semi-algorithm which is based on model-checking and counterexample guided abstraction finement. Given a no-instance of the verification problem, the method is guaranteed to terminate. From an order-n PMRS and an input set generated by a regular tree grammar, our method constructs an order-n weak PMRS which over-approximates only the first-order pattern-matching behaviour, whilst remaining completely faithful to the higher-order control flow. Using a variation of Kobayashi's type-based approach, we show that the (trivial automaton) model-checking problem for weak PMRS is decidable. When a violation of the property is detected in the abstraction which does not correspond to a violation in the model, the abstraction is automatically refined by 'unfolding' the pattern-matching rules in the program to give successively more and more accurate weak PMRS models. Copyright © 2011 ACM