The trivial-automaton model checking problem for higher-order recursion schemes has become a widely studied object in connection with the automatic verification of higher-order programs. The problem is formidably hard: despite considerable progress in recent years, no decision procedures have been demonstrated to scale robustly beyond recursion schemes that comprise more than a few hundred rewrite rules. We present a new, fixed-parameter polynomial time algorithm, based on a novel, type directed form of abstraction refinement in which behaviours of a scheme are distinguished by the abstraction according to the intersection types that they inhabit (the properties that they satisfy). Unlike other intersection type approaches, our algorithm reasons both about acceptance by the property automaton and acceptance by its dual, simultaneously, in order to minimize the amount of work done by converging on the solution to a problem instance from both sides. We have constructed Preface, a prototype implementation of the algorithm, and assembled an extensive body of evidence to demonstrate empirically that the algorithm readily scales to recursion schemes of several thousand rules, well beyond the capabilities of current state-of-the-art higher-order model checkers.

Higher-order model checking is the problem of model checking (possibly) infinite trees generated by higher-order recursion schemes (HORS). HORS are a natural abstract model of functional programs, and HORS model checkers play a similar role to checkers of Boolean programs in the imperative setting. Most research effort so far has focused on checking safety properties specified using trivial tree automata i.e. Buechi tree automata all of whose states are final. Building on our previous work, we present a higher-order model checker, TravMC2, which supports properties specified using alternating parity tree automata (or equivalently monadic second order logic). Our experimental results offer an encouraging comparison with an existing checker, TrecS-APT.

Higher-order model checking - the model checking of trees generated by higher-order recursion schemes (HORS) - is a natural generalisation of finite-state and pushdown model checking. Recent work has shown that it can serve as a basis for software model checking for functional languages such as ML and Haskell. In this paper, we introduce higher-order recursion schemes with cases (HORSC), which extend HORS with a definition-by-cases construct (to express program branching based on data) and non-determinism (to express abstractions of behaviours). This paper is a study of the universal HORSC model checking problem for deterministic trivial automata: does the automaton accept every tree in the tree language generated by the given HORSC? We first characterise the model checking problem by an intersection type system extended with a carefully restricted form of union types. We then present an algorithm for deciding the model checking problem, which is based on the notion of traversals induced by the fully abstract game semantics of these schemes, but presented as a goal-directed construction of derivations in the intersection and union type system. We view HORSC model checking as a suitable backend engine for an approach to verifying functional programs. We have implemented the algorithm in a tool called TravMC, and demonstrated its effectiveness on a test suite of programs, including abstract models of functional programs obtained via an abstraction-refinement procedure from pattern-matching recursion schemes.