A Fragment of ML Decidable by Visibly Pushdown Automata

Abstract

Abstract. The simply-typed, call-by-value language, RML, may be viewed as a canonical restriction of Standard ML to ground-type references, augmented by a "bad variable" construct in the sense of Reynolds. By a short type, we mean a type of order at most 2 and arity at most 1. We consider the O-strict fragment of (finitary) RML, RML O-Str , consisting of terms-in-context x1 : θ1, · · · , xn : θn M : θ such that θ is short, and every argument type of every θi is short. RML O-Str is surprisingly expressive; it includes several instances of (in)equivalence in the literature that are challenging to prove using methods based on (state-based) logical relations. We show that it is decidable whether a given pair of RML O-Str terms-in-context is observationally equivalent. Using the fully abstract game semantics of RML, our algorithm reduces the problem to the language equivalence of visibly pushdown automata. When restricted to terms in canonical form, the problem is EXPTIME-complete.

Hector: An Equivalence Checker for a Higher-Order Fragment of ML

“…We will make use of the fully abstract game semantics of RML [7]. This model views program execution as the playing of a game between the program and its environment.…”

“…In [7] we identified the O-strict fragment of RML. This is the fragment for which the justification pointers from O-moves are always uniquely reconstructible from the underlying move sequence (although those from P-moves can still be ambiguous).…”

“…This gives them highly desirable closure properties; in particular, language equivalence is decidable in polynomial time. Our translation from strategies to VPA allowed us to show that observational equivalence for O-strict terms is EXPTIME-complete [7].…”

“…Our VPA are constructed inductively over the normal forms of the language, following [7]. Given two such VPA, using a product construction [3], it is easy to construct another to accept their symmetric difference.…”

Game Semantic Analysis of Equivalence in IMJ

“…In our case, the potential for automation stems from the compositionality of the underlying semantics, which allows for a compositional translation of terms into automata in the decidable cases. Previous work based on games-based verification was mainly concerned with various fragments of ML equipped with storage of ground-type values [5,10]. In contrast, in this paper we tackle richer interactions of objects equipped with methods.…”