Fragments of ML Decidable by Nested Data Class Memory Automata

Abstract

Abstract. The 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. We consider the fragment of (finitary) RML terms of order at most 1 with free variables of order at most 2, and identify two subfragments of this for which we show observational equivalence to be decidable. The first subfragment, RML P-Str 2⊢1 , consists of those terms in which the P-pointers in the game semantic representation are determined by the underlying sequence of moves. The second subfragment consists of terms in which the O-pointers of moves corresponding to free variables in the game semantic representation are determined by the underlying moves. These results are shown using a reduction to a form of automata over data words in which the data values have a tree-structure, reflecting the tree-structure of the threads in the game semantic plays. In addition we show that observational equivalence is undecidable at every third-or higher-order type, every second-order type which takes at least two firstorder arguments, and every second-order type (of arity greater than one) that has a first-order argument which is not the final argument.

ML and Extended Branching VASS

“…However, like CMA, weakness does help with the closure properties of the languages recognised. The closure properties of these automata are the same as for normal CMA [9]: weak deterministic VPCMA are closed under union, intersection and complementation; similarly for SVPCMA. We show that the complete plays in the game semantics of each RML EBVASS term-in-context are representable as a weak deterministic SVPCMA (Lemma 14).…”

“…Like the earlier results [15,9], Theorem 1 and Theorem 23 are proved by appealing to the game semantics for RML [3,13], which is fully abstract, i.e., the equational theory induced by the semantics coincides with observational equivalence. In game semantics [1,16], player P takes the viewpoint of the term-in-context, and player O takes the viewpoint of the program context or environment.…”

“…In recent work [9,8], we considered finitary RML terms-in-context with types of shape I (see Fig. 1).…”

“…1 by listing for each fragment the shapes of types allowable on the LHS and RHS of the turnstile, where β is a base type. 4 Note that (the RHS type) θ of shape I ranges over all first-order types; and θ Shape LHS Type, θi RHS Type, θ I [9] (β → β) → · · · → (β → β) → β β → · · · → β II [15] ((β → · · · → β) → β) → · · · → ((β → · · · → β) → β) → β (β → · · · → β) → β of shape II admits the simplest second-order types. Because [9] also establishes undecidability for the second-order type θ = (unit → unit) → unit → unit and the simplest third-order type θ = ((unit → unit) → unit) → unit, as far as closed terms are concerned, the only unclassified cases are second-order types of the shape…”

Algorithmic games for full ground references

“…A closely related language, called RML [1] (integer storage but with bad references) was studied in [8,14,27], but no full classification has emerged yet. Interestingly, closed terms of first-order type become decidable in this setting [8], in contrast to unit → unit → unit for GRef. Finally, the approach presented herein was pursued for Interface Middleweight Java [21] and implemented in the equivalence checker Conneqct [20].…”