Algorithmic Games for Full Ground References

Abstract

We present a full classification of decidable and undecidable cases for contextual equivalence in a finitary ML-like language equipped with full ground storage (both integers and reference names can be stored). The simplest undecidable type is unit → unit → unit. At the technical level, our results marry game semantics with automata-theoretic techniques developed to handle infinite alphabets. On the automata-theoretic front, we show decidability of the emptiness problem for register pushdown automata extended with fresh-symbol generation.

Game Semantic Analysis of Equivalence in IMJ

“…The automata are also equipped with a visible pushdown stack which can also store names; these names are communicated between the registers and the stack via push/pop operations. They are therefore pushdown extensions of Fresh-Register Automata [15,10].…”

“…As variants of pushdown fresh-register automata, IMJ-automata have decidable emptiness problem [10]. Moreover, the number of compatible triples (m Σ , Φ, ρ) is bounded with respect to Γ, ∆ modulo name-permutations.…”

“…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.…”

A Contextual Equivalence Checker for IMJ*

“…The particular model for Java [10], that we take advantage of in this paper, is built over an infinite set of names (used to model object references), suggesting the use of automata over infinite alphabets as a framework for representing the denotations. Indeed, in the companion research paper [8], we have identified a fragment of Java, called IMJ*, whose game semantics can be captured using (visibly) pushdown register automata over infinite alphabets with fresh-symbol generation [9]. The automata are equipped with a finite set of registers for storing elements of the infinite alphabet as well as a pushdown store.…”

“…the impossibility of reaching a final state, the particular letter that is read is irrelevant, which explains why the degenerate transition is an epsilon transition. The result of this fixed point computation is a fresh pushdown-register automaton (FP-DRA) [9], that is, a register automaton [11] (RA) with global-fresh transitions and a pushdown stack.…”