Full abstraction for Reduced ML

Abstract

Abstract. We present the first effectively presentable fully abstract model for Stark's Reduced ML, the paradigmatic higher-order programming language combining call-by-value evaluation and integer-valued references. The model is constructed using techniques of nominal game semantics. Its distinctive feature is the presence of carefully restricted information about the store in plays, combined with conditions concerning the participants' ability to distinguish reference names. This leads to an explicit characterization of program equivalence.

Algorithmic Nominal Game Semantics

“…In this section we briefly recapitulate the game semantics of Reduced ML [12], insofar as it concerns modelling RedML β→β fin . We present it in a more concrete way, specialized to the types of RedML β→β fin , along with examples that motivate the respective technical conditions.…”

“…RML had a definite advantage though, as the associated game model was based on a finite set of moves. Equivalence in Reduced ML turns out much more subtle and the corresponding fully abstract game model [12] is unsuited to finite-alphabet representations. It so happens that the presence of bad variables does not change the induced observational equivalence in the call-by-name case of Idealized Algol [8], where a complete map of decidable fragments already exists [11] and increases in complexity (of deciding equivalence) are tightly linked to type-theoretic order.…”

“…To enable a systematic and computer-aided verification of equivalence between such terms, we translate them into a special class of automata over infinite alphabets in such a way that languages accepted by the automata will be faithful representations of their fully abstract game semantics [12].…”

Isomorphisms of types in the presence of higher-order references (extended version)

“…We chose to include bad variables in the language we consider for two reasons. Firstly, the games models that allow bad variables are notably simpler than those which do not [24], for which our methods do not directly apply. Secondly, the impact of allowing bad variables on our result will be reduced by the following proposition: To see how, we consider an extension L + of L + where we add a type constructor gvar for good variables, so that L + has both types var for bad variables and gvar for good variables.…”

Nominal Game Semantics

“…Since 2004, the nominal approach has led to a series of new full abstraction results. The languages covered are the ν-calculus [3] (purely functional language with names), λν [25] (a higher-order language with storage of untyped names), Reduced ML [31] (a higher-order language with integer-valued storage), RefML [32] (higher-order references) and Middleweight Java [34]. Nominal game semantics has also been used to model Concurrent ML [26] and exceptions [34].…”