History-Register Automata

Programs with dynamic allocation are able to create and use an unbounded number of fresh resources, such as references, objects, files, etc. We propose History-Register Automata (HRA), a new automata-theoretic formalism for modelling such programs. HRAs extend the expressiveness of previous approaches and bring us to the limits of decidability for reachability checks. The distinctive feature of our machines is their use of unbounded memory sets (histories) where input symbols can be selectively stored and compared with symbols to follow. In addition, stored symbols can be consumed or deleted by reset. We show that the combination of consumption and reset capabilities renders the automata powerful enough to imitate counter machines, and yields closure under all regular operations apart from complementation. We moreover examine weaker notions of HRAs which strike different balances between expressiveness and effectiveness.Comment: LMCS (improved version of FoSSaCS

Fragments of ML Decidable by Nested Data Class Memory Automata

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 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 first-order arguments, and every second-order type (of arity greater than one) that has a first-order argument which is not the final argument

Fragments of ML decidable by nested data class memory automata

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, RMLP−Str2⊢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 first-order arguments, and every second-order type (of arity greater than one) that has a first-order argument which is not the final argument

Game semantic analysis of equivalence in IMJ

Using game semantics, we investigate the problem of verifying contextual equivalences in Interface Middleweight Java (IMJ), an imperative object calculus in which program phrases are typed using interfaces. Working in the setting where data types are non-recursive and restricted to finite domains, we identify the frontier between decidability and undecidability by reference to the structure of interfaces present in typing judgments. In particular, we show how to determine the decidability status of problem instances (over a fixed type signature) by examining the position of methods inside the term type and the types of its free identifiers. Our results build upon the recent fully abstract game semantics of IMJ. Decidability is proved by translation into visibly pushdown register automata over infinite alphabets with fresh-input recognition

Saturating automata for game semantics

Saturation is a fundamental game-semantic property satisfied by strategies that interpret higher-order concurrent programs. It states that the strategy must be closed under certain rearrangements of moves, and corresponds to the intuition that program moves (P-moves) may depend only on moves made by the environment (O-moves). We propose an automata model over an infinite alphabet, called saturating automata, for which all accepted languages are guaranteed to satisfy a closure property mimicking saturation. We show how to translate the finitary fragment of Idealized Concurrent Algol (FICA) into saturating automata, confirming their suitability for modelling higher-order concurrency. Moreover, we find that, for terms in normal form, the resultant automaton has linearly many transitions and states with respect to term size, and can be constructed in polynomial time. This is in contrast to earlier attempts at finding automata-theoretic models of FICA, which did not guarantee saturation and involved an exponential blow-up during translation, even for normal forms.Comment: Presented at MFPS 202

Fully abstract models for effectful λ-calculi via category-theoretic logical relations

We present a construction which, under suitable assumptions, takes a model of Moggi’s computational λ-calculus with sum types, effect operations and primitives, and yields a model that is adequate and fully abstract. The construction, which uses the theory of fibrations, categorical glueing, ⊤⊤-lifting, and ⊤⊤-closure, takes inspiration from O’Hearn & Riecke’s fully abstract model for PCF. Our construction can be applied in the category of sets and functions, as well as the category of diffeological spaces and smooth maps and the category of quasi-Borel spaces, which have been studied as semantics for differentiable and probabilistic programming

A Complete Normal-Form Bisimilarity for State

International audienceWe present a sound and complete bisimilarity for an untyped $λ-calculus$ with higher-order local references. Our relation compares values by applying them to a fresh variable, like normal-form bisimilarity, and it uses environments to account for the evolving store. We achieve completeness by a careful treatment of evaluation contexts comprising open stuck terms. This work improves over Støvring and Lassen's incomplete environment-based normal-form bisimilarity for the $λρ-calculus$, and confirms, in relatively elementary terms, Jaber and Tabareau's result , that the state construct is discriminative enough to be characterized with a bisimilarity without any quantification over testing arguments