Game Semantics Research Articles

A Game Semantics of Concurrent Separation Logic

In this paper, we develop a game-theoretic account of concurrent separation logic. To every execution trace of the Code confronted to the Environment, we associate a specification game where Eve plays for the Code, and Adam for the Environment. The purpose of Eve and Adam is to decompose every intermediate machine state of the execution trace into three pieces: one piece for the Code, one piece for the Environment, and one piece for the available shared resources. We establish the soundness of concurrent separation logic by interpreting every derivation tree of the logic as a winning strategy of this specification game.

On Dialogue Games and Graph Games

Dialogue games were introduced by Melliès as an attempt to unify two historical paradigms of game semantics: concrete data structures and arena games. The definition of dialogue games relies on the idea that a move m of an arena game can be decomposed as a pair m=(α,v) consisting of a cell α and of a value v. Consequently, a dialogue game is defined as a quadripartite forest whose nodes are separated into four classes: Opponent cells, Opponent values, Player cells, Player values. Although the translation from arena games to dialogue games is essentially immediate, the relationship between dialogue games and concrete data structures is more intricate. In order to clarify it, we study the relationship between dialogue games and graph games which were introduced by Hyland and Schalk to provide a graph-theoretic account of Berry and Curien's sequential algorithm model. We construct a fully faithful functor from a category of dialogue games to the category of graph games and conflict-free strategies. This leads us to an alternative definition of conflict-free strategies in graph games as balanced and bi-invariant strategies in dialogue games.

What do introduction sections tell us about the intent of scholarly work: A contribution on contributions

The paper marks a rare empirical examination of the semantics of contribution claims in the introduction sections of journal articles. The framing of contribution claims is a significantly under-examined area of scholarly activity which underpins the methodical act of communicating the value of research to an audience. The paper presents a systematic review of 538 papers in the three leading industrial marketing journals, Industrial Marketing Management, the Journal of Business & Industrial Marketing and the Journal of Business-to-Business Marketing and uses a phased approach to categorize contribution claims made by authors in their introductions. The paper identifies four main categories of contribution, defined as Incremental, revelatory, replicatory and consolidatory − proposing sub-categorizations within them, and reports on the proportionality of these strategies in the sample whilst capturing the semantic games played by authors in pursuit of these claims. Findings are of interest to industrial marketers but the conceptual framework and systematic methods presented in the paper are offered as transferable to any discipline or body of work, and therefore have broader disciplinary appeal. Findings are of interest to authors, reviewers and editors in coalescing fragmented understanding of contribution strategies into a coherent framework for action.

Game semantics for dependent types

We present a model of dependent type theory (DTT) with Π-, 1-, Σ- and intensional Id-types, which is based on a slight variation of the (call-by-name) category of AJM-games and history-free winning well-bracketed strategies. The model satisfies Streicher's criteria of intensionality and refutes function extensionality. The principle of uniqueness of identity proofs is satisfied.We show it contains a submodel as a full subcategory which gives a faithful interpretation of DTT with Π-, 1-, Σ- and intensional Id-types and, additionally, finite inductive type families. This smaller model is fully (and faithfully) complete with respect to the syntax at the type hierarchy built without Id-types, as well as at the more general class of types where we allow for one strictly positive occurrence of an Id-type. Definability for the full type hierarchy with Id-types remains to be investigated.

A Logical Analysis of Monty Hall and Sleeping Beauty

Hintikka and Sandu’s independence-friendly (IF) logic is a conservative extension of first-order logic that allows one to consider semantic games with imperfect information. In the present article, we first show how several variants of the Monty Hall problem can be modeled as semantic games for IF sentences. In the process, we extend IF logic to include semantic games with chance moves and dub this extension stochastic IF logic. Finally, we use stochastic IF logic to analyze the Sleeping Beauty problem, leading to the conclusion that the thirders are correct while identifying the main error in the halfers’ argument.

Game Semantics in Teaching of Classical First Order Predicate Logic to Stu- dents of Computer Science

Game-theoretical interpretation of logical semantics has been first suggested in 1950s. Although it is well know among professionals, it is often excluded from introductory courses in logic. In this paper we examine whether and to what extent would including game semantics in the syllabus of an introductory course of predicate and propositional logic prove to be beneficial to students’ understanding of the subject. To that purpose, we have exposed two groups of undergraduate students of computer science to game semantics as part of recitations for the course Propositional and Predicate Logic. We focus mainly on how this affects students’ mastery of two particular topics – alternating quantification and restricted quantification. Observations were made during the recitations to evaluate any changes in students’ approach to solving relevant problems in classroom. Moreover, before the final test of the recitations one of the groups was given a brief review of game semantics and the other was not. We have measured the difference in performance of these two groups and the results support the hypothesis that game semantics helps students master certain topics of predicate logic. We reference some of the later research in psychology and pedagogy to help us explain this effect.

O cerco rompido. (Leitura de O círculo, de Alina Paim)

The text aims to draw attention to a novel that has gone completely unnoticed by criticism, relegated to oblivion. Even though its author, a Northeastern woman, was known at that time for another aspect of her work, the more political one, the novel, published in 1965, has not yet received the reading worthy of what it raises. This may be explained by the engendering of the questions posed, both concerning the feminine condition, and the narrative and discursive relationships. I try, then, to underline the labyrinthine exploration of time, semantic games, metaphorical inflections, improprieties, metonymic employment, etc., which could have given it the prominent place of a vanguard narrative.

On the Learnability of Programming Language Semantics

Game semantics is a powerful method of semantic analysis for programming languages. It gives mathematically accurate models ("fully abstract") for a wide variety of programming languages. Game semantic models are combinatorial characterisations of all possible interactions between a term and its syntactic context. Because such interactions can be concretely represented as sets of sequences, it is possible to ask whether they can be learned from examples. Concretely, we are using long short-term memory neural nets (LSTM), a technique which proved effective in learning natural languages for automatic translation and text synthesis, to learn game-semantic models of sequential and concurrent versions of Idealised Algol (IA), which are algorithmically complex yet can be concisely described. We will measure how accurate the learned models are as a function of the degree of the term and the number of free variables involved. Finally, we will show how to use the learned model to perform latent semantic analysis between concurrent and sequential Idealised Algol.

Verifying annotated program families using symbolic game semantics

Many software systems are today built as program families. They permit users to derive a custom program (variant) by selecting suitable configuration options at compile time according to their requirements. Many such program families are safety critical. However, most existing verification techniques are designed to work on the level of single programs. Their application to program families would require to verify each variant in isolation, in a brute force fashion. This approach does not scale in practice due to the (potentially) huge number of possible variants.In this paper, we propose an efficient game semantics based approach for verification of open program families, i.e. program families with undefined components (identifiers). We use symbolic representation of algorithmic game semantics, where symbolic values for inputs are used instead of concrete ones. In this way, we can compactly represent program families with infinite integers as so-called (finite state) featured symbolic automata. Specifically designed model checking algorithms are then employed to uniformly verify safety of all programs (variants) from a family at once using a single compact model and to pinpoint those programs that are unsafe (respectively, safe). We present a prototype tool implementing this approach, and we illustrate its practicality with several examples.

Probabilistic Analysis Based On Symbolic Game Semantics and Model Counting

Probabilistic program analysis aims to quantify the probability that a given program satisfies a required property. It has many potential applications, from program understanding and debugging to computing program reliability, compiler optimizations and quantitative information flow analysis for security. In these situations, it is usually more relevant to quantify the probability of satisfying/violating a given property than to just assess the possibility of such events to occur. In this work, we introduce an approach for probabilistic analysis of open programs (i.e. programs with undefined identifiers) based on game semantics and model counting. We use a symbolic representation of algorithmic game semantics to collect the symbolic constraints on the input data (context) that lead to the occurrence of the target events (e.g. satisfaction/violation of a given property). The constraints are then analyzed to quantify how likely is an input to satisfy them. We use model counting techniques to count the number of solutions (from a bounded integer domain) that satisfy given constraints. These counts are then used to assign probabilities to program executions and to assess the probability for the target event to occur at the desired level of confidence. Finally, we present the results of applying our approach to several interesting examples and illustrate the benefits they may offer.

Algorithmic games for full ground references

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 mathsf {unit}rightarrow mathsf {unit}rightarrow mathsf {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.

Cooperation in Games and Epistemic Readings of Independence-Friendly Sentences

In the literature on logics of imperfect information it is often stated, incorrectly, that the Game-Theoretical Semantics of Independence-Friendly (IF) quantifiers captures the idea that the players of semantical games are forced to make some moves without knowledge of the moves of other players. We survey here the alternative semantics for IF logic that have been suggested in order to enforce this “epistemic reading” of sentences. We introduce some new proposals, and a more general logical language which distinguishes between “independence from actions” and “independence from strategies”. New semantics for IF logic can be obtained by choosing embeddings of the set of IF sentences into this larger language. We compare all the semantics proposed and their purported game-theoretical justifications, and disprove a few claims that have been made in the literature.

GAME SEMANTICS AND THE GEOMETRY OF BACKTRACKING: A NEW COMPLEXITY ANALYSIS OF INTERACTION

AbstractWe present abstract complexity results about Coquand and Hyland–Ong game semantics, that will lead to new bounds on the length of first-order cut-elimination, normalization, interaction between expansion trees and any other dialogical process game semantics can model and apply to. In particular, we provide a novel method to bound the length of interactions between visible strategies and to measure precisely the tower of exponentials defining the worst-case complexity. Our study improves the old estimates on average by several exponentials.

Block structure vs scope extrusion: between innocence and omniscience

We study the semantic meaning of block structure using game semantics. To that end, we introduce the notion of block-innocent strategies and characterise call-by-value computation with block-allocated storage through soundness, finite definability and universality results. This puts us in a good position to conduct a comparative study of purely functional computation, computation with block storage as well as that with dynamic memory allocation. For example, we can show that dynamic variable allocation can be replaced with block-allocated variables exactly when the term involved (open or closed) is of base type and that block-allocated storage can be replaced with purely functional computation when types of order two are involved. To illustrate the restrictive nature of block structure further, we prove a decidability result for a finitary fragment of call-by-value Idealized Algol for which it is known that allowing for dynamic memory allocation leads to undecidability.

Game semantics approach to higher-order complexity

Game semantics was initially defined and used to characterize pcf functionals. We use this approach to propose a definition of complexity for such higher-order functions, as well as a class of polynomial time computable higher-order functions.

Linear β-reduction

Linear head reduction is a key tool for the analysis of reduction machines for lambda-calculus and for game semantics. Its definition requires a notion of redex at a distance named primary redex in the literature. Nevertheless, a clear and complete syntactic analysis of this rule is missing. We present here a general notion of beta-reduction at a distance and of linear reduction (i.e., not restricted to the head variable), and we analyse their relations and properties. This analysis rests on a variant of the so-called sigma-equivalence that is more suitable for the analysis of reduction machines, since the position along the spine of primary redexes is not permuted. We finally show that, in the simply typed case, the proof of strong normalisation of linear reduction can be obtained by a trivial tuning of Gandy's proof for strong normalisation of beta-reduction.

Jump from parallel to sequential proofs: exponentials

In previous works, by importing ideas from game semantics (notably Faggian–Maurel–Curien'sludics nets), we defined a new class of multiplicative/additive polarized proof nets, calledJ-proof nets. The distinctive feature of J-proof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by usingjumps(that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work, we extend J-proof nets to the multiplicative/exponential fragment, in order to take into account structural rules: More precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (calledcone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can bepartially overlapping. Moreover, we define cut-elimination for exponential J-proof nets, proving, by a variant of Gandy's method, that even in case of ‘superposed’ cones, reduction enjoys confluence and strong normalization.

A micrological study of negation

Tensorial logic is a primitive logic of tensor and negation which refines linear logic by relaxing the hypothesis that linear negation is involutive. Thanks to this mild modification, tensorial logic provides a type-theoretic account of game semantics where innocent strategies are portrayed as temporal refinements of traditional proof-nets in linear logic. In this paper, we study the algebraic and combinatorial structure of negation in a non-commutative variant of tensorial logic. The analysis is based on a 2-categorical account of dialogue categories, which unifies tensorial logic with linear logic, and discloses a primitive symmetry between proofs and anti-proofs. The micrological analysis of tensorial negation reveals that it can be decomposed into a series of more elementary components: an adjunction L⊣R between the left and right negation functors L and R; a pair of linear distributivity laws κ▪ and κ▪ which refines the linear distributivity law between ⊗ and ⅋ in linear logic, and generates the Opponent and Proponent views of innocent strategies between dialogue games; a pair of axiom and cut combinators adapted from linear logic; an involutive change of frame (−)⁎ reversing the point of view of Prover and of Denier on the logical dispute, and reversing the polarity of moves in the dialogue game associated to the tensorial formula.

Combining control effects and their models: Game semantics for a hierarchy of static, dynamic and delimited control effects

Computational effects which provide access to the flow of control (such as first-class continuations, exceptions and delimited continuations) are important features of higher-order programming languages. There are fundamental differences between them in terms of operational behaviour, expressiveness and implementation, so that understanding how they combine and relate to each other is a challenging objective, with a key role for semantics in making this precise.This paper develops operational and denotational semantics for a hierarchy of programming languages which include combinations of locally declared control prompts to which a program can escape, with first-class continuations which may either capture their enclosing prompts, or be delimited by them. We describe two different hierarchies of models, both based on categories of games and strategies with a computational monad, but obtained using different methodologies. By relaxing combinations of behavioural constraints on strategies with control flow represented by annotation with control pointers we are able to give direct and explicit characterizations of control operators and their effects, including examples characterizing their macro-expressiveness. By constructing a parallel hierarchy of models by applying sequences of monad transformers, and relating these to the direct interpretation of control effects, we obtain games interpretations of higher-level abstractions such as continuations and exceptions, which can be used as the basis for equational reasoning about programs.

Game semantics for non-monotonic intensional logic programming

Intensional logic programming is an extension of logic programming based on intensional logic, which includes as special cases both temporal and modal logic programming. In [13], M. Orgun and W.W. Wadge provided a general framework for capturing the semantics of intensional logic programming languages. They demonstrated that if the intensional operators of a language obey some simple semantic properties, then the programs of the language are guaranteed to have a minimum model semantics. One key property involved in the construction of [13] is the monotonicity of intensional operators. In this paper we consider intensional logic programming from a game-theoretic perspective. In particular we define a two-person game and demonstrate that it can be used in order to define a model for any given intensional program of the class introduced in [13]. Moreover, this model is shown to be identical to the minimum model constructed in [13]. More importantly, we demonstrate that the game is even applicable to intensional languages with non-monotonic operators. In this way we provide the first (to our knowledge) general framework for capturing the semantics of non-monotonic intensional logic programming.