Game Semantics Research Articles

Games for Topological Fixpoint Logic

Topological fixpoint logics are a family of logics that admits topological models and where the fixpoint operators are defined with respect to the topological interpretations. Here we consider a topological fixpoint logic for relational structures based on Stone spaces, where the fixpoint operators are interpreted via clopen sets. We develop a game-theoretic semantics for this logic. First we introduce games characterising clopen fixpoints of monotone operators on Stone spaces. These fixpoint games allow us to characterise the semantics for our topological fixpoint logic using a two-player graph game. Adequacy of this game is the main result of our paper. Finally, we define bisimulations for the topological structures under consideration and use our game semantics to prove that the truth of a formula of our topological fixpoint logic is bisimulation-invariant.

An invitation to game semantics

Game semantics is a flexible semantic theory that has led in recent years to an unprecedented number of full abstraction results for various programming paradigms. We present a gentle introduction to the subject, focussing on high-level ideas and examples with a view to providing a bridge to more technical literature.

Semantics column

In this quarter's column, Andrzej Murawski and Nikos Tzevelekos offer us an invitation to game semantics. This approach to giving semantics originated in work of the philosopher and mathematician Paul Lorenzen in the late 1950s [10], who viewed the meaning of a proposition ϕ as a dialogue between two players, a proponent who asserts ϕ, and an opponent, who denies ϕ. A proof of ϕ is then a winning strategy for proponent. The approach was used by Andreas Blass [5] to give a semantics to Girard's linear logic [7]. Noting some shortcomings in Blass's model, Abramsky and Jagadeesan [2] refined the model to devise a more expressive game semantics for multiplicative linear logic.

A Game Semantics for System P

In this paper we introduce a game semantics for System P, one of the most studied axiomatic systems for non-monotonic reasoning, conditional logic and belief revision. We prove soundness and completeness of the game semantics with respect to the rules of System P, and show that an inference is valid with respect to the game semantics if and only if it is valid with respect to the standard order semantics of System P. Combining these two results leads to a new completeness proof for System P with respect to its order semantics. Our approach allows us to construct for every inference either a concrete proof of the inference from the rules in System P or a countermodel in the order semantics. Our results rely on the notion of a witnessing set for an inference, whose existence is a concise, necessary and sufficient condition for validity of an inferences in System P. We also introduce an infinitary variant of System P and use the game semantics to show its completeness for the restricted class of well-founded orders.

A Game-Based Approach for PCTL* Stochastic Model Checking with Evidence

Stochastic model checking is a recent extension and generalization of the classical model checking, which focuses on quantitatively checking the temporal property of a system model. PCTL* is one of the important quantitative property specification languages, which is strictly more expressive than either PCTL (probabilistic computation tree logic) or LTL (linear temporal logic) with probability bounds. At present, PCTL* stochastic model checking algorithm is very complicated, and cannot provide any relevant explanation of why a formula does or does not hold in a given model. For dealing with this problem, an intuitive and succinct approach for PCTL* stochastic model checking with evidence is put forward in this paper, which includes: presenting the game semantics for PCTL* in release-PNF (release-positive normal form), defining the PCTL* stochastic model checking game, using strategy solving in game to achieve the PCTL* stochastic model checking, and refining winning strategy as the evidence to certify stochastic model checking result. The soundness and the completeness of game-based PCTL* stochastic model checking are proved, and its complexity matches the known lower and upper bounds. The game-based PCTL* stochastic model checking algorithm is implemented in a visual prototype tool, and its feasibility is demonstrated by an illustrative example.

Aristotle on Universal Quantification: A Study from the Point of View of Game Semantics

In this paper we provide an interpretation of Aristotle's rule for the universal quantifier in Topics Θ 157a34–37 and 160b1–6 in terms of Paul Lorenzen's dialogical logic. This is meant as a contribution to the rehabilitation of the role of dialectic within the Organon. After a review of earlier views of Aristotle on quantification, we argue that this rule is related to the dictum de omni in Prior Analytics A 24b28–29. This would be an indication of the dictum’s origin in the context of dialectical games. One consequence of our approach is a novel explanation of the doctrine of the existential import of the quantifiers in dialectical terms. After a brief survey of Lorenzen's dialogical logic, we offer a set of rules for dialectical games based on previous work by Castelnérac and Marion, to which we add here the rule for the universal quantifier, as interpreted in terms of its counterpart in dialogical logic. We then give textual evidence of the use of that rule in Plato's dialogues, thus showing that Aristotle only made explicit a rule already implicit in practice, while providing a new interpretation of ‘epagogic’ arguments. Finally, we show how a proper understanding of that rule involves further rules concerning counterexamples and delaying tactics, stressing again the parallels with dialogical logic.

R2P and the Syrian crisis: when semantics becomes a matter of life or death

The proposition advanced here is that the failure of the international community to implement R2P respecting the Syrian situation is itself an ‘act of aggression' against the sovereignty of the Syrian State which derives from the sovereignty of the people and does not reside in the Assad regime or any particular Syrian regime. The framing of the responsibility to protect (R2P) and its potential implementation in any particular context, such as Syria, is tragically most often reduced to a semantic game version of Roulette with life or death consequences, with the latter more probable the longer the members of the UN Security Council with veto power play the game and stall on action by force where no peaceful options can end the mass atrocities. Hence human rights advocates must play the semantic game more deftly if they are to be positioned to contribute to saving civilian lives. Here follows then a tentative, but perhaps in some ways audacious attempt at just such semantic gamesmanship in support of R2P as not only a viable principle; but one which goes to the heart of the international rule of law, the intent and meaning of the UN Charter and the very credibility of the UN as a mechanism for the promotion of human rights and the achievement of international peace and security.

Game Theoretical Semantic for Relevant Logic

In 1979 D.E. Over proposed game theoretical semantics for first-degree entailment formulated by Anderson and Belnap. In order to extend this approach to include other systems of relevant logc (e.g., $\boldsymbol{R}$) we have two promoting facts. Firstly, there is Routley- Meyer’s situational semantic for system$\boldsymbol{R}$ of relevant logic. Secondly, this semantics shows some resemblance with W__ojcicki’s situational semantic of non-fregean logic for which the situational game semantics was developed by author exploiting essentially the notion of non-fregean games. In the paper an attempt is done to give a partial account of these results and some conception of situational games developed which laid down into foundation of the game theoretical semantics of relevant logic $\boldsymbol{R}$.

The Rényi–Ulam games and many-valued logics

In this paper we introduce complete game semantics for Product, Gödel, BL and SBL logics (game semantics for Łukasiewicz logic are well-known). For each of these logics we introduce a variant of the Rényi–Ulam game whose states are equipped in a natural way with an algebraic structure. Moreover we prove that each logic is complete with respect to the algebras of the states for the corresponding game.

Multigames and strategies, coalgebraically

Coalgebraic games have been recently introduced as a generalization of Conway games and other notions of games arising in different contexts. Using coalgebraic methods, games can be viewed as elements of a final coalgebra for a suitable functor, and operations on games can be analyzed in terms of (generalized) coiteration schemata. Coalgebraic games are sequential in nature, i.e., at each step either the Left (L) or the Right (R) player moves (global polarization); moreover, only a single move can be performed at each step. Recently, in the context of Game Semantics, concurrent games have been introduced, where global polarization is abandoned, and multiple moves are allowed. In this paper, we introduce coalgebraic multigames, which are situated half-way between traditional sequential games and concurrent games: global polarization is still present, however multiple moves are possible at each step, i.e., a team of L/R players moves in parallel. Coalgebraic operations, such as sum and negation, can be naturally defined on multigames. Interestingly, sum on coalgebraic multigames turns out to be related to Conway's selective sum on games, rather than the usual (sequential) disjoint sum. Selective sum has a parallel nature, in that at each step the current player performs a move in at least one component of the sum game, while on disjoint sum the current player performs a move in exactly one component at each step. A presentation of strategies on coalgebraic games is given via a final coalgebra of a pair of mutually recursive functors, and a suitable notion of simulation. A monoidal closed category of coalgebraic multigames in the vein of a Joyal category of Conway games is then built. The relationship between coalgebraic multigames and games is then formalized via an equivalence of the multigame category and a monoidal closed category of coalgebraic games where tensor is selective sum.

Bounding linear head reduction and visible interaction through skeletons

In this paper, we study the complexity of execution in higher-order programming languages. Our study has two facets: on the one hand we give an upper bound to the length of interactions between bounded P-visible strategies in Hyland-Ong game semantics. This result covers models of programming languages with access to computational effects like non-determinism, state or control operators, but its semantic formulation causes a loose connection to syntax. On the other hand we give a syntactic counterpart of our semantic study: a non-elementary upper bound to the length of the linear head reduction sequence (a low-level notion of reduction, close to the actual implementation of the reduction of higher-order programs by abstract machines) of simply-typed lambda-terms. In both cases our upper bounds are proved optimal by giving matching lower bounds. These two results, although different in scope, are proved using the same method: we introduce a simple reduction on finite trees of natural numbers, hereby called interaction skeletons. We study this reduction and give upper bounds to its complexity. We then apply this study by giving two simulation results: a semantic one measuring progress in game-theoretic interaction via interaction skeletons, and a syntactic one establishing a correspondence between linear head reduction of terms satisfying a locality condition called local scope and the reduction of interaction skeletons. This result is then generalized to arbitrary terms by a local scopization transformation.

A short note on type-inhabitation: Formula-trees vs. game semantics

This short note compares two different methods for exploring type-inhabitation in the simply typed lambda-calculus, highlighting their similarities.

On the use (and abuse) of Logic in Game Theory

A quick glance at the opening paragraphs in many of the classic logic textbooks reveals a common view: Logical methods highlight the reasoning patterns of a single (idealized) agent engaged in some form of mathematical thinking.1 However, this traditional view of the “subject matter” of logic is expanding. There is a growing literature using phrases such as “rational interaction” or “information flow” to describe its subject matter while still employing traditional logical methods. The clearest example of this can be found in the work of Johan van Benthem and others on logical dynamics (van Benthem, 1996; van Ditmarsch et al., 2007; Pacuit, 2013b); Rohit Parikh and others on social software (Parikh, 2002; van Eijck and Vergrugge, 2009); Samson Abramsky and others on game semantics for linear logic (Abramsky, 2007), and Mike Wooldridge, Valentin Goranko and others on logics for multiagent systems (Wooldridge, 2000; Goranko and Jamroga, 2004; van der Hoek and Wooldridge, 2003). There are many issues driving this shift in thinking about what logic is about (see van Benthem, 2005, for a discussion). One important reason for this shift is the close connection between logic and game theory. For centuries, logicians have used game-theoretic notions in their analyses. (See Hodges (2013); Gradel (2011) for an overview of applications of game theory in logic.) In the past 20 years or so, the influence has started to run in the other direction. There is now a very active research area focused on adapting existing logical systems and developing new ones to reason about different aspects of a game situation. Full coverage of this fascinating research area would require much more space than I have in this short article. So, I will not attempt a comprehensive overview of the use of logic in game theory2 and, instead, will focus on just a few key topics. The first topic is the question of motivation: Why should game theorists learn some logic, and, vice versa, why should logicians learn some game theory? There is a very simple reason why logicians have become interested in game theory: The mathematical models that game theorists use to describe a game situation are natural models for many existing logical languages (e.g., epistemic, doxastic and preference logics, as well as conditional logics, temporal logics and logics of action). Thus, game theory provides a fertile testing ground for a variety of logical systems. The following

On the Relation of Interaction Semantics to Continuations and Defunctionalization

In game semantics and related approaches to programming language semantics, programs are modelled by interaction dialogues. Such models have recently been used in the design of new compilation methods, e.g. for hardware synthesis or for programming with sublinear space. This paper relates such semantically motivated non-standard compilation methods to more standard techniques in the compilation of functional programming languages, namely continuation passing and defunctionalization. We first show for the linear {\lambda}-calculus that interpretation in a model of computation by interaction can be described as a call-by-name CPS-translation followed by a defunctionalization procedure that takes into account control-flow information. We then establish a relation between these two compilation methods for the simply-typed {\lambda}-calculus and end by considering recursion.

Combinatorial structure of type dependency

We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of generalised algebraic theories in the sense of Cartmell, and exhibiting it as the category of algebras for a monad on a presheaf category. The objects of the presheaf category encode the basic judgements of a dependent sequent calculus, while the action of the monad encodes the deduction rules; so by giving an explicit description of the monad, we obtain an explicit account of the combinatorics of type dependency. We find that this combinatorics is controlled by a particular kind of decorated ordered tree, familiar from computer science and from innocent game semantics. Furthermore, we find that the monad at issue is of a particularly well-behaved kind: it is local right adjoint in the sense of Street–Weber. In future work, we will use this fact to describe nerves for dependent type theories, and to study the coherence problem for dependent type theory using the tools of two-dimensional monad theory.

Some Turing-Complete Extensions of First-Order Logic

We introduce a natural Turing-complete extension of first-order logic FO. The extension adds two novel features to FO. The first one of these is the capacity to add new points to models and new tuples to relations. The second one is the possibility of recursive looping when a formula is evaluated using a semantic game. We first define a game-theoretic semantics for the logic and then prove that the expressive power of the logic corresponds in a canonical way to the recognition capacity of Turing machines. Finally, we show how to incorporate generalized quantifiers into the logic and argue for a highly natural connection between oracles and generalized quantifiers.

Krivine nets

We define a new approach to compilation to distributed architectures based on networks of abstract machines. Using it we can implement a generalised and fully transparent form of Remote Procedure Call that supports calling higher-order functions across node boundaries, without sending actual code. Our starting point is the classic Krivine machine, which implements reduction for untyped call-by-name PCF. We successively add the features that we need for distributed execution and show the correctness of each addition. Then we construct a two-level operational semantics, where the high level is a network of communicating machines, and the low level is given by local machine transitions. Using these networks, we arrive at our final system, the Krivine Net . We show that Krivine Nets give a correct distributed implementation of the Krivine machine, which preserves both termination and non-termination properties. All the technical results have been formalised and proved correct in Agda. We also implement a prototype compiler which we compare with previous distributing compilers based on Girard's Geometry of Interaction and on Game Semantics.

Denotational Semantics with Nominal Scott Domains

When defining computations over syntax as data, one often runs into tedious issues concerning α -equivalence and semantically correct manipulations of binding constructs. Here we study a semantic framework in which these issues can be dealt with automatically by the programming language. We take the user-friendly “nominal” approach in which bound objects are named. In particular, we develop a version of Scott domains within nominal sets and define two programming languages whose denotational semantics are based on those domains. The first language, λν -PCF, is an extension of Plotkin’s PCF with names that can be swapped, tested for equality and locally scoped; although simple, it already exposes most of the semantic subtleties of our approach. The second language, PNA, extends the first with name abstraction and concretion so that it can be used for metaprogramming over syntax with binders. For both languages, we prove a full abstraction result for nominal Scott domains analogous to Plotkin’s classic result about PCF and conventional Scott domains: two program phrases have the same observable operational behaviour in all contexts if and only if they denote equal elements of the nominal Scott domain model. This is the first full abstraction result we know of for languages combining higher-order functions with some form of locally scoped names which uses a domain theory based on ordinary extensional functions, rather than using the more intensional approach of game semantics. To obtain full abstraction, we need to add two functionals, one for existential quantification over names and one for “definite description” over names. Only adding one of them is not enough, as we give counter-examples to full abstraction in both cases.

A graphical foundation for interleaving in game semantics

In 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notion they called a ⊸-schedule, and the similar notion of ⊗-schedule, both structures describing interleavings of plays in games. Their definition was combinatorial in nature, but researchers often draw pictures when describing schedules in practice. Moreover, several proofs of key properties, such as that the composition of ⊸-schedules is associative, involve cumbersome combinatorial detail, whereas in terms of pictures the proof is straightforward, reflecting the geometry of the plane. Here, we give a geometric formulation of ⊸-schedules and ⊗-schedules, prove that they are isomorphic to Harmer et al.'s definitions, and illustrate their value by giving such geometric proofs. Harmer et al.'s notions may be combined to describe plays in multi-component games, and researchers have similarly developed intuitive graphical representations of plays in these games. We give a characterisation of these diagrams and explicitly describe how they relate to the underlying schedules, finally using this relation to provide new, intuitive proofs of key categorical properties.

Knowledge Spaces and the Completeness of Learning Strategies

We propose a theory of learning aimed to formalize some ideas underlying Coquand's game semantics and Krivine's realizability of classical logic. We introduce a notion of knowledge state together with a new topology, capturing finite positive and negative information that guides a learning strategy. We use a leading example to illustrate how non-constructive proofs lead to continuous and effective learning strategies over knowledge spaces, and prove that our learning semantics is sound and complete w.r.t. classical truth, as it is the case for Coquand's and Krivine's approaches.