Towards Functorial Language-Games

In categorical compositional semantics of natural language one studies functors from a category of grammatical derivations (such as a Lambek pregroup) to a semantic category (such as real vector spaces). We compositionally build game-theoretic semantics of sentences by taking the semantic category to be the category whose morphisms are open games. This requires some modifications to the grammar category to compensate for the failure of open games to form a compact closed category. We illustrate the theory using simple examples of Wittgenstein's language-games.Comment: In Proceedings CAPNS 2018, arXiv:1811.0270

Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus

The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambek's pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambek's previous type-logic, the Lambek calculus, which has been extensively used to formalise and reason about various linguistic phenomena. The apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings.Comment: 29 pages, pending publication in Annals of Pure and Applied Logi

The Structure of First-Order Causality

Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages

The game semantics of game theory

We use a reformulation of compositional game theory to reunite game theory with game semantics, by viewing an open game as the System and its choice of contexts as the Environment. Specifically, the system is jointly controlled by $n \geq 0$ noncooperative players, each independently optimising a real-valued payoff. The goal of the system is to play a Nash equilibrium, and the goal of the environment is to prevent it. The key to this is the realisation that lenses (from functional programming) form a dialectica category, which have an existing game-semantic interpretation. In the second half of this paper, we apply these ideas to build a compact closed category of `computable open games' by replacing the underlying dialectica category with a wave-style geometry of interaction category, specifically the Int-construction applied to the cartesian monoidal category of directed-complete partial orders

Inversion, Iteration, and the Art of Dual Wielding

The humble $\dagger$ ("dagger") is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two operations are usually considered separately from one another, the emergence of reversible notions of computation shows the need to consider how the two ought to interact. In the present paper, we wield both of these daggers at once and consider dagger categories enriched in domains. We develop a notion of a monotone dagger structure as a dagger structure that is well behaved with respect to the enrichment, and show that such a structure leads to pleasant inversion properties of the fixed points that arise as a result. Notably, such a structure guarantees the existence of fixed point adjoints, which we show are intimately related to the conjugates arising from a canonical involutive monoidal structure in the enrichment. Finally, we relate the results to applications in the design and semantics of reversible programming languages.Comment: Accepted for RC 201

Coalgebraic fixpoint logic:Expressivity and completeness results

This dissertation studies the expressivity and completeness of the coalgebraic μ-calculus. This logic is a coalgebraic generalization of the standard μ-calculus, which creates a uniform framework to study different modal fixpoint logics. Our main objective is to show that several important results, such as uniform interpolation, expressive completeness and axiomatic completeness of the standard μ-calculus can be generalized to the level of coalgebras. To achieve this goal we develop automata and game-theoretic tools to study properties of coalgebraic μ-calculus.In Chapter 3, we prove a uniform interpolation theorem for the coalgebraic μ-calculus. This theorem generalizes a result by D’Agostino and Hollenberg (2000) to a wider class of fixpoint logics including the monotone μ-calculus, which is the extension of monotone modal logic with fixpoint operators. In Chapter 4, we generalize the Janin-Walukiewicz theorem (1996), which states that the modal μ-calculus captures exactly the bisimulation invariant fragment of monadic second-order logic, to the level of coalgebras. We obtain a partly new proof of the Janin-Walukiewicz theorem, bisimulation invariance results for the bag functor (graded modal logic), and all exponential polynomial functors. We also derive a characterization theorem for the monotone modal μ-calculus, with respect to a natural monadic second-order language for monotone neighborhood models. In Chapter 5, we prove an axiomatic completeness result for the coalgebraic μ-calculus. Applying ideas from automata theory and coalgebra, we generalize Walukiewicz’ proof of completeness for the modal μ-calculus (2000) to the level of coalgebras. Our main contribution is to bring automata explicitly into the proof theory and distinguish two key aspects of the coalgebraic μ-calculus (and the standard μ-calculus): the one-step dynamic encoded in the semantics of the modal operators, and the combinatorics involved in dealing with nested fixpoints. We provide a generalization of Walukiewicz’ main technical result, which states that every formula of the modal μ-calculus provably implies the translation of a disjunctive automaton, to the level of coalgebras. From this the completeness theorem is almost immediate

Coalgebraic fixpoint logic:Expressivity and completeness results

This dissertation studies the expressivity and completeness of the coalgebraic μ-calculus. This logic is a coalgebraic generalization of the standard μ-calculus, which creates a uniform framework to study different modal fixpoint logics. Our main objective is to show that several important results, such as uniform interpolation, expressive completeness and axiomatic completeness of the standard μ-calculus can be generalized to the level of coalgebras. To achieve this goal we develop automata and game-theoretic tools to study properties of coalgebraic μ-calculus.In Chapter 3, we prove a uniform interpolation theorem for the coalgebraic μ-calculus. This theorem generalizes a result by D’Agostino and Hollenberg (2000) to a wider class of fixpoint logics including the monotone μ-calculus, which is the extension of monotone modal logic with fixpoint operators. In Chapter 4, we generalize the Janin-Walukiewicz theorem (1996), which states that the modal μ-calculus captures exactly the bisimulation invariant fragment of monadic second-order logic, to the level of coalgebras. We obtain a partly new proof of the Janin-Walukiewicz theorem, bisimulation invariance results for the bag functor (graded modal logic), and all exponential polynomial functors. We also derive a characterization theorem for the monotone modal μ-calculus, with respect to a natural monadic second-order language for monotone neighborhood models. In Chapter 5, we prove an axiomatic completeness result for the coalgebraic μ-calculus. Applying ideas from automata theory and coalgebra, we generalize Walukiewicz’ proof of completeness for the modal μ-calculus (2000) to the level of coalgebras. Our main contribution is to bring automata explicitly into the proof theory and distinguish two key aspects of the coalgebraic μ-calculus (and the standard μ-calculus): the one-step dynamic encoded in the semantics of the modal operators, and the combinatorics involved in dealing with nested fixpoints. We provide a generalization of Walukiewicz’ main technical result, which states that every formula of the modal μ-calculus provably implies the translation of a disjunctive automaton, to the level of coalgebras. From this the completeness theorem is almost immediate