[Paper Review] Resource modalities in game semantics

Paul-André Melliès, Nicolas Tabareau|May 3, 2007

Logic, programming, and type systems26 references4 citations

TL;DR

This paper repositions game semantics as more foundational than linear logic, introducing tensorial logic to unify resource modalities—linear, affine, and exponential—within a categorical framework. By extending bracketing to multi-bracketing and using a tensorial negation, it constructs a fully abstract model of PCF with resource-aware strategies, enabling a compositional semantics for computational effects and resources.

ABSTRACT

The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of a misleading conception: the belief that linear logic is more primitive than game semantics. We advocate instead the contrary: that game semantics is conceptually more primitive than linear logic. Starting from this revised point of view, we design a categorical model of resources in game semantics, and construct an arena game model where the usual notion of bracketing is extended to multi- bracketing in order to capture various resource policies: linear, affine and exponential.

Motivation & Objective

• To resolve the foundational schism between game semantics and linear logic by repositioning game semantics as more primitive.
• To introduce tensorial logic as a generalization of linear logic, replacing involutive negation with a tensorial negation to better model resource policies.
• To extend the notion of well-bracketing in arena games to multi-bracketing, capturing linear, affine, and exponential resource policies.
• To construct a categorical model of resources in game semantics that supports affine and exponential modalities via adjunctions and free coproduct completion.
• To provide a fully abstract, compositional model of a linear PCF with resource modalities using the category $Fam( ext{B}^{ullet})$.

Proposed method

• Introduce tensorial logic by replacing the involutive negation of linear logic with a tensorial negation, defined via a linear continuation monad.
• Define pointed games with a distinguished Proponent move that initiates queries, enabling the construction of duals and negations.
• Extend bracketing to multi-bracketing by allowing multiple levels of response nesting, modeling resource consumption policies.
• Construct the affine modality $!_\bullet^w A$ by removing all queries initiated by the first move in a pointed game $A$.
• Define the exponential modality $!_\bullet^e A$ as the composition $!(!_\bullet^w A)$, combining the affine and linear modalities.
• Use the free coproduct completion $Fam(\mathcal{B}^\bullet)$ to enrich the category $\mathcal{B}^\bullet$ with coproducts, preserving monoidal and modal structures.

Experimental results

Research questions

• RQ1 How can resource modalities (linear, affine, exponential) be systematically modeled within game semantics?
• RQ2 What categorical structure underlies a fully abstract game model that supports resource-aware computation?
• RQ3 How can the well-bracketing condition in arena games be generalized to capture richer resource policies?
• RQ4 In what sense is game semantics more primitive than linear logic, and how can this be formalized categorically?
• RQ5 Can a model of tensorial logic be constructed that supports fixpoints and resource modalities, enabling full PCF semantics?

Key findings

• Tensorial logic, with its non-involutive tensorial negation, provides a more foundational framework than linear logic for modeling resources in game semantics.
• The category $\mathcal{B}^\bullet$ of pointed games supports affine and exponential modalities via adjunctions, with $!_\bullet^w A$ and $!_\bullet^e A$ defined as $!(!_\bullet^w A)$.
• Multi-bracketing generalizes standard bracketing to model complex resource policies, such as those in affine and exponential contexts.
• The free coproduct completion $Fam(\mathcal{B}^\bullet)$ inherits symmetric monoidal and modal structures, forming a model of tensorial logic.
• The category $Fam(\mathcal{B}^\bullet)$ supports a fixpoint operator on singleton families, enabling interpretation of a linear PCF variant with resource modalities.
• The construction establishes a fully abstract, compositional game model for PCF with resource modalities, reconciling game semantics and linear logic through tensorial logic.