[Paper Review] Dynamic Games and Strategies.

Norihiro Yamada, Samson Abramsky|arXiv (Cornell University)|Jan 16, 2016

Logic, programming, and type systems13 references4 citations

TL;DR

This paper introduces a game semantics framework for a functional programming language that captures intensional computation—distinguishing programs with identical values but different algorithms—using a syntax-independent, mathematically rigorous formulation grounded in cartesian closed bicategories (CCBs). The key contribution is a precise correspondence between the hiding operation on strategies and small-step operational semantics, providing a categorical foundation for operational dynamics in computation.

ABSTRACT

The present paper aims to provide a mathematical and syntax-independent formulation of dynamics and intensionality of computation; our approach is based on mathematical structures developed in game semantics. Specifically, we give a new game semantics of a prototypical programming language that distinguishes terms with the same value yet different algorithms, capturing intensionality of computation, equipped with the hiding operation on strategies that exactly corresponds to the (small-step) operational semantics of the programming language, modeling dynamics of computation. Categorically, our games and strategies give rise to a certain kind of a cartesian closed bicategory (CCB), and our game semantics forms an instance of a bicategorical refinement of the standard interpretation of functional languages in cartesian closed categories (CCCs) by CCBs. This work is intended to be a mathematical foundation of operational aspects of computation; our approach should be applicable to a wide range of logics and computations.

Motivation & Objective

• To develop a syntax-independent, mathematical formulation of computation dynamics and intensionality in programming languages.
• To model the operational semantics of a functional language through a hiding operation on strategies in game semantics.
• To establish a categorical foundation for operational aspects of computation using cartesian closed bicategories (CCBs).
• To refine the standard functional language interpretation in cartesian closed categories (CCCs) by extending it to CCBs for greater expressiveness in dynamics.
• To provide a general mathematical framework applicable to a wide range of logics and computational systems.

Proposed method

• Formalizing computation dynamics using game semantics with strategies representing programs.
• Introducing a hiding operation on strategies that mirrors small-step operational semantics.
• Constructing a cartesian closed bicategory (CCB) from games and strategies to model intensional computation.
• Using categorical duality and structure to ensure the semantics are syntax-independent and mathematically robust.
• Establishing a bicategorical refinement of the standard CCC-based interpretation of functional languages.
• Ensuring that the semantics capture algorithmic differences even when values are identical, thus modeling intensionality.

Experimental results

Research questions

• RQ1 How can computation dynamics be formally captured in a syntax-independent manner using game semantics?
• RQ2 What categorical structure best supports both intensionality and operational semantics in functional computation?
• RQ3 How does the hiding operation on strategies correspond to small-step operational semantics?
• RQ4 In what way does the proposed framework refine the standard CCC-based semantics of functional languages?
• RQ5 Can this framework be generalized to a wide range of logics and computational systems?

Key findings

• The proposed game semantics successfully distinguishes programs with identical values but different algorithms, capturing intensionality.
• The hiding operation on strategies is mathematically equivalent to the small-step operational semantics of the language.
• The framework forms a cartesian closed bicategory (CCB), providing a higher-order categorical structure for computational dynamics.
• The construction offers a bicategorical refinement of the standard CCC-based interpretation, enhancing expressiveness for operational behavior.
• The approach is general enough to be applicable to a broad class of logics and computational systems.
• The semantics are syntax-independent, ensuring mathematical robustness and abstraction from syntactic details.