Uniform Substitution for Differential Game Logic

TL;DR

This paper introduces a uniform substitution calculus for differential game logic, simplifying its axiomatization and implementation by replacing schema variables with concrete axioms and enabling modular theorem prover integration.

Contribution

It generalizes uniform substitutions to differential game logic, replacing axiom schemata with a finite set of axioms, and proves soundness and completeness of this approach.

Findings

Axiomatization of dGL using uniform substitutions is sound and complete.

Simplifies implementation of differential game logic in theorem provers.

Reduces reliance on schema variables and complex side conditions.

Abstract

This paper presents a uniform substitution calculus for differential game logic (dGL). Church's uniform substitutions substitute a term or formula for a function or predicate symbol everywhere. After generalizing them to differential game logic and allowing for the substitution of hybrid games for game symbols, uniform substitutions make it possible to only use axioms instead of axiom schemata, thereby substantially simplifying implementations. Instead of subtle schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many axiom schema instances to sound ones, the resulting axiomatization adopts only a finite number of ordinary dGL formulas as axioms, which uniform substitutions instantiate soundly. This paper proves soundness and completeness of uniform substitutions for the monotone modal logic dGL. The resulting axiomatization admits a straightforward modular implementation of dGL in theorem provers.