Overview
Research establishes the equiexpressiveness and equivalence between first-order game logic (GL) and the first-order modal mu-calculus ($L_\mu$). This unification extends to their differential counterparts: differential game logic (dGL) and differential modal mu-calculus. The findings include the existence of semantics-preserving and provability-preserving translations between these logic systems, applicable to both the propositional and first-order cases, distinguishing the first-order scenario from its propositional predecessor. A significant consequence is the complete axiomatization of ordinary differential equations (ODEs) and the decidability of infinitesimally robust properties of ODEs via proof search. Furthermore, rational gameplay is shown to simplify multi-player games into single-player games, yielding a strong arithmetical completeness theorem for dGL when applied to rational-time ODEs.
Research Context
The study addresses the relationship between game theory and fixpoint theory within the domain of logic. Specifically, it investigates game logic (GL) and the modal mu-calculus ($L_\mu$). Historically, the propositional forms of these logics exhibited differing expressive powers, with game logic being strictly less expressive than the modal mu-calculus when sabotage games were not included. This research revisits this relationship in a first-order setting and extends it to logics incorporating continuous dynamics, represented by differential equations.
Approach
The core of the research involves demonstrating mutual expressiveness and equivalence through a series of formal proofs. These proofs establish the existence of bidirectional translations between the logical systems. For the first-order case, semantics-preserving translations from GL to $L_\mu$ and from $L_\mu$ to GL are presented. Crucially, these translations are also provability-preserving, indicating that logical inferences made in one system are preserved in the other. The study also proves the equivalence of roundtrip translations (there-and-back-again) within both calculi, reinforcing the robustness of the established equivalences.
The methodology is then extended to systems that incorporate differential equations:
- Differential game logic (dGL)
- Differential modal mu-calculus
The approach for these extensions similarly involves proving their equiexpressiveness and equivalence through semantics-preserving and provability-preserving translations.
Findings
- First-Order Equivalence: First-order game logic (GL) and the first-order modal mu-calculus ($L_\mu$) are proved to be equiexpressive and equivalent. This encompasses both their expressive and deductive power, enabling full alignment between the two systems.
- Translation Properties: Semantics-preserving translations exist from GL to $L_\mu$ and vice versa. These translations are also provability-preserving.
- Roundtrip Equivalence: The equivalence of there-and-back-again roundtrip translations is provable in both GL and $L_\mu$.
- Distinction from Propositional Case: This first-order result contrasts with the propositional case, where game logic without sabotage games is strictly less expressive than the modal mu-calculus.
- Differential Extensions Equivalence: The extensions of these logics with differential equations, namely differential game logic (dGL) and differential modal mu-calculus, are also proved equiexpressive and equivalent.
- ODE Axiomatization: The continuous dynamics of ordinary differential equations (ODEs) can be defined using fixpoints or games. This leads to the complete axiomatization of ODEs.
- Decidability of Robust Properties: As a consequence of complete ODE axiomatization, infinitesimally robust properties of ODEs can be decided through proof search.
- Rational Gameplay Collapse: Rational gameplay provably collapses complex games into single-player games.
- Arithmetical Completeness: This collapse facilitates a strong arithmetical completeness theorem for dGL when applied to rational-time ODEs.
Why This Matters
The established equivalences between game logic and the modal mu-calculus, particularly in their first-order and differential forms, provide a unified framework for reasoning about strategic interactions and continuous system dynamics. The capability to completely axiomatize ODEs and decide their robust properties through proof search offers a formal method for verifying continuous systems. The simplification of complex games via rational gameplay to derive arithmetical completeness has implications for the analysis of game-theoretic systems with timed dynamics.