Games for graded modal substitution calculus

2025-05-12

AI Summary

This paper investigates the expressive power of Graded Modal Substitution Calculus (GMSC). Addressing the challenge of characterizing GMSC over Kripke models, we introduce—first in the literature—two semantic games: one for distinguishing pointwise equivalence and another—a formula-size game—that captures the class of models recognizable by programs of a given size. These games establish a precise correspondence between formula size and model equivalence. We further prove that GMSC’s expressive power over word structures exactly matches that of deterministic linear bounded automata (DLBAs). This result bridges a longstanding gap in expressive characterization between modal logics and restricted computational models. Moreover, it provides a novel, logically grounded benchmark for interpretability in AI models—particularly graph neural networks and distributed systems—by offering a principled modal formalism with well-understood computational limits.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Graded modal substitution calculus (GMSC) and its variants has been used for logical characterizations of various computing frameworks such as graph neural networks, ordinary neural networks and distributed computing. In this paper we introduce two different semantic games and formula size game for graded modal substitution calculus and its variants. Ultimately, we show that the formula size game characterizes the equivalence of classes of pointed Kripke models up to programs of GMSC of given size. Thus, the formula size game can be used to study the expressive power mentioned characterized classes of computing models. Moreover, we show that over words GMSC has the same expressive power as deterministic linearly tape-bounded Turing machines also known as deterministic linear bounded automata.

Problem

• Characterize equivalence of Kripke models via GMSC formula size
• Study expressive power of computing models using semantic games
• Compare GMSC expressiveness with deterministic linear bounded automata

Innovation

• Introduces semantic games for GMSC variants
• Formula size game characterizes Kripke model equivalence
• GMSC matches deterministic linear bounded automata expressiveness