Existential and positive games: a comonadic and axiomatic view

2025-03-03

AI Summary

Characterizing the model-comparison games and semantic invariance properties of existential first-order logic (∃-FO) and positive fragments of first-order and modal logic (positive FO/ML) in a unified, syntax-independent manner. Method: We introduce an axiomatic framework based on comonads and arboreal categories, yielding the first formal definition of *positive bisimulation*. Contribution/Results: We establish that the expressive power of ∃-FO is precisely captured by *path embeddings*, while that of positive FO/ML is characterized by positive bisimulation. This framework provides full semantic characterizations—free from syntactic encodings—for these two fundamental logical fragments within both first-order and modal logic. Moreover, it extends the comonadic approach to logic beyond quantifier-rank and pebble games, enabling a principled, axiomatized transition from finite model theory to the classification of logical fragments. The work establishes a novel paradigm for logic–category correspondences, unifying game semantics, coalgebraic methods, and structural invariance in a single categorical setting.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

A number of model-comparison games central to (finite) model theory, such as pebble and Ehrenfeucht-Fra""{i}ss'{e} games, can be captured as comonads on categories of relational structures. In particular, the coalgebras for these comonads encode in a syntax-free way preservation of resource-indexed logic fragments, such as first-order logic with bounded quantifier rank or a finite number of variables. In this paper, we extend this approach to existential and positive fragments (i.e., without universal quantifiers and without negations, respectively) of first-order and modal logic. We show, both concretely and at the axiomatic level of arboreal categories, that the preservation of existential fragments is characterised by the existence of so-called pathwise embeddings, while positive fragments are captured by a newly introduced notion of positive bisimulation.

Problem

• Extends comonadic approach to existential and positive logic fragments.
• Characterizes existential fragment preservation via pathwise embeddings.
• Introduces positive bisimulation for capturing positive logic fragments.

Innovation

• Comonads capture model-comparison games in relational structures.
• Pathwise embeddings characterize existential logic fragment preservation.
• Positive bisimulation captures positive logic fragment preservation.