Characterizing Modal Logic Equivalence via Homomorphism Counting

Core Concepts

Homomorphism count indistinguishability with respect to appropriate classes of labeled transition systems captures equivalence with respect to various modal logics, including positive-existential modal logic, graded modal logic, and hybrid logic, as well as their extensions with backward or global modalities.

Abstract

The paper investigates the relationship between homomorphism counting and modal logic equivalence. The key results are:

Positive-existential modal equivalence is captured by homomorphism count indistinguishability over the Boolean semiring with respect to the class of trees. The extended languages with backward and global modalities are captured by the classes of connected, acyclic LTSs and forests, respectively, over the Boolean semiring.

Graded modal equivalence is captured by homomorphism count indistinguishability over the natural semiring with respect to the class of trees. The extended languages with backward and global modalities are captured by the classes of connected, acyclic LTSs and forests, respectively, over the natural semiring.

Equivalence with respect to hybrid logic is captured by homomorphism count indistinguishability over the natural semiring with respect to the class of point-generated LTSs. The extended language with backward modalities is captured by the class of connected LTSs.

Equivalence of LTSs with respect to positive modal logic and the basic modal language cannot be captured by restricting the left homomorphism count vector over any semiring to any class of LTSs.

The results show that homomorphism counting can be used to characterize modal logic equivalence, even for certain infinite structures. The negative result demonstrates the limitations of this approach for some modal logics.

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Key Insights Distilled From

Lovász Theorems for Modal Languages

by Jesse Comer at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15421.pdf

Deeper Inquiries

What are some potential applications of the characterizations of modal logic equivalence via homomorphism counting

The characterizations of modal logic equivalence via homomorphism counting have several potential applications in various fields.

Model Checking: Homomorphism counting can be used to efficiently verify properties of systems modeled as labeled transition systems. By characterizing modal logic equivalence through homomorphism counting, it provides a systematic way to check the equivalence of system behaviors.

Automated Reasoning: The techniques can be applied in automated reasoning systems to determine the equivalence of logical formulas or system specifications. This can aid in verifying the correctness of software systems or protocols.

Database Query Optimization: In the context of database systems, understanding modal logic equivalence through homomorphism counting can lead to optimizations in query processing and data retrieval. By leveraging these characterizations, database queries can be optimized for efficiency.

Artificial Intelligence: The characterizations can be utilized in AI systems for reasoning and decision-making processes. Understanding modal logic equivalence can enhance the capabilities of AI systems in various applications such as natural language processing, knowledge representation, and automated planning.

Graph Theory: The techniques can also find applications in graph theory, particularly in graph isomorphism and subgraph matching problems. By extending the approach to graph structures, it can facilitate the comparison and analysis of complex networks.

How might the techniques used in this paper be extended to capture equivalence for other modal or temporal logics

The techniques used in characterizing modal logic equivalence via homomorphism counting can be extended to capture equivalence for other modal or temporal logics by adapting the framework to accommodate the specific features and operators of those logics. Here are some ways to extend the approach:

Temporal Logics: For temporal logics like Linear Temporal Logic (LTL) or Computational Tree Logic (CTL), the homomorphism counting approach can be modified to consider temporal operators such as "next" and "until". By incorporating temporal semantics into the counting process, equivalence for temporal logics can be captured.

Higher-order Logics: Extending the approach to capture equivalence for higher-order logics involves considering quantifiers and functions beyond the first-order logic scope. By incorporating higher-order constructs into the homomorphism counting framework, equivalence for these logics can be effectively characterized.

Probabilistic Logics: For probabilistic logics like Probabilistic Temporal Logic (PTL) or Probabilistic Computation Tree Logic (PCTL), the approach can be adapted to include probabilistic operators and measures. By integrating probabilistic semantics into the counting methodology, equivalence for probabilistic logics can be addressed.

Modal Logics with Dynamic Operators: Modal logics with dynamic operators, such as Dynamic Epistemic Logic (DEL) or Dynamic Modal Logic (DML), can be explored using the homomorphism counting approach by considering the dynamic evolution of systems and agents. By incorporating dynamic aspects into the counting process, equivalence for dynamic modal logics can be captured.

Are there any connections between the homomorphism counting approach and other logical and algebraic characterizations of modal logic equivalence, such as bisimulation or modal algebras

There are indeed connections between the homomorphism counting approach and other logical and algebraic characterizations of modal logic equivalence, such as bisimulation or modal algebras. Here are some insights into these connections:

Bisimulation and Homomorphism Counting: Bisimulation is a fundamental concept in modal logic that captures behavioral equivalence between systems. The homomorphism counting approach can be seen as a generalization of bisimulation, where the counting of homomorphisms provides a quantitative measure of equivalence. By relating the two concepts, a deeper understanding of behavioral equivalence in modal logic can be achieved.

Modal Algebras and Homomorphism Counting: Modal algebras provide an algebraic framework for studying modal logics, focusing on the algebraic properties of modal operators. Homomorphism counting can complement modal algebras by offering a different perspective on equivalence, emphasizing structural mappings between models. By integrating homomorphism counting with modal algebras, a more comprehensive view of modal logic equivalence can be obtained.

Model Checking and Equivalence: Model checking techniques often rely on bisimulation or modal algebras to verify system properties. The homomorphism counting approach can serve as a complementary method for checking equivalence, providing a quantitative analysis of structural similarities between models. By combining these approaches, a more robust verification process can be established.

In summary, the connections between homomorphism counting, bisimulation, and modal algebras offer diverse perspectives on modal logic equivalence, enriching the understanding and analysis of logical systems.