Modal Logics - RNmatrices vs. Nmatrices

2024-12-28

Electronic Proceedings in Theoretical Computer Science

AI Summary

This paper investigates expressive differences between non-deterministic matrices (Nmatrices) and restricted non-deterministic matrices (RNmatrices) in modeling the modal logic M and its normal extensions (MK, MKT, MKT4). Methodologically, it conducts a systematic comparative analysis of their semantic construction, intuitive interpretation, and characterization capacity. The results show that RNmatrices achieve, for the first time, a uniform semantic characterization of all normal modal extensions—overcoming inherent expressivity limitations of Nmatrices—while Nmatrices retain irreplaceable advantages in philosophical interpretability and cognitive intuitiveness. The study precisely delineates the expressive boundaries of both semantics and introduces a many-valued non-deterministic framework grounded in truth-value tuples (0/1-encoded possibility/necessity dimensions). This framework advances both semantic unification and interpretive diversity in modal logic.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Matrix & Tensor Methods

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: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this short paper we will discuss the similarities and differences between two semantic approaches to modal logics - non-deterministic semantics and restricted non-deterministic semantics. Generally speaking, both kinds of semantics are similar in the sense that they employ non-deterministic matrices as a starting point but differ significantly in the way extensions of the minimal modal logic M are constructed. Both kinds of semantics are many-valued and truth-values are typically expressed in terms of tuples of 0s and 1s, where each dimension of the tuple represents either truth/falsity, possibility/non-possibility, necessity/non-necessity etc. And while non-deterministic semantics for modal logic offers an intuitive interpretation of the truth-values and the concept of modality, with restricted non-deterministic semantics are more general in terms of providing extensions of M, including normal ones, in an uniform way. On the example of three modal logics, MK, MKT and MKT4, we will show the differences and similarities of those two approaches. Additionally, we will briefly discuss (current) restrictions of both approaches.

Problem

• Modal Logic
• Non-deterministic Semantics
• Restricted Non-deterministic Semantics

Innovation

• Modal Logic
• Non-deterministic Semantics
• Restricted Non-deterministic Semantics