Modal operator

About: Modal operator is a research topic. Over the lifetime, 1151 publications have been published within this topic receiving 22865 citations. The topic is also known as: modal connective.

A Note on the Complexity of the Satisfiability Problem for Graded Modal Logics

Abstract: Graded modal logic is the formal language obtained from ordinary (propositional) modal logic by endowing its modal operators with cardinality constraints. Under the familiar possible-worlds semantics, these augmented modal operators receive interpretations such as "It is true at no fewer than 15 accessible worlds that...", or "It is true at no more than 2 accessible worlds that...". We investigate the complexity of satisfiability for this language over some familiar classes of frames. This problem is more challenging than its ordinary modal logic counterpart--especially in the case of transitive frames, where graded modal logic lacks the tree-model property. We obtain tight complexity bounds for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property.

A temporalised belief logic for specifying the dynamics of trust for multi-agent systems

Abstract: Temporalisation is a methodology for combining logics whereby a given logic system can be enriched with temporal features to create a new logic system. TML (Typed Modal Logic) extends classical first-order logic with typed variables and multiple belief modal operators; it can be applied to the description of, and reasoning about, trust for multi-agent systems. Without the introduction of a temporal dimension, this logic may not be able to express the dynamics of trust. In this paper, adopting the temporalisation method, we combine TML with a temporal logic to obtain a new logic, so that the users can specify the dynamics of trust and model evolving theories of trust for multi-agent systems.

Algebraic Notions of Termination

Abstract: Five algebraic notions of termination are formalised, analysed and compared: wellfoundedness or Noetherity, L\"ob's formula, absence of infinite iteration, absence of divergence and normalisation. The study is based on modal semirings, which are additively idempotent semirings with forward and backward modal operators. To model infinite behaviours, idempotent semirings are extended to divergence semirings, divergence Kleene algebras and omega algebras. The resulting notions and techniques are used in calculational proofs of classical theorems of rewriting theory. These applications show that modal semirings are powerful tools for reasoning algebraically about the finite and infinite dynamics of programs and transition systems.

Categories: How I Learned to Stop Worrying and Love Two Sorts

Abstract: RS-frames were introduced by Gehrke as relational semantics for substructural logics. They are two-sorted structures, based on RS-polarities with additional relations used to interpret modalities. We propose an intuitive, epistemic interpretation of RS-frames for modal logic, in terms of categorization systems and agents' subjective interpretations of these systems. Categorization systems are a key to any decision-making process and are widely studied in the social and management sciences. A set of objects together with a set of properties and an incidence relation connecting objects with their properties forms a polarity which can be `pruned' into an RS-polarity. Potential categories emerge as the Galois-stable sets of this polarity, just like the concepts of Formal Concept Analysis. An agent's beliefs about objects and their properties (which might be partial) is modelled by a relation which gives rise to a normal modal operator expressing the agent's beliefs about category membership. Fixed-points of the iterations of the belief modalities of all agents are used to model categories constructed through social interaction.