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In theoretical computer science, the modal μ-calculus (Lμ, L μ, sometimes just μ-calculus, although this can have a more general meaning) is an extension of propositional modal logic (with many modalities) by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic.
It is a normal modal logic, and one of the oldest systems of modal logic of any kind. It is formed with propositional calculus formulas and tautologies, and inference apparatus with substitution and modus ponens, but extending the syntax with the modal operator necessarily and its dual possibly . [1] [2]
Modal predicate logic is one widely used variant which includes formulas such as (). In systems of modal logic where {\displaystyle \Box } and {\displaystyle \Diamond } are duals , ϕ {\displaystyle \Box \phi } can be taken as an abbreviation for ¬ ¬ ϕ {\displaystyle \neg \Diamond \neg \phi } , thus eliminating the need for a separate ...
Modal and temporal properties of processes. Springer. pp. 32–39. ISBN 978-0-387-98717-0. Sören Holmström. 1988. "Hennessy-Milner Logic with Recursion as a Specification Language, and a Refinement Calculus based on It". In Proceedings of the BCS-FACS Workshop on Specification and Verification of Concurrent Systems, Charles Rattray (Ed ...
A formula is logically valid (or simply valid) if it is true in every interpretation. [22] These formulas play a role similar to tautologies in propositional logic. A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.
The formula () denotes the global modality of the modal formula , which means that holds true in all worlds in a neighbourhood model. For logic E , the resolution calculus consists of LRES, GRES, G2L, LERES and GERES rules.
Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g. C. I. Lewis's S4 and S5, are normal (and hence are extensions of K). However a number of deontic and epistemic logics, for example, are non-normal, often because they give up the Kripke schema. Every normal modal logic is regular and hence classical.
It can be obtained by adding the modal version of Löb's theorem to the logic K (or K4). Namely, the axioms of GL are all tautologies of classical propositional logic plus all formulas of one of the following forms: Distribution axiom: (p → q) → ( p → q); Löb's axiom: ( p → p) → p. And the rules of inference are: