Concept
Modal μ-calculus
Overview
In theoretical computer science, the modal μ-calculus (Lμ, Lμ, or propositional mu-calculus, 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.
The (propositional, modal) μ-calculus originates with Dana Scott and Jaco de Bakker, and was further developed by Dexter Kozen into the version most used nowadays. It is used to describe properties of labelled transition systems and for verifying these properties. Many temporal logics can be encoded in the μ-calculus, including CTL* and its widely used fragments—linear temporal logic and computational tree logic.
An algebraic view is to see it as an algebra of monotonic functions over a complete lattice, with operators consisting of functional composition plus the least and greatest fixed point operators; from this viewpoint, the modal μ-calculus is over the lattice of a power set algebra. The game semantics of μ-calculus is related to two-player games with perfect information, particularly infinite parity games.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
On the other hand an event-particle is defined so as to exhibit this character of being a route of approximation marked out by entities posited in sense-awareness. A definite event-particle is defined in reference to a definite punct in the following manner: Let the condition {sigma} mean the property of covering all the abstractive elements which are members of that punct; so that an abstractive set which satisfies…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
States what Modal μ-calculus is.
This is nothing else than the necessity of abstract logic, which is the presupposition involved in the very fact of interrelated existence as disclosed in each immediate occasion of experience. The key to the pattern means this fact:—that from a select set of those general conditions, exemplified in any one and the same occasion, a pattern involving an infinite variety of other such conditions, also exemplified in…
Alfred North Whitehead, Science and the modern world
States what Modal μ-calculus is.
If a relational theory of instants is constructed, in which an "instant" is defined as a group of events simultaneous with each other and not all simultaneous with any event outside the group, then if our resulting series of instants is to be compact, it must be possible, if _x_ wholly precedes _y_, to find an event _z_, simultaneous with part of _x_, which wholly precedes some event which wholly precedes _y_. Now…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Modal μ-calculus is.
Related ideas
- First-order logic
- Dana Scott
- Propositional logic
- Lambda calculus
- Model checking
- Temporal logic
- Theoretical computer science
- Modal logic
- Linear temporal logic
- Satisfiability
- CTL*
Place in the tradition
Part of Temporal logic.