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Fair computational tree logic is conventional computational tree logic studied with explicit fairness constraints.
This declares conditions such as all processes execute infinitely often. If you consider the processes to be Pi, then the condition becomes:
Here, if a process is requesting a resource infinitely often (R), it should be allowed to get the resource (C) infinitely often:
Consider a Kripke model with set of states F. A path is considered a fair path, if and only if the path includes all members of F infinitely often.
Fair CTL model checking restricts the checks to only fair paths. There are two kinds of fair quantifiers:
A fair state is one from which at least one fair path originates. This translates to Mf, s |= EGtrue.
A strongly connected component (SCC) of a directed graph is a maximal strongly connected subgraph—all the nodes are reachable from each other. A fair SCC is one that has an edge into at least one node for each of the fair conditions.
To check for fair EG for any formula,
The fix point characterization of Exist Globally is given by: [EGφ] = νZ .([φ] ∩ [EXZ ]), which is basically the limit applied according to Kleene's theorem. To fair paths, it becomes [Ef Gφ] = νZ .([φ] ∩Fi ∈FT [EX[E(Z U(Z ∧ Fi ))]), which means the formula holds in the current state and the next states and the next to next states until it meets all the members of the fair conditions. This means that, the condition is equivalent to a sort of accepting point where the accepting condition is the entire set of Fair conditions.
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced the diagrams in 1948. The interaction of subatomic particles can be complex and difficult to understand; Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams are applied primarily to quantum field theory, they can also be used in other areas of physics, such as solid-state theory. Frank Wilczek wrote that the calculations that won him the 2004 Nobel Prize in Physics "would have been literally unthinkable without Feynman diagrams, as would [Wilczek's] calculations that established a route to production and observation of the Higgs particle."
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space H. It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung".
In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. It is sometimes also used to refer to tense logic, a modal logic-based system of temporal logic introduced by Arthur Prior in the late 1950s, with important contributions by Hans Kamp. It has been further developed by computer scientists, notably Amir Pnueli, and logicians.
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.
In logic, linear temporal logic or linear-time temporal logic (LTL) is a modal temporal logic with modalities referring to time. In LTL, one can encode formulae about the future of paths, e.g., a condition will eventually be true, a condition will be true until another fact becomes true, etc. It is a fragment of the more complex CTL*, which additionally allows branching time and quantifiers. LTL is sometimes called propositional temporal logic, abbreviated PTL. In terms of expressive power, linear temporal logic (LTL) is a fragment of first-order logic.
Computation tree logic (CTL) is a branching-time logic, meaning that its model of time is a tree-like structure in which the future is not determined; there are different paths in the future, any one of which might be an actual path that is realized. It is used in formal verification of software or hardware artifacts, typically by software applications known as model checkers, which determine if a given artifact possesses safety or liveness properties. For example, CTL can specify that when some initial condition is satisfied, then all possible executions of a program avoid some undesirable condition. In this example, the safety property could be verified by a model checker that explores all possible transitions out of program states satisfying the initial condition and ensures that all such executions satisfy the property. Computation tree logic belongs to a class of temporal logics that includes linear temporal logic (LTL). Although there are properties expressible only in CTL and properties expressible only in LTL, all properties expressible in either logic can also be expressed in CTL*.
In logic, finite model theory, and computability theory, Trakhtenbrot's theorem states that the problem of validity in first-order logic on the class of all finite models is undecidable. In fact, the class of valid sentences over finite models is not recursively enumerable.
In theoretical computer science, the modal μ-calculus is an extension of propositional modal logic by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic.
CTL* is a superset of computational tree logic (CTL) and linear temporal logic (LTL). It freely combines path quantifiers and temporal operators. Like CTL, CTL* is a branching-time logic. The formal semantics of CTL* formulae are defined with respect to a given Kripke structure.
Probabilistic Computation Tree Logic (PCTL) is an extension of computation tree logic (CTL) that allows for probabilistic quantification of described properties. It has been defined in the paper by Hansson and Jonsson.
In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The particle follows the path of a semicircle from to where it cannot escape, because the potential from to is infinite. Instead there is total reflection, meaning the particle bounces back and forth between to . The Schrödinger equation for a free particle which is restricted to a semicircle is
In statistical mechanics of continuous systems, a potential for a many-body system is called H-stable if the potential energy per particle is bounded below by a constant that is independent of the total number of particles. In many circumstances, if a potential is not H-stable, it is not possible to define a grand canonical partition function in finite volume, because of catastrophic configurations with infinite particles located in a finite space.
Metric temporal logic (MTL) is a special case of temporal logic. It is an extension of temporal logic in which temporal operators are replaced by time-constrained versions like until, next, since and previous operators. It is a linear-time logic that assumes both the interleaving and fictitious-clock abstractions. It is defined over a point-based weakly-monotonic integer-time semantics.
In model checking, the Metric Interval Temporal Logic (MITL) is a fragment of Metric Temporal Logic (MTL). This fragment is often preferred to MTL because some problems that are undecidable for MTL become decidable for MITL.
In model checking, a subfield of computer science, a clock is a mathematical object used to model time. More precisely, a clock measures how much time passed since a particular event occurs, in this sense, a clock is more precisely an abstraction of a stopwatch. In a model of some particular program, the value of the clock may either be the time since the program was started, or the time since a particular event occurred in the program. Those clocks are used in the definition of timed automaton, signal automaton, timed propositional temporal logic and clock temporal logic. They are also used in programs such as UPPAAL which implement timed automata.
In model checking, a field of computer science, Timed Propositional Temporal Logic (TPTL) is an extension of Linear Temporal Logic (LTL) in which variables are introduced to measure times between two events. For example, while LTL allows to state that each event p is eventually followed by an event q, TPTL furthermore allows to give a time limit for q to occur.
In automata theory, an alternating timed automaton (ATA) is a mix of both timed automaton and alternating finite automaton. That is, it is a sort of automata which can measure time and in which there exists universal and existential transition.
In model checking, a branch of computer science, linear time properties are used to describe requirements of a model of a computer system. Example properties include "the vending machine does not dispense a drink until money has been entered" or "the computer program eventually terminates". Fairness properties can be used to rule out unrealistic paths of a model. For instance, in a model of two traffic lights, the liveness property "both traffic lights are green infinitely often" may only be true under the unconditional fairness constraint "each traffic light changes colour infinitely often".
