Temporal logic

Temporal logic

In logic, the term temporal logic is used to describe 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 particular modal logic-based system of temporal logic introduced by Arthur Prior in the 1960s. Subsequently it has been developed further by computer scientists, notably Amir Pnueli, and logicians.

Temporal logic was first studied in depth by Aristotle, whose writings are filled with a crude form of first-order temporal modal binary logic. Any logic which uses the existential quantifier or the universal quantifier is said to be a predicate logic. Any logic which views time as a sequence of states is a temporal logic, and any logic which uses only two truth values is a binary logic.

Consider the statement: "I am hungry." Though its meaning is constant in time, the truth value of the statement can vary in time. Sometimes the statement is true, and sometimes the statement is false, but the statement is never true and false simultaneously. In a temporal logic, statements can have a truth value which can vary in time. Contrast this with an atemporal logic, which can only handle statements whose truth value is constant in time.

In a temporal logic we can then express statements like "I am "always" hungry", "I will "eventually" be hungry", or "I will be hungry "until" I eat something".

Temporal logic has found an important application in formal verification, where it is used to state requirements of hardware or software systems. For instance, one may wish to say that "whenever" a request is made, access to a resource is "eventually" granted, but it is "never" granted to two requestors simultaneously." Such a statement can conveniently be expressed in a temporal logic.

Temporal logic always has the ability to reason about a time line. So called linear time logics are restricted to this type of reasoning. Branching logics, however, can reason about multiple time lines. This presupposes an environment that may act unpredictably.To continue the example, in a branching logic we may state that "there is a possibility that I will stay hungry forever." We may also state that "there is a possibility that eventually I am no longer hungry." If we do not know whether or not I will ever get fed, these statements are both true.

Two early contenders in formal verifications were Linear Temporal Logic (a linear time logic by Amir Pnueli and Zohar Manna) and Computation Tree Logic, a branching time logic by Edmund Clarke and E. Allen Emerson. The fact that the second logic is more efficient than the first does not reflect on branching and linear logics in general, as has sometimes been argued. Rather, Emerson and Lei show that any linear logic can be extended to a branching logic that can be decided with the same complexity.

Temporal operators

Temporal logic has two kinds of operators: logical operators and modal operators [http://plato.stanford.edu/entries/logic-temporal/] . Logical operators are usual truth-functional operators ($eg,or,and, ightarrow$). The modal operators used in Linear Temporal Logic and Computation Tree Logic are defined as follows.

Alternate symbols:
* operator R is sometimes denoted by V
* The operator W is the "weak until" operator: $f W g$ is equivalent to $f U g or G f$

Unary opearators are well-formed formulas whenever B($phi$) is well-formed. Binary operators are well-formed formulas whenever B($phi$) and C($phi$) are well-formed.

In some logics, some operators cannot be expressed. For example, N operator cannot be expressed in Temporal Logic of Actions.

Temporal logics

Temporal logics include
* Interval temporal logic (ITL)
* μ calculus. which includes as a subset
** Hennessy-Milner logic (HML)
** CTL*, which includes as a subset
*** Computational tree logic (CTL)
*** Linear temporal logic (LTL)

ee also

* HPO formalism
* Duration calculus (DC)
* Hybrid logic
* Temporal logic in finite-state verification
* Temporal logic of actions (TLA)
* Important publications in formal verification (including the use of temporal logic in formal verification)

References

*Venema, Yde, 2001, "Temporal Logic," in Goble, Lou, ed., "The Blackwell Guide to Philosophical Logic". Blackwell.

*E. A. Emerson and C. Lei, modalities for model checking: branching time logic strikes back, in "Science of Computer Programming" 8, p 275-306, 1987.

*E.A. Emerson, Temporal and modal logic, "Handbook of Theoretical Computer Science", Chapter 16, the MIT Press, 1990

* Stanford Encyclopedia of Philosophy: " [http://plato.stanford.edu/entries/logic-temporal/ Temporal Logic] " -- by Anthony Galton.
* [http://staff.science.uva.nl/~yde/papers/TempLog.pdf Temporal Logic] by Yde Venema, formal description of syntax and semantics, questions of axiomatization. Treating also Kamp's dyadic temporal operators (since, until)
* [http://www.doc.ic.ac.uk/~imh/papers/sa.ps.gz Notes on games in temporal logic] by Ian Hodkinson, including a formal description of first-order temporal logic
* [http://www.inrialpes.fr/vasy/cadp CADP - provides generic model checkers for various temporal logic]

Wikimedia Foundation. 2010.

Look at other dictionaries:

• temporal logic — noun A form of symbolic logic used to reason about properties of statements related to order and duration …   Wiktionary

• Temporal logic of actions — (TLA) is a logic developed by Leslie Lamport, which combines temporal logic with a logic of actions.It is used to describe behaviours of concurrent systems.Statements in temporal logic of the form [A] t, where A is an action and t contains a… …   Wikipedia

• Temporal logic in finite-state verification — In finite state verification, model checkers examine finite state machines representing concurrent software systems looking for errors in design. Errors are defined as violations of requirements expressed as properties of the system. In the event …   Wikipedia

• Temporal Logic of Actions — Dieser Artikel oder Abschnitt bedarf einer Überarbeitung. Näheres ist auf der Diskussionsseite angegeben. Hilf mit, ihn zu verbessern, und entferne anschließend diese Markierung. Die Temporale Logik der Aktionen (TLA) ist eine Weiterentwicklung… …   Deutsch Wikipedia

• Interval temporal logic — (also interval logic) is a temporal logic for representing both propositional and first order logical reasoning about periods of time that is capable of handling both sequential and parallel composition. Instead of dealing with infinite sequences …   Wikipedia

• Linear temporal logic — (LTL) is a modal temporal logic with modalities referring to time. In LTL, one can encode formulae about the future of paths such as that a condition will eventually be true, that a condition will be true until another fact becomes true,… …   Wikipedia

• Linear temporal logic — Lineare temporale Logik (LTL oder Linear temporal logic) ist ein Modell temporaler Logik mit zeitlichen Modalitäten. In LTL, können Formeln über die Zukunft von Pfaden aufgestellt werden, wie dass eine Bedingung irgendwann wahr wird, eine… …   Deutsch Wikipedia

• Linear Time Temporal Logic — Die Computation Tree Logic (kurz CTL) ist eine Temporale Logik, die speziell zur Spezifikation und Verifikation von Computersystemen dient. Meist wird sie auch mit CTL* bezeichnet. CTL bezeichnet dann eine spezielle Teilmenge der CTL* Formeln.… …   Deutsch Wikipedia

• Temporal — can refer to: * of or relating to time ** Temporal database, a database recording aspects of time varying values ** The Temporal power of the Popes of the Roman Catholic Church ** a Lord Temporal, secular member of the House of Lords ** Temporal… …   Wikipedia

• Logic in computer science — describes topics where logic is applied to computer science and artificial intelligence. These include:*Investigations into logic that are guided by applications in computer science. For example: Combinatory logic and Abstract interpretation;… …   Wikipedia