Temporal logic
In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of 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 is sometimes also used to refer to tense logic, a particular modal logicbased system of temporal logic introduced by Arthur Prior in the late 1950s, and important results were obtained by Hans Kamp. Subsequently it has been developed further by computer scientists, notably Amir Pnueli, and logicians.
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.
Contents
Motivation
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 discuss statements whose truth value is constant in time. This treatment of truth values over time differentiates temporal logic from computational verb logic.
Temporal logic always has the ability to reason about a time line. Socalled 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 sometimes.
History
Although Aristotle's logic is almost entirely concerned with the theory of the categorical syllogism, there are passages in his work that are now seen as anticipations of temporal logic, and may imply an early, partially developed form of firstorder temporal modal binary logic. Aristotle was particularly concerned with the problem of future contingents, where he could not accept that the principle of bivalence applies to statements about future events, i.e. that we can presently decide if a statement about a future event is true or false, such as "there will be a sea battle tomorrow".^{[1]}
There was little development for millennia, Charles Sanders Peirce noted in the 19th century:^{[2]}
“  Time has usually been considered by logicians to be what is called 'extralogical' matter. I have never shared this opinion. But I have thought that logic had not yet reached the state of development at which the introduction of temporal modifications of its forms would not result in great confusion; and I am much of that way of thinking yet.  ” 
Arthur Prior was concerned with the philosophical matters of free will and predestination. According to his wife, he first considered formalizing temporal logic in 1953. He gave lectures on the topic at the University of Oxford in 1955–6, and in 1957 published a book, Time and Modality, in which he introduced a propositional modal logic with two temporal connectives (modal operators), F and P, corresponding to "sometime in the future" and "sometime in the past". In this early work, Prior considered time to be linear. In 1958 however, he received a letter from Saul Kripke, who pointed out that this assumption is perhaps unwarranted. In a development that foreshadowed a similar one in computer science, Prior took this under advisement, and developed two theories of branching time, which he called "Ockhamist" and "Peircean".^{[2]}^{[clarification needed]} Between 1958 and 1965 Prior also corresponded with Charles Leonard Hamblin, and a number of early developments in the field can be traced to this correspondence, for example Hamblin implications. Prior published his most mature work on the topic, the book Past, Present, and Future in 1967. He died two years later.^{[3]}
The binary temporal operators Since and Until were introduced by Hans Kamp in his 1968 Ph.D. thesis,^{[4]} which also contains an important result relating temporal logic to firstorder logic—a result now known as Kamp's theorem.^{[5]}^{[2]}^{[6]}
Two early contenders in formal verifications were linear temporal logic (a linear time logic by Amir Pnueli) and computation tree logic, a branching time logic by Mordechai BenAri, Zohar Manna and Amir Pnueli. An almost equivalent formalism to CTL was suggested around the same time by E. M. Clarke and E. A. Emerson. The fact that the second logic can be decided more efficiently 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.
Prior's tense logic (TL)
The sentential tense logic introduced in Time and Modality has four (nontruthfunctional) modal operators (in addition to all usual truthfunctional operators in firstorder propositional logic.^{[7]}
 P: "It was the case that..." (P stands for "past")
 F: "It will be the case that..." (F stands for "future")
 G: "It always will be the case that..."
 H: "It always was the case that..."
From P and F one can define G and H, and vice versa:
Syntax and semantics
A minimal syntax for TL is specified with the following BNF grammar:
where a is some atomic formula.^{[8]}
Kripke models are used to evaluate the truth of sentences in TL. A pair (T, <) of a set T and a binary relation < on T (called "precedence") is called a frame. A model is given by triple (T, <, V) of a frame and a function V called a valuation that assigns to each pair (a, u) of an atomic formula and a time value some truth value. The notion "ϕ is true in a model U=(T, <, V) at time u" is abbreviated U⊨ϕ[u]. With this notation,^{[9]}
Statement  ... is true just when 

U⊨a[u]  V(a,u)=true 
U⊨¬ϕ[u]  not U⊨ϕ[u] 
U⊨(ϕ∧ψ)[u]  U⊨ϕ[u] and U⊨ψ[u] 
U⊨(ϕ∨ψ)[u]  U⊨ϕ[u] or U⊨ψ[u] 
U⊨(ϕ→ψ)[u]  U⊨ψ[u] if U⊨ϕ[u] 
U⊨Gϕ[u]  U⊨ϕ[v] for all v with u<v 
U⊨Hϕ[u]  U⊨ϕ[v] for all v with v<u 
Given a class F of frames, a sentence ϕ of TL is
 valid with respect to F if for every model U=(T,<,V) with (T,<) in F and for every u in T, U⊨ϕ[u]
 satisfiable with respect to F if there is a model U=(T,<,V) with (T,<) in F such that for some u in T, U⊨ϕ[u]
 a consequence of a sentence ψ with respect to F if for every model U=(T,<,V) with (T,<) in F and for every u in T, if U⊨ψ[u], then U⊨ϕ[u]
Many sentences are only valid for a limited class of frames. It is common to restrict the class of frames to those with a relation < that is transitive, antisymmetric, reflexive, trichotomic, irreflexive, total, dense, or some combination of these.
A minimal axiomatic logic
Burgess outlines a logic that makes no assumptions on the relation <, but allows for meaningful deductions, based on the following axiom schema:^{[10]}
 A where A is a tautology of firstorder logic
 G(A→B)→(GA→GB)
 H(A→B)→(HA→HB)
 A→GPA
 A→HFA
with the following rules of deduction:
 given A→B and A, deduce B (modus ponens)
 given a tautology A, infer GA
 given a tautology A, infer HA
One can derive the following rules:
 Becker's rule: given A→B, deduce TA→TB where T is a tense, any sequence made of G, H, F, and P.
 Mirroring: given a theorem A, deduce its mirror statement A^{§}, which is obtained by replacing G by H (and so F by P) and vice versa.
 Duality: given a theorem A, deduce its dual statement A*, which is obtained by interchanging ∧ with ∨, G with F, and H with P.
Translation to predicate logic
Burgess gives a Meredith translation from statements in TL into statements in firstorder logic with one free variable x_{0} (representing the present moment). This translation M is defined recursively as follows:^{[11]}
where is the sentence with all variable indices incremented by 1 and is a oneplace predicate defined by .
Temporal operators
Temporal logic has two kinds of operators: logical operators and modal operators [1]. Logical operators are usual truthfunctional operators (). The modal operators used in linear temporal logic and computation tree logic are defined as follows.
Textual  Symbolic  Definition  Explanation  Diagram 

Binary operators  
U  Until: holds at the current or a future position, and has to hold until that position. At that position does not have to hold any more. 


R  Release: releases if is true until the first position in which is true (or forever if such a position does not exist). 


Unary operators  
N  Next: has to hold at the next state. (X is used synonymously.) 


F  Future: eventually has to hold (somewhere on the subsequent path). 


G  Globally: has to hold on the entire subsequent path. 


A  All: has to hold on all paths starting from the current state.  
E  Exists: there exists at least one path starting from the current state where holds. 
Alternate symbols:
 operator R is sometimes denoted by V
 The operator W is the weak until operator: is equivalent to
Unary operators are wellformed formulas whenever B() is wellformed. Binary operators are wellformed formulas whenever B() and C() are wellformed.
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)
 Linear temporal logic (LTL)
 Computation tree logic (CTL)
 Signal temporal logic (STL)^{[12]}
 Timestamp temporal logic (TTL)^{[13]}
 Property specification language (PSL)
 CTL* which generalizes LTL and CTL
 Hennessy–Milner logic (HML)
 Modal μcalculus which includes as a subset HML and CTL*
 Metric temporal logic (MTL)^{[14]}
 Metric interval temporal logic (MITL)^{[12]}
A variation, closely related to temporal or chronological or tense logics, are modal logics based upon "topology", "place", or "spatial position".^{[15]}^{[16]}
See also
 HPO formalism
 Kripke structure
 Automata theory
 Chomsky grammar
 State transition system
 Duration calculus (DC)
 Hybrid logic
 Temporal logic in finitestate verification
 Temporal logic of actions (TLA)
 Important publications in formal verification (including the use of temporal logic in formal verification)
 Reo Coordination Language
 Modal logic
 Research Materials: Max Planck Society Archive
Notes
 ^ Vardi 2008, p. 153
 ^ ^{a} ^{b} ^{c} Vardi 2008, p. 154
 ^ Peter Øhrstrøm; Per F. V. Hasle (1995). Temporal logic: from ancient ideas to artificial intelligence. Springer. ISBN 9780792335863. pp. 176–178, 210
 ^ "Temporal Logic (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 20140730.
 ^ Walter Carnielli; Claudio Pizzi (2008). Modalities and Multimodalities. Springer. p. 181. ISBN 9781402085895.
 ^ Sergio Tessaris; Enrico Franconi; Thomas Eiter (2009). Reasoning Web. Semantic Technologies for Information Systems: 5th International Summer School 2009, BrixenBressanone, Italy, August 30 – September 4, 2009, Tutorial Lectures. Springer. p. 112. ISBN 9783642037535.
 ^ Prior, Arthur Norman (2003). Time and modality: the John Locke lectures for 1955–6, delivered at the University of Oxford. Oxford: The Clarendon Press. ISBN 9780198241584. OCLC 905630146.
 ^ Goranko, Valentin; Galton, Antony (2015). Zalta, Edward N., ed. The Stanford Encyclopedia of Philosophy (Winter 2015 ed.). Metaphysics Research Lab, Stanford University.
 ^ Müller, Thomas (2011). "Tense or temporal logic". In Horsten, Leon. The continuum companion to philosophical logic. A&C Black. p. 329.
 ^ Burgess, John P. (2009). Philosophical logic. Princeton, New Jersey: Princeton University Press. p. 21. ISBN 9781400830497. OCLC 777375659.
 ^ Burgess, John P. (2009). Philosophical logic. Princeton, New Jersey: Princeton University Press. p. 17. ISBN 9781400830497. OCLC 777375659.
 ^ ^{a} ^{b} Maler, O.; Nickovic, D. (2004). "Monitoring temporal properties of continuous signals". doi:10.1007/9783540302063_12.
 ^ https://asu.pure.elsevier.com/en/publications/timestamptemporallogicttlfortestingthetimingofcyberphys
 ^ Koymans, R. (1990). "Specifying realtime properties with metric temporal logic", RealTime Systems 2(4): 255–299. doi:10.1007/BF01995674.
 ^ Nicholas Rescher, James Garson, "Topological Logic" in The Journal of Symbolic Logic, 33(4):537–548, December, 1968
 ^ Georg Henrik von Wright, "A Modal Logic of Place", in E. Sosa (Editor), pp. 65–73, The Philosophy of Nicholas Rescher: Discussion and Replies, D. Reidel, Dordrecht, Holland, 1979
References
 Mordechai BenAri, Zohar Manna, Amir Pnueli: The Temporal Logic of Branching Time. POPL 1981: 164–176
 Amir Pnueli: The Temporal Logic of Programs FOCS 1977: 46–57
 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, pp. 275–306, 1987.
 E. A. Emerson, "Temporal and modal logic", Handbook of Theoretical Computer Science, Chapter 16, the MIT Press, 1990
 A Practical Introduction to PSL, Cindy Eisner, Dana Fisman
 Vardi, Moshe Y. (2008). "Church". In Orna Grumberg; Helmut Veith. 25 years of model checking: history, achievements, perspectives. Springer. ISBN 9783540698494. preprint. Historical perspective on how seemingly disparate ideas came together in computer science and engineering. (The mention of Church in the title of this paper is a reference to a littleknown 1957 paper, in which Church proposed a way to perform hardware verification.)
Further reading
 Peter Øhrstrøm; Per F. V. Hasle (1995). Temporal logic: from ancient ideas to artificial intelligence. Springer. ISBN 9780792335863.
External links
Wikimedia Commons has media related to Temporal logic. 
 Stanford Encyclopedia of Philosophy: "Temporal Logic"  by Anthony Galton.
 Temporal Logic by Yde Venema, formal description of syntax and semantics, questions of axiomatization. Treating also Kamp's dyadic temporal operators (since, until)
 Notes on games in temporal logic by Ian Hodkinson, including a formal description of firstorder temporal logic
 CADP – provides generic model checkers for various temporal logic
 PAT is a powerful free model checker, LTL checker, simulator and refinement checker for CSP and its extensions (with shared variable, arrays, wide range of fairness).