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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 logic]]-based 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 [[logician]]s.
 
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.
 
==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|computational verb 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 some times.
 
==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 first-order 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".<ref>Vardi 2008, p. 153</ref>
 
There was little development for millennia, [[Charles Sanders Peirce]] noted in the 19th century:<ref name=v154>Vardi 2008, p. 154</ref>
{{cquote|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 logic|propositional]] modal logic with two temporal connectives ([[modal operator]]s), 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".<ref name=v154/>{{Clarify|date=April 2011}} 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.<ref>{{cite book|author1=Peter Øhrstrøm|author2=Per F. V. Hasle|title=Temporal logic: from ancient ideas to artificial intelligence|year=1995|publisher=Springer|isbn=978-0-7923-3586-3}} pp. 176-178, 210</ref>
 
The binary temporal operators ''Since'' and ''Until'' were introduced by [[Hans Kamp]] in his 1968 Ph. D. thesis,<ref>http://plato.stanford.edu/entries/logic-temporal/M</ref> which also contains an important result relating temporal logic to [[first order logic]]—a result now known as [[Kamp's theorem]].<ref name="CarnielliPizzi2008">{{cite book|author1=Walter Carnielli|author2=Claudio Pizzi|title=Modalities and Multimodalities|url=http://books.google.com/books?id=XpAFM04G6BAC&pg=PA181|year=2008|publisher=Springer|isbn=978-1-4020-8589-5|page=181}}</ref><ref name=v154>Vardi 2008, p. 154</ref><ref name="TessarisFranconi2009">{{cite book|author1=Sergio Tessaris|author2=Enrico Franconi|author3=Thomas Eiter|title=Reasoning Web. Semantic Technologies for Information Systems: 5th International Summer School 2009, Brixen-Bressanone, Italy, August 30 - September 4, 2009, Tutorial Lectures|url=http://books.google.com/books?id=JdyeU7zs4-AC&pg=PA112|year=2009|publisher=Springer|isbn=978-3-642-03753-5|page=112}}</ref>
 
Two early contenders in formal verifications were [[Linear Temporal Logic]] (a linear time logic by [[Amir Pnueli]]) and [[Computational tree logic|Computation Tree Logic]], a branching time logic by Mordechai Ben-Ari, [[Zohar Manna]] and [[Amir Pnueli]]. An almost equivalent formalism to CTL was suggested around the same time by [[Edmund M. Clarke|E.M. Clarke]] and [[E. Allen Emerson|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.
 
==Temporal operators==
Temporal logic has two kinds of operators: [[logical operator]]s and [[modal operator]]s [http://plato.stanford.edu/entries/logic-temporal/]. Logical operators are usual [[truth-functional]] operators (<math>\neg,\or,\and,\rightarrow</math>). The modal operators used in Linear Temporal Logic and Computation Tree Logic are defined as follows.
 
{| class="wikitable"
|-
! Textual
! Symbolic
! Definition
! Explanation
! Diagram
|-
! colspan="4" | [[Binary operator]]s
|-
|<math>\phi</math> '''U''' <math>\psi</math>
|<math>\phi ~\mathcal{U}~ \psi</math>
|<math>\begin{matrix}(B\,\mathcal{U}\,C)(\phi)= \\ (\exists i:C(\phi_i)\land(\forall j<i:B(\phi_j)))\end{matrix}</math>
|'''U'''ntil: <math>\psi</math> holds at the current or a future position, and <math>\phi</math> has to hold until that position. At that position <math>\phi</math> does not have to hold any more.
|<timeline>
ImageSize = width:240 height:94
PlotArea = left:30 bottom:30 top:0 right:20
DateFormat = x.y
Period = from:0 till:6
TimeAxis = orientation:horizontal
AlignBars = justify
ScaleMajor = gridcolor:black increment:1 start:0
ScaleMinor = gridcolor:black increment:1 start:0
 
PlotData=
bar:p color:red width:10 align:left fontsize:S
from:1 till:3
 
bar:q color:red width:10 align:left fontsize:S
from:3 till:5
 
bar:pUq color:red width:10 align:left fontsize:S
from:1 till:5
</timeline>
|-
|<math>\phi</math> '''R''' <math>\psi</math>
|<math>\phi ~\mathcal{R}~ \psi</math>
|<math>\begin{matrix}(B\,\mathcal{R}\,C)(\phi)= \\ (\forall i:C(\phi_i)\lor(\exists j<i:B(\phi_j)))\end{matrix}</math>
|'''R'''elease: <math>\phi</math> releases <math>\psi</math> if <math>\psi</math> is true until the first position in which <math>\phi</math> is true (or forever if such a position does not exist).
|<timeline>
ImageSize = width:240 height:100
PlotArea = left:30 bottom:30 top:0 right:20
DateFormat = x.y
Period = from:0 till:8
TimeAxis = orientation:horizontal
AlignBars = justify
ScaleMajor = gridcolor:black increment:1 start:0
ScaleMinor = gridcolor:black increment:1 start:0
 
PlotData=
bar:p color:red width:10 align:left fontsize:S
from:2 till:4
from:6 till:8
 
bar:q color:red width:10 align:left fontsize:S
from:1 till:3
from:5 till:6
from:7 till:8
 
bar:pRq color:red width:10 align:left fontsize:S
from:1 till:3
from:7 till:8
</timeline>
|-
! colspan="4" | [[Unary operator]]s
|-
|'''N''' <math>\phi</math>
|<math>\bigcirc \phi</math>
|<math>\mathcal{N}B(\phi_i)=B(\phi_{i+1})</math>
|'''N'''ext: <math>\phi</math> has to hold at the next state. ('''X''' is used synonymously.)
|<timeline>
ImageSize = width:240 height:60
PlotArea = left:30 bottom:30 top:0 right:20
DateFormat = x.y
Period = from:0 till:6
TimeAxis = orientation:horizontal
AlignBars = justify
ScaleMajor = gridcolor:black increment:1 start:0
ScaleMinor = gridcolor:black increment:1 start:0
 
PlotData=
bar:p color:red width:10 align:left fontsize:S
from:2 till:3
from:5 till:6
 
bar:Np color:red width:10 align:left fontsize:S
from:1 till:2
from:4 till:5
</timeline>
|-
|'''F''' <math>\phi</math>
|<math>\Diamond \phi</math>
|<math>\mathcal{F}B(\phi)=(true\,\mathcal{U}\,B)(\phi)</math>
|'''F'''uture: <math>\phi</math> eventually has to hold (somewhere on the subsequent path).
|<timeline>
ImageSize = width:240 height:60
PlotArea = left:30 bottom:30 top:0 right:20
DateFormat = x.y
Period = from:0 till:6
TimeAxis = orientation:horizontal
AlignBars = justify
ScaleMajor = gridcolor:black increment:1 start:0
ScaleMinor = gridcolor:black increment:1 start:0
 
PlotData=
bar:p color:red width:10 align:left fontsize:S
from:2 till:3
from:4 till:5
 
bar:Fp color:red width:10 align:left fontsize:S
from:0 till:5
</timeline>
|-
|'''G''' <math>\phi</math>
|<math>\Box \phi</math>
|<math>\mathcal{G}B(\phi)=\neg\mathcal{F}\neg B(\phi)</math>
|'''G'''lobally: <math>\phi</math> has to hold on the entire subsequent path.
|<timeline>
ImageSize = width:240 height:60
PlotArea = left:30 bottom:30 top:0 right:20
DateFormat = x.y
Period = from:0 till:6
TimeAxis = orientation:horizontal
AlignBars = justify
ScaleMajor = gridcolor:black increment:1 start:0
ScaleMinor = gridcolor:black increment:1 start:0
 
PlotData=
bar:p color:red width:10 align:left fontsize:S
from:1 till:3
from:4 till:6
 
bar:Gp color:red width:10 align:left fontsize:S
from:4 till:6
</timeline>
|-
|'''A''' <math>\phi</math>
|<math>\forall \phi</math>
|<math>\begin{matrix}(\mathcal{A}B)(\psi)= \\ (\forall \phi:\phi_0=\psi\to B(\phi))\end{matrix}</math>
|'''A'''ll: <math>\phi</math> has to hold on all paths starting from the current state.
|
|-
|'''E''' <math>\phi</math>
|<math>\exists \phi</math>
|<math>\begin{matrix}(\mathcal{E}B)(\psi)= \\ (\exists \phi:\phi_0=\psi\land B(\phi))\end{matrix}</math>
|'''E'''xists: there exists at least one path starting from the current state where <math>\phi</math> holds.
|
|}
 
Alternate symbols:
 
* operator '''R''' is sometimes denoted by '''V'''
* The operator '''W''' is the ''weak until'' operator: <math>f W g</math> is equivalent to <math>f U g \or G f</math>
 
Unary operators are [[well-formed formula]]s whenever B(<math>\phi</math>) is well-formed. Binary operators are well-formed formulas whenever B(<math>\phi</math>) and C(<math>\phi</math>) 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)
* [[mu calculus|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)
*** [[Metric Interval Temporal Logic]] (MITL) <ref> O. Maler, D. Nickovic, "Monitoring temporal properties of continuous signals", 2004</ref>
*** [[Signal Temporal Logic]] (STL) <ref> O. Maler, D. Nickovic, "Monitoring temporal properties of continuous signals", 2004</ref>
 
A variation, closely related to Temporal or Chronological or Tense logics, are Modal logics based upon "topology", "place", or "spatial position".<ref>Nicholas Rescher, James Garson, "Topological Logic" in The Journal of Symbolic Logic, 33(4):537-548, December, 1968</ref><ref>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</ref>
 
==See also==
{{Portal|Logic}}
* [[HPO formalism]]
* [[Kripke structure]]
* [[Automata theory]]
* [[Chomsky grammar]]
* [[State transition system]]
* [[Duration calculus]] (DC)
* [[Hybrid logic]]
* [[Temporal logic in finite-state verification]]
* [[Temporal logic of actions]] (TLA)
* [[List of important publications in computer science#Formal verification|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==
{{reflist}}
 
==References==
* Mordechai Ben-Ari, 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, p 275-306, 1987.
* E.A. Emerson, Temporal and modal logic, ''Handbook of Theoretical Computer Science'', Chapter 16, the MIT Press, 1990
* {{cite book|editor=Orna Grumberg, Helmut Veith|title=25 years of model checking: history, achievements, perspectives|year=2008|publisher=Springer|isbn=978-3-540-69849-4|chapter=From [[Alonzo Church|Church]] and Prior to [[Property Specification Language|PSL]]|author=Moshe Y. Vardi}} [http://www.cs.rice.edu/~vardi/papers/25mc.ps.gz preprint] Historical perspective on how seemingly disparate ideas came together in computer science and engineering. (The reference to Church is to a little known 1957 in which he proposed a way to perform hardware verification.)
 
==Further reading==
* {{cite book|author1=Peter Øhrstrøm|author2=Per F. V. Hasle|title=Temporal logic: from ancient ideas to artificial intelligence|year=1995|publisher=Springer|isbn=978-0-7923-3586-3}}
 
==External links==
*[[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]
*[http://www.comp.nus.edu.sg/~pat/ 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).
 
{{Logic}}
 
[[Category:Temporal logic| ]]

Revision as of 09:26, 14 December 2013

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 logic-based 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.

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. 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 some times.

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 first-order 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]

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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]Template:Clarify 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 first order 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 Ben-Ari, 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.

Temporal operators

Temporal logic has two kinds of operators: logical operators and modal operators [1]. Logical operators are usual truth-functional operators (¬,∨,∧,→). The modal operators used in Linear Temporal Logic and Computation Tree Logic are defined as follows.

Textual Symbolic Definition Explanation Diagram
Binary operators
ϕ U ψ ϕ𝒰ψ (B𝒰C)(ϕ)=(∃i:C(ϕi)∧(∀j<i:B(ϕj))) Until: ψ holds at the current or a future position, and ϕ has to hold until that position. At that position ϕ does not have to hold any more. <timeline>

ImageSize = width:240 height:94 PlotArea = left:30 bottom:30 top:0 right:20 DateFormat = x.y Period = from:0 till:6 TimeAxis = orientation:horizontal AlignBars = justify ScaleMajor = gridcolor:black increment:1 start:0 ScaleMinor = gridcolor:black increment:1 start:0

PlotData= 

bar:p color:red width:10 align:left fontsize:S
from:1 till:3

bar:q color:red width:10 align:left fontsize:S
from:3 till:5

bar:pUq color:red width:10 align:left fontsize:S
from:1 till:5

</timeline>

ϕ R ψ ϕℛψ (BℛC)(ϕ)=(∀i:C(ϕi)∨(∃j<i:B(ϕj))) Release: ϕ releases ψ if ψ is true until the first position in which ϕ is true (or forever if such a position does not exist). <timeline>

ImageSize = width:240 height:100 PlotArea = left:30 bottom:30 top:0 right:20 DateFormat = x.y Period = from:0 till:8 TimeAxis = orientation:horizontal AlignBars = justify ScaleMajor = gridcolor:black increment:1 start:0 ScaleMinor = gridcolor:black increment:1 start:0

PlotData= 

bar:p color:red width:10 align:left fontsize:S
from:2 till:4
from:6 till:8

bar:q color:red width:10 align:left fontsize:S
from:1 till:3
from:5 till:6
from:7 till:8

bar:pRq color:red width:10 align:left fontsize:S
from:1 till:3
from:7 till:8

</timeline>

Unary operators
N ϕ ◯ϕ 𝒩B(ϕi)=B(ϕi+1) Next: ϕ has to hold at the next state. (X is used synonymously.) <timeline>

ImageSize = width:240 height:60 PlotArea = left:30 bottom:30 top:0 right:20 DateFormat = x.y Period = from:0 till:6 TimeAxis = orientation:horizontal AlignBars = justify ScaleMajor = gridcolor:black increment:1 start:0 ScaleMinor = gridcolor:black increment:1 start:0

PlotData= 

bar:p color:red width:10 align:left fontsize:S
from:2 till:3
from:5 till:6

bar:Np color:red width:10 align:left fontsize:S
from:1 till:2
from:4 till:5

</timeline>

F ϕ ◊ϕ ℱB(ϕ)=(true𝒰B)(ϕ) Future: ϕ eventually has to hold (somewhere on the subsequent path). <timeline>

ImageSize = width:240 height:60 PlotArea = left:30 bottom:30 top:0 right:20 DateFormat = x.y Period = from:0 till:6 TimeAxis = orientation:horizontal AlignBars = justify ScaleMajor = gridcolor:black increment:1 start:0 ScaleMinor = gridcolor:black increment:1 start:0

PlotData= 

bar:p color:red width:10 align:left fontsize:S
from:2 till:3
from:4 till:5

bar:Fp color:red width:10 align:left fontsize:S
from:0 till:5

</timeline>

G ϕ ◻ϕ 𝒢B(ϕ)=¬ℱ¬B(ϕ) Globally: ϕ has to hold on the entire subsequent path. <timeline>

ImageSize = width:240 height:60 PlotArea = left:30 bottom:30 top:0 right:20 DateFormat = x.y Period = from:0 till:6 TimeAxis = orientation:horizontal AlignBars = justify ScaleMajor = gridcolor:black increment:1 start:0 ScaleMinor = gridcolor:black increment:1 start:0

PlotData= 

bar:p color:red width:10 align:left fontsize:S
from:1 till:3
from:4 till:6

bar:Gp color:red width:10 align:left fontsize:S
from:4 till:6

</timeline>

A ϕ ∀ϕ (𝒜B)(ψ)=(∀ϕ:ϕ0=ψ→B(ϕ)) All: ϕ has to hold on all paths starting from the current state.
E ϕ ∃ϕ (ℰB)(ψ)=(∃ϕ:ϕ0=ψ∧B(ϕ)) Exists: there exists at least one path starting from the current state where ϕ holds.

Alternate symbols:

  • operator R is sometimes denoted by V
  • The operator W is the weak until operator: fWg is equivalent to fUg∨Gf

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

A variation, closely related to Temporal or Chronological or Tense logics, are Modal logics based upon "topology", "place", or "spatial position".[9][10]

See also

Sportspersons Hyslop from Nicolet, usually spends time with pastimes for example martial arts, property developers condominium in singapore singapore and hot rods. Maintains a trip site and has lots to write about after touring Gulf of Porto: Calanche of Piana.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • Mordechai Ben-Ari, 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, p 275-306, 1987.
  • E.A. Emerson, Temporal and modal logic, Handbook of Theoretical Computer Science, Chapter 16, the MIT Press, 1990
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 preprint Historical perspective on how seemingly disparate ideas came together in computer science and engineering. (The reference to Church is to a little known 1957 in which he proposed a way to perform hardware verification.)

Further reading

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

Template:Logic

  1. ↑ Vardi 2008, p. 153
  2. ↑ 2.0 2.1 2.2 Vardi 2008, p. 154
  3. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 pp. 176-178, 210
  4. ↑ http://plato.stanford.edu/entries/logic-temporal/M
  5. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  6. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  7. ↑ O. Maler, D. Nickovic, "Monitoring temporal properties of continuous signals", 2004
  8. ↑ O. Maler, D. Nickovic, "Monitoring temporal properties of continuous signals", 2004
  9. ↑ Nicholas Rescher, James Garson, "Topological Logic" in The Journal of Symbolic Logic, 33(4):537-548, December, 1968
  10. ↑ 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
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