Temporal logic

Consider the statement "I am hungry". Though its meaning is constant in time, the statement's truth value can vary in time. Sometimes it is true, and sometimes false, but never simultaneously true and false. In a temporal logic, a statement can have a truth value that varies in time—in contrast with an atemporal logic, which applies only to statements whose truth values are constant in time. This treatment of truth value over time differentiates temporal logic from computational verb logic.

Temporal logic always has the ability to reason about a timeline. So-called linear "time logics" are restricted to this type of reasoning. Branching logics, however, can reason about multiple timelines. 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", and that "there is a possibility that eventually I am no longer hungry". If we do not know whether or not I will ever be fed, these statements can both be true.

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]

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] 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.

The sentential tense logic introduced in Time and Modality has four (non-truth-functional) modal operators (in addition to all usual truth-functional operators in first-order 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..."

These can be combined if we let π be an infinite path[8]:

• ${\displaystyle \pi \vDash FG\phi }$: "At a certain point, ${\displaystyle \phi }$ is true at all future states of the path"
• ${\displaystyle \pi \vDash GF\phi }$: "${\displaystyle \phi }$ is true at infinitely many states on the path"

From P and F one can define G and H, and vice versa:

${\displaystyle {\begin{aligned}F&\equiv \lnot G\lnot \\P&\equiv \lnot H\lnot \end{aligned}}}$

Temporal logic has two kinds of operators: logical operators and modal operators . Logical operators are usual truth-functional operators (${\displaystyle \neg ,\lor ,\land ,\rightarrow }$). The modal operators used in linear temporal logic and computation tree logic are defined as follows.

Textual	Symbolic	Definition	Explanation	Diagram
Binary operators
${\displaystyle \phi }$ U ${\displaystyle \psi }$	${\displaystyle \phi ~{\mathcal {U}}~\psi }$	${\displaystyle {\begin{matrix}(B\,{\mathcal {U}}\,C)(\phi )=\\(\exists i:C(\phi _{i})\land (\forall j<i:B(\phi _{j})))\end{matrix}}}$	Until: ${\displaystyle \psi }$ holds at the current or a future position, and ${\displaystyle \phi }$ has to hold until that position. At that position ${\displaystyle \phi }$ does not have to hold any more.
${\displaystyle \phi }$ R ${\displaystyle \psi }$	${\displaystyle \phi ~{\mathcal {R}}~\psi }$	${\displaystyle {\begin{matrix}(B\,{\mathcal {R}}\,C)(\phi )=\\(\forall i:C(\phi _{i})\lor (\exists j<i:B(\phi _{j})))\end{matrix}}}$	Release: ${\displaystyle \phi }$ releases ${\displaystyle \psi }$ if ${\displaystyle \psi }$ is true up until and including the first position in which ${\displaystyle \phi }$ is true (or forever if such a position does not exist).
Unary operators
N ${\displaystyle \phi }$	${\displaystyle \bigcirc \phi }$	${\displaystyle {\mathcal {N}}B(\phi _{i})=B(\phi _{i+1})}$	Next: ${\displaystyle \phi }$ has to hold at the next state. (X is used synonymously.)
F ${\displaystyle \phi }$	${\displaystyle \Diamond \phi }$	${\displaystyle {\mathcal {F}}B(\phi )=(true\,{\mathcal {U}}\,B)(\phi )}$	Future: ${\displaystyle \phi }$ eventually has to hold (somewhere on the subsequent path).
G ${\displaystyle \phi }$	${\displaystyle \Box \phi }$	${\displaystyle {\mathcal {G}}B(\phi )=\neg {\mathcal {F}}\neg B(\phi )}$	Globally: ${\displaystyle \phi }$ has to hold on the entire subsequent path.
A ${\displaystyle \phi }$	${\displaystyle \forall \phi }$	${\displaystyle {\begin{matrix}({\mathcal {A}}B)(\psi )=\\(\forall \phi$ :\phi _{0}=\psi \to B(\phi ))\end{matrix}}}	All: ${\displaystyle \phi }$ has to hold on all paths starting from the current state.
E ${\displaystyle \phi }$	${\displaystyle \exists \phi }$	${\displaystyle {\begin{matrix}({\mathcal {E}}B)(\psi )=\\(\exists \phi$ :\phi _{0}=\psi \land B(\phi ))\end{matrix}}}	Exists: there exists at least one path starting from the current state where ${\displaystyle \phi }$ holds.

Alternate symbols:

• operator R is sometimes denoted by V
• The operator W is the weak until operator: ${\displaystyle fWg}$ is equivalent to ${\displaystyle fUg\lor Gf}$

Unary operators are well-formed formulas whenever B(${\displaystyle \phi }$) is well-formed. Binary operators are well-formed formulas whenever B(${\displaystyle \phi }$) and C(${\displaystyle \phi }$) 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 include

• Interval temporal logic (ITL)
• Linear temporal logic (LTL)
• Computation tree logic (CTL)
• Signal temporal logic (STL)[13]
• Timestamp temporal logic (TTL)[14]
• Property specification language (PSL)
• CTL* which generalizes LTL and CTL
• Hennessy–Milner logic (HML)
• Modal μ-calculus which includes as a subset HML and CTL*
• Metric temporal logic (MTL)[15]
• Metric interval temporal logic (MITL)[13]
• Timed propositional temporal logic (TPTL)
• Truncated Linear Temporal Logic (TLTL)[16]

A variation, closely related to temporal or chronological or tense logics, are modal logics based upon "topology", "place", or "spatial position".[17][18]

• 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)
• 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

• 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, 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). "From Church and Prior to PSL". In Orna Grumberg; Helmut Veith (eds.). 25 years of model checking: history, achievements, perspectives. Springer. ISBN 978-3-540-69849-4. 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 little-known 1957 paper, in which Church proposed a way to perform hardware verification.)

• Peter Øhrstrøm; Per F. V. Hasle (1995). Temporal logic: from ancient ideas to artificial intelligence. Springer. ISBN 978-0-7923-3586-3.

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 first-order 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).