# 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 an early, partially-developed 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[1]. Logical operators are usual truth-functional operators ($\neg,\or,\and,\rightarrow$). 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 $\phi ~\mathcal{U}~ \psi$ $\begin{matrix}(B\,\mathcal{U}\,C)(\phi)= \\ (\exists i:C(\phi_i))\land(\forall j 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. φ R ψ $\phi ~\mathcal{R}~ \psi$ $\begin{matrix}(B\,\mathcal{R}\,C)(\phi)= \\ (\forall i:C(\phi_i))\lor(\exists j 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). Unary operators N φ $\circ \phi$ $\mathcal{N}B(\phi_i)=B(\phi_{i+1})$ Next: φ has to hold at the next state. (X is used synonymously.) F φ $\Diamond \phi$ $\mathcal{F}B(\phi)=(true\,\mathcal{U}\,B)(\phi)$ Future: φ eventually has to hold (somewhere on the subsequent path). G φ $\Box \phi$ $\mathcal{G}B(\phi)=\neg\mathcal{F}\neg B(\phi)$ Globally: φ has to hold on the entire subsequent path. A φ $\begin{matrix}(\mathcal{A}B)(\psi)= \\ (\forall \phi:\phi_0=\psi\to B(\phi))\end{matrix}$ All: φ has to hold on all paths starting from the current state. E φ $\begin{matrix}(\mathcal{E}B)(\psi)= \\ (\exists \phi:\phi_0=\psi\land B(\phi))\end{matrix}$ Exists: there exists at least one path starting from the current state where φ holds.

Alternate symbols:

• operator R is sometimes denoted by V
• The operator W is the weak until operator: fWg is equivalent to $f U g \or G f$

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