In logic, linear temporal logic or linear-time temporal logic (LTL) is a modal temporal logic with modalities referring to time. In LTL, one can encode formulae about the future of paths, e.g., a condition will eventually be true, a condition will be true until another fact becomes true, etc. It is a fragment of the more complex CTL**, which additionally allows branching time and quantifiers. Subsequently LTL is sometimes called propositional temporal logic, abbreviated PTL. Linear temporal logic (LTL) is a fragment of S1S (monadic second-order logic of one successor).
Contents
LTL was first proposed for the formal verification of computer programs by Amir Pnueli in 1977.
Syntax
LTL is built up from a finite set of propositional variables AP, the logical operators ¬ and ∨, and the temporal modal operators X (some literature uses O or N) and U. Formally, the set of LTL formulas over AP is inductively defined as follows:
X is read as next and U is read as until. Other than these fundamental operators, there are additional logical and temporal operators defined in terms of the fundamental operators to write LTL formulas succinctly. The additional logical operators are ∧, →, ←→, true, and false. Following are the additional temporal operators.
Semantics
An LTL formula can be satisfied by an infinite sequence of truth evaluations of variables in AP. These sequences can be viewed as a word on a path of a Kripke structure (an ω-word over alphabet 2AP). Let w = a0,a1,a2,... be such an ω-word. Let w(i) = ai. Let wi = ai,ai+1,..., which is a suffix of w. Formally, the satisfaction relation
We say an ω-word w satisfies an LTL formula ψ when w
The additional logical operators are defined as follows:
The additional temporal operators R, F, and G are defined as follows:
Some authors also define a weak until binary operator, denoted W, with semantics similar to that of the until operator but the stop condition is not required to occur (similar to release). It is sometimes useful since both U and R can be defined in terms of the weak until:
The semantics for the temporal operators are pictorially presented as follows.
†The symbols are used in the literature to denote these operators.
Equivalences
Let Φ, ψ, and ρ be LTL formulas. The following tables list some of the useful equivalences which extend standard equivalences among the usual logical operators.
Negation normal form
All the formulas of LTL can be transformed into negation normal form, where
Using the above equivalences for negation propagation, it is possible to derive the normal form. This normal form allows R, true, false, and ∧ to appear in the formula, which are not fundamental operators of LTL. Note that the transformation to the negation normal form does not blow up the size of the formula. This normal form is useful in translation from LTL to Büchi automaton.
Relations with other logics
LTL can be shown to be equivalent to the monadic first-order logic of order, FO[<]—a result known as Kamp's theorem— or equivalently star-free languages.
Computation tree logic (CTL*) and linear temporal logic (LTL) are both a subset of CTL*, but are incomparable. For example,
However, a subset of CTL* exists that is a proper superset of both CTL and LTL.