Linear temporal logic

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".
In terms of expressive power, linear temporal logic (LTL) is a fragment of first-order logic.
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 2"AP").
Let "w" = a0,a1,a2... be such an ω-word. Let "w"(i) = ai. Let "w"i = ai,ai+1..., which is a suffix of "w". Formally, the satisfaction relation formula_1 between a word and an LTL formula is defined as follows:
We say an ω-word "w" satisfies an LTL formula ψ when "w" formula_1 ψ. 
The ω-language "L"(ψ) defined by ψ is {"w" | "w" formula_1 ψ}, which is the set of ω-words that satisfy ψ.
A formula ψ is "satisfiable" if there exist an ω-word "w" such that "w" formula_1 ψ. 
A formula ψ is "valid" if for each ω-word "w" over alphabet 2"AP", "w" formula_1 ψ.
The additional logical operators are defined as follows:
The additional temporal operators R, F, and G are defined as follows:
Weak until and strong release.
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 "strong release" binary operator, denoted M, is the dual of weak until. It is defined similar to the until operator, so that the release condition has to hold at some point. Therefore, it is stronger than the release operator.
The semantics for the temporal operators are pictorially presented as follows.
Equivalences.
Let φ, ψ, and ρ be LTL formulas. The following tables list some of the useful equivalences that 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.
Computational problems.
Model checking and satisfiability against an LTL formula are PSPACE-complete problems. LTL synthesis and the problem of verification of games against an LTL winning condition is 2EXPTIME-complete.
Extensions.
Parametric linear temporal logic extends LTL with variables on the until-modality.