Timed propositional temporal logic

In model checking, a field of computer science, timed propositional temporal logic is an extension of propositional temporal logic">Temporal logic">temporal logic in which variables are introduced to measure times between two events. For example, while LTL allows to state that each event p is eventually followed by an event q, TPTL furthermore allows to give a time limit for q to occur.

Syntax

The future fragment of TPTL is defined similarly to linear temporal logic, in which furthermore, clock variables can be introduced and compared to constants. Formally, given a set of clocks, is built up from:

a finite set of propositional variables AP,
the logical operators ¬ and ∨, and
the temporal modal operator U,
a clock comparison, with, a number and a comparison operator such as <, ≤, =, ≥ or >.
a freeze quantification operator, for a TPTL formula with set of clocks.

Furthermore, for an interval, is considered as an abbreviation for ; and similarly for every other kind of intervals.
The logic TPTL+Past is built as the future fragment of and also contains

the temporal modal operator S.

The next operator is not considered to be a part of syntax. It will instead be defined from other operators.
A closed formula is a formula over an empty set of clocks.

Models

Let, which intuitively represents a set of times. Let a function that associates to each moment a set of propositions from AP. A model of a TPTL formula is such a function. Usually, is either a timed word or a signal. In those cases, is either a discrete subset or an interval containing 0.

Semantics

Let and be as above. Let be a set of clocks. Let .
We are now going to explain what it means for a TPTL formula to hold at time for a valuation. This is denoted by.
Let and be two formulas over the set of clocks, a formula over the set of clocks,,, a number and being a comparison operator such as <, ≤, =, ≥ or >:
We first consider formulas whose main operator also belongs to LTL:

holds if,
holds if
holds if either or, or both
holds if there exists such that and and for each with,,
holds if there exists such that and and for each with,,
holds if,
holds if holds.

Metric temporal logic

Metric temporal logic is another extension of LTL that allows measurement of time. Instead of adding variables, it adds an infinity of operators and for an interval of non-negative numbers. The semantics of the formula at some time is essentially the same than the semantics of the formula, with the constraints that the time at which must hold occurs in the interval.
TPTL is as least as expressive as MTL. Indeed, the MTL formula is equivalent to the TPTL formula where is a new variable.
It follows that any other operator introduced in the page MTL, such as and can also be defined as TPTL formulas.
TPTL is strictly more expressive than MTL both over timed words and over signals. Over timed words, no MTL formula is equivalent to. Over signal, there are no MTL formula equivalent to, which states that the last atomic proposition before time point 1 is an.

Comparison with LTL

A standard infinite word is a function from to. We can consider such a word using the set of time, and the function. In this case, for an arbitrary LTL formula, if and only if, where is considered as a TPTL formula with non-strict operator, and is the only function defined on the empty set.