Metric temporal logic

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Metric temporal logic (MTL) is a special case of temporal logic. It is an extension of temporal logic in which temporal operators are replaced by time-constrained versions like until, next, since and previous operators. It is a linear-time logic that assumes both the interleaving and fictitious-clock abstractions. It is defined over a point-based weakly-monotonic integer-time semantics.

MTL has been described as a prominent specification formalism for real-time systems.[1] Full MTL over infinite timed words is undecidable.[2]

Syntax

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The full metric temporal logic is defined similarly to linear temporal logic, where a set of non-negative real number is added to temporal modal operators U and S. Formally, MTL is built up from:

When the subscript is omitted, it is implicitly equal to  .

Note that the next operator N is not considered to be a part of MTL syntax. It will instead be defined from other operators.

Past and Future

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The past fragment of metric temporal logic, denoted as past-MTL is defined as the restriction of the full metric temporal logic without the until operator. Similarly, the future fragment of metric temporal logic, denoted as future-MTL is defined as the restriction of the full metric temporal logic without the since operator.

Depending on the authors, MTL is either defined as the future fragment of MTL, in which case full-MTL is called MTL+Past.[1][3] Or MTL is defined as full-MTL.

In order to avoid ambiguity, this article uses the names full-MTL, past-MTL and future-MTL. When the statements holds for the three logic, MTL will simply be used.

Model

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Let   intuitively represent a set of points in time. Let   a function which associates a letter to each moment  . A model of a MTL 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

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Let   and   as above and let   some fixed time. We are now going to explain what it means that a MTL formula   holds at time  , which is denoted  .

Let   and  . We first consider the formula  . We say that   if and only if there exists some time   such that:

  •   and
  • for each   with  ,  .

We now consider the formula   (pronounced "  since in    .") We say that   if and only if there exists some time   such that:

  •   and
  • for each   with  ,  .

The definitions of   for the values of   not considered above is similar as the definition in the LTL case.

Operators defined from basic MTL operators

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Some formulas are so often used that a new operator is introduced for them. These operators are usually not considered to belong to the definition of MTL, but are syntactic sugar which denote more complex MTL formula. We first consider operators which also exists in LTL. In this section, we fix   MTL formulas and  .

Operators similar to the ones of LTL

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Release and Back to

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We denote by   (pronounced "  release in  ,  ") the formula  . This formula holds at time   if either:

  • there is some time   such that   holds, and   hold in the interval  .
  • at each time  ,   holds.

The name "release" come from the LTL case, where this formula simply means that   should always hold, unless   releases it.

The past counterpart of release is denote by   (pronounced "  back to in  ,  ") and is equal to the formula  .

Finally and Eventually

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We denote by   or   (pronounced "Finally in  ,  ", or "Eventually in  ,  ") the formula  . Intuitively, this formula holds at time   if there is some time   such that   holds.

We denote by   or   (pronounced "Globally in  ,  ",) the formula  . Intuitively, this formula holds at time   if for all time  ,   holds.

We denote by   and   the formula similar to   and  , where   is replaced by  . Both formula has intuitively the same meaning, but when we consider the past instead of the future.

Next and previous

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This case is slightly different from the previous ones, because the intuitive meaning of the "Next" and "Previously" formulas differs depending on the kind of function   considered.

We denote by   or   (pronounced "Next in  ,  ") the formula  . Similarly, we denote by  [4] (pronounced "Previously in  ,  ) the formula  . The following discussion about the Next operator also holds for the Previously operator, by reversing the past and the future.

When this formula is evaluated over a timed word  , this formula means that both:

  • at the next time in the domain of definition  , the formula   will holds.
  • furthermore, the distance between this next time and the current time belong to the interval  .
  • In particular, this next time holds, thus the current time is not the end of the word.

When this formula is evaluated over a signal  , the notion of next time does not makes sense. Instead, "next" means "immediately after". More precisely   means:

  •   contains an interval of the form   and
  • for each  ,  .

Other operators

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We now consider operators which are not similar to any standard LTL operators.

Fall and Rise

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We denote by   (pronounced "rise  "), a formula which holds when   becomes true. More precisely, either   did not hold in the immediate past, and holds at this time, or it does not hold and it holds in the immediate future. Formally   is defined as  .[5]

Over timed words, this formula always hold. Indeed   and   always hold. Thus the formula is equivalent to  , hence is true.

By symmetry, we denote by   (pronounced "Fall  ), a formula which holds when   becomes false. Thus, it is defined as  .

History and Prophecy

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We now introduce the prophecy operator, denoted by  . We denote by  [6] the formula  . This formula asserts that there exists a first moment in the future such that   holds, and the time to wait for this first moment belongs to  .

We now consider this formula over timed words and over signals. We consider timed words first. Assume that   where   and   represents either open or closed bounds. Let   a timed word and   in its domain of definition. Over timed words, the formula   holds if and only if   also holds. That is, this formula simply assert that, in the future, until the interval   is met,   should not hold. Furthermore,   should hold sometime in the interval  . Indeed, given any time   such that   hold, there exists only a finite number of time   with   and  . Thus, there exists necessarily a smaller such  .

Let us now consider signal. The equivalence mentioned above does not hold anymore over signal. This is due to the fact that, using the variables introduced above, there may exists an infinite number of correct values for  , due to the fact that the domain of definition of a signal is continuous. Thus, the formula   also ensures that the first interval in which   holds is closed on the left.

By temporal symmetry, we define the history operator, denoted by  . We define   as  . This formula asserts that there exists a last moment in the past such that   held. And the time since this first moment belongs to  .

Non-strict operator

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The semantic of operators until and since introduced do not consider the current time. That is, in order for   to holds at some time  , neither   nor   has to hold at time  . This is not always wanted, for example in the sentence "there is no bug until the system is turned-off", it may actually be wanted that there are no bug at current time. Thus, we introduce another until operator, called non-strict until, denoted by  , which consider the current time.

We denote by   and   either:

  • the formulas   and   if  , and
  • the formulas   and   otherwise.

For any of the operators   introduced above, we denote   the formula in which non-strict untils and sinces are used. For example   is an abbreviation for  .

Strict operator can not be defined using non-strict operator. That is, there is no formula equivalent to   which uses only non-strict operator. This formula is defined as  . This formula can never hold at a time   if it is required that   holds at time  .

Example

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We now give examples of MTL formulas. Some more example can be found on article of fragments of MITL, such as metric interval temporal logic.

  •   states that each letter   is followed exactly one time unit later by a letter  .
  •   states that no two successive occurrences of   can occur at exactly one time unit from each other.

Comparison with LTL

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A standard (untimed) 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 MTL formula with non-strict operator and   subscript. In this sense, MTL is an extension of LTL.[clarification needed]

For this reason, a formula using only non-strict operator with   subscript is called an LTL formula. This is because the[further explanation needed]

Algorithmic complexity

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The satisfiability of ECL over signals is EXPSPACE-complete.[6]

Fragments of MTL

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We now consider some fragments of MTL.

MITL

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An important subset of MTL is the Metric Interval Temporal Logic (MITL). This is defined similarly to MTL, with the restriction that the sets  , used in   and  , are intervals which are not singletons, and whose bounds are natural numbers or infinity.

Some other subsets of MITL are defined in the article MITL.

Future Fragments

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Future-MTL was already introduced above. Both over timed-words and over signals, it is less expressive than Full-MTL[3]: 3 .

Event-Clock Temporal Logic

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The fragment Event-Clock Temporal Logic[6] of MTL, denoted EventClockTL or ECL, allows only the following operators:

  • the boolean operators, and, or, not
  • the untimed until and since operators.
  • The timed prophecy and history operators.

Over signals, ECL is as expressive as MITL and as MITL0. The equivalence between the two last logics is explained in the article MITL0. We sketch the equivalence of those logics with ECL.

If   is not a singleton and   is a MITL formula,   is defined as a MITL formula. If   is a singleton, then   is equivalent to   which is a MITL-formula. Reciprocally, for   an ECL-formula, and   an interval whose lower bound is 0,   is equivalent to the ECL-formula  .

The satisfiability of ECL over signals is PSPACE-complete.[6]

Positive normal form

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A MTL-formula in positive normal form is defined almost as any MTL formula, with the two following change:

  • the operators Release and Back are introduced in the logical language and are not considered anymore to be notations for some other formulas.
  • negations can only be applied to letters.

Any MTL formula is equivalent to formula in normal form. This can be shown by an easy induction on formulas. For example, the formula   is equivalent to the formula  . Similarly, conjunctions and disjunctions can be considered using De Morgan's laws.

Strictly speaking, the set of formulas in positive normal form is not a fragment of MTL.

See also

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References

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  1. ^ a b J. Ouaknine and J. Worrell, "On the decidability of metric temporal logic," 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05), 2005, pp. 188-197.
  2. ^ Ouaknine J., Worrell J. (2006) On Metric Temporal Logic and Faulty Turing Machines. In: Aceto L., Ingólfsdóttir A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2006. Lecture Notes in Computer Science, vol 3921. Springer, Berlin, Heidelberg
  3. ^ a b Bouyer, Patricia; Chevalier, Fabrice; Markey, Nicolas (2005). "On the Expressiveness of TPTL and MTL". In Sundar Sarukkai; Sandeep Sen (eds.). FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science, Proceedings. 25th International Conference, Hyderabad, India, December 15–18, 2005. Lecture Notes in Computer Science. Vol. 3821. p. 436. doi:10.1007/11590156_35. ISBN 978-3-540-32419-5.
  4. ^ Maler, Oded; Nickovic, Dejan; Pnueli, Amir (2008). "Checking temporal properties of discrete, timed and continuous behaviors". Pillars of computer science. ACM. p. 478. ISBN 978-3-540-78126-4.
  5. ^ Nickovic, Dejan (31 August 2009). "3". Checking Timed and Hybrid Properties: Theory and Applications (Thesis).
  6. ^ a b c d Henzinger, T.A.; Raskin, J.F.; Schobbens, P.-Y. (1998). "The regular real-time languages". Automata, Languages and Programming. Lecture Notes in Computer Science. Vol. 1443. p. 590. doi:10.1007/BFb0055086. ISBN 978-3-540-64781-2.