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Metric Temporal Logic
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.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. Full MTL over infinite timed words is undecidable.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 Syntax The full metr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temporal Logic
In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I eat something"). It is sometimes also used to refer to tense logic, a modal logic-based system of temporal logic introduced by Arthur Prior in the late 1950s, with important contributions by Hans Kamp. It has been further developed by computer scientists, notably Amir Pnueli, and logicians. 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. Motivation Consider the statement "I am hungry". Though its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Interval Temporal Logic
In model checking, the Metric Interval Temporal Logic (MITL) is a fragment of Metric Temporal Logic (MTL). This fragment is often preferred to MTL because some problems that are undecidable for MTL become decidable for MITL. Definition A MITL formula is an MTL formula, such that each set of reals used in subscript are intervals, which are not singletons, and whose bounds are either a natural number or are infinite. Difference from MTL MTL can express a statement such as the sentence S: "P held exactly ten time units ago". This is impossible in MITL. Instead, MITL can say T: "P held between 9 and 10 time units ago". Since MITL can express T but not S, in a sense, MITL is a restriction of MTL which allows only less precise statements. Problems that MITL avoids One reason to want to avoid a statement such as S is that its truth value may change an arbitrary number of times in a single time unit. Indeed, the truth value of this statement may change as many times as the truth v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Morgan's Laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. The rules can be expressed in English as: * The negation of a disjunction is the conjunction of the negations * The negation of a conjunction is the disjunction of the negations or * The complement of the union of two sets is the same as the intersection of their complements * The complement of the intersection of two sets is the same as the union of their complements or * not (A or B) = (not A) and (not B) * not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning ''at least'' one of A or B rather than an "exclusive or" that means ''exactly'' one of A or B. In set theory and Boolean algebra, these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE-complete
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (polynomial space) and if every other problem that can be solved in polynomial space can be transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. Problems known to be PSPACE-complete include determining properties of regular expressions and context-sensitive grammars, determining the truth of quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization problems, and many puzzles and games. Theory A problem is defined to be PSPACE-complete if it can be solved using a polynomial amount of memory (it belongs to PSPACE) and every problem in PSPACE can be tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MITL0
In model checking, the Metric Interval Temporal Logic (MITL) is a fragment of Metric Temporal Logic (MTL). This fragment is often preferred to MTL because some problems that are undecidable for MTL become decidable for MITL. Definition A MITL formula is an MTL formula, such that each set of reals used in subscript are intervals, which are not singletons, and whose bounds are either a natural number or are infinite. Difference from MTL MTL can express a statement such as the sentence S: "P held exactly ten time units ago". This is impossible in MITL. Instead, MITL can say T: "P held between 9 and 10 time units ago". Since MITL can express T but not S, in a sense, MITL is a restriction of MTL which allows only less precise statements. Problems that MITL avoids One reason to want to avoid a statement such as S is that its truth value may change an arbitrary number of times in a single time unit. Indeed, the truth value of this statement may change as many times as the truth v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MITL
In model checking, the Metric Interval Temporal Logic (MITL) is a fragment of Metric Temporal Logic (MTL). This fragment is often preferred to MTL because some problems that are undecidable for MTL become decidable for MITL. Definition A MITL formula is an MTL formula, such that each set of reals used in subscript are intervals, which are not singletons, and whose bounds are either a natural number or are infinite. Difference from MTL MTL can express a statement such as the sentence S: "P held exactly ten time units ago". This is impossible in MITL. Instead, MITL can say T: "P held between 9 and 10 time units ago". Since MITL can express T but not S, in a sense, MITL is a restriction of MTL which allows only less precise statements. Problems that MITL avoids One reason to want to avoid a statement such as S is that its truth value may change an arbitrary number of times in a single time unit. Indeed, the truth value of this statement may change as many times as the truth v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete (complexity)
In computational complexity theory, a computational problem is complete for a complexity class if it is, in a technical sense, among the "hardest" (or "most expressive") problems in the complexity class. More formally, a problem ''p'' is called hard for a complexity class ''C'' under a given type of reduction if there exists a reduction (of the given type) from any problem in ''C'' to ''p''. If a problem is both hard for the class and a member of the class, it is complete for that class (for that type of reduction). A problem that is complete for a class ''C'' is said to be C-complete, and the class of all problems complete for ''C'' is denoted C-complete. The first complete class to be defined and the most well known is NP-complete, a class that contains many difficult-to-solve problems that arise in practice. Similarly, a problem hard for a class ''C'' is called C-hard, e.g. NP-hard. Normally, it is assumed that the reduction in question does not have higher computational co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
EXPSPACE
In computational complexity theory, is the set of all decision problems solvable by a deterministic Turing machine in exponential space, i.e., in O(2^) space, where p(n) is a polynomial function of n. Some authors restrict p(n) to be a linear function, but most authors instead call the resulting class . If we use a nondeterministic machine instead, we get the class , which is equal to by Savitch's theorem. A decision problem is if it is in , and every problem in has a polynomial-time many-one reduction to it. In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer. problems might be thought of as the hardest problems in . is a strict superset of , , and and is believed to be a strict superset of . Formal definition In terms of and , :\mathsf = \bigcup_ \mathsf\left(2^\right) = \bigcup_ \mathsf\left(2^\right) Examples of problems An example of an problem is the problem of recognizing wheth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syntactic Sugar
In computer science, syntactic sugar is syntax within a programming language that is designed to make things easier to read or to express. It makes the language "sweeter" for human use: things can be expressed more clearly, more concisely, or in an alternative style that some may prefer. Syntactic sugar is usually a shorthand for a common operation that could also be expressed in an alternate, more verbose, form: The programmer has a choice of whether to use the shorter form or the longer form, but will usually use the shorter form since it is shorter and easier to type and read. For example, many programming languages provide special syntax for referencing and updating array elements. Abstractly, an array reference is a procedure of two arguments: an array and a subscript vector, which could be expressed as get_array(Array, vector(i,j)). Instead, many languages provide syntax such as Array ,j/code>. Similarly an array element update is a procedure consisting of three arguments, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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: * if p ∈ ''AP'' then p is an LTL formula; * if Ï ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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: * if p ∈ ''AP'' then p is an LTL formula; * if Ï ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal (model Checking)
In model checking, a subfield of computer science, a signal or timed state sequence is an extension of the notion of words in a formal language, in which letters are continuously emitted. While a word is traditionally defined as a function from a set of non-negative integers to letters, a signal is a function from a set of real numbers to letters. This allow the use of formalisms similar to the ones of automata theory to deal with continuous signals. Example Consider an elevator. What is formally called a letter could be in fact information such as "someone is pressing the button on the 2nd floor", or "the doors are currently open on the third floor". In this case, a signal indicates, at each time, which is the current state of the elevator and its buttons. The signal can then be analyzed using formal methods to check whether a property such that "each time the elevator is called, it arrives in less than three minutes, assuming that no one held the door for more than fifteen sec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
