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	~~Friends call him Royal Seyler~~. ~~Managing people~~ is ~~what I do~~ in ~~my day occupation~~. ~~Alabama~~ has ~~usually been his home~~. The ~~preferred pastime~~ for ~~him~~ and ~~his kids~~ is ~~to drive~~ and ~~now he~~ is ~~attempting~~ to ~~earn cash~~ with it.<br><br>~~Also visit my webpage~~ :: [http://~~Vozdemujer~~.~~net~~/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~5527~~ http://~~Vozdemujer~~.~~net~~]		'''CTL''' is a superset of [[computational tree logic]] (CTL) and [[linear temporal logic]] (LTL). It freely combines path quantifiers and temporal operators. Like CTL, CTL is a branching time logic. The formal semantics of CTL* formulae are defined with respect to a given [[Kripke structure]].

			== History ==

			LTL has been proposed for the verification of computer programs first by [[Amir Pnueli]] in 1977. Four years later in 1981 [[E. M. Clarke]] and [[E. A. Emerson]] invented CTL and CTL model checking. CTL* was defined by [[E. A. Emerson]] and [[Joseph Halpern\|Joseph Y. Halpern]] in 1986.

			Interestingly, CTL and LTL have been developed independently before CTL. Both sublogics have become very important in the [[model checking]] community, while CTL is not yet of practical importance. This is surprising because the [[computational complexity]] of model checking in CTL* is not worse than that of LTL: they both lie in [[PSPACE]].

			== Syntax ==
			The [[formal language\|language]] of [[well formed formula\|well-formed CTL* formulae]] is generated by the following unambiguous (wrt bracketing) [[context-free grammar]]:

			:<math>\Phi::=\bot \mid \top \mid p \mid (\neg\Phi) \mid (\Phi\and\Phi) \mid (\Phi\or\Phi) \mid
			(\Phi\Rightarrow\Phi) \mid (\Phi\Leftrightarrow\Phi) \mid A\phi \mid E\phi</math>
			:<math>\phi::=\Phi \mid (\neg\phi) \mid (\phi\and\phi) \mid (\phi\or\phi) \mid
			(\phi\Rightarrow\phi) \mid (\phi\Leftrightarrow\phi) \mid X\phi \mid F\phi \mid G\phi \mid [\phi U \phi]
			</math>

			where <math>p</math> ranges over a set of [[atomic formula]]s. Valid CTL*-formulae are built using the nonterminal <math>\Phi</math>. These formulae are called ''state formulae'', while those created by the symbol <math>\phi</math> are called ''path formulae''. (The above grammar contains some redundancies; for example <math>\Phi\or\Phi</math> as well as implication and equivalence can be defined as just for [[Boolean algebras]] (or [[propositional logic]]) from negation and conjunction, and the temporal operators '''X''' and '''U''' are [[Temporal logic#Temporal operators\|sufficient to define the other two]].)

			The operators basically are the same as in [[Computation tree logic#Operators\|CTL]]. However, in CTL, every temporal operator (<math>X, F, G, U</math>) has to be directly preceded by a quantifier, while in CTL* this is not required. The universal path quantifier may be defined in CTL* in the same way as for classical [[predicate calculus]] <math>A\phi = \neg E \neg \phi</math>, although this in not possible in the CTL fragment.

			=== Examples of formulae ===

			* CTL* formula that is neither in LTL or in CTL: <math>EX(p) \and AFG(p)</math>
			* LTL formula that is not in CTL: <math>\ AFG(p)</math>
			* CTL formula that is not in LTL: <math>\ EX(p)</math>
			* CTL* formula that is in CTL and LTL: <math>\ AG(p)</math>

			Remark: When taking LTL as subset of CTL*, any LTL formula is implicitly prefixed with the universal path quantifier <math>\ A</math>

			== Semantics ==
			The semantics of CTL* are defined with respect to some [[Kripke structure]]. As the names imply, state formulae are interpreted with respect to the states of this structure, while path formulae are interpreted over paths on it.

			=== State formulae ===
			If a state <math>s</math> of the Kripke structure satisfies a state formula <math>\Phi</math> it is denoted <math>s\models\Phi</math>. This relation is defined inductively as follows:

			# <math>\Big( (\mathcal{M}, s) \models \top \Big) \land \Big( (\mathcal{M}, s) \not\models \bot \Big)</math>
			# <math>\Big( (\mathcal{M}, s) \models p \Big) \Leftrightarrow \Big( p \in L(s) \Big)</math>
			# <math>\Big( (\mathcal{M}, s) \models \neg\Phi \Big) \Leftrightarrow \Big( (\mathcal{M}, s) \not\models \Phi \Big)</math>
			# <math>\Big( (\mathcal{M}, s) \models \Phi_1 \land \Phi_2 \Big) \Leftrightarrow \Big( \big((\mathcal{M}, s) \models \Phi_1 \big) \land \big((\mathcal{M}, s) \models \Phi_2 \big) \Big)</math>
			# <math>\Big( (\mathcal{M}, s) \models \Phi_1 \lor \Phi_2 \Big) \Leftrightarrow \Big( \big((\mathcal{M}, s) \models \Phi_1 \big) \lor \big((\mathcal{M}, s) \models \Phi_2 \big) \Big)</math>
			# <math>\Big( (\mathcal{M}, s) \models \Phi_1 \Rightarrow \Phi_2 \Big) \Leftrightarrow \Big( \big((\mathcal{M}, s) \not\models \Phi_1 \big) \lor \big((\mathcal{M}, s) \models \Phi_2 \big) \Big)</math>
			# <math>\bigg( (\mathcal{M}, s) \models \Phi_1 \Leftrightarrow \Phi_2 \bigg) \Leftrightarrow \bigg( \Big( \big((\mathcal{M}, s) \models \Phi_1 \big) \land \big((\mathcal{M}, s) \models \Phi_2 \big) \Big) \lor \Big( \neg \big((\mathcal{M}, s) \models \Phi_1 \big) \land \neg \big((\mathcal{M}, s) \models \Phi_2 \big) \Big) \bigg)</math>
			# <math>\Big( (\mathcal{M}, s) \models A\phi \Big) \Leftrightarrow \Big(\pi\models\phi</math> for all paths <math>\ \pi</math> starting in <math>s\Big)</math>
			# <math>\Big( (\mathcal{M}, s) \models E\phi \Big) \Leftrightarrow \Big(\pi\models\phi</math> for some path <math>\ \pi</math> starting in <math>s\Big)</math>

			=== Path formulae ===
			The satisfaction relation <math>\pi\models\phi</math> for path formulae <math>\ \phi</math> and a path <math>\pi = s_0 \to s_1 \to \cdots</math> is also defined inductively. For this, let <math>\ \pi[n]</math> denote the sub-path <math>s_n \to s_{n+1} \to \cdots</math>:

			# <math>\Big( \pi \models \Phi \Big) \Leftrightarrow \Big((\mathcal{M}, s_0) \models \Phi\Big)</math>
			# <math>\Big( \pi \models \neg\phi \Big) \Leftrightarrow \Big( \pi \not\models \phi \Big)</math>
			# <math>\Big( \pi \models \phi_1 \land \phi_2 \Big) \Leftrightarrow \Big( \big(\pi \models \phi_1 \big) \land \big(\pi \models \phi_2 \big) \Big)</math>
			# <math>\Big( \pi \models \phi_1 \lor \phi_2 \Big) \Leftrightarrow \Big( \big(\pi \models \phi_1 \big) \lor \big(\pi \models \phi_2 \big) \Big)</math>
			# <math>\Big( \pi \models \phi_1 \Rightarrow \phi_2 \Big) \Leftrightarrow \Big( \big(\pi \not\models \phi_1 \big) \lor \big(\pi \models \phi_2 \big) \Big)</math>
			# <math>\bigg( \pi \models \phi_1 \Leftrightarrow \phi_2 \bigg) \Leftrightarrow \bigg( \Big( \big(\pi \models \phi_1 \big) \land \big(\pi \models \phi_2 \big) \Big) \lor \Big( \neg \big(\pi \models \phi_1 \big) \land \neg \big(\pi \models \phi_2 \big) \Big) \bigg)</math>
			# <math>\Big( \pi \models X\phi \Big) \Leftrightarrow \Big( \pi[1] \models \phi \Big)</math>
			# <math>\Big( \pi \models F\phi \Big) \Leftrightarrow \Big( \exists n\geqslant 0: \pi[n] \models \phi \Big)</math>
			# <math>\Big( \pi \models G\phi \Big) \Leftrightarrow \Big( \forall n\geqslant 0: \pi[n] \models \phi \Big)</math>
			# <math>\Big( \pi \models [\phi_1U\phi_2] \Big) \Leftrightarrow \Big( \exists n\geqslant 0: \big(\pi[n] \models \phi_2 \land \forall 0\leqslant k < n:~ \pi[k] \models \phi_1 \big)\Big)</math>

			== See also ==

			* [[Temporal logic]]
			* [[Kripke structure]]
			* [[Model checking]]
			* [[Reo Coordination Language]]

			== References ==

			* [[Amir Pnueli]]: The temporal logic of programs. Proceedings of the 18th Annual Symposium on Foundations of Computer Science (FOCS), 1977, 46–57. DOI= 10.1109/SFCS.1977.32
			* [[E. Allen Emerson]], [[Joseph Halpern\|Joseph Y. Halpern]]: "Sometimes" and "not never" revisited: on branching versus linear time temporal logic. J. ACM 33, 1 (Jan. 1986), 151–178. DOI= http://doi.acm.org/10.1145/4904.4999
			* Ph. Schnoebelen: The Complexity of Temporal Logic Model Checking. Advances in Modal Logic 2002: 393–436

			== External links ==
			*[http://www.inf.unibz.it/~artale/FM/slide4.pdf CTL Teaching slides] of professor [[Alessandro Artale]] at the [[Free University of Bozen-Bolzano]]

			[[Category:Logic in computer science]]
			[[Category:Temporal logic]]

14.139.59.131 at 22:04, 18 December 2011

2011-12-18T22:04:16Z

New page

Friends call him Royal Seyler. Managing people is what I do in my day occupation. Alabama has usually been his home. The preferred pastime for him and his kids is to drive and now he is attempting to earn cash with it.<br><br>Also visit my webpage :: [http://Vozdemujer.net/index.php?mod=users&action=view&id=5527 http://Vozdemujer.net]

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en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q4115391

14.139.59.131 at 22:04, 18 December 2011