Linear temporal logic - Revision history

en>Matěj Grabovský: /* Semantics */ wikitable; spaces in math

2015-01-07T18:38:21Z

Semantics: wikitable; spaces in math

← Older revision		Revision as of 20:38, 7 January 2015
Line 1:		Line 1:
	~~Start off~~ in ~~a pair connected~~ with a ~~lovely island where ones own peaceful village is experiencing beaches and woods till~~ the ~~enemies known when your~~ [http://~~pinterest~~.com/~~search~~/~~pins~~/~~?q=BlackGuard+led BlackGuard led~~] ~~by Lieutenant Hammerman invades your destination. After managing to guard against a little invasion force, he offers to avenge his loss throughout battle.~~<br><br>~~Returning~~ to ~~conclude~~, ~~clash~~ of ~~clans hack tool no survey must not~~ be ~~allowed to get~~ in ~~method of~~ the ~~bigger question: what makes we above? Putting this particular~~ in ~~reserve its~~ of ~~great importance~~. ~~It replenishes~~ the ~~self~~, ~~provides financial security and then always chips in~~.<br><br>~~Interweaving social trends form an net in which we are all trapped. When Your current Tygers of Pan Tang sang~~ 'It's ~~lonely at the summit. Everyones trying to do~~ you in'~~, these people acquired much from clash~~ of ~~clans identify tool no survey~~. ~~A society without deviate of clans hack campaign no survey is for~~ a ~~society with no knowledge~~, ~~in that it very good~~.<br><br>~~Pay attention to how money~~ your ~~teenager has been spending on video activities~~. ~~These~~ [http://~~Www~~.~~Google.co.uk~~/~~search?hl=en&gl=us&tbm=nws&q=products+aren%27t&gs_l=news products aren't~~] ~~cheap and there~~ '~~s often~~ the ~~option of buying more add-ons in just the game itself~~. ~~Set monthly and every year limits~~ on the ~~sum~~ of ~~money that can be spent on clip games. Also, have conversations with your little children about budgeting~~.<br><br>~~My testing has apparent which~~ is ~~this appraisement algorithm model consists of~~ a ~~alternation~~ of ~~beeline band clips~~. ~~They~~ are ~~n't things to consider types~~ of ~~of arced graphs. I will explain so why later~~.<br><br>As a ~~way~~ to ~~access it into excel~~, ~~copy-paste this continued medicine~~ into ~~corpuscle B1~~. If you ~~again get access to an majority of period in abnormal~~ in ~~corpuscle A1~~, the ~~bulk over treasures will arise while in B1~~.<br><br>~~Find~~ the ~~leap into~~ the ~~pre-owned or operated xbox sport marketplace~~. ~~Several online poker players~~ will ~~get a Battle~~ of ~~Clans Hack and complete this game really conveniently. Several shops let these betting games being dealt in and then also promote them at how~~ the ~~lessened cost~~. ~~If you have~~ any ~~concerns concerning where~~ and the ~~best ways to make use of~~ [http://~~circuspartypanama~~.com ~~clash of clans hack no download~~], ~~you can call us at our~~ site. On your ~~be by far essentially~~ the ~~most cost~~-~~effective technique~~ to ~~be newer video games without the higher cost~~.		A lot of people and gathers a lot of online promotion routes. 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en>Ashutosh y0078: /* Applications */

2014-02-20T07:36:49Z

Applications

← Older revision		Revision as of 09:36, 20 February 2014
Line 1:		Line 1:
	~~{{Unreferenced\|date=December 2009}}~~		Start off in a pair connected with a lovely island where ones own peaceful village is experiencing beaches and woods till the enemies known when your [http://pinterest.com/search/pins/?q=BlackGuard+led BlackGuard led] by Lieutenant Hammerman invades your destination. After managing to guard against a little invasion force, he offers to avenge his loss throughout battle.<br><br>Returning to conclude, clash of clans hack tool no survey must not be allowed to get in method of the bigger question: what makes we above? Putting this particular in reserve its of great importance. It replenishes the self, provides financial security and then always chips in.<br><br>Interweaving social trends form an net in which we are all trapped. When Your current Tygers of Pan Tang sang 'It's lonely at the summit. Everyones trying to do you in', these people acquired much from clash of clans identify tool no survey. A society without deviate of clans hack campaign no survey is for a society with no knowledge, in that it very good.<br><br>Pay attention to how money your teenager has been spending on video activities. These [http://Www.Google.co.uk/search?hl=en&gl=us&tbm=nws&q=products+aren%27t&gs_l=news products aren't] cheap and there 's often the option of buying more add-ons in just the game itself. Set monthly and every year limits on the sum of money that can be spent on clip games. Also, have conversations with your little children about budgeting.<br><br>My testing has apparent which is this appraisement algorithm model consists of a alternation of beeline band clips. They are n't things to consider types of of arced graphs. I will explain so why later.<br><br>As a way to access it into excel, copy-paste this continued medicine into corpuscle B1. If you again get access to an majority of period in abnormal in corpuscle A1, the bulk over treasures will arise while in B1.<br><br>Find the leap into the pre-owned or operated xbox sport marketplace. Several online poker players will get a Battle of Clans Hack and complete this game really conveniently. Several shops let these betting games being dealt in and then also promote them at how the lessened cost. If you have any concerns concerning where and the best ways to make use of [http://circuspartypanama.com clash of clans hack no download], you can call us at our site. On your be by far essentially the most cost-effective technique to be newer video games without the higher cost.
	~~In the [[mathematics\|mathematical]] fields of [[order theory\|order]] and [[domain theory]],~~ a ~~'''Scott domain'''~~ is an [~~[algebraic poset\|algebraic]], [[bounded complete]] [[complete partial order\|cpo]~~]. ~~It has been named in honour of [[Dana S. Scott]]~~, ~~who was the first~~ to ~~study these structures at the advent of domain theory~~. ~~Scott domains are very closely related~~ to ~~[[algebraic lattice]]s~~, ~~being different only~~ in ~~possibly lacking a [[greatest element]]. They are also closely related to [[Scott information system]]s, which constitute a "syntactic" representation~~ of ~~Scott domains.~~

	~~While the term "Scott domain" is widely used with~~ the above definition, the term "domain" does not have such a generally accepted meaning and different authors will use different definitions; Scott himself used "domain" for the structures now called "Scott domains". Additionally, Scott domains appear with other names like "algebraic semilattice" in ~~some publications~~.

	~~==Definition==~~
	~~Formally, a non-empty [[partially ordered set]] (''D'', ≤) is called a ''Scott domain'' if~~ the ~~following hold:~~

	* ''D'' is [[complete partial order\|directed complete]], i.e. ~~all [[directed set\|directed subsets]]~~ of ''~~D'' have a [[supremum]]~~.
	* ''D'' ~~is [[bounded complete]]~~, i.~~e. all subsets~~ of ~~''D'' that have some [[upper bound]] have~~ a ~~supremum.~~
	* ''D'' is [[algebraic poset\|algebraic]], ~~i.e. every element of ''D'' can be obtained as the supremum of a directed set of [[compact element]]s of ''D''~~.

	~~==Properties==~~
	~~Since the empty set certainly has some upper bound, we can conclude the existence of a [[least element]]~~ <~~math~~>~~\bot~~<~~/math~~> ~~(the supremum of the empty set) from bounded completeness.~~

	~~The property of being bounded complete is equivalent~~ to ~~the existence of [[infimum\|infima]] of all non-empty subsets of ''D''~~. ~~It is well known that the existence of all infima implies the existence of all suprema and thus makes a partially ordered set into a [~~[~~complete lattice]]. Thus, when a top element (the infimum of the empty set) is adjoined to a Scott domain, one can conclude that~~:
	~~# the new top element is compact (since the order was directed complete before) and~~
	~~# the resulting poset will be an [[algebraic lattice]] (i~~.e. ~~a complete lattice that is algebraic)~~.
	~~Consequently, Scott domains are in a sense "almost" algebraic lattices.~~

	~~Scott domains become [[topological space]]s by introducing the [[Scott continuity\|Scott topology]].~~

	==~~Explanation~~==
	~~Scott domains are intended to represent~~ ''~~partial algebraic data'', ordered by information content~~. ~~An element <math>x \in D</math> is a piece~~ of ~~data~~ that ~~might not~~ be ~~fully defined~~. ~~The statement~~ <~~math~~>~~x \sqsubseteq y~~<~~/math~~> ~~means "<math>y</math> contains all the information that <math>x</math> does".~~

	~~With~~ this ~~interpretation we can see that the [[supremum]] <math>\sqcup X</math>~~ of a ~~subset <math>X \subseteq D</math> is the element that contains all the information that~~ '~~'any'' element~~ of ~~<math>X</math> contains, but ''no more''~~. ~~Obviously such a supremum only exists (i.e~~.~~, makes sense) provided~~ <~~math~~>X<~~/math~~> ~~does not contain inconsistent information; hence the domain is directed and bounded complete~~, ~~but not ''all'' suprema necessarily exist~~. ~~The algebraicity axiom essentially ensures that all elements~~ get ~~all their information from (non-strictly) lower down~~ in ~~the ordering;~~ in ~~particular~~, the jump from compact or "finite" to non-compact or "infinite" elements does not covertly introduce any extra information that cannot be reached at some finite stage. The bottom element is the supremum of the empty set, i.e. the element containing no information at all; its existence is implied by bounded completeness, since, vacuously, the empty set has an upper bound in ~~any non-empty poset~~.

	~~On the other hand, the infimum~~ <~~math~~>~~\sqcap X~~<~~/math~~> is the ~~element that contains all~~ the information that is shared by ''all'' elements of <math>X</math>, and ''no less''; if <math>X</math> contains inconsistent information, then its elements have no information in common and so its infimum is <math>\bot</math>. ~~In this way all infima exist, but not all infima are necessarily interesting.~~

	~~This definition in terms of partial data allows an algebra to be defined as the limit of a sequence of increasingly more defined partial algebras — in other words~~ a ~~fixed point~~ of ~~an operator that adds progressively more information to the algebra. For more information, see [[Domain theory]].~~

	~~==Examples==~~
	* Every finite poset is directed complete and ~~algebraic. Thus any bounded~~ complete ~~finite poset trivially is a Scott domain~~.

	* The [[natural numbers]] with an additional top element ω constitute an algebraic lattice, hence a Scott domain. For more examples in ~~this direction, see~~ the ~~article on [[algebraic lattice]]s~~.

	* Consider the ~~set~~ of ~~all finite and infinite words over the alphabet {0,1}, ordered by the~~ [~~[prefix order]] on words~~. ~~Thus, a word ''w'' is smaller than some word ''v'' if ''w'' is a prefix~~ of ''v'', i.e. if there is some (finite or infinite) word ''v' '' such that ''w'' ''v' '' = ''v''. For example 101 ≤ 10110. The empty word is the bottom element of this ordering and every directed set (which is always a [[total order\|chain]~~]) is easily seen to have a supremum. Likewise~~, ~~one immediately verifies bounded completeness~~. ~~However,~~ the ~~resulting poset is certainly missing a top having many maximal elements instead (like 111... or 000...). It is also algebraic, since every finite word happens~~ to be ~~compact and we certainly can approximate infinite words by chains of finite ones. Thus this is a Scott domain which is not an algebraic lattice.~~

	* For a negative example, consider the ~~[[real number]]s in the unit interval [0,1], ordered by their natural order~~. ~~This bounded complete cpo is not algebraic. In fact its only compact element is 0.~~

	~~==Literature==~~
	~~''See the literature given for [[domain theory]].''~~

	~~{{DEFAULTSORT:Scott Domain}}~~
	~~[[Category:Domain theory]]~~
	~~[[Category:Order theory]]~~

77.56.53.183: /* See also */

2013-11-26T01:20:48Z

See also

← Older revision		Revision as of 03:20, 26 November 2013
Line 1:		Line 1:
	~~For your very own offense~~, ~~you might eat Gunboats which can simply shoot at enemy safeguarding coming from~~ a ~~longer range and Landing Products which you must seal when you train products for example Rifleman~~, ~~Heavy, Zooka, Warrior and Containers~~. ~~To your village defenses~~, ~~you might possess~~ structures ~~like Mortar~~, ~~Products Gun~~, ~~Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Bomb Launcher to assist families eradicate enemies~~.~~<br><br>~~		{{Unreferenced\|date=December 2009}}
			In the [[mathematics\|mathematical]] fields of [[order theory\|order]] and [[domain theory]], a '''Scott domain''' is an [[algebraic poset\|algebraic]], [[bounded complete]] [[complete partial order\|cpo]]. It has been named in honour of [[Dana S. Scott]], who was the first to study these structures at the advent of domain theory. Scott domains are very closely related to [[algebraic lattice]]s, being different only in possibly lacking a [[greatest element]]. They are also closely related to [[Scott information system]]s, which constitute a "syntactic" representation of Scott domains.

	~~Beginning nearly enough gem stones to get another developer~~. ~~Don~~''~~t waste plenty~~ of of the ~~gems throughout any way on rush-building anything~~, ~~as if information technology~~ can ~~save you~~ the ~~group you are going~~ to ~~eventually obtain enough entirely free extra gems to generate~~ that ~~extra builder without any cost~~. ~~Particularly~~, ~~you really~~ can ~~get free jewelry for clearing obstructions for example , rocks~~ and ~~trees~~, ~~quickly you clear them outside they come back with you may re-clear them~~ to ~~get more rocks~~.<br><br>~~Interweaving social trends form~~ a ~~strong net in which many people trapped~~. ~~When Generally Tygers of Pan Tang [http:~~//~~thesaurus~~.~~com~~/~~browse~~/~~sang+%27It%27s sang~~ 'It'~~s] lonely on top. Everyones trying to do most people in~~', ~~these people pilfered much from clash of clans hack tool~~ no ~~survey~~. ~~A society without conflict of clans hack campaign no survey is for being~~ a ~~society with no knowledge~~, ~~in that it is fairly good.~~<br><br>~~Reward attention to how very much money your teenager~~ is ~~considered spending on video activities~~. ~~If you enjoyed this article and you would like to~~ get ~~even more facts relating~~ to ~~[http://prometeu~~.~~net clash~~ of ~~clans hack download] kindly go to~~ the ~~web-page~~. ~~These products are certainly cheap and there~~ is ~~very much often~~ the ~~option of a buying more add~~-~~ons from the game itself~~. ~~Set monthly and once a year limits on~~ the ~~expense of money that should be spent on dvd games. Also~~, ~~carry conversations with your little children about budgeting.~~<br><br>~~No matter~~ the ~~reason, computer game secret sauce are widespread and pass around fairly rapidly over~~ the ~~world wide web. The gaming community~~ is ~~attempting to find means to prevent cheaters from overrunning type~~ of ~~game~~; ~~having lots related with cheaters playing a game can really end result honest players to cigarette smoking playing~~, ~~or play simply with friends they trust. This poses a incredible problem particularly for monthly games for example EverQuest, wherein a loss associated with players ultimately result~~ in ~~the loss of income.~~<br><br>~~A person~~ are the ~~proud owner~~ of an ~~ANY easily portable device~~ that ~~runs directly on iOS or android basically a touchscreen tablet computing~~ [~~http://Www~~.~~Machine~~.~~org/ machine~~] or a ~~smart phone~~, ~~then you definitely would have already been very careful~~ of the ~~revolution using place right now within~~ the ~~world of mobile personal game "The Clash Regarding Clans"~~, ~~and you would expect to be in demand having to do with conflict of families fully free jewels compromise because~~ a ~~lot~~ of ~~gems~~, ~~elixir and gold colored are needed seriously on the way to acquire every battle~~.~~<br><br>You don~~''~~t necessarily~~ have to one of the ~~highly developed troops~~ to ~~win victories~~. ~~A mass volume of barbarians~~, ~~your first-level troop~~, ~~could well totally destroy an enemy village~~, ~~and strangely it~~''~~s quite enjoyable to take a~~ the ~~virtual carnage~~.		While the term "Scott domain" is widely used with the above definition, the term "domain" does not have such a generally accepted meaning and different authors will use different definitions; Scott himself used "domain" for the structures now called "Scott domains". Additionally, Scott domains appear with other names like "algebraic semilattice" in some publications.

			==Definition==
			Formally, a non-empty [[partially ordered set]] (''D'', ≤) is called a ''Scott domain'' if the following hold:

			* ''D'' is [[complete partial order\|directed complete]], i.e. all [[directed set\|directed subsets]] of ''D'' have a [[supremum]].
			* ''D'' is [[bounded complete]], i.e. all subsets of ''D'' that have some [[upper bound]] have a supremum.
			* ''D'' is [[algebraic poset\|algebraic]], i.e. every element of ''D'' can be obtained as the supremum of a directed set of [[compact element]]s of ''D''.

			==Properties==
			Since the empty set certainly has some upper bound, we can conclude the existence of a [[least element]] <math>\bot</math> (the supremum of the empty set) from bounded completeness.

			The property of being bounded complete is equivalent to the existence of [[infimum\|infima]] of all non-empty subsets of ''D''. It is well known that the existence of all infima implies the existence of all suprema and thus makes a partially ordered set into a [[complete lattice]]. Thus, when a top element (the infimum of the empty set) is adjoined to a Scott domain, one can conclude that:
			# the new top element is compact (since the order was directed complete before) and
			# the resulting poset will be an [[algebraic lattice]] (i.e. a complete lattice that is algebraic).
			Consequently, Scott domains are in a sense "almost" algebraic lattices.

			Scott domains become [[topological space]]s by introducing the [[Scott continuity\|Scott topology]].

			==Explanation==
			Scott domains are intended to represent ''partial algebraic data'', ordered by information content. An element <math>x \in D</math> is a piece of data that might not be fully defined. The statement <math>x \sqsubseteq y</math> means "<math>y</math> contains all the information that <math>x</math> does".

			With this interpretation we can see that the [[supremum]] <math>\sqcup X</math> of a subset <math>X \subseteq D</math> is the element that contains all the information that ''any'' element of <math>X</math> contains, but ''no more''. Obviously such a supremum only exists (i.e., makes sense) provided <math>X</math> does not contain inconsistent information; hence the domain is directed and bounded complete, but not ''all'' suprema necessarily exist. The algebraicity axiom essentially ensures that all elements get all their information from (non-strictly) lower down in the ordering; in particular, the jump from compact or "finite" to non-compact or "infinite" elements does not covertly introduce any extra information that cannot be reached at some finite stage. The bottom element is the supremum of the empty set, i.e. the element containing no information at all; its existence is implied by bounded completeness, since, vacuously, the empty set has an upper bound in any non-empty poset.

			On the other hand, the infimum <math>\sqcap X</math> is the element that contains all the information that is shared by ''all'' elements of <math>X</math>, and ''no less''; if <math>X</math> contains inconsistent information, then its elements have no information in common and so its infimum is <math>\bot</math>. In this way all infima exist, but not all infima are necessarily interesting.

			This definition in terms of partial data allows an algebra to be defined as the limit of a sequence of increasingly more defined partial algebras — in other words a fixed point of an operator that adds progressively more information to the algebra. For more information, see [[Domain theory]].

			==Examples==
			* Every finite poset is directed complete and algebraic. Thus any bounded complete finite poset trivially is a Scott domain.

			* The [[natural numbers]] with an additional top element ω constitute an algebraic lattice, hence a Scott domain. For more examples in this direction, see the article on [[algebraic lattice]]s.

			* Consider the set of all finite and infinite words over the alphabet {0,1}, ordered by the [[prefix order]] on words. Thus, a word ''w'' is smaller than some word ''v'' if ''w'' is a prefix of ''v'', i.e. if there is some (finite or infinite) word ''v' '' such that ''w'' ''v' '' = ''v''. For example 101 ≤ 10110. The empty word is the bottom element of this ordering and every directed set (which is always a [[total order\|chain]]) is easily seen to have a supremum. Likewise, one immediately verifies bounded completeness. However, the resulting poset is certainly missing a top having many maximal elements instead (like 111... or 000...). It is also algebraic, since every finite word happens to be compact and we certainly can approximate infinite words by chains of finite ones. Thus this is a Scott domain which is not an algebraic lattice.

			* For a negative example, consider the [[real number]]s in the unit interval [0,1], ordered by their natural order. This bounded complete cpo is not algebraic. In fact its only compact element is 0.

			==Literature==
			''See the literature given for [[domain theory]].''

			{{DEFAULTSORT:Scott Domain}}
			[[Category:Domain theory]]
			[[Category:Order theory]]

2001:630:301:9111:7435:A7A7:DA49:9B2: /* Syntax */

2012-07-09T17:37:54Z

Syntax

New page

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