en>Tomruen: 5-polytope

2014-12-27T15:08:58Z

5-polytope

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	~~{{mergeto\|Model checking\|date=January 2011}}~~		Ed is what people call me and my wife doesn't like it at all. For years he's been residing in Mississippi and he doesn't strategy on altering it. Since he was eighteen he's been working as an information officer but he plans on altering it. To climb is some thing I really appreciate doing.<br><br>My homepage - online reader ([http://www.sirudang.com/siroo_Notice/2110 www.sirudang.com])

	~~'''Fair computational tree logic'~~'~~' is conventional [[computational tree logic]] studied with explicit fairness constraints.~~

	~~==Weak fairness / justice==~~
	~~This declares conditions such as all processes are executing infinitely often. If you consider the processes to be P<sub>i</sub>, then the condition becomes:~~

	~~:<math>\bigwedge GFP_{i}</math>~~

	~~==Strong fairness / compassion==~~
	~~Here, if a process is requesting a resource infinitely often (R),~~ it ~~should be allowed to get the resource (C) infinitely often:~~

	~~:<math>\bigwedge( GFR \longrightarrow GFC)</math>~~

	~~==Model checking for fair CTL==~~
	~~If you consider a Kripke Model, the fair Kripke Model has a set of States F. A path <math>\pi = s_o, s_1 \dots</math> is considered a fair path, if and~~
	~~only if the path includes~~ all ~~members of F infinitely often~~. ~~<br/>~~
	~~Fair CTL model checking restricts the checks to only fair paths. There are two kinds:~~

	~~:1. M<sub>f</sub>,~~s~~<sub>i</sub> \|= A<math>\phi</math> if~~ and ~~only if <math>\phi</math> holds in ALL fair paths~~.

	~~: 2. M<sub>f</sub>,~~s~~<sub>i</sub> \|= E<math>\phi</math> if and only if <math>\phi</math> holds in one or more fair paths~~.

	~~A fair state~~ is ~~one from which at least one fair path originates~~. ~~This translates to M~~<~~sub~~>f<~~/sub~~>~~,s \|= EGtrue.~~

	~~==SCC~~-~~based approach==~~
	~~The~~ [[strongly connected component]] (SCC) of a directed graph is a maximally connected graph - all the nodes are reachable from each other. A fair SCC is one that has an edge into at least one node for each of the fair conditions.

	~~To check for fair EG for any formula,~~

	~~# Compute what is called the ''denotation'' of the formula. Basically all the states such that M, s \|= formula.~~
	~~# restrict the model to the denotation.~~
	~~# Find the fair SCC.~~
	~~# Obtain the union of all 3(above).~~
	~~# Compute the states that can reach the union.~~

	~~==Emerson Lei algorithm==~~
	~~The fix point characterization of Exist Globally is given by~~: ~~[EGφ] = νZ .([φ] ∩ [EXZ ]), which is basically the limit applied according to Kleene's theorem~~. ~~To fair paths, it becomes [Ef Gφ] = νZ~~ .~~([φ] ∩<sub>Fi ∈FT<~~/sub> [EX[E(Z U(Z ∧ Fi ))]) which means the formula holds in the current state and the next states and the next to next states until it meets all the members of the fair conditions. This means that, the condition is equivalent to a sort of accepting point where the accepting condition is the entire set of Fair conditions.

	~~==References==~~
	* {{cite journal \| author=Emerson, E. A. and Halpern, J. Y. \| title=Decision procedures and expressiveness in the temporal logic of branching time \| journal=Journal of Computer and System Sciences\| year=1985\| volume=30 \| issue=1 \| pages=1–24 \| doi=10.1016/~~0022-0000(85)90001-7}}~~
	* {{cite journal \| author=Clarke, E. M., Emerson, E. A., and Sistla, A. P. \| title=Automatic verification of finite-state concurrent systems using temporal logic specifications \| journal=ACM Transactions on Programming Languages and Systems\| year=1986\| volume=8 \| issue=2 \| pages=244–263 \| doi=10.~~1145/5397~~.~~5399}}~~

	~~[[Category:Temporal logic]~~]

en>Double sharp: /* References */

2012-08-11T14:11:21Z

References

New page

{{mergeto|Model checking|date=January 2011}}

'''Fair computational tree logic''' is conventional [[computational tree logic]] studied with explicit fairness constraints.

==Weak fairness / justice==
This declares conditions such as all processes are executing infinitely often. If you consider the processes to be Pi, then the condition becomes:

:<math>\bigwedge GFP_{i}</math>

==Strong fairness / compassion==
Here, if a process is requesting a resource infinitely often (R), it should be allowed to get the resource (C) infinitely often:

:<math>\bigwedge( GFR \longrightarrow GFC)</math>

==Model checking for fair CTL==
If you consider a Kripke Model, the fair Kripke Model has a set of States F. A path <math>\pi = s_o, s_1 \dots</math> is considered a fair path, if and
only if the path includes all members of F infinitely often. 
Fair CTL model checking restricts the checks to only fair paths. There are two kinds:

:1. Mf,si |= A<math>\phi</math> if and only if <math>\phi</math> holds in ALL fair paths.

: 2. Mf,si |= E<math>\phi</math> if and only if <math>\phi</math> holds in one or more fair paths.

A fair state is one from which at least one fair path originates. This translates to Mf,s |= EGtrue.

==SCC-based approach==
The [[strongly connected component]] (SCC) of a directed graph is a maximally connected graph - all the nodes are reachable from each other. A fair SCC is one that has an edge into at least one node for each of the fair conditions.

To check for fair EG for any formula,

# Compute what is called the ''denotation'' of the formula. Basically all the states such that M, s |= formula.
# restrict the model to the denotation.
# Find the fair SCC.
# Obtain the union of all 3(above).
# Compute the states that can reach the union.

==Emerson Lei algorithm==
The fix point characterization of Exist Globally is given by: [EGφ] = νZ .([φ] ∩ [EXZ ]), which is basically the limit applied according to Kleene's theorem. To fair paths, it becomes [Ef Gφ] = νZ .([φ] ∩Fi ∈FT [EX[E(Z U(Z ∧ Fi ))]) which means the formula holds in the current state and the next states and the next to next states until it meets all the members of the fair conditions. This means that, the condition is equivalent to a sort of accepting point where the accepting condition is the entire set of Fair conditions.

==References==
* {{cite journal | author=Emerson, E. A. and Halpern, J. Y. | title=Decision procedures and expressiveness in the temporal logic of branching time | journal=Journal of Computer and System Sciences| year=1985| volume=30 | issue=1 | pages=1–24 | doi=10.1016/0022-0000(85)90001-7}}
* {{cite journal | author=Clarke, E. M., Emerson, E. A., and Sistla, A. P. | title=Automatic verification of finite-state concurrent systems using temporal logic specifications | journal=ACM Transactions on Programming Languages and Systems| year=1986| volume=8 | issue=2 | pages=244–263 | doi=10.1145/5397.5399}}

[[Category:Temporal logic]]

5-simplex - Revision history

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en>Double sharp: /* References */