Fair computational tree logic: Difference between revisions

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'''Fair Computationalcomputational tree logic''' is conventional [http://en.wikipedia.org/wiki/Computational_tree_logic Computational[computational tree logic]] studied with explicit fairness constraints.

==Weak fairness / Justicejustice==

This declares conditions such as all processes are executing infinitely often. If you consider the processes to be P_i, then the condition becomes: <br /><math>\bigwedge GFP_{i}</math>

==Strong fairness / Compassion==

Here, if a process is requesting a resource infinitely often (T), it should be allowed to get the resource (C)infinitely often <br />

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

==Strong fairness / Compassioncompassion==

Here, if a process is requesting a resource infinitely often (T), it should be allowed to get the resource (C) infinitely often: <brmath>\bigwedge( GFR \longrightarrow GFC)</math>

==Model checking for fair CTL==

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A fair state is one from which atleast one fair path originates. The is translatable to, M_f,s |= EGtrue <br />

==SCC -based approach==

The [http://en.wikipedia.org/wiki/Strongly_connected_component Strongly[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 atleast one node for each of the fair conditions.

<br /><br />

To check for fair EG for any formula - <br/>

1. Compute what is called the denotation of the formula. Basically all the states such that M, s |= formula. <br />

2. restrict the Model to the denotation. <br />

3. Find the fair SCC <br />

4. Obtain the union of all 3(above). <br />

5. Compute the states that can reach the union. <br/>

To check for fair EG for any formula - <br/>,

1.# Compute what is called the ''denotation'' of the formula. Basically all the states such that M, s |= formula. <br />

2.# restrict the Modelmodel to the denotation. <br />

3.# Find the fair SCC <br />.

4.# Obtain the union of all 3(above). <br />

5.# Compute the states that can reach the union. <br/>

==Emersion Lei Algorithmalgorithm==

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.

The fix point characterization of Exist Globally is given by:- <br/>

[EGφ] = νZ .([φ] ∩ [EXZ ]) , which is basically the limit applied according to Kleene's theorem. <br/>

To fair paths, it becomes - <br/>

[Ef Gφ] = νZ .([φ] ∩_{Fi ∈FT} [EX[E(Z U(Z ∧ Fi ))])<br/>

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}}

* {{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}}

{{Uncategorized|January 2007}}

[[Category:Logic in computer science]]

[[Category:Modal logic]]