Étale cohomology: Difference between revisions

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{{about|transition systems as used in [[operational semantics]]|an [[automata theory|automata]]-theoretic view|semiautomaton}}
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In [[theoretical computer science]], a '''state transition system''' is an [[abstract machine]] used in the study of [[computation]]. The machine consists of a set of [[state (computer science)|states]] and transitions between states, which may be labeled with labels chosen from a set; the same label may appear on more than one transition. If the label set is a [[Singleton (mathematics)|singleton]], the system is essentially unlabeled, and a simpler definition that omits the labels is possible.
 
State transition systems coincide mathematically with [[abstract rewriting system]]s (as explained further in this article). State transition systems differ however from [[finite state automata]] in several ways:
 
* In a state transition system the set of states is not necessarily finite, or even countable.
* In a state transition system the set of transitions is not necessarily finite, or even countable.
* A finite-state automaton distinguishes a special "start" state and a set of special "final" states.
 
State transition systems can be represented as [[directed graph]]s.
 
== Formal definition ==
 
Formally, a state transition system is a pair (''S'', &rarr;) where ''S'' is a set (of states) and &rarr; &sube; ''S'' &times; ''S'' is a [[binary relation]] over ''S'' (of transitions). If ''p'', ''q'' &isin; ''S'', (''p'', ''q'') &isin; &rarr; is usually written as ''p'' &rarr; ''q''. This represents the fact that there is a transition from state ''p'' to state ''q''.
 
A labelled transition system is a tuple (''S'', &Lambda;, &rarr;) where ''S'' is a set (of states), &Lambda; is a set (of labels) and &rarr; &sube; ''S'' &times; &Lambda; &times; ''S'' is a [[ternary relation]] (of labelled transitions). If ''p'', ''q'' &isin; ''S'' and &alpha; &isin; &Lambda;, then (''p'',&alpha;,''q'') &isin; &rarr; is written as
 
: <math>
p \overset{\alpha}{\rightarrow} q. \,
</math>
 
This represents the fact that there is a transition from state ''p'' to state ''q'' with label &alpha;. Labels can represent different things depending on the language of interest. Typical uses of labels include representing input expected, conditions that must be true to trigger the transition, or actions performed during the transition.
 
If, for any given ''p'' and &alpha;, there exists only a single tuple (''p'',&alpha;,''q'') in &rarr;, then one says that &alpha; is ''deterministic'' (for ''p''). If, for any given ''p'' and &alpha;, there exists at least one tuple (''p'',&alpha;,''q'') in &rarr;, then one says that &alpha; is ''executable'' (for ''p'').
 
== Relation between labelled and unlabelled transition systems. ==
 
There are many relations between these concepts. Some are simple, such as observing that a labelled transition system where the set of labels consists of only one element is equivalent to an unlabelled transition system. However not all these relations are equally trivial.
 
== Comparison with abstract rewriting systems ==
As a mathematical object, an unlabeled state transition system is identical with an (unindexed) [[abstract rewriting system]]. If we consider the rewriting relation as an indexed set of relations, as some authors do, then a labeled state transition system is equivalent to an abstract rewriting system with the indices being the labels. The focus of the study and the terminology are different however. In a state transition system one is interested in interpreting the labels as actions, whereas in an abstract rewriting system the focus is on how objects may be transformed (rewritten) into others.<ref>Marc Bezem, J. W. Klop, Roel de Vrijer ("Terese"), ''Term rewriting systems'', Cambridge University Press, 2003, ISBN 0-521-39115-6. p. 7-8</ref>
 
== Extensions ==
In [[model checking]], a state transition system is sometimes defined to include an additional labeling function for the states as well, resulting in a notion that encompasses that of [[Kripke structure]].<ref name="BaierKatoen2008">{{cite book|author1=Christel Baier|author2=Joost-Pieter Katoen|title=Principles of model checking|publisher=The MIT Press|isbn=978-0-262-02649-9|page=20}}</ref>
 
[[Action language]]s are a special case of transition systems, adding a set of ''fluents'' ''F'', a set of values ''V'', and a function that maps ''F'' &times; ''S'' to ''V''.<ref>Micheal Gelfond, Vladimir Lifschitz (1998) "Action Languages", ''Linköping Electronic Articles in Computer and Information Science'', vol '''3''', nr ''16''.</ref>
 
== See also ==
 
* [[Simulation preorder]]
* [[Bisimulation]]
* [[Operational semantics]]
* [[Kripke structure]]
* [[State Machine]]
 
== References ==
{{reflist}}
 
[[Category:Models of computation]]

Latest revision as of 16:07, 19 December 2014

If you should accelerate your PC then we have come to the appropriate region. I will show we, now, five quick methods that you can drastically improve the computer's performance.

The PC registry begins to receive errors and fragmented the more we utilize the computer because we enter more data each time, as well as create changes in the systems and setup. When the registry starts to receive overloaded and full of errors, your computer usually eventually crash. It can be done to fix it on your own yet pretty dangerous, particularly should you have no extensive experience in doing this. Therefore, do NOT even attempt to do this oneself.

The error is basically a result of issue with Windows Installer package. The Windows Installer is a tool chosen to install, uninstall and repair the most programs on your computer. Let us discuss a few items that helped a lot of individuals that facing the similar issue.

In purchase to remove the programs on the computer, Windows Installer should be inside a healthy state. If its installation is corrupted we may obtain error 1721 inside Windows 7, Vista plus XP throughout the program removal task. Just re-registering its component files would solve the issue.

So to fix this, you merely have to be able to make all the registry files non-corrupted again. This can dramatically speed up the loading time of your computer and may allow you to a large amount of elements on it again. And fixing these files couldn't be easier - you merely have to employ a tool called a tuneup utilities.

Another element is registry. It is one of the most crucial part in a Windows XP, Vista operating systems. When Windows start up, it read associated information from registry and load into computer RAM. This takes up a big part of the startup time. After the data is all loaded, computer runs the startup programs.

The disk needs area in purchase to run smoothly. By freeing up several space from a disk, you will be capable to speed up the PC a bit. Delete all file inside the temporary internet files folder, recycle bin, well-defined shortcuts plus icons from the desktop which we do not use and remove programs we never use.

So in summary, when comparing registry cleaning, ensure that the one you choose offers we the following.A backup plus restore center, quick operation, automatic deletion facility, start-up administration, an effortless way of contact along with a money back guarantee.

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