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| '''Configuration graphs''' are a theoretical tool used in [[computational complexity theory]] to prove a relation between [[Graph (mathematics)|graph]] [[reachability]] and [[Computational_complexity_theory#Complexity_classes|complexity classes]].
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| == Definition ==
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| A theoretical computational model, like [[Turing machine]] or [[finite automata]], explains how to do a computation. The model explains both what is an initial configuration of the machine and which steps can be taken to continue the computation, until we eventually stop. A ''configuration'', also called an ''Instantaneous Description(ID)'' is a finite representation of the machine at a given time. For example, for a finite automata and a given input, the configuration will be the current state and the number of read letters, for a Turing machine it will be the state, the content of the tape and the position of the head. A configuration graph is a directed [[labeled graph]] where the label of the vertices are the possible configurations of the models and where there is an edge from one configuration to another if it correspond to a computational step of the model.
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| The initial and accepting configuration(s) of the machine are special vertices of the configuration graph. The computation accepts if and only if there is a path from an initial vertex to an accepting vertex.
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| ==Useful property==
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| If the computation is deterministic then from any configuration there is at most one possible step, so the graph is of out-degree 1, and there is exactly one initial state.
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| Once we add a dummy initial vertex with an edge to every initial vertex and a dummy accepting vertex with an edge from every accepting vertex, checking if there is an accepting computation only requires to check if there is a path from the initial vertex to the accepting vertex, which is the [[reachability]] problem.
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| A cycle in the graph means that there is a possible infinite loop in the computation.
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| ==Size of the graph==
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| The computational graph can be of infinite size if there are no restrictions on possible configurations; indeed, it is easy to see that there are Turing machines which can reach arbitrarily large configurations.
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| It is also possible to have finite graphs: on [[Deterministic finite automaton]] with <math>s</math> sates, for a given word of size <math>n</math> the configuration is composed of the position of the head and the current state. So the graph is of size <math>(n+1)s</math>, and the accessible part from the intitial state is of size <math>n+1</math>.
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| == Use of this object ==
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| This notion is useful because it reduces computational problems to graph [[reachability]] problems.
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| For example, since [[reachability]] is in [[NL (complexity)|NL]] when we can represent configurations in space which is logarithmic in the size of the input, and since the configuration of the Turing Machine in [[NL (complexity class)|NL]] is indeed of logarithmic size, then graph-reachability is [[Complete (complexity)|complete]] for NL.<ref name="papa">[[Christos Papadimitriou|Papadimitriou, Christos H.]] (1994). ''Computational Complexity'', Reading, Massachusetts: Addison-Wesley. ISBN 0-201-53082-1.</ref>
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| In the other direction, it helps to verify the complexity of a computation model; the decision problem for a (deterministic) model whose configuration are of space which is logarithmic in the size of the input is in ([[L (complexity)|L]]) [[NL (complexity)|NL]]. This is for example the case of finite automata and finite automata with one counter.
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| == References ==
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| {{Reflist}}
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| * {{cite book|author = [[Sanjeev Arora]] and [[Boaz Barak]] | year = 2009 | title = Computational complexity, a modern approach | publisher= Cambridge University Press | isbn = 978-0-521-42426-4}} Section 4.3: NL-completeness, p. 87.
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| {{DEFAULTSORT:Computational Complexity Theory}}
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| [[Category:Computational complexity theory|*]]
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Name: Lorenza Boynton
Age: 27
Country: Germany
City: Ebermannsdorf
ZIP: 92263
Address: Schaarsteinweg 50
Feel free to surf to my page Info size bike mountain bike sizing.