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<title>Configuration graph</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Configuration graph</span></span>

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<p><b>Configuration graphs</b> are a theoretical tool used in <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a> to prove a relation between <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> <a href="Reachability" title="Reachability">reachability</a> and <a href="Computational_complexity_theory#Complexity_classes" title="Computational complexity theory">complexity classes</a>.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>A theoretical computational model, like <a href="Turing_machine" title="Turing machine">Turing machine</a> or <a href="Finite_automata" class="mw-redirect" title="Finite automata">finite automata</a>, 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 <i>configuration</i>, also called an <i>instantaneous description</i> (<i>ID</i>), 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 <a href="Labeled_graph" class="mw-redirect" title="Labeled graph">labeled graph</a> 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 corresponds to a computational step of the model.

</p><p>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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<div class="mw-heading mw-heading2"><h2 id="Useful_property">Useful property</h2></div>

<p>If there exists exactly one initial state, then a computation is deterministic if and only if from any configuration there is at most one possible step, so if and only if the graph is of out-degree 1.

</p><p>Once a dummy initial vertex with an edge to every initial vertex and a dummy accepting vertex with an edge from every accepting vertex are added, 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 <a href="Reachability" title="Reachability">reachability</a> problem.

</p><p>A computation is said to be unambiguous if there exists at most one path from an initial vertex to an accepting vertex.

</p><p>A cycle in the graph corresponds to an infinite loop in the computation.

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<div class="mw-heading mw-heading2"><h2 id="Size_of_the_graph">Size of the graph</h2></div>

<p>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.

</p><p>It is also possible to have finite graphs: on <a href="Deterministic_finite_automaton" title="Deterministic finite automaton">Deterministic finite automaton</a> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> states, for a given word of size <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> the configuration is composed of the position of the head and the current state. So the graph is of size <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)s}">

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<annotation encoding="application/x-tex">{\displaystyle (n+1)s}</annotation>

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</math></span><img src="./0983a21635d6bd320ecbec895c2b8c34dba8c217.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.297ex; height:2.843ex;" alt="{\displaystyle (n+1)s}" loading="lazy"></span>, and the accessible part from the initial state is of size <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

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<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

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</math></span><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Use_of_this_object">Use of this object</h2></div>

<p>This notion is useful because it reduces computational problems to graph <a href="Reachability" title="Reachability">reachability</a> problems.

</p><p>For example, since reachability is in <a href="NL_(complexity)" title="NL (complexity)">NL</a> when we can represent configurations in space which is logarithmic in the size of the input, and since the configuration of a Turing Machine in <a href="NL_(complexity_class)" class="mw-redirect" title="NL (complexity class)">NL</a> is indeed of logarithmic size, it follows that graph-reachability is <a href="Complete_(complexity)" title="Complete (complexity)">complete</a> for NL.<sup id="cite_ref-papa_1-0" class="reference"><a href="#cite_note-papa-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>In the other direction, it helps to verify the complexity of a computation model; the decision problem for a (deterministic) model whose configurations are of space which is logarithmic in the size of the input is in (<a href="L_(complexity)" title="L (complexity)">L</a>) <a href="NL_(complexity)" title="NL (complexity)">NL</a>. This is for example the case of finite automata and finite automata with one counter.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-papa-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-papa_1-0">^</a></b></span> <span class="reference-text"><a href="Christos_Papadimitriou" title="Christos Papadimitriou">Papadimitriou, Christos H.</a> (1994). <i>Computational Complexity</i>, Reading, Massachusetts: Addison-Wesley. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-53082-1</bdi>.</span>

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<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFAroraBarak2009" class="citation book cs1"><a href="Sanjeev_Arora" title="Sanjeev Arora">Arora, Sanjeev</a>; Barak, Boaz (2009). <i>Computational complexity, a modern approach</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-42426-4</bdi>.</cite> Section 4.3: NL-completeness, p. 87.</li></ul></div><div><div class="zim-footer">

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