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| In [[computational complexity theory]], non-deterministic space or '''NSPACE''' is the [[computational resource]] describing the [[memory space]] for a [[non-deterministic Turing machine]]. It is the non-deterministic counterpart of '''[[DSPACE]]'''.
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| ==Complexity classes==
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| The measure '''NSPACE''' is used to define the [[complexity class]] whose solutions can be determined by a [[non-deterministic Turing machine]]. The [[complexity class]] '''NSPACE(''f''(''n''))''' is the set of [[decision problem]]s that can be solved by a [[non-deterministic Turing machine]], ''M'', using space ''O''(''f''(''n'')), where ''f''(''n'') is the maximum number of tape cells that ''M'' scans on any input of length ''n''.<ref>{{cite book|last=Sipser|first=Michael|title=Introduction to the Theory of Computation (2nd ed.)|year=2006|publisher=Course Technology|isbn=978-0-534-95097-2|pages=303–304}}</ref>
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| Several important complexity classes can be defined in terms of '''NSPACE'''. These include:
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| * '''[[regular language|REG]]''' = '''DSPACE'''(''O''(1)) = '''NSPACE'''(''O''(1)), where '''REG''' is the class of [[regular language]]s (nondeterminism does not add power in constant space).
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| * '''[[NL (complexity)|NL]]''' = '''NSPACE'''(''O''(log ''n''))
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| * '''[[context-sensitive language|CSL]]''' = '''NSPACE'''(''O''(''n'')), where '''CSL''' is the class of [[context-sensitive language]]s.
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| * '''[[PSPACE]]''' = '''NPSPACE''' = <math>\bigcup_{k\in\mathbb{N}} \mbox{NSPACE}(n^k)</math>
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| * '''[[EXPSPACE]]''' = '''NEXPSPACE''' = <math>\bigcup_{k\in\mathbb{N}} \mbox{NSPACE}(2^{n^k})</math>
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| The [[Immerman–Szelepcsényi theorem]] states that '''NSPACE'''(''s''(''n'')) is closed under complement for every function {{nowrap|''s''(''n'') ≥ log ''n''.}} | |
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| A further generalization is '''ASPACE''', defined with [[alternation (complexity)|alternating Turing machines]].
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| ==Relation with other complexity classes==
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| ===DSPACE===
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| '''NSPACE''' is the non-deterministic counterpart of '''[[DSPACE]]''', the class of [[memory space]] on a [[deterministic Turing machine]]. By [[Savitch's theorem]], we have that:
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| <blockquote>
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| <math>\mbox{DSPACE}[s(n)] \subseteq \mbox{NSPACE}[s(n)] \subseteq \mbox{DSPACE}[(s(n))^2].</math>
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| </blockquote> | |
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| ===Time===
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| '''NSPACE''' can also be used to determine the time complexity of a [[deterministic Turing machine]] by the following theorem:
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| <blockquote>
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| If a language ''L'' is decided in space ''S(n)'' (where ''S(n)'' ≥ log ''n'') by a non-deterministic TM, then there exists a constant ''C'' such that ''L'' is decided in time ''O''(''C''<sup>''S''(''n'')</sup>) by a deterministic one.<ref>{{cite book|last=Goddard|first=Wayne|title=Introducing the Theory of Computation|year=2008|publisher=Jones and Bartlett Publishers, Inc.|isbn=978-0-7637-4125-9|pages=183}}</ref>
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| </blockquote>
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| ==Limitations==
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| The measure of [[space complexity]] in terms of '''[[DSPACE]]''' is useful because it represents the total amount of memory that an actual computer would need to solve a given [[computational problem]] with a given [[algorithm]]. The reason is that '''DSPACE''' describes the space complexity used by [[deterministic Turing machine]]s, which can represent actual computers. On the other hand, '''NSPACE''' describes the space complexity of [[non-deterministic Turing machine]]s, which are not useful when trying to represent actual computers. For this reason, '''NSPACE''' is limited in its usefulness to real-world applications.
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| ==References==
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| {{Reflist}}
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| ==External links==
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| {{ComplexityZoo|NSPACE(''f''(''n''))|N#nspace}}.
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| {{ComplexityClasses}}
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| {{DEFAULTSORT:Nspace}}
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| [[Category:Complexity classes]]
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| [[Category:Computational resources]]
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| {{Comp-sci-theory-stub}}
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