In computational complexity theory complexity theory , EXPSPACE is the Set mathematics set of all decision problem s solvable by a deterministic Turing machine in big O notation O 2 sup p n sup space, where p n is a polynomial function of n . Some authors restrict p n to be a linear function , but most authors instead call the resulting class ESPACE . If we use a nondeterministic machine instead, we get the class NEXPSPACE , which is equal to EXPSPACE by Savitch s theorem . In terms of DSPACE and NSPACE , math mbox EXPSPACE bigcup k in mathbb N mbox DSPACE 2 n k bigcup k in mathbb N mbox NSPACE 2 n k math A decision problem is EXPSPACE complete if it is in EXPSPACE , and every problem in EXPSPACE has a polynomial time many one reduction to it. In other words, there is a polynomial time algorithm that transforms instances of one to instances of the other with the same answer. EXPSPACE complete problems might be thought of as the hardest problems in EXPSPACE . EXPSPACE is a strict superset of PSPACE , NP complexity NP , and P complexity P and is believed to be a strict superset of EXPTIME . An example of an EXPSPACE complete problem is the problem of recognizing whether two regular expression s represent different languages, where the expressions are limited to four operators union, concatenation , the Kleene star zero or more copies of an expression , and squaring two copies of an expression . ref Meyer, A.R. and Larry Stockmeyer L. Stockmeyer . http people.csail.mit.edu meyer rsq.pdf The equivalence problem for regular expressions with squaring requires exponential space . 13th IEEE Symposium on Switching and Automata Theory , Oct 1972, pp.125&ndash 129. ref If the Kleene ... is in EXPSPACE . See also Game complexity References references L. Berman The complexity of logical ... of regular expressions with exponentiation is EXPSPACE complete. ComplexityClasses Category Complexity classes es EXPSPACE ko EXPSPACE it EXPSPACE ja EXPSPACE pt EXPSPACE zh EXPSPACE ... more details
For the car, see Renault Espace In computational complexity theory , the complexity class ESPACE is the set of decision problem s that can be solved by a deterministic Turing machine in space 2 sup Big O notation O n sup . See also EXPSPACE . External links CZoo Class ESPACE E espace Category Complexity classes comp sci theory stub es ESPACE ja ESPACE pt ESPACE ... more details
In computational complexity theory , the complexity class 2 EXPTIME sometimes called 2 EXP is the Set mathematics set of all decision problem s solvable by a deterministic Turing machine in big O notation O 2 sup 2 sup p n sup sup time, where p n is a polynomial function of n . In terms of DTIME , math mbox 2 EXPTIME bigcup k in mathbb N mbox DTIME left 2 2 n k right . math We know P complexity P math subseteq math NP complexity NP math subseteq math PSPACE math subseteq math EXPTIME math subseteq math NEXPTIME math subseteq math EXPSPACE math subseteq math 2 EXPTIME math subseteq math ELEMENTARY . 2 EXPTIME can also be reformulated as the space class AEXPSPACE, the problems that can be solved by an alternating Turing machine in exponential space. This is one way to see that EXPSPACE math subseteq math 2 EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine. ref Christos Papadimitriou, Computational Complexity 1994 , ISBN 978 0 201 53082 7. Section 20.1, corollary 3, page 495. ref 2 EXPTIME is one class in a hierarchy of complexity classes with increasingly higher time bounds. The class 3 EXPTIME is defined similarly to 2 EXPTIME but with a triply exponential time bound math 2 2 2 n k math . This can be generalized to higher and higher time bounds. 2 EXPTIME complete problems Generalizations of many fully observable games are EXPTIME complete. These games can be viewed as particular instance of a class of transition systems defined in terms of a set of state variables and actions events that change the values of the state variables, together with the question of whether a winning strategy exists. A generalization of this class of fully observable problems to partially observable problems lifts the complexity from EXPTIME complete to 2 EXPTIME complete. ref cite journal author Jussi Rintanen title Complexity of Planning with Partial Observability journal Proceedings of International Conference on Automated Planning and ... more details
NL , P complexity P , NP complexity NP , PH complexity PH , EXPTIME and EXPSPACE note that math ... mbox PH subseteq mbox PSPACE math math mbox PSPACE subseteq mbox EXPTIME subseteq mbox EXPSPACE math math mbox NL subsetneq mbox PSPACE subsetneq mbox EXPSPACE math It is known that in the first ... more details
Unreferenced date December 2009 In computational complexity theory , 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 It is the non deterministic counterpart of DSPACE . Several important complexity classes can be defined in terms of NSPACE . These include 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 . NL complexity NL NSPACE O log n context sensitive language CSL NSPACE O n , where CSL is the class of context sensitive language s. PSPACE NPSPACE math bigcup k in mathbb N mbox NSPACE n k math EXPSPACE NEXPSPACE math bigcup k in mathbb N mbox NSPACE 2 n k math The last two results above follow from Savitch s theorem , which states that for any function f n log n , NSPACE f n DSPACE f sup 2 sup n . The Immerman Szelepcs nyi theorem states that NSPACE s n is closed under complement for every function nowrap s n log n . NSPACE can be related to DTIME as follows. For any space constructible function s n , math mbox NSPACE s n subseteq bigcup k geq 1 mbox DTIME 2 k cdot s n math A further generalization is ASPACE , defined with alternation complexity alternating Turing machines . References Reflist External Links ComplexityZoo NSPACE f n N nspace . ComplexityClasses DEFAULTSORT Nspace Category Complexity classes Category Computational resources Comp sci theory stub de NSPACE es NSPACE ja NSPACE pt NSPACE zh NSPACE ... more details
EXP redirects here for other uses, see exp . Infobox Complexity Class class EXPTIME, also called EXP image File Complexity subsets pspace.svg 200px long name Exponential time description wheredefined external urls http qwiki.stanford.edu index.php Complexity Zoo E exp Complexity Zoo complete class EXPTIME complete complement class self proper supersets EXPSPACE ref name spacehierarchy Space hierarchy theorem ref In computational complexity theory , the complexity class EXPTIME sometimes called EXP is the Set mathematics set of all decision problem s solvable by a deterministic Turing machine in big O notation O 2 sup p n sup time, where p n is a polynomial function of n . In terms of DTIME , math mbox EXPTIME bigcup k in mathbb N mbox DTIME left 2 n k right . math We know P complexity P math subseteq math NP complexity NP math subseteq math PSPACE math subseteq math EXPTIME math subseteq math NEXPTIME math subseteq math EXPSPACE and also, by the time hierarchy theorem and the space hierarchy theorem , that P math subsetneq math EXPTIME nbsp 2 and nbsp 2 NP math subsetneq math NEXPTIME nbsp 2 and nbsp 2 PSPACE math subsetneq math EXPSPACE so at least one of the first three inclusions and at least one of the last three inclusions must be proper, but it is not known which ones are. Most experts Who date December 2010 believe all the inclusions are proper. It s also known that if nowrap P NP problem P NP , then nowrap EXPTIME NEXPTIME , the class of problems solvable in exponential time by a nondeterministic Turing machine . ref cite book author Christos Papadimitriou title Computational Complexity publisher Addison Wesley year 1994 isbn 0 201 53082 1 authorlink Christos Papadimitriou Section 20.1, page 491. ref More precisely, EXPTIME NEXPTIME if and only if there exist sparse language s in NP that are not in P . ref Juris Hartmanis, Neil Immerman, Vivian Sewelson. Sparse Sets in NP P EXPTIME versus NEXPTIME. Information and Control , volume 65, issue 2 3, pp.158 181. ... more details
In theoretical computer science , a context sensitive language is a formal language that can be defined by a context sensitive grammar . That is one of the four types of grammars in the Chomsky hierarchy . Of the four, this is the least often used, in both theory and practice. Computational properties Computationally, a context sensitive language is equivalent with a linear bounded nondeterministic Turing machine , also called a linear bounded automaton . That is a non deterministic Turing machine with a tape of only kn cells, where n is the size of the input and k is a constant associated with the machine. This means that every formal language that can be decided by such a machine is a context sensitive language, and every context sensitive language can be decided by such a machine. This set of languages is also known as NLIN SPACE , because they can be accepted using linear space on a non deterministic Turing machine. The class LIN SPACE is defined the same, except using a Deterministic automaton deterministic Turing machine. Clearly LIN SPACE is a subset of NLIN SPACE , but it is not known whether LIN SPACE NLIN SPACE . It is widely suspected they are not equal. Examples An example of a context sensitive language that is not context free is L a sup p sup p is a prime number . L can be shown to be a context sensitive language by constructing a linear bounded automaton which accepts L . The language can easily be shown to be neither regular language regular nor context free language context free by applying the respective pumping lemma s for each of the language classes to L . An example of recursive language that is not context sensitive is any recursive language whose decision is an EXPSPACE hard problem, say, the set of pairs of equivalent regular expression s with exponentiation. Properties of context sensitive languages The union, intersection, concatenation and kleene star of two context sensitive languages is context sensitive. The complement of a context se ... more details
For digital repositories DSpace Refimprove date October 2009 In computational complexity theory , DSPACE or SPACE is the computational resource describing the resource of memory space for a deterministic Turing machine . It represents the total amount of memory space that a normal physical computer would need to solve a given computational problem with a given algorithm . It is one of the most well studied complexity measures, because it corresponds so closely to an important real world resource the amount of physical Computer storage computer memory needed to run a given program. Complexity classes The measure DSPACE is used to define complexity class es, sets of all of the decision problem s which can be solved using a certain amount of memory space. For each function f n , there is a complexity class SPACE f n , the set of decision problem s which can be solved by a deterministic Turing machine using space O f n . There is no restriction on the amount of computation time which can be used, though there may be restrictions on some other complexity measures like Alternation complexity alternation . Several important complexity classes are defined in terms of DSPACE . These include regular language REG DSPACE O 1 , where REG is the class of regular language s. In fact, REG DSPACE o log log n that is, log log n space is required to recognize any nonregular language . L complexity L DSPACE O log n PSPACE math bigcup k in mathbb N mbox DSPACE n k math EXPSPACE math bigcup k in mathbb N mbox DSPACE 2 n k math Machine models DSPACE is traditionally measured on a deterministic Turing machine . Several important space complexity classes are sublinear , that is, smaller than the size of the input. Thus, charging the algorithm for the size of the input, or for the size of the output, would not truly capture the memory space used. This is solved by defining the multi string Turing machine with input and output , which is a standard multi tape Turing ... more details
poly n EXPSPACE Deterministic Turing machine Space 2 sup poly n sup NSPACE f n Non deterministic Turing ... n sup It turns out that PSPACE NPSPACE and EXPSPACE NEXPSPACE by Savitch s theorem . Other important ... 0 border 1 style background lightGreen width 100 height 100 tr td style text align center EXPSPACE ... more details
of the problem EXPSPACE problems on von Neumann architecture von Neumann machines , it still grows exponentially with the size of the problem on DNA machines. For very large EXPSPACE problems, the amount ... more details
machine Space poly n EXPSPACE Deterministic Turing machine Space 2 sup poly n sup NSPACE f n Non ... Space 2 sup poly n sup It turns out that PSPACE NPSPACE and EXPSPACE NEXPSPACE by Savitch s theorem ... more details
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