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April P is an important complexity class of counting problems not decision problems. April Main article: Reduction complexity Many complexity classes are defined using the concept of a reduction.
A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y.
There are many different types of reductions, based on the method of reduction, such as Cook reductionsKarp reductions and Levin reductionsand the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.
The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers.
This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.
This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C.
Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used.
In particular, the set of problems that are hard for NP is the set of NP-hard problems. This means that X is the hardest problem in C Since there could be many problems which are equally hard, one might say that X is one of the hardest problems in C.
Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Closure properties of classes[ edit ] Complexity classes have a variety of closure properties; for example, decision classes may be closed under negationdisjunctionconjunctionor even under all Boolean operations.
Moreover, they might also be closed under a variety of quantification schemes. P, for instance, is closed under all Boolean operations, and under quantification over polynomially sized domains.
However, it is most likely not closed under quantification over exponential sized domains. Each class X that is not closed under negation has a complement class co-Y, which consists of the complements of the languages contained in X.
Similarly one can define the Boolean closure of a class, and so on; this is however less commonly done. One possible route to separating two complexity classes is to find some closure property possessed by one and not by the other.
One central question of complexity theory is whether nondeterminism adds significant power to a computational model.
UGSpace is the institutional repository of the University of Ghana. UGSpace is an open access electronic archive for the collection, preservation and distribution of digital materials. CYCLONE P&E Microcomputer Systems Automated, Stand-Alone Production Programmer, Debug & Test. CYCLONE FX P&E Microcomputer Systems Automated, Stand-Alone Production Programmer, Debug & Test. DCI-GSI2 dSPACE dSPACE DCI-GSI2 is a low latency and high bandwidth on-chip debug interface for bypassing. Boloka: NWU Institutional Repository Theses and dissertations completed at the NWU since are available electronically via Boloka.. Theses and dissertations before will be considered for digitization upon request from users.; Access to the complete theses (print) collection is available via the online Library Catalogue (Search tip: Change "Material Type" to "Thesis").
This is central to the open P versus NP problem in the context of time. Savitch's theorem shows that for space, nondeterminism does not add significantly more power, where "significant" means the difference between polynomial and superpolynomial resource requirements or, for EXPSPACE, the difference between exponential and superexponential.
For example, Savitch's theorem proves that no problem that requires exponential space for a deterministic Turing machine can be solved by a nondeterministic polynomial space Turing machine.CYCLONE P&E Microcomputer Systems Automated, Stand-Alone Production Programmer, Debug & Test.
CYCLONE FX P&E Microcomputer Systems Automated, Stand-Alone Production Programmer, Debug & Test. DCI-GSI2 dSPACE dSPACE DCI-GSI2 is a low latency and high bandwidth on-chip debug interface for bypassing.
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In computational complexity theory, a complexity class is a set of problems of related resource-based complexity.
A typical complexity class has a definition of the form: the set of problems that can be solved by an abstract machine M using O(f(n)) of resource R, where n is the size of the input. Combinatorial game theory has several ways of measuring game benjaminpohle.com article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity.
UGSpace is the institutional repository of the University of Ghana. UGSpace is an open access electronic archive for the collection, preservation and distribution of digital materials. Measures of game complexity State-space complexity. The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game..
When this is too hard to calculate, an upper bound can often be computed by including illegal positions or positions that can never arise in the course of a game..
Game tree size. The game tree size is the total number.