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Many-one Reducible
In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction which converts instances of one decision problem L_1 into instances of a second decision problem L_2 where the instance reduced to is in the language L_2 if the initial instance was in its language L_1 and is not in the language L_2 if the initial instance was not in its language L_1. Thus if we can decide whether L_2 instances are in the language L_2, we can decide whether L_1 instances are in its language by applying the reduction and solving L_2. Thus, reductions can be used to measure the relative computational difficulty of two problems. It is said that L_1 reduces to L_2 if, in layman's terms L_2 is harder to solve than L_1. That is to say, any algorithm that solves L_2 can also be used as part of a (otherwise relatively simple) program that solves L_1. Many-one reductions are a special case and stronger form of Turing reductions. With many-one red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P (complexity)
In computational complexity theory, P, also known as PTIME or DTIME(''n''O(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or " tractable". This is inexact: in practice, some problems not known to be in P have practical solutions, and some that are in P do not, but this is a useful rule of thumb. Definition A language ''L'' is in P if and only if there exists a deterministic Turing machine ''M'', such that * ''M'' runs for polynomial time on all inputs * For all ''x'' in ''L'', ''M'' outputs 1 * For all ''x'' not in ''L'', ''M'' outputs 0 P can also be viewed as a uniform family of boolean circuits. A language ''L'' is in P if and only if there exists a polynomial-time uniform family of boolean circuits \, such that * For all n \in \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an algorithm that enumerates the members of ''S''. That means that its output is simply a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm runs forever, and no information is returned. A set that is "completely decidable" is a computable set. The second condition suggests why ''computably enumerable'' is used. The abbreviations c.e. and r.e. are oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halting Problem
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program–input pairs cannot exist. For any program that might determine whether programs halt, a "pathological" program , called with some input, can pass its own source and its input to ''f'' and then specifically do the opposite of what ''f'' predicts ''g'' will do. No ''f'' can exist that handles this case. A key part of the proof is a mathematical definition of a computer and program, which is known as a Turing machine; the halting problem is '' undecidable'' over Turing machines. It is one of the first cases of decision problems proven to be unsolvable. This proof is significant to practical computing efforts, defining a class of applications which no programming inventi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Powerset
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . The notation , meaning the set of all functions from S to a given set of two elements (e.g., ), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set. Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, and a symmetric preorder is an equivalence relation. The name comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol \,\leq\, can be used as the notational device for the relation. However, because they are not necessarily antisymmetric, some of the ordinary intuition associated to the symbol \,\leq\, may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Relation
In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. Definitions Let R be a binary relation on a set X, which by definition is just a subset of X \times X. For any x, y \in X, the notation x R y means that (x, y) \in R while "not x R y" means that (x, y) \not\in R. The relation R is called if x R x for every x \in X or equivalently, if \operatorname_X \subseteq R where \operatorname_X := \ denotes the identity relation on X. The of R is the union R \cup \operatorname_X, which can equivalently be defined as the smallest (with respect to \subseteq) reflexive relation on X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Relation
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie. On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can ''never'' be the birth parent of Claire. "Is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (mathematics)
In mathematics, a relation on a set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a minimal element with respect to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE
In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. Formal definition If we denote by SPACE(''t''(''n'')), the set of all problems that can be solved by Turing machines using ''O''(''t''(''n'')) space for some function ''t'' of the input size ''n'', then we can define PSPACE formally asArora & Barak (2009) p.81 :\mathsf = \bigcup_ \mathsf(n^k). PSPACE is a strict superset of the set of context-sensitive languages. It turns out that allowing the Turing machine to be nondeterministic does not add any extra power. Because of Savitch's theorem,Arora & Barak (2009) p.85 NPSPACE is equivalent to PSPACE, essentially because a deterministic Turing machine can simulate a non-deterministic Turing machine without needing much more space (even though it may use much more time).Arora & Barak (2009) p.86 Also, the complements of all problems in PSPACE are also in PSPACE, meaning tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-NP
In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP precisely if only ''no''-instances have a polynomial-length " certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate. That is, co-NP is the set of decision problems where there exists a polynomial ''p(n)'' and a polynomial-time bounded Turing machine ''M'' such that for every instance ''x'', ''x'' is a ''no''-instance if and only if: for some possible certificate ''c'' of length bounded by ''p(n)'', the Turing machine ''M'' accepts the pair (''x'', ''c''). Complementary Problems While an NP problem asks whether a given instance is a ''yes''-instance, its ''complement'' asks whether an instance is a ''no''-instance, which means the complement is in co-NP. Any ''yes''-instance for the original NP ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
