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RE (complexity)
In computability theory and computational complexity theory, RE (recursively enumerable) is the class of decision problems for which a 'yes' answer can be verified by a Turing machine in a finite amount of time. Informally, it means that if the answer to a problem instance is 'yes', then there is some procedure that takes finite time to determine this, and this procedure never falsely reports 'yes' when the true answer is 'no'. However, when the true answer is 'no', the procedure is not required to halt; it may go into an " infinite loop" for some 'no' cases. Such a procedure is sometimes called a semi-algorithm, to distinguish it from an algorithm, defined as a complete solution to a decision problem. Similarly, co-RE is the set of all languages that are complements of a language in RE. In a sense, co-RE contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever. Equivalent definition Equivalently, RE is the class ... [...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 definable set, 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 (mathematics), 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 computational complexity theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-one Reduction
In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction that converts instances of one decision problem (whether an instance is in L_1) to another decision problem (whether an instance is in L_2) using a computable function. The reduced instance is in the language L_2 if and only if the initial instance is 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 the language L_1 by applying the reduction and solving for 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 at least as hard to solve as L_1. This means that 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Post Correspondence Problem
The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the halting problem In computability theory (computer science), 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 for ... and the ''Entscheidungsproblem'' it is often used in proofs of undecidability. Definition of the problem Let A be an alphabet with at least two symbols. The input of the problem consists of two finite lists \alpha_, \ldots, \alpha_ and \beta_, \ldots, \beta_ of words over A. A solution to this problem is a sequence of indices (i_k)_ with K \ge 1 and 1 \le i_k \le N for all k, such that : \alpha_ \ldots \alpha_ = \beta_ \ldots \beta_. The decision problem then is to decide whether such a solution exists or not. Alternative definition :g: (i_1,\ldots,i_K) \mapsto \alpha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Validity (logic)
In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be truth, true and the conclusion nevertheless to be False (logic), false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formula, well-formed formulas (also called ''wffs'' or simply ''formulas''). The validity of an argument can be tested, proved or disproved, and depends on its logical form. Arguments In logic, an argument is a set of related statements expressing the ''premises'' (which may consists of non-empirical evidence, empirical evidence or may contain some axiomatic truths) and a ''necessary conclusion based on the relationship of the premises.'' An argument is ''valid'' if and only if it would be contradicto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Grammar
A formal grammar is a set of Terminal and nonterminal symbols, symbols and the Production (computer science), production rules for rewriting some of them into every possible string of a formal language over an Alphabet (formal languages), alphabet. A grammar does not describe the semantics, meaning of the strings — only their form. In applied mathematics, formal language theory is the discipline that studies formal grammars and languages. Its applications are found in theoretical computer science, theoretical linguistics, Formal semantics (logic), formal semantics, mathematical logic, and other areas. A formal grammar is a Set_(mathematics), set of rules for rewriting strings, along with a "start symbol" from which rewriting starts. Therefore, a grammar is usually thought of as a language generator. However, it can also sometimes be used as the basis for a "recognizer"—a function in computing that determines whether a given string belongs to the language or is grammatical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unrestricted Grammar
In automata theory, the class of unrestricted grammars (also called semi-Thue, type-0 or phrase structure grammars) is the most general class of grammars in the Chomsky hierarchy. No restrictions are made on the productions of an unrestricted grammar, other than each of their left-hand sides being non-empty. This grammar class can generate arbitrary recursively enumerable languages. Formal definition An unrestricted grammar is a formal grammar G = (N, T, P, S), where * N is a finite set of nonterminal symbols, * T is a finite set of terminal symbols with N and T disjoint,Actually, T\cap N=\emptyset is not strictly necessary since unrestricted grammars make no real distinction between the two. The designation exists purely so that one knows when to stop generating sentential forms of the grammar; more precisely, the language L(G) recognized by G is restricted to strings of terminal symbols. * P is a finite set of production rules of the form \alpha \to \beta , where \alpha an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word Problem For Groups
A word is a basic element of language that carries meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguists on its definition and numerous attempts to find specific criteria of the concept remain controversial. Different standards have been proposed, depending on the theoretical background and descriptive context; these do not converge on a single definition. Some specific definitions of the term "word" are employed to convey its different meanings at different levels of description, for example based on phonological, grammatical or orthographic basis. Others suggest that the concept is simply a convention used in everyday situations. The concept of "word" is distinguished from that of a morpheme, which is the smallest unit of language that has a meaning, even if it cannot stand on its own. Words are made out of at least one morpheme. Morphemes can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): , or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative, so is not necessarily equal to ; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a ''commutative semigroup' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word Problem (mathematics)
In computational mathematics, a word problem is the decision problem, problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. Some deep results of computational theory concern the undecidable problem, undecidability of this question in many important cases. Background and motivation In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a ''solution to the word problem''. For example, imagine that x,y,z are symbols representing real numbers - then a relevant solution to the word problem would, given the input (x \cdot y)/z \mathrel (x/z)\cdot y, produ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Creative Set
In computability theory, productive sets and creative sets are types of sets of natural numbers that have important applications in mathematical logic. They are a standard topic in mathematical logic textbooks such as and . Definition and example For the remainder of this article, assume that \varphi_i is an admissible numbering of the computable functions and ''W''''i'' the corresponding numbering of the recursively enumerable sets. A set ''A'' of natural numbers is called productive if there exists a total recursive (computable) function f so that for all i \in \mathbb, if W_i \subseteq A then f(i) \in A \setminus W_i. The function f is called the productive function for A. A set ''A'' of natural numbers is called creative if ''A'' is recursively enumerable and its complement \mathbb\setminus A is productive. Not every productive set has a recursively enumerable complement, however, as illustrated below. The archetypal creative set is K = \, the set representing the halting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
