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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 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_ \ldots \alpha_ :h: (i_1,\ldots,i_K) \mapsto \beta_ \ldots \beta_. This gives rise to an equivalent alternative definition often found in the literature, according to which any two homomorphisms g,h with a common domain and a common codomain form an in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Degree
In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set. Overview The concept of Turing degree is fundamental in computability theory, where sets of natural numbers are often regarded as decision problems. The Turing degree of a set is a measure of how difficult it is to solve the decision problem associated with the set, that is, to determine whether an arbitrary number is in the given set. Two sets are Turing equivalent if they have the same level of unsolvability; each Turing degree is a collection of Turing equivalent sets, so that two sets are in different Turing degrees exactly when they are not Turing equivalent. Furthermore, the Turing degrees are partially ordered, so that if the Turing degree of a set ''X'' is less than the Turing degree of a set ''Y'', then any (noncomputable) procedure that correctly decides whether numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid Morphism
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for automa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Computation
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or to what degree (e.g., approximation algorithms, approximate solutions versus precise ones). The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: ''"What are the fundamental capabilities and limitations of computers?".'' In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. There are several models in use, but the most commonly examined is the Turing machine. Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to prove results, and because it represents what many con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chomsky Hierarchy
In formal language theory, computer science and linguistics, the Chomsky hierarchy (also referred to as the Chomsky–Schützenberger hierarchy) is a containment hierarchy of classes of formal grammars. This hierarchy of grammars was described by Noam Chomsky in 1956. It is also named after Marcel-Paul Schützenberger, who played a crucial role in the development of the theory of formal languages. Formal grammars A formal grammar of this type consists of a finite set of '' production rules'' (''left-hand side'' → ''right-hand side''), where each side consists of a finite sequence of the following symbols: * a finite set of '' nonterminal symbols'' (indicating that some production rule can yet be applied) * a finite set of ''terminal symbols'' (indicating that no production rule can be applied) * a ''start symbol'' (a distinguished nonterminal symbol) A formal grammar provides an axiom schema for (or ''generates'') a ''formal language'', which is a (usually infinite) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substring
In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, "''the best of''" is a substring of "''It was the best of times''". In contrast, "''Itwastimes''" is a subsequence of "''It was the best of times''", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S is a substring of S that occurs at the beginning of S; likewise, a suffix of a string S is a substring that occurs at the end of S. The substrings of the string "''apple''" would be: "''a''", "''ap''", "''app''", "''appl''", "''apple''", "''p''", "''pp''", "''ppl''", "''pple''", "''pl''", "''ple''", "''l''", "''le''" "''e''", "" (note the empty string at the end). Substring A string u is a substring (or factor) of a string t if there exists two strings p and s such that t = pus. In particular, the empty string is a substring of every string. Example: The string u=ana is equal to substrings (and subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
EXPTIME
In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable by a deterministic Turing machine in exponential time, i.e., in O(2''p''(''n'')) time, where ''p''(''n'') is a polynomial function of ''n''. EXPTIME is one intuitive class in an exponential hierarchy of complexity classes with increasingly more complex oracles or quantifier alternations. For example, the class 2-EXPTIME is defined similarly to EXPTIME but with a doubly exponential time bound. This can be generalized to higher and higher time bounds. EXPTIME can also be reformulated as the space class APSPACE, the set of all problems that can be solved by an alternating Turing machine in polynomial space. EXPTIME relates to the other basic time and space complexity classes in the following way: P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE. Furthemore, by the time hierarchy theorem and the space hierarc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Satisfiability Problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called ''satisfiable''. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. SAT is the first problem that was proved to be NP-complet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Guide To The Theory Of NP-Completeness
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version can be written in two forms: the double-storey a and single-storey ɑ. The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English grammar, " a", and its variant " an", are indefinite articles. History The earliest certain ancestor of "A" is aleph (also written 'aleph), the first letter of the Phoenician alphabet, which consisted entirely of consonants (for that reason, it is also called an abjad to distinguish it f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a dete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugacy
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under b = gag^. for all elements g in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set). Functions that are constant for members of the same conjugacy class are called class functions. Definition Let G be a group. Two elements a, b \in G are conjugate if there exists an element g \in G such that gag^ = b, in which case b is called of a and a is called a conjugate of b. In the case of the general linear group \operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite State Machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of '' states'' at any given time. The FSM can change from one state to another in response to some inputs; the change from one state to another is called a ''transition''. An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types— deterministic finite-state machines and non-deterministic finite-state machines. A deterministic finite-state machine can be constructed equivalent to any non-deterministic one. The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions depending on a sequence of events with which they are presented. Simple examples are vending machines, which dispense p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undecidable Problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run. Background A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns ''yes''. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
