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Nonelementary Problem
In computational complexity theory, a nonelementary problem is a problem that is not a member of the class ELEMENTARY. As a class it is sometimes denoted as NONELEMENTARY. Examples of nonelementary problems that are nevertheless decidable include: * the problem of regular expression equivalence with complementation * the decision problem for monadic second-order logic over trees (see S2S) * the decision problem for term algebras * satisfiability of W. V. O. Quine's fluted fragment of first-order logic * deciding β-convertibility of two closed terms in typed lambda calculus * reachability in vector addition systems; it is Ackermann-complete. * reachability in Petri nets A Petri net, also known as a place/transition net (PT net), is one of several mathematical modeling languages for the description of distributed systems. It is a class of discrete event dynamic system. A Petri net is a directed bipartite grap ...; it is Ackermann-complete. References Complexity clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ELEMENTARY
Elementary may refer to: Arts, entertainment, and media Music * ''Elementary'' (Cindy Morgan album), 2001 * ''Elementary'' (The End album), 2007 * ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977 Other uses in arts, entertainment, and media * ''Elementary'' (TV series), a 2012 American drama television series * "Elementary, my dear Watson", a catchphrase of Sherlock Holmes Education * Elementary and Secondary Education Act, US * Elementary education, or primary education, the first years of formal, structured education * Elementary Education Act 1870, England and Wales * Elementary school, a school providing elementary or primary education Science and technology * ELEMENTARY, a class of objects in computational complexity theory * Elementary, a widget set based on the Enlightenment Foundation Libraries * Elementary abelian group, an abelian group in which every nontrivial element is of prime order * Elementary algebra * Elementary arithmetic * Elementary charge, ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Expression
A regular expression (shortened as regex or regexp), sometimes referred to as rational expression, is a sequence of characters that specifies a match pattern in text. Usually such patterns are used by string-searching algorithms for "find" or "find and replace" operations on strings, or for input validation. Regular expression techniques are developed in theoretical computer science and formal language theory. The concept of regular expressions began in the 1950s, when the American mathematician Stephen Cole Kleene formalized the concept of a regular language. They came into common use with Unix text-processing utilities. Different syntaxes for writing regular expressions have existed since the 1980s, one being the POSIX standard and another, widely used, being the Perl syntax. Regular expressions are used in search engines, in search and replace dialogs of word processors and text editors, in text processing utilities such as sed and AWK, and in lexical analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monadic Second-order Logic
In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. It is particularly important in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second-order formulas over graphs of bounded treewidth. It is also of fundamental importance in automata theory, where the Büchi–Elgot–Trakhtenbrot theorem gives a logical characterization of the regular languages. Second-order logic allows quantification over Predicate (mathematical logic), predicates. However, MSO is the Fragment (logic), fragment in which second-order quantification is limited to monadic predicates (predicates having a single argument). This is often described as quantification over "sets" because monadic predicates are equivalent in expressive power to sets (the set of elements for which the predicate is true). Variants Monadic second-order logic come ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S2S (mathematics)
In mathematics, S2S is the monadic second order theory with two successors. It is one of the most expressive natural decidable theories known, with many decidable theories interpretable in S2S. Its decidability was proved by Rabin in 1969. Basic properties The first order objects of S2S are finite binary strings. The second order objects are arbitrary sets (or unary predicates) of finite binary strings. S2S has functions ''s''→''s''0 and ''s''→''s''1 on strings, and predicate ''s''∈''S'' (equivalently, ''S''(''s'')) meaning string ''s'' belongs to set ''S''. Some properties and conventions: * By default, lowercase letters refer to first order objects, and uppercase to second order objects. * The inclusion of sets makes S2S second order, with "monadic" indicating absence of ''k''-ary predicate variables for ''k''>1. * Concatenation of strings ''s'' and ''t'' is denoted by ''st'', and is ''not'' generally available in S2S, not even ''s''→0''s''. The prefix relation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term Algebra
Term may refer to: Language *Terminology, context-specific nouns or compound words **Technical term (or ''term of art''), used by specialists in a field ***Scientific terminology, used by scientists *Term (argumentation), part of an argument in debate theory Law and finance *Contractual term, a provision in a contract **Credit repayment terms **Payment terms, "net ''D''" on a trade invoice **Purchase order#Legal, Purchase order, invoice terms more generally *Term life insurance Lengths of time *Academic term, part of a year at school or university *Term of office, a set period a person serves in an elected office *Term of patent, the period of enforcement of patent rights *Term of a pregnancy *Prison sentence Mathematics and physics *Term (logic), a component of a logical or mathematical expression (not to be confused with term logic, or Aristotelian logic) **Ground term, a term with no variables *Term (arithmetic), or addend, an operand to the addition operator **Term of a summ ... [...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] |
Typed Lambda Calculus
A typed lambda calculus is a typed formalism that uses the lambda symbol (\lambda) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus, but from another point of view, they can also be considered the more fundamental theory and ''untyped lambda calculus'' a special case with only one type. Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. Typed lambda calculi play an important role in the design of type systems for programming languages; here, typability usually captures desirable properties of the program (e.g., the program will not cause a memory acce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science (journal)
''Theoretical Computer Science'' (''TCS'') is a computer science journal published by Elsevier, started in 1975 and covering theoretical computer science. The journal publishes 52 issues a year. It is abstracted and indexed by Scopus and the Science Citation Index. According to the Journal Citation Reports, its 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... is 0.827. References Computer science journals Elsevier academic journals Academic journals established in 1975 {{comp-sci-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reachability Problem
Reachability is a fundamental problem which can be formulated as follows: ''Given a computational system with a set of allowed rules or transformations, decide whether a certain state of a system is reachable from a given initial state of the system.'' It appears in several different contexts: finite- and infinite-state concurrent systems, cellular automata and Petri nets, program analysis, discrete and continuous systems, time critical systems, hybrid systems, rewriting systems, probabilistic and parametric systems, and open systems modelled as games. Variants of the reachability problem may result from additional constraints on the initial or final states, specific requirement for reachability paths as well as for iterative reachability or changing the questions into analysis of winning strategies in infinite games or unavoidability of some dynamics. Typically, for a fixed system description given in some form (reduction rules, systems of equations, logical formulas, et ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
