|
Omega-regular Languages
The ω-regular languages are a class of ω-languages that generalize the definition of regular languages to infinite words. Formal definition An ω-language ''L'' is ω-regular if it has the form * ''A''ω where ''A'' is a regular language not containing the empty string * ''AB'', the concatenation of a regular language ''A'' and an ω-regular language ''B'' (Note that ''BA'' is ''not'' well-defined) * ''A'' ∪ ''B'' where ''A'' and ''B'' are ω-regular languages (this rule can only be applied finitely many times) The elements of ''A''ω are obtained by concatenating words from ''A'' infinitely many times. Note that if ''A'' is regular, ''A''ω is not necessarily ω-regular, since ''A'' could be for example , the set containing only the empty string, in which case ''A''ω=''A'', which is not an ω-language and therefore not an ω-regular language. It is a straightforward consequence of the definition that the ω-regular languages are precisely the ω-languages of the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omega Language
In formal language theory within theoretical computer science, an infinite word is an infinite-length sequence (specifically, an ω-length sequence) of symbols, and an ω-language is a set of infinite words. Here, ω refers to the first ordinal number, the set of natural numbers. Formal definition Let Σ be a set of symbols (not necessarily finite). Following the standard definition from formal language theory, Σ* is the set of all ''finite'' words over Σ. Every finite word has a length, which is a natural number. Given a word ''w'' of length ''n'', ''w'' can be viewed as a function from the set → Σ, with the value at ''i'' giving the symbol at position ''i''. The infinite words, or ω-words, can likewise be viewed as functions from \mathbb to Σ. The set of all infinite words over Σ is denoted Σω. The set of all finite ''and'' infinite words over Σ is sometimes written Σ∞ or Σ≤ω. Thus an ω-language ''L'' over Σ is a subset of Σω. Operations Some common ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expressions engines, which are augmented with features that allow recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognized by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by Type-3 grammars. Formal definition The collection of regular languages over an alphabet Σ is defined recursively as follows: * The empty language Ø is a regular language. * For each ''a'' ∈ Σ (''a'' belongs to Σ), the singleton language is a regular language. * If ''A'' is a regular language, ''A''* ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty String
In formal language theory, the empty string, or empty word, is the unique string of length zero. Formal theory Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special case where the sequence has length zero, so there are no symbols in the string. There is only one empty string, because two strings are only different if they have different lengths or a different sequence of symbols. In formal treatments, the empty string is denoted with ε or sometimes Λ or λ. The empty string should not be confused with the empty language ∅, which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string. The empty string has several properties: * , ε, = 0. Its string length is zero. * ε ⋅ s = s ⋅ ε = s. The empty string is the identity element of the concatenation operation. The set of all strings forms a free monoid with respect to ⋅ and ε. * εR = ε. Reversal o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Büchi Automaton
In computer science and automata theory, a deterministic Büchi automaton is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machine should move to from its current state when it reads the next input character. Some states are accepting states and one state is the start state. The machine accepts an input if and only if it will pass through an accepting state infinitely many times as it reads the input. A non-deterministic Büchi automaton, later referred to just as a Büchi automaton, has a transition function which may have multiple outputs, leading to many possible paths for the same input; it accepts an infinite input if and only if some possible path is accepting. Deterministic and non-deterministic Büchi automata generalize deterministic finite automata and nondeterministic finite automata to infinite inputs. Each are types of ω-automata. Büchi automata rec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Büchi Automata
Buchi can mean: __NOTOC__ Items *Bachi, special Japanese drumsticks *Butsi, the Hispanised term for jin deui (pastry made from glutinous rice) in the Philippines *Büchi automaton, finite state automata extended to infinite inputs *Büchi arithmetic, a mathematical logical fragment People Given names *Buchi Atuonwu, Nigerian reggae gospel artist *Buchi (comedian), stage name of Onyebuchi Ojieh, Nigerian comedian *Buchi Emecheta, (d. 2017) Nigerian British writer Family names *George Büchi (1921–1998), an organic chemist *Julius Richard Büchi (1924–1984), developer of the Büchi automaton *Hernán Büchi (born 1949), Finance Minister of Chile (1985–1989) *Albert Büchi (1907–1988), a Swiss professional road bicycle racer Nicknames *Yutaka Izubuchi, anime designer and director *Nigerian Igbo first names such as Onyebuchi, Nnabuchi, Maduabuchi, a suffix that translates as "...is God." Fictional characters *Buchi in ''One Piece ''One Piece'' (stylized in all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julius Richard Büchi
Julius Richard Büchi (1924–1984) was a Swiss logician and mathematician. He received his Dr. sc. nat. in 1950 at ETH Zurich under the supervision of Paul Bernays and Ferdinand Gonseth. Shortly afterwards he went to Purdue University in Lafayette, Indiana. He and his first student Lawrence Landweber had a major influence on the development of theoretical computer science. Together with his friend Saunders Mac Lane, a student of Paul Bernays as well, Büchi published numerous celebrated works. He invented what is now known as the Büchi automaton, a finite-state machine accepting certain sets of infinite sequences of characters known as omega-regular languages. The "''n'' squares' problem", known also as Büchi's problem, is an open problem from number theory, closely related to Hilbert's tenth problem Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a gene ... [...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 predicates. However, MSO is the 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 comes in two variants. In the variant considered over str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Van Leeuwen
Jan van Leeuwen (born December 17, 1946, in Waddinxveen) is a Dutch computer scientist and Emeritus professor of computer science at the Department of Information and Computing Sciences at Utrecht University.Curriculum vitae retrieved 2011-03-27. Education and career Van Leeuwen completed his undergraduate studies in mathematics at in 1967 and received a PhD in mathematics in 1972 from the same institution under the supervision of .. After postdoctoral studies at the |
