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ω-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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omega-regular Language
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] |
Ludwig Staiger
Ludwig Staiger is a German mathematician and computer scientist at the Martin Luther University of Halle-Wittenberg. He received his Ph.D. in mathematics from the University of Jena in 1976; Staiger wrote his doctoral thesis, ''Zur Topologie der regulären Mengen'', under the direction of and Rolf Lindner. Previously he held positions at the Academy of Sciences in Berlin (East), the Central Institute of Cybernetics and Information Processes, the Karl Weierstrass Institute for Mathematics and the Technical University Otto-von-Guericke Magdeburg. He was a visiting professor at RWTH Aachen University, the universities Dortmund, Siegen, and Cottbus in Germany and the Technical University Vienna, Austria. He is a member of the Managing Committee of the Georg Cantor Association and an external researcher of the Center for Discrete Mathematics and Theoretical Computer Science at the University of Auckland, New Zealand. He co-invented with Klaus Wagner the Staiger–Wagner auto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Time Property
In model checking, a branch of computer science, linear time properties are used to describe requirements of a model of a computer system. Example properties include "the vending machine does not dispense a drink until money has been entered" (a safety property) or "the computer program eventually terminates" (a liveness property). Fairness properties can be used to rule out unrealistic paths of a model. For instance, in a model of two traffic lights, the liveness property "both traffic lights are green infinitely often" may only be true under the unconditional fairness constraint "each traffic light changes colour infinitely often" (to exclude the case where one traffic light is "infinitely faster" than the other). Formally, a linear time property is an ω-language over the power set of "atomic propositions". That is, the property contains sequences of sets of propositions, each sequence known as a "word". Every property can be rewritten as "''P'' and ''Q'' both occur" for some s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...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 r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or '' well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or semantics. In computational compl ... [...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 Dirk van Dalen.. After postdoctoral studies at the |
Arto Salomaa
Arto K. Salomaa (born 6 June 1934) is a Finnish mathematician and computer scientist. His research career, which spans over forty years, is focused on formal languages and automata theory. Early life and education Salomaa was born in Turku, Finland on June 6, 1934. He earned a Bachelor's degree from the University of Turku in 1954 and a PhD from the same university in 1960. Salomaa's father was a professor of philosophy at the University of Turku. Salomaa was introduced to the theory of automata and formal languages during seminars at Berkeley given by John Myhill in 1957. Career In 1965, Salomaa became a professor of mathematics at the University of Turku, a position he retired from in 1999. He also spent two years in the late 1960s at the University of Western Ontario in London, Ontario, Canada, and two years in the 1970s at Aarhus University in Aarhus, Denmark.. Salomaa was president of the European Association for Theoretical Computer Science from 1979 until 1985. Publicat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grzegorz Rozenberg
Grzegorz Rozenberg (born 14 March 1942, Warsaw) is a Polish and Dutch computer scientist. His primary research areas are natural computing, formal language and automata theory, graph transformations, and concurrent systems. He is referred to as the guru of natural computing, as he was promoting the vision of natural computing as a coherent scientific discipline already in the 1970s, gave this discipline its current name, and defined its scope. His research career spans over forty five years. He is a professor at the Leiden Institute of Advanced Computer Science of Leiden University, The Netherlands and adjoint professor at the Department of Computer Science, University of Colorado at Boulder, USA. Rozenberg is also a performing magician, with the artist name Bolgani and specializing in close-up illusions. He is the father of well-known Dutch artist Dadara. Education and career Rozenberg received his Master and Engineer degrees in computer science from the Warsaw Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Éric Pin
Jean-Éric Pin is a French mathematician and theoretical computer scientist known for his contributions to the algebraic automata theory and semigroup theory. He is a CNRS research director. Biography Pin earned his undergraduate degree from ENS Cachan in 1976 and his doctorate (Doctorat d'état) from the Pierre and Marie Curie University in 1981. Since 1988 he has been a CNRS research director at Paris Diderot University. In the years 1992–2006 he was a professor at École Polytechnique. Pin is a member of the Academia Europaea The Academia Europaea is a pan-European Academy of Humanities, Letters, Law, and Sciences. The Academia was founded in 1988 as a functioning Europe-wide Academy that encompasses all fields of scholarly inquiry. It acts as co-ordinator of Europea ... (2011) and an EATCS fellow (2014). In 2018, Pin became the first recipient of the Salomaa Prize in Automata Theory, Formal Languages, and Related Topics. References External links Personal page ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Checking
In computer science, model checking or property checking is a method for checking whether a finite-state model of a system meets a given specification (also known as correctness). This is typically associated with hardware or software systems, where the specification contains liveness requirements (such as avoidance of livelock) as well as safety requirements (such as avoidance of states representing a system crash). In order to solve such a problem algorithmically, both the model of the system and its specification are formulated in some precise mathematical language. To this end, the problem is formulated as a task in logic, namely to check whether a structure satisfies a given logical formula. This general concept applies to many kinds of logic and many kinds of structures. A simple model-checking problem consists of verifying whether a formula in the propositional logic is satisfied by a given structure. Overview Property checking is used for verification when two d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
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 we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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