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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 infinite ordinal number, modeling a 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language
In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also called "words"). Words that belong to a particular formal language are sometimes called ''well-formed words''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar. 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 meanings or semantics. In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that can be parsed by machines with limited computational power. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omega-regular Language
In computer science and formal language theory, the ω-regular languages are a class of ω-languages that generalize the definition of regular languages to infinite words. As regular languages accept finite strings (such as strings beginning in an ''a'', or strings alternating between ''a'' and ''b''), ω-regular languages accept infinite words (such as, infinite sequences beginning in an ''a'', or infinite sequences alternating between ''a'' and ''b''). 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Van Leeuwen
Jan van Leeuwen (born 17 December 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 Kustaa Salomaa (6 June 1934 – 26 January 2025) was a Finnish mathematician and computer scientist. His research career, which spanned over 40 years, was focused on formal languages and automata theory. Early life and education Salomaa was born in Turku, Finland on 6 June 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 ... [...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 Univ ... [...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] |
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 (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. Notable Work Pin is the author of the prominent textbook ''Varieties of Formal Languages'' on automata theory and formal language theory In logic, mathematics, computer science, and linguistics, a formal language is a set of string (computer science), strings whose symbols are taken from a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominique Perrin
Dominique Pierre Perrin (b. 1946) is a French mathematician and theoretical computer scientist known for his contributions to coding theory and to combinatorics on words. He is a professor of the University of Marne-la-Vallée and currently serves as the President of ESIEE Paris. Biography Perrin earned his PhD from Paris 7 University in 1975. In his early career, he was a CNRS researcher (1970–1977) and taught at the University of Chile (1972–1973). Later, he worked as a professor at the University of Rouen (1977–1983), Paris 7 University (1983–1993), and École Polytechnique (1982–2002). Since 1993, Perrin is a professor at the University of Marne-la-Vallée, and since 2004, he is the President of ESIEE Paris. Perrin is a member of Academia Europaea since 1989. Scientific contributions Perrin has been a member of the Lothaire group of mathematicians that developed the foundations of combinatorics on words. He has co-authored three scientific monographs: "Theo ... [...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 ... [...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 Omega language, 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 occ ... [...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 . 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 well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...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] |
