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Language Equation
Language equations are mathematical statements that resemble equation, numerical equations, but the variables assume values of formal languages rather than numbers. Instead of arithmetic operations in numerical equations, the variables are joined by language operations. Among the most common operations on two languages ''A'' and ''B'' are the set union ''A'' ∪ ''B'', the set intersection ''A'' ∩ ''B'', and the Concatenation#Concatenation_of_sets_of_strings, concatenation ''A''⋅''B''. Finally, as an operation taking a single operand, the set ''A''* denotes the Kleene star of the language ''A''. Therefore language equations can be used to represent formal grammars, since the languages generated by the grammar must be the solution of a system of language equations. Language equations and context-free grammars Seymour Ginsburg, Ginsburg and Henry Gordon Rice, Rice gave an alternative definition of context-free grammars by language equations. To every context-free grammar G = (V, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Franz Baader
Franz Baader (15 June 1959, Spalt) is a German computer scientist at Dresden University of Technology. He received his PhD in Computer Science in 1989 from the University of Erlangen-Nuremberg, Germany, where he was a teaching and research assistant for 4 years. In 1989, he went to the German Research Centre for Artificial Intelligence (DFKI) as a senior researcher and project leader. In 1993 he became associate professor for computer science at RWTH Aachen, and in 2002 full professor for computer science at TU Dresden. He received the Herbrand Award The Herbrand Award for Distinguished Contributions to Automated Reasoning is an award given by the Conference on Automated Deduction (CADE), Inc., (although it predates the formal incorporation of CADE) to honour persons or groups for important cont ... for the year 2020 "in recognition of his significant contributions to unification theory, combinations of theories and reasoning in description logics". Works * * * * References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Constraint
In mathematics and theoretical computer science, a set constraint is an equation or an inequation between sets of terms. Similar to systems of ( in)equations between numbers, methods are studied for solving systems of set constraints. Different approaches admit different operators (like "∪", "∩", "\", and function application)If ''f'' is an ''n''-ary function symbol admitted in a term, then "''f''(''E''1,...,''E''''n'')" is a set expression denoting the set , where ''E''1,...,''E''''n'' are set expressions in turn. on sets and different (in)equation relations (like "=", "⊆", and "⊈") between set expressions. Systems of set constraints are useful to describe (in particular infinite) sets of ground terms.This is similar to describing e.g. a rational number as a solution to an equation ''a''⋅''x'' + ''b'' = 0, with integer coefficients ''a'', ''b''. They arise in program analysis, abstract interpretation, and type inference. Relation to regular tree grammars Each regular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arden's Rule
In theoretical computer science, Arden's rule, also known as Arden's lemma, is a mathematical statement about a certain form of language equations. Background A (formal) language is simply a set of strings. Such sets can be specified by means of some language equation, which in turn is based on operations on languages. Language equations are mathematical statements that resemble numerical equations, but the variables assume values of formal languages rather than numbers. Among the most common operations on two languages ''A'' and ''B'' are the set union ''A'' ∪ ''B'', and their concatenation ''A''⋅''B''. Finally, as an operation taking a single operand, the set ''A''* denotes the Kleene star of the language ''A''. Statement of Arden's rule Arden's rule states that the set ''A''*⋅''B'' is the smallest language that is a solution for ''X'' in the linear equation ''X'' = ''A''⋅''X'' ∪ ''B'' where ''X'', ''A'', ''B'' are sets of strings. Moreover, if the set ''A'' does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Grammar
Boolean grammars, introduced by , are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context-free grammars, with conjunction and negation operations. Besides these explicit operations, Boolean grammars allow implicit disjunction represented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction and negation can be used, in particular, to specify intersection and complement of languages. An intermediate class of grammars known as conjunctive grammars allows conjunction and disjunction, but not negation. The rules of a Boolean grammar are of the form A \to \alpha_1 \And \ldots \And \alpha_m \And \lnot\beta_1 \And \ldots \And \lnot\beta_n where A is a nonterminal, m+n \ge 1 and \alpha_1, ..., \alpha_m, \beta_1, ..., \beta_n are strings formed of symbols in \Sigma and N. Informally, such a rule asserts that every string w over \Sigma that satis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunctive Grammar
Conjunctive grammars are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context-free grammars, with a conjunction operation. Besides explicit conjunction, conjunctive grammars allow implicit disjunction represented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction can be used, in particular, to specify intersection of languages. A further extension of conjunctive grammars known as Boolean grammars additionally allows explicit negation. The rules of a conjunctive grammar are of the form :A \to \alpha_1 \And \ldots \And \alpha_m where A is a nonterminal and \alpha_1, ..., \alpha_m are strings formed of symbols in \Sigma and V (finite sets of terminal and nonterminal symbols respectively). Informally, such a rule asserts that every string w over \Sigma that satisfies each of the syntactical conditions represented by \alpha_1, ..., \alpha_m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert R
Albert may refer to: Companies * Albert (supermarket), a supermarket chain in the Czech Republic * Albert Heijn, a supermarket chain in the Netherlands * Albert Market, a street market in The Gambia * Albert Productions, a record label * Albert Computers, Inc., a computer manufacturer in the 1980s Entertainment * ''Albert'' (1985 film), a Czechoslovak film directed by František Vláčil * ''Albert'' (2015 film), a film by Karsten Kiilerich * ''Albert'' (2016 film), an American TV movie * ''Albert'' (Ed Hall album), 1988 * "Albert" (short story), by Leo Tolstoy * Albert (comics), a character in Marvel Comics * Albert (''Discworld''), a character in Terry Pratchett's ''Discworld'' series * Albert, a character in Dario Argento's 1977 film ''Suspiria'' Military * Battle of Albert (1914), a WWI battle at Albert, Somme, France * Battle of Albert (1916), a WWI battle at Albert, Somme, France * Battle of Albert (1918), a WWI battle at Albert, Somme, France People * Albert (given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Halpern
Joseph Yehuda Halpern (born 1953) is an Israeli-American professor of computer science at Cornell University. Most of his research is on reasoning about knowledge and uncertainty. Biography Halpern graduated in 1975 from University of Toronto with a B.S. in mathematics. He went on to earn a Ph.D. in mathematics from Harvard University in 1981 under the supervision of Albert R. Meyer and Gerald Sacks. He has written three books, ''Actual Causality'', ''Reasoning about Uncertainty,'' and ''Reasoning About Knowledge'' and is a winner of the 1997 Gödel Prize in theoretical computer science and the 2009 Dijkstra Prize in distributed computing. From 1997 to 2003 he was editor-in-chief of the Journal of the ACM. In 2002 he was inducted as a Fellow of the Association for Computing Machinery and in 2012 he was selected as an IEEE Fellow. In 2011 he was awarded a Senior Fellowship of the Zukunftskolleg at the University of Konstanz. In 2019, Halpern was elected a member of the National ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ashok K
Ashoka (, ; also ''Asoka''; 304 – 232 BCE), popularly known as Ashoka the Great, was the third emperor of the Maurya Empire of Indian subcontinent during to 232 BCE. His empire covered a large part of the Indian subcontinent, stretching from present-day Afghanistan in the west to present-day Bangladesh in the east, with its capital at Pataliputra. A patron of Buddhism, he is credited with playing an important role in the spread of Buddhism across ancient Asia. Much of the information about Ashoka comes from his Brahmi edicts, which are among the earliest long inscriptions of ancient India, and the Buddhist legends written centuries after his death. Ashoka was son of Bindusara, and a grandson of the dynasty's founder Chandragupta. During his father's reign, he served as the governor of Ujjain in central India. According to some Buddhist legends, he also suppressed a revolt in Takshashila as a prince, and after his father's death, killed his brothers to ascend t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rohit Parikh
Rohit Jivanlal Parikh (born November 20, 1936) is an Indian-American mathematician, logician, and philosopher who has worked in many areas in traditional logic, including recursion theory and proof theory. He is a Distinguished Professor at Brooklyn College at the City University of New York (CUNY). Research Parikh worked on topics like vagueness, ultrafinitism, belief revision, logic of knowledge, game theory and social software (social procedure). This last area seeks to combine techniques from logic, computer science (especially logic of programs) and game theory to understand the structure of social algorithms. Personal life and politics Rohit Parikh was married from 1968 to 1994 to Carol Parikh (née Geris), who is best known for her stories and biography of Oscar Zariski, ''The Unreal Life of Oscar Zariski''. Parikh is a nontheist opposing abortions. To fight abortions he joined the Atheist and Agnostic Pro-Life League. In 2018, a Facebook post by Parikh, called fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michal Kunc
Michal (; he, מיכל , gr, Μιχάλ) was, according to the first Book of Samuel, a princess of the United Kingdom of Israel; the younger daughter of King Saul, she was the first wife of David (), who later became king, first of Judah, then of all Israel. In the Bible identifies Saul's elder daughter as Merab and younger daughter as Michal. Michal's story is recorded in the first Book of Samuel, where it is said in and that Michal loved David. The narrative does not indicate whether this is reciprocated. After David's success in battle against the Philistine giant Goliath, Merab was given in marriage to Adriel. Later, after Merab had married Adriel the Meholathite, Saul invited David to marry Michal. David replied, "I am a poor and lightly esteemed man", meaning that he was unable to provide a bride price. Saul then advised him that no bride price was required except for the foreskins of 100 Philistines. David took part in a further battle, killed 200 Philistines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
