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Brzozowski Derivative
In theoretical computer science, in particular in formal language theory, the Brzozowski derivative u^S of a set (mathematics), set S of word (formal languages), strings and a string u is the set of all strings obtainable from a string in S by cutting off the prefix (computer science), prefix u. Formally: :u^S = \. For example, :\text^\ = \. The Brzozowski derivative was introduced under various different names since the late 1950s. Today it is named after the computer scientist Janusz Brzozowski (computer scientist), Janusz Brzozowski who investigated its properties and gave an algorithm to compute the derivative of a generalized regular expression. Definition Even though originally studied for regular expressions, the definition applies to arbitrary formal languages. Given any formal language S over an alphabet \Sigma and any string u \in \Sigma^*, the derivative of S with respect to u is defined as: :u^S = \ The Brzozowski derivative is a special case of quotient of a form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (set Theory)
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only plants that are usable as lumber, or only plants above a specified height. But wider definitions include taller palms, tree ferns, bananas, and bamboos. Trees are not a monophyletic taxonomic group but consist of a wide variety of plant species that have independently evolved a trunk and branches as a way to tower above other plants to compete for sunlight. The majority of tree species are angiosperms or hardwoods; of the rest, many are gymnosperms or softwoods. Trees tend to be long-lived, some trees reaching several thousand years old. Trees evolved around 400 million years ago, and it is estimated that there are around three trillion mature trees in the world currently. A tree typically has many secondary branches supported clear of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Earley Parser
In computer science, the Earley parser is an algorithm for parsing strings that belong to a given context-free language, though (depending on the variant) it may suffer problems with certain nullable grammars. The algorithm, named after its inventor Jay Earley, is a chart parser that uses dynamic programming; it is mainly used for parsing in computational linguistics. It was first introduced in his dissertation in 1968 (and later appeared in abbreviated, more legible form in a journal). Earley parsers are appealing because they can parse all context-free languages, unlike LR parsers and LL parsers, which are more typically used in compilers but which can only handle restricted classes of languages. The Earley parser executes in cubic time in the general case (n^3), where ''n'' is the length of the parsed string, quadratic time for unambiguous grammars (n^2), and linear time for all deterministic context-free grammars. It performs particularly well when the rules are written l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context-free Language
In formal language theory, a context-free language (CFL), also called a Chomsky type-2 language, is a language generated by a context-free grammar (CFG). Context-free languages have many applications in programming languages, in particular, most arithmetic expressions are generated by context-free grammars. Background Context-free grammar Different context-free grammars can generate the same context-free language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar by comparing multiple grammars that describe the language. Automata The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing. Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct. Examples An e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parsing
Parsing, syntax analysis, or syntactic analysis is a process of analyzing a String (computer science), string of Symbol (formal), symbols, either in natural language, computer languages or data structures, conforming to the rules of a formal grammar by breaking it into parts. The term ''parsing'' comes from Latin ''pars'' (''orationis''), meaning Part of speech, part (of speech). The term has slightly different meanings in different branches of linguistics and computer science. Traditional Sentence (linguistics), sentence parsing is often performed as a method of understanding the exact meaning of a sentence or word, sometimes with the aid of devices such as sentence diagrams. It usually emphasizes the importance of grammatical divisions such as subject (grammar), subject and predicate (grammar), predicate. Within computational linguistics the term is used to refer to the formal analysis by a computer of a sentence or other string of words into its constituents, resulting in a par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context-free Grammar
In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules can be applied to a nonterminal symbol regardless of its context. In particular, in a context-free grammar, each production rule is of the form : A\ \to\ \alpha with A a ''single'' nonterminal symbol, and \alpha a string of terminals and/or nonterminals (\alpha can be empty). Regardless of which symbols surround it, the single nonterminal A on the left hand side can always be replaced by \alpha on the right hand side. This distinguishes it from a context-sensitive grammar, which can have production rules in the form \alpha A \beta \rightarrow \alpha \gamma \beta with A a nonterminal symbol and \alpha, \beta, and \gamma strings of terminal and/or nonterminal symbols. A formal grammar is essentially a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the first rule in the picture, : \lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Myhill–Nerode Theorem
In the theory of formal languages, the Myhill–Nerode theorem provides a necessary and sufficient condition for a language to be regular. The theorem is named for John Myhill and Anil Nerode, who proved it at the University of Chicago in 1957 . Statement Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that exactly one of the two strings xz and yz belongs to L. Define a relation \sim_L on strings as x\; \sim_L\ y if there is no distinguishing extension for x and y. It is easy to show that \sim_L is an equivalence relation on strings, and thus it divides the set of all strings into equivalence classes. The Myhill–Nerode theorem states that a language L is regular if and only if \sim_L has a finite number of equivalence classes, and moreover, that this number is equal to the number of states in the minimal deterministic finite automaton (DFA) accepting L. Furthermore, every minimal DFA for the language is isomorphic t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deterministic Finite Automaton
In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite-state machine (DFSM), or deterministic finite-state automaton (DFSA)—is a finite-state machine that accepts or rejects a given string of symbols, by running through a state sequence uniquely determined by the string. ''Deterministic'' refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943. The figure illustrates a deterministic finite automaton using a state diagram. In this example automaton, there are three states: S0, S1, and S2 (denoted graphically by circles). The automaton takes a finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene Closure
In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a set to generate a set of all finite-length strings that are composed of zero or more repetitions of members from . It was named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize automata for regular expressions. In mathematics, it is more commonly known as the free monoid construction. Definition Given a set V, define :V^=\ (the set consists only of the empty string), :V^=V, and define recursively the set :V^=\ for each i>0. V^i is called the i-th power of V, it is a shorthand for the concatenation of V by itself i times. That is, ''V^i'' can be understood to be the set of all strings that can be represented as the concatenation of i members from V. The definition of Kleene star on V is : V^*=\bigcup_V^i = V^0 \cup V^1 \cup V^2 \cup V^3 \cup V^4 \cup \cdots. Kleene plus In some fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alphabet
An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from another in a given language. Not all writing systems represent language in this way: a syllabary assigns symbols to spoken syllables, while logographies assign symbols to words, morphemes, or other semantic units. The first letters were invented in Ancient Egypt to serve as an aid in writing Egyptian hieroglyphs; these are referred to as Egyptian uniliteral signs by lexicographers. This system was used until the 5th century AD, and fundamentally differed by adding pronunciation hints to existing hieroglyphs that had previously carried no pronunciation information. Later on, these phonemic symbols also became used to transcribe foreign words. The first fully phonemic script was the Proto-Sinaitic script, also descending from Egyptian hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite-tree Automaton
In computer science and mathematical logic, an infinite-tree automaton is a state machine that deals with infinite tree structures. It can be seen as an extension of top-down finite-tree automata to infinite trees or as an extension of infinite-word automata to infinite trees. A finite automaton which runs on an infinite tree was first used by Michael Rabin for proving decidability of S2S, the monadic second-order theory with two successors. It has been further observed that tree automata and logical theories are closely connected and it allows decision problems in logic to be reduced into decision problems for automata. Definition Infinite-tree automata work on \Sigma-labeled trees. There are many slightly different definitions; here is one. A (nondeterministic) infinite-tree automaton is a tuple A = (\Sigma, D, Q, q_0, \delta, F ) with the following components. * \Sigma is an alphabet. This alphabet is used to label nodes of an input tree. * D\subset \mathbb is a finite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
