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Brzozowski Derivative
In theoretical computer science, in particular in formal language theory, the Brzozowski derivative u^S of a set S of strings and a string u is the set of all strings obtainable from a string in S by cutting off the prefix u, as illustrated in the figure. Formally: u^S = \. It was introduced under various different names since the late 1950s. Today it is named after the 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 left quotient by a singleton set containing only u: \ u^S = \ \;\backslash\; S. Equivalently, for all u,v \in \Sigma^*: :v \in u^S \;\Leftrightarro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brzozowski Derivative
In theoretical computer science, in particular in formal language theory, the Brzozowski derivative u^S of a set S of strings and a string u is the set of all strings obtainable from a string in S by cutting off the prefix u, as illustrated in the figure. Formally: u^S = \. It was introduced under various different names since the late 1950s. Today it is named after the 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 left quotient by a singleton set containing only u: \ u^S = \ \;\backslash\; S. Equivalently, for all u,v \in \Sigma^*: :v \in u^S \;\Leftrightarro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (set Theory)
In set theory, a tree is a partially ordered set (''T'', <) such that for each ''t'' ∈ ''T'', the set is by the relation <. Frequently trees are assumed to have only one root (i.e. ), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees. Definition A tree is a (poset) (''T'', <) such that for each ''t'' ∈ ''T'', the set is by the ...[...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 an 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Complexity
In 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 generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context-free Language
In formal language theory, a context-free language (CFL) 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 example context-free language is L = \, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parsing
Parsing, syntax analysis, or syntactic analysis is the process of analyzing a string of symbols, either in natural language, computer languages or data structures, conforming to the rules of a formal grammar. The term ''parsing'' comes from Latin ''pars'' (''orationis''), meaning part (of speech). The term has slightly different meanings in different branches of linguistics and computer science. Traditional 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 and 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 parse tree showing their syntactic relation to each other, which may also contain semantic and other information (p-values). Some parsing algor ... [...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 are 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). A formal grammar is "context-free" if its production rules can be applied regardless of the context of a nonterminal. No matter which symbols surround it, the single nonterminal on the left hand side can always be replaced by the right hand side. This is what distinguishes it from a context-sensitive grammar. 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, :\langle\text\rangle \to \langle\text\rangle = \langle\text\rangle ; replaces \langle\text\rangle with \langle\text\rangle = \langle\text\rangle ;. There can be multiple replacement rules for a given nonterminal symbol. The ... [...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 1958 . 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 _L on strings as x\; _L\ y iff there is no distinguishing extension for x and y. It is easy to show that _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 _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) recognizing L. In particular, this implies that there is a ''unique'' minimal DFA for each ... [...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. Hopcroft 2001: ''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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene Closure
In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics, it is more commonly known as the free monoid construction. The application of the Kleene star to a set V is written as ''V^*''. It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterize certain automata, where it means "zero or more repetitions". # If V is a set of strings, then ''V^*'' is defined as the smallest superset of V that contains the empty string \varepsilon and is closed under the string concatenation operation. # If V is a set of symbols or characters, then ''V^*'' is the set of all strings over symbols in V, including the empty string \varepsilon. The set ''V^*'' can also be described as the set containing the empty string and all finite-length strings that can be generated by concatenating arbitrary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alphabet
An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllable, and logographic systems use characters to represent words, morphemes, or other semantic units. The first fully phonemic script, the Proto-Sinaitic script, later known as the Phoenician alphabet, is considered to be the first alphabet and is the ancestor of most modern alphabets, including Arabic, Cyrillic, Greek, Hebrew, Latin, and possibly Brahmic. It was created by Semitic-speaking workers and slaves in the Sinai Peninsula (as the Proto-Sinaitic script), by selecting a small number of hieroglyphs commonly seen in their Egyptian surroundings to describe the sounds, as opposed to the semantic values of the Canaanite languages. However, Peter T. Daniels distinguishes an abugida, a set of graphemes that represent consonantal base ... [...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] |
