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String Operations
In computer science, in the area of formal language theory, frequent use is made of a variety of string functions; however, the notation used is different from that used for computer programming, and some commonly used functions in the theoretical realm are rarely used when programming. This article defines some of these basic terms. Strings and languages A string is a finite sequence of characters. The empty string is denoted by \varepsilon. The concatenation of two string s and t is denoted by s \cdot t, or shorter by s t. Concatenating with the empty string makes no difference: s \cdot \varepsilon = s = \varepsilon \cdot s. Concatenation of strings is associative: s \cdot (t \cdot u) = (s \cdot t) \cdot u. For example, (\langle b \rangle \cdot \langle l \rangle) \cdot (\varepsilon \cdot \langle ah \rangle) = \langle bl \rangle \cdot \langle ah \rangle = \langle blah \rangle. A language is a finite or infinite set of strings. Besides the usual set operations like union, inters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Substitution
In computer science, in the area of formal language theory, frequent use is made of a variety of string functions; however, the notation used is different from that used for computer programming, and some commonly used functions in the theoretical realm are rarely used when programming. This article defines some of these basic terms. Strings and languages A string is a finite sequence of characters. The empty string is denoted by \varepsilon. The concatenation of two string s and t is denoted by s \cdot t, or shorter by s t. Concatenating with the empty string makes no difference: s \cdot \varepsilon = s = \varepsilon \cdot s. Concatenation of strings is associative: s \cdot (t \cdot u) = (s \cdot t) \cdot u. For example, (\langle b \rangle \cdot \langle l \rangle) \cdot (\varepsilon \cdot \langle ah \rangle) = \langle bl \rangle \cdot \langle ah \rangle = \langle blah \rangle. A language is a finite or infinite set of strings. Besides the usual set operations like union, inters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syntactic Monoid
In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the minimal monoid that recognizes the language L. By the Myhill–Nerode theorem, the syntactic monoid is unique up to unique isomorphism. Syntactic quotient An alphabet is a finite set. The free monoid on a given alphabet is the monoid whose elements are all the strings of zero or more elements from that set, with string concatenation as the monoid operation and the empty string as the identity element. Given a subset S of a free monoid M, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient of S by an element m from M is the set :S \ / \ m=\. Similarly, the left quotient is :m \setminus S=\. Syntactic equivalence The syntactic quotient induces an equivalence relation on M, called the syntactic relation, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-state Automaton
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of ''states'' at any given time. The FSM can change from one state to another in response to some inputs; the change from one state to another is called a ''transition''. An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types— deterministic finite-state machines and non-deterministic finite-state machines. For any non-deterministic finite-state machine, an equivalent deterministic one can be constructed. The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions depending on a sequence of events with which they are presented. Simple examples are: vending machines, which dispens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expression engines, which are Regular expression#Patterns for non-regular languages, augmented with features that allow the recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognised by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by regular grammar, Type-3 grammars. Formal definition The collection of regular languages over an Alphabet (formal languages), alphabet Σ is defined recursively as follows: * The empty language ∅ is a regular language. * For each ''a'' ∈ Σ (''a'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syntactic Relation
In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the minimal monoid that recognizes the language L. By the Myhill–Nerode theorem, the syntactic monoid is unique up to unique isomorphism. Syntactic quotient An alphabet is a finite set. The free monoid on a given alphabet is the monoid whose elements are all the strings of zero or more elements from that set, with string concatenation as the monoid operation and the empty string as the identity element. Given a subset S of a free monoid M, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient of S by an element m from M is the set :S \ / \ m=\. Similarly, the left quotient is :m \setminus S=\. Syntactic equivalence The syntactic quotient induces an equivalence relation on M, called the syntactic relation, or syn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Expression
A regular expression (shortened as regex or regexp), sometimes referred to as rational expression, is a sequence of characters that specifies a match pattern in text. Usually such patterns are used by string-searching algorithms for "find" or "find and replace" operations on strings, or for input validation. Regular expression techniques are developed in theoretical computer science and formal language theory. The concept of regular expressions began in the 1950s, when the American mathematician Stephen Cole Kleene formalized the concept of a regular language. They came into common use with Unix text-processing utilities. Different syntaxes for writing regular expressions have existed since the 1980s, one being the POSIX standard and another, widely used, being the Perl syntax. Regular expressions are used in search engines, in search and replace dialogs of word processors and text editors, in text processing utilities such as sed and AWK, and in lexical analysis ... [...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 (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] |
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] |
Projection In Relational Algebra
In relational algebra, a projection is a unary operation written as \Pi_( R ), where R is a relation and a_1,...,a_n are attribute names. Its result is defined as the set obtained when the components of the tuples in R are restricted to the set \ – it ''discards'' (or ''excludes'') the other attributes. In practical terms, if a relation is thought of as a table, then projection can be thought of as picking a subset of its columns. For example, if the attributes are (name, age), then projection of the relation onto attribute list (age) yields – we have discarded the names, and only know what ages are present. Projections may also modify attribute values. For example, if R has attributes a, b, c, where the values of b are numbers, then \Pi_( R ) is like R, but with all b-values halved.http://www.csee.umbc.edu/~pmundur/courses/CMSC661-02/rel-alg.pdf ''See Problem 3.8.B on page 3'' Related concepts The closely related concept in set theory (see: projection (set theory)) differs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ASCII
ASCII ( ), an acronym for American Standard Code for Information Interchange, is a character encoding standard for representing a particular set of 95 (English language focused) printable character, printable and 33 control character, control characters a total of 128 code points. The set of available punctuation had significant impact on the syntax of computer languages and text markup. ASCII hugely influenced the design of character sets used by modern computers; for example, the first 128 code points of Unicode are the same as ASCII. ASCII encodes each code-point as a value from 0 to 127 storable as a seven-bit integer. Ninety-five code-points are printable, including digits ''0'' to ''9'', lowercase letters ''a'' to ''z'', uppercase letters ''A'' to ''Z'', and commonly used punctuation symbols. For example, the letter is represented as 105 (decimal). Also, ASCII specifies 33 non-printing control codes which originated with ; most of which are now obsolete. The control cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
