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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] |
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] |
Proof By Cases
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: # A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases. # A proof of each of the cases. The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance. Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Languages
In logic, mathematics, computer science, and linguistics, a formal language is a set of string (computer science), strings whose symbols are taken from a set called "#Definition, 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 Formal language#Definition, ''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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Armed Services Technical Information Agency
Armed (May, 1941–1964) was an American Thoroughbred gelding race horse who was the American Horse of the Year in 1947 and Champion Older Male Horse in both 1946 and 1947. He was inducted into the National Museum of Racing and Hall of Fame in 1963. Background Armed was sired by the great stakes winner Bull Lea, the sire of Citation. His dam was Armful, whose sire was Belmont Stakes winner Chance Shot and whose grandsire was the great Fair Play. Besides being small for his age and very headstrong, Armed had the habits of biting and kicking hay out of his handler's pitchfork. Since he was also practically untrainable, his trainer, Ben A. Jones, sent him back to Calumet Farm to be gelded and turned out to grow up. He returned to the track late in his two-year-old season and resumed training. Racing career His first start was as a three-year-old the following February, and he won at Hialeah Park by eight lengths. He won again less than a week later but then won only on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. The journal is devoted to shorter research articles. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * [...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] |
Pumping Lemma For Regular Languages
In the theory of formal languages, the pumping lemma for regular languages is a Lemma (mathematics), lemma that describes an essential property of all regular languages. Informally, it says that all sufficiently long string (computer science), strings in a regular language may be ''pumped''—that is, have a middle section of the string repeated an arbitrary number of times—to produce a new string that is also part of the language. The pumping lemma is useful for proving that a specific language is not a regular language, by showing that the language does not have the property. Specifically, the pumping lemma says that for any regular language L, there exists a constant p such that any string w in L with length at least p can be split into three substrings x, y and z (w = xyz, with y being non-empty), such that the strings xz, xyz, xyyz, xyyyz, ... are also in L. The process of repeating y zero or more times is known as "pumping". Moreover, the pumping lemma guarantees that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree Automata
A tree automaton is a type of state machine. Tree automata deal with tree structures, rather than the strings of more conventional state machines. The following article deals with branching tree automata, which correspond to regular languages of trees. As with classical automata, finite tree automata (FTA) can be either a deterministic automaton or not. According to how the automaton processes the input tree, finite tree automata can be of two types: (a) bottom up, (b) top down. This is an important issue, as although non-deterministic (ND) top-down and ND bottom-up tree automata are equivalent in expressive power, deterministic top-down automata are strictly less powerful than their deterministic bottom-up counterparts, because tree properties specified by deterministic top-down tree automata can only depend on path properties. (Deterministic bottom-up tree automata are as powerful as ND tree automata.) Definitions A bottom-up finite tree automaton over ''F'' is defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else. Overview In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term ''corollary'', rather than ''proposition'' or ''theorem'', is intrinsically subjective. More formally, proposition ''B'' is a corollary of proposition ''A'', if ''B'' can be readily deduced from ''A'' or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary to that of the theorem. In particular, ''B'' is unlikely to be te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty String
In formal language theory, the empty string, or empty word, is the unique String (computer science), string of length zero. Formal theory Formally, a string is a finite, ordered sequence of character (symbol), characters such as letters, digits or spaces. The empty string is the special case where the sequence has length zero, so there are no symbols in the string. There is only one empty string, because two strings are only different if they have different lengths or a different sequence of symbols. In formal treatments, the empty string is denoted with ε or sometimes Λ or λ. The empty string should not be confused with the empty language ∅, which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string. The empty string has several properties: * , ε, = 0. Its string (computer science)#Formal theory, string length is zero. * ε ⋅ s = s ⋅ ε = s. The empty string is the identity element of the concatenation operation. The set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
