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K-synchronized Sequence
In mathematics and theoretical computer science, a ''k''-synchronized sequence is an infinite sequence of terms ''s''(''n'') characterized by a finite automaton taking as input two strings ''m'' and ''n'', each expressed in some fixed base ''k'', and accepting if ''m'' = ''s''(''n''). The class of ''k''-synchronized sequences lies between the classes of ''k''-automatic sequences and ''k''-regular sequences. Definitions As relations Let Σ be an alphabet of ''k'' symbols where ''k'' ≥ 2, and let 'n''sub>''k'' denote the base-''k'' representation of some number ''n''. Given ''r'' ≥ 2, a subset ''R'' of \mathbb^ is ''k''-synchronized if the relation is a right-synchronized rational relation over Σ∗ × ... × Σ∗, where (''n''1, ..., ''n''''r'') \in ''R''.Carpi & Maggi (2010) Language-theoretic Let ''n'' ≥ 0 be a natural number and let ''f'': \mathbb \rightarrow \mathbb be a map, where both ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's ACM SIGACT, 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 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 n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-state Machine
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. A deterministic finite-state machine can be constructed equivalent to any non-deterministic one. 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] |
Radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9. In any standard positional numeral system, a number is conventionally written as with ''x'' as the string of digits and ''y'' as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four. Etymology ''Radix'' is a Latin word for "root". ''Root'' can be considered a synonym for ''base,'' in the arithmetical sense. In numeral systems In the system with radix 13, for example, a string of digits such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automatic Sequence
In mathematics and theoretical computer science, an automatic sequence (also called a ''k''-automatic sequence or a ''k''-recognizable sequence when one wants to indicate that the base of the numerals used is ''k'') is an infinite sequence of terms characterized by a finite automaton. The ''n''-th term of an automatic sequence ''a''(''n'') is a mapping of the final state reached in a finite automaton accepting the digits of the number ''n'' in some fixed base ''k''.Allouche & Shallit (2003) p. 152Berstel et al (2009) p. 78 An automatic set is a set of non-negative integers ''S'' for which the sequence of values of its characteristic function χ''S'' is an automatic sequence; that is, ''S'' is ''k''-automatic if χ''S''(''n'') is ''k''-automatic, where χ''S''(''n'') = 1 if ''n'' \in ''S'' and 0 otherwise. Definition Automatic sequences may be defined in a number of ways, all of which are equivalent. Four common definitions are as follows. Automata-theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-regular Sequence
In mathematics and theoretical computer science, a ''k''-regular sequence is a sequence satisfying linear recurrence equations that reflect the Positional notation, base-''k'' representations of the integers. The class of ''k''-regular sequences generalizes the class of automatic sequence, ''k''-automatic sequences to alphabets of infinite size. Definition There exist several characterizations of ''k''-regular sequences, all of which are equivalent. Some common characterizations are as follows. For each, we take ''R''′ to be a commutative Noetherian ring and we take ''R'' to be a ring (mathematics), ring containing ''R''′. ''k''-kernel Let ''k'' ≥ 2. The ''k-kernel'' of the sequence s(n)_ is the set of subsequences :K_(s) = \. The sequence s(n)_ is (''R''′, ''k'')-regular (often shortened to just "''k''-regular") if the R'-module generated by ''K''''k''(''s'') is a Finitely generated module, finitely-generated ''R''′-module (mathematics), module.Allouche and Sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Set
In computer science, more precisely in automata theory, a rational set of a monoid is an element of the minimal class of subsets of this monoid that contains all finite subsets and is closed under union, product and Kleene star. Rational sets are useful in automata theory, formal languages and algebra. A rational set generalizes the notion of rational (regular) language (understood as defined by regular expressions) to monoids that are not necessarily free. Definition Let (N,\cdot) be a monoid with identity element e. The set \mathrm(N) of rational subsets of N is the smallest set that contains every finite set and is closed under * union: if A,B\in \mathrm(N) then A\cup B\in \mathrm(N) * product: if A,B\in \mathrm(N) then A\cdot B=\\in\mathrm(N) * Kleene star: if A\in \mathrm(N) then A^*=\bigcup_^\infty A^i \in\mathrm(N) where A^0=\ is the singleton containing the identity element, and where A^=A^n \cdot A. This means that any rational subset of N can be obtained by taking a fini ... [...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 expressions engines, which are augmented with features that allow recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognized 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 Type-3 grammars. Formal definition The collection of regular languages over an alphabet Σ is defined recursively as follows: * The empty language Ø is a regular language. * For each ''a'' ∈ Σ (''a'' belongs to Σ), the singleton language is a regular language. * If ''A'' is a regular language, ''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substring
In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, "''the best of''" is a substring of "''It was the best of times''". In contrast, "''Itwastimes''" is a subsequence of "''It was the best of times''", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S is a substring of S that occurs at the beginning of S; likewise, a suffix of a string S is a substring that occurs at the end of S. The substrings of the string "''apple''" would be: "''a''", "''ap''", "''app''", "''appl''", "''apple''", "''p''", "''pp''", "''ppl''", "''pple''", "''pl''", "''ple''", "''l''", "''le''" "''e''", "" (note the empty string at the end). Substring A string u is a substring (or factor) of a string t if there exists two strings p and s such that t = pus. In particular, the empty string is a substring of every string. Example: The string u=ana is equal to substrings (and subseq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Palindrome
A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a plan, a canal – Panama". The 19-letter Finnish word ''saippuakivikauppias'' (a soapstone vendor), is the longest single-word palindrome in everyday use, while the 12-letter term ''tattarrattat'' (from James Joyce in '' Ulysses'') is the longest in English. The word ''palindrome'' was introduced by English poet and writer Henry Peacham in 1638.Henry Peacham, ''The Truth of our Times Revealed out of One Mans Experience'', 1638p. 123/ref> The concept of a palindrome can be dated to the 3rd-century BCE, although no examples survive; the first physical examples can be dated to the 1st-century CE with the Latin acrostic word square, the Sator Square (contains both word and sentence palindromes), and the 4th-century Greek Byzantine sentence palindrome '' nipson ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
