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Semiautomaton
In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set ''Q'' of states, a set Σ called the input alphabet, and a function ''T'': ''Q'' × Σ → ''Q'' called the transition function. Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states ''Q''. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of ''Q''. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category theory, semiautomata essentially are functors. Transformation semigroups and monoid acts : A transformation semigroup or transformation monoid is a pair (M,Q) consisting of a set ''Q'' (often called the "set of states") and a semigroup or monoid ''M'' of funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup Action
In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are ''acting'' as transformations of the set. From an algebraic perspective, a semigroup action is a generalization of the notion of a group action in group theory. From the computer science point of view, semigroup actions are closely related to automata: the set models the state of the automaton and the action models transformations of that state in response to inputs. An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set. From a category theoretic point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Finite Automata
In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They provide a mathematical abstraction of real-world quantum computers. Several types of automata may be defined, including ''measure-once'' and ''measure-many'' automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata. The automata work by receiving a finite-length string \sigma=(\sigma_0,\sigma_1,\cdots,\sigma_k) of letters \sigma_i from a finite alphabet \Sigma, and assigning to each such string a probability \operatorname(\sigma) indicating the probability of the automaton being in an accept state; that is, indicating whether the automaton accepted or rejected the string. The languages accepted by QFAs are not the regular languages of deter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid Action
In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are ''acting'' as transformations of the set. From an algebraic perspective, a semigroup action is a generalization of the notion of a group action in group theory. From the computer science point of view, semigroup actions are closely related to automata: the set models the state of the automaton and the action models transformations of that state in response to inputs. An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set. From a category theoret ... [...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] |
Accept State
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 dispense p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Monoid
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set ''A'' is usually denoted ''A''∗. The free semigroup on ''A'' is the subsemigroup of ''A''∗ containing all elements except the empty string. It is usually denoted ''A''+./ref> More generally, an abstract monoid (or semigroup) ''S'' is described as free if it is isomorphic to the free monoid (or semigroup) on some set. As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The st ... [...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 dispense ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alphabet (computer Science)
In formal language theory, an alphabet is a non-empty set of symbols/glyphs, typically thought of as representing letters, characters, or digits but among other possibilities the "symbols" could also be a set of phonemes (sound units). Alphabets in this technical sense of a set are used in a diverse range of fields including logic, mathematics, computer science, and linguistics. An alphabet may have any cardinality ("size") and depending on its purpose maybe be finite (e.g., the alphabet of letters "a" through "z"), countable (e.g., \), or even uncountable (e.g., \). Strings, also known as "words", over an alphabet are defined as a sequence of the symbols from the alphabet set. For example, the alphabet of lowercase letters "a" through "z" can be used to form English words like "iceberg" while the alphabet of both upper and lower case letters can also be used to form proper names like "Wikipedia". A common alphabet is , the binary alphabet, and a "00101111" is an example of a bi ... [...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] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function, so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic properties If is any function, then we h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene Star
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 arbitr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
