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Büchi Automaton
In computer science and automata theory, a deterministic Büchi automaton is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machine should move to from its current state when it reads the next input character. Some states are accepting states and one state is the start state. The machine accepts an input if and only if it will pass through an accepting state infinitely many times as it reads the input. A non-deterministic Büchi automaton, later referred to just as a Büchi automaton, has a transition function which may have multiple outputs, leading to many possible paths for the same input; it accepts an infinite input if and only if some possible path is accepting. Deterministic and non-deterministic Büchi automata generalize deterministic finite automata and nondeterministic finite automata to infinite inputs. Each are types of ω-automata. Büchi automata r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automate De Buchi2
Automation describes a wide range of technologies that reduce human intervention in processes, namely by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines. Automation has been achieved by various means including mechanical, hydraulic, pneumatic, electrical, electronic devices, and computers, usually in combination. Complicated systems, such as modern factories, airplanes, and ships typically use combinations of all of these techniques. The benefit of automation includes labor savings, reducing waste, savings in electricity costs, savings in material costs, and improvements to quality, accuracy, and precision. Automation includes the use of various equipment and control systems such as machinery, processes in factories, boilers, and heat-treating ovens, switching on telephone networks, steering, and stabilization of ships, aircraft, and other applications and vehicles with reduced human ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omega Language
In formal language theory within theoretical computer science, an infinite word is an infinite-length sequence (specifically, an ω-length sequence) of symbols, and an ω-language is a set of infinite words. Here, ω refers to the first ordinal number, the set of natural numbers. Formal definition Let Σ be a set of symbols (not necessarily finite). Following the standard definition from formal language theory, Σ* is the set of all ''finite'' words over Σ. Every finite word has a length, which is a natural number. Given a word ''w'' of length ''n'', ''w'' can be viewed as a function from the set → Σ, with the value at ''i'' giving the symbol at position ''i''. The infinite words, or ω-words, can likewise be viewed as functions from \mathbb to Σ. The set of all infinite words over Σ is denoted Σω. The set of all finite ''and'' infinite words over Σ is sometimes written Σ∞ or Σ≤ω. Thus an ω-language ''L'' over Σ is a subset of Σω. Operations Some common ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-deterministic Büchi Automaton
In automata theory, a semi-deterministic Büchi automaton (also known as Büchi automaton deterministic in the limit, or limit-deterministic Büchi automaton) is a special type of Büchi automaton. In such an automaton, the set of states can be partitioned into two subsets: one subset forms a deterministic automaton and also contains all the accepting states. For every Büchi automaton, a semi-deterministic Büchi automaton can be constructed such that both recognize the same ω-language. But, a deterministic Büchi automaton may not exist for the same ω-language. Motivation In standard model checking against linear temporal logic (LTL) properties, it is sufficient to translate an LTL formula into a non-deterministic Büchi automaton. But, in probabilistic model checking, LTL formulae are typically translated into deterministic Rabin automata (DRA), as for instance in the PRISM tool. However, a fully deterministic automaton is not needed. Indeed, semi-deterministic Büchi a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Büchi Automaton
In computer science and automata theory, a Weak Büchi automaton is a formalism which represents a set of infinite words. A Weak Büchi automaton is a modification of Büchi automaton such that for all pair of states q and q' belonging to the same strongly connected component, q is accepting if and only if q' is accepting. A Büchi automaton In computer science and automata theory, a deterministic Büchi automaton is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machi ... accepts a word w if there exists a run, such that at least one state occurring infinitely often in the final state set F. For Weak Büchi automata, this condition is equivalent to the existence of a run which ultimately stays in the set of accepting states. Weak Büchi automata are strictly less-expressive than Büchi automata and than Co-Büchi automata. Properties The deterministic Weak Bà ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-Büchi Automaton
In automata theory, a co-Büchi automaton is a variant of Büchi automaton. The only difference is the accepting condition: a Co-Büchi automaton accepts an infinite word w if there exists a run, such that all the states occurring infinitely often in the run are in the final state set F. In contrast, a Büchi automaton accepts a word w if there exists a run, such that at least one state occurring infinitely often in the final state set F. (Deterministic) Co-Büchi automata are strictly weaker than (nondeterministic) Büchi automata. Formal definition Formally, a deterministic co-Büchi automaton is a tuple \mathcal = (Q,\Sigma,\delta,q_0,F) that consists of the following components: * Q is a finite set. The elements of Q are called the ''states'' of \mathcal. * \Sigma is a finite set called the ''alphabet'' of \mathcal. * \delta : Q \times \Sigma \rightarrow Q is the ''transition function'' of \mathcal. * q_0 is an element of Q, called the initial state. * F\subseteq Q is the ''fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimizing Deterministic Finite Automaton
In automata theory (a branch of theoretical computer science), DFA minimization is the task of transforming a given deterministic finite automaton (DFA) into an equivalent DFA that has a minimum number of states. Here, two DFAs are called equivalent if they recognize the same regular language. Several different algorithms accomplishing this task are known and described in standard textbooks on automata theory. Minimal DFA For each regular language, there also exists a minimal automaton that accepts it, that is, a DFA with a minimum number of states and this DFA is unique (except that states can be given different names). The minimal DFA ensures minimal computational cost for tasks such as pattern matching. There are two classes of states that can be removed or merged from the original DFA without affecting the language it accepts. * Unreachable states are the states that are not reachable from the initial state of the DFA, for any input string. These states can be removed. * Dead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimizing Nondeterministic Finite Automaton
Minimisation or minimization may refer to: * Minimisation (psychology), downplaying the significance of an event or emotion * Minimisation (clinical trials) * Minimisation (code) or Minification, removing unnecessary characters from source code * Structural risk minimization * Boolean minimization, a technique for optimizing combinational digital circuits * Cost-minimization analysis, in pharmacoeconomics * Expenditure minimization problem, in microeconomics * Waste minimisation * Harm reduction * Maxima and minima, in mathematical analysis * Minimal element of a partial order, in mathematics * Minimax approximation algorithm * Minimisation operator ("μ operator"), the add-on to primitive recursion to obtain μ-recursive functions in computer science See also *Optimization (mathematics) *Minimal (other) *Minimalism (other) * Minification (other) * Maximisation (other) * Magnification * Plateau's problem In mathematics, Plateau's problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Depth-first Search
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking. Extra memory, usually a stack, is needed to keep track of the nodes discovered so far along a specified branch which helps in backtracking of the graph. A version of depth-first search was investigated in the 19th century by French mathematician Charles Pierre Trémaux as a strategy for solving mazes. Properties The time and space analysis of DFS differs according to its application area. In theoretical computer science, DFS is typically used to traverse an entire graph, and takes time where , V, is the number of vertices and , E, the number of edges. This is linear in the size of the graph. In these applications it also uses space O(, V, ) in the worst case to store the stack of vertices on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Connected Component
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(''V'' + ''E'')). Definitions A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first. In a directed graph ''G'' that may not itself be strongly connected, a pair of vertices ''u'' and ''v'' are said to be strongly connected to each other if there is a path in each direction between them. The binary relation of being strongly connected is an equivalence relation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rabin Automaton
Rabin is a Hebrew surname. It originates from the Hebrew word ''rav'' meaning Rabbi, or from the name of the specific Rabbi Abin. The most well known bearer of the name was Yitzhak Rabin, prime minister of Israel and Nobel Peace prize Laureate. People with surname Rabin * Al Rabin (1936–2012), American soap opera producer * Beatie Deutsch (née Rabin; born 1989), Haredi Jewish American-Israeli marathon runner * Chaim Menachem Rabin, German-Israeli semitic-linguist * Eve Queler (née Rabin), American conductor * Leah Rabin, wife of Yitzhak Rabin * Matthew Rabin, American professor and researcher in economics * Michael Rabin (1936–1972), American violin virtuoso * Michael O. Rabin, Israeli computer scientist and Turing Award recipient * Nathan Rabin, American film and music critic * John James Audubon (born Jean Rabin, 1785–1851), American ornithologist * Oscar Rabin (1899–1958), Latvian-born British band leader and musician * Oscar Rabin (1928–2018), Russian pain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Muller Automaton
In automata theory, a Muller automaton is a type of an ω-automaton. The acceptance condition separates a Muller automaton from other ω-automata. The Muller automaton is defined using a Muller acceptance condition, i.e. the set of all states visited infinitely often must be an element of the acceptance set. Both deterministic and non-deterministic Muller automata recognize the ω-regular languages. They are named after David E. Muller, an American mathematician and computer scientist, who invented them in 1963. Formal definition Formally, a deterministic Muller-automaton is a tuple ''A'' = (''Q'',Σ,δ,''q''0,F) that consists of the following information: * ''Q'' is a finite set. The elements of ''Q'' are called the ''states'' of ''A''. * Σ is a finite set called the ''alphabet'' of ''A''. * δ: ''Q'' × Σ → ''Q'' is a function, called the ''transition function'' of ''A''. * ''q''0 is an element of ''Q'', called the initial state. * F is a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
