With deterministic finite automata
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In theoretical computer science and formal language theory, the complementation of an automaton [ clarify ] is the problem of computing an automaton that accepts precisely the words rejected by another automaton. Formally, given an automaton A which recognizes a regular language L, we want to compute an automaton recognizing precisely the words that are not in L, i.e., the complement of L.
Several questions on the complementation operation are studied, such as:
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With a nondeterministic finite automaton, the state complexity of the complement automaton may be exponential. [1] Lower bounds are also known in the case of unambiguous automata. [2]
Complementation has also been studied for two-way automata. [3]
In theoretical computer science and formal language theory, a regular language is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science.
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science with close connections to mathematical logic. The word automata comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving". An automaton is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton (FA) or Finite-State Machine (FSM). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. This automaton consists of states and transitions. As the automaton sees a symbol of input, it makes a transition to another state, according to its transition function, which takes the previous state and current input symbol as its arguments.
In mathematics and computer science, the Krohn–Rhodes theory is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined in a feedback-free manner.
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. 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.
In automata theory, a finite-state machine is called a deterministic finite automaton (DFA), if
In theoretical computer science, more precisely in the theory of formal languages, the star height is a measure for the structural complexity of regular expressions and regular languages. The star height of a regular expression equals the maximum nesting depth of stars appearing in that expression. The star height of a regular language is the least star height of any regular expression for that language. The concept of star height was first defined and studied by Eggan (1963).
In computer science, a linear bounded automaton is a restricted form of Turing machine.
In computer science, in particular in automata theory, a two-way finite automaton is a finite automaton that is allowed to re-read its input.
ACC0, sometimes called ACC, is a class of computational models and problems defined in circuit complexity, a field of theoretical computer science. The class is defined by augmenting the class AC0 of constant-depth "alternating circuits" with the ability to count; the acronym ACC stands for "AC with counters". Specifically, a problem belongs to ACC0 if it can be solved by polynomial-size, constant-depth circuits of unbounded fan-in gates, including gates that count modulo a fixed integer. ACC0 corresponds to computation in any solvable monoid. The class is very well studied in theoretical computer science because of the algebraic connections and because it is one of the largest concrete computational models for which computational impossibility results, so-called circuit lower bounds, can be proved.
In formal language theory, deterministic context-free languages (DCFL) are a proper subset of context-free languages. They are the context-free languages that can be accepted by a deterministic pushdown automaton. DCFLs are always unambiguous, meaning that they admit an unambiguous grammar. There are non-deterministic unambiguous CFLs, so DCFLs form a proper subset of unambiguous CFLs.
Language equations are mathematical statements that resemble numerical equations, but the variables assume values of formal languages rather than numbers. Instead of arithmetic operations in numerical equations, the variables are joined by language operations. Among the most common operations on two languages A and B are the set union A ∪ B, the set intersection A ∩ B, and the concatenation A⋅B. Finally, as an operation taking a single operand, the set A* denotes the Kleene star of the language A. Therefore language equations can be used to represent formal grammars, since the languages generated by the grammar must be the solution of a system of language equations.
In automata theory, complementation of a Büchi automaton is the task of complementing a Büchi automaton, i.e., constructing another automaton that recognizes the complement of the ω-regular language recognized by the given Büchi automaton. Existence of algorithms for this construction proves that the set of ω-regular languages is closed under complementation.
In mathematics, a local language is a formal language for which membership of a word in the language can be determined by looking at the first and last symbol and each two-symbol substring of the word. Equivalently, it is a language recognised by a local automaton, a particular kind of deterministic finite automaton.
In theoretical computer science and formal language theory, a weighted automaton or weighted finite-state machine is a generalization of a finite-state machine in which the edges have weights, for example real numbers or integers. Finite-state machines are only capable of answering decision problems; they take as input a string and produce a Boolean output, i.e. either "accept" or "reject". In contrast, weighted automata produce a quantitative output, for example a count of how many answers are possible on a given input string, or a probability of how likely the input string is according to a probability distribution. They are one of the simplest studied models of quantitative automata.
In automata theory, an unambiguous finite automaton (UFA) is a nondeterministic finite automaton (NFA) such that each word has at most one accepting path. Each deterministic finite automaton (DFA) is an UFA, but not vice versa. DFA, UFA, and NFA recognize exactly the same class of formal languages. On the one hand, an NFA can be exponentially smaller than an equivalent DFA. On the other hand, some problems are easily solved on DFAs and not on UFAs. For example, given an automaton A, an automaton A′ which accepts the complement of A can be computed in linear time when A is a DFA, whereas it is known that this cannot be done in polynomial time for UFAs. Hence UFAs are a mix of the worlds of DFA and of NFA; in some cases, they lead to smaller automata than DFA and quicker algorithms than NFA.
Kai Tapani Salomaa is a Finnish Canadian theoretical computer scientist, known for his numerous contributions to the state complexity of finite automata. His highly cited 1994 joint paper with Yu and Zhuang laid the foundations of the area. He has published over 100 papers in scientific journals on various subjects in formal language theory. Salomaa is a full professor at Queen's University.
Viliam Geffert is a Slovak theoretical computer scientist known for his contributions to the computational complexity theory in sublogarithmic space and to the state complexity of two-way finite automata. He has also developed new in-place sorting algorithms. He is a professor and the head of the computer science department at the P. J. Šafárik University in Košice.
Eero Urho Juhani Karhumäki is a Finnish mathematician and theoretical computer scientist known for his contributions to automata theory. He is a professor at the University of Turku.
State complexity is an area of theoretical computer science dealing with the size of abstract automata, such as different kinds of finite automata. The classical result in the area is that simulating an -state nondeterministic finite automaton by a deterministic finite automaton requires exactly states in the worst case.
Giovanni Pighizzini is an Italian theoretical computer scientist known for his work in formal language theory and particularly in state complexity of two-way finite automata. He earned his PhD in 1993 from the University of Milan, where he is a full professor since 2001. Pighizzini serves as the Steering Committee Chair of the annual Descriptional Complexity of Formal Systems academic conference since 2006.
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