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A finite-state machine (FSM) or finite-state automaton, 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.
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
In mathematical logic and computer science, the Kleene star 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 is written as . 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 is a set of strings, then is defined as the smallest superset of that contains the empty string and is closed under the string concatenation operation.
- If is a set of symbols or characters, then is the set of all strings over symbols in , including the empty string .
A regular expression is a sequence of characters that specifies a search pattern. Usually such patterns are used by string-searching algorithms for "find" or "find and replace" operations on strings, or for input validation. It is a technique developed in theoretical computer science and formal language theory.
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.
In computer science, string-searching algorithms, sometimes called string-matching algorithms, are an important class of string algorithms that try to find a place where one or several strings are found within a larger string or text.
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree. The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".
In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenation theory, also called string theory, string concatenation is a primitive notion.
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. 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).
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem.
In computer programming, glob patterns specify sets of filenames with wildcard characters. For example, the Unix Bash shell command mv *.txt textfiles/ moves all files with names ending in .txt from the current directory to the directory textfiles. Here, * is a wildcard standing for "any string of characters" and *.txt is a glob pattern. The other common wildcard is the question mark (?), which stands for one character. For example, mv ?.txt shorttextfiles/ will move all files named with a single character followed by .txt from the current directory to directory shorttextfiles, while ??.txt would match all files whose name consists of 2 characters followed by .txt.
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In automata theory, a finite-state machine is called a deterministic finite automaton (DFA), if
In the theory of computation, a generalized nondeterministic finite automaton (GNFA), also known as an expression automaton or a generalized nondeterministic finite state machine, is a variation of a nondeterministic finite automaton (NFA) where each transition is labeled with any regular expression. The GNFA reads blocks of symbols from the input which constitute a string as defined by the regular expression on the transition. There are several differences between a standard finite state machine and a generalized nondeterministic finite state machine. A GNFA must have only one start state and one accept state, and these cannot be the same state, whereas a NFA or DFA both may have several accept states, and the start state can be an accept state. A GNFA must have only one transition between any two states, whereas a NFA or DFA both allow for numerous transitions between states. In a GNFA, a state has a single transition to every state in the machine, although often it is a convention to ignore the transitions that are labelled with the empty set when drawing generalized nondeterministic finite state machines.
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. 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, countable, or even uncountable.
In theoretical computer science, Arden's rule, also known as Arden's lemma, is a mathematical statement about a certain form of language equations.
In computer science, Thompson's construction algorithm, also called the McNaughton–Yamada–Thompson algorithm, is a method of transforming a regular expression into an equivalent nondeterministic finite automaton (NFA). This NFA can be used to match strings against the regular expression. This algorithm is credited to Ken Thompson.
In computational learning theory, induction of regular languages refers to the task of learning a formal description of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way, approaches have been investigated for a variety of subclasses. They are sketched in this article. For learning of more general grammars, see Grammar induction.
In theoretical computer science, in particular in formal language theory, Kleene's algorithm transforms a given nondeterministic finite automaton (NFA) into a regular expression. Together with other conversion algorithms, it establishes the equivalence of several description formats for regular languages. Alternative presentations of the same method include the "elimination method" attributed to Brzozowski and McCluskey, the algorithm of McNaughton and Yamada, and the use of Arden's lemma.
References
- 1 2 Linz, Peter (2006). "Theorem 4.1". An Introduction to Formal Languages and Automata. Jones & Bartlett Learning. pp. 100–101. ISBN 9780763737986.
- ↑ Fitzgerald, Michael (2012). "Alternation". Introducing Regular Expressions: Unraveling Regular Expressions, Step-by-Step. O'Reilly Media. pp. 43–45. ISBN 9781449338893.
- ↑ "Alternation with The Vertical Bar". regular-expressions.info. Retrieved 2021-11-11.
- ↑ Cooper, Keith; Torczon, Linda (2011). Engineering a Compiler (2nd ed.). Elsevier. p. 41. ISBN 9780080916613.
