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. 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 for both 0 and 1. Upon reading a symbol, a DFA jumps deterministically from one state to another by following the transition arrow. For example, if the automaton is currently in state S0 and the current input symbol is 1, then it deterministically jumps to state S1. A DFA has a start state (denoted graphically by an arrow coming in from nowhere) where computations begin, and a set of accept states (denoted graphically by a double circle) which help define when a computation is successful. A DFA is defined as an abstract mathematical concept, but is often implemented in hardware and software for solving various specific problems such as lexical analysis and pattern matching. For example, a DFA can model software that decides whether or not online user input such as email addresses are syntactically valid. DFAs have been generalized to nondeterministic finite automata (NFA) which may have several arrows of the same label starting from a state. Using the powerset construction method, every NFA can be translated to a DFA that recognizes the same language. DFAs, and NFAs as well, recognize exactly the set of regular languages. == Formal definition == A deterministic finite automaton M is a 5-tuple, (Q, Σ, δ, q0, F), consisting of a finite set of states Q a finite set of input symbols called the alphabet Σ a transition function δ : Q × Σ → Q an initial (or start) state q 0 ∈ Q {\displaystyle q_{0}\in Q} a set of accepting (or final) states F ⊆ Q {\displaystyle F\subseteq Q} Let w = a1a2...an be a string over the alphabet Σ. The automaton M accepts the string w if a sequence of states, r0, r1, ..., rn, exists in Q with the following conditions: r0 = q0 ri+1 = δ(ri, ai+1), for i = 0, ..., n − 1 r n ∈ F {\displaystyle r_{n}\in F} . In words, the first condition says that the machine starts in the start state q0. The second condition says that given each character of string w, the machine will transition from state to state according to the transition function δ. The last condition says that the machine accepts w if the last input of w causes the machine to halt in one of the accepting states. Otherwise, it is said that the automaton rejects the string. The set of strings that M accepts is the language recognized by M and this language is denoted by L(M). A deterministic finite automaton without accept states and without a starting state is known as a transition system or semiautomaton. For more comprehensive introduction of the formal definition see automata theory. == Example == The following example is of a DFA M, with a binary alphabet, which requires that the input contains an even number of 0s. M = (Q, Σ, δ, q0, F) where Q = {S1, S2} Σ = {0, 1} q0 = S1 F = {S1} and δ is defined by the following state transition table: The state S1 represents that there has been an even number of 0s in the input so far, while S2 signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, M will finish in state S1, an accepting state, so the input string will be accepted. The language recognized by M is the regular language given by the regular expression (1) (0 (1) 0 (1)), where is the Kleene star, e.g., 1 denotes any number (possibly zero) of consecutive ones. == Variations == === Complete and incomplete === According to the above definition, deterministic finite automata are always complete: they define from each state a transition for each input symbol. While this is the most common definition, some authors use the term deterministic finite automaton for a slightly different notion: an automaton that defines at most one transition for each state and each input symbol; the transition function is allowed to be partial. When no transition is defined, such an automaton halts. === Local automata === A local automaton is a DFA, not necessarily complete, for which all edges with the same label lead to a single vertex. Local automata accept the class of local languages, those for which membership of a word in the language is determined by a "sliding window" of length two on the word. A Myhill graph over an alphabet A is a directed graph with vertex set A and subsets of vertices labelled "start" and "finish". The language accepted by a Myhill graph is the set of directed paths from a start vertex to a finish vertex: the graph thus acts as an automaton. The class of languages accepted by Myhill graphs is the class of local languages. === Randomness === When the start state and accept states are ignored, a DFA of n states and an alphabet of size k can be seen as a digraph of n vertices in which all vertices have k out-arcs labeled 1, ..., k (a k-out digraph). It is known that when k ≥ 2 is a fixed integer, with high probability, the largest strongly connected component (SCC) in such a k-out digraph chosen uniformly at random is of linear size and it can be reached by all vertices. It has also been proven that if k is allowed to increase as n increases, then the whole digraph has a phase transition for strong connectivity similar to Erdős–Rényi model for connectivity. In a random DFA, the maximum number of vertices reachable from one vertex is very close to the number of vertices in the largest SCC with high probability. This is also true for the largest induced sub-digraph of minimum in-degree one, which can be seen as a directed version of 1-core. == Closure properties == If DFAs recognize the languages that are obtained by applying an operation on the DFA recognizable languages then DFAs are said to be closed under the operation. The DFAs are closed under the following operations. For each operation, an optimal construction with respect to the number of states has been determined in state complexity research. Since DFAs are equivalent to nondeterministic finite automata (NFA), these closures may also be proved using closure properties of NFA. == As a transition monoid == A run of a given DFA can be seen as a sequence of compositions of a very general formulation of the transition function with itself. Here we construct that function. For a given input symbol a ∈ Σ {\displaystyle a\in \Sigma } , one may construct a transition function δ a : Q → Q {\displaystyle \delta _{a}:Q\rightarrow Q} by defining δ a ( q ) = δ ( q , a ) {\displaystyle \delta _{a}(q)=\delta (q,a)} for all q ∈ Q {\displaystyle q\in Q} . (This trick is called currying.) From this perspective, δ a {\displaystyle \delta _{a}} "acts" on a state in Q to yield another state. One may then consider the result of function composition repeatedly applied to the various functions δ a {\displaystyle \delta _{a}} , δ b {\displaystyle \delta _{b}} , and so on. Given a pair of letters a , b ∈ Σ {\displaystyle a,b\in \Sigma } , one may define a new function δ ^ a b = δ a ∘ δ b {\displaystyle {\widehat {\delta }}_{ab}=\delta _{a}\circ \delta _{b}} , where ∘ {\displaystyle \circ } denotes function composition. Clearly, this process may be recursively continued, giving the following recursive definition of δ ^ : Q × Σ ⋆ → Q {\displaystyle {\widehat {\delta }}:Q\times \Sigma ^{\star }\rightarrow Q} : δ ^ ( q , ϵ ) = q {\displaystyle {\widehat {\delta }}(q,\epsilon )=q} , where ϵ {\displaystyle \epsilon } is the empty string and δ ^ ( q , w a ) = δ a ( δ ^ ( q , w ) ) {\displaystyle {\widehat {\delta }}(q,wa)=\delta _{a}({\widehat {\delta }}(q,w))} , where w ∈ Σ ∗ , a ∈ Σ {\displaystyle w\in \Sigma ^{},a\in \Sigma } and q ∈ Q {\displaystyle q\in Q} . δ ^ {\displaystyle {\widehat {\delta }}} is defined for all words w ∈ Σ ∗ {\displaystyle w\in \Sigma ^{}} . A run of the DFA is a sequence of compositions of δ ^ {\displaystyle {\widehat {\delta }}} with itself. Repeated function composition forms a monoid. For the transition functions, this monoid is known as the transition monoid, or sometimes the transformation semigroup. The construction can also be reversed: given a δ ^ {\displaystyle {\wide
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