In thetheory of computation, a branch oftheoretical computer science, adeterministic finite automaton (DFA)—also known asdeterministic finite acceptor (DFA),deterministic finite-state machine (DFSM), ordeterministic finite-state automaton (DFSA)—is afinite-state machine that accepts or rejects a givenstring of symbols, by running through a state sequence uniquely determined by the string.[1]Deterministic refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines,Warren McCulloch andWalter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943.[2][3]
The figure illustrates a deterministic finite automaton using astate diagram. In this example automaton, there are three states: S0, S1, and S2 (denoted graphically by circles). The automaton takes a finitesequence 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 jumpsdeterministically 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 astart state (denoted graphically by an arrow coming in from nowhere) where computations begin, and aset ofaccept 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 aslexical analysis andpattern matching. For example, a DFA can model software that decides whether or not online user input such as email addresses are syntactically valid.[4]
DFAs have been generalized tonondeterministic finite automata (NFA) which may have several arrows of the same label starting from a state. Using thepowerset construction method, every NFA can be translated to a DFA that recognizes the same language. DFAs, and NFAs as well, recognize exactly the set ofregular languages.[1]
A deterministic finite automatonM is a 5-tuple,(Q, Σ,δ,q0,F), consisting of
Letw =a1a2...an be a string over the alphabetΣ. The automatonM accepts the stringw if a sequence of states,r0,r1, ...,rn, exists inQ with the following conditions:
In words, the first condition says that the machine starts in the start stateq0. The second condition says that given each character of stringw, the machine will transition from state to state according to the transition functionδ. The last condition says that the machine acceptsw if the last input ofw causes the machine to halt in one of the accepting states. Otherwise, it is said that the automatonrejects the string. The set of strings thatM accepts is thelanguagerecognized byM and this language is denoted byL(M).
A deterministic finite automaton without accept states and without a starting state is known as atransition system orsemiautomaton.
For more comprehensive introduction of the formal definition seeautomata theory.
The following example is of a DFAM, with a binary alphabet, which requires that the input contains an even number of 0s.
M = (Q, Σ,δ,q0,F) where
0 | 1 | |
| S1 | S2 | S1 |
| S2 | S1 | S2 |
The stateS1 represents that there has been an even number of 0s in the input so far, whileS2 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 stateS1, an accepting state, so the input string will be accepted.
The language recognized byM is theregular language given by theregular expression(1*) (0 (1*) 0 (1*))*, where* is theKleene star, e.g.,1* denotes any number (possibly zero) of consecutive ones.
According to the above definition, deterministic finite automata are alwayscomplete: 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 definesat most one transition for each state and each input symbol; the transition function is allowed to bepartial.[5] When no transition is defined, such an automaton halts.
Alocal 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 oflocal languages, those for which membership of a word in the language is determined by a "sliding window" of length two on the word.[6][7]
AMyhill graph over an alphabetA is adirected graph withvertex setA 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.[6] The class of languages accepted by Myhill graphs is the class of local languages.[8]
When the start state and accept states are ignored, a DFA ofn states and an alphabet of sizek can be seen as adigraph ofn vertices in which all vertices havek out-arcs labeled1, ...,k (ak-out digraph). It is known that whenk ≥ 2 is a fixed integer, with high probability, the largeststrongly connected component (SCC) in such ak-out digraph chosen uniformly at random is of linear size and it can be reached by all vertices.[9] It has also been proven that ifk is allowed to increase asn increases, then the whole digraph has a phase transition for strong connectivity similar toErdős–Rényi model for connectivity.[10]
In a random DFA, the maximum number of vertices reachable from one vertex is very close to the number of vertices in the largestSCC with high probability.[9][11] This is also true for the largestinduced sub-digraph of minimum in-degree one, which can be seen as a directed version of1-core.[10]
If DFAs recognize the languages that are obtained by applying an operation on the DFA recognizable languages then DFAs are said to beclosed 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 instate complexity research.Since DFAs areequivalent tonondeterministic finite automata (NFA), these closures may also be proved using closure properties of NFA.
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, one may construct a transition function by defining for all. (This trick is calledcurrying.) From this perspective, "acts" on a state in Q to yield another state. One may then consider the result offunction composition repeatedly applied to the various functions,, and so on. Given a pair of letters, one may define a new function, where denotes function composition.
Clearly, this process may be recursively continued, giving the following recursive definition of:
is defined for all words. A run of the DFA is a sequence of compositions of with itself.
Repeated function composition forms amonoid. For the transition functions, this monoid is known as thetransition monoid, or sometimes thetransformation semigroup. The construction can also be reversed: given a, one can reconstruct a, and so the two descriptions are equivalent.
DFAs are one of the most practical models of computation, since there is a trivial linear time, constant-space,online algorithm to simulate a DFA on a stream of input. Also, there are efficient algorithms to find a DFA recognizing:
Because DFAs can be reduced to acanonical form (minimal DFAs), there are also efficient algorithms to determine:
DFAs are equivalent in computing power tonondeterministic finite automata (NFAs). This is because, firstly any DFA is also an NFA, so an NFA can do what a DFA can do. Also, given an NFA, using thepowerset construction one can build a DFA that recognizes the same language as the NFA, although the DFA could have exponentially larger number of states than the NFA.[15][16] However, even though NFAs are computationally equivalent to DFAs, the above-mentioned problems are not necessarily solved efficiently also for NFAs. The non-universality problem for NFAs isPSPACE complete since there are small NFAs with shortest rejecting word in exponential size. A DFA is universal if and only if all states are final states, but this does not hold for NFAs. The Equality, Inclusion and Minimization Problems are also PSPACE complete since they require forming the complement of an NFA which results in an exponential blow up of size.[17]
On the other hand, finite-state automata are of strictly limited power in the languages they can recognize; many simple languages, including any problem that requires more than constant space to solve, cannot be recognized by a DFA. The classic example of a simply described language that no DFA can recognize is bracket orDyck language, i.e., the language that consists of properly paired brackets such as word "(()())". Intuitively, no DFA can recognize the Dyck language because DFAs are not capable of counting: a DFA-like automaton needs to have a state to represent any possible number of "currently open" parentheses, meaning it would need an unbounded number of states. Another simpler example is the language consisting of strings of the formanbn for some finite but arbitrary number ofa's, followed by an equal number ofb's.[18]
Given a set ofpositive words and a set ofnegative words one can construct a DFA that accepts all words from and rejects all words from: this problem is calledDFA identification (synthesis, learning).Whilesome DFA can be constructed in linear time, the problem of identifying a DFA with the minimal number of states is NP-complete.[19]The first algorithm for minimal DFA identification has been proposed by Trakhtenbrot and Barzdin[20] and is called theTB-algorithm.However, the TB-algorithm assumes that all words from up to a given length are contained in either.
Later, K. Lang proposed an extension of the TB-algorithm that does not use any assumptions about and, theTraxbar algorithm.[21]However, Traxbar does not guarantee the minimality of the constructed DFA.In his work[19] E.M. Gold also proposed a heuristic algorithm for minimal DFA identification.Gold's algorithm assumes that and contain acharacteristic set of the regular language; otherwise, the constructed DFA will be inconsistent either with or.Other notable DFA identification algorithms include the RPNI algorithm,[22] the Blue-Fringe evidence-driven state-merging algorithm,[23] and Windowed-EDSM.[24]Another research direction is the application ofevolutionary algorithms: the smart state labeling evolutionary algorithm[25] allowed to solve a modified DFA identification problem in which the training data (sets and) isnoisy in the sense that some words are attributed to wrong classes.
Yet another step forward is due to application ofSAT solvers byMarjin J. H. Heule and S. Verwer: the minimal DFA identification problem is reduced to deciding the satisfiability of a Boolean formula.[26] The main idea is to build an augmented prefix-tree acceptor (atrie containing all input words with corresponding labels) based on the input sets and reduce the problem of finding a DFA with states tocoloring the tree vertices with states in such a way that when vertices with one color are merged to one state, the generated automaton is deterministic and complies with and.Though this approach allows finding the minimal DFA, it suffers from exponential blow-up of execution time when the size of input data increases.Therefore, Heule and Verwer's initial algorithm has later been augmented with making several steps of the EDSM algorithm prior to SAT solver execution: the DFASAT algorithm.[27]This allows reducing the search space of the problem, but leads to loss of the minimality guarantee.Another way of reducing the search space has been proposed by Ulyantsev et al.[28] by means of new symmetry breaking predicates based on thebreadth-first search algorithm:the sought DFA's states are constrained to be numbered according to the BFS algorithm launched from the initial state. This approach reduces the search space by by eliminating isomorphic automata.
Read-only right-moving Turing machines are a particular type ofTuring machine that only moves right; theseare almost exactly equivalent to DFAs.[29]The definition based on a singly infinite tape is a 7-tuple
where
The machine always accepts a regular language. There must exist at least one element of the setF (aHALT state) for the language to be nonempty.
| Current stateA | Current stateB | Current stateC | |||||||
| tape symbol | Write symbol | Move tape | Next state | Write symbol | Move tape | Next state | Write symbol | Move tape | Next state |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | R | B | 1 | R | A | 1 | R | B |
| 1 | 1 | R | C | 1 | R | B | 1 | N | HALT |