Formal Definition Of a Finite Automata

In the theory of computation and automata theory, a deterministic finite state machine—also known as deterministic finite automaton (DFA)—is a finite state machine accepting finite strings of symbols. For each state, there is a transition arrow leading out to a next state for each symbol. Upon reading a symbol, a DFA jumps deterministically from a state to another by following the transition arrow. Deterministic means that there is only one outcome (i.e. move to next state when the symbol matches (S0 -> S1) or move back to the same state (S0 -> S0)). 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.

DFAs recognize exactly the set of regular languages which are, among other things, useful for doing lexical analysis and pattern matching. ^[1] A DFA can be used in either an accepting mode to verify that an input string is indeed part of the language it represents, or a generating mode to create a list of all the strings in the language.

DFA is defined as an abstract mathematical concept, but due to the deterministic nature of DFA, it is implementable in hardware and software for solving various specific problems. For example, a software state machine that decides whether or not online user-input such as phone numbers and email addresses are valid. ^[2] Another example in hardware is the digital logic circuitry that controls whether an automatic door is open or closed, using input from motion sensors or pressure pads to decide whether or not to perform a state transition (see: finite state machine).

In the theory of computation, a nondeterministic finite state machine or nondeterministic finite automaton (NFA) is a finite state machine where for each pair of state and input symbol there may be several possible next states. This distinguishes it from the deterministic finite automaton (DFA), where the next possible state is uniquely determined. Although the DFA and NFA have distinct definitions, it may be shown in the formal theory that they are equivalent, in that, for any given NFA, one may construct an equivalent DFA, and vice-versa: this is the powerset construction. Both types of automata recognize only regular languages. Non-deterministic finite state machines are sometimes studied by the name subshifts of finite type. Non-deterministic finite state machines are generalized by probabilistic automata, which assign a probability to each state transition.

Nondeterministic finite automata were introduced in 1959 by Michael O. Rabin and Dana Scott,^[1] who also showed their equivalence to deterministic finite automata.

Two similar types of NFAs are commonly defined: the NFA and the NFA with ε-moves. The ordinary is defined as a 5-tuple, (Q, Σ, T, q₀, F), consisting of

• a finite set of states Q
• a finite set of input symbols Σ
• a transition function T : Q × Σ → P(Q).
• an initial (or start) state q₀ ∈ Q
• a set of states F distinguished as accepting (or final) states F ⊆ Q.

Here, P(Q) denotes the power set of Q. The NFA with ε-moves (also sometimes called NFA-epsilon or NFA-lambda) replaces the transition function with one that allows the empty string ε as a possible input, so that one has instead

T : Q × (Σ ∪{ε}) → P(Q).

It can be shown that ordinary NFA and NFA with epsilon moves are equivalent, in that, given either one, one can construct the other, which recognizes the same language.

The machine starts in the specified initial state and reads in a string of symbols from its alphabet. The automaton uses the state transition function T to determine the next state using the current state, and the symbol just read or the empty string. However, "the next state of an NFA depends not only on the current input event, but also on an arbitrary number of subsequent input events. Until these subsequent events occur it is not possible to determine which state the machine is in".^[2] If, when the automaton has finished reading, it is in an accepting state, the NFA is said to accept the string, otherwise it is said to reject the string.

The set of all strings accepted by an NFA is the language the NFA accepts. This language is a regular language.

For every NFA a deterministic finite state machine (DFA) can be found that accepts the same language. Therefore it is possible to convert an existing NFA into a DFA for the purpose of implementing a (perhaps) simpler machine. This can be performed using the powerset construction, which may lead to an exponential rise in the number of necessary states. A formal proof of the powerset construction is given here