There are several types of finite state machines:
Acceptors produce a "yes or no" answer to the input; either accepting the input or not. Recognizers categorise the input. Transducers are used to generate an output from a given input.
Finite automata may operate on languages of finite words (the standard case), infinite words (Rabin automata, Büchi automata[?]), or various types of trees (tree automata), to name the most important cases.
A further distinction is between deterministic and non-deterministic automata. In deterministic automata, for each state there is at most one transition for each possible input. In non-deterministic automata, there can be more than one transition from a given state for a given possible input. Non-deterministic automata are usually implemented by converting them to deterministic automata - in the worst case, the generated deterministic automaton is exponentially bigger than the non-deterministic automaton (although it can usually be substantially optimised).
The standard acceptance condition for non-deterministic automata requires that some computation accepts the input. Alternating automata[?] also provide a dual notion, where for acceptance all non-deterministic computations must accept.
Apart from theory, finite state machines occur also in hardware circuits, where the input, the state and the output are bit vectors of fixed size (Moore and Mealy machines).
Mealy machines have actions (outputs) associated with transitions and Moore machines have actions associated with states.
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Formal definitions
Deterministic finite automaton
Formally, a deterministic finite automaton (DFA) consists of
- an alphabet (Σ),
- a set of states (S), one of which is chosen as a start state and zero or more as accepting states, and
- a transition function (T : S × Σ -> S).
The machine starts in the start state and reads in a string of symbols from its alphabet. It uses the transition function T to determine the next state using the current state and the symbol just read. If, when it has finished reading, it is in an accepting state, it is said to accept the string, otherwise it is said to reject the string. The set of strings it accepts form a language, which is the language the DFA recognises.
Example of a deterministic finite state machine
The following example explains a deterministic finite state machine with a binary alphabet, which determines if the input contains an even number of 0s.
- The alphabet Σ = {0, 1} (the two binary digits)
- The machine has two states, S1 and S2
- S1 is both the start state and the only accepting state
- The transition function T is visualised by the directed graph shown on the right, and defined as follows:
- T(S1, 0) = S2
- T(S1, 1) = S1
- T(S2, 0) = S1
- T(S2, 1) = S2
Simply put, 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.
Non-deterministic finite automaton
A non-deterministic finite automaton (NFA) consists of
- an alphabet (Σ)
- a set of states (S), one or more of which are identified as the start state(s) and zero or more as the accepting state(s)
- a transition relation (T : S × Σ -> S)
The machine starts in all of the start states and reads in a string of symbols from its alphabet. It uses the transition relation T to determine the next state(s) using the current state(s) and the symbol just read. If, when it has finished reading, it is in an accepting state, it is said to accept the string, otherwise it is said to reject the string. The set of strings it accepts form a language, which is the language the NFA recognises.
Optimization and Canonicalisation
The problem of optimizing an FSM (finding the machine with the least number of states that performs the same function) is decidable, unlike the same problem for more computationally powerful machines. Furthermore, it is possible to construct a canonical version of any FSM, in order to test for equality. Both of these problems can be solved using a colouring algorithm.
Computational power[?]
FSMs can only recognize regular languages, and hence they are less computationally powerful than Turing machines - there are decidable problems that are not computable using a FSM.
For each non-deterministic FSM a deterministic FSM of equal computational power can be constructed with an algorithm.
Representation
A FSM may be represented using a state transition table or a state diagram.
Implementation
A finite state machine can be implemented in software with a state transition matrix (in some cases a sparse matrix[?] implemented with linked lists or a huge switch-statement for detecting the internal state and then individual switch statements for decoding the input symbol.In hardware a FSM may be built from a programmable logic device.
See also
References