Concepts and vocabulary

Entry action: which is performed when entering the state ;Exit action: which is performed when exiting the state

Input action: which is performed depending on present state and input conditions ;Transition action: which is performed when performing a certain transition
   A FSM can be represented using a state diagram (or state transition diagram) as in figure 1 above. Besides this, several state transition table types are used. The most common representation is shown below: the combination of current state (B) and condition (Y) shows the next state (C). The complete actions information can be added only using footnotes. An FSM definition including the full actions information is possible using state tables (see also VFSM).

Classification

Acceptors and recognizers

Start state

Accept state

Transducers

Moore machine: The FSM uses only entry actions, for example output depends only on the state. The advantage of the Moore model is a simplification of the behaviour. The example in figure 1 shows a Moore FSM of an elevator door. The state machine recognizes two commands: "command_open" and "command_close" which trigger state changes. The entry action (E:) in state "Opening" starts a motor opening the door, the entry action in state "Closing" starts a motor in the other direction closing the door. States "Opened" and "Closed" don't perform any actions. They signal to the outside world (for example to other state machines) the situation: "door is open" or "door is closed".

Mealy machine: The FSM uses only input actions, for example output depends on input and state. The use of a Mealy FSM leads often to a reduction of the number of states. The example in figure 4 shows a Mealy FSM implementing the same behaviour as in the Moore example (the behaviour depends on the implemented FSM execution model and will work for example for virtual FSM but not for event driven FSM). There are two input actions (I:): "start motor to close the door if command_close arrives" and "start motor in the other direction to open the door if command_open arrives". In practice mixed models are often used.
   More details about the differences and usage of Moore and Mealy models, including an executable example, can be found in the external technical note "Moore or Mealy model?" A further distinction is between deterministic (DFA) and non-deterministic (NDFA, GNFA) automata. In deterministic automata, for each state there's exactly one transition for each possible input. In non-deterministic automata, there can be none or more than one transition from a given state for a given possible input. This distinction is relevant in practice, but not in theory, as there exists an algorithm which can transform any NDFA into an equivalent DFA, although this transformation typically significantly increases the complexity of the automaton.
   The FSM with only one state is called a combinatorial FSM and uses only input actions. This concept is useful in cases where a number of FSM are required to work together, and where it's convenient to consider a purely combinatorial part as a form of FSM to suit the design tools.

FSM logic

Mathematical model

• $Sigma$ is the input alphabet (a finite, non-empty set of symbols).
• $S$ is a finite, non-empty set of states.
• $s\_0$ is an initial state, an element of $S$. In a Nondeterministic finite state machine, $s\_0$ is a set of initial states.
• $delta$ is the state-transition function: $delta:\; S\; imes\; Sigma\; ightarrow\; S$.
• $F$ is the set of final states, a (possibly empty) subset of $S$.

$omega$ is the output function. If the output function is a function of a state and input alphabet ($omega:\; S\; imes\; Sigma\; ightarrow\; Gamma$) that definition corresponds to the Mealy model, and can be modelled as a Mealy machine. If the output function depends only on a state ($omega:\; S\; ightarrow\; Gamma$) that definition corresponds to the Moore model, and can be modelled as a Moore machine. A finite-state machine with no output function at all is known as a semiautomaton or transition system.

Optimization

Optimizing an FSM means finding the machine with the minimum number of states that performs the same function. One possibility is by using an Implication table or the Moore reduction procedure. Another possibility is bottom-up algorithm for Acyclic FSAs.

Implementation

Hardware applications

In a digital circuit, an FSM may be built using a programmable logic device, a programmable logic controller, logic gates and flip flops or relays. More specifically, a hardware implementation requires a register to store state variables, a block of combinational logic which determines the state transition, and a second block of combinational logic that determines the output of an FSM. One of the classic hardware implementations is the Richard's Controller.

Software applications

The following concepts are commonly used to build software applications with finite state machines: