Finite state machine

Fig.1 Example of a Finite State Machine

A finite state machine (FSM) or finite state automaton (plural: automata) or simply a state machine, is a model of behavior composed of a finite number of states, transitions between those states, and actions. A finite state machine is an abstract model of a machine with a primitive internal memory.

Concepts and vocabulary

A current state is determined by past states of the system. As such, it can be said to record information about the past, i.e. it reflects the input changes from the system start to the present moment. A transition indicates a state change and is described by a condition that would need to be fulfilled to enable the transition. An action is a description of an activity that is to be performed at a given moment. There are several action types:

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

An 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).

State transition table
Current State → Condition	State A	State B	State C
Condition X	...	...	...
Condition Y	...	State C	...
Condition Z	...	...	...

In addition to their use in modeling reactive systems presented here, finite state automata are significant in many different areas, including electrical engineering, linguistics, computer science, philosophy, biology, mathematics, and logic. A complete survey of their applications is outside the scope of this article. Finite state machines are a class of automata studied in automata theory and the theory of computation. In computer science, finite state machines are widely used in modeling of application behavior, design of hardware digital systems, software engineering, compilers, network protocols, and the study of computation and languages.

Classification

There are two different groups: Acceptors/Recognizers and Transducers.

Acceptors and recognizers

Fig. 2 Acceptor FSM: parsing the word "nice"

Acceptors and recognizers (also sequence detectors) produce a binary output, saying either yes or no to answer whether the input is accepted by the machine or not. All states of the FSM are said to be either accepting or not accepting. At the time when all input is processed, if the current state is an accepting state, the input is accepted; otherwise it is rejected. As a rule the input are symbols (characters); actions are not used. The example in figure 2 shows a finite state machine which accepts the word "nice". In this FSM the only accepting state is number 7.

The machine can also be described as defining a language, which would contain every word accepted by the machine but none of the rejected ones; we say then that the language is accepted by the machine. By definition, the languages accepted by FSMs are the regular languages - that is, a language is regular if there is some FSM that accepts it.

Start state

The start state is usually shown drawn with an arrow "pointing at it from nowhere" (Sipser (2006) p.34).

Accept state

Fig. 3: A finite state machine that determines if a binary number has an odd or even number of 0s.

An accept state (sometimes referred to as an accepting state) is a state at which the machine has successfully performed its procedure. It is usually represented by a double circle.

An example of an accepting state appears on the left in this diagram of a deterministic finite automaton (DFA) which determines if the binary input contains an even number of 0s.

S₁ (which is also the start state) indicates the state at which an even number of 0s has been input and is therefore defined as an accepting state. This machine will give a correct end state if the binary number contains an even number of zeros including a string with no zeros. Examples of strings accepted by this DFA are epsilon (the empty string), 1, 11, 11..., 00, 010, 1010, 10110 and so on.

Transducers

Transducers generate output based on a given input and/or a state using actions. They are used for control applications and in the field of computational linguistics. Here two types are distinguished:

Moore machine: The FSM uses only entry actions, i.e. 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 (e.g. to other state machines) the situation: "door is open" or "door is closed".

Fig. 4 Transducer FSM: Mealy model example

Mealy machine: The FSM uses only input actions, i.e. 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 e.g. 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 including the predicate transition state machine invented by Andrew Wightman.

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 is exactly one transition for each possible input. In non-deterministic automata, there can be none, one, 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 is convenient to consider a purely combinatorial part as a form of FSM to suit the design tools.

FSM logic

Fig. 5 FSM Logic (Mealy)

The next state and output of an FSM is a function of the input and of the current state. The FSM logic is shown in Figure 5.

Mathematical model

Depending on the type there are several definitions.

A deterministic finite state machine is a quintuple , where:

$\Sigma$ 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 $S$ .
$\delta$ is the state-transition function: $\delta: S \times \Sigma \rightarrow S$ . In a Nondeterministic finite state machine, $\delta: S \times \Sigma \rightarrow \mathcal{P}(S)$ , $\delta$ returns a set of states.
$F$ is the set of final states, a (possibly empty) subset of $S$ .

An acceptor finite-state machine is a quintuple , where:

$\Sigma$ 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 $S$ .
$\delta$ is the state-transition function: $\delta: S \times \Sigma \rightarrow S$ .
$F$ is the set of final states, a (possibly empty) subset of $S$ .

A finite state transducer is a sextuple , where:

$\Sigma$ is the input alphabet (a finite non empty set of symbols).
$\Gamma$ is the output alphabet (a finite, non-empty set of symbols).
$S$ is a finite, non-empty set of states.
$s_0$ is the 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: $\delta: S \times \Sigma \rightarrow S$ .
$\omega$ is the output function.

If the output function is a function of a state and input alphabet ( $\omega: S \times \Sigma \rightarrow \Gamma$ ) that definition corresponds to the Mealy model, and can be modelled as a Mealy machine. If the output function depends only on a state ( $\omega: S \rightarrow \Gamma$ ) 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. The fastest known algorithm doing this is the Hopcroft minimization algorithm.^[1]^[2]. Other techniques include using an Implication table, or the Moore reduction procedure. Additionally, acyclic FSAs can be optimized using a simple bottom up algorithm.

Implementation

Hardware applications

Fig. 6 The circuit diagram for a 4 bit TTL counter, a type of state machine

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:

event driven FSM
virtual FSM (VFSM)
Automata-based programming

External links

Description from the Free On-Line Dictionary of Computing
NIST Dictionary of Algorithms and Data Structures entry
Hierarchical State Machines
COGs - Cogs is a open source, small FSM and HSM ( hierarchical state machine) library for use with Flash / Flex Actionscript 3.0 projects. An introductory talk here
FSM Generator - FSMGenerator is a turn-key solution for FSM (Finite State Machine) automatic generation and integration within user's software
FSM Designer - An open-source tool for the graphical development of FSMs and automatic Verilog code generation
Round-trip Engineering State Machines
Using state machines in practical applications
sinelaboreRT - generates human readable c-code from state-charts especially targeting embedded real-time systems
libfa - An open-source C library for automata computations
The State Machine Compiler - A tool to model and compile statemachines for a number of languages; java, C# etc. As well as display them as graphically.
Quantum Leaps - Provider of open source, state machine-based frameworks. Includes several examples and C/C++ code.
gFSM - An open source implementation of FSM (Finite State Machine) programming framework in C.
Vaucanson - An open source C++ finite state machine manipulation platform, consisting of a library and tools implemented on top of it.
UniMod - An open source plugin for Eclipse for automata-based programming.
PureMVC StateMachine Utility - A simple but effective FSM implementation integrated seamlessly into the PureMVC Framework.

Automata theory: formal languages and formal grammars

Chomsky hierarchy	Grammars	Languages	Minimal automaton

Type-0 n/a Type-1 n/a n/a Type-2 n/a Type-3 n/a	Unrestricted (no common name) Context-sensitive Indexed Tree-adjoining etc. Context-free Deterministic context-free Regular n/a	Recursively enumerable Recursive Context-sensitive Indexed (Mildly context-sensitive) Context-free Deterministic context-free Regular Star-free	Turing machine Decider Linear-bounded Nested stack Embedded pushdown Nondeterministic pushdown Deterministic pushdown Finite Aperiodic finite

Each category of languages or grammars is a proper subset of the category directly above it;
and any automaton in each category has an equivalent automaton in the category directly above it.

Finite state machine

Contents

Concepts and vocabulary

Classification

Acceptors and recognizers

Start state

Accept state

Transducers

FSM logic

Mathematical model

Optimization

Implementation

Hardware applications

Software applications

See also

Further reading

General

Finite state machines (automata theory) in theoretical computer science

Abstract state machines in theoretical computer science

Machine learning using finite-state algorithms

Hardware engineering: state minimization and synthesis of sequential circuits

Finite Markov chain processes

External links