Finite state machine

A finite state machine (FSM) or finite automaton is a model of behavior composed of states, transitions and actions. A state stores 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: execute the action when entering the state
Exit action: execute the action when exiting the state
Input action: execute the action dependent on present state and input conditions
Transition action: execute the action when performing a certain transition

FSM can be represented using a state diagram (or state transition diagram) as in figure 1. 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).

Caption | 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 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 one type of the automata studied in automata theory and the theory of computation. In computer science, finite state machines are widely used in modelling of application behaviour, design of hardware digital systems, software engineering, compilers, and the study of computation and languages.

Classification

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

Acceptors and recognizers

This kind of machine gives 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. If when all input is processed 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.

Accept State

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 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 inputted and is therefore defined as an accepting state.

Transducers

Transducers generate output based on a given input and/or a state using actions. They are used for control applications. 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 3 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".

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.

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 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

The next state and output of a 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. An acceptor finite state machine is a quintuple (Σ, S, s₀, δ, F), where:

Σ is the input alphabet (a finite non empty set of symbols).
S is a finite non empty set of states.
s₀ is an initial state, an element of S. In a Nondeterministic finite state machine, s₀ is a set of initial states.
δ is the state transition function: δ: S x Σ → S.
F is the set of final states, a (possibly empty) subset of S.

A transducer finite state machine is a six tuple (Σ, Γ, S, s₀, δ, ω), where:

Σ is the input alphabet (a finite non empty set of symbols).
Γ is the output alphabet (a finite non empty set of symbols).
S is a finite non empty set of states.
s₀ is the initial state, an element of S. In a Nondeterministic finite state machine, s₀ is a set of initial states.
δ is the state transition function: δ: S x Σ → S x Γ.
ω is the output function.

If the output function is a function of a state and input alphabet (ω: S x Σ → Γ) that definition corresponds to the Mealy model. If the output function depends only on a state (ω: S → Γ) that definition corresponds to the Moore model.

Optimization

Optimizing an FSM means finding the machine with the minimum number of states that performs the same function. This problem can be solved using a coloring algorithm.

Implementation

Hardware applications

In a digital circuit, a 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 a FSM.

Software applications

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

Tools

AT&T FSM Library™ ^*
AutoFSM ^*
AsmL ^*
Bandera ^*
Boost Statechart Library ^*
CAZE - FSM-based .NET authorization library ^*
Cellogica - State machine for gene expression ^*
Covered ^*
Concurrent Hierarchical State Machine ^*
DescoGUI ^*
Dynamic Attachment Finite State Machine (DAFSM) ^*
Exorciser ^*
Finite State Kernel Creator ^*
Finite State Machine Editor ^*
Finite State Machine Explorer ^*

FIRE Engine, Station and Works ^*
FSMGenerator ^*
Grail+ ^*
FSA Utilities ^*
Libero ^*
Java Finite Automata ^*
JFLAP ^*
jFAST ^*
jrexx-Lab ^*
JSpasm ^*
Kara ^*
MetaAuto ^*
Nunni FSM Generator ^*
Petrify ^*
PyFSA ^*
Qfsm ^*
Quantum-Leaps ^*

Ragel ^*
RWTH FSA Toolkit ^*
SCXML (State Chart XML) ^*
SFST, the Stuttgart Finite State Transducer Tools ^*
Statestep ^*
StateWORKS ^*
State Machine Compiler ^*
Steed.net ^*
SMC - Finite State Machine Compiler (Java, C++) ^*
Supremica ^*
UniMod ^*
Visio2Switch ^*
visualSTATE ^*
WhatOS ^*
Xerox Finite-State Software Tools ^*
XTND (XML Transition Network Definition) ^*

References

Wagner, F., "Modeling Software with Finite State Machines: A Practical Approach", Auerbach Publications, 2006, ISBN 0-8493-8086-3.
Cassandras, C., Lafortune, S., "Introduction to Discrete Event Systems". Kluwer, 1999, ISBN 0-7923-8609-4.
Timothy Kam, Synthesis of Finite State Machines: Functional Optimization. Kluwer Academic Publishers, Boston 1997, ISBN 0-7923-9842-4
Tiziano Villa, Synthesis of Finite State Machines: Logic Optimization. Kluwer Academic Publishers, Boston 1997, ISBN 0-7923-9892-0
Carroll, J., Long, D. , Theory of Finite Automata with an Introduction to Formal Languages. Prentice Hall. Englewood Cliffs, 1989.
Hopcroft, J.E., Ullman, J.D., Introduction to Automata Theory, Languages and Computation. Addison -Wesley, 1979.
Kohavi, Z., Switching and Finite Automata Theory. McGraw-Hill, 1978.
Gill, A., Introduction to the Theory of Finite-state Machines. McGraw-Hill, 1962.
Ginsburg, S., An Introduction to Mathematical Machine Theory. Addison-Wesley, 1962.

External links

Description from the Free On-Line Dictionary of Computing
NIST Dictionary of Algorithms and Data Structures entry
Hierarchical State Machines

Computational models | Digital electronics | Formal methods

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