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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.

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).

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.

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.

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.

S_{1} (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.

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, 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.

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.

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

A finite state transducer is a sextuple $(Sigma,\; Gamma,\; S,\; s\_0,\; delta,\; omega)$, 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.

- 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
- Tool for design and visualization of Finite State Machines
- 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.
- bottom-up algorithm for Acyclic FSAs - FSM minimization algorithms.
- 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.

- Wagner, F., "Modeling Software with Finite State Machines: A Practical Approach", Auerbach Publications, 2006, ISBN 0-8493-8086-3.
- Samek, M., Practical Statecharts in C/C++, CMP Books, 2002, ISBN 1-57820-110-1.
- Samek, M., Practical UML Statecharts in C/C++, 2nd Edition, Newnes, 2008, ISBN 0-75068-706-1.
- Gardner, T., Advanced State Management, 2007 (video at 360Flex 2008)
- 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.
- 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.

- Arbib, Michael A. (1969).
*Theories of Abstract Automata*. 1st ed., Englewood Cliffs, N.J.: Prentice-Hall, Inc.. - Bobrow, Leonard S.; Michael A. Arbib (1974).
*Discrete Mathematics: Applied Algebra for Computer and Information Science*. 1st ed., Philadelphia: W. B. Saunders Company, Inc.. - Booth, Taylor L. (1967).
*Sequential Machines and Automata Theory*. 1st, New York: John Wiley and Sons, Inc.. Library of Congress Card Catalog Number 67-25924. Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes. - Boolos, George; Richard Jeffrey (1989, 1999).
*Computability and Logic*. 3rd ed., Cambridge, England: Cambridge University Press. ISBN 0-521-20402-X. Excellent. Has been in print in various editions and reprints since 1974 (1974, 1980, 1989, 1999). - Brookshear, J. Glenn (1989).
*Theory of Computation: Formal Languages, Automata, and Complexity*. Redwood City, California: Benjamin/Cummings Publish Company, Inc.. ISBN 0-8053-0143-7. Approaches Church-Turing thesis from three angles: levels of finite automata as acceptors of formal languages, primitive and partial recursive theory, and power of bare-bones programming languages to implement algorithms, all in one slim volume. - Davis, Martin; Ron Sigal, Elaine J. Weyuker (1994).
*Second Edition: Computability, Complexity, and Languages and Logic: Fundamentals of Theoretical Computer Science*. 2nd ed., San Diego: Academic Press, Harcourt, Brace & Company. - Hopcroft, John E (1971). An n log n algorithm for minimizing states in a finite automaton
- Hopcroft, John; Jeffrey Ullman (1979).
*Introduction to Automata Theory, Languages and Computation*. 1st ed., Reading Mass: Addison-Wesley. ISBN 0-201-02988-X. A difficult book centered around the issues of machine-interpretation of "languages", NP-Completeness, etc. - Hopcroft, John E.; Rajeev Motwani, Jeffrey D. Ullman (2001).
*Introduction to Automata Theory, Languages, and Computation*. 2nd ed., Reading Mass: Addison-Wesley. Distinctly different and less intimidating than the first edition. - Hopkin, David; Barbara Moss (1976).
*Automata*. New York: Elsevier North-Holland. ISBN 0-444-00249-9. - Kozen, Dexter C. (1997).
*Automata and Computability*. 1st ed., New York: Springer-Verlag. ISBN 0-387-94907-0. - Lewis, Harry R.; Christos H. Papadimitriou (1998).
*Elements of the Theory of Computation*. 2nd, Upper Saddle River, New Jersey: Prentice-Hall. ISBN 0-13-262478-8. - Linz, Peter (2006).
*Formal Languages and Automata*. 4th, Sudbury, MA: Jones and Bartlett. ISBN-13: 978-0-7637-3798-6. - Minsky, Marvin (1967).
*Computation: Finite and Infinite Machines*. 1st, New Jersey: Prentice-Hall. Minsky spends pages 11-20 defining what a “state” is in context of FSMs. His state diagram convention is unconventional. Excellent, i.e. relatively readable, sometimes funny. - Christos Papadimitriou (1993).
*Computational Complexity*. 1st edition, Addison Wesley. ISBN 0-201-53082-1. - Pippenger, Nicholas (1997).
*Theories of Computability*. 1st, Cambridge, England: Cambridge University Press. 0-521-55380-6 (hc). Abstract algebra is at the core of the book, rendering it advanced and less accessible than other texts. - Rodger, Susan; Thomas Finley (2006).
*JFLAP: An Interactive Formal Languages and Automata Package*. 1st, Sudbury, MA: Jones and Bartlett. ISBN-10: 0763738344. - Sipser, Michael (2006).
*Introduction to the Theory of Computation, Second Edition*. 2nd, Boston Mass: Thomson Course Technology. ISBN-10: 0-534-95097-3. cf Finite state machines (finite automata) in chapter 29. - Wood, Derick (1987).
*Theory of Computation*. 1st, New York: Harper & Row, Publishers, Inc.. ISBN-10: 0-06-047208-1. - Almeida, Marco; Moreira, Nelma; Reis, Rogerio (2007). On the performance of automata minimization algrotims.

- Yuri Gurevich (2000), Sequential Abstract State Machines Capture Sequential Algorithms, ACM Transactions on Computational Logic, vl. 1, no. 1 (July 2000), pages 77-111. http://research.microsoft.com/~gurevich/Opera/141.pdf

- Mitchell, Tom M. (1997).
*Machine Learning*. 1st, New York: WCB/McGraw-Hill Corporation. ISBN 0-07-042807-7. A broad brush but quite thorough and sometimes difficult, meant for computer scientists and engineers. Chapter 13 Reinforcement Learning deals with robot-learning involving state-machine-like algorithms.

- Booth, Taylor L. (1967).
*Sequential Machines and Automata Theory*. 1st, New York: John Wiley and Sons, Inc.. Library of Congress Card Catalog Number 67-25924. Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes. - Booth, Taylor L. (1971).
*Digital Networks and Computer Systems*. 1st, New York: John Wiley and Sons, Inc.. ISBN 0-471-08840-4. Meant for electrical engineers. More focused, less demanding than his earlier book. His treatment of computers is out-dated. Interesting take on definition of ‘algorithm’. - McCluskey, E. J. (1965).
*Introduction to the Theory of Switching Circuits*. 1st, New York: McGraw-Hill Book Company, Inc.. Library of Congress Card Catalog Number 65-17394. Meant for hardware electrical engineers. With detailed explanations of state minimization techniques and synthesis techniques for design of combinatory logic circuits. - Hill, Fredrick J.; Gerald R. Peterson (1965).
*Introduction to the Theory of Switching Circuits*. 1st, New York: McGraw-Hill Book Company. Library of Congress Card Catalog Number 65-17394. Meant for hardware electrical engineers. Excellent explanations of state minimization techniques and synthesis techniques for design of combinatory and sequential logic circuits.

- "We may think of a Markov chain as a process that moves successively through a set of states s1, s2, ..., sr. ... if it is in state si it moves on to the next stop to state sj with probability pij. These probabilities can be exhibited in the form of a transition matrix" (Kemeny (1959), p. 384)

- Booth, Taylor L. (1967).
*Sequential Machines and Automata Theory*. 1st, New York: John Wiley and Sons, Inc.. Library of Congress Card Catalog Number 67-25924. Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes. - Kemeny, John G.; Hazleton Mirkil, J. Laurie Snell, Gerald L. Thompson (1959).
*Finite Mathematical Structures*. 1st, Englewood Cliffs, N.J.: Prentice-Hall, Inc.. Library of Congress Card Catalog Number 59-12841. Classical text . cf Chapter 6 ‘’Finite Markov Chains”.

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Last updated on Wednesday October 01, 2008 at 19:37:08 PDT (GMT -0700)

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