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.

Concepts and vocabulary

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

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).

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

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 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 not give a correct end state if the binary number does not contain any zeros.

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.

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

Fig. 5 FSM Logic

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. An acceptor finite-state machine is a quintuple Failed to parse (Missing texvc executable; please see math/README to configure.): (\Sigma, S, s_0, \delta, F) , where:

• Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma

is the input alphabet (a finite, non-empty set of symbols).

• Failed to parse (Missing texvc executable; please see math/README to configure.): S

is a finite, non-empty set of states.

• Failed to parse (Missing texvc executable; please see math/README to configure.): s_0

is an initial state, an element of Failed to parse (Missing texvc executable; please see math/README to configure.): S

. In a Nondeterministic finite state machine, Failed to parse (Missing texvc executable; please see math/README to configure.): s_0

is a set of initial states.

• Failed to parse (Missing texvc executable; please see math/README to configure.): \delta

is the state-transition function: Failed to parse (Missing texvc executable; please see math/README to configure.): \delta: S \times \Sigma \rightarrow S

.

• Failed to parse (Missing texvc executable; please see math/README to configure.): F

is the set of final states, a (possibly empty) subset of Failed to parse (Missing texvc executable; please see math/README to configure.): S

.

A transducer finite-state machine is a sextuple Failed to parse (Missing texvc executable; please see math/README to configure.): (\Sigma, \Gamma, S, s_0, \delta, \omega) , where:

• Failed to parse (Missing texvc executable; please see math/README to configure.): \Sigma

is the input alphabet (a finite non empty set of symbols).

• Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma

is the output alphabet (a finite, non-empty set of symbols).

• Failed to parse (Missing texvc executable; please see math/README to configure.): S

is a finite, non-empty set of states.

• Failed to parse (Missing texvc executable; please see math/README to configure.): s_0

is the initial state, an element of Failed to parse (Missing texvc executable; please see math/README to configure.): S

. In a Nondeterministic finite state machine, Failed to parse (Missing texvc executable; please see math/README to configure.): s_0

is a set of initial states.

• Failed to parse (Missing texvc executable; please see math/README to configure.): \delta

is the state-transition function: Failed to parse (Missing texvc executable; please see math/README to configure.): \delta: S \times \Sigma \rightarrow S

.

• Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

is the output function.

If the output function is a function of a state and input alphabet (Failed to parse (Missing texvc executable; please see math/README to configure.): \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 (Failed to parse (Missing texvc executable; please see math/README to configure.): \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. 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:

• event driven FSM
• virtual FSM (VFSM)
• Automata-Based Programming

History

Starting in the 1970s, Leslie Lamport, an early leader within the distributed systems research community, used finite state machines as the basis for an algorithm he called state machine replication. In this approach, a deterministic computer program or service is replaced with a set of replicas that use some form of atomic broadcast to perform operations in a manner tolerant of failures.

See also

External links

• Description from the Free On-Line Dictionary of Computing
• NIST Dictionary of Algorithms and Data Structures entry
• Hierarchical State Machines
• 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
• Flash based demonstration of Finite State Machines being used in regular expressions

References

General

• 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.
• 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.

Finite state machines (automata theory) in theoretical computer science

• 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; 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. 

Abstract state machines in theoretical computer science

• 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

Machine learning using finite-state algorithms

• 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.

Hardware engineering: state minimization and synthesis of sequential circuits

• 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.

Finite Markov chain processes

"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)

Finite Markov-chain processes are also known as subshifts of finite type.

• 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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