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Finite state machine

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Fig.1 Finite State Machine

A finite state machine (FSM) or finite state automaton (plural: automata) is a model of behavior composed of a finite number of states, transitions between those states, 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..

The concept of the FSM is at the center of theory of computing, as it begins with the basic processes by which finite bits of properly encoded information could theoretically be handled intelligently by a machine.

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).

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

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 modelling of application behaviour, design of hardware digital systems, software engineering, compilers, network protocols, and the study of computation and languages.

Contents

[edit] Classification

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

[edit] Acceptors and recognizers

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

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 (cf. Kleene's Theorem).

[edit] Start State

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

[edit] 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:

Image:DFAexample.svg

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.

[edit] 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:

Fig. 3 Transducer FSM: Moore model example
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".
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.

[edit] 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.

[edit] Mathematical model

Depending on the type there are several definitions. An acceptor finite state machine is a quintuple <math>(\Sigma, S, s_0, \delta, F)</math>, where:

  • <math>\Sigma</math> is the input alphabet (a finite non empty set of symbols).
  • <math>S</math> is a finite non empty set of states.
  • <math>s_0</math> is an initial state, an element of <math>S</math>. In a Nondeterministic finite state machine, <math>s_0</math> is a set of initial states.
  • <math>\delta</math> is the state transition function: <math>\delta: S \times \Sigma \rightarrow S</math>.
  • <math>F</math> is the set of final states, a (possibly empty) subset of <math>S</math>.

A transducer finite state machine is a six tuple <math>(\Sigma, \Gamma, S, s_0, \delta, \omega)</math>, where:

  • <math>\Sigma</math> is the input alphabet (a finite non empty set of symbols).
  • <math>\Gamma</math> is the output alphabet (a finite non empty set of symbols).
  • <math>S</math> is a finite non empty set of states.
  • <math>S_0</math> is the initial state, an element of <math>S</math>. In a Nondeterministic finite state machine, <math>s_0</math> is a set of initial states.
  • <math>\delta</math> is the state transition function: <math>\delta: S \times \Sigma \rightarrow S</math>.
  • <math>\omega</math> is the output function.

If the output function is a function of a state and input alphabet (<math>\omega: S \times \Sigma \rightarrow \Gamma</math>) that definition corresponds to the Mealy model. If the output function depends only on a state (<math>\omega: S \rightarrow \Gamma</math>) that definition corresponds to the Moore model.

[edit] Optimization

Optimizing an FSM means finding the machine with the minimum number of states that performs the same function. An excellent description of this is here: [1].

[edit] Implementation

[edit] Hardware applications

Image:4 bit counter.png
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. One of the classic hardware implementations is the Richard's Controller

[edit] Software applications

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

[edit] Tools

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  • AsmL [2]
  • AT&T FSM Library™ [3]
  • AutoFSM [4]
  • Automata [5]
  • Bandera [6]
  • Boost Statechart Library [7]
  • CAZE - FSM-based .NET authorization library [8]
  • Cellogica - State machine for gene expression [9]
  • Concurrent Hierarchical State Machine [10]
  • Covered [11]
  • DescoGUI [12]
  • dk.brics.automaton [13]
  • dRegAut [14]
  • Dynamic Attachment Finite State Machine (DAFSM) [15]
  • Finite State Kernel Creator [16]
  • Finite State Machine Editor [17]
  • Finite State Machine Explorer [18]

[edit] 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.
  • 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.

[edit] References: 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).
  • 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.
  • 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.
  • 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.
  • 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.

[edit] References: Abstract State Machines in Theoretical Computer Science

[edit] References: 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.

[edit] References: 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.

[edit] References: 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 sj 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”.

[edit] See also

[edit] External links

Automata theory: formal languages and formal grammars
Chomsky
hierarchy 		Grammars Languages Minimal
automaton
Type-0 Unrestricted Recursively enumerable Turing machine
n/a (no common name) Recursive Decider
Type-1 Context-sensitive Context-sensitive Linear-bounded
Type-2 Context-free Context-free Pushdown
Type-3 Regular Regular Finite
Each category of languages or grammars is a proper subset of the category directly above it.
bg:Краен автомат

ca:Autòmat finit cs:Konečný automat de:Endlicher Automat es:Autómata finito fr:Automate fini it:Automa a stati finiti he:אוטומט סופי nl:Eindige toestandsautomaat nn:Automatteori ja:有限オートマトン pl:Maszyna stanów skończonych pt:Máquina de estado finito ro:Automat finit ru:Конечный автомат sr:Коначан аутомат fi:Äärellinen automaatti sv:Ändlig automat uk:Автомат скінченний zh:有限状态自动机

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