05-12-2023-1629 - A finite-state machine (FSM) or finite-state automaton (FSA, plural: automata), finite automaton, or simply a state machine ; Automata_theory (draft)

A finite-state machine (FSM) or finite-state automaton (FSA, plural: automata), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of states at any given time. The FSM can change from one state to another in response to some inputs; the change from one state to another is called a transition.^[1] An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types—deterministic finite-state machines and non-deterministic finite-state machines.^[2] A deterministic finite-state machine can be constructed equivalent to any non-deterministic one.

The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions depending on a sequence of events with which they are presented. Simple examples are: vending machines, which dispense products when the proper combination of coins is deposited; elevators, whose sequence of stops is determined by the floors requested by riders; traffic lights, which change sequence when cars are waiting; combination locks, which require the input of a sequence of numbers in the proper order.

The finite-state machine has less computational power than some other models of computation such as the Turing machine.^[3] The computational power distinction means there are computational tasks that a Turing machine can do but an FSM cannot. This is because an FSM's memory is limited by the number of states it has. A finite-state machine has the same computational power as a Turing machine that is restricted such that its head may only perform "read" operations, and always has to move from left to right. FSMs are studied in the more general field of automata theory.

Example: coin-operated turnstile

State diagram for a turnstile

A turnstile

An example of a simple mechanism that can be modeled by a state machine is a turnstile.^[4][5] A turnstile, used to control access to subways and amusement park rides, is a gate with three rotating arms at waist height, one across the entryway. Initially the arms are locked, blocking the entry, preventing patrons from passing through. Depositing a coin or token in a slot on the turnstile unlocks the arms, allowing a single customer to push through. After the customer passes through, the arms are locked again until another coin is inserted.

Considered as a state machine, the turnstile has two possible states: Locked and Unlocked.^[4] There are two possible inputs that affect its state: putting a coin in the slot (coin) and pushing the arm (push). In the locked state, pushing on the arm has no effect; no matter how many times the input push is given, it stays in the locked state. Putting a coin in – that is, giving the machine a coin input – shifts the state from Locked to Unlocked. In the unlocked state, putting additional coins in has no effect; that is, giving additional coin inputs does not change the state. However, a customer pushing through the arms, giving a push input, shifts the state back to Locked.

The turnstile state machine can be represented by a state-transition table, showing for each possible state, the transitions between them (based upon the inputs given to the machine) and the outputs resulting from each input:

Current State	Input	Next State	Output
Locked	coin	Unlocked	Unlocks the turnstile so that the customer can push through.
	push	Locked	None
Unlocked	coin	Unlocked	None
	push	Locked	When the customer has pushed through, locks the turnstile.

The turnstile state machine can also be represented by a directed graph called a state diagram (above). Each state is represented by a node (circle). Edges (arrows) show the transitions from one state to another. Each arrow is labeled with the input that triggers that transition. An input that doesn't cause a change of state (such as a coin input in the Unlocked state) is represented by a circular arrow returning to the original state. The arrow into the Locked node from the black dot indicates it is the initial state.

Concepts and terminology

A state is a description of the status of a system that is waiting to execute a transition. A transition is a set of actions to be executed when a condition is fulfilled or when an event is received. For example, when using an audio system to listen to the radio (the system is in the "radio" state), receiving a "next" stimulus results in moving to the next station. When the system is in the "CD" state, the "next" stimulus results in moving to the next track. Identical stimuli trigger different actions depending on the current state.

In some finite-state machine representations, it is also possible to associate actions with a state:

• an entry action: performed when entering the state, and
• an exit action: performed when exiting the state.

Representations

Fig. 1 UML state chart example (a toaster oven)

Fig. 2 SDL state machine example

Fig. 3 Example of a simple finite-state machine

For an introduction, see State diagram.

State/Event table

Several state-transition table types are used. The most common representation is shown below: the combination of current state (e.g. B) and input (e.g. Y) shows the next state (e.g. C). The complete action's information is not directly described in the table and can only be added using footnotes.^{[further explanation needed]} An FSM definition including the full action's information is possible using state tables (see also virtual finite-state machine).

State-transition table

Current state Input	State A	State B	State C
Input X	...	...	...
Input Y	...	State C	...
Input Z	...	...	...

UML state machines

The Unified Modeling Language has a notation for describing state machines. UML state machines overcome the limitations^{[citation needed]} of traditional finite-state machines while retaining their main benefits. UML state machines introduce the new concepts of hierarchically nested states and orthogonal regions, while extending the notion of actions. UML state machines have the characteristics of both Mealy machines and Moore machines. They support actions that depend on both the state of the system and the triggering event, as in Mealy machines, as well as entry and exit actions, which are associated with states rather than transitions, as in Moore machines.^{[citation needed]}

SDL state machines

The Specification and Description Language is a standard from ITU that includes graphical symbols to describe actions in the transition:

• send an event
• receive an event
• start a timer
• cancel a timer
• start another concurrent state machine
• decision

SDL embeds basic data types called "Abstract Data Types", an action language, and an execution semantic in order to make the finite-state machine executable.^{[citation needed]}

Other state diagrams

There are a large number of variants to represent an FSM such as the one in figure 3.

Usage

In addition to their use in modeling reactive systems presented here, finite-state machines are significant in many different areas, including electrical engineering, linguistics, computer science, philosophy, biology, mathematics, video game programming, and logic. 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 (control theory), design of hardware digital systems, software engineering, compilers, network protocols, and computational linguistics.

Classification

Finite-state machines can be subdivided into acceptors, classifiers, transducers and sequencers.^[6]

Acceptors

Fig. 4: Acceptor FSM: parsing the string "nice".

Fig. 5: Representation of an acceptor; this example shows one that determines whether a binary number has an even number of 0s, where S₁ is an accepting state and S₂ is a non accepting state.

Acceptors (also called detectors or recognizers) produce binary output, indicating whether or not the received input is accepted. Each state of an acceptor is either accepting or non accepting. Once all input has been received, if the current state is an accepting state, the input is accepted; otherwise it is rejected. As a rule, input is a sequence of symbols (characters); actions are not used. The start state can also be an accepting state, in which case the acceptor accepts the empty string. The example in figure 4 shows an acceptor that accepts the string "nice". In this acceptor, the only accepting state is state 7.

A (possibly infinite) set of symbol sequences, called a formal language, is a regular language if there is some acceptor that accepts exactly that set. For example, the set of binary strings with an even number of zeroes is a regular language (cf. Fig. 5), while the set of all strings whose length is a prime number is not.^{[7]: 18, 71}

An acceptor could also be described as defining a language that would contain every string accepted by the acceptor but none of the rejected ones; that language is accepted by the acceptor. By definition, the languages accepted by acceptors are the regular languages.

The problem of determining the language accepted by a given acceptor is an instance of the algebraic path problem—itself a generalization of the shortest path problem to graphs with edges weighted by the elements of an (arbitrary) semiring.^{[8][9][jargon]}

An example of an accepting state appears in Fig. 5: a deterministic finite automaton (DFA) that detects whether the binary input string 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. S₁ is therefore an accepting state. This acceptor will finish in an accept state, if the binary string contains an even number of 0s (including any binary string containing no 0s). Examples of strings accepted by this acceptor are ε (the empty string), 1, 11, 11..., 00, 010, 1010, 10110, etc.

Classifiers

Classifiers are a generalization of acceptors that produce n-ary output where n is strictly greater than two.^[10]

Transducers

Main article: Finite-state transducer

Fig. 6 Transducer FSM: Moore model example

Fig. 7 Transducer FSM: Mealy model example

Transducers produce 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.

In control applications, two types are distinguished:

Moore machine

The FSM uses only entry actions, i.e., output depends only on state. The advantage of the Moore model is a simplification of the behaviour. Consider 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" stop the motor when fully opened or closed. 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 also uses 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 7 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". The "opening" and "closing" intermediate states are not shown.

Sequencers

Sequencers (also called generators) are a subclass of acceptors and transducers that have a single-letter input alphabet. They produce only one sequence, which can be seen as an output sequence of acceptor or transducer outputs.^[6]

Determinism

A further distinction is between deterministic (DFA) and non-deterministic (NFA, GNFA) automata. In a deterministic automaton, every state has exactly one transition for each possible input. In a non-deterministic automaton, an input can lead to one, more than one, or no transition for a given state. The powerset construction algorithm can transform any nondeterministic automaton into a (usually more complex) deterministic automaton with identical functionality.

A finite-state machine with only one state is called a "combinatorial FSM". It only allows actions upon transition into a state. This concept is useful in cases where a number of finite-state machines are required to work together, and when it is convenient to consider a purely combinatorial part as a form of FSM to suit the design tools.^[11]

Alternative semantics

There are other sets of semantics available to represent state machines. For example, there are tools for modeling and designing logic for embedded controllers.^[12] They combine hierarchical state machines (which usually have more than one current state), flow graphs, and truth tables into one language, resulting in a different formalism and set of semantics.^[13] These charts, like Harel's original state machines,^[14] support hierarchically nested states, orthogonal regions, state actions, and transition actions.^[15]

Mathematical model

In accordance with the general classification, the following formal definitions are found.

A deterministic finite-state machine or deterministic finite-state acceptor is a quintuple ${\displaystyle (\mathrm{\Sigma },S,s_0,\delta ,F)}$, where:

• ${\displaystyle \mathrm{\Sigma }}$ is the input alphabet (a finite non-empty set of symbols);
• ${\displaystyle S}$ is a finite non-empty set of states;
• ${\displaystyle s_0}$ is an initial state, an element of ${\displaystyle S}$;
• ${\displaystyle \delta }$ is the state-transition function: ${\displaystyle \delta :S\times \mathrm{\Sigma }\to S}$ (in a nondeterministic finite automaton it would be ${\displaystyle \delta :S\times \mathrm{\Sigma }\to \mathcal{P}(S)}$, i.e. ${\displaystyle \delta }$ would return a set of states);
• ${\displaystyle F}$ is the set of final states, a (possibly empty) subset of ${\displaystyle S}$.

For both deterministic and non-deterministic FSMs, it is conventional to allow ${\displaystyle \delta }$ to be a partial function, i.e. ${\displaystyle \delta (s,x)}$ does not have to be defined for every combination of ${\displaystyle s\in S}$ and ${\displaystyle x\in \mathrm{\Sigma }}$. If an FSM ${\displaystyle M}$ is in a state ${\displaystyle s}$, the next symbol is ${\displaystyle x}$ and ${\displaystyle \delta (s,x)}$ is not defined, then ${\displaystyle M}$ can announce an error (i.e. reject the input). This is useful in definitions of general state machines, but less useful when transforming the machine. Some algorithms in their default form may require total functions.

A finite-state machine has the same computational power as a Turing machine that is restricted such that its head may only perform "read" operations, and always has to move from left to right. That is, each formal language accepted by a finite-state machine is accepted by such a kind of restricted Turing machine, and vice versa.^[16]

A finite-state transducer is a sextuple ${\displaystyle (\mathrm{\Sigma },\mathrm{\Gamma },S,s_0,\delta ,\omega )}$, where:

• ${\displaystyle \mathrm{\Sigma }}$ is the input alphabet (a finite non-empty set of symbols);
• ${\displaystyle \mathrm{\Gamma }}$ is the output alphabet (a finite non-empty set of symbols);
• ${\displaystyle S}$ is a finite non-empty set of states;
• ${\displaystyle s_0}$ is the initial state, an element of ${\displaystyle S}$;
• ${\displaystyle \delta }$ is the state-transition function: ${\displaystyle \delta :S\times \mathrm{\Sigma }\to S}$;
• ${\displaystyle \omega }$ is the output function.

If the output function depends on the state and input symbol (${\displaystyle \omega :S\times \mathrm{\Sigma }\to \mathrm{\Gamma }}$) that definition corresponds to the Mealy model, and can be modelled as a Mealy machine. If the output function depends only on the state (${\displaystyle \omega :S\to \mathrm{\Gamma }}$) 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.

If we disregard the first output symbol of a Moore machine, ${\displaystyle \omega (s_0)}$, then it can be readily converted to an output-equivalent Mealy machine by setting the output function of every Mealy transition (i.e. labeling every edge) with the output symbol given of the destination Moore state. The converse transformation is less straightforward because a Mealy machine state may have different output labels on its incoming transitions (edges). Every such state needs to be split in multiple Moore machine states, one for every incident output symbol.^[17]

Optimization

Main article: DFA minimization

Optimizing an FSM means finding a machine with the minimum number of states that performs the same function. The fastest known algorithm doing this is the Hopcroft minimization algorithm.^[18][19] Other techniques include using an implication table, or the Moore reduction procedure.^[20] Additionally, acyclic FSAs can be minimized in linear time.^[21]

Implementation

Hardware applications

Fig. 9 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 that 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 Richards controller.

In a Medvedev machine, the output is directly connected to the state flip-flops minimizing the time delay between flip-flops and output.^[22][23]

Through state encoding for low power state machines may be optimized to minimize power consumption.

Software applications

The following concepts are commonly used to build software applications with finite-state machines:

• Automata-based programming
• Event-driven finite-state machine
• Virtual finite-state machine
• State design pattern

Finite-state machines and compilers

Finite automata are often used in the frontend of programming language compilers. Such a frontend may comprise several finite-state machines that implement a lexical analyzer and a parser. Starting from a sequence of characters, the lexical analyzer builds a sequence of language tokens (such as reserved words, literals, and identifiers) from which the parser builds a syntax tree. The lexical analyzer and the parser handle the regular and context-free parts of the programming language's grammar.^[24]

See also

• Abstract state machines
• Alternating finite automaton
• Communicating finite-state machine
• Control system
• Control table
• Decision tables
• DEVS
• Hidden Markov model
• Petri net
• Pushdown automaton
• Quantum finite automaton
• SCXML
• Semiautomaton
• Semigroup action
• Sequential logic
• State diagram
• Synchronizing word
• Transformation semigroup
• Transition system
• Tree automaton
• Turing machine
• UML state machine

References

1. 
   

Wang, Jiacun (2019). Formal Methods in Computer Science. CRC Press. p. 34. ISBN 978-1-4987-7532-8.

"Finite State Machines – Brilliant Math & Science Wiki". brilliant.org. Retrieved 2018-04-14.

Belzer, Jack; Holzman, Albert George; Kent, Allen (1975). Encyclopedia of Computer Science and Technology. Vol. 25. USA: CRC Press. p. 73. ISBN 978-0-8247-2275-3.

Koshy, Thomas (2004). Discrete Mathematics With Applications. Academic Press. p. 762. ISBN 978-0-12-421180-3.

Wright, David R. (2005). "Finite State Machines" (PDF). CSC215 Class Notes. David R. Wright website, N. Carolina State Univ. Archived from the original (PDF) on 2014-03-27. Retrieved 2012-07-14.

Keller, Robert M. (2001). "Classifiers, Acceptors, Transducers, and Sequencers" (PDF). Computer Science: Abstraction to Implementation (PDF). Harvey Mudd College. p. 480.

John E. Hopcroft and Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Reading/MA: Addison-Wesley. ISBN 978-0-201-02988-8.

Pouly, Marc; Kohlas, Jürg (2011). Generic Inference: A Unifying Theory for Automated Reasoning. John Wiley & Sons. Chapter 6. Valuation Algebras for Path Problems, p. 223 in particular. ISBN 978-1-118-01086-0.

Jacek Jonczy (Jun 2008). "Algebraic path problems" (PDF). Archived from the original (PDF) on 2014-08-21. Retrieved 2014-08-20., p. 34

Felkin, M. (2007). Guillet, Fabrice; Hamilton, Howard J. (eds.). Quality Measures in Data Mining - Studies in Computational Intelligence. Vol. 43. Springer, Berlin, Heidelberg. pp. 277–278. doi:10.1007/978-3-540-44918-8_12. ISBN 978-3-540-44911-9.

Brutscheck, M., Berger, S., Franke, M., Schwarzbacher, A., Becker, S.: Structural Division Procedure for Efficient IC Analysis. IET Irish Signals and Systems Conference, (ISSC 2008), pp.18–23. Galway, Ireland, 18–19 June 2008. [1]

"Tiwari, A. (2002). Formal Semantics and Analysis Methods for Simulink Stateflow Models" (PDF). sri.com. Retrieved 2018-04-14.

Hamon, G. (2005). A Denotational Semantics for Stateflow. International Conference on Embedded Software. Jersey City, NJ: ACM. pp. 164–172. CiteSeerX 10.1.1.89.8817.

"Harel, D. (1987). A Visual Formalism for Complex Systems. Science of Computer Programming, 231–274" (PDF). Archived from the original (PDF) on 2011-07-15. Retrieved 2011-06-07.

"Alur, R., Kanade, A., Ramesh, S., & Shashidhar, K. C. (2008). Symbolic analysis for improving simulation coverage of Simulink/Stateflow models. International Conference on Embedded Software (pp. 89–98). Atlanta, GA: ACM" (PDF). Archived from the original (PDF) on 2011-07-15.

Black, Paul E (12 May 2008). "Finite State Machine". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Archived from the original on 13 October 2018. Retrieved 2 November 2016.

Anderson, James Andrew; Head, Thomas J. (2006). Automata theory with modern applications. Cambridge University Press. pp. 105–108. ISBN 978-0-521-84887-9.

Hopcroft, John E. (1971). An n log n algorithm for minimizing states in a finite automaton (PDF) (Technical Report). Vol. CS-TR-71-190. Stanford Univ.^{[permanent dead link]}

Almeida, Marco; Moreira, Nelma; Reis, Rogerio (2007). On the performance of automata minimization algorithms (PDF) (Technical Report). Vol. DCC-2007-03. Porto Univ. Archived from the original (PDF) on 17 January 2009. Retrieved 25 June 2008.

Edward F. Moore (1956). C.E. Shannon and J. McCarthy (ed.). "Gedanken-Experiments on Sequential Machines". Annals of Mathematics Studies. Princeton University Press. 34: 129–153. Here: Theorem 4, p.142.

Revuz, D. (1992). "Minimization of Acyclic automata in Linear Time". Theoretical Computer Science. 92: 181–189. doi:10.1016/0304-3975(92)90142-3.

Kaeslin, Hubert (2008). "Mealy, Moore, Medvedev-type and combinatorial output bits". Digital Integrated Circuit Design: From VLSI Architectures to CMOS Fabrication. Cambridge University Press. p. 787. ISBN 978-0-521-88267-5.

Slides Archived 18 January 2017 at the Wayback Machine, Synchronous Finite State Machines; Design and Behaviour, University of Applied Sciences Hamburg, p.18

24. Aho, Alfred V.; Sethi, Ravi; Ullman, Jeffrey D. (1986). Compilers: Principles, Techniques, and Tools (1st ed.). Addison-Wesley. ISBN 978-0-201-10088-4.

Further reading

General

• Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge University Press. ISBN 978-0-521-84425-3. Zbl 1188.68177.
• Wagner, F., "Modeling Software with Finite State Machines: A Practical Approach", Auerbach Publications, 2006, ISBN 0-8493-8086-3.
• ITU-T, Recommendation Z.100 Specification and Description Language (SDL)
• 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 0-7506-8706-1.
• Gardner, T., Advanced State Management Archived 2008-11-19 at the Wayback Machine, 2007
• 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. ISBN 978-0-13-913368-8.
• Bobrow, Leonard S.; Arbib, Michael A. (1974). Discrete Mathematics: Applied Algebra for Computer and Information Science (1st ed.). Philadelphia: W. B. Saunders Company, Inc. ISBN 978-0-7216-1768-8.
• Booth, Taylor L. (1967). Sequential Machines and Automata Theory (1st ed.). New York: John Wiley and Sons, Inc. Library of Congress Card Catalog Number 67-25924.
• Boolos, George; Jeffrey, Richard (1999) [1989]. Computability and Logic (3rd ed.). Cambridge, England: Cambridge University Press. ISBN 978-0-521-20402-6.
• Brookshear, J. Glenn (1989). Theory of Computation: Formal Languages, Automata, and Complexity. Redwood City, California: Benjamin/Cummings Publish Company, Inc. ISBN 978-0-8053-0143-4.
• Davis, Martin; Sigal, Ron; Weyuker, Elaine J. (1994). Computability, Complexity, and Languages and Logic: Fundamentals of Theoretical Computer Science (2nd ed.). San Diego: Academic Press, Harcourt, Brace & Company. ISBN 978-0-12-206382-4.
• Hopcroft, John; Ullman, Jeffrey (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Reading Mass: Addison-Wesley. ISBN 978-0-201-02988-8.
• Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D. (2001). Introduction to Automata Theory, Languages, and Computation (2nd ed.). Reading Mass: Addison-Wesley. ISBN 978-0-201-44124-6.
• Hopkin, David; Moss, Barbara (1976). Automata. New York: Elsevier North-Holland. ISBN 978-0-444-00249-5.
• Kozen, Dexter C. (1997). Automata and Computability (1st ed.). New York: Springer-Verlag. ISBN 978-0-387-94907-9.
• Lewis, Harry R.; Papadimitriou, Christos H. (1998). Elements of the Theory of Computation (2nd ed.). Upper Saddle River, New Jersey: Prentice-Hall. ISBN 978-0-13-262478-7.
• Linz, Peter (2006). Formal Languages and Automata (4th ed.). Sudbury, MA: Jones and Bartlett. ISBN 978-0-7637-3798-6.
• Minsky, Marvin (1967). Computation: Finite and Infinite Machines (1st ed.). New Jersey: Prentice-Hall.
• Papadimitriou, Christos (1993). Computational Complexity (1st ed.). Addison Wesley. ISBN 978-0-201-53082-7.
• Pippenger, Nicholas (1997). Theories of Computability (1st ed.). Cambridge, England: Cambridge University Press. ISBN 978-0-521-55380-3.
• Rodger, Susan; Finley, Thomas (2006). JFLAP: An Interactive Formal Languages and Automata Package (1st ed.). Sudbury, MA: Jones and Bartlett. ISBN 978-0-7637-3834-1.
• Sipser, Michael (2006). Introduction to the Theory of Computation (2nd ed.). Boston Mass: Thomson Course Technology. ISBN 978-0-534-95097-2.
• Wood, Derick (1987). Theory of Computation (1st ed.). New York: Harper & Row, Publishers, Inc. ISBN 978-0-06-047208-5.

Abstract state machines in theoretical computer science

• Gurevich, Yuri (July 2000). "Sequential Abstract State Machines Capture Sequential Algorithms" (PDF). ACM Transactions on Computational Logic. 1 (1): 77–111. CiteSeerX 10.1.1.146.3017. doi:10.1145/343369.343384. S2CID 2031696.

Machine learning using finite-state algorithms

• Mitchell, Tom M. (1997). Machine Learning (1st ed.). New York: WCB/McGraw-Hill Corporation. ISBN 978-0-07-042807-2.

Hardware engineering: state minimization and synthesis of sequential circuits

• Booth, Taylor L. (1967). Sequential Machines and Automata Theory (1st ed.). New York: John Wiley and Sons, Inc. Library of Congress Card Catalog Number 67-25924.
• Booth, Taylor L. (1971). Digital Networks and Computer Systems (1st ed.). New York: John Wiley and Sons, Inc. ISBN 978-0-471-08840-0.
• McCluskey, E. J. (1965). Introduction to the Theory of Switching Circuits (1st ed.). New York: McGraw-Hill Book Company, Inc. Library of Congress Card Catalog Number 65-17394.
• Hill, Fredrick J.; Peterson, Gerald R. (1965). Introduction to the Theory of Switching Circuits (1st ed.). New York: McGraw-Hill Book Company. Library of Congress Card Catalog Number 65-17394.

Finite Markov chain processes

"We may think of a Markov chain as a process that moves successively through a set of states s₁, s₂, …, s_r. … if it is in state s_i it moves on to the next stop to state s_j with probability p_ij. 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 ed.). New York: John Wiley and Sons, Inc. Library of Congress Card Catalog Number 67-25924.
• Kemeny, John G.; Mirkil, Hazleton; Snell, J. Laurie; Thompson, Gerald L. (1959). Finite Mathematical Structures (1st ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc. Library of Congress Card Catalog Number 59-12841. Chapter 6 "Finite Markov Chains".

External links

• Finite State Automata at Curlie
• Modeling a Simple AI behavior using a Finite State Machine Example of usage in Video Games
• Free On-Line Dictionary of Computing description of Finite-State Machines
• NIST Dictionary of Algorithms and Data Structures description of Finite-State Machines
• A brief overview of state machine types, comparing theoretical aspects of Mealy, Moore, Harel & UML state machines.

v t e Automata theory: formal languages and formal grammars

v t e Digital electronics

Categories:

• Finite automata
• Management cybernetics

 

https://en.wikipedia.org/wiki/Finite-state_machine

Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word automata comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving". An automaton (automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton (FA) or Finite-State Machine (FSM). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. This automaton consists of states (represented in the figure by circles) and transitions (represented by arrows). As the automaton sees a symbol of input, it makes a transition (or jump) to another state, according to its transition function, which takes the previous state and current input symbol as its arguments.

Automata theory is closely related to formal language theory. In this context, automata are used as finite representations of formal languages that may be infinite. Automata are often classified by the class of formal languages they can recognize, as in the Chomsky hierarchy, which describes a nesting relationship between major classes of automata. Automata play a major role in the theory of computation, compiler construction, artificial intelligence, parsing and formal verification.

History

The theory of abstract automata was developed in the mid-20th century in connection with finite automata.^[1] Automata theory was initially considered a branch of mathematical systems theory, studying the behavior of discrete-parameter systems. Early work in automata theory differed from previous work on systems by using abstract algebra to describe information systems rather than differential calculus to describe material systems.^[2] The theory of the finite-state transducer was developed under different names by different research communities.^[3] The earlier concept of Turing machine was also included in the discipline along with new forms of infinite-state automata, such as pushdown automata.

1956 saw the publication of Automata Studies, which collected work by scientists including Claude Shannon, W. Ross Ashby, John von Neumann, Marvin Minsky, Edward F. Moore, and Stephen Cole Kleene.^[4] With the publication of this volume, "automata theory emerged as a relatively autonomous discipline".^[5] The book included Kleene's description of the set of regular events, or regular languages, and a relatively stable measure of complexity in Turing machine programs by Shannon.^[6] In the same year, Noam Chomsky described the Chomsky hierarchy, a correspondence between automata and formal grammars,^[7] and Ross Ashby published An Introduction to Cybernetics, an accessible textbook explaining automata and information using basic set theory.

The study of linear bounded automata led to the Myhill–Nerode theorem,^[8] which gives a necessary and sufficient condition for a formal language to be regular, and an exact count of the number of states in a minimal machine for the language. The pumping lemma for regular languages, also useful in regularity proofs, was proven in this period by Michael O. Rabin and Dana Scott, along with the computational equivalence of deterministic and nondeterministic finite automata.^[9]

In the 1960s, a body of algebraic results known as "structure theory" or "algebraic decomposition theory" emerged, which dealt with the realization of sequential machines from smaller machines by interconnection.^[10] While any finite automaton can be simulated using a universal gate set, this requires that the simulating circuit contain loops of arbitrary complexity. Structure theory deals with the "loop-free" realizability of machines.^[5] The theory of computational complexity also took shape in the 1960s.^[11][12] By the end of the decade, automata theory came to be seen as "the pure mathematics of computer science".^[5]

Automata

What follows is a general definition of an automaton, which restricts a broader definition of a system to one viewed as acting in discrete time-steps, with its state behavior and outputs defined at each step by unchanging functions of only its state and input.^[5] 0'.'|01,"|",1|

Informal description

An automaton runs when it is given some sequence of inputs in discrete (individual) time steps (or just steps). An automaton processes one input picked from a set of symbols or letters, which is called an input alphabet. The symbols received by the automaton as input at any step are a sequence of symbols called words. An automaton has a set of states. At each moment during a run of the automaton, the automaton is in one of its states. When the automaton receives new input it moves to another state (or transitions) based on a transition function that takes the previous state and current input symbol as parameters. At the same time, another function called the output function produces symbols from the output alphabet, also according to the previous state and current input symbol. The automaton reads the symbols of the input word and transitions between states until the word is read completely, if it is finite in length, at which point the automaton halts. A state at which the automaton halts is called the final state.

To investigate the possible state/input/output sequences in an automaton using formal language theory, a machine can be assigned a starting state and a set of accepting states. Then, depending on whether a run starting from the starting state ends in an accepting state, the automaton can be said to accept or reject an input sequence. The set of all the words accepted by an automaton is called the language recognized by the automaton. A familiar example of a machine recognizing a language is an electronic lock, which accepts or rejects attempts to enter the correct code.

Formal definition

Automaton

An automaton can be represented formally by a 5-tuple ${\displaystyle M=⟨\mathrm{\Sigma },\mathrm{\Gamma },Q,\delta ,\lambda ⟩}$, where:

• ${\displaystyle \mathrm{\Sigma }}$ is a finite set of symbols, called the input alphabet of the automaton,
• ${\displaystyle \mathrm{\Gamma }}$ is another finite set of symbols, called the output alphabet of the automaton,
• ${\displaystyle Q}$ is a set of states,
• ${\displaystyle \delta }$ is the next-state function or transition function ${\displaystyle \delta :Q\times \mathrm{\Sigma }\to Q}$ mapping state-input pairs to successor states,
• ${\displaystyle \lambda }$ is the next-output function ${\displaystyle \lambda :Q\times \mathrm{\Sigma }\to \mathrm{\Gamma }}$ mapping state-input pairs to outputs.

If ${\displaystyle Q}$ is finite, then ${\displaystyle M}$ is a finite automaton.^[5]

Input word

An automaton reads a finite string of symbols ${\displaystyle a_1{a}_2...a_n}$, where ${\displaystyle a_i\in \mathrm{\Sigma }}$, which is called an input word. The set of all words is denoted by ${\displaystyle {\mathrm{\Sigma }}^{\ast }}$.

Run

A sequence of states ${\displaystyle q_0,q_1,...,q_n}$, where ${\displaystyle q_i\in Q}$ such that ${\displaystyle q_i=\delta (q_{i-1},a_i)}$ for ${\displaystyle 0<i\le n}$, is a run of the automaton on an input ${\displaystyle a_1{a}_2...a_n\in {\mathrm{\Sigma }}^{\ast }}$ starting from state ${\displaystyle q_0}$. In other words, at first the automaton is at the start state ${\displaystyle q_0}$, and receives input ${\displaystyle a_1}$. For ${\displaystyle a_1}$ and every following ${\displaystyle a_i}$ in the input string, the automaton picks the next state ${\displaystyle q_i}$ according to the transition function ${\displaystyle \delta (q_{i-1},a_i)}$, until the last symbol ${\displaystyle a_n}$ has been read, leaving the machine in the final state of the run, ${\displaystyle q_n}$. Similarly, at each step, the automaton emits an output symbol according to the output function ${\displaystyle \lambda (q_{i-1},a_i)}$.

The transition function ${\displaystyle \delta }$ is extended inductively into ${\displaystyle \overline{\delta }:Q\times {\mathrm{\Sigma }}^{\ast }\to Q}$ to describe the machine's behavior when fed whole input words. For the empty string ${\displaystyle \epsilon }$, ${\displaystyle \overline{\delta }(q,\epsilon )=q}$ for all states ${\displaystyle q}$, and for strings ${\displaystyle wa}$ where ${\displaystyle a}$ is the last symbol and ${\displaystyle w}$ is the (possibly empty) rest of the string, ${\displaystyle \overline{\delta }(q,wa)=\delta (\overline{\delta }(q,w),a)}$.^[10] The output function ${\displaystyle \lambda }$ may be extended similarly into ${\displaystyle \overline{\lambda }(q,w)}$, which gives the complete output of the machine when run on word ${\displaystyle w}$ from state ${\displaystyle q}$.

Acceptor

In order to study an automaton with the theory of formal languages, an automaton may be considered as an acceptor, replacing the output alphabet and function ${\displaystyle \mathrm{\Gamma }}$ and ${\displaystyle \lambda }$ with

• ${\displaystyle q_0\in Q}$, a designated start state, and
• ${\displaystyle F}$, a set of states of ${\displaystyle Q}$ (i.e. ${\displaystyle F\subseteq Q}$) called accept states.

This allows the following to be defined:

Accepting word

A word ${\displaystyle w=a_1{a}_2...a_n\in {\mathrm{\Sigma }}^{\ast }}$ is an accepting word for the automaton if ${\displaystyle \overline{\delta }(q_0,w)\in F}$, that is, if after consuming the whole string ${\displaystyle w}$ the machine is in an accept state.

Recognized language

The language ${\displaystyle L\subseteq {\mathrm{\Sigma }}^{\ast }}$ recognized by an automaton is the set of all the words that are accepted by the automaton, ${\displaystyle L=\{w\in {\mathrm{\Sigma }}^{\ast }|\overline{\delta }(q_0,w)\in F\}}$.^[13]

Recognizable languages

The recognizable languages are the set of languages that are recognized by some automaton. For finite automata the recognizable languages are regular languages. For different types of automata, the recognizable languages are different.

Variant definitions of automata

Automata are defined to study useful machines under mathematical formalism. So the definition of an automaton is open to variations according to the "real world machine" that we want to model using the automaton. People have studied many variations of automata. The following are some popular variations in the definition of different components of automata.

Input

• Finite input: An automaton that accepts only finite sequences of symbols. The above introductory definition only encompasses finite words.
• Infinite input: An automaton that accepts infinite words (ω-words). Such automata are called ω-automata.
• Tree input: The input may be a tree of symbols instead of sequence of symbols. In this case after reading each symbol, the automaton reads all the successor symbols in the input tree. It is said that the automaton makes one copy of itself for each successor and each such copy starts running on one of the successor symbols from the state according to the transition relation of the automaton. Such an automaton is called a tree automaton.
• Infinite tree input : The two extensions above can be combined, so the automaton reads a tree structure with (in)finite branches. Such an automaton is called an infinite tree automaton.

States

• Single state: An automaton with one state, also called a combinational circuit, performs a transformation which may implement combinational logic.^[10]
• Finite states: An automaton that contains only a finite number of states.
• Infinite states: An automaton that may not have a finite number of states, or even a countable number of states. Different kinds of abstract memory may be used to give such machines finite descriptions.
• Stack memory: An automaton may also contain some extra memory in the form of a stack in which symbols can be pushed and popped. This kind of automaton is called a pushdown automaton.
• Queue memory: An automaton may have memory in the form of a queue. Such a machine is called queue machine and is Turing-complete.
• Tape memory: The inputs and outputs of automata are often described as input and output tapes. Some machines have additional working tapes, including the Turing machine, linear bounded automaton, and log-space transducer.

Transition function

• Deterministic: For a given current state and an input symbol, if an automaton can only jump to one and only one state then it is a deterministic automaton.
• Nondeterministic: An automaton that, after reading an input symbol, may jump into any of a number of states, as licensed by its transition relation. The term transition function is replaced by transition relation: The automaton non-deterministically decides to jump into one of the allowed choices. Such automata are called nondeterministic automata.
• Alternation: This idea is quite similar to tree automata but orthogonal. The automaton may run its multiple copies on the same next read symbol. Such automata are called alternating automata. The acceptance condition must be satisfied on all runs of such copies to accept the input.

Acceptance condition

• Acceptance of finite words: Same as described in the informal definition above.
• Acceptance of infinite words: an ω-automaton cannot have final states, as infinite words never terminate. Rather, acceptance of the word is decided by looking at the infinite sequence of visited states during the run.
• Probabilistic acceptance: An automaton need not strictly accept or reject an input. It may accept the input with some probability between zero and one. For example, quantum finite automata, geometric automata and metric automata have probabilistic acceptance.

Different combinations of the above variations produce many classes of automata.

Automata theory is a subject matter that studies properties of various types of automata. For example, the following questions are studied about a given type of automata.

• Which class of formal languages is recognizable by some type of automata? (Recognizable languages)
• Are certain automata closed under union, intersection, or complementation of formal languages? (Closure properties)
• How expressive is a type of automata in terms of recognizing a class of formal languages? And, their relative expressive power? (Language hierarchy)

Automata theory also studies the existence or nonexistence of any effective algorithms to solve problems similar to the following list:

• Does an automaton accept at least one input word? (Emptiness checking)
• Is it possible to transform a given non-deterministic automaton into a deterministic automaton without changing the language recognized? (Determinization)
• For a given formal language, what is the smallest automaton that recognizes it? (Minimization)

Types of automata

The following is an incomplete list of types of automata.

Automaton	Recognizable languages
Nondeterministic/Deterministic finite-state machine (FSM)	regular languages
Deterministic pushdown automaton (DPDA)	deterministic context-free languages
Pushdown automaton (PDA)	context-free languages
Linear bounded automaton (LBA)	context-sensitive languages
Turing machine	recursively enumerable languages
Deterministic Büchi automaton	ω-limit languages
Nondeterministic Büchi automaton	ω-regular languages
Rabin automaton, Streett automaton, Parity automaton, Muller automaton
Weighted automaton

Discrete, continuous, and hybrid automata

Normally automata theory describes the states of abstract machines but there are discrete automata, analog automata or continuous automata, or hybrid discrete-continuous automata, which use digital data, analog data or continuous time, or digital and analog data, respectively.

Hierarchy in terms of powers

The following is an incomplete hierarchy in terms of powers of different types of virtual machines. The hierarchy reflects the nested categories of languages the machines are able to accept.^[14]

Automaton

Deterministic Finite Automaton (DFA) -- Lowest Power (same power)    ${\displaystyle ||}$   (same power) Nondeterministic Finite Automaton (NFA) (above is weaker)    ${\displaystyle \cap }$    (below is stronger) Deterministic Push Down Automaton (DPDA-I) with 1 push-down store ${\displaystyle \cap }$ Nondeterministic Push Down Automaton (NPDA-I) with 1 push-down store ${\displaystyle \cap }$ Linear Bounded Automaton (LBA) ${\displaystyle \cap }$ Deterministic Push Down Automaton (DPDA-II) with 2 push-down stores ${\displaystyle ||}$ Nondeterministic Push Down Automaton (NPDA-II) with 2 push-down stores ${\displaystyle ||}$ Deterministic Turing Machine (DTM) ${\displaystyle ||}$ Nondeterministic Turing Machine (NTM) ${\displaystyle ||}$ Probabilistic Turing Machine (PTM) ${\displaystyle ||}$ Multitape Turing Machine (MTM) ${\displaystyle ||}$ Multidimensional Turing Machine

Applications

Each model in automata theory plays important roles in several applied areas. Finite automata are used in text processing, compilers, and hardware design. Context-free grammar (CFGs) are used in programming languages and artificial intelligence. Originally, CFGs were used in the study of human languages. Cellular automata are used in the field of artificial life, the most famous example being John Conway's Game of Life. Some other examples which could be explained using automata theory in biology include mollusk and pine cone growth and pigmentation patterns. Going further, a theory suggesting that the whole universe is computed by some sort of a discrete automaton, is advocated by some scientists. The idea originated in the work of Konrad Zuse, and was popularized in America by Edward Fredkin. Automata also appear in the theory of finite fields: the set of irreducible polynomials that can be written as composition of degree two polynomials is in fact a regular language.^[15] Another problem for which automata can be used is the induction of regular languages.

Automata simulators

Automata simulators are pedagogical tools used to teach, learn and research automata theory. An automata simulator takes as input the description of an automaton and then simulates its working for an arbitrary input string. The description of the automaton can be entered in several ways. An automaton can be defined in a symbolic language or its specification may be entered in a predesigned form or its transition diagram may be drawn by clicking and dragging the mouse. Well known automata simulators include Turing's World, JFLAP, VAS, TAGS and SimStudio.^[16]

Connection to category theory

One can define several distinct categories of automata^[17] following the automata classification into different types described in the previous section. The mathematical category of deterministic automata, sequential machines or sequential automata, and Turing machines with automata homomorphisms defining the arrows between automata is a Cartesian closed category,^[18] it has both categorical limits and colimits. An automata homomorphism maps a quintuple of an automaton A_i onto the quintuple of another automaton A_j. Automata homomorphisms can also be considered as automata transformations or as semigroup homomorphisms, when the state space, S, of the automaton is defined as a semigroup S_g. Monoids are also considered as a suitable setting for automata in monoidal categories.^[19][20][21]

Categories of variable automata

One could also define a variable automaton, in the sense of Norbert Wiener in his book on The Human Use of Human Beings via the endomorphisms ${\displaystyle A_i\to A_i}$. Then one can show that such variable automata homomorphisms form a mathematical group. In the case of non-deterministic, or other complex kinds of automata, the latter set of endomorphisms may become, however, a variable automaton groupoid. Therefore, in the most general case, categories of variable automata of any kind are categories of groupoids or groupoid categories. Moreover, the category of reversible automata is then a 2-category, and also a subcategory of the 2-category of groupoids, or the groupoid category.

See also

• Boolean differential calculus

References

1. 
   

Mahoney, Michael S. "The Structures of Computation and the Mathematical Structure of Nature". The Rutherford Journal. Retrieved 2020-06-07.

Booth, Taylor (1967). Sequential Machines and Automata Theory. New York: John Wiley & Sons. p. 1-13. ISBN 0-471-08848-X.

Ashby, William Ross (1967-01-15). "The Place of the Brain in the Natural World" (PDF). Currents in Modern Biology. 1 (2): 95–104. doi:10.1016/0303-2647(67)90021-4. PMID 6060865.: "The theories, now well developed, of the "finite-state machine" (Gill, 1962), of the "noiseless transducer" (Shannon and Weaver, 1949), of the "state-determined system" (Ashby, 1952), and of the "sequential circuit", are essentially homologous."

Ashby, W. R.; et al. (1956). C.E. Shannon; J. McCarthy (eds.). Automata Studies. Princeton, N.J.: Princeton University Press.

Arbib, Michael (1969). Theories of Abstract Automata. Englewood Cliffs, N.J.: Prentice-Hall.

Li, Ming; Paul, Vitanyi (1997). An Introduction to Kolmogorov Complexity and its Applications. New York: Springer-Verlag. p. 84.

Chomsky, Noam (1956). "Three models for the description of language" (PDF). IRE Transactions on Information Theory. 2 (3): 113–124. doi:10.1109/TIT.1956.1056813.

Nerode, A. (1958). "Linear Automaton Transformations". Proceedings of the American Mathematical Society. 9 (4): 541. doi:10.1090/S0002-9939-1958-0135681-9.

Rabin, Michael; Scott, Dana (Apr 1959). "Finite Automata and Their Decision Problems" (PDF). IBM Journal of Research and Development. 3 (2): 114–125. doi:10.1147/rd.32.0114. Archived from the original on 2010-12-14.

Hartmanis, J.; Stearns, R.E. (1966). Algebraic Structure Theory of Sequential Machines. Englewood Cliffs, N.J.: Prentice-Hall.

Hartmanis, J.; Stearns, R. E. (1964). "Computational complexity of recursive sequences" (PDF).

Fortnow, Lance; Homer, Steve (2002). "A Short History of Computational Complexity" (PDF).

Moore, Cristopher (2019-07-31). "Automata, languages, and grammars". arXiv:1907.12713 [cs.CC].

Yan, Song Y. (1998). An Introduction to Formal Languages and Machine Computation. Singapore: World Scientific Publishing Co. Pte. Ltd. pp. 155–156. ISBN 9789810234225.

Ferraguti, A.; Micheli, G.; Schnyder, R. (2018), Irreducible compositions of degree two polynomials over finite fields have regular structure, The Quarterly Journal of Mathematics, vol. 69, Oxford University Press, pp. 1089–1099, arXiv:1701.06040, doi:10.1093/qmath/hay015, S2CID 3962424

Chakraborty, P.; Saxena, P. C.; Katti, C. P. (2011). "Fifty Years of Automata Simulation: A Review". ACM Inroads. 2 (4): 59–70. doi:10.1145/2038876.2038893. S2CID 6446749.

Jirí Adámek and Věra Trnková. 1990. Automata and Algebras in Categories. Kluwer Academic Publishers:Dordrecht and Prague

Mac Lane, Saunders (1971). Categories for the Working Mathematician. New York: Springer. ISBN 9780387900360.

http://www.math.cornell.edu/~worthing/asl2010.pdf James Worthington.2010.Determinizing, Forgetting, and Automata in Monoidal Categories. ASL North American Annual Meeting, 17 March 2010

Aguiar, M. and Mahajan, S.2010. "Monoidal Functors, Species, and Hopf Algebras".

21. Meseguer, J., Montanari, U.: 1990 Petri nets are monoids. Information and Computation 88:105–155

Further reading

• John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman (2000). Introduction to Automata Theory, Languages, and Computation (2nd ed.). Pearson Education. ISBN 978-0-201-44124-6.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 978-0-534-94728-6. Part One: Automata and Languages, chapters 1–2, pp. 29–122. Section 4.1: Decidable Languages, pp. 152–159. Section 5.1: Undecidable Problems from Language Theory, pp. 172–183.
• Elaine Rich (2008). Automata, Computability and Complexity: Theory and Applications. Pearson. ISBN 978-0-13-228806-4.
• Salomaa, Arto (1985). Computation and automata. Encyclopedia of Mathematics and Its Applications. Vol. 25. Cambridge University Press. ISBN 978-0-521-30245-6. Zbl 0565.68046.
• Anderson, James A. (2006). Automata theory with modern applications. With contributions by Tom Head. Cambridge: Cambridge University Press. ISBN 978-0-521-61324-8. Zbl 1127.68049.
• Conway, J.H. (1971). Regular algebra and finite machines. Chapman and Hall Mathematics Series. London: Chapman & Hall. Zbl 0231.94041.
• John M. Howie (1991) Automata and Languages, Clarendon Press ISBN 0-19-853424-8 MR1254435
• Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge University Press. ISBN 978-0-521-84425-3. Zbl 1188.68177.
• James P. Schmeiser, David T. Barnard (1995). Producing a top-down parse order with bottom-up parsing. Elsevier North-Holland.
• Igor Aleksander, F.Keith Hanna (1975). Automata Theory : An Engineering Approach. New York: Crane Russak. ISBN 978-0-8448-0657-0.
• Marvin Minsky (1967). Computation : Finite and infinite machines. Princeton, N.J.: Prentice Hall.
• John C. Martin (2011). Introduction to Languages and The Theory of Computation. New York: McGraw Hill. ISBN 978-0-07-319146-1.

External links

• dk.brics.automaton
• libfa

v t e Automata theory: formal languages and formal grammars

v t e Computer science

v t e Industrial and applied mathematics

Authority control

Category:

• Automata (computation)

 

https://en.wikipedia.org/wiki/Automata_theory

Probability theory

• Distributions (random variables)
• Stochastic processes / analysis
• Path integral
• Stochastic variational calculus

 

https://en.wikipedia.org/wiki/Automata_theory

 https://en.wikipedia.org/wiki/Supersymmetric_theory_of_stochastic_dynamics

A torus, one of the most frequently studied objects in algebraic topology

https://en.wikipedia.org/wiki/Algebraic_topology

 

In the mathematical field of topology, a section (or cross section)^[1] of a fiber bundle ${\displaystyle E}$ is a continuous right inverse of the projection function ${\displaystyle \pi }$. In other words, if ${\displaystyle E}$ is a fiber bundle over a base space, ${\displaystyle B}$:

${\displaystyle \pi :E\to B}$

then a section of that fiber bundle is a continuous map,

${\displaystyle \sigma :B\to E}$

such that

${\displaystyle \pi (\sigma (x))=x}$ for all ${\displaystyle x\in B}$.

A section is an abstract characterization of what it means to be a graph. The graph of a function ${\displaystyle g:B\to Y}$ can be identified with a function taking its values in the Cartesian product ${\displaystyle E=B\times Y}$, of ${\displaystyle B}$ and ${\displaystyle Y}$:

${\displaystyle \sigma :B\to E,\sigma (x)=(x,g(x))\in E.}$

Let ${\displaystyle \pi :E\to B}$ be the projection onto the first factor: ${\displaystyle \pi (x,y)=x}$. Then a graph is any function ${\displaystyle \sigma }$ for which ${\displaystyle \pi (\sigma (x))=x}$.

The language of fibre bundles allows this notion of a section to be generalized to the case when ${\displaystyle E}$ is not necessarily a Cartesian product. If ${\displaystyle \pi :E\to B}$ is a fibre bundle, then a section is a choice of point ${\displaystyle \sigma (x)}$ in each of the fibres. The condition ${\displaystyle \pi (\sigma (x))=x}$ simply means that the section at a point ${\displaystyle x}$ must lie over ${\displaystyle x}$. (See image.)

For example, when ${\displaystyle E}$ is a vector bundle a section of ${\displaystyle E}$ is an element of the vector space ${\displaystyle E_x}$ lying over each point ${\displaystyle x\in B}$. In particular, a vector field on a smooth manifold ${\displaystyle M}$ is a choice of tangent vector at each point of ${\displaystyle M}$: this is a section of the tangent bundle of ${\displaystyle M}$. Likewise, a 1-form on ${\displaystyle M}$ is a section of the cotangent bundle.

Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space ${\displaystyle B}$ is a smooth manifold ${\displaystyle M}$, and ${\displaystyle E}$ is assumed to be a smooth fiber bundle over ${\displaystyle M}$ (i.e., ${\displaystyle E}$ is a smooth manifold and ${\displaystyle \pi :E\to M}$ is a smooth map). In this case, one considers the space of smooth sections of ${\displaystyle E}$ over an open set ${\displaystyle U}$, denoted ${\displaystyle C^{\mathrm{\infty }}(U,E)}$. It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., ${\displaystyle C^k}$ sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces). 

 

https://en.wikipedia.org/wiki/Section_(fiber_bundle)

 https://en.wikipedia.org/wiki/Sobolev_space