Finite-state machine

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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 external inputs and/or a condition is satisfied; the change from one state to another is called a transition.  An FSM is defined by a list of its states, its initial state, and the conditions for each transition. Finite state machines are of two types – deterministic finite state machines and non-deterministic finite state machines.  A deterministic finite-state machine can be constructed equivalent to any non-deterministic one.

In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology.

An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system used in automata theory. Abstraction of computing processes is used in both the computer science and computer engineering disciplines and usually assumes a discrete time paradigm.

In information technology and computer science, a system is described as stateful if it is designed to remember preceding events or user interactions; the remembered information is called the state of the system.

Contents

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, and combination locks, which require the input of a sequence of numbers in the proper order. A vending machine is an automated machine that provides items such as snacks, beverages, cigarettes and lottery tickets to consumers after money, a credit card, or a specially designed card is inserted into the machine. The first modern vending machines were developed in England in the early 1880s and dispensed postcards. Vending machines exist in many countries, and in more recent times, specialized vending machines that provide less common products compared to traditional vending machine items have been created. An elevator or lift is a type of vertical transportation device that moves people or goods between floors of a building, vessel, or other structure. Elevators are typically powered by electric motors that drive traction cables and counterweight systems like a hoist, although some pump hydraulic fluid to raise a cylindrical piston like a jack. Traffic lights, also known as traffic signals, traffic lamps, traffic semaphore, signal lights, stop lights, robots, and traffic control signals, are signalling devices positioned at road intersections, pedestrian crossings, and other locations to control flows of traffic.

The finite state machine has less computational power than some other models of computation such as the Turing machine.  The computational power distinction means there are computational tasks that a Turing machine can do but a FSM cannot. This is because a FSM's memory is limited by the number of states it has. FSMs are studied in the more general field of automata theory. A Turing machine is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, given any computer algorithm, a Turing machine capable of simulating that algorithm's logic can be constructed. In computing, memory refers to the computer hardware integrated circuits that store information for immediate use in a computer; it is synonymous with the term "primary storage". Computer memory operates at a high speed, for example random-access memory (RAM), as a distinction from storage that provides slow-to-access information but offers higher capacities. If needed, contents of the computer memory can be transferred to secondary storage; a very common way of doing this is through a memory management technique called "virtual memory". An archaic synonym for memory is store. 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 and discrete mathematics. The word automata comes from the Greek word αὐτόματα, which means "self-making".

Example: coin-operated turnstile

An example of a simple mechanism that can be modeled by a state machine is a turnstile.   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. A turnstile, also called a baffle gate, is a form of gate which allows one person to pass at a time. It can also be made so as to enforce one-way traffic of people, and in addition, it can restrict passage only to people who insert a coin, a ticket, a pass, or similar. Thus a turnstile can be used in the case of paid access, for example to access public transport, a pay toilet, or to restrict access to authorized people, for example in the lobby of an office building. In numismatics, token coins or trade tokens are coin-like objects used instead of coins. The field of token coins is part of exonumia and token coins are token money. Tokens have a denomination either shown or implied by size, color or shape. "Tokens" are often made of cheaper metals: copper, pewter, aluminium, brass and tin were commonly used, while bakelite, leather, porcelain, and other less durable materials are also known.

Considered as a state machine, the turnstile has two possible states: Locked and Unlocked.  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:

In automata theory and sequential logic, a state transition table is a table showing what state a finite semiautomaton or finite state machine will move to, based on the current state and other inputs. A state table is essentially a truth table in which some of the inputs are the current state, and the outputs include the next state, along with other outputs.

Current StateInputNext StateOutput
LockedcoinUnlockedUnlocks the turnstile so that the customer can push through.
pushLockedNone
UnlockedcoinUnlockedNone
pushLockedWhen 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

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. A FSM definition including the full actions information is possible using state tables (see also virtual finite-state machine).

State transition table
Current
state
Input
State AState BState C
Input X
Input YState C
Input Z

UML state machines

The Unified Modeling Language has a notation for describing state machines. UML state machines overcome the limitations 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, 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, design of hardware digital systems, software engineering, compilers, network protocols, and the study of computation and languages.

Classification

Finite state machines can be subdivided into transducers, acceptors, classifiers and sequencers. 

Acceptors (recognizers)

Acceptors (also called recognizers and sequence detectors), produce binary output, indicating whether or not the received input is accepted. Each state of an FSM is either "accepting" or "not 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 example in figure 4 shows a finite state machine that accepts the string "nice". In this FSM, the only accepting state is state 7.

A (possibly infinite) set of symbol sequences, aka. formal language, is called a regular language if there is some Finite State Machine 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.  :18,71

A machine could also be described as defining a language, that would contain every string accepted by the machine but none of the rejected ones; that language is "accepted" by the machine. By definition, the languages accepted by FSMs are the regular languages—; a language is regular if there is some FSM that accepts it.

The problem of determining the language accepted by a given finite state 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.   [ jargon ] Fig. 5: Representation of a finite-state machine; this example shows one that determines whether a binary number has an even number of 0s, where S1{\displaystyle S_{1}} is an accepting state.

The start state can also be an accepting state, in which case the automaton accepts the empty string.

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 machine 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 DFA are ε (the empty string), 1, 11, 11…, 00, 010, 1010, 10110, etc.

Classifiers

A classifier is a generalization of a finite state machine that, similar to an acceptor, produces a single output on termination but has more than two terminal states.[ citation needed ]

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.

In control applications, 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. 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.

Generators

Sequencers, or generators, are a subclass of the acceptor and transducer types 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.[ citation needed ]

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. 

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.  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.  These charts, like Harel's original state machines,  support hierarchically nested states, orthogonal regions, state actions, and transition actions. 

Mathematical model

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

• A deterministic finite state machine or acceptor deterministic finite state machine is a quintuple $(\Sigma ,S,s_{0},\delta ,F)$ , where:
• $\Sigma$ is the input alphabet (a finite, non-empty set of symbols).
• $S$ is a finite, non-empty set of states.
• $s_{0}$ is an initial state, an element of $S$ .
• $\delta$ is the state-transition function: $\delta :S\times \Sigma \rightarrow S$ (in a nondeterministic finite automaton it would be $\delta :S\times \Sigma \rightarrow {\mathcal {P}}(S)$ , i.e., $\delta$ would return a set of states).
• $F$ is the set of final states, a (possibly empty) subset of $S$ .

For both deterministic and non-deterministic FSMs, it is conventional to allow $\delta$ to be a partial function, i.e. $\delta (q,x)$ does not have to be defined for every combination of $q\in S$ and $x\in \Sigma$ . If an FSM $M$ is in a state $q$ , the next symbol is $x$ and $\delta (q,x)$ is not defined, then $M$ 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. 

• A finite-state transducer is a sextuple $(\Sigma ,\Gamma ,S,s_{0},\delta ,\omega )$ , where:
• $\Sigma$ is the input alphabet (a finite non-empty set of symbols).
• $\Gamma$ is the output alphabet (a finite, non-empty set of symbols).
• $S$ is a finite, non-empty set of states.
• $s_{0}$ is the initial state, an element of $S$ . In a nondeterministic finite automaton, $s_{0}$ is a set of initial states.
• $\delta$ is the state-transition function: $\delta :S\times \Sigma \rightarrow S$ .
• $\omega$ is the output function.

If the output function is a function of a state and input alphabet ($\omega :S\times \Sigma \rightarrow \Gamma$ ) that definition corresponds to the Mealy model, and can be modelled as a Mealy machine. If the output function depends only on a state ($\omega :S\rightarrow \Gamma$ ) that definition corresponds to the Moore model, and can be modelled as a Moore machine. A finite-state machine with no output function at all is known as a semiautomaton or transition system.

If we disregard the first output symbol of a Moore machine, $\omega (s_{0})$ , 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. 

Optimization

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.   Other techniques include using an implication table, or the Moore reduction procedure. Additionally, acyclic FSAs can be minimized in linear time. 

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.  

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:

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. 

Related Research Articles In the theory of computation, a branch of theoretical computer science, a pushdown automaton (PDA) is a type of automaton that employs a stack. A state diagram is a type of diagram used in computer science and related fields to describe the behavior of systems. State diagrams require that the system described is composed of a finite number of states; sometimes, this is indeed the case, while at other times this is a reasonable abstraction. Many forms of state diagrams exist, which differ slightly and have different semantics.

In the theory of computation, a Mealy machine is a finite-state machine whose output values are determined both by its current state and the current inputs. This is in contrast to a Moore machine, whose (Moore) output values are determined solely by its current state. A Mealy machine is a deterministic finite-state transducer: for each state and input, at most one transition is possible.

In computer science and automata theory, a Büchi automaton is a type of ω-automaton, which extends a finite automaton to infinite inputs. It accepts an infinite input sequence if there exists a run of the automaton that visits one of the final states infinitely often. Büchi automata recognize the omega-regular languages, the infinite word version of regular languages. It is named after the Swiss mathematician Julius Richard Büchi who invented this kind of automaton in 1962.

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem. In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite state machine (DFSM), or deterministic finite state automaton (DFSA)—is a finite-state machine that accepts or rejects strings of symbols and only produces a unique computation of the automaton for each input string. Deterministic refers to the uniqueness of the computation. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943.

In automata theory, a finite state machine is called a deterministic finite automaton (DFA), if

In automata theory, an alternating finite automaton (AFA) is a nondeterministic finite automaton whose transitions are divided into existential and universal transitions. For example, let A be an alternating automaton.

A finite-state transducer (FST) is a finite-state machine with two memory tapes, following the terminology for Turing machines: an input tape and an output tape. This contrasts with an ordinary finite-state automaton, which has a single tape. An FST is a type of finite-state automaton that maps between two sets of symbols. An FST is more general than a finite-state automaton (FSA). An FSA defines a formal language by defining a set of accepted strings, while an FST defines relations between sets of strings.

In automata theory, a permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states.

In automata theory, a deterministic pushdown automaton is a variation of the pushdown automaton. The class of deterministic pushdown automata accepts the deterministic context-free languages, a proper subset of context-free languages.

In computer science, in particular in automata theory, a two-way finite automaton is a finite automaton that is allowed to re-read its input.

In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They are related to quantum computers in a similar fashion as finite automata are related to Turing machines. Several types of automata may be defined, including measure-once and measure-many automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata.

A queue machine or queue automaton is a finite state machine with the ability to store and retrieve data from an infinite-memory queue. It is a model of computation equivalent to a Turing machine, and therefore it can process the same class of formal languages.

A read-only Turing machine or Two-way deterministic finite-state automaton (2DFA) is class of models of computability that behave like a standard Turing machine and can move in both directions across input, except cannot write to its input tape. The machine in its bare form is equivalent to a Deterministic finite automaton in computational power, and therefore can only parse a regular language.

In computer science, more specifically in automata and formal language theory, nested words are a concept proposed by Alur and Madhusudan as a joint generalization of words, as traditionally used for modelling linearly ordered structures, and of ordered unranked trees, as traditionally used for modelling hierarchical structures. Finite-state acceptors for nested words, so-called nested word automata, then give a more expressive generalization of finite automata on words. The linear encodings of languages accepted by finite nested word automata gives the class of visibly pushdown languages. The latter language class lies properly between the regular languages and the deterministic context-free languages. Since their introduction in 2004, these concepts have triggered much research in that area.

In computer science and mathematical logic, an infinite-tree automaton is a state machine that deals with infinite tree structures. It can be seen as an extension of top-down finite-tree automata to infinite trees or as an extension of infinite-word automata to infinite trees.

In automata theory, a branch of theoretical computer science, an ω-automaton is a variation of finite automatons that runs on infinite, rather than finite, strings as input. Since ω-automata do not stop, they have a variety of acceptance conditions rather than simply a set of accepting states.

In automata theory, a self-verifying finite automaton (SVFA) is a special kind of a nondeterministic finite automaton (NFA) with a symmetric kind of nondeterminism introduced by Hromkovič and Schnitger. Generally, in self-verifying nondeterminism, each computation path is concluded with any of the three possible answers: yes, no, and I do not know. For each input string, no two paths may give contradictory answers, namely both answers yes and no on the same input are not possible. At least one path must give answer yes or no, and if it is yes then the string is considered accepted. SVFA accept the same class of languages as deterministic finite automata (DFA) and NFA but have different state complexity.

1. Wang, Jiacun (2019). Formal Methods in Computer Science. CRC Press. p. 34. ISBN   978-1-4987-7532-8.
2. "Finite State Machines - Brilliant Math & Science Wiki". brilliant.org. Retrieved 2018-04-14.
3. Belzer, Jack; Holzman, Albert George; Kent, Allen (1975). Encyclopedia of Computer Science and Technology. 25. USA: CRC Press. p. 73. ISBN   978-0-8247-2275-3.
4. Koshy, Thomas (2004). Discrete Mathematics With Applications. Academic Press. p. 762. ISBN   978-0-12-421180-3.
5. 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.
6. Keller, Robert M. (2001). "Classifiers, Acceptors, Transducers, and Sequencers" (PDF). Computer Science: Abstraction to Implementation (PDF). Harvey Mudd College. p. 480.
7. 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.
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.
9. Jacek Jonczy (June 2008). "Algebraic path problems" (PDF). Archived from the original (PDF) on 21 August 2014. Retrieved 20 August 2014., p. 34
10. 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.
11. "Tiwari, A. (2002). Formal Semantics and Analysis Methods for Simulink Stateflow Models" (PDF). sri.com. Retrieved 2018-04-14.
12. Hamon, G. (2005). A Denotational Semantics for Stateflow. International Conference on Embedded Software. Jersey City, NJ: ACM. pp. 164–172. CiteSeerX  .
13. 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.
14. Anderson, James Andrew; Head, Thomas J. (2006). Automata theory with modern applications. Cambridge University Press. pp. 105–108. ISBN   978-0-521-84887-9.
15. Hopcroft, John E. (1971). An n log n algorithm for minimizing states in a finite automaton (PDF) (Technical Report). CS-TR-71-190. Stanford Univ.
16. Almeida, Marco; Moreira, Nelma; Reis, Rogerio (2007). On the performance of automata minimization algorithms (PDF) (Technical Report). DCC-2007-03. Porto Univ. Archived from the original (PDF) on 17 January 2009. Retrieved 25 June 2008.
17. Revuz, D. (1992). "Minimization of Acyclic automata in Linear Time". Theoretical Computer Science. 92: 181–189. doi:10.1016/0304-3975(92)90142-3.
18. 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.
19. Slides Archived 18 January 2017 at the Wayback Machine , Synchronous Finite State Machines; Design and Behaviour, University of Applied Sciences Hamburg, p.18
20. Aho, Alfred V.; Sethi, Ravi; Ullman, Jeffrey D. (1986). Compilers: Principles, Techniques, and Tools (1st ed.). Addison-Wesley. ISBN   978-0-201-10088-4.