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Encyclopedia > Finite state machine
Fig.1 Example of a Finite State Machine

A finite state machine (FSM) or finite state automaton (plural: automata) or simply a state machine, is a model of behavior composed of a finite number of states, transitions between those states, and actions. A finite state machine is an abstract model of a machine with a primitive internal memory. In theoretical computer science, a state transition system is an abstract machine used in the study of computation. ... In information processing, a state is the complete set of properties (for example, its energy level, etc. ...

A state stores information about the past, i.e. it reflects the input changes from the system start to the present moment. A transition indicates a state change and is described by a condition that would need to be fulfilled to enable the transition. An action is a description of an activity that is to be performed at a given moment. There are several action types:

Entry action
which is performed when entering the state
Exit action
which is performed when exiting the state
Input action
which is performed depending on present state and input conditions
Transition action
which is performed when performing a certain transition

A FSM can be represented using a state diagram (or state transition diagram) as in figure 1 above. Besides this, several state transition table types are used. The most common representation is shown below: the combination of current state (B) and condition (Y) shows the next state (C). The complete actions information can be added only using footnotes. An FSM definition including the full actions information is possible using state tables (see also VFSM). State diagrams are used to graphically represent finite state machines. ... In Automata Theory, a state transition table is a table describing the transition function T of a finite automaton. ... Figure 1: VFSM in the Virtual Environment The virtual finite state machine (VFSM) is a concept promoted by SW Software and implemented in their StateWORKS product. ... Figure 1: VFSM in the Virtual Environment A finite state machine (FSM) defined in a virtual environment is called virtual finite state machine (VFSM). ...

 Current State → Condition State A State B State C Condition X ... ... ... Condition Y ... State C ... Condition Z ... ... ...

In addition to their use in modeling reactive systems presented here, finite state automata are significant in many different areas, including electrical engineering, linguistics, computer science, philosophy, biology, mathematics, and logic. A complete survey of their applications is outside the scope of this article. Finite state machines are a class of automata studied in automata theory and the theory of computation. In computer science, finite state machines are widely used in modeling of application behavior, design of hardware digital systems, software engineering, compilers, network protocols, and the study of computation and languages. Electrical Engineers design power systems. ... For the journal, see Linguistics (journal). ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... For other uses, see Philosophy (disambiguation). ... For other uses, see Biology (disambiguation). ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... â€œAutomataâ€ redirects here. ... The theory of computation is the branch of computer science that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm. ...

## Classification

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

### Acceptors and recognizers

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

Acceptors and recognizers (also sequence detectors) produce a binary output, saying either yes or no to answer whether the input is accepted by the machine or not. All states of the FSM are said to be either accepting or not accepting. At the time when all input is processed, if the current state is an accepting state, the input is accepted; otherwise it is rejected. As a rule the input are symbols (characters); actions are not used. The example in figure 2 shows a finite state machine which accepts the word "nice". In this FSM the only accepting state is number 7.

The machine can also be described as defining a language, which would contain every word accepted by the machine but none of the rejected ones; we say then that the language is accepted by the machine. By definition, the languages accepted by FSMs are the regular languages - that is, a language is regular if there is some FSM that accepts it. In theoretical computer science, a regular language is a formal language (i. ...

#### Start state

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

#### Accept state

Fig. 3: A finite state machine that determines if a binary number has an odd or even number of 0s.

An accept state (sometimes referred to as an accepting state) is a state at which the machine has successfully performed its procedure. It is usually represented by a double circle. Image File history File links DFAexample. ... Image File history File links DFAexample. ...

An example of an accepting state appears on the left in this diagram of a deterministic finite automaton (DFA) which determines if the binary input contains an even number of 0s. In the theory of computation, a deterministic finite state machine or deterministic finite automaton (DFA) is a finite state machine where for each pair of state and input symbol there is a deterministic next state. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ...

S1 (which is also the start state) indicates the state at which an even number of 0s has been input and is therefore defined as an accepting state. This machine will give a correct end state if the binary number contains an even number of zeros including a string with no zeros. Examples of strings accepted by this DFA are epsilon (the empty string), 1, 11, 11..., 00, 010, 1010, 10110 and so on.

### Transducers

Transducers generate output based on a given input and/or a state using actions. They are used for control applications and in the field of computational linguistics. Here two types are distinguished: A finite state transducer (FST) is a finite state machine with two tapes. ... Computational linguistics is an interdisciplinary field dealing with the statistical and logical modeling of natural language from a computational perspective. ...

Moore machine
The FSM uses only entry actions, i.e. output depends only on the state. The advantage of the Moore model is a simplification of the behaviour. The example in figure 1 shows a Moore FSM of an elevator door. The state machine recognizes two commands: "command_open" and "command_close" which trigger state changes. The entry action (E:) in state "Opening" starts a motor opening the door, the entry action in state "Closing" starts a motor in the other direction closing the door. States "Opened" and "Closed" don't perform any actions. They signal to the outside world (e.g. to other state machines) the situation: "door is open" or "door is closed".
Fig. 4 Transducer FSM: Mealy model example
Mealy machine
The FSM uses only input actions, i.e. output depends on input and state. The use of a Mealy FSM leads often to a reduction of the number of states. The example in figure 4 shows a Mealy FSM implementing the same behaviour as in the Moore example (the behaviour depends on the implemented FSM execution model and will work e.g. for virtual FSM but not for event driven FSM). There are two input actions (I:): "start motor to close the door if command_close arrives" and "start motor in the other direction to open the door if command_open arrives".

In practice mixed models are often used. Moore model: control of an elevator door In the theory of computation, a Moore machine is a finite state automaton where the outputs are determined by the current state alone (and not on the input). ... fsm example of a mealy model: control of an elevator door This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... fsm example of a mealy model: control of an elevator door This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... In the theory of computation, a Mealy machine is a finite state machine where the outputs are determined by the current state and the input. ... Figure 1: VFSM in the Virtual Environment The virtual finite state machine (VFSM) is a concept promoted by SW Software and implemented in their StateWORKS product. ... A finite state machine (FSM) is event driven if any incoming input has to be consumed immediately, i. ...

More details about the differences and usage of Moore and Mealy models, including an executable example, can be found in the external technical note "Moore or Mealy model?"

A further distinction is between deterministic (DFA) and non-deterministic (NDFA, GNFA) automata. In deterministic automata, for each state there is exactly one transition for each possible input. In non-deterministic automata, there can be none or more than one transition from a given state for a given possible input. This distinction is relevant in practice, but not in theory, as there exists an algorithm which can transform any NDFA into an equivalent DFA, although this transformation typically significantly increases the complexity of the automaton. In the theory of computation, a deterministic finite state machine or deterministic finite automaton (DFA) is a finite state machine where for each pair of state and input symbol there is one and only one transition to a next state. ... In the theory of computation, a nondeterministic finite state machine or nondeterministic finite automaton (NFA) is a finite state machine where for each pair of state and input symbol there may be several possible next states. ... In the theory of computation, a generalized nondeterministic finite state machine or generalized nondeterministic finite automaton (GNFA) is a NFA where each transition may be labeled with any regular expression. ...

The FSM with only one state is called a combinatorial FSM and uses only input actions. This concept is useful in cases where a number of FSM are required to work together, and where it is convenient to consider a purely combinatorial part as a form of FSM to suit the design tools.

## FSM logic

Fig. 5 FSM Logic

The next state and output of an FSM is a function of the input and of the current state. The FSM logic is shown in Figure 5. finite state machine definition File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... finite state machine definition File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

## Mathematical model

Depending on the type there are several definitions. An acceptor finite-state machine is a quintuple (Σ,S,s0,δ,F), where: In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...

• Σ is the input alphabet (a finite, non-empty set of symbols).
• S is a finite, non-empty set of states.
• s0 is an initial state, an element of S. In a Nondeterministic finite state machine, s0 is a set of initial states.
• δ is the state-transition function: $delta: S times Sigma rightarrow S$.
• F is the set of final states, a (possibly empty) subset of S.

A finite state transducer is a sextuple (Σ,Γ,S,s0,δ,ω), where: In computer science, an alphabet is a finite set of characters or digits. ... In the theory of computation, a nondeterministic finite state machine or nondeterministic finite automaton (NFA) is a finite state machine where for each pair of state and input symbol there may be several possible next states. ... A finite state transducer (FST) is a finite state machine with two tapes. ...

• Σ is the input alphabet (a finite non empty set of symbols).
• Γ is the output alphabet (a finite, non-empty set of symbols).
• S is a finite, non-empty set of states.
• s0 is the initial state, an element of S. In a Nondeterministic finite state machine, s0 is a set of initial states.
• δ is the state-transition function: $delta: S times Sigma rightarrow S$.
• ω 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. In computer science, an alphabet is a finite set of characters or digits. ... In the theory of computation, a nondeterministic finite state machine or nondeterministic finite automaton (NFA) is a finite state machine where for each pair of state and input symbol there may be several possible next states. ... In the theory of computation, a Mealy machine is a finite state machine where the outputs are determined by the current state and the input. ... Moore model: control of an elevator door In the theory of computation, a Moore machine is a finite state automaton where the outputs are determined by the current state alone (and not on the input). ... It has been suggested that state transition system be merged into this article or section. ... It has been suggested that state transition system be merged into this article or section. ...

## Optimization

Optimizing an FSM means finding the machine with the minimum number of states that performs the same function. The fastest known algorithm doing this is the Hopcroft minimization algorithm.[1][2]. Other techniques include using an Implication table, or the Moore reduction procedure. Additionally, acyclic FSAs can be optimized using a simple bottom up algorithm[3]. An implication table is a tool used to faciliate the minimization of states in a state machine. ... The Moore reduction procedure is a method used for minimizing states in a state machine. ...

## Implementation

### Hardware applications

Fig. 6 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 which determines the state transition, and a second block of combinational logic that determines the output of an FSM. One of the classic hardware implementations is the Richard's Controller. Download high resolution version (1712x943, 21 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1712x943, 21 KB) Wikipedia does not have an article with this exact name. ... The circuit diagram for a 4 bit TTL counter, a type of state machine A circuit diagram (also known as an electrical diagram, elementary diagram, or electronic schematic) is a simplified conventional pictorial representation of an electrical circuit. ... A Motorola 68000-based computer with various TTL chips. ... Digital circuits are electric circuits based on a number of discrete voltage levels. ... A programmable logic device or PLD is an electronic component used to build digital circuits. ... PLC & input/output arrangements A programmable logic controller (PLC), or programmable controller is a digital computer used for automation of industrial processes, such as control of machinery on factory assembly lines. ... A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. ... In digital circuits, a flip-flop is a bistable multivibrator, an electronic circuit that has two stable states and thereby is capable of serving as one bit of memory. ... Automotive style miniature relay A relay is an electrical switch that opens and closes under the control of another electrical circuit. ... In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to frequently used valuesâ€”typically, these values are involved in multiple expression evaluations occurring within a small region on the program. ... The Richardâ€™s Controller is a clever method of implementing a finite state machine using simple integrated circuits and combinational logic. ...

### Software applications

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

A finite state machine (FSM) is event driven if any incoming input has to be consumed immediately, i. ... Figure 1: VFSM in the Virtual Environment The virtual finite state machine (VFSM) is a concept promoted by SW Software and implemented in their StateWORKS product. ... Automata-Based Programming is a programming paradigm. ...

## History

Starting in the 1970s, Leslie Lamport, an early leader within the distributed systems research community, used finite state machines as the basis for an algorithm he called state machine replication. In this approach, a deterministic computer program or service is replaced with a set of replicas that use some form of atomic broadcast to perform operations in a manner tolerant of failures. Leslie Lamport Dr. Leslie Lamport (born 1941) is an American computer scientist. ...

An abstract state machine is most often used as a synonym for a finite state machine with regards to abstract algorithms (for sorting, etc. ... Decision tables are a precise yet compact way to model complicated logic. ... In a conventional finite state machine, the transition is associated with a set of input Boolean conditions & a set of output Boolean functions. ... A Petri net is a mathematical representation of discrete distributed systems. ... In automata theory, a pushdown automaton (PDA) is a finite automaton that can make use of a stack containing data. ... In quantum computing, quantum finite automata or QFA are a quantum analog of probabilistic automata. ... In digital circuit theory, sequential logic is a type of logic circuit whose output depends not only on the present input but also on the history of the input. ... Statecharts originally invented by David Harel in 1984 are an extension to state machines. ... It has been suggested that state transition system be merged into this article or section. ... For the test of artificial intelligence, see Turing test. ... State transitions in a hidden Markov model (example) x â€” hidden states y â€” observable outputs a â€” transition probabilities b â€” output probabilities A hidden Markov model (HMM) is a statistical model in which the system being modeled is assumed to be a Markov process with unknown parameters, and the challenge is to... SCXML stands for State Chart XML : State Machine Notation for Control Abstraction. ...

Results from FactBites:

 AutoFSM - Automated Finite State Machine (573 words) Finite state machines can be constructed in a variety of ways. Here is the list of attributes that various flavors of software finite state machines may have. The finite state machine may be either the master, driving the other parts of the program, or act as a subroutine keeping track of state between calls.
 Finite state machine (830 words) Finite state machines are studied in automata theory, a subfield of theoretical computer science. Finite automata may operate on languages of finite words (the standard case), infinite words (Rabin automata, Büchi automata[?]), or various types of trees (tree automata), to name the most important cases. Apart from theory, finite state machines occur also in hardware circuits, where the input, the state and the output are bit vectors of fixed size (Moore and Mealy machines).
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