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. In Computer science and Automata theory, a state is a unique configuration of information in a program or machine A finite state machine is an abstract model of a machine with a primitive internal memory.

## Concepts and vocabulary

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

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

A FSM can be represented using a state diagram (or state transition diagram) as in figure 1 above. State diagrams is a Diagram used in the field of Computer science, representing the behavior of a system which is composed of a finite number of states Besides this, several state transition table types are used. In Automata theory and Sequential logic, a state transition table is a table showing what state (or states in the case of a nondeterministic finite automaton 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). The "virtual finite state machine" (VFSM is a concept promoted by SW Software and implemented in their StateWORKS product The "virtual finite state machine" (VFSM is a concept promoted by SW Software and implemented in their StateWORKS product

 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. Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of Linguistics is the scientific study of Language, encompassing a number of sub-fields Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Foundations of modern biology There are five unifying principles Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Logic is the study of the principles of valid demonstration and Inference. 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. 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 In computer science, finite state machines are widely used in modeling of application behavior, design of hardware digital systems, software engineering, compilers, network protocols, and the study of computation and languages.

## Classification

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

### Acceptors and recognizers

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.

#### Start state

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

#### Accept state

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

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

An example of an accepting state appears on the left in this diagram of a deterministic finite automaton (DFA) which determines if the binary input contains an even number of 0s. In the Theory of computation, a deterministic finite state machine (also known as deterministic finite state automaton (DFSA or deterministic finite 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. A finite-state transducer ( FST) is a Finite state machine with two tapes an input tape and an output tape They are used for control applications and in the field of computational linguistics. Computational linguistics is an Interdisciplinary field dealing with the statistical and/or rule-based modeling of Natural language from a computational Here two types are distinguished:

Moore machine
The FSM uses only entry actions, i. In the Theory of computation, a Moore machine is a Finite state automaton where the outputs are determined by the current state 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. In the Theory of computation, a Mealy machine is a Finite state machine (and more accurately a Finite state transducer) that generates an output based 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). The "virtual finite state machine" (VFSM is a concept promoted by SW Software and implemented in their StateWORKS product In Computation, a Finite state machine (FSM is event driven if the creator of the FSM intends to think of the machine as consuming events or messages There are two input actions (I:): "start motor to close the door if command_close arrives" and "start motor in the other direction to open the door if command_open arrives".

In practice mixed models are often used.

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

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

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

## FSM logic

Fig. 5 FSM Logic

The next state and output of an FSM is a function of the input and of the current state. The FSM logic is shown in Figure 5.

## Mathematical model

Depending on the type there are several definitions. An acceptor finite-state machine is a quintuple (Σ,S,s0,δ,F), where:

• Σ is the input alphabet (a finite, non-empty set of symbols). In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Computer science, an alphabet is a usually finite set of characters or digits
• 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. In the Theory of computation, a nondeterministic finite state machine or nondeterministic finite automaton (NFA is a Finite state machine where for each
• 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 transducer finite-state machine is a sextuple (Σ,Γ,S,s0,δ,ω), where:

• Σ is the input alphabet (a finite non empty set of symbols). In Computer science, an alphabet is a usually finite set of characters or digits
• 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. In the Theory of computation, a nondeterministic finite state machine or nondeterministic finite automaton (NFA is a Finite state machine where for each
• 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. In the Theory of computation, a Mealy machine is a Finite state machine (and more accurately a Finite state transducer) that generates an output based 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. In the Theory of computation, a Moore machine is a Finite state automaton where the outputs are determined by the current state A finite-state machine with no output function at all is known as a semiautomaton or transition system. In Theoretical computer science, a semiautomaton is is an automaton having only an input and no output In Theoretical computer science, a state transition system is an Abstract machine used in the study of Computation.

## Optimization

Optimizing an FSM means finding the machine with the minimum number of states that performs the same function. One possibility is by using an Implication table or the Moore reduction procedure. An implication table is a tool used to facilitate the minimization of states in a State machine. The Moore reduction procedure is a method used for minimizing states in a State machine. Another possibility is bottom-up algorithm for Acyclic FSAs.

## 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. A circuit diagram (also known as an electrical diagram Wiring diagram, elementary diagram or electronic Schematic) is a simplified conventional pictorial representation Transistor–transistor logic ( TTL) is a class of Digital circuits built from Bipolar junction transistors (BJT and Resistors It is called Digital electronics are Electronics systems that use Digital signals Digital electronics are representations of Boolean algebra also see A programmable logic device or PLD is an electronic component used to build reconfigurable Digital circuits Unlike a Logic gate, which has a A programmable logic controller ( PLC) or programmable controller is a Digital computer used for Automation of industrial processes such as 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 term referring to an Electronic circuit (a Bistable Multivibrator) that has two stable states and thereby A relay is an electrical Switch that opens and closes under the control of another Electrical circuit. 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. In Computer architecture, a processor register is a small amount of storage available on the CPU whose contents can be accessed more quickly than storage One of the classic hardware implementations is the Richard's Controller. The Richard’s Controller is a 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:

## 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 Computation, a Finite state machine (FSM is event driven if the creator of the FSM intends to think of the machine as consuming events or messages 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 in which the program or its part is thought of as a model of a Finite state machine or any other (often more Dr Leslie Lamport (born February 7, 1941 in New York City) is an American computer scientist. Introduction from Schneider's 1990 survey "Distributed software is often structured in terms of clients and services 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.