Finite-state machine
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] For any non-deterministic finite-state machine, an equivalent deterministic one can be constructed.
"State machine" redirects here. For infinite-state machines, see Transition system. For fault-tolerance methodology, see State machine replication.
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.
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:
Usage[edit]
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.
Alternative semantics[edit]
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]
is the input (a finite non-empty set of symbols);
alphabet
is a finite non-empty set of states;
is an initial state, an element of ;
is the state-transition function: (in a it would be , i.e. would return a set of states);
nondeterministic finite automaton
is the set of final states, a (possibly empty) subset of .
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 , where:
For both deterministic and non-deterministic FSMs, it is conventional to allow to be a partial function, i.e. does not have to be defined for every combination of and . If an FSM is in a state , the next symbol is and is not defined, then 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 , where:
If the output function depends on the state and input symbol () that definition corresponds to the Mealy model, and can be modelled as a Mealy machine. If the output function depends only on the state () 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, , 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]
