automaton
An automaton is a semiautomaton with two types of special states: starting states, and final states. Specifically, an automaton $A$ is is a fivetuple $(S,\mathrm{\Sigma},\delta ,I,F)$, where

1.
$(S,\mathrm{\Sigma},\delta )$ is a semiautomaton,

2.
a nonempty set $I\subseteq S$ of starting states, and

3.
a set $F\subseteq S$ of final states or terminating states.
A triple $(s,a,t)$ is called a transition if $t\in \delta (s,a)$, and is written $s\stackrel{a}{\u27f6}t$. The set $\delta (s,a)$ may contain more than one element (or none at all), which is why an automaton is also said to be nondeterministic. If on the other hand $\delta (s,a)$ is a singleton for every $(s,a)$, and $I$ is a singleton, then $A$ is said to be deterministic^{}. In a deterministic automaton, $\delta $ can be viewed as a function from $S\times \mathrm{\Sigma}$ to $S$.
If $S$ and $\mathrm{\Sigma}$ are both finite, then $A$ is called a finitestate automaton, or FSA for short.
The state diagram^{} ${G}_{A}$ of a finitestate automaton $A$ is constructed as if $A$ is being treated as a semiautomaton. In addition^{}, a vertex corresponding to a starting state has an incoming arrow without a source, and a vertex corresponding to a final state has an outgoing arrow without a destination (alternatively, it may be represented by a double circle). This is illustrated in the following example:
Let $A$ be given by $S=\{\sigma ,s,t\}$, where $\sigma $ is the starting state, and $t$ the final state, $\mathrm{\Sigma}=\{a,b\}$, with the transition function given by the following table
Title  automaton 
Canonical name  Automaton 
Date of creation  20130322 12:26:34 
Last modified on  20130322 12:26:34 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  42 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03D05 
Classification  msc 68Q45 
Synonym  nextstate function 
Synonym  terminating state 
Synonym  FSA 
Synonym  start state 
Synonym  recognizer 
Related topic  DeterministicFiniteAutomaton 
Related topic  NonDeterministicFiniteAutomaton 
Related topic  PushdownAutomaton 
Related topic  Semiautomaton 
Defines  finitestate automaton 
Defines  transition function 
Defines  starting state 
Defines  final state 
Defines  configuration^{} 
Defines  acceptor 
Defines  automata 
Defines  deterministic automaton 
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