simplified automaton

Word acceptance is the primary function of an automaton. When a word is fed into an automaton at an initial state, it is accepted exactly when a final state is reached upon reading the last symbol of the word. So when a state does not contribute in anyway to word acceptance, it may be tossed out without changing the accepting power of the automaton.

Definition. Let $A=(S,\mathrm{\Sigma },\delta ,I,F)$ be an automaton. A state $s\in S$ is said to be reachable from a state $t\in S$, if for some word $u$ over $\mathrm{\Sigma }$, $t\in \delta (s,u)$. A state is said to be accessible if it is reachable from a start state. A state is live or coaccessible if a final state can be reached from it. If a state is both accessible and live, then it is called useful.

Clearly, every start state is accessible, and every final state is live. The following is the state diagram of an automaton that illustrates the concepts defined above.

States $\sigma ,p,r,s$ are accessible, while states $\sigma ,p,q$ are live. So the useful states are $\sigma$ and $p$.

The automaton $A$ is said to be accessible if every state in $S$ is accessible, and is live if every state is live. $A$ is called simplified or trim if it is both accessible and live (every state is useful).

Proposition 1.

An automaton is equivalent (http://planetmath.org/EquivalentAutomata) to a simplified automaton iff it accepts at least one word.

Proof.

Suppose an automaton $A$ is equivalent to a simplified automaton $A^{\prime }$. Then the set of start states of $A^{\prime }$ is non-empty, say containing a state $q$. So $q$ is live, which means there is a word $u$ such that $\delta (p,u)$ contains a final state, or that $u\in L(A^{\prime })=L(A)$.

Conversely, suppose an automaton $A=(S,\mathrm{\Sigma },\delta ,I,F)$ accepts a word $u$. This means that there is a start state $q$ and a final state $r$ such that $r\in \delta (q,u)$, so that both $q$ and $r$ are useful. Now, let $S^{\prime }$ be the subset (of $S$) all useful states in $A$, and let $I^{\prime }=I\cap S^{\prime }$, $F^{\prime }=F\cap S^{\prime }$. Then neither $I^{\prime }$ nor $F^{\prime }$ is empty. Finally, let ${\delta }^{\prime }$ be the restriction of $\delta$ on the set $S^{\prime }\times \mathrm{\Sigma }$, such that

$${\delta }^{\prime }(s,a):=\delta (s,a)\cap S^{\prime }.$$

Then the five-tuple $A^{\prime }:=(S^{\prime },\mathrm{\Sigma },{\delta }^{\prime },I^{\prime },F^{\prime })$ is an automaton. Obviously, $L(A^{\prime })\subseteq L(A)$. On the other hand, the word $u$ above is accepted by $A^{\prime }$. Since $u$ is arbitrary, we see that $L(A^{\prime })=L(A)$ as a result. ∎

Remark. If we allow the start state set of an automaton to be empty, then every automaton can be simplified, whether or not it accepts any words.

In the example above, the automaton is equivalent to the following simplified automaton:

However, the construction in the proof shows that, given a deterministic automaton $A$, the simplified automaton $A^{\prime }$ may not be determinstic. In the example above, $A^{\prime }$ is not deterministic because $\delta (\sigma ,b)=\mathrm{\varnothing }$. With respect to deterministic automata, the best we can do is the following:

Proposition 2.

A deterministic automaton is equivalent (http://planetmath.org/EquivalentAutomata) to a deterministic accessible automaton.

The proof of this is very similar to the one given above, except $S^{\prime },I^{\prime }$, and $F^{\prime }$ now consist of all accessible states, and ${\delta }^{\prime }$ is just a restriction on the domain of $\delta$, since if a state $s$ is accessible, so is any state in $\delta (s,a)$ for any $a\in \mathrm{\Sigma }$.

Again, referring the example above, $A$ is deterministic, and it is equivalent to the following deterministic accessible automaton:

The language accepted by the automaton is represented by the regular expression $a{(ab)}^{*}$.

Title	simplified automaton
Canonical name	SimplifiedAutomaton
Date of creation	2013-03-22 19:01:51
Last modified on	2013-03-22 19:01:51
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03D05
Classification	msc 68Q45
Synonym	trim automaton
Synonym	coaccessible
Related topic	ReducedAutomaton
Defines	reachable
Defines	accessible
Defines	live
Defines	useful