reduced automaton
Besides eliminating useless states that do not contribute to word acceptance, what else can we do to “clean up” an automaton? In other words, we would like to reduce the number of states in an automaton to as small as possible without affecting its power of accepting words.
The next thing we can do is to look to find states that do the same thing and combine them into one state. Two states do the same thing if, acting as starting states, when an arbitrary word is fed into the automaton, they lead to the same conclusion: either both accept the word, or both reject it. Arbitrary automata are too general to ensure definite conclusions, so we restrict our attention to deterministic finite automata.
Definition. Let be an automaton with a single start state . For any state , define to be the automaton obtained from by replacing by . In other words, . Two states are said to be indistinguishable if .
Let us write to mean that and are indistinguishable. Then is an equivalence relation on . For each , write the equivalence class containing . Let be the set of equivalence classes.
If is a DFA, is the same as saying that, for all words over , iff . For the following discussion, we shall assume that is a DFA, unless otherwise specified.
Here are two basic observations:
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if , then : Let and . Given any over , we have that , which is in iff is in . Hence .
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if and , then : as shows that . As a result of this, iff .
Define as follows: , , , and , for any .
Now, is a well-defined function from the first observation above, so that is a well-defined automaton. It has the following property: for any word over :
This can be proved by induction on the length of :
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suppose , then for any ,
It also has the property that for if two states are indistinguishable, they are in fact the same state: suppose , then for every over , iff , or iff , which is the same as saying iff , or , or .
Definition. An finite automaton with a single start state is said to be reduced or minimal if implies , where is the indistinguishable relation on its state set.
Thus constructed above is reduced.
Below are state diagrams of an automaton (not deterministic) and its reduction :
Note that , as , and are combined into one state . Similarly, , as , and are combined into . Finally, note that .
With respect to DFA’s, the following is true:
Proposition 1.
Every DFA is equivalent to a reduced DFA . Furthermore, if is accessible (or simplified), so is . Finally, two equivalent reduced accessible automata are isomorphic.
Proof.
We have already shown that given a DFA , the automaton constructed above is reduced. Next, we see that iff iff iff iff . Therefore, .
Next, suppose is accessible, we want to show that is accessible for any . Since is accessible, for some word over . So is accessible as well. This proves that is accessible.
Suppose now that is simplified. Pick any , we want to show that is useful. Since is useful, there are words over such that and , or equivalently and , so that is useful. This shows that is simplified.
Finally, if and are both accessible and reduced such that . Then the Myhill-Nerode relations induced by and are both equal to the Nerode equivalence of . As a result, and are both isomorphic to the automaton generated by . ∎
Remark. The proposition above is not true in general for non-deterministic finite automata: two equivalent simplified reduced NDFA may not be isomorphic. However, there exist reduction techniques for NDFA such that the reduced automata produced using these techniques are unique up to isomorphism.
Title | reduced automaton |
Canonical name | ReducedAutomaton |
Date of creation | 2013-03-22 19:01:56 |
Last modified on | 2013-03-22 19:01:56 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 14 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 03D10 |
Classification | msc 68Q42 |
Classification | msc 68Q05 |
Synonym | minimal automaton |
Related topic | SimplifiedAutomaton |
Defines | indistinguishable |
Defines | reduced |
Defines | minimal |