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Equivalence and Minimization of Deterministic Finite Automata

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**1. **Equivalence and Minimization of Deterministic Finite Automata Joseph Southern
CPSC 627 Spring 2003

**3. **Introduction Basic Idea of Minimization of DFAs
We can take any DFA and find an equivalent DFA that has the minimum number of states.
In fact, this DFA is essentially unique: given any two minimum state DFAs that are equivalent, we can always find a way to rename the states so that the two DFA become the same.

**4. **Introduction Basic Idea
Find equivalent states
Use table-filling algorithm
Partition equivalent states derived from table-filling algorithm into blocks and condense DFA.

**5. **Definitions Equivalent When two distinct states p and q can be replaced by a single state that behaves like both p and q.
Distinguishable When two states are not equivalent. That is, state p is distinguishable from state q if there is at least one string ? such that one of ?(p,?) and ?(q,?) is accepting, and the other is not accepting.

**6. **Table-filling Algorithm Initialize all entries as unmarked and with no dependencies.
Mark all pairs of a accepting and non-accepting state.
For each unmarked pair p,q and input symbol a:
Let r = ?(p,a), s = ?(q,a).
If (r,s) unmarked, add (p,q) to (r,s)s dependencies,
Otherwise mark (p,q), and recursively mark all dependencies of newly-marked entries.

**7. **Table-filling Example

**25. **Minimization of DFA Partition equivalent states together.

**28. **Citations J. E. Hopcroft, R. Motwani, and J. D. Ullman. "Introduction to Automata Theory, Language, and Computation". Addison--Wesley, 2nd edition, 2001.
http://www.cs.yorku.ca/course_archive/2001-02/S/2001B/lectures/gh.pdf

**29. **Questions?