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state-minimize

# state-minimize - 8b State Minimization One obvious way to...

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8b: State Minimization One obvious way to reduce the number of states is to eliminate any states that are not reachable from the start state. Hereafter, we shall assume that all states in the given DFA M are reachable from the start state. Let M = ( K. Σ , δ, s, F ) be a DFA. Define a relation A M K × Σ as follows: ( q, w ) A M if and only if w drives M from q to an accepting state. That is, ( q, w ) A M if and only if ( q, w ) M ( f, e ) for some f F . Definition Two states q, p K are equivalent, denoted q p , if the following holds: for all z Σ , ( q, z ) A M if and only if ( p, z ) A M . i.e., all strings z drive M from both p and q to some final state(s) or both to some non-final state(s). Note that is an equivalence relation (i.e., reﬂexive, sym- metric, and transitive) thus it induces a partition on the 1

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set K . 2
Constructing minimum-state automata Given a DFA M , there exists an algorithm that construct a minimum-state automaton M such that L ( M ) = L ( M ).

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