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Doubt in reduction of automata transition relation

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I did not understand, how is the transition relation reduced to what we have in the given automaton formulas. Can someone please explain?

in * TF "Emb. Sys. and Rob." by (360 points)

1 Answer

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The transition relation r0&!r1&!a | !r0&r1&a | r0&r1 is equivalent to r0&!a | r1&a:

    r0&!r1&!a | !r0&r1&a | r0&r1
    = (a ? r0&!r1&!1 | !r0&r1&1 | r0&r1 : r0&!r1&!0 | !r0&r1&0 | r0&r1)
    = (a ? !r0&r1 | r0&r1 : r0&!r1 | r0&r1)
    = (a ? (!r0|r0)&r1 : r0&(!r1|r1))
    = (a ? r1 : r0)
    = a&r1 | !a&r0
by (166k points)

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