Automata, Languages, and Machines/Part A: v. A by Samuel Eilenberg

By Samuel Eilenberg

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Extra resources for Automata, Languages, and Machines/Part A: v. A

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5n 5N} + 1 states. Construct such an automaton with ex- 3. Operations on Automata When we consider two automata and A it will be silently assumed that their sets of states Q,, and 0, do not overlap. This can always be achieved by relabeling the states in one of them. We shall now pass in review a number of basic operations on automata and analyze their behaviors. , u T u . 3. This yields ~ d u . a ~ = / d ~ u ~ L 2 ? ~ . 2? , define the Zautomaton L d X I&= 29 by setting Q~~= Q . x~ Q ~ ~ rc, = r, x I ~ , T~ = T .

Show that for any subset A of erties are equivalent: P the following prop- If st E A , then s E A . (i) all states are terminal. d, A is the belzavior of a deterministic automaton in which all states are (iii) t er in ina 1. Show that this class of sets i s closed under intersection. 2. For each (non-deterministic) 2-automaton let d ' denote the result I$ the accessible (zero-free) subset construction, let d " be the completion of d', and let be the reversal of d.

1) = 1- for all X E Qu Indeed 1 X s E TO} = {S I SK'XE T " } = {s 1 1 E s-'*Y} P'TO = {S = * {sIs€X}=-Y For any subset ,4 of Z* we may consider the (infinite) complete 2automaton d . 1) I LdAdo I =A Let = (Q, i, T) be any deterministic 2-automaton with behavior A. 2) p': d-+ Ld2,0 is a state-mapping. s T u l E q - ' T u l E q p u q p E TO Indeed, since d', is"complete, the last line shows that the state-mapping rp is proper; however, this fact will not be used. Ill. Deterministic A u t o m a t a 46 Let Ld-l = dltbe the trim part of d ' , O .

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