Formal Languages and Their Relation to Automata by John E. Hopcroft, Jeffrey D. Ullman

By John E. Hopcroft, Jeffrey D. Ullman

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Three cases arise' 1. ~(p~, a) = (q, S). Here M will again scan cell j, this time in state q. So P~+i = q. t Do not confuse R in the range of 4, which means reject, with R in the range of which means move right. 7 43 2. 3(p, a ) = (q, R). Here, we have our answer. M has moved to cell j + 1 for the first time and entered state q. , Az(pl) = q. 3. 3(p,, a) = (q, L). Here, M next moves left t o cell j - 1. We consult A1 to see what happens next. If ,51(q) = R, then M will never again scan cell j, so it cannot scan j + 1 either.

Then since R' is right invariant, for each z in Z*, xzR'yz, and thus yz is in L if and only if xz is in L. Thus xRy, and hence, the equivalence class of x in R' is contained in the equivalence class of x in R. We conclude that each equivalence class of R' is contained within some equivalence class of R. (3) ~ (1). Assume that xRy. Then for each w and z in Z*, xwz is in L if and only if ywz is in L. Thus xwRyw, and R is right invariant. Now let K' be the finite set of equivalence classes of R and x the element of K' containing x.

The reader can simplify grammar G1 so that its equivalence to G is readily observable. 5 PROPERTIES OF TYPE 3 L A N G U A G E S Since the class of languages generated by type 3 grammars is equivalent to the class of sets accepted by finite automata, we shall use both formulations in establishing the properties of the class of type 3 languages. First we intend to show that the type 3 languages form a Boolean algebra1" of sets. 1. The class of type 3 languages is closed under union. Proof Two proofs are possible.

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