By Almansa J.A., et al.

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**Example text**

And G\, G2,... be, respectively, the standard enumeration of Turing machines, (TM's), nondeterministic polynomial-time bounded TM's, deterministic polynomial-time bounded TM's and context-free grammars. The Kleene Hierarchy provides an elegant classification of undecidable problems. e. sets and it can be characterized as the set of languages of the form L = {x\(3y)R[x,y]}, where R is a recursive predicate. Ill is the corresponding class with a universal quantifier over a recursive predicate, L={x\(Vy)R[x,y}}.

E. list of names A^, Ai2,... such that 26 {L(Aii)\j>l} = {L(Ai)\AimA}. e. list of names Ail,Ai2,... write A as a E2 set: A = {Ai\ (3Aij)(Wx)[Aij(x) = then we could Mx)}}. This is a contradiction to the assumption that A is a Il2-hard set. e. e. e. sets for which the property is true and sets for which it is false. e. sets is recursively undecidable for Turing machines. The situation is quite different in automata theory. There are many decidable and undecidable problems of various difficulty and different varieties of incompleteness results.

From the above it follows that for each M, we can effectively construct such that L{Ga{i)) = £* - VAL(Mi) = Ga^ VAL(Mi). We observe that L(Mi) = 0 iff VAL{Mi) = 0 iff VAL(Mi) = E*. Since VAL(Mi) = L(GtT^i-j) and it is recursively undecidable if L{Mi) — 0, we see that for context-free grammars it is recursively undecidable if L(G,) = E*. Recall that it is recursively decidable if L(d) = 0 and if L(Gi) is infinite. 2> • • •, it is recursively undecidable if L{Di) = 0 and if L(D{) is infinite. Every undecidability result implies a Godel-like incompleteness result.