Question
2. Reducibility (16 points) (1) (10 points) For each of the following languages either prove it is undecid- able using a reducibility proof or show
2. Reducibility (16 points)
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(1) (10 points) For each of the following languages either prove it is undecid- able using a reducibility proof or show it is decidable by giving pseudocode for a TM that decides it:
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(a) Let FINITETM = {M | M is a TM and M accepts a finite number
of strings}
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(b) Let ALLTM = {M | M is a TM and M accepts every string}
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(c) Let HALTLBA = {B,w | B is an LBA and B halts on input w}
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(d) Let MORECFG = {G1,G2 | G1 and G2 are CFGs where |G1| < |G2|
(i.e., G2 accepts strictly more strings than G1}
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(e) Let EMPTYCFG = {G | G is a CFG and G generates }
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(2) (2 points) Show that EQCF G = {G1, G2 | G1 and G2 are CFGs where L(G1) = L(G2} is co-Turing-recognizable.
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(3) (4 points) A useless state in a Turing machine is one that is never entered on any input string. Consider the problem of testing whether a given state in a Turing machine is a useless state. Formulate this problem as a language and show that it is undecidable.
Hint: Consider the language ETM and whether the accept state is a useless state.
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