For each of the following languages either prove it is undecid-able using a reducibility proof or…

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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 it is decidable by giving pseudocode for a TM that decides it: (a) Let FINITE M IM is a TM and M accepts a finite number of strings (b) Let ALLTAr (M) I M is a TM and M accepts every string) (c) Let HALT B, w) I B is an LBA and B halts on input wh LBA (d) Let MORE G1, G2) I G1 and G2 are CFGs where IG1 IG CFG (i.e., G2 accepts strictly more strings than G1) (e) Let EMPTY (G) I G is a CFG and G generates E) (2) (2 points) Show that CFG (G1, G2) G1 and G2 are CFGs where L(G1) L(G2 is co-Turing-recognizable. (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

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: (a) Let FINITE_TM = {(M) | M is a TM and M accepts a finite number of strings} (b) Let ALL_TM = {(M) | M is a TM and M accepts every string} (c) Let H ALT_LBA = {(B, w) | B is an LBA and B halts on input w} (d) Let MORE_CFG = {(G_1, G_2) | G_1 and G_2 are CFGs where |G_1| < |G_2| (i.e., G_2 accepts strictly more strings than G_1} (e) Let EMPTY_CFG = {(G) | G is a CFG and G generates epsilon} Show that EQ_CFG = {(G_1, G_2) | G_1 and G_2 are CFGs where L(G_1) = L(G_2} is co-Turing-recognizable. 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.

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