Prove that L /∈R

Languages and Automata 2020 #11 Decidable and recognizable languages

1. Prove that the language of all prime numbers (in binary representation) is recursive (Turing-decidable).

(You can use tha fact, that arithmetic operations can be performed by Turing machines.)

Solution: The Turing machine checks if the input can be divided by any number smaller than itself.

If yes, it rejects. If not, it accepts.

2. Prove that (a) L_d 6∈R (b) L_d ∈RE

Solution: A Turing machine for L_d:

(a) checkw - if it is not a code for a TM, accept (b) otherwise runM_w on w

(c) if it accepts, stop and accept, otherwise reject or run forever

If L_d would be recursive, then R =RE would hold. But there are languages inRE\R, so L_d∈/ R.

3. Let L={w#s:w∈L_d, and s6∈L(M_w)}. Prove that L /∈R.

Solution: IfL∈R, then there is a Turing machine that stops on all inputs, and decides if w#s ∈L.

If we give that Turing machine w#w as an input, we have a Turing machine for L_d. But we know, that there can not be one, so L /∈R.

4. Let L be the language of descriptions w of Turing machines where M_w accepts the prefix of w of length 5 (i.e. the string formed by the first 5 characters ofw). Show that this language is recognizable (it is in RE).

Solution: A Turing machine for L:

(a) checkw - if it is not a code for a TM, reject (b) otherwise runMw on the first five letter of w

(c) if it accepts, stop and accept, otherwise reject or run forever 5. Let L={w:w is a decription of a TM and L(M_w) =∅}.

(a) Prove that L /∈R.

(b) Is it true that L∈RE?

(c) Is it true that L∈co RE?

Solution:

(a) Rice’s theorem. L₁ =∅ ∈RE has this property,L₂ = Σ^∗ ∈RE does not have this property.

(c) It is in coRE. L={w|wis not a description of a TM, or L(M_w) accepts at least one word}

A Turing machine for L:

(a) checkw - if it is not a code for a TM, accept

(b) otherwise check all possible words withM_w using the diagonal method (c) if there is one input it accepts, stop and accept

(b) Because L∈coRE\R, it cannot be in RE.

6. The languageLconsists of Turing machine descriptionsw, such that M_w accepts only strings of odd length (but not necessarily accepts all of these strings). Prove that

(a) L /∈R.

(b) L∈RE.

Solution:

(a) Rice’s theorem: L₁ =}w| |w|= 3} ∈RE has this property,L₂ = Σ^∗ ∈REdoes not have this property.

(b) L={w|w is not a description of a TM, or L(Mw) accepts at least one even length word}

A Turing machine for L:

(a) checkw - if it is not a code for a TM, accept

(b) otherwise check all possible even length words with M_w using the diagonal method (c) if there is one input it accepts, stop and accept