Exam 13: A: Modeling Computation

Construct a Turing machine that computes $f \left( n _ { 1 } , n _ { 2 } \right) = n _ { 2 } + 1 , \text { where } n _ { 1 } , n _ { 2 } \geq 0$

Suppose a phrase-structure grammar has productions S → S11, S → 0A, S → A1, A → 0. Find a derivation of 01.

Find a grammar for the set $\left\{ 0 ^ { 2 n } 1 ^ { n } \mid n \geq 0 \right\}$

Suppose a phrase-structure grammar has productions S → 1S0, S → 0A, A → 0. Find a derivation of 00.

Construct a finite-state automaton that recognizes all strings that end with 11.

Find the Kleene closure of A={0,1,2} .

What language is generated by the phrase-structure grammar if the productions are $S \rightarrow S 11 , S \rightarrow \lambda$ where $S$ is the start symbol?

Suppose a phrase-structure grammar has productions S → S11, S → 0A, S → A1, A → 0. Find a derivation of 0011.

Suppose A = {0, 1}. Describe all strings belonging to A∗.

Determine if 1101 belongs to the regular set 1(10)*1*.

For the following Turing machines T , find the final tape when T is run on the following tape, beginning in the initial position (the first nonzero entry from the left): \@cdots 0 0 0 1 0 \@cdots ,0,,0,R , ,1,,0,R , ,0,,1,R , ,1,,1,L , ,B,,1,L

let $V = \{ S , A , B , 0,1 \} \text { and } T = \{ 0,1 \} \text {. For each set of productions determine whether the }$ resulting grammar G is (i) type 0 grammar, but not type 1 , (ii) type 1 grammar, but not type 2 , (iii) type 2 grammar, but not type 3 , (iv) type 3 grammar. - $S \rightarrow B , A \rightarrow B , B \rightarrow A$

Find a production of the form "A → ____" such that S → 1S , S → 0A, A → ____ produces {1ⁿ00 | n ≥ 0}.

Suppose a phrase-structure grammar has productions S → 1S0, S → 0A, A → 0. Find a derivation of

For the following Turing machine T , find the final tape when T is run on the following tape, beginning in the initial position (the first nonzero entry from the left): \@cdots 0 0 0 1 0 \@cdots ,0,,0,R , ,B,,1,R , ,0,,1,R , ,0,,1,L , ,1,,1,R

Determine if 1101 belongs to the regular set (11)*0*(11)*.

Find the set recognized by this deterministic finite-state machine.

What is the language generated by the grammar with productions $S \rightarrow S A , S \rightarrow 0 , A \rightarrow 1 A$ , and $A \rightarrow 1$ , where S is the start symbol?

Which strings belong to the regular set represented by the regular expression (1∗01∗0) ?

Consider the Turing machine T : $\left( s _ { 0 } , 0 , s _ { 1 } , 1 , R \right) , \left( s _ { 0 } , 1 , s _ { 1 } , 1 , R \right) , \left( s _ { 1 } , 0 , s _ { 0 } , 1 , L \right) , \left( s _ { 1 } , 1 , s _ { 0 } , 0 , R \right) , \left( s _ { 0 } , B , s _ { 1 } , 1 , R \right)$