Question 1

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

Accepted Answer

$$\left( s _ { 0 } , 1 , s _ { 1 } , B , R \right) , \left( s _ { 1 } , 1 , s _ { 1 } , B , R \right) , \left( s _ { 1 } , * , s _ { 2 } , 1 , R \right)$$

Question 2

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

Accepted Answer

$$S \Rightarrow A 1 \Rightarrow 01$$

Question 3

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

Accepted Answer

Use the grammar with productions $S \rightarrow 00 S 1$ and $S \rightarrow \lambda$ , where S is the start symbol.

Question 4

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

Accepted Answer

The answer of Suppose a phrase-structure grammar has productions S...

Question 5

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

Accepted Answer

The answer of Construct a ﬁnite-state automaton that recognizes all...

Question 6

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

Accepted Answer

All string

Question 7

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?

Accepted Answer

The language generated is the

Question 8

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

Accepted Answer

The answer of Suppose a phrase-structure grammar has productions S...

Question 9

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

Accepted Answer

A∗ consists of all s

Question 10

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

Accepted Answer

The answer of Determine if 1101 belongs to the regular...

Question 11

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): $$\begin{array}{l}
\begin{array} { l | l | l | l | l | l | l | l | l | l | l | l } 
\hline \cdots & \mathrm { B } & \mathrm { B } & 0 & 0 & 0 & 1 & \mathrm {~B} & 0 & \mathrm {~B} & \mathrm {~B} & \cdots \
\hline
\end{array}\
\left( s _ { 0 } , 0 , s _ { 0 } , 0 , R \right) , \left( s _ { 0 } , 1 , s _ { 1 } , 0 , R \right) , \left( s _ { 1 } , 0 , s _ { 1 } , 1 , R \right) , \left( s _ { 1 } , 1 , s _ { 2 } , 1 , L \right) , \left( s _ { 1 } , B , s _ { 1 } , 1 , L \right)
\end{array}$$

Accepted Answer

\[\begin{array} { l | l | l |

Question 12

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.
-$$S \rightarrow B , A \rightarrow B , B \rightarrow A$$

Accepted Answer

The answer of let \(V = \{ S , A...

Question 13

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

Accepted Answer

Question 14

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

Accepted Answer

The answer of Suppose a phrase-structure grammar has productions S...

Question 15

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): $$\begin{array}{l}
\begin{array} { l | l | l | l | l | l | l | l | l | l | l | l } 
\hline \cdots & \mathrm { B } & \mathrm { B } & 0 & 0 & 0 & 1 & \mathrm {~B} & 0 & \mathrm {~B} & \mathrm {~B} & \cdots \
\hline
\end{array}\
\left( s _ { 0 } , 0 , s _ { 2 } , 0 , R \right) , \left( s _ { 0 } , B , s _ { 0 } , 1 , R \right) , \left( s _ { 1 } , 0 , s _ { 2 } , 1 , R \right) , \left( s _ { 2 } , 0 , s _ { 1 } , 1 , L \right) , \left( s _ { 2 } , 1 , s _ { 0 } , 1 , R \right)
\end{array}$$

Accepted Answer

\[\begin{array} { c | c | c |

Question 16

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

Accepted Answer

The answer of Determine if 1101 belongs to the regular...

Question 17

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

Accepted Answer

The set re

Question 18

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?

Accepted Answer

The set of all bit s

Question 19

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

Accepted Answer

The strings containi

Question 20

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

Accepted Answer

\[\begin{array} { c | c | c |

Exam 13: A: Modeling Computation

Construct a Turing machine that computes f(n1,n2)=n2+1, where n1,n2≥0f \left( n _ { 1 } , n _ { 2 } \right) = n _ { 2 } + 1 , \text { where } n _ { 1 } , n _ { 2 } \geq 0f(n1​,n2​)=n2​+1, where n1​,n2​≥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 {02n1n∣n≥0}\left\{ 0 ^ { 2 n } 1 ^ { n } \mid n \geq 0 \right\}{02n1n∣n≥0}

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→S11,S→λS \rightarrow S 11 , S \rightarrow \lambdaS→S11,S→λ where SSS 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

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→SA,S→0,A→1AS \rightarrow S A , S \rightarrow 0 , A \rightarrow 1 AS→SA,S→0,A→1A , and A→1A \rightarrow 1A→1 , where S is the start symbol?

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

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

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

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

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?

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

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

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

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?