Zecyel/LyTOC · Datasets at Hugging Face

homework string	exercise_number string	content string	full_id string
hw1	1	Prove $$\frac{1}{2\ln 2} + \frac{1}{3\ln 3} + \ldots + \frac{1}{n\ln n} = \ln\ln n + O(1)$$ and $$\frac{1}{1^{1.1}} + \frac{1}{2^{1.1}} + \ldots \frac{1}{n^{1.1}} = O(1).$$	hw1_ex1
hw1	2	Order the following functions by asymptotic growth. Place $f$ before $g$ if $f(n) = O(g(n))$. * $n^n$ * $\sqrt{n}$ * $\log\log n$ * $0.001n^7$ * $5000n$ * $e^n$ * $2^{\log^2 n}$ * $10^{10^{10}}$ * $n!$ * $2^{2^n}$	hw1_ex2
hw1	3	Let $\Sigma = \{0, 1\}$. Let language $$L = \{w \in \{0, 1\}^* : w \text{ has an unequal number of 0's and 1's}\}.$$ Prove $L^* = \Sigma^*$.	hw1_ex3
hw10	1	Use first-order Peano arithmetic formulas to express "$m = 2^n$".	hw10_ex1
hw10	2	Prove P is closed under Kleene star. In other words, if a language $L$ belongs to P, then so does $L^*$.	hw10_ex2
hw10	3	Let $$L_1 = \{(\alpha, x) : \text{TM } M_\alpha \text{ accepts input } x\},$$ and $$L_2 = \{\alpha : \text{TM } M_\alpha \text{ halts on the empty input}\}.$$ Prove that $L_1 \leq_T L_2$.	hw10_ex3
hw11	1	Define the problem QUADEQ as follows: given m quadratic equations in n variables over the finite field GF(2), $$p_1(x_1, x_2, \ldots, x_n) = 0,$$ $$p_2(x_1, x_2, \ldots, x_n) = 0,$$ $$\vdots$$ $$p_m(x_1, x_2, \ldots, x_n) = 0,$$ determine whether there exists at least one solution. Prove that QUADEQ ∈ NP.	hw11_ex1
hw11	2	Prove if L is NP-hard and L ∈ P, then P = NP.	hw11_ex2
hw11	3	Prove that $L_1 \leq_P L_2$ and $L_2 \in$ NP, then $L_1 \in$ NP.	hw11_ex3
hw13	1	Construct a function $T(n)$ that satisfies $n \leq T(n) \leq n^2$ and is not time-constructible.	hw13_ex1
hw13	2	Define the low-degree clique (LDC) problem as follows: Given a graph $G = (V, E)$ and an integer $k$, determine whether $G$ contains a clique of size at least $k$ consisting solely of vertices whose degree does not exceed the median vertex degree of $G$. Prove that LDC is NP-complete.	hw13_ex2
hw13	3	Let $$\text{DOUBLE-SAT} = \{\langle \varphi \rangle \mid \text{formula } \varphi \text{ has at least two satisfying assignments}\}.$$ Prove that DOUBLE-SAT is NP-complete.	hw13_ex3
hw2	1	Design a finite automaton that accepts the following language $$L = \left\{ w \in \{0, 1\}^* \mid w = w_1w_2 \cdots w_n \text{ such that } \sum_{i=1}^{n} w_i 2^{n-i} \equiv 0 \pmod{5} \right\}.$$ In other words, the language $L$ is the set of all binary strings that represent multiples of 5, written with the most significant bit first (MSB). Draw the state diagram of the finite automaton.	hw2_ex1
hw2	2	Design a finite automaton that recognizes the following language $$L = \{w \in \{0, 1\}^* : w \text{ does not contain substring } 10\}.$$ Draw its state diagram.	hw2_ex2
hw2	3	Use the definition of big-O notation to prove the following: 1) If $f_1 = O(g_1)$ and $f_2 = O(g_2)$, then $f_1f_2 = O(g_1g_2)$. 2) If $f_1 = O(g)$ and $f_2 = O(g)$, then $f_1 + f_2 = O(g)$.	hw2_ex3
hw3	1	Define a backward-read DFA as a deterministic finite automaton that processes its input string from right to left. Prove that a language is regular if and only if it can be accepted by some backward-read DFA.	hw3_ex1
hw3	2	Give a regular expression for language $$L = \{w \in \{0, 1\}^* : \text{ the number of 1's in } w \text{ is even, and the number of 0's is even}\}.$$	hw3_ex2
hw3	3	Construct NFAs corresponding to the following regular expressions: * $0^+ \cup (01)^$; $(0 \cup 1^+)0^*1^+$.	hw3_ex3
hw5	1	The input is two binary numbers, $x$ and $y$ (with the most significant bit on the left), separated by a symbol #, for example, 100#100⊔. (Assume there are no leading zeros unless the number is zero.) Accept if $x$ and $y$ are equal. Draw the state diagram of the corresponding Turing machine and submit its source code on eLearning.	hw5_ex1
hw5	2	Let language $$L = \{0^n : n \text{ is a Fibonacci number}\}.$$ Design a Turing machine that decides L. Draw the state diagram of the corresponding Turing machine and submit its source code on eLearning.	hw5_ex2
hw5	3	In the definition of the Turing machine, the head can move left or right. A Turing machine whose head can move left, right, or stay in place is called a Turing machine with a stationary move, or more formally: $$\delta : Q \times \Gamma \to Q \times \Gamma \times \{L, R, S\}.$$ Prove that if a language $L$ can be decided by a Turing machine with stationary moves, then $L$ can also be decided by a standard Turing machine.	hw5_ex3
hw6	1	Define a two-dimensional Turing machine to be a TM where each of its tapes is an infinite grid (and the machine can move not only Left and Right but also Up and Down). Show that if language L can be decided in time T(n) by a two-dimensional TM then L ∈ DTIME(T(n)<sup>2</sup>).	hw6_ex1
hw6	2	Define a one-tape two-head Turing machine as a Turing machine equipped with a single tape but two independent heads operating on it. Each head can read, write, and move simultaneously and independently of the other. Both heads may read the same tape cell at the same time. Conflicts are resolved as follows: * If both heads attempt to write to the same cell simultaneously, the first head goes first, and the second head overwrites any previous content. * If the first head tries to write while the second head tries to read the same cell, the first head goes first, and the second head reads the updated content. * If the first head tries to read while the second head tries to write the same cell, the first head goes first, reading the old content, and then the second head updates the content. Show that if language L can be decided in time T(n) by a one-tape two-head Turing machine, then L ∈ DTIME(T(n)<sup>2</sup>).	hw6_ex2
hw6	3	Prove the number of multitape Turing machines is countable.	hw6_ex3
hw9	1	Prove that the Post Correspondence Problem is decidable over the unary alphabet Σ = {1}.	hw9_ex1
hw9	2	Prove that the Post Correspondence Problem is undecidable over the binary alphabet Σ = {0, 1}, assuming that the PCP is undecidable over an arbitrary finite alphabet.	hw9_ex2
hw9	3	Express the Twin Prime Conjecture in first-order Peano Arithmetic using only the symbols 0, S, +, ×, = and quantifiers ∀, ∃. (a) Define a formula Prime(x) expressing that x is a prime number. (b) Define a formula TwinPrime(p) expressing that p is the smaller number of a twin prime pair. (c) Write a single first-order PA formula that states: for every number x, there exists a larger twin prime p > x.	hw9_ex3

hw1

1

Prove $$\frac{1}{2\ln 2} + \frac{1}{3\ln 3} + \ldots + \frac{1}{n\ln n} = \ln\ln n + O(1)$$ and $$\frac{1}{1^{1.1}} + \frac{1}{2^{1.1}} + \ldots \frac{1}{n^{1.1}} = O(1).$$

hw1_ex1

hw1

2

Order the following functions by asymptotic growth. Place $f$ before $g$ if $f(n) = O(g(n))$. * $n^n$ * $\sqrt{n}$ * $\log\log n$ * $0.001n^7$ * $5000n$ * $e^n$ * $2^{\log^2 n}$ * $10^{10^{10}}$ * $n!$ * $2^{2^n}$

hw1_ex2

hw1

3

Let $\Sigma = \{0, 1\}$. Let language $$L = \{w \in \{0, 1\}^* : w \text{ has an unequal number of 0's and 1's}\}.$$ Prove $L^* = \Sigma^*$.

hw1_ex3

hw10

1

Use first-order Peano arithmetic formulas to express "$m = 2^n$".

hw10_ex1

hw10

2

Prove P is closed under Kleene star. In other words, if a language $L$ belongs to P, then so does $L^*$.

hw10_ex2

hw10

3

Let $$L_1 = \{(\alpha, x) : \text{TM } M_\alpha \text{ accepts input } x\},$$ and $$L_2 = \{\alpha : \text{TM } M_\alpha \text{ halts on the empty input}\}.$$ Prove that $L_1 \leq_T L_2$.

hw10_ex3

hw11

1

Define the problem QUADEQ as follows: given m quadratic equations in n variables over the finite field GF(2), $$p_1(x_1, x_2, \ldots, x_n) = 0,$$ $$p_2(x_1, x_2, \ldots, x_n) = 0,$$ $$\vdots$$ $$p_m(x_1, x_2, \ldots, x_n) = 0,$$ determine whether there exists at least one solution. Prove that QUADEQ ∈ NP.

hw11_ex1

hw11

2

Prove if L is NP-hard and L ∈ P, then P = NP.

hw11_ex2

hw11

3

Prove that $L_1 \leq_P L_2$ and $L_2 \in$ NP, then $L_1 \in$ NP.

hw11_ex3

hw13

1

Construct a function $T(n)$ that satisfies $n \leq T(n) \leq n^2$ and is not time-constructible.

hw13_ex1

hw13

2

Define the *low-degree clique* (LDC) problem as follows: Given a graph $G = (V, E)$ and an integer $k$, determine whether $G$ contains a clique of size at least $k$ consisting solely of vertices whose degree does not exceed the median vertex degree of $G$. Prove that LDC is NP-complete.

hw13_ex2

hw13

3

Let $$\text{DOUBLE-SAT} = \{\langle \varphi \rangle \mid \text{formula } \varphi \text{ has at least two satisfying assignments}\}.$$ Prove that DOUBLE-SAT is NP-complete.

hw13_ex3

hw2

1

Design a finite automaton that accepts the following language $$L = \left\{ w \in \{0, 1\}^* \mid w = w_1w_2 \cdots w_n \text{ such that } \sum_{i=1}^{n} w_i 2^{n-i} \equiv 0 \pmod{5} \right\}.$$ In other words, the language $L$ is the set of all binary strings that represent multiples of 5, written with the most significant bit first (MSB). Draw the state diagram of the finite automaton.

hw2_ex1

hw2

2

Design a finite automaton that recognizes the following language $$L = \{w \in \{0, 1\}^* : w \text{ does not contain substring } 10\}.$$ Draw its state diagram.

hw2_ex2

hw2

3

Use the definition of big-O notation to prove the following: 1) If $f_1 = O(g_1)$ and $f_2 = O(g_2)$, then $f_1f_2 = O(g_1g_2)$. 2) If $f_1 = O(g)$ and $f_2 = O(g)$, then $f_1 + f_2 = O(g)$.

hw2_ex3

hw3

1

Define a *backward-read DFA* as a deterministic finite automaton that processes its input string from right to left. Prove that a language is regular if and only if it can be accepted by some backward-read DFA.

hw3_ex1

hw3

2

Give a regular expression for language $$L = \{w \in \{0, 1\}^* : \text{ the number of 1's in } w \text{ is even, and the number of 0's is even}\}.$$

hw3_ex2

hw3

3

Construct NFAs corresponding to the following regular expressions: * $0^+ \cup (01)^*$; * $(0 \cup 1^+)0^*1^+$.

hw3_ex3

hw5

1

The input is two binary numbers, $x$ and $y$ (with the most significant bit on the left), separated by a symbol #, for example, 100#100⊔. (Assume there are no leading zeros unless the number is zero.) Accept if $x$ and $y$ are equal. Draw the state diagram of the corresponding Turing machine and submit its source code on eLearning.

hw5_ex1

hw5

2

Let language $$L = \{0^n : n \text{ is a Fibonacci number}\}.$$ Design a Turing machine that decides L. Draw the state diagram of the corresponding Turing machine and submit its source code on eLearning.

hw5_ex2

hw5

3

In the definition of the Turing machine, the head can move left or right. A Turing machine whose head can move left, right, or stay in place is called a *Turing machine with a stationary move*, or more formally: $$\delta : Q \times \Gamma \to Q \times \Gamma \times \{L, R, S\}.$$ Prove that if a language $L$ can be decided by a Turing machine with stationary moves, then $L$ can also be decided by a standard Turing machine.

hw5_ex3

hw6

1

Define a *two-dimensional* Turing machine to be a TM where each of its tapes is an infinite grid (and the machine can move not only Left and Right but also Up and Down). Show that if language L can be decided in time T(n) by a two-dimensional TM then L ∈ DTIME(T(n)<sup>2</sup>).

hw6_ex1

hw6

2

Define a *one-tape two-head Turing machine* as a Turing machine equipped with a single tape but two independent heads operating on it. Each head can read, write, and move simultaneously and independently of the other. Both heads may read the same tape cell at the same time. Conflicts are resolved as follows: * If both heads attempt to write to the same cell simultaneously, the first head goes first, and the second head overwrites any previous content. * If the first head tries to write while the second head tries to read the same cell, the first head goes first, and the second head reads the updated content. * If the first head tries to read while the second head tries to write the same cell, the first head goes first, reading the old content, and then the second head updates the content. Show that if language L can be decided in time T(n) by a one-tape two-head Turing machine, then L ∈ DTIME(T(n)<sup>2</sup>).

hw6_ex2

hw6

3

Prove the number of multitape Turing machines is countable.

hw6_ex3

hw9

1

Prove that the Post Correspondence Problem is decidable over the unary alphabet Σ = {1}.

hw9_ex1

hw9

2

Prove that the Post Correspondence Problem is undecidable over the binary alphabet Σ = {0, 1}, assuming that the PCP is undecidable over an arbitrary finite alphabet.

hw9_ex2

hw9

3

Express the *Twin Prime Conjecture* in first-order Peano Arithmetic using only the symbols 0, S, +, ×, = and quantifiers ∀, ∃. (a) Define a formula Prime(x) expressing that x is a prime number. (b) Define a formula TwinPrime(p) expressing that p is the smaller number of a twin prime pair. (c) Write a single first-order PA formula that states: *for every number x, there exists a larger twin prime p > x*.

hw9_ex3