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Questions tagged [proof-techniques]

Questions about general methods and techniques for proving multiple theorems. When asking about the proof of a single statement, use tags relating to what the proof is about instead.

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28 views

How do I prove or disprove the following statements?

I have to prove or disprove the following statements. a) ≈(L(X)) ⊆ ≈(L(Y)) ⇒ ~(X) ⊆ ~(Y) b) L(X) ⊆ L(Y) ⇒ ~(X) ⊆ ~(Y) ...
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23 views

Why can't non-deterministic pushdown automata always be expressed as deterministic PDAs?

Why can't non-deterministic pushdown automata always be expressed as deterministic PDAs?
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45 views

On the complexity of existential and universal quantifiers

I'm trying to analyze the time complexities of the two former kind of quantifiers, I need help figuring out if I'm following the right path or if I'm making mistakes, here's what I've produced so far: ...
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37 views

How to prove a recursive's function Big-Theta without using repeated substitution, master theorem, or having the closed form?

I have a function defined: $V(j, k)$ where $j, k \in \mathbb{N}$ and $t > 0 \in \mathbb{N}$ and $1 \leq q \leq j - 1$. Note $\mathbb{N}$ includes $0$. $V(j, k) = \begin{cases} tj & k \leq 2 \\...
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32 views

Weaker conjectures to prove in order to arrive at P =/= NP

We know we have a long way to go before we come to a proof of P $\neq$ NP. We also know that this road is studded with minor conjectures that will have to be proved/disproved in order to arrive at the ...
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2answers
98 views

Prove that the following language is not regular: $\{0^i1^j : i \neq j\}$ [duplicate]

I was trying to approach this proof, after multiple reads and attempts I am getting nowhere. If someone could help me out that would be great. Should I use the pumping lemma, if so how show I start, ...
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3answers
255 views

Is {a^n: n is a product of exactly two primes} regular?

I am struggling to prove the following question. $L_1 = \{a^n: n \text{ is a product of exactly two primes}\}$ I feel like the language is not regular but I am having trouble proving it. I tried ...
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1answer
58 views

Codeword constructed by Huffman's algorithm has average length of at most log n

I am interested in the following question: Prove that the average length of a codeword constructed by Huffman's algorithm has average length at most $\log n$, where $n$ is the number of codewords. ...
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1answer
71 views

Proving with co-induction principles

I'm going through Adam Chlipala's "Certified Programming with Dependent Types" (available here for convenience), and I'm a bit stuck at internalizing the introduction of co-induction principle for the ...
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1answer
69 views

How to justify $f(n) = O(g(n))$ [duplicate]

The following question is in my homework: Is the statement $f(n) = O(g(n))$ true, when $f(n) = n/2 + 4$ and $g(n) = \sqrt{n} + 2\log_2 n + 3$? I understand how $f(n)$ is the upper bound of $g(n)$. ...
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115 views

Proof by reduction and Turing machines [closed]

This is a practice question I have, but I can't wrap my head around it. ............. Let L = {M | M is a TM that halts with exactly two words on its tape in the form Bw1Bw2B}. B = Blank Position the ...
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1answer
92 views

How to prove by contradiction that every nonempty hereditary language contains the empty string?

A language L is called hereditary if it has the following property: For every nonempty string x in L, there is a character in x which can be deleted from x to give another string in L. Prove by ...
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2answers
22 views

Isn't ZKP is a reduction to a hard problem, rather than true zero knowledge?

Take for example "Hamiltonian cycle for a large graph". The proof works by starting with a graph G that contains a hamiltonian cycle, then constructing an isomorphic graph H, and then either showing ...
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2answers
47 views

Randomly built binary search trees

In Introduction to Algorithms (CLRS) 3rd Edition, page 299, the section attempts to prove: The expected height of a randomly built binary search tree on $n$ distinct keys is $O(\lg n)$. We define "...
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1answer
101 views

The law of excluded middle and decidability

I will divide my question in two parts. The first part I am sure that there is a objective answer, but I am not sure about the second part. First part: Is it all (decisions) problems trivial to prove ...
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1answer
40 views

It is decidable whether a pushdown automaton will accept a word? [duplicate]

I'm asking myself if the problem of decide whether a push down automaton will accept a word is decidable. I would say that you can simulate a push down automaton with a Turing Machine and, if it ...
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3answers
3k views

Show that there are infinitely more problems than we will ever be able to compute

I was looking at this reading of MIT on computational complexity and on minute 15:00 Erik Demaine embarks on a demonstration to show what is stated in the title of this question. However I cannot ...
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2answers
89 views

Prove that $T(n) \leq 8n^2$ or find value of $n$ when statement is not true (recurrence relation)

We have a function $T: \mathbb{N}\to\mathbb{N}$ defined recurrently: $$T(n)=\begin{cases} 0 &\text{ if } n=0,\\ 3T(\lfloor{n/2}\rfloor) + 2n^2 &\text{otherwise.} \end{cases}$$ Prove that for ...
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1answer
101 views

Could we define the decimal notation of a natural number as a series of operations applied to 0? [closed]

People have made mistakes. For that reason, there probably is a demand by some people for a computer program that's easier to verify makes the calculations properly. I think that according to the ...
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1answer
61 views

How to prove $L \notin \texttt{DSPACE}(f)$

I want to prove that a language $L$ is not in $\texttt{DSPACE}(f(n))$, the class of languages that a deterministic Turing machine can decide with fixed tape length of $f(n)$ (wiki). That is, I want to ...
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1answer
53 views

How to reduce $\{w \mid |T(M_w)| \geq 42\}$ to the halting problem?

For a string $w$, $M_w$ denotes the Turing machine whose encoding is $w$. I want to reduce the language $L=\{w \mid |T(M_w)| \geq 42\}$ to $H_0 = \{w \mid M_w \text{ halts on } \epsilon\}$, but I ...
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37 views

Proof that this sorting algorithm sorts the input

I'm given this "sorting" algorithm and now I'm supposed to prove, that if given an array of integers of length $n$, sort(A,0,n-1) will sort it. ...
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73 views

Red-black tree trinode restructuring after insertion and deletion

When performing an insertion/deletion on a red-black tree, how can be argued or proved that it requires at most one/two trinode restructuring(s) respectively? My thoughts so far were: after inserting ...
2
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1answer
60 views

How do we prove the time complexity of this simple problem in probabilistic inference on a Bayesian network?

Suppose we have a simple Bayesian network with two rows of nodes: $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_n$. Each node $x_k$ takes a state of either 0 or 1 with equal probability. Each ...
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1answer
52 views

How to correctly describe this action, deleting an edge that “shortcut” some vertices

Haven't written a proof in years, not sure how to describe an algorithm like this ? Let us what we have a graph. like this below: 1). How to describe edge removal of{ (0, 1),(3,4), (1,2) }done in ...
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59 views

proving DFA stuck

This DFA fulfills: Define a function $diff: \{0,1\}^*\to\Bbb Z$, for $w \in\{0,1\}^*$, $diff(w)=($# of 1's in $w)- ($# of 0's in $w$). Thus, $diff(\epsilon)=0$; $diff(0)=−1$; $diff(1)=1$. Let $L = ...
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34 views

dfa subtract multiple of 3 [duplicate]

Define a function 𝑑𝑖𝑓𝑓 ∈{0,1}→ℤ so: for everything w ∈{0,1}, diff w = # of 1's in w- # of 0's in w. Thus: 𝑑𝑖𝑓𝑓 𝜀=0; 𝑑𝑖𝑓𝑓 0=−1; 𝑑𝑖𝑓𝑓 0=−1; Let 𝐿 = {𝑤∈ {0,1} * | 𝑑𝑖𝑓𝑓 𝑤 = 3𝑚 ...
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2answers
224 views

prove by induction that the complete recursion tree for computing the nth Fibonacci number has n leaves

I have referenced this similar question: Prove correctness of recursive Fibonacci algorithm, using proof by induction *Edit: my professor had a significant typo in this assignment, I have attempted ...
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0answers
43 views

Can we ignore the postcondition in the Hoare conditional rule when there is a return statement?

I'm proving the correctness of naive string matching using Hoare logic. I have the following pseudocode: ...
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4answers
400 views

Deriving lower and upper bounds for T(n) = T(n-1) + T(n-2) + 10

The solution is to find the upper and lower bounds from: 2T(n-2) < T(n) < 2T(n-1) + 10 So I have to find ...
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1answer
39 views

Must a function in lambda-calculus which inputs a boolean function be defined in a certian way?

This question is my best attempt to get at a more general question about what one can get from terms in the lambda calculus. Using the church encoding, we define booleans by $\texttt{true} = \lambda ...
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1answer
33 views

Finding maximum takes at least $\lceil n/2 \rceil$ comparisons

We are given an array $A$ with $n$ elements, $n \in \mathbb{N}$ and all elements are in the set $\{1,2,3, \cdots, n \}$. I want to prove that finding the maximum in $A$ (that is, outputting the index ...
2
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1answer
130 views

Union of infinitely many regular languages [duplicate]

I need to prove or disprove the following statement. If $A_n ⊆ \Sigma^*$ is regular for each $n \in \mathbb{N}$ then $\bigcup\limits_{n=0}^{\infty} A_n$ is regular. I know that if two languages ...
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1answer
96 views

Proving the singleton language {x} is regular for all x ∈ Σ*

So I'm aware that the singleton language is in fact regular for all x ∈ Σ*, but I do not understand why it is. A formal proof would help, but also getting some intuition as to why it is regular would ...
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2answers
110 views

Proving that Every Full Prefix-Free Language is Maximal

I'm practicing a problem where I need to prove that every full prefix-free language is maximal. I know a prefix-free language A is maximal if it is not a proper subset of any prefix-free language, ...
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1answer
88 views

Proving that A(B ∩ C) ⊆ AB ∩ AC

A(B ∩ C) = { UV | U ∈ A, V ∈ B and V ∈ C } for the left part. ΑΒ = { UV | U ∈ A, V ∈ B }, ΑC = { UV | U ∈ A, V ∈ c }, AB ∩ AC = { UV | U ∈ AB and AC, V ∈ AB and AC } for the right part. How can I ...
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1answer
81 views

Proof of an Infinite Binary Sequence

I have a problem where given an infinite binary sequence S ∈ {0, 1}∞ to be "prefix-repetitive" if there are infinitely many strings w ∈ {0, 1}* such that ww is a prefix of S. I need to prove that if ...
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4answers
84 views

Proving B* = B on a given set

I have the set: B = {x ∈ {0,1}* | there is an equal number of 0's and 1's in x} and therefore, B* = {e,01,10,0011,0101,0110,1100,1010,1001,....etc} I need to either prove or disprove that B*=B I ...
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0answers
52 views

Sum of long geometric progression [closed]

Finding sum of a geometric progression is simple when we just need to report the sum, but when some modulo or multiplicative inverse is asked of that sum the task become tedious for me. I have a ...
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0answers
49 views

Can Pareto Optimality be compared to Nash Equilibrium?

Given a state $s$, and a value function $v^i$ that determines the expected payoff for the i-th agent in that state, can the two definitions below, one of Nash equilibrium and another of Pareto ...
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0answers
216 views

Proof that G is a Tree After DFS and BFS form the same tree T [closed]

Let G be a connected, undirected graph containing some vertex s. let's say that BFS and DFS are both run on G starting at s and that the breadth first search and depth first search ...
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1answer
111 views

Prove Halting on all Inputs is not in RE simulation

I don't understand why when proving if Halting on all inputs problem si not in RE using the complement of the halting problem, I have to take a turing machine and simulate the machine M(the machine ...
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0answers
188 views

Could you show the intractibility of SAT by showing that the number of variables contributing to an arbitrary unsatisfied clause is not constant? [closed]

Preface: This is not an attempted proof at P vs NP Starting with some CNF Boolean expression ϕ, by the rules of logical disjunction, a clause is only unsatisfied if each of the literals in it are ...
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1answer
45 views

Why proving the solution of a problem is polynomial time is sufficient enough to say that it is a NP prolbem? [duplicate]

Why proving that we can verify the solution of a problem is polynomial time is sufficient enough to say that the problem is nondeterministic polynomial time? Please note: this is not a question on how ...
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0answers
82 views

Correctness proof: induction on sequence of steps, need a stronger claim?

Im trying to prove the correctness of the construction proposed in this site answer: a two stack PDA that simulates a Turing Machine. By "correctness" i mean to prove more or less formally that we can ...
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2answers
43 views

Decidability of $SEQ_{CFG} = \{⟨G,H⟩ \mid \text{$G,H$ are CFGs and $L(G) ⊆ L(H)$}\}$

How can I prove that $SEQ_{CFG} = \{⟨G,H⟩ \mid \text{$G,H$ are CFGs and $L(G) ⊆ L(H)$}\}$ is decidable ? I know that $EQ_{CFG} = \{⟨G, H⟩ \mid \text{$G,H$ are CFGs and $L(G) = L(H)$}\}$ is not.
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4answers
80 views

Prooving by Pigeonhole principle

I've been given a question to solve: ...
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2answers
105 views

How to generate an instance for an NP_hard proof, where each element has two inputs?

I want to prove the NP-hardness of an scheduling problem. The problem seems to be NP-hard in the ordinary sense, so I am trying with the Partition Problem, precisely the Equal Cardinality Partition (...
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20 views

How to prove that models of indirect and direct RAM machines are equivalent?

as in the title, I am looking for a formal proof how to show that models of indirect and direct RAM (random-access) machines are equivalent. I would really appreciate your help.
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1answer
37 views

Is this a valid induction proof example ?

Learning induction proof now, found a "simple" example, which is a bit confusing to me (not sure if it is a valid example). If so, why the IH( suppose a root of rank k has at least $2^k$ vertices in ...