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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21 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
28 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
84 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
30 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
86 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
99 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
57 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
50 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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33 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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0answers
56 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
50 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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55 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, $diff(\epsilon)=0$; $diff(0)=โˆ’1$; $diff(1)=1$. Let $L = ...
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34 views

dfa subtract multiple of 3 [duplicate]

Define a function ๐‘‘๐‘–๐‘“๐‘“ โˆˆ{0,1}โ†’โ„ค so: for everything w โˆˆ{0,1}, diff w = # of 1's in w- # of 0's in w. Thus: ๐‘‘๐‘–๐‘“๐‘“ ๐œ€=0; ๐‘‘๐‘–๐‘“๐‘“ 0=โˆ’1; ๐‘‘๐‘–๐‘“๐‘“ 0=โˆ’1; Let ๐ฟ = {๐‘คโˆˆ {0,1} * | ๐‘‘๐‘–๐‘“๐‘“ ๐‘ค = 3๐‘š ...
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2answers
155 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
42 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
170 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
38 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
31 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 ...
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1answer
80 views

Union of infinitely many regular languages [duplicate]

I need to prove or disprove the following statement. If $A_n โŠ† \Sigma^*$ 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
89 views

Proving the singleton language {x} is regular for all x โˆˆ ฮฃ*

So I'm aware that the singleton language is in fact regular for all x โˆˆ ฮฃ*, 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
95 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
61 views

Proving that A(B โˆฉ C) โŠ† AB โˆฉ AC

A(B โˆฉ C) = { UV | U โˆˆ A, V โˆˆ B and V โˆˆ C } for the left part. ฮ‘ฮ’ = { UV | U โˆˆ A, V โˆˆ B }, ฮ‘C = { UV | U โˆˆ A, V โˆˆ c }, AB โˆฉ AC = { UV | U โˆˆ AB and AC, V โˆˆ AB and AC } for the right part. How can I ...
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1answer
65 views

Proof of an Infinite Binary Sequence

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

Proving B* = B on a given set

I have the set: B = {x โˆˆ {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
51 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
48 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
111 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
77 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 ฯ•, 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
63 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
42 views

Decidability of $SEQ_{CFG} = \{โŸจG,HโŸฉ \mid \text{$G,H$ are CFGs and $L(G) โŠ† L(H)$}\}$

How can I prove that $SEQ_{CFG} = \{โŸจG,HโŸฉ \mid \text{$G,H$ are CFGs and $L(G) โŠ† L(H)$}\}$ is decidable ? I know that $EQ_{CFG} = \{โŸจG, HโŸฉ \mid \text{$G,H$ are CFGs and $L(G) = L(H)$}\}$ is not.
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4answers
77 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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0answers
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
35 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 ...
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1answer
172 views

Turing machine with semi infinite tape - Prove by construction

I'm studying constrained Turing Machines. There's a theorem that proves that both infinite and semi-infinite tape TM have the same computational power. The theorem that proves this by emulating a TM1 ...
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1answer
64 views

Trouble understanding this proof about the minimum number of states of a deterministic finite automaton

So I was browsing online looking for the general structure to proving a DFA has a minimum of $n$ states for some $n$ and most of them use contradiction. However, I'm having a hard time understanding ...
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3answers
56 views

Intuitive proof for a tree with n nodes, has n-1 edges

I am interested in an intuitive proof for "any binary tree with $n$ nodes has $n-1$ edges", that goes beyond proof by strong induction.
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1answer
52 views

Alternate proof of the Caro-Wei theorem for lower bounding the independence number

Let $G$ be a graph on $n$ vertices whose degree sequence is $d_1,d_2,...,d_n$. Let $\alpha(G)$ denote the size of maximum independent set of $G$, i.e., the size of a maximum subset of vertices of $G$ ...
2
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1answer
126 views

Proving problem NP-completeness [duplicate]

I am studying computational complexity and i am trying to solve this problem. We are given a (non-bipartite) complete graph: G = (V, W, E) where the vertices can be divided in two classes V and W ...
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2answers
163 views

Proving a value is the result of the execution of an algorithm

Assuming an algorithm $A$ known to both Alice and Bob. Alice runs the algorithm and gets a result $R$. How can Alice prove to Bob that $R$ is the result of the execution of $A$ and not some random ...
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1answer
77 views

Proof for Turing Machines being able to simulate any algorithm in the same time complexity

I have always read that Turing machines can simulate any algorithm, without changing the time complexity of the algorithm, and hence it is easier to study the Turing machine equivalent of the ...
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1answer
60 views

Proof By Contradiction - Hamiltonian Paths and Cycles

Was hoping if anyone had any way to prove the following claim using proof by contradiction Let $G = (V, E)$ be a simple graph with at least one vertex, and let $G'$ be the graph formed by adding a ...
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1answer
27 views

Showing $2^x$ is a lower bound

How do I show that $2^x - x^2 \in \Omega(2^x)$? Basically, I know that this means that $\exists a, x_0 \in \mathbb{R^+}, \forall x \in \mathbb{N}, a.2^x \leq 2^x - x^2$. I worked around a bit with ...
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0answers
47 views

How do we know that Icosoku always has solutions?

This is a continuation of a question I asked here. The puzzle Icosoku is now described by Wikipedia as: "The puzzle frame is a blue plastic icosahedron, and the pieces are 20 white equilateral-...
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3answers
292 views

Is there a *simple* proof that the intersection of a CFL and a regular language is a CFL?

I am following a course on complexity theory where languages are a part of the course. There is a proof that no matter how hard I try to understand, it is till so complex that I cannot make it to half ...
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1answer
22 views

There could not be an edge from u to v in a DAG, if w is before v in a topological order

I am trying to prove that given a DAG. There exists a valid topological ordering that has v in front of u iff there is no path from u to v. The proof is related to the fact that reverse DFS post ...