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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Proving Minimum Sequence Disjoint Set

Prove that 9 is the minimum number of calls to make-set, union-set, find-set such that a disjoint set union using weight (number of nodes) by path compression and disjoint set union using rank (upper ...
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33 views

Difficulty in understanding the proof of “Every context-sensitive language L is recursive” as given in the Peter Linz text

I was going through the automata text by Peter Linz. There I came across the proof the theorem below. I could not quite get the portion of the proof in bolds. Every context-sensitive language L is ...
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37 views

Arguing that $L= \{a^n | n\geq 0\} \cup \{a^nb^n| n\geq 0\}$ is not $LL(k)$ for any $k$

$\text{Consider the language $L= \{a^n | n\geq 0\} \cup \{a^nb^n| n\geq 0\}$ }$ $\text{and the following statements.}$ $\quad\quad\text{I. $L$ is deterministic context-free.}$ $\quad\quad\text{II. $L$...
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How to use butterfly network for data copy?

I know butterfly networks (and benes as well) allow routing a packet from any input to any output node. Congestion is $\sqrt{n}$ but with bene it can be $1$. Now assume that in a butterfly network, ...
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32 views

Given CFG generates all words with equally many 0s and 1s

Here is an exercise from an introduction to computation class: Show that the following context-free grammar $G$ generates the language $L$ of words over $\{0,1\}$ with an equal number of $0$s and $1$...
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1answer
9 views

Inductive sequence of words in a biprefix code

Let $X = X_1 \cup X_2$ a code on an alphabet $A$, with $X_1$ a biprefix code and $X_2$ a uniform code, with $m(X_1) < m(X_2)$, i.e. the maximal length of the first is strictly lower than the second....
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45 views

Pumping lemma for regular languages. Proof

Please help me understand the following $L = \{ a | a ∈ \{0, 1\}^∗, |a| = k ≥ 4, a = a_1a_2...a_{k−1}a_k, ∃i ∈ N, 1 ≤ i < k : a_i = a_{i+1} \}$ To prove: The language $L$ has regular pumping ...
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44 views

Towers of Hanoi with sufficiently many stacks, show that $T_k(n)=\Theta(n)$ for all $k\geq 2 + \frac{n-1}{2}$

I'm trying to show that for the following Towers of Hanoi general algorithm that $T_k(n)=\Theta(n)$ for all $k\geq 2 + \frac{n-1}{2}$, I'm not sure how to incorporate the restriction on $k$ into my ...
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65 views

Understanding the language

Could you please help me understand the following Language $L = \{ a | a ∈ \{0, 1\}^∗, |a| = k ≥ 4, a = a_1a_2...a_{k−1}a_k, ∃i ∈ N, 1 ≤ i < k : a_i = a_{i+1} \}$ what does $a_i = a_{i+1}$ mean? ...
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56 views

Pumping Lemma Proof (Type of wcw language)

I have the language $L = \{ dkd\space \mid d \in \{a,b\}^*, k \in \{a,b\} \}$ and i have to show that it's non-regular using the pumping lemma. The structure of the language i think can be explained ...
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18 views

Proving using a SAT solver that KB entails D

Suppose you're given this KB: $$KB = (A, A ⇒ B, A ⇒ C, B ∧ C ⇒ D)$$ How would you show using a SAT solver that $KB \vDash D$?
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Proof that CSL ⊊ REC

I'm trying to prove that a context sensitive language ⊊ Turing-acceptable language. I was thinking of working out the complement of the language $A$, where $A$ consists of all words $w$ such that $M_w$...
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Gale-Shapley Stable matching where one man's preference list changes

Given $n>1$ women and men. Let $M$ be the stable matching given by the Gale-Shapley algorithm with men proposing. Is there a stable matching instance such that: changing one man's preference list ...
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Prove the correctness of divide algorithm

This is the pseudocode for the algorithm: function divide($y,z$) comment Return $q,r \in \mathbf{N}$ such that $y=qz+r$ and $r<z$ where $y,z \in \mathbf{N} $ $r:=y$, $q:=0$, $w:=z$; while $w \...
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Constant terms at each level of a recursion tree

In CLRS, exercise 4.4-5 the following question is asked: Use a recursion tree to determine a good asymptotic upper bound on the recurrence $$T(n) = T(n-1) + T(n/2) + n$$ In my recursion tree, the ...
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37 views

Proving big-theta complexity with constants in $f(n)$

I am working through a problem in which I have to prove that a particular $f(n) = \Theta(g(n))$. I know that for this to be true there need to exist positive constants $c_1$, $c_2$, and $n_0$ such ...
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Big theta notation in substitution proofs for recurrences

Often in CLRS, when proving recurrences via substitution, $\Theta(f(n))$ is replaced with $cf(n)$. For example, on page 91, the recurrence $$ T(n) = 3T(⌊n/4⌋) + \Theta(n^2) $$ is written like so in ...
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Equivalence of two approximation algorithms for min Steiner tree

I learned two approximation algorithms for the min Steiner tree: The first algorithm: 1- Compute the metric closure G' of G. 2- Compute a min spanning tree T' of G' 3- Construct the union U of the ...
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180 views

Algorithm for solving a mixed integer programming problem in polynomial time?

I have the following mixed integer programming (MIP) problem: $$ \begin{array}{rll} \text{Maximize } & z=k \\ \text{subject to } & a_ik - m_i \geq 0 & (i=1,\dots,n) \\ & b_ik - m_i \...
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Decidability of whether $w \in L(M_1) \setminus L(M_2)$

I'm studying for my finals and I came across this question from past exams: Is the following language decidable? $$ L = \{ \langle M_1,M_2,w \rangle \mid w \in L(M_1) \setminus L(M_2) \}. $$ How can ...
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123 views

How can I simulate nested WHILE loops in a theoretical programming language to show Turing completeness?

PRE-WORK-POST is a theoretical programming language with the following structure, where P,Q and R are LOOP program: $$\text{PRE} \ P \ \text{WORK} \ Q \ \text{POST} \ R \ \text{END}$$ First $P$ is ...
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53 views

Proving equivalence of two substitutions by induction

I'm trying to prove the following reduction: $$ t\{x:=u\}\{y:=v\} = t\{y:=v\}\{x:=u\{y:=v\}\} $$ under the following assumptions: $x \neq y$ $x$ is not a free variable of $v$ (in symbols, $x \...
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How to specify mutated types mathematically?

Say I have an object which I pass to a method, and the method returns that same object, just mutated. So it goes like this: ...
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1answer
48 views

How to show a language is Partially Decidable?

I am trying to solve some questions on partial decidability of languages and I am getting confused in how to construct proper arguments through the idea of Universal Turing Machine. I am not posting ...
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42 views

How to create mathematical proof of TSP and SLAP equivalence?

In my thesis, I'm dealing with SLAP (storage location assignment problem) -- which is finding optimal distribution of products to location slots in a generic warehouse. My aim was to implement ...
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163 views

reduction from ALLTM to ETM

I am trying to understand why this "proof" of ETM undecidability is wrong. ALLTM={ < M >|M is a TM, L(M)=∑*} ETM={< M >|M is a TM, L(M)=∅} We know that ALLTM is undecidable, lets ...
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1answer
28 views

Time complexity of binary search

Proposition: The binary search algorithm runs in $O(\log n)$ time for a sorted sequence with $n$ elements. When justifying this claim, first we say that with each recursive call the number of ...
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1answer
88 views

Early Deadline First (EDF) scheduling in real-time systems feasibility test proof

I am trying to prove Theorem 6.2 on page 127 of the book Real-Time Systems by Jane W. S. Liu: http://www.cse.hcmut.edu.vn/~thai/books/2000%20_%20Liu-%20Real%20Time%20Systems.pdf It is based on Early ...
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How to prove a statement in regular expression?

I cannot figure out how to go about proving this statement in regular expression. $$ L(R_1) \subseteq L(R_2) \subseteq L(R_3) \implies L(R_1^*+R_3)^* \subseteq L(R_2^*+R_3^*) $$ Here's what I have ...
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Having trouble understanding how to prove a language context free? [duplicate]

I've been studying the theory of automata. I came across this problem in the book and unable to understand how to solve this. I've solved some examples using the Pumping lemma but this one uses ...
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49 views

When are existential quantifiers in the intuitionistic propositional calculus eliminable and when not

I am so ignorant I don't even know where should I ask this - on FOM? On mathoverflow? On cstheory? So please consider as sort of a meta-question readdressing me in case you think this is a wrong site. ...
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88 views

How would I prove that nondeterministic Turing machines are undecidable?

How would I go about proving that the language: $$A_{NTM }= \{\langle N, w\rangle | N \text{ is a nondeterministic TM and } N \text{ accepts }w\}$$ is undecidable? I looked at the proof for $A_{TM}$ ...
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Prove (p → ¬q) is equivalent to ¬(p ∧ q)

I need to prove the above sequent using natural deduction. I did the first half already i.e. I proved $(p\rightarrow\neg q)\rightarrow \neg (p \wedge q)$, but I'm stuck on where to start for the ...
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1answer
46 views

Why discriminate the base case allows me to complete the induction proof?

I have a successful completed proof which used induction. but I essentially proved the goal on the base case by tactic discriminate. Why is this induction proof ...
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1answer
33 views

Converse proof for random coding capacity of AVC

I want to see the converse proof for the random coding (shared randomness) capacity of AVC. All I can find online is Csiszar Narayan's AVC paper which looks at deterministic coding. Further, the proof ...
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49 views

Are different assignments allowed for the implication graph proof of 2-SAT being in P?

One proof for $2-SAT$ being in $P$ uses the implication graph, i.e. one creates 2 vertices per variable $a$, one for each possible literal ($a$ and $\neg a$). One then adds 2 arcs per clause $(a \lor ...
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Can every sentence of first-order logic be converted into an equisatisfiable equation in Boolean algebra?

There may be some theoretical literature, unknown to me, that addresses this question. If possible, I would like a practical approach to this problem. My attempt involves the use of an equational ...
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37 views

why is behaviour of simpl differ so much after a commutative operation and how to inspect simpl?

In Coq, while trying to prove a lemma mult_n_Sm for mult_comm, I have this equation in a proof: ...
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1answer
28 views

Why isn't plus_assoc rewriting correctly?

First I have plus_assoc ready. Theorem plus_assoc : forall n m p : nat, n + (m + p) = (n + m) + p. for simplicity we omit the ...
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1answer
25 views

How do you prove these string/number radix encoding/decoding algorithms work?

A while back I learned of these great algorithms: ...
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2answers
47 views

Closed form of recurrence with two inputs

This question comes from a relatively simple coding challenge at Codesignal, but represents an interesting CS/math puzzle. The question states: "When a candle finishes burning it leaves a ...
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1answer
631 views

Is every unambiguous grammar regular?

While searching for an answer to this question I found out that there is an unambiguous grammar for every regular language. But is there a regular language for every unambiguous grammar? How can I ...
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67 views

Hardness of maximizing difference of functions

Suppose that the problem of maximizing a real function $f$ over a certain domain $D$ is NP_HARD. What can be said about the problem of maximizing $f-g$, with $g$ being another function over $D$? Is it ...
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Constant in Substitution method for recurrence

The solution for solving the following recurrence with the substitution method involves adding the a constant inside the recurrence, which is confusing to me. This is question 4.3-2 in the CLRS ...
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42 views

Proof using tableau algorithm

I have spent an hour finding the answer to this problem but can't do it, this is the problem: Determining whether the following semantic entailment holds or not by using the tableau algorithm ...
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1answer
57 views

Why doesn't diagonalization require taking a limit?

When we quantify infinite sums, we do so by taking the limit as $i$ goes to infinity. For example, we look at $\lim_{n\rightarrow \infty}\sum_{n\in \mathbb{N}}n$, and then we say that this diverges ...
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1answer
94 views

On the proof techniques of Udi Manber

I was familiar with the approach of first coming up with an algorithm, and then proving the loop invariant to come up with an algorithm as elucidated in CLRS (Introduction to algorithms, Thomas H. ...
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50 views

what is the relevance of computability when applying diagonallization?

When thinking about diagonalization, I've always glossed over whether or not the enumeration, or the diagonal is computable or not. When does it matter? Say for example, that have an enumeration of ...
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Understanding the proof of “DFS of undirected graph $G$, yields either tree edge or back edge” better with graph for each statement in proof

I was going through the edge classification section by $\text{DFS}$ algorithm on an undirected graph from the text Introduction to Algorithms by Cormen et. al. where I came across the following proof. ...

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