Questions tagged [induction]

Questions about mathematical induction and inductive proofs.

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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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27 views

Recursive definition for the length of a string?

I found a couple of answers online but I don't quite understand why the answer is right: The length of a string is: If a string has no characters, then its length is 0. Otherwise, the length of the ...
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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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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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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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Inductive proof on Quicksort with Explicit Stacking

Prove by induction that if Quicksort with Explicit Stacking is modified so that the end-points of the larger sublist are stacked, and the other sublist is sorted first, then the maximum stack size is $...
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323 views

Is my proof of my greedy algorithm to find subsequence correct?

Credit to KleinBerg and Taros Book Some of your friends have gotten into the burgeoning field of time-series data mining, in which one looks for patterns in sequences of events that occur over time. ...
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193 views

Iterative Fibonacci algorithm correctness proof, finding loop invariants

The algorithm take in an integer $n$ and outputs the $n$th number in the Fibonacci sequence ($F_n$). The sequence starts with $F_0$. I am trying to prove the correctness assuming valid input: ...
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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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1answer
75 views

Induction on strings (words)

Given is an alphabet $\Sigma = \{ 0, 1, 2 \}$ and a function quer to calculate the cross sum of a word. $quer : \Sigma^*\to \Bbb N$ with: $$quer(w)=\begin{cases} 0, &\text{when } w=\epsilon\\ ...
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How does one know what statements in Coq require Induction?

I was trying to learn Coq using the famous book Software Foundations. In it I found the following: ...
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How do I show that an iterative solution to Tower of Hanoi performs the same exact steps as a recursive solution? [duplicate]

So given the typical recursive solution to the Tower of Hanoi problem wherein you reduce the n-disk tower to two instances of an (n-1)-disk tower i.e move (n-1) disks from start to auxiliary. move ...
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1answer
124 views

Induction to prove equivalence of a recursive and iterative algorithm for Towers of Hanoi

Using induction how do you prove that two algorithm implementations, one recursive and the other iterative, of the Towers of Hanoi perform identical move operations? The implementations are as follows....
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209 views

Proof of QuickSort algorithm correctness

Recently I’ve studied QuickSort and understood its general idea. Basically, we do the following: Pick an element from the array (no matter which one and how in this context) Rearrange elements in ...
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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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How to prove that the predecessor of each node in Dijkstra form a tree?

Prove that the array prev[.] computed by Dijkstra’s algorithm, the edges (v, prev[v]) for all v ∈ V , form a tree In order to prove this I used induction. Lemma :...
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Why does the denotational semantics for a while loop have a existence quantifier?

I was going through these notes and they have the following operator on partial functions: $$ \mathcal F^{k}(\bot)(\sigma) = \left\{ \begin{array}{ll} \alpha( [\![s]\!]\sigma ) & [\![b]\!]\...
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PL: How can I prove the type of something using “Inversion for Typing”?

I'm currently going through this book about programming languages, and in section 4.2, Lemma 4.2 it says this: Lemma 4.2 (Inversion for Typing). Suppose that $\Gamma \vdash e : \tau$. If $e = \...
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How to prove even with structural induction, for expression

E::= zero|two|expression + expression|expression*expression, E element of expression how do i prove E to be even. I have no clue on how to go about tackling this.
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How to prove that a string is made up of subsequences occurring some arbitrary number of times using concatenation?

How to prove that a string, s is made up of n > 1 subsequences occurring some arbitrary number of times using concatenation ...
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1answer
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Induction on typing derivation in refinement types system

From the text Principles of Type Refinement page 14: The author introduces in definition 2.2.7 the rule: $$ \dfrac{\Pi \vdash t : R \qquad R \le S}{\Pi \vdash t : S} $$ and gives the following ...
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158 views

Is Loop Invariant Proof a form of Induction?

As far as I see, what computer scientists refer to as loop invariant proofs are exact replicas of induction proof. Is it true? Can I state that loop invariant proof implies an induction? Is there a ...
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1answer
85 views

Proving time-complexity analysis for merge-sort-like algorihtm

I have this algorithm, which is exactly like merge-sort, but instead of halving the array each recursion, it actually splits it into $1/4$ and $3/4$ parts. Other then that, it does exactly the same ...
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89 views

Proving the correctness of a square summing algorithm

int sumHelper(int n, int a) { if (n==0) return a; else return sumHelper(n-1, a + n*n); } int sumSqr(int n) { return sumHelper(n, 0); } I am supposed ...
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129 views

When proof by induction on length string is not possible?

I found out an exercise where you have to prove the correctness of the following CFG: Let $L=\{ 0^i 1^j|2i \leq j \leq 3i \}\:$ and $\: G: S\rightarrow 0S11 | 0S111| \epsilon$ claim: Every string $w ...
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Proving that $(A \cup B)^* = A^*(BA^*)^*$

I would like to prove that $(A \cup B)^* = A^*(BA^*)^*$, where * means the Kleene star. I would like to use induction to prove this equality but I do not how to proceed and how is the best way to set ...
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1answer
50 views

Prove that the number of full nodes plus one is qual to the number of leaves in a nonempty binary tree

I'm trying to write an induction statement to prove a full node in a tree but I have no idea how to do that. I've always been terrible when it comes to logic. Where do I even start with this? I know ...
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Induction / (machine learning) of Resource Grammars for Grammatical Framework?

Grammatical Framework is based in Abstract Categorial Grammars. It is known that Combinatory Categorial Grammars have grammar induction/learning capabilities see e.g. https://link.springer.com/article/...
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Inductive proof that $n^2 + bn + d$ is $O(n^2)$ using definition of big O

Given that $T(n) = n^2 + bn + d$ then it's $O(n^2)$ if I can prove that: $O(n^2) = \{T(n): \text{there exist positive constants } c, n_0 \text{ such that } \forall n \geq n_0, 0 \leq T(n) \leq cn^2 \}$...
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3answers
306 views

Prove correctness for computing the nth Fibonacci number for the pseudo code

How do we prove the correctness of this pseudo code by induction? ...
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1answer
36 views

Complex property of sparse horner polynomials by induction

I'm following this article to do a formal proof on elliptic curve cryptography. My question here addresses only a property "easily proved by induction". Definitions (the important part is the ...
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1answer
94 views

Can you give an inductive definition to define the length of a list L?

Having trouble understanding what it means to define something inductively in the following context. Can you give an inductive to define the length of a list L? a. The total number of items in ...
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2answers
328 views

Induction rules for reflexive, transitive closure

I'm trying to solve an exercise on inductive definitions, the premiss is: Let $\to$ be a relation on $A$ and $\to^*$ its reflexive, transitive closure, which is defined by following two rules: $a \...
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347 views

How to use structural induction to prove law on lists

I want to prove that the following equation holds using structural induction on (finite) lists ...
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1answer
129 views

Where I can find example how prove red black tree?

I need prove that any red-black tree with at least two elements obtained through the insertion algorithm has at least one red node. For this, I need use Induction. I don't understand how apply ...
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104 views

Proving laws of take and drop functions using structural induction on lists

I'm trying to prove the following laws using structural induction on (finite) lists: ...
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1answer
269 views

Prove the equivalence between a CFG and a Context free language

I have to prove that the language $L=\{a^ib^j:2i=3j+1\}$ and the CFG G with the following rewrite rules: $S\rightarrow a^2Tb$ $T\rightarrow a^3Tb^2 |\epsilon$ are equivalent to each other. I'm ...
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108 views

Using Induction to prove that $n^3log^4n=O(n^4)$

I need to prove the following asymptotic relation for the purpose of cacluating a recurrence relation: $$n^3log^4n=O(n^4)$$ I tried and failed to do it with induction, which, if possible using basic ...
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311 views

Proving property of a term using Induction

This is one of the example lemma that has been proved in TAPL book which I'm unable to grasp. The objective is to prove |Consts(t)| <= size(t) where ...
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Can you apply the induction hypothesis to its outcome?

Assume a well-founded relation $<$ over a set $S$ and a property $P$ on $S$ such that: $P$ holds for all minimal elements of $S$. For every $b \in P$ and $a < b$ we have: if $P(a)$ then there ...
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1answer
177 views

Proof by mathematical induction

Before we start, my textbook declares lgn as base two. I only have one question, how did log2 become n (both highlighted in yellow in the picture)? Is it because log2 = 1, which is too small to even ...
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2answers
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Proof of correctness of algorithms (induction)

I am reading Algorithm's Design Manual by S.Skiena and I have a hard time understanding and proving the correctness of algorithms. I should use proof by induction and when we talk about summations and ...
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2answers
282 views

How to prove with induction [duplicate]

So far I have learned how to write proofs by induction and it went fine until I got this recursive problem, which I'm not quite sure how to begin and how to prove that with induction. ...
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2answers
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Prove correctess of recurrence result by mathematical induction

I have the following recurrence relation which I already solved using repeated substitution. $T(n) = \begin{Bmatrix} 1 & if & n = 1\\ 4T(\frac{n}2) + n & if & n >= 2 \end{Bmatrix} ...
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108 views

Is structural induction on terms applicable when a function is involved?

Assume an evaluation-relation on terms $t \Downarrow v$. If I want to prove correctness of a function $\phi$ w.r.t. evaluation, I have to show that the following implication always holds: $$\frac{\...
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What is induction-induction?

What is induction-induction? The resources I found are: the HoTT book, at the end of chapter 5.7. nLab's article a paper called Inductive-inductive definitions this blog post also mentions inductive-...
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Structural induction in non-local program transformation

Assume a functional language and a specialization operation (pulling out sub-expressions): let f x y = (h 23 x) + (g 42 y) becomes ...
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7k views

Trying to understand this Quicksort Correctness proof

This proof is a proof by induction, and goes as follows: P(n) is the assertion that "Quicksort correctly sorts every input array of length n." Base case: every input array of length 1 is already ...
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1answer
380 views

Show that $T(n) = 2T(\lfloor n/2\rfloor) + n$ is $\Omega(n\log n)$ using substitution

I have to solve this using the substitution method. Floor functions cannot be skipped. IH: Assume that $T(k) \geq ck\log(k) $ for all $k \leq n$, where c is a constant. IS: Must prove $T(k) \geq ...