Questions about methods and techniques for proving theorems.

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32
votes
2answers
2k views

Is there a system behind the magic of algorithm analysis?

There are lots of algorithm-analysis questions around. Many are similar, for instance those asking for an analysis of nested loops or divide & conquer algorithms, but most answers seem to be ...
3
votes
2answers
46 views

Constructively deciding whether a decidable predicate holds universally

I am trying to obtain the proof of the proposition: $(\forall x \in \mathbb{N}, P(x)) \vee (\neg \forall x, P(x))$ given that the property $P$ is decidable for every $x \in \mathbb{N}$, i.e. ...
0
votes
1answer
12 views

How to prove the following properties of Small-step semantics?

I have to prove the following 2 properties of the Small-step semantics of the WHILE programming language: If $\langle C_1; C_2, s\rangle \rightarrow^k s'$ then there is a state $s''$ and natural ...
0
votes
3answers
140 views

How to find whether a grammar's language is finite or infinite?

I have this context-free grammar and I want to find out whether its language is finite or infinite. ...
2
votes
1answer
52 views

Proving that the continuation of a non-regular language is not ω-regular

I want to prove that a language is not $\omega$-regular. The language I'm working with can be defined as: $$L = \{ a_1 \dots a_n x^\omega ~ | ~ n > 0, a_1 \dots a_n \in L^\prime \}$$ where ...
1
vote
4answers
79 views

How can I prove $P \rightarrow (Q \rightarrow R)$ is equivalent to $(P \wedge Q) \rightarrow R$

I'm a freshman CS student at my university and i'm struggling with understanding my professor through his thick accent. I've asked him to explain the proof for this multiple times and still have ...
3
votes
1answer
36 views

Polynomial Reduction 3SAT to K-Clique

I am reading the reduction given by Sipser in his textbook "Introduction to the Theory of Computation," on page 303. The reduction is: \begin{equation} 3SAT \leq_p KCLIQUE \end{equation} I am really ...
7
votes
0answers
79 views

Proof of PCP theorem

I am reading the proof of PCP theorem in Proof Verication and Hardness of Approximation Problems. The following paragraph appears in section 3 (page 4), "Outline of the Proof of the Main Theorem". ...
3
votes
0answers
29 views

What are methods for showing that concurrent objects are not linearizable?

Linearizability is a well-known correctness condition for concurrent objects. It provides the illusion that each operation applied by concurrent processes takes effect instantaneously at some point ...
-1
votes
2answers
58 views

Induction proof, base case not working but induction step works? [closed]

$1+3+5+...+(2n+3)=n^2+4n$ For this series using induction proof. Base case 1,2,3,.. not working. But induction step works well. Base case is not given in question.
2
votes
0answers
36 views

Proof of contention of the wait-free consensus algorithm

I know that this is a known theorem but I can't find its proof. The theorem is: The write-contention of any $n$-process wait-free consensus algorithm (implemented from any read-modify-write ...
2
votes
2answers
106 views

How to apply the substitution method to n/2?

I recently was introduced to solving recurrence bounds by substitution but there's something i don't understand about it. In standard induction proofs you prove a base case, assume it holds for n ...
4
votes
0answers
28 views

What are the properties of the unsided fold?

Foldl and folr are 2 very important functions for FP and Haskell, but I have never heard much about the unsided fold: fold f [a,b,c,d] = (f (f a b) (f c d)) That ...
0
votes
0answers
41 views

Equivalency of two pushdown automata

How would one go about proving that one PDA that may only pop one symbol from its stack per transition, is equivalent to a second PDA that is allowed to pop any number of symbols? That is a PDA with ...
3
votes
1answer
109 views

Euclidean Algorithm in Coq

Question How do I write more intuitive proofs of the two following results in Coq? ...
5
votes
1answer
145 views

How to prove that a language is context-free?

There are many techniques to prove that a language is not context-free, but how do I prove that a language is context-free? What techniques are there to prove this? Obviously, one way is to exhibit ...
-1
votes
2answers
50 views

BST Successor Proof

I'm studying for my CS final and I can't seem to get the anywhere with one of the questions. This is the question: Prove that if a node in a BST has a successor, but has no right child, then its ...
2
votes
1answer
89 views

How to prove that the minimum square partition of a 3X2 rectangle has 3 squares

This question is motivated by an older question about tiling an orthogonal polygon with squares.         Given a $3\times 2$ rectangle like the first image, the ...
1
vote
1answer
102 views

Is my proof for a context free language correct? Same number of a's as b's

I have the following grammar G: $$ \begin{align*} &S \to aB|bA \\ &A \to a|aS|bAA \\ &B \to b|bS|aBB \end{align*} $$ I am going to prove that this language L(G) consists of words with the ...
0
votes
1answer
66 views

What is wrong in this proof? [closed]

I stumbled upon this wikipage that has got this proof : I rechecked sum rule of differentiation. And i can not understand where is this wrong. Any Tips ? I think that the second line x^2 = x + ...
1
vote
1answer
60 views

Prove language is not regular? [duplicate]

I know how to use the pumping lemma to do so, but I don't think that can be used for this language: $$L = \{x \in \{0,1\}^* : \text{no prefix of $x$ has more $1$'s than $0$'s}\}. $$ What other ...
2
votes
1answer
71 views

Recommendations for machine learning book?

I am a microbiologist and I am currently self-studying machine learning from some open lecture videos. I am finding it pretty difficult to understand proofs that are somewhat "obvious", my poor ...
3
votes
2answers
370 views

Proof that there is unambigous grammar for every regular language

How can I prove, or where can I find proof that for every regular language there is unambigous grammar?
0
votes
1answer
67 views

Reference for an undecidability proof [duplicate]

I'm searching for a reference of an undecidability proof that is as simple as possible and starts "from scratch". With "from scratch" I mean that it does not use some other undecidable problem to ...
0
votes
1answer
62 views

trouble with bijection definition [closed]

I have a bijection problem that I cannot get my head around. It goes like this: let f: A -> B and g: B -> C be functions such that g o f is a bijection. Prove that f must be one-to-one and that g ...
3
votes
0answers
47 views

How to prove that the composite strategy is prefix-closed and respects the alternation condition?

I'm doing some research on game semantics using these notes. Currently I'm trying to prove that the definition of composite-strategy is indeed a strategy. I have already proved all the conditions of ...
1
vote
1answer
333 views

Proving the language $L= \{0^n 1^m \space | \space m \equiv 0 \space mod \space n, \space n \geq 2 \}$ is not regular using the pumping lemma

I am trying to learn about applying the pumping lemma and I'm not really sure how to go about proving this language isn't regular with the pumping lemma: $L= \{0^n 1^m \space | \space m \equiv 0 ...
1
vote
1answer
42 views

Help with recurrence solutions

We started learning recurrences and I am having trouble with some of the problems. Our professor is having us substitute in $n=2^m$ and $S(m)=T(2^m)$ then writing down equations and summing them all ...
5
votes
3answers
109 views

Is it possible to prove thread safety?

Given a program consisting of variables and instructions which modify these variables, and a synchronization primitive (a monitor, mutex, java's synchronized or C#'s lock), is it possible to prove ...
0
votes
2answers
1k views

Solving recurrences using substitution method

I already have a solution for this problem but it's just not making sense to me. Here is the problem (It's from Introduction to Algorithms by CLRS found in CH.4): Show $T(n) = 2T(\lfloor n/2 ...
3
votes
3answers
141 views

Does a proof using the well-ordering principle need a base case?

For proofs by well-ordering principle the general template is to consider the negation of some predicate $P(n)$. Then assume the set of all elements that fulfill $\lnot P(n)$, i.e. $\qquad N = \{ n ...
6
votes
1answer
245 views

Proof of Ramsey's theorem: the number of cliques or anti cliques in a graph

Ramsey's theorem states that every graph with $n$ nodes contains either a clique or an independent set with at least $\frac{1}{2}\log_2 n$ nodes. I tried to look it up at a few places (including ...
0
votes
1answer
57 views

Prove that the syntax is equivalent

I don't "see" why the following syntax is equivalent to the second syntax1: E -> E + T E -> T syntax2: ...
0
votes
1answer
344 views

Use Rice's theorem to show that the language of optimisable Turing machines is undecidable

I have an assignment to do and I'm quite stuck with the following question : Use Rice's theorem to show that $ \qquad L' = \{ \langle M \rangle \mid \; (\exists \text{ TM } M') \; [ L(M') = ...
0
votes
1answer
49 views

Coordinated Attack Problem Different Requirements

There is a famous Coordinated Attack Problem. Let define a simple message-passing system $S$ with requirements Uniform Agreement: No two processes decide dierently. Validity: (a) If all processes ...
3
votes
2answers
81 views

Why is this sequence of recurrence relevant?

I am learning how to solve the time complexity for the recurrence relation $$ T(n) = 2T(n - 1) + n^2\text{, where }T(1) = 1 $$ The solution notes that I should begin by considering the following ...
2
votes
3answers
166 views

Howto formally go about proving that two LTL formulas are equivalent?

Do they need to "unwind" exactly to the same set of paths or does it suffice when one set is contained in the other ? Or is it sufficient to argue that M,s satisfies both LTL formulas for any ...
5
votes
2answers
120 views

Binary Search Tree: Replace $k$ min elements with their average

Given a valid binary search tree whose keys are unique real numbers, and a set of $k$ pointers to the $k$ minimum elements in the tree, will the BST property be maintained if I replace all $k$ ...
1
vote
1answer
123 views

r-regular graph and hamiltonian path

I am having some issues proving a problem I am working on. I have been sketching out examples but the proof is not jumping out at me. Question: Let $G = (V,E)$ be an undirected $r$-regular graph ...
3
votes
2answers
240 views

Analysis of algorithms, 'big O' question

The main question is, how exactly is the big O analysis calculated on routines? Is there a specific formula that relates what each function in a program does to a big O calculation? Also, what about ...
0
votes
0answers
53 views

Prove a bisimulation relation

I need to prove a bisimulation relation on $CA_{\tau}(N)$ (communication algebra with tau-steps) and names $N$. It need to prove that $p!d.x||p?d.y$ is bisimular with $p!d.(x||p?d.y)+p?d.(p!d.x||y)$ ...
13
votes
4answers
500 views

Do undecidable languages exist in constructivist logic?

Constructivist logic is a system which removes the Law of the Excluded Middle, as well as Double Negation, as axioms. It's described on Wikipedia here and here. In particular, the system doesn't ...
3
votes
2answers
87 views

MAX 10-SAT Algorithm

The MAX k-SAT problem is: “Given a set of clauses C1,…,Ck, each of length k, over a set of variables x1,…,xn, find a truth assignment that satisfies as many of the clauses as possible.” I'm ...
5
votes
2answers
118 views

Understanding why the polynomial $p(n) = \sum_{i=0}^{k} a_in^i$ is in $\Theta(n^k)$

Hi I've read this lemma in my book: Lemma 2.1. Let $p(n) = \sum_{i=0}^{k} a_in^i$ denote any polynomial and assume $a_k > 0$. Then $p(n) \in \Theta(n^k)$ Proof. It suffices to show that ...
3
votes
2answers
173 views

Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular

Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular. These are the things that I would use to start. As $L_1L_2$ is regular, then the homomorphism $h(L_1L_2)$ is regular. Let ...
1
vote
1answer
44 views

Showing transitivity of PSPACE?

For the following question: If B is an element of PSPACE and A is an element of PSPACE-Complete, and A polynomial reduces to B, then B is an element of PSPACE-Complete. I am trying to prove this, ...
3
votes
1answer
384 views

How to show that L = L(G)?

Specifying formal languages by giving formal grammars is a frequent task: we need grammars not only to describe languages, but also to parse them, or even do proper science. In all cases, it is ...
16
votes
4answers
537 views

What are common techniques for reducing problems to each other?

In computability and complexity theory (and maybe other fields), reductions are ubiquitous. There are many kinds, but the principle remains the same: show that one problem $L_1$ is at least as hard as ...
6
votes
2answers
964 views

How to show that a function is not computable?

I know that there exist a Turing Machine, if a function is computable. Then how to show that the function is not computable or there aren't any Turing Machine for that. Is there anything like a ...
5
votes
2answers
410 views

A pumping lemma for deterministic context-free languages?

The pumping lemma for regular languages can be used to prove that certain languages are not regular, and the pumping lemma for context-free languages (along with Ogden's lemma) can be used to prove ...