Questions about methods and techniques for proving theorems.

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1answer
23 views

How to prove strict space lower bounds using crossing sequences in Turing machines?

I understand the notion of crossing sequences when talking about time, however how are they used to actually prove strict lower bounds for some decision/search problems? For example, suppose that you ...
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1answer
50 views

Is an algorithm in pseudocode a reasonable way to establish complexity?

We define the language $$ L = \{a^nb^n : n\geq0 \} $$ and we want to prove the following $$ L = \mathrm{DSPACE}(\log n)\,. $$ So we have to prove that by using $\log n$ space on the work tape of ...
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1answer
26 views

Relationship between an NP-hard problems with the subsets of them (part 2)? [duplicate]

I asked two questions about NP-hard problems here Relationship between an NP-hard problems with the subsets of them? and here Does this manner of proof for being NP-hard is true? but unfortunately ...
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0answers
38 views

Does this manner of proof for being NP-hard is true? [duplicate]

I have a problem and I want to prove the problem is NP-hard. Thus, I considered a subset of the problem that has a minimum answer (NP-complete problem). Afterward, I proved there is not any solution ...
1
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1answer
28 views

How do I solve interdependent recurrence relations?

I have three functions with values given as $$\begin{align*} P(0) &= 0 \quad & P(i+1) &= 5M(i)\\ M(0) &= 1 \quad & M(i+1) &= R(i) + 2P(i)\\ R(0) &= 3 ...
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2answers
32 views

Hoare Calculus Incorrect Assignment Axiom

I'm currently preparing for an exam and recently came across the following exercise and would like to know whether my solution would be correct. Exercise: Prove that the following axiom is not ...
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1answer
75 views

What is the min # of moves to sort an array from 1 to n?

Problem: You are required to sort an array with numbers from 1 to n. You can do a "move", which means choosing one element and moving it to any place you want (insert to any place, not swap). Prove ...
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2answers
198 views

Direct NP-Complete proofs

I'm just starting to learn about NP-completeness. While I understand that reducibility plays a key role in this, I'm astonished how few problems I've been able to find who's proof that they are ...
2
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1answer
86 views

Propositional logic — syntactical completeness

Lets consider propositional logic. We say a proof system for propositional logic is syntactically (negation) complete if for every $\alpha$, either $\alpha$ or $\neg \alpha$ are provable within the ...
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0answers
19 views

How to conduct time complexity analysis for an implemented algorithm [duplicate]

Main task In my bachelor degree's thesis I've developed an algorithm for recommender systems which uses personalized PageRank with some particular features as nodes. In the recommender systems' ...
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2answers
66 views

NP-hardness of an optimization problem with real value

I have an optimization problem, whose answer is a real value, not an integer such as vertex cover and set cover. Therefore, the decision version of my problem is given an input and a real value $r$. ...
2
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1answer
75 views

Proof technique in complexity theory

I have a (stupid ?) question about complexity theory. It's about a "proof technique". I want to compare 2 models of computation. I want to prove that for each langage recognized in polynomial time by ...
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1answer
32 views

Why does reduction from vertex cover to subset sum use base-4? [closed]

Why does reduction from vertex cover to subset sum use base-4? 30.13 Subset Sum (from Vertex Cover)
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1answer
38 views

How to prove that a predicate is prefix closed

Suppose we have the predicate $\qquad A.p.q ≡ (∀i \mid p≤i≤j<q : X.i≤X.j)$ which says that $X[p..q)$ is ascending. Apparently, the predicate holds for empty segments, is prefix closed and is ...
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1answer
30 views

In what context does “definition” arise? [closed]

If I am correct, the concept of a "theorem" doesn't appear until the concept of a formal system is introduced. "Definition" is used more often than "theorem" is. So I wonder if the concept of ...
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0answers
22 views

How to guess solutions for “divide and conquer” type of recurrences? [duplicate]

I am trying to solve "divide and conquer" recurrence relations of the type $T(n) = a T(n/b) + f(n)$ using the Substitution/Inductive Methods ( NOT the Master Theorem ). Every site I see on the web ...
1
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1answer
92 views

Prove that an $n$-length string has $2^n$ i18n-style abbreviations

An i18n-style abbreviation is one in which multiple letters are shortened to that number of letters. E.g., "internationalization" -> "i18n" ("nternationalizatio" is substituted with its length 18). ...
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2answers
87 views

Proof of an Optimal substructure in Dynammic Programming?

Could someone please explain how exactly the proof of optimal substructure property in dynamic programing problems works?, they usually say that " let's say the global optimal solution is A, and B is ...
0
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1answer
71 views

A DFA recognizing my name

How can I know if my DFA is implemented correctly? For example, I need to build a DFA, and then minimize it which will recognize my name. Language which describe my name is: L = {pustai, marius} I ...
0
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1answer
109 views

Concept used in the proof [closed]

In the paper "Resolution for Quantified Boolean Formulas", I am unable to understand the proof of theorem 3.4. Please help me with the basic concept used on page 4: The concept that I am referring ...
3
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2answers
115 views

Need Help Reducing Subset Sum to Show a Problem is NP-Complete

I want to show that the following problem is NP-Complete: For a set of vectors $v_1,\ldots,v_n \in \mathbb{N}^d$ and an integer $k$, does there exist a subset $S \subseteq \{v_1,\ldots,v_n\}$, such ...
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2answers
66 views

if (dis)proving a conjecture on graph theory can be done just by a counter example then can every (dis)proof be mapped actually to a counter-example?

Suppose we have a conjecture on graph theory that can be (dis)proved by means of a counter example, then, is it true that every alternative (dis)proof of the conjecture can be mapped to a counter ...
2
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1answer
38 views

Is proof of totality of a semantic function for binary numbers cicular?

I think the following proof found in a textbook is circular, since the proof for case n=n'0 assumes the case n=n'1 and vice versa. Am I missing something? Proof that the semantic function N is a ...
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1answer
64 views

Prove that a proof system is not complete [on hold]

I want to prove that the proof system A is not complete. A consists of these axioms: ...
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1answer
325 views

Correctness of proof by induction

Suppose a person states the following: $n^2 = (n * n), \forall n > 0$. One can check such equality by saying, via proof by induction, that: for $n := 0:\ 0^2 = (0 * 0)$; for $n := 1:\ 1^2 = (1 * ...
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4answers
882 views

How to prove that problem is not in P

Given some abstract problem how can I prove that this problem is not in P. I mean, what is the method for proving such thesis?
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0answers
16 views

How does supercompilers relate to macro tree transducers?

Supercompilers can be used as a generalisation of deforestation of a functional program. Macro Tree Transducers composition can be used to the same effect, using a completely different approach. What ...
2
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1answer
44 views

Choose $n/2$ vertices and guarantee $3/4$ of edges are accounted for proof

Give a polynomial-time algorithm that finds ceil(V/2) vertices that collectively account for at least three-fourths (3/4) of the edges in an arbitrary undirected graph. The algorithm I have come up ...
4
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1answer
35 views

Should we not reuse constants in tableaux proofs?

I am trying to understand the proof of the following using tableaux: $$ \exists x\forall y.r(x,y) \to \forall x \exists y . r(x,y) $$ This is how it works out: $$ (1) \space \exists x \forall y ...
0
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1answer
37 views

How does one figure out where a class of languages falls under some complexity class? [closed]

I was wondering how can someone prove that one class of languages is of a certain complexity? For example, how could I show the Turing-recognizable languages are in P? Would I have to come up with ...
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1answer
99 views

induction proof for kleene star

i posted this on mathematics stack exchange here before i realised this one existed. i am going through some past exam paper questions on regular languages for some revision, and i am having a bit of ...
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1answer
68 views

Proving that a language does not belong to a language class by using more specific instances of that language

You have a description of a language that you have to prove is regular, context free, or other. In order to prove that it does not belong to a certain class of languages, you might think that it will ...
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1answer
59 views

Prove not context free

How can we prove that: $$ L = \{ w_1\#w_2 \mid w_1 \in w_2;\; |w_2| > |w_1|;\; w_1 , w_2 \in \{0, 1\}^*\} $$ is not context-free? The language defines $w_1$ as a sub-string of $w_2$, and they ...
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1answer
31 views

Finding a finite model

Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this: Suppose $L$ is a language which includes an infinite list ...
3
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1answer
31 views

LK-$\phi$ proof of $\exists y Pby$

I am having difficulty with the concept of $LK-\Phi$ proofs, here is a question I have been working on: Let $\Phi = \{Pafa\}$, where $P$ is a binary predicate symbol and $f$ is a unary function ...
38
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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
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2answers
62 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
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1answer
15 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 ...
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3answers
259 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
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1answer
57 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 ...
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4answers
105 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
59 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
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0answers
108 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
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0answers
36 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 ...
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2answers
91 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.
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0answers
40 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
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2answers
134 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
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0answers
29 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 ...
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0answers
46 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
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1answer
167 views

Euclidean Algorithm in Coq

Question How do I write more intuitive proofs of the two following results in Coq? ...