# 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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### 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". ...
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### 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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### Weaker conjectures to prove in order to arrive at P =/= NP

We know we have a long way to go before we come to a proof of P $\neq$ NP. We also know that this road is studded with minor conjectures that will have to be proved/disproved in order to arrive at the ...
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### 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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### 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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### 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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### Strategy / Technique for Proving Equivalence between Multiple Program Forms

Wondering how to prove different compiled forms of a program to be "effectively the same" or "equivalent". For example, you can have a program represented as your normal nested function calls, or ...
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### how real example projects apply formal proof techniques

Given that there are examples of formal proofs used to formally verify real-world software applications, I would like to know what these people and teams actually do to create these formal proofs. ...
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### Accounting method vs Potential method for analysing an augmented stack and differences with standard complexity analysis

With reference to chapter 17 of CLRS, (Amortized analysis). I'm trying to understand the differences between the accounting method and the potential method. Let's start with standard analysis of the ...
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### How can I prove impossibility of generalizing a given higher order function from pure to monadic or applicative?

There is a great divide in Haskell between pure and monadic algorithms. While the latter are indistinguishable from their usual imperative counterparts, the former can get much more magical. What this ...
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### Doubt on dovetailing

Let < M > be an encoding of a Turing machine. L = { < M > | M is a Turing machine that accepts a string of length 2014 } Above language is R.E(even though we have infinite TM's) as we have ...
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### What's the polynomial involved in the PCP theorem?

Statements of the PCP theorem always speak of a proof of length $poly(n)$. But what polynomial is that exactly? Could you actually construct the PCP for some mathematical fact in real life?
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### Why does the proof that #SAT is in IP stop after m rounds?

I've been struggling to understand why the interactive proof for #SAT stops after only $m$ rounds, where $m$ is the number of variables in the formula $\phi$. I understand that two polynomials of ...
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### Minimal number of states

In a recent IT class we got the task of creating a finite state automaton that accepts only the words "auto" "automarke" "tomaten" and "automaten" and no others. Basically the whole class had no ...
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### Divide and Conquer a problem into a sub-problem to solve it efficiently

Assume that problem A cannot be solved in O(n^2) time. However, we can transform problem A into a problem B in O(n^2 log n) time, and then solve B, and finally transform the solution of B into the ...
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Consider the following expressions about homomorphisms and show if the statements are true or not. Σ={0,1}, L1 and L2 are Languages ⊆ Σ*, and ᵠ is a homomorphism ᵠ: Σ* → Σ*. ᵠ(L1 ∪ L2) = ᵠ(L1) ∪ ᵠ(...
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### On the complexity of existential and universal quantifiers

I'm trying to analyze the time complexities of the two former kind of quantifiers, I need help figuring out if I'm following the right path or if I'm making mistakes, here's what I've produced so far: ...
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. ...