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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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Why proving the solution of a problem is polynomial time is sufficient enough to say that it is a NP prolbem?

Why proving that we can verify the solution of a problem is polynomial time is sufficient enough to say that the problem is nondeterministic polynomial time? Please note: this is not a question on how ...
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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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2answers
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Decidability of $SEQ_{CFG} = \{⟨G,H⟩ \mid \text{$G,H$ are CFGs and $L(G) ⊆ L(H)$}\}$

How can I prove that $SEQ_{CFG} = \{⟨G,H⟩ \mid \text{$G,H$ are CFGs and $L(G) ⊆ L(H)$}\}$ is decidable ? I know that $EQ_{CFG} = \{⟨G, H⟩ \mid \text{$G,H$ are CFGs and $L(G) = L(H)$}\}$ is not.
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4answers
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Prooving by Pigeonhole principle

I've been given a question to solve: ...
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2answers
85 views

How to generate an instance for an NP_hard proof, where each element has two inputs?

I want to prove the NP-hardness of an scheduling problem. The problem seems to be NP-hard in the ordinary sense, so I am trying with the Partition Problem, precisely the Equal Cardinality Partition (...
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How to prove that models of indirect and direct RAM machines are equivalent?

as in the title, I am looking for a formal proof how to show that models of indirect and direct RAM (random-access) machines are equivalent. I would really appreciate your help.
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1answer
29 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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1answer
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Turing machine with semi infinite tape - Prove by construction

I'm studying constrained Turing Machines. There's a theorem that proves that both infinite and semi-infinite tape TM have the same computational power. The theorem that proves this by emulating a TM1 ...
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1answer
25 views

Trouble understanding this proof about the minimum number of states of a deterministic finite automaton

So I was browsing online looking for the general structure to proving a DFA has a minimum of $n$ states for some $n$ and most of them use contradiction. However, I'm having a hard time understanding ...
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3answers
36 views

Intuitive proof for a tree with n nodes, has n-1 edges

I am interested in an intuitive proof for "any binary tree with $n$ nodes has $n-1$ edges", that goes beyond proof by strong induction.
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1answer
35 views

Alternate proof of the Caro-Wei theorem for lower bounding the independence number

Let $G$ be a graph on $n$ vertices whose degree sequence is $d_1,d_2,...,d_n$. Let $\alpha(G)$ denote the size of maximum independent set of $G$, i.e., the size of a maximum subset of vertices of $G$ ...
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1answer
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Proving problem NP-completeness [duplicate]

I am studying computational complexity and i am trying to solve this problem. We are given a (non-bipartite) complete graph: G = (V, W, E) where the vertices can be divided in two classes V and W ...
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2answers
159 views

Proving a value is the result of the execution of an algorithm

Assuming an algorithm $A$ known to both Alice and Bob. Alice runs the algorithm and gets a result $R$. How can Alice prove to Bob that $R$ is the result of the execution of $A$ and not some random ...
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1answer
30 views

Proof for Turing Machines being able to simulate any algorithm in the same time complexity

I have always read that Turing machines can simulate any algorithm, without changing the time complexity of the algorithm, and hence it is easier to study the Turing machine equivalent of the ...
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1answer
40 views

Proof By Contradiction - Hamiltonian Paths and Cycles

Was hoping if anyone had any way to prove the following claim using proof by contradiction Let $G = (V, E)$ be a simple graph with at least one vertex, and let $G'$ be the graph formed by adding a ...
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1answer
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Showing $2^x$ is a lower bound

How do I show that $2^x - x^2 \in \Omega(2^x)$? Basically, I know that this means that $\exists a, x_0 \in \mathbb{R^+}, \forall x \in \mathbb{N}, a.2^x \leq 2^x - x^2$. I worked around a bit with ...
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0answers
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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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3answers
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Is there a *simple* proof that the intersection of a CFL and a regular language is a CFL?

I am following a course on complexity theory where languages are a part of the course. There is a proof that no matter how hard I try to understand, it is till so complex that I cannot make it to half ...
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1answer
17 views

There could not be an edge from u to v in a DAG, if w is before v in a topological order

I am trying to prove that given a DAG. There exists a valid topological ordering that has v in front of u iff there is no path from u to v. The proof is related to the fact that reverse DFS post ...
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0answers
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Challenging exercises for proof correctness [closed]

I would like to know where I can find challenging exercises that ask to prove the correctness of an algorithm. The invariant of most of the exercises I’ve found on the internet are quite easy (...
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1answer
83 views

lower bound proof with adversary argument

We have to run a song on a Walkman, for that we need 2 full batteries. Let's say we have a mixed set of 30 batteries (15 are empty and and 15 are full) and then only way to test if the battery is full ...
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1answer
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How to reduce a problem?

I am a bit confused on how to reduce a problem. I'll give an example: Let's say there is a problem called HALTEMPTY and we know it is undecidable. $HALTEMPTY_{TM} = \{\langle M\rangle \mid M \text{ ...
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1answer
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Undecidable problem intersection of two DCFL languages is DCFL?

We have two deterministic pushdown automatas A and B, which languages are deterministic context-free. The problem is to decide if there exists a deterministic pushdown automata, which language is an ...
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0answers
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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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1answer
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How to Apply Elementary Axioms from Kleene Star to an Inequality

Axioms For * \begin{align} 1 + aa^* &\leq a^* \\ 1 + a^*a &\leq a^* \\ b + ax &\leq x \to a^*b \leq x \\ b + xa &\leq x \to ba^* \leq x \\ \end{align} Elementary Results \begin{...
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3answers
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Prove that A** = A*, where A is a language over Σ*

Let $\mathcal A$ be an arbitrary language over $\Sigma^*$ Proof. To prove, $\mathcal A^{**} = \mathcal A^* $ $\mathcal A^{**} = \left( \mathcal A^0 \cup \mathcal A^1 \cup {...} \cup \mathcal A^n \...
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1answer
35 views

Any finite Graph G with all V have at least degree of 2, is it true that every vertex is necessarily contained IN a cycle?

As title, (note: this questions is asking weather or not all vertices are contained IN a cycle not asking if the G contains a cycle. My attempt is that: So this graph would be an counter example ...
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1answer
100 views

Relating a proof to a Haskell program

I am trying to relate the following integer square root theorem $\forall x: \mathbb{N}, \exists y : \mathbb{N}((y^2 \leq x) \land (x < (y+1)^2))$ and its proof to its role as a specification of ...
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1answer
109 views

Is it possible to simulate a fair coin with a finite number of tossing of a biased one?

It is a classic problem to simulate a fair coin with a biased one. According to Fair Coin (wiki), John von Neumann gave the following procedure: Toss the coin twice. If the results ...
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1answer
50 views

Does indirect diagonalization a relativize technique?

My main question is can with R.kanon , Fortnow ,... technique that shows lower bounds for SAT seperate P and NP ? Baker-Gill-Solovay showed that $P?=NP$ could not be solved with relativization. Does ...
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1answer
69 views

Removing arithmetic within recurrences

A similar question was asked here: Solving recurrences using substitution method, but I am still somewhat hazy as to how this process works. Say, for $T(n) = T(\lceil n/5 \rceil + 36) + n \log n$ ...
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1answer
49 views

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
41 views

Prove that $f^{-1}:\mathbb{N} \rightarrow \mathbb{N}$ is partial recursive

I'm stuck on this problem: Given $f:\mathbb{N} \rightarrow \mathbb{N}$ a partial recursive function that is also injective and total. Prove that the function $f^{-1}:\mathbb{N} \rightarrow \mathbb{N}$...
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1answer
81 views

Prove that $D =\{x \in \mathbb{N} | \Phi_x(x)\uparrow\}$ is **not** recursively enumerable

So I tried to prove that $D =\{x \in \mathbb{N} | \Phi_x(x)\uparrow\}$ is not recursively enumerable in the following way: let's suppose that $g$ is the computable function that represents $D$ $$g(x) ...
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1answer
38 views

Prove that $L^r$ is context free without alphabet

I'm stuck with this problem: Given $L$ a CFL on the alphabet $\Sigma$. Prove that $L^r=\{x^r|x\in L\}$, where for each $a\in\Sigma$ and $y\in\Sigma^*$, $$\epsilon^r=\epsilon,$$ $$(ay)^r=y^ra,$$ is ...
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1answer
49 views

Why is the graph inside Graham Scan always planar

One of the ways to prove that Graham Scan constructs convex hull in linear time is using planarity of the graph obtained by running the algorithm. This graph is always planar, so according to Euler's ...
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1answer
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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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0answers
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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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1answer
57 views

How CompCert “proves” different things in its codebase

In order to understand examples of formal proofs, I am interested in how CompCert applies "proof" techniques. Specifically, I am wondering what a particular example is of something CompCert "proves" ...
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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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2answers
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An example of something you can formally verify with proofs in Software Development

I have been working on understanding formal verification of software. Formal methods include things like modeling your software with Petri Nets, Automata, or State-Transition Graphs. Other techniques ...
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1answer
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How to Specify the Behavior of Automata for Verification

I am wondering what it takes to "verify" or "prove" that an automaton is correct. What the components are that are required. It seems that an automaton would be an easier thing to formally verify as ...
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0answers
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What you want to “prove” in algorithms

So it seems that you can get pretty far with just type definitions as a formal model of a system. The typed properties verify that the properties will have that type, typed function arguments verify ...
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158 views

NP-completeness of vertex cover

Show that the following language is NP-complete $$ L = \{ \langle G,k \rangle \mid \text{$G$ is a graph with a set $S$ of $k$ vertices hitting every edge of $G$}\}. $$ I know I should reduce the ...
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Is there any proof that Scheduling with a resource constraint works with two resources?

I've been reading this paper: https://ac.els-cdn.com/S0304397511002623/1-s2.0-S0304397511002623-main.pdf?_tid=bc074517-2b81-4d37-bc92-a5c20dba6b23&acdnat=...
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2answers
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How to prove a side effect in a function

I asked a question earlier about Saving to the Database, which was very general and about the requirements for a proof when you go through many layers of non-verified systems such as the network and ...
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1answer
58 views

High-level requirements for a Proof of “Saving to the Database”

This question is about a few sentence description of what a proof would look like (and technologies / logics involved) for a complex api call through many layers. Trying to get a sense of the ...
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1answer
46 views

Understanding Hoare Logic Axioms

Given these 5 axioms of Hoare Logic: \begin{array}{cl} \frac{}{\{\phi([x \leftarrow E])\}\ x := E\ \{\phi(x)\}} & \mathtt{Assignment}\\\\ \frac{\{\phi\}\ P_1\ \{\eta\} \quad \{\eta\}\ P_2\ \{\psi\...
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1answer
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1answer
58 views

If a Triple Graph Grammar rule counts as a Mathematical Proof

I am intrigued by Triple Graph Grammars (TGG) as a potential for formal mathematical proof. Triple Graph Grammars (TGGs) are a technique for defining the correspondence between two different ...