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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.

-1
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0answers
43 views

dfa modulo 6 don't work correctly [closed]

this dfa accepts the multiples of 6 in binary, I have tried it and it works with 0,6,12,18,24,30,36. this dfa accepts 0000, 0110,100,10010,11,000,11110 and 100100. the dfa has only worked correctly ...
2
votes
1answer
42 views

How do we prove the time complexity of this simple problem in probabilistic inference on a Bayesian network?

Perhaps a rather trivial question, but I'm trying to refresh on proof methods in CS... Suppose we have a simple Bayesian network with two rows of nodes: $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots,...
1
vote
1answer
33 views

How to correctly describe this action, deleting an edge that “shortcut” some vertices

Haven't written a proof in years, not sure how to describe an algorithm like this ? Let us what we have a graph. like this below: 1). How to describe edge removal of{ (0, 1),(3,4), (1,2) }done in ...
0
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0answers
53 views

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, $diff(\epsilon)=0$; $diff(0)=βˆ’1$; $diff(1)=1$. Let $L = ...
0
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0answers
34 views

dfa subtract multiple of 3 [duplicate]

Define a function 𝑑𝑖𝑓𝑓 ∈{0,1}β†’β„€ so: for everything w ∈{0,1}, diff w = # of 1's in w- # of 0's in w. Thus: 𝑑𝑖𝑓𝑓 πœ€=0; 𝑑𝑖𝑓𝑓 0=βˆ’1; 𝑑𝑖𝑓𝑓 0=βˆ’1; Let 𝐿 = {π‘€βˆˆ {0,1} * | 𝑑𝑖𝑓𝑓 𝑀 = 3π‘š ...
0
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2answers
91 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 ...
1
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0answers
27 views

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: ...
1
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4answers
105 views

Deriving lower and upper bounds for T(n) = T(n-1) + T(n-2) + 10

The solution is to find the upper and lower bounds from: 2T(n-2) < T(n) < 2T(n-1) + 10 So I have to find ...
2
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1answer
37 views

Must a function in lambda-calculus which inputs a boolean function be defined in a certian way?

This question is my best attempt to get at a more general question about what one can get from terms in the lambda calculus. Using the church encoding, we define booleans by $\texttt{true} = \lambda ...
3
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1answer
30 views

Finding maximum takes at least $\lceil n/2 \rceil$ comparisons

We are given an array $A$ with $n$ elements, $n \in \mathbb{N}$ and all elements are in the set $\{1,2,3, \cdots, n \}$. I want to prove that finding the maximum in $A$ (that is, outputting the index ...
2
votes
1answer
66 views

Union of infinitely many regular languages [duplicate]

I need to prove or disprove the following statement. If $A_n βŠ† \Sigma^*$ is regular for each $n \in \mathbb{N}$ then $\bigcup\limits_{n=0}^{\infty} A_n$ is regular. I know that if two languages ...
0
votes
1answer
77 views

Proving the singleton language {x} is regular for all x ∈ Σ*

So I'm aware that the singleton language is in fact regular for all x ∈ Σ*, but I do not understand why it is. A formal proof would help, but also getting some intuition as to why it is regular would ...
2
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2answers
80 views

Proving that Every Full Prefix-Free Language is Maximal

I'm practicing a problem where I need to prove that every full prefix-free language is maximal. I know a prefix-free language A is maximal if it is not a proper subset of any prefix-free language, ...
0
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1answer
54 views

Proving that A(B ∩ C) βŠ† AB ∩ AC

A(B ∩ C) = { UV | U ∈ A, V ∈ B and V ∈ C } for the left part. ΑΒ = { UV | U ∈ A, V ∈ B }, Ξ‘C = { UV | U ∈ A, V ∈ c }, AB ∩ AC = { UV | U ∈ AB and AC, V ∈ AB and AC } for the right part. How can I ...
1
vote
1answer
59 views

Proof of an Infinite Binary Sequence

I have a problem where given an infinite binary sequence S ∈ {0, 1}∞ to be "prefix-repetitive" if there are infinitely many strings w ∈ {0, 1}* such that ww is a prefix of S. I need to prove that if ...
1
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4answers
81 views

Proving B* = B on a given set

I have the set: B = {x ∈ {0,1}* | there is an equal number of 0's and 1's in x} and therefore, B* = {e,01,10,0011,0101,0110,1100,1010,1001,....etc} I need to either prove or disprove that B*=B I ...
1
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0answers
51 views

Sum of long geometric progression [closed]

Finding sum of a geometric progression is simple when we just need to report the sum, but when some modulo or multiplicative inverse is asked of that sum the task become tedious for me. I have a ...
2
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0answers
46 views

Can Pareto Optimality be compared to Nash Equilibrium?

Given a state $s$, and a value function $v^i$ that determines the expected payoff for the i-th agent in that state, can the two definitions below, one of Nash equilibrium and another of Pareto ...
1
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0answers
61 views

Proof that G is a Tree After DFS and BFS form the same tree T [closed]

Let G be a connected, undirected graph containing some vertex s. let's say that BFS and DFS are both run on G starting at s and that the breadth first search and depth first search ...
1
vote
1answer
47 views

Prove Halting on all Inputs is not in RE simulation

I don't understand why when proving if Halting on all inputs problem si not in RE using the complement of the halting problem, I have to take a turing machine and simulate the machine M(the machine ...
1
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0answers
186 views

Could you show the intractibility of SAT by showing that the number of variables contributing to an arbitrary unsatisfied clause is not constant? [closed]

Preface: This is not an attempted proof at P vs NP Starting with some CNF Boolean expression Ο•, by the rules of logical disjunction, a clause is only unsatisfied if each of the literals in it are ...
0
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1answer
43 views

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

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 ...
0
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0answers
43 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 ...
2
votes
2answers
40 views

Decidability of $SEQ_{CFG} = \{⟨G,H⟩ \mid \text{$G,H$ are CFGs and $L(G) βŠ† L(H)$}\}$

How can I prove that $SEQ_{CFG} = \{⟨G,H⟩ \mid \text{$G,H$ are CFGs and $L(G) βŠ† L(H)$}\}$ is decidable ? I know that $EQ_{CFG} = \{⟨G, H⟩ \mid \text{$G,H$ are CFGs and $L(G) = L(H)$}\}$ is not.
2
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4answers
69 views

Prooving by Pigeonhole principle

I've been given a question to solve: ...
1
vote
2answers
100 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 (...
0
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0answers
17 views

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.
0
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1answer
32 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 ...
2
votes
1answer
76 views

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 ...
1
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1answer
41 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 ...
1
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3answers
45 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.
2
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1answer
38 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 $G$ ...
1
vote
1answer
94 views

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 ...
4
votes
2answers
160 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 ...
1
vote
1answer
47 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 ...
1
vote
1answer
48 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 ...
0
votes
1answer
23 views

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 ...
1
vote
0answers
32 views

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-...
1
vote
3answers
156 views

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 ...
0
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1answer
18 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 ...
2
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0answers
32 views

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 (...
1
vote
1answer
127 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 ...
2
votes
1answer
57 views

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{ ...
3
votes
1answer
145 views

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 ...
1
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0answers
62 views

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 :...
2
votes
1answer
48 views

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{...
6
votes
3answers
2k views

Prove that A** = A*, where A is a language over Ξ£*

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 \...
3
votes
1answer
39 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 ...
3
votes
1answer
138 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 ...
5
votes
1answer
126 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 ...