Questions about methods and techniques for proving theorems.

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3
votes
1answer
43 views

Why add a +1 to the constant proving an O(n) bound?

I have calculated a running-time function $T(n) = 4 + 4n$, which is $O(n)$. To determine the constant $C$ given by the relation $|T(n)| < C \cdot g(n)$, we take $\qquad\displaystyle \lim_{n \to ...
0
votes
2answers
43 views
0
votes
0answers
9 views

Proving the Multiway cut problem is NP Complete [duplicate]

Problem Statement: Given k nodes: $$ u_1, u_2, u_3..., u_k $$ remove edges of total minimum weight that separates $u_i$ from $u_j$ for all $i != j$ for all k >= 3 I just need some help identifying ...
3
votes
2answers
64 views

Determining states in a DFA

In computational theory for deterministic finite automata, there is a 5-tuple representation containing $Q$, $\Sigma$, $\Delta$, $Q_0$ and $F$. I am wondering how one understands thinking about the ...
0
votes
0answers
23 views

Is there a general form of polynomial reductions in complexity theory? [duplicate]

While reading Sipser, in computability I read about many to one mapping reducibility and Turing reducibility,the latter one being a more general form of reducibility. But in the introductory chapter ...
4
votes
2answers
708 views

Is a single string enough to prove regular expressions inequivalent?

Which of the following regular expressions generate a language that is different from the rest? (a+b)$^*$a(a+b)$^*$(a+b)$^*$ b$^*$ab$^*$a(a+b)$^*$ (a+b)$^*$ab$^*$ab$^*$ ...
2
votes
1answer
53 views

Prove that every thin AVL tree may be converted to red-black tree

Let's define thin AVL tree as AVL tree $t'$ such that contains minimal possible number of nodes among all AVL tree $t$ such that $height(t)=height(t')$. I am trying to prove that every thin AVL ...
1
vote
2answers
49 views

Why does the proof of undecidability of $A_{TM}$ require the universal TM to take input $\langle M,\langle M\rangle\rangle$?

I've read a proof explaining why $A_{\mathrm{TM}}$ is undecidable, and I don't seem to understand why we need to give the opposite of $H$ function $D$ itself as input. Here's the copy-paste of that ...
3
votes
2answers
64 views

Choosing nonzero entries from an array so no pair in same row or column

Suppose we have an $n\times n$ array $A$ of non-negative real numbers in which the sum of each row and each column is $1$. We want to find $n$ entries of the array $(x_1,y_1), \dots, ...
-1
votes
1answer
27 views

Tools or techniques for studying the language a CFG produces? [duplicate]

When developing a CFG, I find that one can be confused about whether the grammar is correct, i.e. whether it recognizes only the required strings and not other strings. But this can be hard to see? ...
21
votes
2answers
372 views

Are there any specific problems known to be undecidable for reasons other than diagonalization, self-reference, or reducibility?

Every undecidable problem that I know of falls into one of the following categories: Problems that are undecidable because of diagonalization (indirect self-reference). These problems, like the ...
1
vote
1answer
53 views

How Splitting Summation method works

I'm reading Cormen, Leiserson, Rivest and Stein, Introduction to Algorithms, Appendix A, page 1152. They discuss a method called "Splitting Summations", where they split the summation and ...
0
votes
3answers
77 views

Proving/Disproving that language L is non-regular/CFL

Here are three examples of questions I run into. I'm not looking for solutions. If $L$ is CFL then $L' = \{ ww^R | w \in L \}$ is non-regular. If $L$ is non-regular then $L' = \{ ww^R | w \in ...
-1
votes
2answers
40 views

How to simplify boolean expression? [closed]

I am having trouble simplifying logical expressions to a much simpler form, can someone provide me some insight on how to approach the problem? Let's assume i have the following expression: $ABCD + ...
1
vote
1answer
60 views

Do polynomial reduction functions work both ways?

For example to prove 3-Sat ≤p Independent Set do I just have to prove this theorem: Theorem- Formula F is satisfiable IFF graph has an independent set. If I have to prove it this way does this also ...
9
votes
3answers
411 views

How do I verify that a DFA is equivalent to a NFA?

I'm learning how to convert NFAs to DFAs and I want to make sure I'm doing it right. Obviously, going back in the other direction isn't a thing. Does anyone know of an algorithm to check that a DFA is ...
2
votes
1answer
214 views

Why do puzzles like Masyu lie in NP?

The puzzle is made up of (n x n) squares so when taking the problem the input size would be n. Rules of Masyu: The goal is to draw a single continuous non-intersecting loop that properly passes ...
0
votes
1answer
103 views

Languages that are not subset, but are union

Are there examples of regular languages $L_1$ and $L_2$, where $L_1$ and $L_2$ is not a subset of each other but that $(L_1 \cup L_2)^* = L_1^* \cup L_2^*$ ?
0
votes
0answers
50 views

Why if $G$ has two spanning trees $A$ and $A'$, then every edge of $A'\cup \{e_i\}\in A'$

Theorem: Let be $G$ a weighted graph in which every edge has a different weight. Suppose that $G$ has two spanning trees $A$ and $A'$. Let be $i$ the first index such that $e_i\ne e'_i$ ...
0
votes
2answers
18 views

Why does if A is a spanning tree which doesn't have $e_1$ then $A\bigcup\{e_1\}$ has a unique cycle?

I am studying the algorithm of Sollin and we recently studied a lemma: Let be G a graph which values are diffferent on the edges. We sort the edges $e_1,e_2,...e_m$ such as $v(e_i)<v(e_j)$ ...
0
votes
1answer
41 views

Understanding the proof of the halting problem [closed]

I came across the following example that proves that the blank tape halting problem is not decidable. I understand the proof technique, but I just don't see how the blank tape problem is shown to ...
0
votes
0answers
21 views

Method of inductive statements for proving partial correctness of block-schemes

I'm trying to find an explanation and more information on a method and some example problems with solutions using that method. The method doesn't seem to translate well in english (I'm from a ...
0
votes
1answer
94 views

How do I prove that a language is deletion closed?

For example, how could I prove that the following language is deletion closed: {$a^k$$b^j$ : $j$, $k$ $\geqslant$ 0} The reason seems obvious to me, I just can't see a way to prove it.
1
vote
1answer
90 views

Proving grammar only generate strings that is multiple of 3

Hello I have an exercise for homework and I was hopping to get some hints in order to solve it. num-> 11 | 10 num' 01 | num 0 | num num num'-> 00 num' | 1 num' | ε I need to prove that my ...
2
votes
1answer
35 views

how to prove the periodity of an LFSR

everywhere I've searched it says that the minimal period of an LFSR given by a characteristic polynomial $c(x)$ is the least number $r \in \mathbb{N}$ that: $$c(x)|(x^r-1)$$ but how do I prove it's ...
0
votes
1answer
88 views

if $\delta(q, a) = q$ for all symbols $a$, show that $\delta^* (q, w) = q$ is true for all strings $w$

Let $A$ be a DFA and $q$ a particular state of $A$, such that $\delta(q, a) = q$ for all input symbols $a$. Show by induction on the length of input that for all strings $w$, $\delta^* (q, w) = ...
24
votes
5answers
6k views

Proof that dead code cannot be detected by compilers

I'm planning to teach a winter course on a varying number of topics, one of which is going to be compilers. Now, I came across this problem while thinking of assignments to give throughout the ...
3
votes
1answer
73 views

Solving recurrences by substitution method: why can I introduce new constants?

I am solving an exercise from the book of Cormen et al. (Introduction To Algorithms). The task is: Show that solution of $T(n) = T(\lceil n/2\rceil) + 1$ is $O(\lg n)$ So, by big-O definition I ...
1
vote
1answer
29 views

Optimal prefix code: full binary tree existence

I'm ok following the passages generally used to prove this theorem, like in this question concerning the same subject. Anyway i've the unpleasant feeling that the proof lacks something to be ...
4
votes
1answer
152 views

How rule 110 would be proven to be universal if the tag system did not exist?

I was reading about Cellular Automata and I read in this question that Matthew Cook proved that rule 110 is universal, and that his proof relied upon showing how rule 110 can simulate a tag system. ...
3
votes
1answer
63 views

Proving that picking odd-numbered symbols can create the universal set from a non-regular language

Let's define the following operations: $odd(string) = $ odd characters of $string$, $even(string) = $ even characters of $string$ Now say we have some language $L$, we will then define the ...
2
votes
0answers
49 views

How to prove a Language is neither a Computably enumerable nor Co-Computably enumerable?

What would be the general approach for that? And what are the things that generally overlooked while proving such things? For example, I have a Language, L ={e:$L(M_e)$ such that it accepts only 'a ...
3
votes
1answer
95 views

How to prove closure property of regular languages using regular expressions?

I know that we can prove closure of two regular languages under operations like union, intersection, concatenation etc. by constructing NFAs for them but how to do the same thing using regular ...
1
vote
1answer
36 views

Proof that there are same number rotation moves in any binary tree with both children compulsory

I am working on this project where I am required to find the theoretical proof for following. I have a particular type of binary trees, where 1) each internal node will definitely have two children. ...
0
votes
1answer
29 views

Big O Proof for Logarithmic Function [duplicate]

I am an undergraduate student in Computer Engineering and going through one of the textbook examples, I am asked to prove that $T(n)$ is $O(\log{}n)$ Where $T(n)= 5\log_{2} 2n +7$. I understand ...
2
votes
5answers
63 views

How to prove a set has infinite cardinality?

Set S is a set consisting of all string of one or more a or b such as "a, b, ab, ba, abb, bba..." and how to prove set S is a infinity set. I have tried proving set S as one to one corresponding to ...
1
vote
0answers
70 views

Proving NP-Completeness by reduction

I'm given a more restricted version of 3-SAT called 3-SAT-M: Problem: 3-SAT-M INPUT: A set of clauses C {c1,...,ck} over n boolean variables {x1,...,xn}, where every clause contains ...
0
votes
0answers
10 views

method of proving solution with dynamic programming [duplicate]

Is there any method of proving solution with dynamic programming ? Maybe induction ? I don't have idea. Help me please.
0
votes
2answers
23 views

Show that a function is not a PRFs

I am studying cryptography and having some trouble with the PRFs How do I show that $f_{k}(x) =x+k^2$ mod $2^n$ is not a Psuedo Random function? I know that I need to show that the distignuisher can ...
4
votes
1answer
71 views

Proving that a language is not Recursive

I have the following language: T = {M | there exists w such that M accepts w within |w| steps} I am trying to prove that this language is not recursive and that it is recursive-enumerable. To prove ...
1
vote
1answer
46 views

What can be concluded from a full application of resolution?

I know that resolution is refutation complete, but what can we conclude if a resolution procedure leads to a situation with no more chance to operate the resolution? Given a propositional formula ...
1
vote
1answer
29 views

How to determine approximability of a problem when we don't know how good a solution is?

As far as I have learned, an approximation algorithm for an optimization problem Runs in polynomial time, and Whose cost can be bounded by a function of input in terms of distance from the optimal ...
0
votes
0answers
65 views

Proving a dynamic programming recurrence for coin exchange correct

Suppose I have $n$ kinds of coins $c_1, c_2, \dots, c_n$. I'm given: $S$, an amount of money I should construct with minimum number of coins. I came into the following formula: $$ T(n,S) = ...
4
votes
1answer
64 views

How to prove $0\neq1$ using the J rule?

Suppose I have a simple dependent type theory with bottom, unit, sums, dependent pairs, dependent functions, natural numbers and homogeneous identity with J-elimination. Is there a way to prove $(0 = ...
5
votes
1answer
148 views

What's wrong with my pumping lemma proof?

The language $L = \{0^{2n} \space |\space n \ge 0 \}$ is obviously regular – for example, it matches the regular expression $(00)^*$. But the following pumping lemma argument seems to show it's ...
2
votes
1answer
49 views

Chomsky hierarchy type determined by language

I have some modified automata and the task is to give the type of Chomsky hierarchy to it. All task is between type 3 and 0 noninclusive. For regular languages there are lot of tools and I can check ...
4
votes
0answers
50 views

Proof that P is closed against switching between polynomially related encodings

Lemma 34.1 Let $Q$ be an abstract decision problem on an instance set $I$, and let $e_1$ and $e_2$ be polynomially related encodings on $I$. Then, $e_1(Q)\in \mathrm{P}$ if and ...
2
votes
0answers
20 views

Computational models - proving language is decidable [duplicate]

I tried to prove that the following language is recursive/decidable/in R: for $\Sigma=\{0,1\}$, $k$ a positive integer: $$ L_k= H_\text{TM,epsilon}\cap \Sigma^k $$ where $H_\text{TM,epsilon}=\{\langle ...
7
votes
3answers
156 views

Constructive proof of decidability of finite Halting-problem-style set that does not use table lookup

I tried to prove that the following language is recursive: for $\Sigma=\{0,1\}$, $k$ a positive integer: $$ L_k= H_{\mathrm{TM},\varepsilon}\cap \Sigma^k $$ where ...
4
votes
1answer
46 views

Proof by induction over rules for mutually recursive relations

Consider the (big-step) semantics of a language ($a, e$ terms, $v$ values), defined by two mutually recursive relations, $\downarrow$ and $\Downarrow$, given by a set of rule-schemata (simplified): ...