Questions about methods and techniques for proving theorems.

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3answers
141 views

What is wrong with this reasoning that finding the genus of a degree 3 bipartite graph is NP-complete?

Finding genus of a biparite graph is $NP$-complete and finding genus of a degree $3$ graph is $NP$-complete and so finding genus of a degree $3$ bipartite graph is $NP$-complete. Though implication ...
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0answers
9 views

Proof with closure properties for regular languages [duplicate]

Hi how to prove using properties closure for regular languajes that: $L = \{w|w \in \{a,b\}^* \wedge |w|_b = 2|w|_a \}$ is not regular. Thanks
2
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0answers
32 views

Proving weak simulation

I want to prove something but I am not sure if it is the right way to do it. I have two LTS that define different semantics. A=($Q_a,Λ,\to)$, and B=$(Q_b,Λ\cup\{\beta\},\leadsto)$, where $\beta$ is ...
3
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1answer
62 views

How to come up the number of nodes on a given level in heaps?

CLRS asked it's readers to prove that there are at most $\lceil n/2^{h+1} \rceil$ nodes of height $h$ in any n-element heap as an exercise. The principle of Mathematical Induction can be used to prove ...
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0answers
24 views

How would one prove that the following scheme definition is an ordered stream of integers

How would one prove that the following scheme definition is an ordered stream of integers (define integers (cons-stream 1 (add-streams ones integers)))
1
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0answers
45 views

Modal logic axiom S4, transitive and reflexive frame, tableaux solver

I have a difficult problem to solve which as mentioned in the title is related to modal logic axiom S4. So, here is some background knowledge that can be useful: S4 axiom is a class of transitive ...
0
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0answers
33 views

simple iterate algorithm proof by induction

Suppose I have a function where it calculates which bit is larger called LargerBinary. Let's say I have an input 110111;101001, the output will be 110111 and if the input is 110110:110110, the output ...
1
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1answer
69 views

Proving a language is neither Recursively Enumerable nor co-Recursively Enumerable

$$L = \{ \langle M \rangle \mid \text{\(M\) is a Turing Machine and \(|L(M)| = 1\)} \}$$ I have to prove that this is not R.E. and not co-R.E. I know how to approach these kind of problems. For ...
1
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3answers
277 views

Can I simplify log(n+1) before showing that it is in O(log n)?

Had a question about the following: $$\log (n+1) \in O(\log n)$$ Can the left side be simplified any further or do I need to just go ahead and find a c and n that hold?
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0answers
22 views

How do I prove a language is regular? [duplicate]

I've done a lot of research on this topic, but still don't feel very confident about it. Let's say the example is: For a language L over an Σ, define N(L)={w∈Σ∗: wk∈L for some k∈Σ∗}. Prove that, if L ...
3
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1answer
120 views

How can I check constraints on my state machine behaviour?

My background is fairly practical rather than theoretical, so this question may be a bit basic. I have a state machine with events, and events may optionally trigger action functions to be called as ...
0
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2answers
39 views

Big-O Justification Question

I am trying to justify the big-O order of a runtime complexity by finding a $c$ and $n_0$ that hold for it. Does the left side of the justification need to be one or higher, or can it be any value so ...
0
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0answers
69 views

How not to prove that P ≠ NP implies NP ≠ PSPACE

Let's define the two variants of the Travelling salesmen problem: $TSP_{opt}$ : Give me the shortest tour $TSP_{dec}$ : Is there a tour of $l$ or shorther (Yes/No) Now assume $P \neq NP$: Since ...
3
votes
1answer
78 views

Is the NP-hardness Proof with One Way Implication Correct and Why?

A problem $\Pi$ is NP-hard if I can prove this: a known NP-hard problem $\Pi'$ reduces to $\Pi$ in polynomial time; and $f(x) \in \Pi\iff$ $x \in \Pi'$. If I can show only one way implication, ...
1
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1answer
20 views

Can I change the input of my reductionduring the proof?

To prove that a problem $\Pi_2$ is NP-hard one has to: select a known NP-hard problem $\Pi_1$; from an arbitrary instance of $\Pi_1$, create an instance of $\Pi_2$ in polynomial-time; and show ...
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2answers
80 views

proof using induction of automaton

How I can explain this. Consider the following automaton, $A$. Prove using the method of induction that every word/string $w\in L(A)$ contains an odd number(length) of $1$'s. Show that there are ...
1
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2answers
106 views

Show that any monotone Boolean function is computable by a circuit containing only AND and OR gates

A Boolean function $f : \{0, 1\}^n → \{0, 1\}$ is called monotone if changing any of the $n$ input bits $x_1, \ldots , x_n$ from $0$ to $1$ can only ever change the output $f(x_1, \ldots ,x_n)$ from ...
0
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1answer
49 views

Proving following regular expressions equal to one another?

How would I go about proving the following two regular expressions are equal to one another: ( a + b )* a ( a + b )* b( a + b )* = (a + b)* ab(a + b)* I can ...
3
votes
1answer
48 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
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2answers
62 views
0
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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
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2answers
79 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
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0answers
24 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
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2answers
726 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
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1answer
56 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 ...
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2answers
58 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
68 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, ...
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1answer
28 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? ...
23
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2answers
391 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
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1answer
54 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
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3answers
100 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 ...
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2answers
97 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
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1answer
77 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 ...
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3answers
499 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
243 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
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1answer
108 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 $(L_1 \cup L_2)^* = L_1^* \cup L_2^*$ ?
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0answers
51 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
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2answers
19 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
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1answer
47 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
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0answers
22 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
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1answer
95 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.
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1answer
124 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
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1answer
37 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
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1answer
90 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) = ...
26
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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 ...
4
votes
1answer
81 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
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1answer
44 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 ...
5
votes
1answer
170 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
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1answer
66 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
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0answers
51 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 ...