All mathematical questions about computer science, including theoretical computer science, formal methods, verification, and artificial intelligence. For questions about Turing computability, please use the (computability) tag instead. For numerical analysis, use the (numerical-methods) tag. For ...

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35 views

Prove that $L=\{a^nb^nc^md^m \mid m,n >=0\}$ is context free language

I'm trying to write the grammar of this language, in order to prove that it is CFL but I'm stuck because m or n could be 0. The language is: $L=\{a^nb^nc^md^m \mid m,n >=0\}$ . If they were ...
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58 views

AoPS Intermediate Algebra vs. Higher Algebra by Hall and Knight? And some more questions about learning math.

Ok. I'm learning algebra at the level of AoPS algebra 2, and I want to quickly progress through math. Allow me to explain the situation. I am highly interested in artificial intelligence/computer ...
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0answers
40 views

Turing Machine That Accepts Machines With Undecidable Languages

So I'm reviewing my Computability notes for my final, and I understand how reduction arguments work, but I'm having trouble framing one for the following Turing machine: Undecidable TM = { ⟨M⟩ | L(M) ...
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1answer
30 views

Why the only binary MDS codes are trivial ones?

Why the only binary MDS codes are trivial ones? I have been thinking how to draw a contradiction by assuming the MDS code is not trivial. Thank you very much!
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2answers
18 views

Proving that median of list $[x_1,x_2,…,x_n]$ minimises the sum $\sum_{i=1}^{i=n} |x_i-m|$ where $m$ is some number

The problem is in the title. Here is a detailed description: Let's say we have list $[x_i]_{i=1}^{i=n}$ where $x_i\in\Bbb{N}$. I want to pick such $m\in\Bbb{N}$ which minimises the sum ...
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1answer
101 views

What kind of edge do we have?

In order to find the kind of the edges of a graph, at which we applied the Depth-first search algorithm, we could use this: $$\begin{bmatrix} \text{ tree edges: } x \to y & [d[y],f[y]] \subset ...
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1answer
57 views

Trying to find formula for max number of nodes in a non-Binary tree.

I'm trying to find the max number of nodes in a tree that is defined as follows: The root can have at most $2$ children. Each subtree on the left can have at most $L$ children. Each subtree on the ...
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21 views

Is there a universal constant for size of disjoint clauses in 3-CNF

We are given a 3-CNF formula $\Phi$ on n variables, and a guarantee that at least 1% of $2^n$ possible assignments satisfy all clauses in $\Phi$. Now construct set $S$ of disjoint clauses so that no ...
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35 views

Linear Algebra Book Recommendation [duplicate]

I am taking an undergraduate course in computer science an is in the first year of my college. I like mathematics and am willing to learn Linear Algebra first and then move on to Abstract Algebra and ...
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1answer
41 views

How do I solve for the zeros of a Chebyshev polynomical? (on a computer)

I am working on a computer program and have a method that returns a number for a given $x$, $y$. So $f(x, y) = z$, where $f$ is my method. if I know $y$ and $z$, can I find what $x$ will be, without ...
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27 views

Discreet Math - Given n>= 5 how many times does fib(4) occur?

I have been trying to solve the below problem (and similar problems) but I have no clue how to tackle it. Can please help me tackle this particular problem, and how to attack similar problems? The ...
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1answer
19 views

Urn Probability Combination Problem

I am in my first year of Comp Sci and I am reviewing for my Math Final. There are two Urns $U_1$ and $U_2$. $U_1$ has $10$ red balls and $8$ blue balls. $U_2$ has $16$ red and $4$ blue. Suppose you ...
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22 views

What does it mean for an equivalence class to straddle a set of states?

In the following question, what does it mean for an equivalence class to straddle a set of states? Assume you currently have two equivalence classes Φ and Ψ. In addition you have a set of ...
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1answer
32 views

Special Binary Relations/ Empty Relation, Universal Relation And identity Relation?

The universal relation U = A × A. (Correct me if I'm Wrong). I believe that the Universal Relation is an Equivalence Relation The empty relation E = ∅. From my understanding, a Empty relation on a non ...
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0answers
39 views

Gram matrix of Gaussian kernel is not positive definite

I am developing a machine learning software, where I am trying to apply kernel methods. I have N uniformly sampled scalar values, $\{x_1,\dots,x_N\}$ from a given interval $[a,b]$. My aim is to ...
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3answers
87 views

Can a Turing Machine process an infinite string?

I read in a text book once that a finite state acceptor machine cannot be an acceptor for an infinite language. My question is does this apply to Turing Machines? The implication, it seems to me, ...
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2answers
32 views

c(n,k) equals subdivisions

To compare n files, the total comparison count is: $$ {{n}\choose{k}} = C^k_n = \dfrac{n!}{k! ( n - k )!} $$ with k = 2. Input space is composed by all pairs of files to compare. I want to split ...
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0answers
52 views

How to check homeomorphic embedding relation programmatically?

This is a follow up to this question and Deedlit's answer. I'm looking for a precise definition of the "hem?" (tree A homeomorphically embeddable in tree B?) relation, preferably in terms of a ...
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2answers
41 views

The expected time to sort $n$ elements is bounded below

Prove that the expected time to sort $n$ elements is bounded below by $cn \log n$ for some constant $c$. Could you give me some hints how I could do that?
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1answer
35 views

Is my Turing Machine (Transition Function) correct for finding if a string is of even or odd length?

I've been asked to create a formal Turing Machine (by means of a transition function) to which takes a string $a^n \in$ {a}* and decides whether it is an even or odd length. What I have made is the ...
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1answer
66 views

Calculating of genus of a curve

Let $C$ be a curve over $\mathbb{F}_q$ in projective plane. So $C$ can be done as zeroes of some gomogeneous polynomial $\in \mathbb{F}_q[x,y,z]$ with degree $n$. Whether is there algorithm which is ...
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2answers
36 views

Find both the largest and second largest elements from a set

Consider finding both the largest and second largest elements from a set of $n$ elements by means of comparisons. Prove that $n+\lceil \log n \rceil -2$ comparisons are necessary and sufficient. ...
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1answer
51 views

Can 2 items be added/taken away from a stack in push down automata at once?

Here is a language and 2 ways (I hope) of representing it with a PDA. Can I use the notation (b,a $\to$ ee) or anything of the like, to take away 2 items from the top of a list at once? Such as I ...
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1answer
29 views

Modular arithmetic with huge modulus?

When the dividend is some huge power but the modulus is not so big, I can use modular exponentiation. But how can I compute the residue when the modulus is, for example, $2^{107} - 1$, a Mersenne ...
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2answers
49 views

Construct a PDA to accept the language

construct a PDA that accepts the language: a) $L_1 = \{ a^k b^k c^i \mid k,i \ge 0 \}$ my answer is : $$\begin{align*} &S\to AA\\ &A\to abc \mid ab \mid c \mid \lambda \end{align*}$$ b) ...
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1answer
19 views

complexity question regarding whether it is decision problem

When self teaching complexity theory and seeing arguments that were made online. I get some confusion. In the class, we classify problems into P: can be computed polynomially NP: given a claimed ...
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1answer
41 views

What is the possible maximum/minimum height for a 29 nodes AVL tree?

What is the possible maximum/minimum height for a $29$ nodes AVL tree? Maximum: We've learned in class that $h\le \phi \log_2n$. Where $\phi\approx 1.4404$. So if we calculate for $n=29$ we get ...
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2answers
53 views

How to design a Context-Free Grammar and Pushdown Automaton for the following language:

How would you design a context-free grammar for the following language? $\{p^n \ r^m \ p \ \ b^{m+n} \ \ r^2 ∣ m,n\geq 0\}$ Derive a Pushdown Automaton that accepts the same language as the CFG. ...
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2answers
44 views

Discrete Math: Array-Pointer Representation

I am confused as to how the table is filled in for a pointer-array representation of a graph, and I can't find anything online that talks much about array-pointer representation. My book does not ...
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3answers
60 views

How many comparisons are required?

Let $S$ be a set of $n$ integers. Assume you can perform only addition of elements of $S$ and comparisons between sums. Under these conditions how many comparisons are required to find the maximum ...
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1answer
48 views

A decision tree has an expected depth of at least $\log n!$

I am looking at the proof of the following theorem and I have some questions. The theorem is the following: On the assumption that all permutations of a sequence of $n$ elements are equally ...
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2answers
63 views

The decision tree has height at least $\log n!$

The proof of the theorem Any decision tree that sorts $n$ distinct elements has height at least $\log n!$ is the following: Since the result of sorting $n$ elements can be any one of the $n!$ ...
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0answers
49 views

Is there a name for this constant? (0.0100011011…)

It's the simplest number I could think of that contains any finite binary code in its digits: $$\begin{align} c &= 0.0100011011000001010011100101110111...\\ &= ...
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1answer
34 views

checking boolean logical equivalence

Given two boolean formula (aka. logic circuit), I want to check if they are logically equivalent, namely that they compute the same truth table. Is this an NP-complete problem? What is the proof?
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smallest circuit

Let $SMALLESTCIRCUIT$ be the language consisting of all Boolean Circuits $C$ with the property that there is no smaller circuit $C^{'}$ that has the same truth table as $C$. (smaller means having ...
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0answers
34 views

Size of increments in commutative ring to reach given number

I have the ring $\Bbb Z_q = \{0,1,\ldots,q-1\}$, where $q$ is a prime. Starting from $0$, I want to make exactly $n$ equally sized increments and reach $a\in \Bbb Z_q$, with $n<q-1$. For example if ...
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1answer
18 views

Hexidecimal subtraction when more than one borrow is needed

I wanted to know how I would subtract two hexadecimal values from each other when more than one borrow is needed. Given 0x00000200 - 0x00000004 Here is what I ...
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1answer
66 views

Subsets of a monoid closed under left-multiplication by elements of a submonoid

Let $M, T$ be monoids (or, semigroups) with $M \subset T$. Then we can consider subsets $S$ of $T$ that are closed under left-multiplication by something in $M$, i.e. $$ a \in S, m \in M \implies ma ...
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11 views

Proving if the graph of F is over it's tangent line at all points then Newton Raphson converges.

Here's the problem: Let $F : \mathbb{R} \rightarrow \mathbb{R} \,, F \in \mathbb{C}^1$ so that $F'(x) < 0 \, \forall \, x$ and there's a unique $r$ so that $F(r)=0$ and let $L_{x_0} (x)$ be the ...
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2answers
154 views

Prove correctness of algorithm using induction

Bubblesort(A) int i, j; for i from 1 to n { for j from n-1 downto i { if (A[j] > A[j+1]) swap(A[j], A[j+1]) } } Could ...
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5answers
248 views

Proving formula for sum of squares with binomial coefficient

$$\sum_{k=0}^{n-1}(k^2)= \binom{n}{3} + \binom{n+1}{3}$$ How should I prove that it is the correct formula for sum of squares? Should I use induction to prove the basis? Any help is appreciated.
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39 views

Minimize sum of squared error

I have an array of real numbers, I want to partition them into k sets. In each set, I calculate the sum of squared error. Then, I add up all the sum of squared error for all the set. I want to ...
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33 views

Partition problem of equal size

I have an array S of size 2n, each element in the array is an integer I want to split it into two arrays of size n, and under this condition, minimize the difference of the sum of integers in the two ...
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1answer
62 views

Is a thorough study of algorithms useful for a mathematician?

In my university, there is a core course called "basics computer science for mathematicians". The topics covered range from algorithms, to the bases of programming, to theory of computability. The ...
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0answers
23 views

Deriving a recurrence equation and apply Master's Thrm to it

So for a function such as: function Hi(n) if n > 1 then for j ← 1 to n do print(”Hi”) Hi(n/2) Hi(n/2) Hi(n/2) It's very easy to eyeball this and get a ...
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2answers
125 views

Application of Mergesort

We have $8$ players and we want to sort them in $24$ hours. There is one stadium. Each game lasts one hour. In how many hours can we sort them?? I thought that we could it as followed: ...
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21 views

Application of divide-and-conquer

Good afternoon. Consider a two-position switch with two inputs and two outputs. In one position inputs 1 and 2 are connected to outputs 1 and 2 respectively. Using these switches, design a network ...
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0answers
20 views

Fleury’s Algorithm in case of we have odd-degree nodes

I'm studying Fleury’s Algorithm to find Eulerian tour. I'm confused in case of we have two odd-degree nodes. What should we do in this case? Should we duplicate the path between the two add-degree ...
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33 views

Proof of Correctness: Recursion inside loop

I am trying to prove the correctness of the algorithm in the research paper. It is at page 17 in the pdf. ...