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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0answers
113 views

Related paradigms in Computer Science and Mathematics

I've read in a number of places discussions on whether Mathematics is a branch of Computer Science and vice-versa (see here for example) Having a background in both, I know that the Computer Sciences ...
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1answer
353 views

Context free grammar to pushdown automata…

<expr> −→ <term> | <expr> + <term> <term> −→ <factor> | <term> × <factor> <factor> −→ (<expr>) | b ...
3
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2answers
45 views

$(2^a -1)(2^b -1)=2^{2^c}+1$ has no nonnegative integer solutions

$(2^a -1)(2^b -1)=2^{2^c}+1$ is not possible for a,b,c nonnegative integers. Any solutions using parity Approach: $(2^a -1)(2^b -1)=2^{2^c}+1\Rightarrow$ $2^{a+b}-2^a-2^b=2^{2^c}\Rightarrow$
1
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1answer
270 views

Let $L_{UIUC}$ = $\{ \langle M \rangle$ : $L(M)$ contains the string $UIUC\}$. Prove that $L_{UIUC}$ is undecidable.

Been stumped as to why the following proof works. Note: I have taken this proof directly from here. Proof by reduction from $A_{TM}$. Suppose that $L_{UIUC}$ were decidable and let $R$ be a Turing ...
4
votes
1answer
993 views

Is Cartesian Product same as SQL Full Outer Join?

Is Cartesian Product same as Full Outer Join found in Relational Database SQL? I ask because I am taking a Discrete Mathematics course and I just want a better understanding of how what I am studying ...
2
votes
1answer
67 views

The “computability” of fundamental physical constants

I would like to ask if any of the fundamental physical quantities like the speed of light or plancks constant (all measured according to a common standard of of units) can be classified as computable ...
0
votes
1answer
15 views

xyplus recursive language

For {x, y, +}, the language XYPlus is shown recursively as: (1) x and y are in XYPlus . (2) So if a and b are words in XYPlus ...
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3answers
2k views

Big O notation - Proving that a function is not O(n)

Show that the function, $T(n) = 4n^2$ is NOT $O(n)$. I'm not looking for someone to give me a full answer, I just need some pointers on how to go about starting to show that it is not $O(n)$. Many ...
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1answer
77 views

Help in understanding risch algorithm

This is an excerpt from the paper that explains risch algorithm for symbolic integration. I do not understand how do we get to the residue directly and also how does it help us in the integration. ...
0
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1answer
83 views

Expressing as 1's complement

I have the problem: Express ( -43) as a word-length 1’s complement number 43 is in decimal. I understand how to convert to Hexadecimal and take the 2's ...
1
vote
1answer
114 views

Ambiguous set-builder notation

I apologize in advance for the Python code below. I don't know how to express this question unambiguously in the language of math. Let $\mathcal{F}$ be a family of sets. It seems to me that $$S(x) = ...
2
votes
2answers
117 views

Minimum queens to reach $8 \times 8$ squares as a graph problem

A homework problem asks What is the minimum number of queens to reach all squares on a $8 \times 8$ chess board? We are expected to solve this by somehow casting the problem as a graph problem ...
4
votes
1answer
49 views

Transforming Nested Fixed-Point Formulas into Infinitary Logic Formulas with Finitely many Variables

There is a definition (actually a description of how it could be defined) of a fixed-point logic formula. The formula is in inflationary fixed point logic (IFP) in this case but it could also be ...
4
votes
2answers
109 views

Have action/predicate systems (or similar) been considered in the literature?

Question. Has the following concept, or anything similar, been considered in the literature? Definition. An action/predicate system consists of sets $A$ (the actions) and $X$ (the predicates) such ...
3
votes
3answers
172 views

Why are the hierarchy theorem proofs called diagonalization?

Proofs of the various hierarchy theorems in theoretical computer science (see e.g. http://www.cs.princeton.edu/theory/complexity/diagchap.pdf) are usually called diagonalization proofs. Why they are ...
2
votes
4answers
4k views

What is the difference between discrete and continuous mathematics?

I am studying computer science and this has me absolutely flummoxed. The definition I can find is that discrete data is countable and that continuous is uncountable. Examples are given stating that ...
1
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1answer
123 views

Designing a turing machine

Suppose you have a tape that has a block of $a$ strokes followed by a space, followed by a block of $b$ strokes, followed by a space, followed by a block of $c$ strokes, and otherwise blank. ...
1
vote
2answers
59 views

Pumping Lemma Help

Can somebody explain or possible solve the following question regarding the pumping lemma? I have a book from Sipser that I took this from and I honestly can't really get a good grasp of the pumping ...
4
votes
1answer
209 views

Solve Recurrence Equation with Induction

Question: Given the recurrence equation for the recursive Fibonacci sequence program: $T(n) = T(n-1) + T(n-2) + b$ $T(0) = T(1) = a$ Using induction, show that $T(n) \leq f(n)$, where $f(n) = c2^n, ...
2
votes
1answer
114 views

How to convert a Turing Machine program to a tiling using Wang Tiles?

To illustrate my question I provide the following example. The website Online Turing Machine provides a Turing Machine simulator. The following program adds 1 to any binary number. q0,1 q0,1,> ...
0
votes
1answer
51 views

Constrained disjoint subsets

How to partition $n$ weighted elements into $m$ disjoint subsets such that the sum of weight of all elements in a subset is less than equals to the capacity of $j$th subset ($c_j$) . It is given that ...
0
votes
2answers
64 views

Shifting Method

I'm taking a Course in Computer Science where we're having a refresher on Calculus 2 material. There is a problem that I don't understand or know how to do. Compute the following using the ...
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1answer
52 views

Finding the wrong regular expression

Which one of the following regular expressions does not define the language of all strings that ends with a. $(a + b)^*a$ $b^*aa^*(bb^*aa^*)^*$ $[a(ba)^* + b(ab)^*](a + b)^*a$ $(b + aa^*b)^*a(a + ...
3
votes
2answers
131 views

Proving that $xy = yx$ where $x$ and $y$ are both strings.

I am to prove that the following holds for any two strings $x, y \in \lbrace 0, 1\rbrace^*$ $xy = yx$ if and only if $\exists z \in \{0,1\}^*$ and $i,j \in \mathbb N$, such that $x = z^i$ and $y = ...
1
vote
1answer
25 views

Interpreting probabilistic time turning machines

I was trying to understand better the definition of a strong PSRG and I came across this expression which I am trying to understand better: $$ Pr_{r \in \{0,1\}^l}[A(r) = "yes"]$$ Where r is a truly ...
142
votes
14answers
32k views

Is computer science a branch of mathematics?

I have been wondering, is computer science a branch of mathematics? No one has ever adequately described it to me. It all seems very math-like to me. My second question is, are there any books about ...
3
votes
2answers
158 views

What will I be doing? [closed]

I'm a freshman studying discrete mathematics B.sc. It's easy to google ''typical mathematician jobs'' and get an idea of what mathematicians do for a living, but what sort of jobs can I expect to work ...
2
votes
1answer
55 views

Number of triangles in a graph

Could anybody explain to my why the asymptotic upper bound for the number of triangles in a graph with n vertices is O(n^3). I could not imagine a graph with n vertices which can contain indeed n^3 ...
2
votes
1answer
3k views

Preventing underflow, log sum exp trick

I have some difficulties with understanding the schema to prevent underflow, which is very often mentioned as The log-sum-exp trick, the partial decription is The log-sum-exp trick. In short, I ...
2
votes
1answer
53 views

Number of trees of a certain size

Given a branching factor $b$ and a tree height $h$, a complete tree has $\sum_{i=0}^h b^i$ nodes. Define a partial tree as a sub-tree of the complete tree, with the same root. How many such partial ...
3
votes
4answers
226 views

Algorithm Analysis: How to simplify a summation leading up to a maximal term?

Okay so I have a summation which goes: $$\sum_{i=1}^{n^3} 3i^2\cdot\log(i)$$ My goal is to find the order of the function, not the exact summation amount. I have found the order of it by writing ...
1
vote
1answer
74 views

What's wrong with this argument for $NP \ne EXP$?

Let $\{M_i\}$ be any enumeration of all Turing machines in which each machine appears an infinite number of times. Consider the language $D = \{i \, | \, M_i(i) \text{ does not accept within ...
6
votes
2answers
228 views

Relatively prime property verification

I am reading a computer science puzzles book. And I get the following question - "You have a five quart jug, a three quart jug and unlimited supply of water. How would you come up with exactly four ...
4
votes
0answers
137 views

Computational hard math problem

Given a square filled randomly with the numbers $1$ to $N$, for instance $$\begin{array}{cccc} 16 &12 & 9 & 1\\ 11 & 3 & 4 & 7\\ 2& 8 & 5&14\\ 6& 10& ...
0
votes
1answer
63 views

Is this language context free and if it is which grammar generates it?

$$L=\{\, w \in\{a,b,c\}^* :w=a^ib^jc^k, j=\max\{i,k\}\,\}$$ I think I proved it not context-free using pumping lema for CFL, but I'm not sure I'm doing it right. So, if someone knows grammar that ...
3
votes
1answer
329 views

An alternate analysis to the (worst-case) run time of the euclidean algorithm

I was trying to figure out the running time of the euclidean algorithm. The analysis that I found on Wikipedia and CLRS both analyze the run time of the euclidean algorithm using the Fibonacci ...
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1answer
56 views

Languages in P that are not P-complete

Why aren't there any languages in P that are not P-complete?
0
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0answers
71 views

How to check if a function is negligible?

Let $\epsilon(x)$ be a negligible function. Let $p$ be a polynomial such that $p(k) \geq 0$ for all $k > 0$. What can we say about $\epsilon(p(k))$? Is this a negligible function? If yes, ...
0
votes
1answer
27 views

Determining if a problem is solvable by a Push-Down Automaton

I have the following language: {0^n 1^n 0^n 1^n | n >=0} And need to find a PDA that recognizes the language. I have devised PDAS which determine ...
1
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0answers
58 views

Theory of Automata concepts

I just started taking Theory of Automata and I'm having a hard time understanding some of the concepts. It's been only a week and the following questions are my homework. I'm not asking you to do my ...
0
votes
3answers
154 views

Distance between pixel centres on a tv

I have a question i am stuck on in my maths class. I am going to change the numbers so i can figure it out this example myself. Need to get the distance between the two pixels calculated. Thanks ...
0
votes
3answers
44 views

Reference for problems without efficient algorithm (in polynomial time)

I'm writing paper and need your help in finding some famous (or not so famous) problems without efficient algorithm, but from logic or computer science. So far, I have: -Boolean satisfiability ...
2
votes
1answer
50 views

Strong primes in cryptography, their relation to complexity theory and security

According to the lecture slide by Shafi Goldwasser a prime is a strong prime if: $$p = 2q + 1$$ for some prime q. For me it, seems a bit arbitrary that is the definition of a strong prime in ...
0
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3answers
197 views

Acyclic graph - source node

How can I prove that a directed acyclic graph has a source node? A node 'a' is called source node if doesn't exists edges like ('b','a').
2
votes
0answers
99 views

Implications of NP=coNP for PSPACE

If NP = coNP, then the Polynomial Hierarchy collapses to its first level (NP). Intuitively, it seems to me that PSPACE should collapse down to NP as well. As a loose heuristic argument, take the ...
2
votes
1answer
157 views

Subsets of all Diophantine's sets

Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable $\Leftrightarrow$ graph of function is Diophantine. Consider some subset $S$ of computable functions (for example some Grzegorczyk's class or ...
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0answers
197 views

what are the advantages and disadvantages of Belief propagation

Belief Propagation cannot solve the graphical model which has cycles. For undirected graphical model for example MRF and CRF in computer vision area, in which cases the model has no cycle ? As far as ...
0
votes
1answer
43 views

classifying problems with reducibility

How can we use a reduction to prove non membership of a class. Can we say if A is reducible to B they are in same class or if we can't reduce A to B. B is not same class as A. Regards,
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1answer
76 views

Group of computable permutations

Why group of computable permutations of natural numbers is not finitely generated? It is obvious for all permutations but why it is also true for computable permutations?
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1answer
53 views

Logic question- its about propositional logic and it asks for a valuation for statisfiability

I dont understand this question that comes from a past paper, so please any help is appreciated. The part that i dont understand is what does it mean (question 3) that it wants me to consider the ...