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GATE 2014 Question 26: Which of these mapping reductions is true?
Here we answer GATE 2014 Question 26 about mapping reductions. This is about simply understanding the properties and definition of mapping reductions.
Easy Theory Website: https://www.easytheory.org
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Gold Supporters: Micah Wood
Silver Supporters: Timmy Gy
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
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https://www.youtube.com/watch?v=V0wyqXM3G8M
Easy Theory Website: https://www.easytheory.org
Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join
Donation (appears on streams): https://streamlabs.com/easytheory1/tip
Paypal: https://paypal.me/easytheory
Patreon: https://www.patreon.com/easytheory
Discord: https://discord.gg/SD4U3hs
Youtube Live Streaming (Sundays) - subscribe for when these occur.
Merch:
Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122
Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang
If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
Gold Supporters: Micah Wood
Silver Supporters: Timmy Gy
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
...
https://www.youtube.com/watch?v=V0wyqXM3G8M
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VIDEO
PUMPI
pumping-lemma-for-linear-languages-proof
Here we prove a "pumping lemma" for linear languages and give an example of a language that is context-free but not linear. A linear grammar is one where every rule has at most one variable on its right side, and a linear language is one that has a linear grammar for it. We have then shown that regular languages are a strict subset of linear languages, which are a strict subset of context-free languages.
Timeline:
0:00 - Intro
0:35 - Example of Linear Language
1:55 - Proof of PL for Linear Languages
14:05 - Pumping Lemma Statement
16:55 - Example for {a^i b^i c^j d^j : i, j at least 0} not linear
Easy Theory Website: https://www.easytheory.org
Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join
Donation (appears on streams): https://streamlabs.com/easytheory1/tip
Paypal: https://paypal.me/easytheory
Patreon: https://www.patreon.com/easytheory
Discord: https://discord.gg/SD4U3hs
#easytheory
Youtube Live Streaming (Sundays) - subscribe for when these occur.
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If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
...
https://www.youtube.com/watch?v=mj19qKj9YEk
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Controlling
VIDEO
LOGAR
logarithmic-space-in-complexity-theory
Here we introduce the notion of "logarithmic space", which is understanding what problems can be solved with a "very small" amount of space. We define what a Turing Machine needs to do in order to achieve that, as well as give some example problems that are in deterministic log space, and nondeterministic log space.
What is space in complexity theory? It is the maximum amount of cells used on a Turing Machine that runs on inputs of size n. See https://www.youtube.com/watch?v=yMhQdU5j6ag&t=8s&ab_channel=EasyTheory for more details.
Timeline:
0:00 - Intro
2:30 - Model for Small Space
7:30 - Definition of Log Space and Nondet Log Space
8:30 - All regular languages are in L
15:05 - {0^n 1^n} is in L
18:25 - PATH vs DPATH, DPATH in NL
Easy Theory Website: https://www.easytheory.org
Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join
Donation (appears on streams): https://streamlabs.com/easytheory1/tip
Paypal: https://paypal.me/easytheory
Patreon: https://www.patreon.com/easytheory
Discord: https://discord.gg/SD4U3hs
Youtube Live Streaming (Sundays) - subscribe for when these occur.
Merch:
Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122
Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang
If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
...
https://www.youtube.com/watch?v=X9R6mvENyQQ
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Controlling
VIDEO
I PRO
i-proved-a-math-conjecture
Original covering arrays video: https://www.youtube.com/watch?v=m1j4OBs1wsY&ab_channel=EasyTheory
arXiv preprint: https://arxiv.org/pdf/2211.01209.pdf
Here we discuss a recent proof of mine of a conjecture in my research area, which has to do with sizes of covering arrays. I first go over what the problem, which is determining the asymptotic sizes (# of rows) of covering arrays of "higher index", which is to say each interaction must appear at least a certain number of times, instead of just once. I give "easy" upper and lower bounds; unlike the prior video's scenario, there is a "gap" between them. Next I give some upper bounds that have been published and are "better", but there is still a small gap. My proof this year is that there is no gap between the "easy" lower bound and the new upper bound I prove, which is logarithmic in the number of columns, and additively linear in the index.
This proof also uses the probabilistic method, but with a new ingredient: the Lambert W function. My proof first uses the Cauchy-Schwarz inequality on the equation coming from the probabilistic method. The C-S inequality relates cross correlations of sums (hard to understand) to self-correlations (easy to understand), but paying a penalty; in other words, sum(a_i * b_i) changes to sum(a_i^2) * sum(b_i^2) * penalty. Then we use simple known upper bounds on the two sums that come out. The final "trick" is to use the Lambert W function, which is the inverse of f(W) = W * e^W. One would have to analyze the equation that comes out, and notice where in the Lambert W function one is; in this case, it is in the negative region. To finish everything off, we use *lower* bounds on W since in this region, W is negative (which translates to *upper* bounds on covering array sizes).
Easy Theory Website: https://www.easytheory.org
Discord: https://discord.gg/SD4U3hs
If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about it. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
The views expressed in this video are not reflective of any of my current or former employers.
...
https://www.youtube.com/watch?v=D3H8yEuCsSk
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VIDEO
WHY M
why-most-p-vs.-np-attempts-fail
Here we give some explanations as to why every P vs NP proof has failed, so far. P refers to problems that are *solvable* in polynomial time, and NP are those that *can be checked* in polynomial time. A central question in complexity theory is: does P = NP? It is sufficient to show that some NP-hard problem is in P (to show that P = NP), or to show that EVERY NP-hard problem is not in P (to show that P != NP). It is also not ruled out yet that the statement "Does P = NP?" is impossible to solve with current mathematical techniques (or any slight variation).
I give a set of reasons for why known "proofs" have failed, so that you can see through why a "proof" is actually false; many people claim to have solved the problem online. I deliberately did not go into details because the video would be insanely long if I did (if there's interest, I can make such videos...eventually...). The truth is: it is unlikely we will see a resolution of P vs NP soon, or even within our lifetimes, because it is so insanely difficult for some of the reasons I go into in the video. The unedited version of this video is here: https://youtu.be/MWZ_p8nXIrQ
The humorous (and sadly outdated) P-versus-NP page: https://www.win.tue.nl/~gwoegi/P-versus-NP.htm
Thanks to the following supporters of the channel for helping support this video. If you want to contribute, links are below. Names are listed in alphabetical order by surname.
Dolev Abuhazira, Simone Glinz, Timmy Gy, Josh Hibschman, Patrik Keinonen, Travis Schnider, Tao Su, Micah Wood
Easy Theory Website: https://www.easytheory.org
Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join
Donation (appears on streams): https://streamlabs.com/easytheory1/tip
Paypal: https://paypal.me/easytheory
Patreon: https://www.patreon.com/easytheory
Discord: https://discord.gg/SD4U3hs
Merch:
Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122
Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang
If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
...
https://www.youtube.com/watch?v=RwcVyWaQ4TQ
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video/mp4
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Controlling
VIDEO
OFFIC
office-hours-5-with-easy-theory-proof-of
Powered by Restream https://restream.io/
Patreon: https://www.patreon.com/easytheory
Twitch: https://www.twitch.tv/easytheory
Mixer: https://mixer.com/easytheory
Discord: https://discord.gg/SD4U3hs
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If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught over 12 courses at Arizona State University, as well as Colgate University, including several sections of undergraduate theory.
▶ABOUT THIS CHANNEL◀
The theory of computation is perhaps the fundamental theory of computer science. It sets out to define, mathematically, what exactly computation is, what is feasible to solve using a computer, and also what is not possible to solve using a computer. The main objective is to define a computer mathematically, without the reliance on real-world computers, hardware or software, or the plethora of programming languages we have in use today. The notion of a Turing machine serves this purpose and defines what we believe is the crux of all computable functions.
This channel is also about weaker forms of computation, concentrating on two classes: regular languages and context-free languages. These two models help understand what we can do with restricted means of computation, and offer a rich theory using which you can hone your mathematical skills in reasoning with simple machines and the languages they define.
However, they are not simply there as a weak form of computation--the most attractive aspect of them is that problems formulated on them are tractable, i.e. we can build efficient algorithms to reason with objects such as finite automata, context-free grammars and pushdown automata. For example, we can model a piece of hardware (a circuit) as a finite-state system and solve whether the circuit satisfies a property (like whether it performs addition of 16-bit registers correctly). We can model the syntax of a programming language using a grammar, and build algorithms that check if a string parses according to this grammar.
On the other hand, most problems that ask properties about Turing machines are undecidable. This Youtube channel will help you see and prove that several tasks involving Turing machines are unsolvable---i.e., no computer, no software, can solve it. For example, you will see that there is no software that can check whether a C program will halt on a particular input. To prove something is possible is, of cours
...
https://www.youtube.com/watch?v=rHmynKxk_aQ
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VIDEO
LET'S
let's-solve-some-problems-3-easy-theory
Donation (appears on streams): https://streamlabs.com/easytheory1/tip
Paypal: https://paypal.me/easytheory
Patreon: https://www.patreon.com/easytheory
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#easytheory #gate #theory
Youtube Live Streaming (Sundays) - subscribe for when these occur.
Social Media:
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Facebook group: https://www.facebook.com/groups/easytheory/
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If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
Gold Supporters: Micah Wood
Silver Supporters: Timmy Gy
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
...
https://www.youtube.com/watch?v=1an8nj4NuwM
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VIDEO
WHY A
why-are-turing-machines-so-powerful
Get 20% off your next adventure with No One with my code DFATM! Here we look at the question of *why* Turing Machines (TMs) are more powerful than deterministic finite automata (DFAs). TMs have the ability to (1) read/write cells, (2) move left and right, and (3) acquire new memory. The question we address here is what of 1, 2, or 3 cause TMs to be more powerful - is it that all three have to work in conjunction to finally escape the regular languages?
(In case it's not obvious - this video is NOT sponsored by any actual entity.)
Thanks to the following supporters of the channel for helping support this video. If you want to contribute, links are below.
Dolev Abuhazira, Josh Hibschman, Micah Wood, Morgan Jones, Patrik Keinonen, Simone Glinz, Tao Su, Timothy Gorden, unit220, Valentine Eben
Easy Theory Website: https://www.easytheory.org
Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join
Donation (appears on streams): https://streamlabs.com/easytheory1/tip
Paypal: https://paypal.me/easytheory
Patreon: https://www.patreon.com/easytheory
Discord: https://discord.gg/SD4U3hs
Merch:
Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122
Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang
If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
...
https://www.youtube.com/watch?v=NQDFUHiTngE
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VIDEO
EQUIV
equivalence-for-turing-machines-is
Here we show that the EQ_TM problem is undecidable. Supposing that it were decidable, we show that the E_TM (emptiness) problem is decidable, but in reality it is not.
What is a Turing Machine? It is a state machine that has a set of states, input, tape alphabet, a start state, exactly one accept state, and exactly one reject state. See https://www.youtube.com/watch?v=j0bIxPqlYLE&ab_channel=EasyTheory for more details.
Easy Theory Website: https://www.easytheory.org
Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join
Donation (appears on streams): https://streamlabs.com/easytheory1/tip
Paypal: https://paypal.me/easytheory
Patreon: https://www.patreon.com/easytheory
Discord: https://discord.gg/SD4U3hs
#easytheory
Youtube Live Streaming (Sundays) - subscribe for when these occur.
Social Media:
Facebook Page: https://www.facebook.com/easytheory/
Facebook group: https://www.facebook.com/groups/easytheory/
Twitter: https://twitter.com/EasyTheory
Merch:
Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122
Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang
If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1
Gold Supporters: Micah Wood
Silver Supporters: Timmy Gy
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.
...
https://www.youtube.com/watch?v=LGbI-mnQKRo
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10 months ago
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video/mp4
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VIDEO
REGUL
regular-languages-closed-under-2
(Easy Theory is in 60 FPS now!) Here we look at the problem of showing that regular languages are closed under subsequences (i.e., a substring but doesn't have to be contiguous). The idea is to "eliminate" transitions in the initial DFA by introducing epsilon transitions for every existing transition. We also give a concrete example of this.
#easytheory #nfa #dfa #gate #gateconcept #theoryofcomputing #turingmachine #nfatoregex #cfg #pda #undecidable #ricestheorem
Contribute:
Patreon: https://www.patreon.com/easytheory
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(Youtube also)
Mixer: https://mixer.com/easytheory
Social Media:
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Twitter: https://twitter.com/EasyTheory
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Silver Supporters: Timmy Gy
Supporters: Yash Singhal
▶ADDITIONAL QUESTIONS◀
1. What about "supersequences"? (i.e., all strings that include some string w in L as a subsequence)
2. What about substrings?
▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com
▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught over 12 courses at Arizona State University, as well as Colgate University, including several sections of undergraduate theory.
▶ABOUT THIS CHANNEL◀
The theory of computation is perhaps the fundamental theory of computer science. It sets out to define, mathematically, what exactly computation is, what is feasible to solve using a computer, and also what is not possible to solve using a computer. The main objective is to define a computer mathematically, without the reliance on real-world computers, hardware or software, or the plethora of programming languages we have in use today. The notion of a Turing machine serves this purpose and defines what we believe is the crux of all computable functions.
This channel is also about weaker forms of computation, concentrating on two classes: regular languages and context-free languages. These two models help understand what we can do with restricted means of computation, and offer a rich theory using which you can hone your mathematical skills in reasoning with simple machines and the languages they define.
However, they are not sim
...
https://www.youtube.com/watch?v=2bPMta_92tI
Transaction
Created
10 months ago
Content Type
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video/mp4
English