Here we look at the quicksort algorithm, which is another sorting algorithm that uses recursion. The idea is that it is very similar to mergesort, but instead of recursively working on the two halves and then merging, we first "split" the array into two pieces, and *then* recursively work on the two pieces. The idea is to pick a "pivot" element, and to move all of the other elements to the "correct side" of this pivot element. The two halves are just the partition created by this pivot.
▶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=bDM-OCeynfQ
Here we address the question of what happens if a DFA has empty alphabet? And we give another proof of a famous result regarding the empty set star.
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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=K7wlqxlTTwo
Here we prove (and state) the pumping lemma for context-free languages (CFL), by observing a parse tree of a CFG in Chomsky Normal Form (CNF). The properties of the parse tree allow us to show that if the string generated is big enough, the longest root-to-leaf path in the parse tree must repeat a variable. We utilize this fact to generate more parse trees (i.e., more strings the CFG makes), and look at the properties of parts of this string.
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▶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=-UH9L2sJpPQ
Easy Theory Website: https://www.easytheory.org
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▶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=anwyKFYbLdQ
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▶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=4cswnw6ECyk
Here we derive a PDA for the infamous non-regular language {0^n 1^n : n at least 0}. We give some tips as well for how to solve other PDAs. The general aspect here is to "match" 0s with 1s by pushing 0s onto the stack, and popping the 1s in tandem. There is some complication with popping an empty stack, so we push a "dummy" character on the stack to detect when the matching has concluded to avoid this problem.
What is a pushdown automaton? It is a finite state machine, where on each transition, items can be pushed or popped off of a stack it has, which has unlimited height. See https://www.youtube.com/watch?v=Br44Zxv84-Q&ab_channel=EasyTheory for more details.
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▶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=SSJ0i-I2VXU
Here we solve GATE 2017 Set 2 question 30, which is solving a recurrence relation. The master theorem cannot be applied to this problem directly, but we use two "domain transformations" to convert the problem into one that can be solved using the master theorem. Then we solve the original problem after substituting the original input size back in.
Easy Theory Website: https://www.easytheory.org
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Youtube Live Streaming (Sundays) - subscribe for when these occur.
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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.
...
https://www.youtube.com/watch?v=kF0vtzMQC1o
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▶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, mos
...
https://www.youtube.com/watch?v=9Gc7DWcNVBs
Here we feed questions from a theoretical computer science exam I have given into ChatGPT, and analyze the output answers it gives.
Timeline:
0:00 - Intro
0:48 - Question 1 (True/False)
15:03 - Question 2
18:24 - Question 3
20:58 - Question 4
22:46 - Question 5
23:11 - Question 6
25:33 - Thoughts on ChatGPT
Easy Theory Website: https://www.easytheory.org
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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
▶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=iwcCMcBXWmg
Here we prove the runtime of the randomized quicksort algorithm using integrals (yes, from calculus). This algorithm picks its pivot randomly, so we have to analyze the algorithm's runtime on average, by taking all hypothetical splits of the pivot, and averaging their runtimes.
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.
Platinum: Micah Wood
Silver: Timmy Gy, Josh Hibschman, Patrik Keinonen, Travis Schnider, and Tao Su
Easy Theory Website: https://www.easytheory.org
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Merch:
Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122
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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=ceG03n8ZLio