Turing Machine Variants -- Intro Theory of Computation Help Session #12
Here we go over various Turing Machine variants and show that they are all equivalent to the standard Turing Machine model. This was recorded on 10 April 2017.
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
▶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=-k_sXZumBqA
Here we look at the language of strings that are ternary (base 3) multiples of 5. The idea is to make a DFA for all possibilities, which means the remainder with respect to 5. Then, all we do is relate each of the states to the others depending on whether we see a 0, 1, or 2. The pro-tip here is that one only needs to look at a single one of the transitions, and immediately the other transitions are determined. Another pro-tip is that this technique works in general, not just for base 3 and not just for multiples of 5!
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▶ADDITIONAL QUESTIONS◀
1. How many transitions would a base-k multiple of n DFA have?
2. Is this the smallest DFA possible for this language?
3. Is this DFA unique?
▶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.
...
https://www.youtube.com/watch?v=iuHbMH7tYr4
Here we introduce the notion of polynomial space and PSPACE-completeness, which is the same as NP-completeness except the language has to be in PSPACE. We also show that NP is a subset of PSPACE because the SAT problem can be solved in polynomial space (actually linear), but if SAT is PSPACE-complete, then NP = PSPACE, an open problem.
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
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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=n-arD88rCOw
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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, 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=h07oJjP40gc
Here we look at one of the most important problems with regards to proving statements via induction: Sipser's problem 0.12, which asks to show why a proposed proof of a statement about "all horses have the same color" is incorrect. It really gets at the heart of why proofs are important, and why needing to justify every step in a proof is also important.
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
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=9Q4R1ROXrg0
Here we show that Busy Beaver numbers are undecidable.
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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=yCl0SGA2hg8
Here we look at Tom Scott's "sorting algorithms and big-O notation" video and assess its accuracy, as well as give some additional insight. His video is here: https://www.youtube.com/watch?v=RGuJga2Gl_k&ab_channel=TomScott. He goes over algorithms to sort items, and why certain algorithms are "slower" than others in a certain sense. The algorithms he mentions are: bubble sort, insertion sort, quicksort, and bogosort (like, why even?), and gives animations to each. I mention several (more esoteric) others. Big-O is important, but only gives a "mostly true" picture of how algorithms behave compared to each other.
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
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If you need help, here are some relevant links:
Big-O: https://www.youtube.com/watch?v=rP5peOkXsbY&ab_channel=EasyTheory
Algorithms Playlist: https://www.youtube.com/watch?v=sT3ImMy0fRE&list=PLylTVsqZiRXNArM--1hbcrlJyZn7EpNJU&index=1&ab_channel=EasyTheory
--------------------------------------------------------------------------------------------------------------
Easy Theory Website: https://www.easytheory.org
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Merch:
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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=YvBw_M6Dboc
Here we give a (faulty) proof that all languages are regular. We use the basic notions and concepts related to regular languages to give a "proof" of this fact. Can you spot the error(s)?
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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
▶ADDITIONAL QUESTIONS◀
1. Can you prove something about the resulting NFA/DFA if you were to construct this using the product construction directly?
▶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
...
https://www.youtube.com/watch?v=gPyBg6FN1pE
Here we look at a GATE 2019 problem for Exam Question Monday, about determining which of four languages is not regular. Two of the languages are Prefixes and Suffixes of a regular language, one is the concatenation of L and L^R, and the last is even-length palindromes over L. We show that regular languages are closed under prefix and suffix, as well as reversal, which yields that the last language is not regular.
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▶ADDITIONAL QUESTIONS◀
1. Is {ww^R : w in {a,b}*} deterministic context free?
2. For what subsets L (not necessarily regular) is {ww^R : w in L} a regular language?
▶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 an
...
https://www.youtube.com/watch?v=9Sl109TmSoU
Here we show that regular languages are closed under the "avoids" operation (Sipser 1.70 solution). This operation is all the strings in A that don't have any substring in B. This isn't the same as all the strings in A not in B. The idea is to build up what the definition of "avoids" means, by first considering all strings that *do* have some substring in B, and then take those strings away from A.
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▶ADDITIONAL QUESTIONS◀
1. How about subsequence avoidance?
▶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 two different universities, including several sections of undergraduate and graduate theory-level classes.
...
https://www.youtube.com/watch?v=y86jpV5tn_0