The Church-Turing Thesis Lecture by H. Munoz-Avila.
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Transcript of The Church-Turing Thesis Lecture by H. Munoz-Avila.
![Page 1: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/1.jpg)
The Church-Turing Thesis
Lecture by H. Munoz-Avila
![Page 2: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/2.jpg)
We have the Notion of Turing Machines
• Transitions: ((p,),(q,R))
• Here is a Turing machine “in action”• http://www.youtube.com/watch?v=FTSAiF9AHN4
![Page 3: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/3.jpg)
The Church-Turing Thesis
Algorithms Turing Machines
![Page 4: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/4.jpg)
Sounds Unbelievable
• We are so used to programming scripting languages• Things like (in tolua, a variant of Lua that allows C++
constructs):
![Page 5: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/5.jpg)
But Actually it is not so “unbelievable”
![Page 6: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/6.jpg)
But Actually it is not so “unbelievable”
Can be translated into C++
(not an actual translation of the code above)
![Page 7: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/7.jpg)
But Actually it is not so “unbelievable”
Can be translated into C
(not an actual translation of the code above)
![Page 8: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/8.jpg)
But Actually it is not so “unbelievable”
can be translated into C kernel
(not an actual translation of the code above)
![Page 9: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/9.jpg)
But Actually it is not so “unbelievable”
can be translated into Assembler
(not an actual translation of the code above)
![Page 10: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/10.jpg)
But Actually it is not so “unbelievable”
Can be ran by the Von Neumann machine
![Page 11: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/11.jpg)
But Actually it is not so “unbelievable”
We have the same basic elements in Turing Machines:
• We can do arithmetic• Control• And a lot of memory!
![Page 12: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/12.jpg)
So Why is it Called a “Thesis”
• There is no precise notion for “algorithm”
• Of course there is a precise notion for a C++ program
• But how does programs will look like 40 years from now?– Think how programs looked like 40 years ago
• So we have a “moving target”
![Page 13: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/13.jpg)
A Bit of History
• Das Entscheidungsproblem (Hilbert, 1928)– Is there a decider for First-order logic?
• Vollständigkeit des Logikkalküls (Gödel, 1929)
• Church developed -calculus and proved that the Entscheidungsproblem cannot be solved (1936)– impossible to prove that two -calculus
are equivalent• Turing proved that the Halting problem can
be reduced to the Entscheidungsproblem – And hence it cannot be solved (1936)
![Page 14: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/14.jpg)
Three Equivalent Formalisms
• -calculus
• Recursive functions
• Turing machines
![Page 15: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/15.jpg)
LISP
(defun palindrome( L )
(cond
((null L) T )
((equal (car L) (car (last L)))
(palindrome (cdr (reverse (cdr L)))))))
(inspired by -calculus)
![Page 16: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/16.jpg)
Prolog
palCheck(List) :- reverse(List,List).
reverse(L1,L2) :- rev(L1,[],L2).
rev([],L,L).
rev([H|L],L2,L3) :- rev(L,[H|L2],L3).
(inspired by recursive functions)
![Page 17: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/17.jpg)
Turing Machine
![Page 18: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/18.jpg)
C Program
![Page 19: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/19.jpg)
Not an Accident
• Any algorithm written in any one of these languages can be written in any of the other ones
• Researchers sometime refer to programming languages having this property as Turing-complete
• Examples of Turing-complete languages: C, C++, java, LISP, Prolog, …
• Examples that are not: Context-free languages, “STRIPS” planning, LOOP
![Page 20: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/20.jpg)
What Comes Next
• Study some difficult problems that are in fact decidable
• Study some harder problems that are:
1. Not decidable but recognizable
2. Problems that are not even recognizable
3. By the Church-Turing thesis, no algorithm exists that solves problems in (1) and (2)
![Page 21: The Church-Turing Thesis Lecture by H. Munoz-Avila.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e9e5503460f94b9f244/html5/thumbnails/21.jpg)
(any non-decidable problem)