The Church-Turing Thesis Lecture by H. Munoz-Avila

We have the Notion of Turing Machines Transitions: ((p,  ),(q,R)) Here is a Turing machine “in action”

The Church-Turing Thesis Algorithms  Turing Machines

Sounds Unbelievable We are so used to programming scripting languages Things like (in tolua, a variant of Lua that allows C++ constructs):

But Actually it is not so “unbelievable”

Can be translated into C++ (not an actual translation of the code above)

But Actually it is not so “unbelievable” Can be translated into C (not an actual translation of the code above)

But Actually it is not so “unbelievable” can be translated into C kernel (not an actual translation of the code above)

But Actually it is not so “unbelievable” can be translated into Assembler (not an actual translation of the code above)

But Actually it is not so “unbelievable” Can be ran by the Von Neumann machine

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!

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 ago40 years ago So we have a “moving target”

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)

Three Equivalent Formalisms -calculus Recursive functions Turing machines

LISP (defun palindrome( L ) (cond ((null L) T ) ((equal (car L) (car (last L))) (palindrome (cdr (reverse (cdr L))))))) (inspired by -calculus)

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)

Turing Machine

C Program

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

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)

(any non-decidable problem)