church turing thesis
TRANSCRIPT
Turing Machine Alan Turing has created Turing Machine Model. This model has computing of
general purpose computer.
The Turing Machine is a collection of following components:
M = (Q, ∑, Г, δ, , Δ or B, F)
1. Q is a finite set of states.
2. Г is finite set of external symbols.
3. ∑ is a finite set of input symbols.
4. Δ or b or B Є Г is a blank symbol majorly used as end marker for input.
5. δ is a transition or a mapping function.
Turing Machine
A Turing Machine (TM) is a theoretical symbol manipulating device.
A TM can simulate any computer algorithm (this is a simple formation of what came to be known as the Church-Turing Thesis, a version of Church’s Thesis)
The combination of the current symbol and the state determines what the device does next.
TMs are useful for simulating and understanding how computer CPUs work.
Church Turing Thesis In 1936, Alonzo Church created a method for defining functions
called the λ-calculus. Within λ-calculus, he defined an encoding of the natural numbers called the Church numerals.
Also in 1936, before learning of Church's work, Alan Turing created a theoretical model for machines, now called Turing machines, that could carry out calculations from inputs by manipulating symbols on a tape.
Church Turing Thesis
A Turing machine is an abstract representation of a computing device.
It is more like a computer hardware than a computer software.
LCMs [Logical Computing Machines: Turing’s expression for Turing machines] were first proposed by Alan Turing, in an attempt to give a mathematically precise definition of "algorithm" or "mechanical procedure".
Church Turing Thesis
The Church-Turing thesis concerns an effective or mechanical method in logic and mathematics.
A method, M, is called ‘effective’ or ‘mechanical’ just in case:
M is set out in terms of a finite number of exact instructions (each instruction being expressed by means of a finite number of symbols);
M will, if carried out without error, always produce the desired result in a finite number of steps;
Church Turing Thesis
M can (in practice or in principle) be carried out by a human being unaided by any machinery except for paper and pencil;
M demands no insight or ingenuity on the part of the human being carrying it out.
They gave an hypothesis which means proposing certain facts.
The Church’s hypothesis or Church’s turing thesis can be stated as:
Church Turing Thesis
The assumption that the intuitive notion of computable functions can be identified with partial recursive functions.
This statement was first formulated by Alonzo Church in the 1930s and is usually referred to as Church’s thesis, or the Church-Turing thesis.
However, this hypothesis cannot be proved.
Church Turing Thesis
The computability of recursive functions is based on following assumptions:
1. Each elementary function is computable.
2. Let f be the computable function & g be the another function which can be obtained by applying the elementary operation to f, then g becomes a computable function.
3. Any function becomes computable if it is obtained by rule 1 & 2.
Example
Construct a TM for language consisting of strings having any number of 0’s and only even numbers of 1’s over the input set ∑ = {0,1} .
The FSM can be draw as:
𝑞0 𝑞1
0 0
1
1
Example
Now the same idea can be used to draw TM.
𝑞1
(0 ,0 ,R)
(0 ,0 ,R)
(1,1 ,𝑅)
(
𝑠𝑡𝑎𝑟𝑡
h𝑎𝑙𝑡(Δ, Δ ,𝐿)
Example
Let us simulate the above TM for the input 110101 which has even number of 1’s.
Thus this input is accepted by TM.
Thank YouThe End.