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Computability

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Mino Basha

on 24 November 2014

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Transcript of Computability

Chapter 1
Computable Function
Sets

Functions
Relations & predicates
Logical notation
Algorithms, effective procedures
Unlimited Register machine
Decidable predicates
Khaled A.
Mina S.
Done By:-
Computability
Titles:-
Sets
Functions
Relations & predicates
logical notation
Algorithms, effective procedures
The URM
URM - computable fuctions
Decidable predicates and problems
Computability on other domains
A
x
x
A
x is member of set A
A
B
A is subset of B
x
A
B
x is intersected between A and B.
A
B = x
x
A & B
A
B
x
x
x
A
OR
x
B
OR
Both A & B
A\B

A-B
The difference between A and B
OR
the relative complement
A's complement
What is the Function?
Dom(A) and Ran(B)

Domain: {1, 2, 3, 4}
Codomain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Range: {3, 5, 7, 9}
Predicates
if
2<3 is true (holds)
5>9 is false (doesn't hold)
Equivalence order
When?
Example on x,y,z A
Reflexivity
R(x,x);
Symmetry
if R(x,y) then R(y,x);
Transitivity
if R(x,y)&R(y,z)thenR(x,z);
Partial order
When?
Ir-reflexivity
not R(x,x);
Transitivity
if R(x,y)&R(y,z) then R(x,z);
Example on x,y,z A
Algorithms
mechanical rule or automatic method or program for performing some mathematical operations
What is it?
Algorithms Types
given n , finding the nth prime number.
differentiating a polynomial.
finding the highest common factor.
if x is multiple of y. [mod]
What Algorithms look like!
URM
What is it?
Abstraction or idealization of computing device
It's characteristics:-

Register
program
instructions
operations
termination
Registers
A URM has a number of locations called Registers which can store nutral numbers {0,1,2,3....}

URM Programs use finite number of registers
Registers are refereed BY R ex: R1 ,R2 ...
the number held bu registers is reffered by r ex: r1 ,r2 ,r3 ...
The Registers are unlimited in the following two Senses:
Although URM uses finit number of Registers, there is no actual bound on how many a program can actually use.
There is no actual bound on the size of the natural numbers that maybe stored in a Register
Program
Finite list of instructions
Instructions are writen in a fixed order
For historical reasons the number of instructions is called line numbers.
Basic Instructions :)
PS: At the End of teh Running the result is found in register R1
Lets Solve an Example
any questions!
Example on Basic Instructions
Example :-
If x is multiple of y
ex: if 18 is multiple of 2 --->1 OR yes
Decidable Predicate
When?
If
function is computable
then it may be decidable or not
BUT
If
function is not computable
then it is not decidable
Ref:
Theory of Computation"St. Joseph’s College of Engineering & Technology Palai"
Elisa Elshamy:How To Think About Algorithms
Nigel Cutland: Computability
John C. Shepherdson, Howard E. Sturgis: Computability of Recursive Functions.
A.I. Mal'tsev:Algorithms and recursive functions
Rockafellar, R. T. : Convex Analysis
Full transcript