# robot vehicle

**Topics:**Theory of computation, Turing machine, Computer

**Pages:**5 (698 words)

**Published:**April 18, 2014

SCHOOL OF COMPUTER SCIENCE, MATHEMATHICS AND INFORMATION TECHNOLOGY

DEPARMENT: COMPUTER SCIENCE A1

GROUP 3

NAMES OF GROUP MEMBERSMATRIC NUMBER

OSO ABAYOMI OMOBOLANLE0907 IT 00989

PRESENTATION TOPIC: MU RECURSIVE FUCNTION

LECTURER NAME: MR OSHIBOGUN

RECURSIVE FUNCTION

Recursive functions are built up from basic functions by some operations. The successor function

Lets get very primitive. Suppose we have 0 defined and want to build the nonnegative integers and entire number system. We define the successor operation S(x) that takes a number x to its successor X+1. This gives one the nonnegative integers N0= {0,1,2….} We formalize the above process primitive recursive functions are built up from three basic functions using two operations these basic functions are: 1. Zero. Z(x)=0

2. Successor. S(x)= x+1.

3. Projection. A projection function selects out one of the argument. Specially P1(x,y)=x and P2(x,y)=y The composition operation

There are two operations that make new functions from old: composition and primitive recursion Composition replaces the arguments of a function by another. For example, one can define a function f by

F(x,y)=g(h1(x,y), h2(x,y))

Where one supplies the functions g1, g2 and h.

Primitive recursion

A typical use of primitive recursion has the following form

Where one supplies the function g1, g2 and h.

For example, in the case of addition, the h is the successor function of the projection of the 2nd argument. A special case of primitive recursion is for some constant number K:

A function is said to be recursive if it can be from by:

Composition function

Primitive function

Application of unbounded minimization to regular functions

DEFINITION

The family of recursive function is defined as follows:

i. The successor, zero and projection function are urecursive. ii. If h is an n-variable u recursive function and g1,. . . .gn are k-variable u recursive function, then f=h (g1,. . . .gn )is recursive. iii. If g and h are n and n+2-variable u recursive functions, then the function f defined from g and h by primitive recursion is u recursive. iv. If p() is a total u recursive predicate, then f () is u recursive. v. A function is u recursive only if it can be obtained from i) by a finite number of application of the rules n ii), iii) and iv). THEOREM

A function is computable by a turing machine if it is μ-recursive. Proof sketch: by construction.

a. Since composition is strict it suffices to show that the turing machine implementation if the u operator exists for the “f” direction. Since this is a search procedure it is clear that a machine can be built for that. b. For the converse we provide a procedure to encode any turing machine as a function based on an enumeration of all possible configurations. Since enumeration of all possible and countable, this is possible and therefore it is possible to construct a function representing a turing machine.

RELATIONSHIP BETWEEN TURNING MACHINE AND A COMPUTATABLE FUNCTION Here we investigate the relationship between turing machines and computable functions. For convenience we will restrict ourselves to only look at numeric computations, this does not reflect any loss of generality since all computational problems can be encoded as number. Kurt Godel used this fact in his famous incompleteness proof. We will show that,

The functions computable by a Turing machine are exactly the u recursive functions. U recursive functions were developed by Godel and Stephen kleene. So between Turing, Church, Godel, and Kleene we obtain the following equivalence relation

Algorithms Turing machines Recursive function calculus In order to work towards a proof of this equivalence we start with primitive recursive function.

Summary

It is not hard to believe that all such functions can be computed by some turning machine. What is a much...

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