Understanding Alan Turing and his Scientific Legacy - PowerPoint Presentation

Slide1

Understanding Alan Turing and his Scientific Legacy

1912-1954Slide2

Mathematical Agenda set by Hilbert

Requirements for the solution of a mathematical problem

It shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must be exactly formulated.Slide3

Whitehead and Russell

Principia Mathematica 2008

Formalized Mathematical Logic

Developed

H

igher Order Logic

L

aid the foundation of Type TheorySlide4

Propositional Logic

Theory of declarative sentences that combine Boolean variables using Boolean connectives.

If monsoon fails then there will be drought.

P: monsoon fails

Q: there will be drought

P

Q

 Slide5

First Order Logic (FOL)

Sentences in FOL contain predicates (functions/relations), quantifiers in addition to symbols permitted in propositional logic.

You can fool some of the people all of the time

Canfool

(

p,t

): you can fool person

p

at time t

 Slide6

One more example

Symmetric Graph

 Slide7

Second Order Logic

Has in addition to notations of FOL has quantifiers with propositional or functional variables as operator variables.

))

Is satisfied when

P

(

x

) is true for the set of even numbers.

 Slide8

Logic for Arithmetic

Arithmetic formulae can be described in sentences of FOL which has functions for addition and multiplication.

Together with the axioms of number theory we have a logical system defining arithmetic .Slide9

Godel’s Theorems 1931

Incompleteness of Arithmetic: There exists

no algorithm with the help of which using the axioms of number theory we can derive precisely the valid sentences of arithmetic

.

Undecidability

of Arithmetic:

There exists no algorithm by the help of which we can decide for every arithmetical sentence in finitely many steps whether it is valid.Slide10

What was Turing’s Agenda

To settle the Entescheidungs Problem (decision problem for FOL)

On Computable Numbers , with an Application to the

Enscheidungs

-Problem,

Proc. London Math. Soc., Ser. 2-42,

230-65.Slide11

Turing’s A Machine

All arguments which can be given are bound to be, fundamentally, appeals to intuition….and for this reason rather unsatisfactorily mathematically….. Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child’s arithmetic book. In elementary arithmetic, 2-dimensional character of the paper is sometimes used. But such a use is alwaysSlide12

Turing’s A machine: cont.

avoidable, and I think it will be agreed that 2-dimensional character of paper is no essential of computation. I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed may be finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrary small extent…It is always possible to use sequencesSlide13

Turing’s A machine: cont.

of symbols in the place of single symbols…..The difference from our point of view between the single and compound symbols, if they are too lengthy, canot be observed at a glance……We cannot tell at a glance whether 999999999 and 9999999999 are the same.

T

he

behaviour

of the computer at any moment by the symbols he is observing, and his “state of mind” at that moment. We may Slide14

Turing’s A machine: cont.

suppose that there is a bound B to the number of symbols on squares which the computer can observe at any moment. If he wishes to use more, he must use successive observations. We will also suppose that the number of states of mind which need to be taken into account is finite. The reasons for this are of the same character as those which restrict the number of symbols…..Let us imagine that the operations Slide15

Turing’s A machine: cont.

performed by the computer are split up into “simple operations”, which are so elementary that it is not easy to imagine them further divided. Every such operation consists of some change of the physical system consisting of the computer and his tape. We know the state of the system if we know the sequence of symbols on the tape, which of those are observed by the Slide16

Turing’s A machine: cont.

computer (possibly with a special order), and the state of mind of the computer. We may suppose that in a simple operation not more than one symbol is altered, [and]…without loss of generality assume that the squares whose symbols are changed are always “observed” squares.

Besides these changes of symbols, the simple operations must include changes of distribution of observed squares. The newSlide17

Turing’s A machine: cont.

observed squares must be immediately recognisable by the computer… Let us say that each of the new observed squares is within

L

squares of an immediately previously observed square.

The simple operations must therefore include:

(a) Changes of the symbol on one of the observed squares.

(b) Changes of one of the squares observed to Slide18

Turing’s A machine: cont.

another square within L

squares of one of the previously observed square.

It may be that some of these changes necessarily involve a change of state of mind… The operation actually performed is determined …by the state of mind of the computer and the observed symbols. In particular they determine the state of the mind of the computer after the operation is carried out.

We may now construct a machine to do the work of this computer…….Slide19

Universal Turing Machine

There exists a Turing machine which when given a coded description of any Turing machine T

and the data

x

on which

T

is supposed to work will output what

T

will output on input x.Slide20

Turing showed

There exists no general procedure by the help of which we can determine in finitely many steps, for any given formula of FOL whether or not the formula is valid.Slide21

Common Knowledge about Turing’s Work

Code Breaking: The Enigma Machine

Artificial Intelligence: Turing Test

Stored Program ComputerSlide22

Turing’s Contributions to Biology

Morphogenesis: Biological process that causes an organism to develop its shape.

In “The Chemical Basis of Morphogenesis”

T

uring laid the mathematical foundation of reaction-diffusion processes that enable stripes, spots, spirals to arise out of homogeneous uniform state.Slide23

Morphogen

– Gradient Model with Two Non-interacting Chemicals

S.

Miyazama

/ScienceSlide24

Turing Patterns on Thin Slabs of Gel

D Virgil, H.

Swinney

, University of Texas

A

ustin 1992Slide25

Turing Patterns in Seashells

Seashells from

Bishougai

-HP, simulations from D. Fowler and H.

Meinhardt

/ScienceSlide26

Turing Patterns around eyes of Popper Fish

Fish by Massimo

B

oyer, simulations from A.R. Sanders et al.Slide27

Turing Patterns in Zebra Fish (a Model Organisation

)

In the leftmost two columns are photographs of juvenile and adult zebra fish marking. In the other two are Turing pattern simulations, developing over time (

K

ondo and

Nakamusu

PNAS)Slide28

Turing Patterns in Cells in Dictyostelium

, or a Slime Mold

Turing patterns can involve not just chemicals, but large complex systems in which each unit-for example a cell - is distributed like molecules of pigment. Image NIHSlide29

Turing’s Legacy

Nondeterminism Complexity of Computation

Cryptography

Notion of Universality, and

The Ultimate Computer: The Internet Slide30

Readings

Alan Turing: The Enigma, by Andrew Hodges

Alan M. Turing, by Sara Turing

Alan Turing: His Work and Impact, by S.

B

arry Cooper (ed.) and J. van

Leeuvan

(ed.)

Turing, by Andrew HodgesThe Universal Computer: The Road from Leibniz to Turing, by Martin DavisTuring Evolved, by

D

avid

KitsonTuring (A Novel about Computation), by Christos H. Papadimitriou