Understanding Alan Turing and his Scientific Legacy 1912-1954.

Understanding Alan Turing and his Scientific Legacy

1912-1954

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.

Whitehead and Russell

Principia Mathematica 2008

Formalized Mathematical LogicDeveloped Higher Order LogicLaid the foundation of Type Theory

Propositional Logic

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

If monsoon fails then there will be drought. P: monsoon failsQ: there will be drought

P Q

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 timeCanfool(p,t): you can fool person p at time t

One more example

Symmetric Graph

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.

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 .

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.

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.

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 always

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 sequences

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.

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

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

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

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 new

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

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…….

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.

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.

Common Knowledge about Turing’s Work

Code Breaking: The Enigma MachineArtificial Intelligence: Turing Test Stored Program Computer

Turing’s Contributions to Biology

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

In “The Chemical Basis of Morphogenesis” Turing laid the mathematical foundation of reaction-diffusion processes that enable stripes, spots, spirals to arise out of homogeneous uniform state.

Morphogen – Gradient Model with Two Non-interacting Chemicals

S. Miyazama/Science

Turing Patterns on Thin Slabs of GelD Virgil, H. Swinney, University of Texas Austin

1992

Turing Patterns in SeashellsSeashells from Bishougai-HP, simulations from D. Fowler and H.

Meinhardt/Science

Turing Patterns around eyes of Popper FishFish by Massimo Boyer, simulations from A.R. Sanders et al.

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 (Kondo and Nakamusu PNAS)

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 NIH

Turing’s Legacy

NondeterminismComplexity of ComputationCryptographyNotion of Universality, and The Ultimate Computer: The Internet

Readings

Alan Turing: The Enigma, by Andrew HodgesAlan M. Turing, by Sara TuringAlan Turing: His Work and Impact, by S. Barry Cooper (ed.) and J. van Leeuvan (ed.)Turing, by Andrew HodgesThe Universal Computer: The Road from Leibniz to Turing, by Martin DavisTuring Evolved, by David KitsonTuring (A Novel about Computation), by Christos H. Papadimitriou