The universal computer

The road from Leibniz to Turing

Instructor: Viola Schiaffonati

March, 30th 2017

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� Processing information in an automatic way

� The birth of computer science (1930s)

� The birth of computer engineering (1940s)

� Different research traditions and their roles

� Thinking and calculating

� Thinking and reasoning

� Calculating, programming, and implementing

Thinking, calculating, programming

3The construction of knowledge

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Constant human tendency to represent

Different research traditions

5Representing ‘external’ aspects

� Heron of Alexandria (I century

A.D.)

� Semiautomatic machines

(autòmatha)

� Water-powered and steam-

powered

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� Raimon Lull (1235-1315)

� Ars Magna: general

principles of human

knowledge represented by

numbers and symbols

composed to obtain further

knowledge

� Ars inveniendi veritatem

Representing ‘internal’ aspects’

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� Discourse on method

� Foundations of knowledge and reduction of any form of

knowledge to scientific rigor

«To divide each of the difficulties under examination into as

many parts as possible, and as might be necessary for its

adequate solution»

(Decartes, second rule)

Reducing for knowing: Decartes (XVII century)

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� First mechanical calculator

� Prodigy and not instrument

Calculating numbers: Pascal (XVII century)

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� Theory of reasoning as a theory of adequate

combinations

Thought and representation: Hobbes (XVII century)

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� Project of mechanizing rationality

� Axiomatic-deductive system

� Characteristica universalis and calculus ratiocinator

Calculating thoughts: Leibniz (XVII century)

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«[...] if controversies were to arise, there would be no more need of

disputation between two philosophers than between two calculators. For

it would suffice for them to take their pencils in their hands and to sit

down at the abacus, and say to each other (and if they so wish also to a

friend called to help): Let us calculate.»

(Leibniz 1666)

Calculemus!

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� An unsolvable problem

� Adequate representation of knowledge

� No efficient characteristica universalis

� No calculus ratiocinator

Leibniz’s dream

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� Difference Engine

� Automatic calculation of logarithmic tables

� Analytic machine

� Memory warehouse

� Control system

More engineering: Babbage (XIX century)

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� Another unsolvable problem

� Lack of financial support

� Analytic machine never practically built

Babbage’s ambitions

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� Boole (1854): algebrization of logic

� Laws constituting the ‘mathematics’ of human

cognition

� Frege (1876): formal system (first order

logic), notion of proof

Leibniz: from dream to reality

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� Reduction of mathematics to logic (1893)

Frege’s dream

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� Russell’s antinomy (1902)

� Let R be the set of all sets which are

not members of themselves. Then R

is neither a member of itself nor not a

member of itself

A broken dream

18Frege’s integrity

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� David Hilbert (1862-1943)

� Hilbert program

� Formal foundations of mathematics in terms of axioms

Another way?

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� International congress of mathematics (1900)

� Logical decision problem

� Problem of finding a general mechanical procedure which,

for any formal axiom system and any formula, can decide

if the formula can be derived from the axioms in the

logical calculus

Entscheidungsproblem!

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� How to demonstrate that a procedure able to perform the requested task in an automatic way has not been invented yet and will not be invented in the future?

� The concept of mechanical process needs to be conceived in a precise and rigorous way

� The Turing Machine

On Computable Numbers (Turing 1936)

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� The notion of algorithm receives a satisfactory account only after Turing (1936) has introduced his machine model of a computer

� This model results from Turing’s analysis of the possible processes a human (‘the computer’) can go through while performing a calculation using paper and pencil applying rules from a given finite set

� The human computer follows the rules ‘blindly’, without using any insight or ingenuity

� Negative answer to the decision problem

Turing machine

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� Logical analysis of notions such as formal system and

formal proof (and also algorithm and computable

function)

� Incredible progress in the engineering of the electronic

components

Two traditions meeting for the first time (1940s)

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� Z3 (1943), Colossus (1943), Eniac (1945)

� No program in memory!

Computer engineering

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� Von Neumann in the USA, Newman in UK

� Manchester baby (1948): first electronic computer

with a program in memory

Turing in engineers hands

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� Davies, M. (2012) The Universal Computer: The Road

from Leibniz to Turing, Taylor & Francis Group

� Frixione, M., Palladino, D. (2004). Funzioni, macchine,

algoritmi. Introduzione alla teoria della computabilità.

Carocci

� McCorduck, P. (2004). Machines Who Think: 25th

anniversary edition. A. K. Peters.

References