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)
9
� Theory of reasoning as a theory of adequate
combinations
Thought and representation: Hobbes (XVII century)
10
� 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
13
� Difference Engine
� Automatic calculation of logarithmic tables
� Analytic machine
� Memory warehouse
� Control system
More engineering: Babbage (XIX century)
14
� Another unsolvable problem
� Lack of financial support
� Analytic machine never practically built
Babbage’s ambitions
15
� 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
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