What Computers Can't Compute Dr Nick Benton
Queens' College &
Microsoft Research
nick@microsoft.com

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David Hilbert
(1862-1943) Hilbert's programme:
To establish the foundations of mathematics, in particular by clarifying and justifying use of the infinite: ``The definitive clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences but for the honour of human understanding itself.'' Aimed to reconstitute infinitistic mathematics in terms of a formal system which could be proved (finitistically) consistent, complete and decidable.

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Consistent: It should be impossible to derive a contradiction (such as 1=2).
Complete: All true statements should be provable.
Decidable: There should be a (definite, finitary, terminating) procedure for deciding whether or not an arbitrary statement is provable. (The Entscheidungsproblem) There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.
Wir müssen wissen, wir werden wissen

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Bertrand Russell
(1872-1970) Alfred Whitehead
(1861-1947) Russell's paradox showed inconsistency of naive foundations such as Frege's: {X | XX}
"The set of sets which are not members of themselves"
Theory of Types and Principia Mathematica (1910,1912,1913)

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Kurt Gödel
(1906-1978) Uber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (1931)
Any sufficiently strong, consistent formal system must be
Incomplete
Unable to prove its own consistency

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Alan Turing
(1912-1954) On computable numbers with an application to the Entscheidungsproblem (1936)
Church, Kleene, Post

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x Turing's Model of a Mathematician Finite state brain
Finite alphabet of symbols
Infinite supply of notebooks

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The Turing Machine A A C G C T T G C 1 Replaces GC with TA

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The Turing Machine A A C G C T T G C 1 Replaces GC with TA

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The Turing Machine A A C G C T T G C 1 Replaces GC with TA

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The Turing Machine A A C G C T T G C 1 Replaces GC with TA

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The Turing Machine A A C G C T T G C 2 Replaces GC with TA

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The Turing Machine A A C G C T T G C 2 Replaces GC with TA

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The Turing Machine A A C G C T T G C 3 Replaces GC with TA

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The Turing Machine A A C G C T T G C 3 Replaces GC with TA

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The Turing Machine A A C T C T T G C 4 Replaces GC with TA

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The Turing Machine A A C T C T T G C 4 Replaces GC with TA

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The Turing Machine A A A T C T T G C 1 Replaces GC with TA

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Another example: Binary Addition

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So particular Turing Machine is specified by
Its alphabet
Its transition table
Each TM then defines a partial function from Tapes to Tapes.
Given a machine M and a tape T, there are two possible things
that can happen when we run machine M on input tape T:
EITHER the machine simply runs forever without stopping, OR
The machine eventually stops with an output tape T’ M T T’

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Since we can represent natural numbers on the tape (using decimal, binary, roman numerals, whatever), we can write TMs to compute (partial) functions from ℕ to ℕ. The word function has at least two senses.
Mathematical. A function is a set of pairs, giving all the (argument, result) combinations together. So the ‘square’ function, for example, looks like {(0,0), (1,1), (2,4), (3,9),…}.
Computational. A function is a procedure, method, algorithm, operation, formula for computing the result from the argument. There’s some kind of causal relation between input and output.
Investigating the relationship between these two views (the denotational and the operational) is central to theoretical computer science.

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Not all mathematical functions
are computable by a Turing machine. (we’ll see an example soon)
But all other notions of computation which people have invented turn out to give exactly the same set of computable functions.
That this is the essential meaning of ‘computable’ is known as the Church-Turing Hypothesis, though this is clearly not a rigorous notion.

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Turing’s first result.
Since a particular TM is specified by a finite amount of information, we can encode it as a finite string of symbols in some alphabet (equivalently as a natural number).
We’ll write M for the code of machine M. (the details of the coding scheme are unimportant…)
But we can write M onto the tape, so one TM can take as input the code of another one (or even itself). There is a Universal Turing Machine, U

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If and only if For any machine M and tapes T and T’

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Turing’s second result The ‘Halting Problem’ is undecidable There is NO machine H which computes whether or not any other machine will halt on a given input: M T H M,T YES NO iff iff

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Proof of the undecidability of the halting problem
We’ll assume that there is such a machine, H, and derive a contradiction. First, we define a copy machine (this is easy):

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COPY H Now modify H so that it goes into a loop instead of printing ‘yes’ …and call the resulting machine H ’ …plug the copy machine into the front What happens when we feed H’ its own code?

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COPY H H’ H’, H’ Machine H’ terminates on input H’ if and only if
The modified H terminates on input H’ ,H’ , which happens if and only if
The original H prints ‘no’ on input H’ ,H’ , which happens if and only if
Machine H’ does not terminate on input H’

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COPY H H’ H’, H’ Hence our original assumption, that H exists, must be false.

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Corollaries of Turing’s result It’s uncomputable whether an arbitrary machine halts when given an empty initial tape.
In fact, all ‘interesting’ properties of computer programs are uncomputable. For example
It’s impossible to write a perfect virus checker.
The `full employment theorem’ for compiler writers.
The Entscheidungsproblem is unsolvable:
Roughly, because ‘Turing machine M halts on tape T’ is expressible as a logical formula which, if true, will be provable (because it only requires a finite demonstration). Hence if there were a decision procedure for the provability of arbitrary propositions, there’d be one for the halting problem.
This is the ‘full employment theorem’ for mathematicians.

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Further developments of Turing’s work Complexity theory.
From ‘what can we compute?’ to `how fast can we compute?’. Turing machines are still a basic concept in this huge area of computer science.
Higher-type recursion theory and synthetic domain theory.
Once we add types, the notion of computable becomes rather more subtle. Developments in this area have led to ‘mathematical universes’ in which computability is built-in from the start, and these have been proposed as good places in which to model and reason about computer programs.

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Philosophy and Artificial Intelligence
Implications of Gödel’s and Turing’s work for the philosophy of mind and the possibility of ‘thinking machines’ are still hotly debated. See for example Roger Penrose’s The Emperor’s New Mind and Shadows of the Mind.
Really crazy stuff…
DNA and restriction enzyme implementation of TMs
It has been suggested that one could compute the uncomputable by sending computers through wormholes in space so that they run for an infinite amount of time in a finite amount of the observer’s time . Other developments

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One can encode the propositions and rules of inference of a formal system as natural numbers, so that statements about the system become statements about arithmetic.
Thus, if the system is sufficiently powerful to prove things about arithmetic, it can talk (indirectly) about itself.
The key idea is then to construct a proposition P which, under this interpretation, asserts P is not provable Then P must be true (for if P were false, P would be provable and hence, by consistency, true - a contradiction!)
So P is true and unprovable, i.e. the system is incomplete. Proof of Gödel's Incompleteness Theorem.

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Further Reading Popular
Alice’s Adventures in Wonderland and Through the Looking Glass (And What Alice Found There). Lewis Carroll.
Godel, Escher, Bach: an Eternal Golden Braid. Douglas R. Hofstadter (Basic Books,1979)
Alan Turing: the Enigma. Andrew Hodges (1983)
http://www.turing.org.uk/
To Mock a Mockingbird and What is the Name of this Book?. Raymond Smullyan
Academic
The Undecidable: Basic papers on undecidable propositions, unsolvable problems and computable functions. Martin Davis (Raven Press,1965)
From Frege to Gödel: A Sourcebook in Mathematical Logic. J. van Heijenoort (Harvard,1967)

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