alan turing s theory of computation

Introduction Computability Decidability Computable numbers

Alan Turing’s theory of computation

Joel David HamkinsProfessor of Logic

Sir Peter Strawson Fellow

University of OxfordUniversity College

Oxford Cambridge Club, London6 June 2019

Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford

Alan Turing (1912–1954)

Sculpture by Stephen Kettle

Enigma

German ‘Enigma’ encryption device

The Bombe

Bombe device, Bletchley Park

On computable numbers, 1936

Turing’s 1936 paper, “On computable numbers...”It is difficult to overstate the importance of Turing’s paper,written while he was a student at Cambridge.

The paper introduced profound, fundamental ideas oncomputability.

Defined Turing machinesProved existence of universal computersIdentified undecidability phenomenonSolved Godel’s entscheidungsproblemIntroduced the computable numbers

Turing’s ideas laid the foundation for the contemporarycomputer era.

Computability

What does it mean to say that a function is computable?

Is there a hierarchy of computational power?

Can every mathematical question be solved in principle bycomputation?

Turing’s computability concept

Turing reflected philosophically on the nature of computation.

In that era, the word ‘computer’ referred not to a machine, butto a person, or more specifically, to an occupation.

Some firms had whole rooms full of computers, the ‘computerroom’, filled with people hired as computers and tasked withvarious computational duties, often in finance or engineering.

In old photos, you can see the computers—mostlywomen—sitting at big wooden desks, with pencils and asufficient supply of paper. They would perform theircomputations by writing on the paper, of course, according tovarious definite computational procedures.

What ‘computers’ do

Turing aimed to model an idealized form of computationalprocesses.

He reflected that ‘computers’ (that is, people working ascomputers) work with pencil and paper, making various marksaccording to a computational procedure; perhaps make markson paper in front of them, or look back at earlier computationalmarks.

Eventually, the procedure may come to completion and theygive an output.

Nature of computation

Turing realized that some simplifying assumptions do notfundamental affect the nature of computation.

We may assume marks appear in a grid of cells.

May assume each cell has only 0 or 1.Two dimensions not important; assume a line of cells.May assume computer has limited memory.Rudimentary actions determined by current ‘state’ of mind.

We may assume marks appear in a grid of cells.May assume each cell has only 0 or 1.

Two dimensions not important; assume a line of cells.May assume computer has limited memory.Rudimentary actions determined by current ‘state’ of mind.

We may assume marks appear in a grid of cells.May assume each cell has only 0 or 1.Two dimensions not important; assume a line of cells.

May assume computer has limited memory.Rudimentary actions determined by current ‘state’ of mind.

We may assume marks appear in a grid of cells.May assume each cell has only 0 or 1.Two dimensions not important; assume a line of cells.May assume computer has limited memory.

Rudimentary actions determined by current ‘state’ of mind.

We may assume marks appear in a grid of cells.May assume each cell has only 0 or 1.Two dimensions not important; assume a line of cells.May assume computer has limited memory.Rudimentary actions determined by current ‘state’ of mind.

Turing machinesThus, Turing derived his machine concept of computation.

1 1 0 0 0 · · ·

A Turing machine has an infinite paper tape, divided into cells,which accommodate 0 and 1.

The head, at any moment in one of finitely many states, readsand writes on the tape, moving according to the rigidinstructions of a program.

(q,a) 7→ (r ,b,d).

When in state q reading symbol a, then change to stater , write symbol b, and move one cell in direction d.

Turing machinesThus, Turing derived his machine concept of computation.

1 1 0 0 0 · · ·

A Turing machine has an infinite paper tape, divided into cells,which accommodate 0 and 1.

The head, at any moment in one of finitely many states, readsand writes on the tape, moving according to the rigidinstructions of a program.

(q,a) 7→ (r ,b,d).

When in state q reading symbol a, then change to stater , write symbol b, and move one cell in direction d.

Computability

Every program computes a corresponding function: on input n,the program runs, and if it eventually reaches the halt state, theoutput is f (n).

A function is computable if there is a program that computes it.

Computability

Every program computes a corresponding function: on input n,the program runs, and if it eventually reaches the halt state, theoutput is f (n).

A function is computable if there is a program that computes it.

Decidability

A set is computably decidable, if there is a computationalmethod to determine whether given inputs x are in the set ornot.

A set A is computably enumerable, in contrast, if there is acomputable algorithm that enumerates all and only theelements of A.

Question

Are these the same?

It turns out, by Turing’s arguments, that they are different.

Decidability

Question

Are these the same?

Decidability

Question

Are these the same?

Decidability

Question

Are these the same?

Puzzling example

Consider the decision problem: on input n, decide whetherthere are n consecutive 8s in the decimal expansion of π.

3.14159265358979323846264338327950288419 · · ·

Question

Is this decision problem computably decidable?

Naive attempt: on input n, begin to search the decimalexpansion of π. If you find n consecutive 8s, then say Yes.

...but when shall we say No?

Puzzling example

3.14159265358979323846264338327950288419 · · ·

Question

Puzzling example

3.14159265358979323846264338327950288419 · · ·

Question

Puzzling example

3.14159265358979323846264338327950288419 · · ·

Question

More sophisticated answer

Decision problem: are there n consecutive 8s in π?

Argue by cases.

Case 1. If arbitrarily long blocks of 8s appear in π, then theanswer is always Yes. And this is computable: just say Yes.

Case 2. Otherwise, there is some longest string of 8s, of somelength N. But now the answer is Yes if n ≤ N and otherwise No.For the particular (unknown but fixed) N, this also iscomputable.

So in any case the problem is computable. We just don’t knowwhich algorithm works.

Argue by cases.

Universal computerThe earliest computational devices each performed a single,specific computational task.

To change the functionality, you had to build or rebuild adifferent machine.

Turing realized that one might design a universal computer.

A universal computer could run any program, if you simply givethat program as input.

This was an enormous conceptual advance.

But also a case where a profound idea through familiaritybecomes banal. We all carry them around in our pockets now.

Mental capacity

The existence of universal Turing machines shows in principlethat one does not need increasingly large mental capacity inorder to undertake arbitrary computational tasks.

The universal computer has a fixed number of states, not verylarge.

The capacity for written memory suffices for arbitrarily complexcomputation.

Mental capacity

Undecidability

Can there be undecidable decision problems?

Is every mathematical question in principle answerable by acomputational procedure?

Turing sought to answer these questions.

Undecidability

Halting problem

Turing considered the halting problem: given a program p,determine whether it will halt.

Is this computably decidable?

Given program p, we could run the program, and wait for it tohalt.

This would half-solve the problem: it gives us the Yes answers.

But can we also compute the No answers?

Halting problem

Halting problem is undecidable

Turing proved that the halting problem is undecidable. There isno computational procedure that correctly determines whethera given program will halt.

A beautiful argument

Suppose toward contradiction that we could solve the haltingproblem. Consider this strange algorithm q: On input p, a program,we ask whether program p would halt if given p itself as input; if yes,then we jump into an infinite loop; if no, then we halt immediately.

Let us run q on input q. It will halt if and only if it doesn’t halt.Contradiction.

So the halting problem is undecidable.

Halting problem is undecidable

Turing proved that the halting problem is undecidable. There isno computational procedure that correctly determines whethera given program will halt.

A beautiful argument

Suppose toward contradiction that we could solve the haltingproblem. Consider this strange algorithm q: On input p, a program,we ask whether program p would halt if given p itself as input; if yes,then we jump into an infinite loop; if no, then we halt immediately.

Let us run q on input q. It will halt if and only if it doesn’t halt.Contradiction.

So the halting problem is undecidable.

Undecidability

The undecidability phenomenon is now known to be pervasivein mathematics.

We have thousands of concrete decision problems, whichcannot in principle be solved by any computational procedure.

Often, one can prove that a problem is undecidable byembedding the halting problem inside it.

The tiling problem

Tiling problem: given finitely many polygonal tile types,determine whether they can tile the plane.

Question. Is the tiling problem decidable?

Answer. No. There is no computational procedure that correctlysolves this problem.

The tiling problem

Computable numbers

Turing introduces the concept of computable numbers.

He defines that a real number is computable, when there is acomputational procedure to enumerate the decimal digits of thenumber.

3.14159265358979323846264338327950288419 · · ·

He proceeds to develop the theory of computable real numbers.

Computable numbers

3.14159265358979323846264338327950288419 · · ·

Computable numbers

3.14159265358979323846264338327950288419 · · ·

Arithmetic

Consider the ordinary arithmetic operations a + b = c.

0.111111111111 . . .

+ 0.222222222222 . . .

0.333333333333 . . .

We would like to compute the sum of two given numbers.

Arithmetic

Consider the ordinary arithmetic operations a + b = c.

0.111111111111 . . .

+ 0.222222222222 . . .

0.333333333333 . . .

We would like to compute the sum of two given numbers.

Arithmetic is more difficult than you expect!But hang on. Consider the following case of a + b.

0.333333333333 . . .

+ 0.666666666666 . . .

0.999999999999 . . .

We can start writing down the answer 0.99999 . . ., but thesedigits will be wrong if we ever find a carry term, since then theanswer should be 1.000000 . . .

But 1.0000000 . . . will be wrong if we ever find a digit place withsum less than 9.

It seems that we cannot start to give the digits of a + b, giventhe initial digits of a and b.

0.333333333333 . . .

+ 0.666666666666 . . .

0.999999999999 . . .

0.333333333333 . . .

+ 0.666666666666 . . .

0.999999999999 . . .

Arithmetic on computable numbers

This is actually a fundamental problem.

There is no computable procedure to compute the digits ofa + b, given programs for computing the digits of a and bseparately.

In this sense, ordinary arithmetic is not a computable operationwith the digits-conception of computable number.

But if one modifies Turing’s concept slightly, however, theneverything works great!

A solution

Namely, logicians today define that a computable real numberis a program that computes rational approximations to a realnumber, as accurately as desired.

As approximations, 0.999999999 and 1.00000000 are veryclose, even though their digits are totally different.

With this modified concept, all the usual operations arecomputable.

a + b ab sin(x) ex

Turing’s computable real number idea turns into the robustsubject known as computable analysis.

A solution

Namely, logicians today define that a computable real numberis a program that computes rational approximations to a realnumber, as accurately as desired.

As approximations, 0.999999999 and 1.00000000 are veryclose, even though their digits are totally different.

With this modified concept, all the usual operations arecomputable.

a + b ab sin(x) ex

Turing’s computable real number idea turns into the robustsubject known as computable analysis.

Summary

Alan Turing’s 1936 paper, “On computable numbers. . . ”

Written while he was a student at Cambridge.

One of the most important papers ever written.Introduced fundamental ideas on computability, layingfoundation for the contemporary computer era.Established the existence of universal computers.Revealed the undecidability phenomenon.Introduced concept of computable numbers.Yet, there is a flaw in that account.But a modified conception leads to the subject ofcomputable analysis.

Summary

Written while he was a student at Cambridge.One of the most important papers ever written.

Introduced fundamental ideas on computability, layingfoundation for the contemporary computer era.Established the existence of universal computers.Revealed the undecidability phenomenon.Introduced concept of computable numbers.Yet, there is a flaw in that account.But a modified conception leads to the subject ofcomputable analysis.

Summary

Written while he was a student at Cambridge.One of the most important papers ever written.Introduced fundamental ideas on computability, layingfoundation for the contemporary computer era.

Established the existence of universal computers.Revealed the undecidability phenomenon.Introduced concept of computable numbers.Yet, there is a flaw in that account.But a modified conception leads to the subject ofcomputable analysis.

Summary

Written while he was a student at Cambridge.One of the most important papers ever written.Introduced fundamental ideas on computability, layingfoundation for the contemporary computer era.Established the existence of universal computers.

Revealed the undecidability phenomenon.Introduced concept of computable numbers.Yet, there is a flaw in that account.But a modified conception leads to the subject ofcomputable analysis.

Summary

Written while he was a student at Cambridge.One of the most important papers ever written.Introduced fundamental ideas on computability, layingfoundation for the contemporary computer era.Established the existence of universal computers.Revealed the undecidability phenomenon.

Introduced concept of computable numbers.Yet, there is a flaw in that account.But a modified conception leads to the subject ofcomputable analysis.

Summary

Written while he was a student at Cambridge.One of the most important papers ever written.Introduced fundamental ideas on computability, layingfoundation for the contemporary computer era.Established the existence of universal computers.Revealed the undecidability phenomenon.Introduced concept of computable numbers.

Yet, there is a flaw in that account.But a modified conception leads to the subject ofcomputable analysis.

Summary

Written while he was a student at Cambridge.One of the most important papers ever written.Introduced fundamental ideas on computability, layingfoundation for the contemporary computer era.Established the existence of universal computers.Revealed the undecidability phenomenon.Introduced concept of computable numbers.Yet, there is a flaw in that account.

But a modified conception leads to the subject ofcomputable analysis.

Summary

Written while he was a student at Cambridge.One of the most important papers ever written.Introduced fundamental ideas on computability, layingfoundation for the contemporary computer era.Established the existence of universal computers.Revealed the undecidability phenomenon.Introduced concept of computable numbers.Yet, there is a flaw in that account.But a modified conception leads to the subject ofcomputable analysis.

Thank you.

Slides and articles available on http://jdh.hamkins.org.

Joel David HamkinsOxford University

alan turing s theory of computation

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