Introduction Computability Decidability Computable numbers Alan Turing’s theory of computation Joel David Hamkins Professor of Logic Sir Peter Strawson Fellow University of Oxford University College Oxford Cambridge Club, London 6 June 2019 Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford
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.
Oxford Cambridge Club London 2019 Joel David Hamkins, Oxford