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Mathematics for Computer Science
What is this course about?
Lecture 1: Overview and review of basic Set Theory
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There are many things that a course called “Mathematics
for Computer Science” could be about. It could be a course
in discrete mathematics, the tool that helps us understand
data structures and algorithmics. Or it could be a course in
probability theory, information theory, and statistics,
topics which are useful for many applications.
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But this is a course that deals with the concepts of
computability, uncomputability and computational
complexity. That is, this course is not just a course in
applied mathematics — rather, its aim is to introduce the
kind of mathematics that tell us what computer science
actually is.
Areces, Blackburn and Figueira
TALARIS team
INRIA Nancy Grand Est
Contact: [email protected]
Course website:
http://www.loria.fr/~blackbur/courses/math
Areces, Blackburn, Figueira: Mathematics for Computer Science
Turing machine
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Church’s thesis
The central concept in this course is the concept of a
Turing machine. This is a very abstract — and incredibly
simple — model of what computation is.
It is also a very natural and plausible notion of
computation, and was inspired by reflecting on what people
do when they actually calculate something.
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Part of the answer is obvious: if we have a robust model of
what computation is (and Turing machines are believed to
be such a model) then we have a mathematical basis for
computer science. That is, the model gives us lots of
positive information about computability.
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Moreover, the concept of a Turing machine splits the
mathematical world into two. A consequence of the model
is that there are many well-defined mathematical problems
that cannot be computed by any means. That is, the
model also gives us lots of negative information about
computability.
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Church’s thesis says that anything at all that we can
compute can be computed by a Turing machine!
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That is, this simple model appears to provide a universal
model of computation.
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And indeed, over the years, many different models of
computation have been proposed. But they have all turned
out to be equivalent to the notion of a Turing machine.
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But these concepts have ramifications way beyond the
confines of theoretical computer science — they are also
relevant to such domains as logic, philosophy, cognitive
science, and artificial intelligence.
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The link with logic is deep and intimate. In fact, it was
logicians such as Kurt G¨
odel, Alonzo Church, Alan Turing,
and Stephen Kleene who first introduced and explored the
concept of computability. The concept grew naturally out
of their logical investigations — and in turn fed back into
logic and revealed its limitations.
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Impact on Cognitive Science and Artificial Intelligence
The impact on philosophy has also been profound. The
idea that certain problems are not computable (that is, not
solvable by any realistic computational device) — and
provably not solvable — was radical at the time, and
affected Anglo-American philosophical culture deeply.
But the area raises many other interesting philosophical
issues. To name just one: what is the status of Church’s
thesis? It is not a mathematical truth, something that can
be proved. But it is not obviously correct to call it an
empirical generalisation either. So what is its status?
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Areces, Blackburn, Figueira: Mathematics for Computer Science
Impact on Philosophy
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As we shall see, at first sight, the concept of a Turing
machine looks far too simple to be interesting. But here
appearances are misleading.
But it is also important for other reasons
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Areces, Blackburn, Figueira: Mathematics for Computer Science
Why is this material important?
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Turing machines were inspired by thinking about the
process of calculation; they have at least some (abstract)
claim to cognitive plausibility.
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Moreover the model they gave rise to — devices with
internal states — provided exactly the kinds of ideas that
helped psychology to break free of behaviorism in a
scientific way, thus giving rise to modern cognitive science.
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It also suggested that intelligence was something that could
not only be modeled, but perhaps even created by
programming. This set the scene for the birth of artificial
intelligence.
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Linking these ideas together another way
Alan Turing (1912–1954)
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The ideas underlying the course can be tied together in
another way — by considering the career of one man,
namely Alan Turing.
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It may seem bold to claim that the ideas underlying Turing
Machines touch on areas as diverse as Philosophy and
Computer Science.
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But Turing himself would not have been surprised: indeed,
he made many of the connections himself.
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So who was Alan Turing . . . ?
Areces, Blackburn, Figueira: Mathematics for Computer Science
Alan Turing (born 23 June 1912, died 7 June 1954) was a
British logician, cryptographer, a key figure in the birth of
theoretical and practical computer science, and a pioneer of
Artificial Intelligence and Cognitive Science (with special
reference to Natural Language). He also did important early
work on the formation of patterns and mathematical biology.
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Alan Turing
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Early life
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He excelled in science — though this was at a time when
science was not in such high regard as it is now. Because
he couldn’t be bothered studying Latin or Greek
intensively, he failed to win a scholarship to Trinity College
Cambridge and went to King’s College Cambridge instead.
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He was an undergraduate there from 1931–1934, and in
1935 was elected to be fellow of the college because of the
strength of his mathematical research.
Born in London, Turing genius was recognized by many of his
teachers from the age of 6 onwards.
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Computability and Incompatibility
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World War II and Cryptography
In his classic 1936 paper “On Computable Numbers, with
an Application to the Entscheidungsproblem” Turing
introduced what are now called Turing machines, simple
formal devices which naturally modeled computation.
He showed convincingly that such machines would be
capable of performing any conceivable mathematical
problem that could be dealt with algorithmically.
These simple machines are still a key concept in the study
of computability — and indeed computational complexity.
In 1937 and 1938 he was in Princeton, working under
Alonso Church. He gained his PhD there in 1938; his thesis
dealt with even more abstract notions of computation,
namely Turing machines equipped with Oracles (a concept
we won’t discuss in this course).
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During the Second World War Turing was the key figure in
the British attempts at Benchley Park to break German
codes, and in particular, the codes used by U-boats.
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His work (which was classified for many years after the
war) was crucial to Britain’s survival; it these codes had
not been broken it seems likely that Britain would have
been forced to negotiate a treaty with Germany.
Areces, Blackburn, Figueira: Mathematics for Computer Science
Enigma Machine
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Turing’s response
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Each German U-boat was equipped with an Enigma
machine, capable of encoding and decoding transmissions.
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Eventually an Enigma machine was captured — but how
could it be understood? U-boat Enigma’s could have up to
1022 states!
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His answer was the creation of the Bombe. This was an
extension of an earlier Polish-designed hand-operated code
breaking machine called Bomba; apparently when a
possible solution was reached, a part would fall onto the
floor, making a noise — and hence the name!
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The Bombe was, in effect, an early computer, though not a
universal computer, that could rapidly test many possible
settings of the Enigma machine. It used logical deduction
to rule out certain settings as impossible.
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The Bombe
The postwar development of the computer
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After the war Turing worked at the National Physical
Laboratory.
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In 1946 he designed one of the first stored-program
computers (that is, one of the first real-existing universal
machines).
The Bombe was a very real machine — and a very useful one —
not an abstraction like his Turing Machines. Turing had the
first Bombe running by March 1940, just 6 months after he
went to Benchley. By the end of the war, 200 were in operation.
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Pilot version of ACE
Pilot ACE (reconstruction, Science Museum London)
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A full version of the machine, which was called the ACE, was
never actually made, though a smaller pilot version was.
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Further pioneering work in theoretical computer science:
for example, the concept of compilation — translating a
higher level programming language down into a machine
language. Though legend has it that Turing thought
compilation was unnecessary — after all, it was so easy to
compute directly in binary code!
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Pioneering work in AI: Turing also wrote the first chess
playing program. Unfortunately, no computer powerful
enough to run it actually existed at the time. However he
did simulate it by hand (taking half an hour to compute
each move) and the program is said to have won some
games.
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The Turing test
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Legacy
The test was pioneering concept in the philosophy of AI
and Cognitive Science; it was introduced in his classic 1950
paper “Computing Machinery and Intelligence”.
In essence, he replaced the question “Can a machine
think?” by a concrete experiment: if a program can chat
with a person (in the modern sense of using MSN, or
Yahoo or Skype . . . ) over an extended session, and at the
end of the session the person cannot decide if he was
chatting with a computer or a machine, then we must
conclude we have a machine that thinks.
Turing concluded: I believe that at the end of the century
the use of words and general educated opinion will have
altered so much that one will be able to speak of machines
thinking without expecting to be contradicted.
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Other work from this period
The Manchester Mark 1 was one of the world’s first fully
general computers (that is, it was a physical embodiment of the
ideas underlying the abstract notion of a Turing machine).
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A pilot version of ACE ran its first program on 10/5/1950.
But by this time, Turing was no longer at the National
physical laboratory, because in 1948 he moved to the
Computer Science Department of Manchester University
where he participated in the development of the
Manchester Mark 1.
Areces, Blackburn, Figueira: Mathematics for Computer Science
Manchester Mark 1 - 1949
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Turing died in 1954, probably by suicide.
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He has been widely recognized and honoured since his
death. For example, the computer science world’s
equivalent of the the Nobel Prize (awarded by the
Association for Computing Machinery) is called the Turing
Prize.
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He was in Time Magazine’s list of 100 most influential
people of the 20th century.
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And much else beside . . .
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Marathon Man
And now, back to the course . . .
What will you get out of this course?
An understanding of the most fundamental concepts of
theoretical computer science, namely that there are
both computable and uncomputable functions. You will
gain deeper insight into what computation actually is,
and see how simply the concept can be modeled.
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Areces, Blackburn, Figueira: Mathematics for Computer Science
And now, back to the course . . .
And now, back to the course . . .
What will you get out of this course?
What will you get out of this course?
You will gain certain concrete mathematical skills. In
particular, we will be paying close attention to how we
prove things, that is, the way we can show general
facts about the concepts we introduce. At the end of
the course you will have seen a number of proofs, often
of a very different nature. In short: this course should
develop your abstract reasoning skills.
Areces, Blackburn, Figueira: Mathematics for Computer Science
You will have seen some deep mathematical logical
results. The ideas discussed in this course are closely
linked with key concepts of modern logic. We will show
you these links, and this should deepen and enrich
your understanding of logic.
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And now, back to the course . . .
And, although it is not the primary aim of the course,
we hope you will also gain deeper insight into how the
concept of computability touches on cognitive science
and natural language. That is, this course should give
you deeper insight into Alan Turing’s legacy.
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Lecture 1 Overview and review of basic set theory.
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Lecture 2 : Functions and equinumerous sets.
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Lecture 3 : Turing machines.
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Lecture 4 : Church’s thesis.
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Lecture 5 : Uncomputability and undecidability I.
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Lecture 6 : Uncomputability and undecidability II.
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Lecture 7 : Undecidability of first-order logic.
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Lecture 8 : Computational complexity I.
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Lecture 9 : Computational complexity II.
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Lecture 10 : Computational complexity III.
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And what about in the rest of this lecture?
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Basic concepts of set theory
In the rest of this first lecture we will be
introducing/revising the basic ideas of Set Theory.
Nowadays set theory is a setting in which almost all
mathematics is conducted. A quick review of basic set
theory will give us the mathematical background we
need for the course — and will lead us to interesting
(and surprising) territory very fast.
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Lectures
What will you get out of this course?
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A set is a collection of ‘things’ — the ‘things’ might be
numbers, concrete objects, or other sets.
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The items in a set are called its elements or its members.
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If x is an element of a set y we say that “x is an element of
y” or “x is a member of y” and write x ∈ y.
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When are two sets equal? Whenever they contain exactly
the same elements.
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To put it the other way around: two sets are different if
and only if they contain at least one different element.
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If set x and set y are equal we write x = y.
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You will often hear it said that sets are extensional. This
simply means that they are completely determined by their
elements.
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Specifying (finite) sets
The Empty Set
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We specify a set (that is, we say what set we are talking
about) by saying what its elements are.
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A very useful set is the empty set; this is the set that
contains no members at all.
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For example, a perfectly good set is the set containing the
the first three positive integers 1, 2, and 3.
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We usually write ∅ (or { }) for the empty set.
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Why is the empty set “useful”? It may seem a bit strange.
In fact it is useful in much the same way that the number 0
is useful in mathematics — it helps give sets an elegant
algebraic structure (it completes the picture, so to speak).
We’ll see some examples soon.
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The empty set is unique. That is, there is only one empty
set. Why is this?
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We could write this set as {1, 2, 3}.
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That is, we can specify a set by listing its elements and
enclosing them in braces, that is, the symbols { and }.
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Of course, this method only works for finite sets (and
indeed, is only practical for small finite sets). As we shall
see soon, there are other ways of specifying sets.
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Areces, Blackburn, Figueira: Mathematics for Computer Science
Subset
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Proper subset
A set x is a subset of a set y if every element of set x is an
element of set y.
The notation x ⊆ y means that x is a subset of y.
It follows from this definition that every set is a subset of
itself. That is, for any set x we have that if x ⊆ x. So ⊆ is
a reflexive relation between sets.
It also follows from this definition that for any sets x , y
and z, if x ⊆ y and y ⊆ z then x ⊆ z. That is, ⊆ is a
transitive relation between sets.
It also follows from this definition that for any sets x and
y, that if x ⊆ y and y ⊆ x then x = y. That is, ⊆ is a
antisymmetric relation between sets.
Do not get confused between ∈ and ⊂! They are very
different concepts!
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Intersection is an operation that takes two sets and makes
a new one; we use ∩ to indicate this operation.
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We define this operation as follows: for any sets x and y,
x ∩ y is the set containing precisely those elements that
belong to x and y.
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If x ∩ y = ∅ then we say that x and y are disjoint.
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A set x is a proper subset of y if x is a subset of y but x is
not equal to y.
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This is usually written as x ( y.
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Union is also an operation that takes two sets and makes a
new one; we use ∪ to indicate this operation.
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We define this operation as follows: for any sets x and y,
x ∪ y is the set containing precisely those elements that
belong to either x or y.
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Difference
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Algebra of sets (I)
These operations induce a rich algebraic structure on sets. For
example, we have that:
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Difference is another operation that takes two sets and
makes a new one; we use \ to indicate this operation.
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We define this operation as follows: for any sets x and y,
x\y is the set containing precisely those elements of x that
do not belong to y.
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Union
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Suppose x and y do not have any elements in common.
Then x ∩ y = ∅. That is, as promised, ∅ is playing and
algebraic role similar to that of the number 0.
Sometimes we speak of one set being a proper subset of
another.
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Intersection
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Commutative rules:
x∪y =y∪x
x∩y =y∩x
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Associative rules:
x ∪ (y ∪ z) = (x ∪ y) ∪ z
x ∩ (y ∩ z) = (x ∩ y) ∩ z
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Algebra of sets (II)
Sets of numbers
Sets of numbers play an important role in mathematics, and
will play an important role in this course. Two important sets
of numbers are:
These operations induce a rich algebraic structure on sets. For
example, we have that:
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Distributive rules:
x ∪ (y ∩ z) = (x ∪ y) ∩ (x ∪ z)
x ∩ (y ∪ z) = (x ∩ y) ∪ (x ∩ z)
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The set of natural numbers is the set whose elements are
the numbers 0, 1, 2, 3, 4, 5, . . . and so on. We use N (or ω)
to specify this (infinite) set.
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The set of integers is the set whose elements are the
numbers . . . −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, . . . and so
on. We use Z to specify this set. Obviously Z is an infinite
set, and obviously N ⊆ Z.
De Morgan rules:
x\(y ∪ z) = (x\y) ∩ (x\z)
x\(y ∩ z) = (x\y) ∪ (x\z)
Remark: in set theory, N is usually taken to be the
fundamental infinite set, and other sets of numbers are defined
in terms of it. For more information on this topic, see the
references at the end of the lecture.
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More sets of numbers
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Specifying richer sets II
The set of positive integers is the set whose elements are
the numbers 1, 2, 3, 4, 5, . . . and so on. We use Z+ to
specify this set. Clearly Z+ = N\{0}.
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The set of negative integers is the set whose elements are
the numbers −1, −2, −3, −4, −5, . . . and so on. We use
Z− to specify this set. Clearly Z = Z− ∪ {0} ∪ Z+ .
The simple “list and enclose in braces method” only works
for finite sets. What about infinite sets, like the set of all
even numbers?
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We specify these sets by describing them. For example, the
set of even numbers is the set
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The set of rational numbers (or fractions) is the set of all
numbers of the from n/m where n ∈ Z and m ∈ Z+ . We
use Q as a name of this set.
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The set of real numbers is the set of all numbers standardly
used in a calculus course. It consists of Q together with all
the transcendental numbers such as π and e. We use R as
a name for this set.
Areces, Blackburn, Figueira: Mathematics for Computer Science
{x ∈ N : for some y, 2 × y = x}
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Areces, Blackburn, Figueira: Mathematics for Computer Science
Power Set
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For any set x, we are free to form another set called the
power set of x.
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The power set of a set x is defined to be the set of all
subsets of x.
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For any set x, we denote the power set of x by P(x).
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For any set x, both x and ∅ belong to P(x).
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What is the powerset of ∅?
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How large are power sets?
As we shall see, power sets are very interesting. Let’s ask a very
basic question about them, namely how large are (finite) power
sets? Here is the answer:
If a finite set contains n elements, then its power set
contain 2n elements.
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As we shall learn, this is a more important result than it at
first appears.
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But that’s for the future. A more basic question is: how do
we prove it. We shall do so using a fundamental proof
technique: mathematical induction.
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Mathematical induction
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It’s just like knocking over dominoes . . .
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Many mathematical problems ask us to show that
something is true for all natural number n.
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We can prove this as follows. First show that the result
holds for 0. This is called proving the base case.
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Then show that if the result holds for some natural number
n > 0, it also holds for n + 1. This is called proving the
inductive step.
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If we can do these two things, we have shown that the
result holds for all natural numbers, and proved the result.
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There are important restrictions on what the description
can be; see the references at the end of the lecture for
further discussion.
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Proving the base case is like knocking over the first domino.
Proving the inductive step is like showing the dominoes are
spaced so that each will hit the next.
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The power set example: base case
The power set example: inductive step
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We want to prove that for every natural number n, if a set
x has n elements then its power set has size 2n .
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So let’s consider the smallest finite set of all, the empty set,
the one containing 0 elements. We want to show that its
power set contains 1 element,
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But we know that this is true. After all, P(∅) is {∅}. And
this has 1 element, which is what we wanted to show.
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So we have proved the base case.
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Areces, Blackburn, Figueira: Mathematics for Computer Science
Summary of Lecture 1
Key concepts:
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Mathematical induction is a proof technique used to show
that some property holds for all natural numbers n.
Mathematical induction consists of showing two things:
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First, that the desired property holds for 0 (the base case).
Second, that if the property holds for n, it also holds for
n + 1 (the inductive step).
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We specify a set by saying what its elements are. For finite
sets, we can do so simply by listing them. For more
complex sets, we can describe how they are constructed out
of sets that we know about, using ∪, ∩, \ and P (that is,
by forming power sets.
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There are many “standard” mathematical sets, and today
we met some of them. The simplest (and smallest) of these
was ∅. We also met a number of standard infinite sets of
numbers, namely N, Z, Z+ , Z− , Q, and R.
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Elements of Set Theory, by Herbert Enderton, Academic
Press 1977. One of the most readable introductions to Set
Theory. Clear, accurate, and covers quite a lot of ground.
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Naive Set Theory, by Paul Halmos, Springer, 1970. This
slim book is a classic mathematics text. It covers a lot of
ground with a minimum of notation.
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The Wikipedia article on mathematical induction is quite
useful; it considers some variants of mathematical
induction, and has some interesting historical remarks.
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Alan Turing: The Enigma of Intelligence, by Andre
Hodges, Burnett Books, 1983. A readable and interesting
biography of Alan Turing.
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A History of Modern Computing, by Paul Ceruzzi, MIT
Press, 1983. An excellent source.
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The Wikipedia article on Alan Turing is also quite
informative and has useful links to related topics.
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References on Set Theory
Areces, Blackburn, Figueira: Mathematics for Computer Science
Sets are a key mathematical notion. A set is determined by
its elements. Two sets are equal if and only if they have
the same elements.
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References on Alan Turing
We also made use of a key proof technique, namely
mathematical induction.
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Summary of Lecture 1 (continued)
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Summary of Lecture 1 (continued)
Key concepts:
I We briefly mentioned some of the key concepts we will be
discussing in this course, such as Turing machines and
Church’s Thesis.
I Although we did not explain these concepts in detail we
explained why they are of such fundamental importance to
computer science: they give a simple model of what it
means for something to be computable and (just as
importantly) show that many things are not computable.
I We also indicated the importance of these topics for other
subjects than computer science, such as Cognitive Science,
Philosophy, Artificial Intelligence and Natural language
Processing.
I Indeed, as we saw, many of these connections were
introduced and explored by Alan Turing himself.
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Now let us suppose that for some natural number n ≥ 0 we
have shown that for any set with n elements, its power set
has 2n elements.
We want to show that this implies that shown that for any
set with n + 1 elements, its power set has 2n+1 elements.
So let S be any set containing n + 1 elements. Let’s write it
as {s1 , s2 , s3 , . . . , sn , sn+1 }. Or to put it another way,
S = T ∪ {sn+1 }, where T is {s1 , s2 , s3 , . . . , sn }
Now the power set of T contains 2n elements. Why?.
And now for the key observation: every element in the
power set of S is either an element p of the power set of T ,
or and has the form p ∪ {sn+1 }. Why?
So there are 2n + 2n elements in P(S). But
2n + 2n = 2 × 2n = 2n+1 , which is what we wanted to show.
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