SEL2211: Contexts
Lecture 5: Computer Science
Computers are everywhere. Our world could not function without them,
and we use them naturally as tools in our daily lives.
Perhaps surprisingly, however, the science underlying the computer
technology which surrounds us is fundamental to contemporary
understanding of mind, cognition, and language.
This lecture outlines the fundamentals of this science.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.1 History
Historically, computation was restricted to
numerical calculation.
The earliest numerical calculation device we
know about was the abacus, invented as
early as c.2000 BC in Mesopotamia:
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.1 History
This was subsequently developed in the
ancient Mediterranean world as well as in
China, and is still in use in some countries
today.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.1 History
The Greeks appear to have developed a
non-abacus type of calculating device by
about 100 BC, which is what the
archaeological find opposite is taken to be.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.1 History
Several types of mechanical
calculating machine were developed in
the Islamic world between about 700
and 1300 AD.
In the West nothing is known until the
17th century, when several mechanical
calculating devices were invented.
The German mathematician Leibniz,
for example, designed and had built
the 'step reckoner' by 1694, and as the
replica opposite shows, things had
come a long way in terms of
sophistication.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.1 History
None of the early computational devices were really computers in the
modern sense.
For these to emerge, several more developments were required:
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.1 History
-- Modern digital computers work not with the base-10 number system
with which humans are most familiar, but with the binary number
system, which has only two digits, 1 and 0.
The binary system was invented by the same Leibniz as he of step
reckoner fame, but it could only be applied to general computation
after George Boole had developed a binary algebra named Boolean
algebra in his honour.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.1 History
-- In the early part of the 20th century, mathematicians became interested in the
question of computability –whether all conceivable mathematical functions
could be calculated, and, if not, which ones could be and which could not.
This work, in combination with Boolean algebra, led directly to the invention of
the Turing Machine, the theoretical model for all subsequent computational
theory and technology, of which more below.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.2 Fundamental ideas
Understanding computation in the contemporary scientific sense requires
familiarity with just a few simple ideas.
Once these ideas are grasped, the principle of computation is perfectly
straightforward.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.2 Fundamental ideas
1.2.1 Symbols and symbol systems
A symbol is any physical thing that represents –in other words, that humans
agree to interpret as standing for something else.
When anyone in the world sees a flag with stars and stripes, for example, it is
immediately recognized as a symbol for the USA. A symbol system is a
collection of related symbols –the flags of the world’s countries, for example,
constitute a symbol system.
The western alphabet is another symbol system: its individual letters are
symbols that stand for phonemes of a language.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.2 Fundamental ideas
1.2.2 Strings
A string is a left-to-right sequence of symbols taken from some symbol system.
Thus, using the alphabet symbol system as an example, the following are
strings:
aaabbb
Xdghjfdsahjll
computer
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.2 Fundamental ideas
1.2.3 Algorithms
An algorithm is a sequence of instructions which is guaranteed to result in some desired
state of affairs.
Say the desired state of affairs is to get to the Student Union from the third floor of the
Percy Building. An algorithm to achieve that might be:
i. Go through the red doors on the landing
ii. Walk down to the ground floor
iii. Go through the front door
iv. Walk down the quad
v. Walk under the arches
vi. Cross the road
vii. Walk 10 metres
viii. Turn right
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.2 Fundamental ideas
1.2.4 Computation
What, in terms of symbols, symbol strings, and algorithms, is computation?
Computation is string transformation according to an algorithm, no more and no less.
Some examples:
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.2 Fundamental ideas
1.2.4 Computation
-- Arithmetic:
The computation is to multiply two numbers together, divide the result by a third
number, and output the result.
The numbers are represented by symbols taken from the symbol system {0..9}, the
arithmetic operators are represented by symbols taken from the symbol system {( ) +
- x /}, and the computation itself is written as a string: (2 x 6) / 3.
An arithmetic algorithm is applied to this string, and that algorithm transforms it into
another string, 4.
SEL2211: Contexts
Lecture 5: Computer Science
1. The nature of computation
1.2 Fundamental ideas
1.2.4 Computation
-- Language translation:
The computation is to translate German-language expressions into English-language
ones.
The German words are represented using symbols from the symbol system of
German words given by, say a German dictionary, and the English ones are
represented using symbols from the symbol system of English words.
The string to be translated is Das rote Buch.
A translation algorithm is applied to it, and the result is another string: The red book.
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
As already noted, in the early part of the 20th century, mathematicians were
interested in the question of computability: whether all conceivable mathematical
functions could be solved, and, if not, which ones could be and which could not.
An important class of functions are those which can be solved in principle but which,
as the size of the problem is increased, become so time-consuming that they
become impractical to solve.
A famous example is the Tower of Hanoi problem.
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
According to legend (actually invented in 1883 by French mathematician Edouard
Lucas), there was a temple with a dome which marked the center of the world in
Benares.
Within the dome, priests moved golden disks between diamond needles.
God had placed 64 gold disks on one needle at the time of creation, and it was said
that, when they completed their task, the universe would come to an end
(http://www.superkids.com/aweb/tools/logic/towers/).
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Lecture 5: Computer Science
2. The Turing Machine
Here's how Wikipedia (http://en.wikipedia.org/wiki/Tower_of_Hanoi) describes what
the monks were doing:
The Tower of Hanoi...is a mathematical game or puzzle. It consists of three rods,
and a number of disks of different sizes which can slide onto any rod. The puzzle
starts with the disks in a neat stack in ascending order of size on one rod, the
smallest at the top, thus making a conical shape.
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
The objective of the puzzle is to move the entire stack to another rod, obeying the
following rules:
•Only one disk may be moved at a time.
•Each move consists of taking the upper disk from one of the rods and sliding it
onto another rod, on top of the other disks that may already be present on that rod.
•No disk may be placed on top of a smaller disk.
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
This seems simple, but anyone attempting it quickly discovers that it is anything but.
There are algorithms to solve it, described in detail at the above Wikipedia page and
elsewhere on the Web, and where the number of discs is relatively small, as above,
following any one of the algorithms leads to a solution in a small number of steps.
But as the number of discs grows the number of steps increases very quickly.
Intuitively, one expects that the number of steps for 6 discs will be double the
number of steps for three, but this is not the case, as the following animation shows:
Mazeworks: http://www.mazeworks.com/hanoi/
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
For 3 discs 7 steps are required, but for 6 the number of steps is not 14 but 63,
nine times as many, and for 9 discs the number of steps is not 21 but 511, which
is 73 times as many.
This dramatic rate of increase in the number of steps relative to modest
increases in the number of discs soon exceeds all reasonable bounds: for 64
discs it takes 264 -1 steps and, assuming one step per second, the solution
would take 585,000,000,000 years to complete
(http://www.superkids.com/aweb/tools/logic/towers/).
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
This type of rather abstruse research question became very relevant to real-world
concerns during the Second World War, when Allied shipping in the North Atlantic
was being sunk by German submarines and Britain was thereby being starved of
resources.
The reason the German submarines were so successful is that they and their
command centre were using a code to send messages that the Allies could not
decipher using existing methods. A team of mathematicians was assembled at
Bletchley Park near Milton Keynes.
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
Their remit was to crack the code; using ideas from research into computability
they were successful, thereby contributing greatly to the Allied victory.
The breakthrough came when one of these mathematicians, Alan Turing, came
up with a design for what he called an 'automatic machine' which, once physically
implemented, was able to break the German code.
This design was to have momentous consequences beyond the immediate one
of affecting the course of the Second World War, because it underlies all present
day computer technology and is the basis of contemporary scientific
understanding in a range of sciences including cognitive science and linguistics.
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
The following diagram shows a Turing Machine.
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
The string to be transformed is written on
the input tape, one symbol per square.
The transformed string which is the result
of the computation is written on the output
tape, one symbol per square.
The memory tape is used to store
symbols used in the course of the
computation
The control contains an algorithm which
orchestrates reading from the input tape
using the read head, writing to the output
tape using the write head, and reading
and writing symbols to from and to the
memory tape using the read/write head.
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
To show how the Turing machine carries out its computations, let’s look at a very
simple example: transformation of the string 1+1 into the string 2
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
SEL2211: Contexts
Lecture 5: Computer Science
2. The Turing Machine
Follow this link to see a physically-implemented Turing machine at work:
This design for a computer, simple as it is, can compute any computable
function, no matter how complex.
SEL2211: Contexts
Lecture 5: Computer Science
3. The Turing Machine and Computer Technology
The Turing Machine as described above is the basis for standard present-day
computer technology.
Turing’s formulation is theoretical, and therefore too simple for practical purposes,
but actual computers differ only in terms of complexity and efficiency –they all
work in principle as described above.
SEL2211: Contexts
Lecture 5: Computer Science
3. The Turing Machine and Computer Technology
The input tape = keyboard of a computer: successive key strokes constitute the
input string to be transformed.
The output tape = the monitor and/or the printer, both of which put out letter
sequences or strings which the user then interprets.
The memory tape = a combination of random access memory (RAM) and hard
disk inside the computer case.
The control of the Turing Machine corresponds to the central processor inside the
computer case
The algorithms / programs that allow the computer to perform its various tasks,
such as word processing, email, Web browsing and so on, are stored as files on
the hard disk. The central processor executes these algorithms by going to the
appropriate file on the disk and then following the instructions.
SEL2211: Contexts
Lecture 5: Computer Science
4. Outline of modern computer technological development
The theory of computation was invented in the 1930s to solve the
mathematical problem of computability: what sorts of problem can
mathematics solve? It had nothing to do with representation of language
The first physical computer was built to break enemy codes during WW2 by
numerical calculation.
The earliest peacetime computers, built soon after WW2, were still used
exclusively for numerical calculation.
In the mid-later 1950s, it was realized that there could be useful applications of
computers to representation and analysis of language. These applications
developed slowly, and the main use of computers was still numerical.
SEL2211: Contexts
Lecture 5: Computer Science
4. Outline of modern computer technological development
Prior to 1975, computers had been huge and very expensive devices, well out
of the reach of all but governments and large businesses. Thereafter,
advances in electronics made possible the construction of smaller and much
less expensive computers, bringing them within financial reach of interested
individuals. These were called 'microcomputers'.
As microcomputers became more popular, some enthusiasts saw their
educational and commercial potential and developed (relatively) easy to use
software; in this period --the late 1970s and early 1980s-- Microsoft and Apple
were established.
A crucial piece of software in this period was the word processor, effectively an
electronic typewriter, which revolutionized language representation by moving
it from print produced by typewriters to electronic text files produced by
computers.
SEL2211: Contexts
Lecture 5: Computer Science
4. Outline of modern computer technological development
Computers had from early on been networked, that is, connected together with
wires to that electronic data could be moved between and among them. Such
early networks were relatively small and slow. From the mid-1980s, as the
number of computers rapidly grew, the networks became much larger and very
much quicker. This formed the basis for the present-day Internet.
As the Internet developed, software that could exploit it was invented. Email
came first, and then Web browsers like Netscape. This network software was
at first used mainly by governments, the military, and academics. These made
it possible to move electronic text quickly and cheaply from computer to
computer.
By the early 1990s, computers and the software that runs on them had
become consumer products. The consequences are observable around us.
SEL2211: Contexts
Lecture 5: Computer Science
5. Computation and meaning
As we shall see next week, computation provides the theoretical framework for
contemporary understanding of cognition in general and language in particular.
There is a major problem with this, however.
Human cognition includes meaning, and linguistic expressions more
specifically have meaning.
There is no meaning in a Turing Machine-based computational system.
On the face of it, therefore, computation would appear inadequate as a
theoretical framework for understanding of cognition and language.
Much depends, however, on the nature of meaning, as we shall see.