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Turing’s Legacy
Minds & Machines
October 14, 2004
Turing’s Legacy
• Turing’s legacy consists of 2 parts:
– Turing Machines
– Turing Test
Formal Logic
The housemaid or the butler did it
If the housemaid did it, the alarm
would have gone off
1. H  B
A.
2.
HA
A.
3.
~A
A.
4. ~H
2, 3 MT
5.
1, 4 DS
B
The alarm did not go off
… therefore …
The butler did it!
Algorithms
• An algorithm is a systematic, step-by-step
procedure:
– Steps: Algorithms take discrete steps
– Precision: Each step is precisely defined
– Systematicity: After each step it is clear which step to
take next
• Examples:
– Cookbook recipe
– Filling out tax forms (ok, maybe not)
– Long division
Computations
• Computations are where the ideas of
formal logic and algorithms come together.
• A computation is a symbol-manipulation
algorithm.
– Example: long division.
• Not every algorithm is a computation
– Example: furniture assembly instructions
Computers
• A ‘computer’ is something that computes, i.e.
something that performs a computation, i.e.
something that follows a systematic procedure to
transform input symbol strings into output
symbol strings.
• As long as the procedure is effective, humans
can take the role of a computer by following that
procedure. Indeed, some 60 years ago, a
‘computer’ was understood to be a human
being!
• By mechanizing this process, we obtain
computers as we now know them.
The Scope and Limits of
Effective Computation I
• An algorithm or procedure that we humans are
able to follow or execute is called ‘effective’.
• In 1936, Turing wrote a paper in which he
explored the scope and limits of effective
computation.
• Turing tried to find the basic elements (the
atomic components) of such a process.
The Scope and Limits of
Effective Computation II
• Take the example of multiplication: we make
marks on any place on the paper, depending on
what other marks there already are, and on what
‘stage’ in the algorithm we are (we can be in the
process of multiplying two digits, adding a bunch
of digits, carrying over).
• So, when going through an algorithm we go
through a series of stages or states that indicate
what we should do next (we should multiply two
digits, we should write a digit, we should carry
over a digit, we should add digits, etc).
The Scope and Limits of
Effective Computation III
• The stages we are in vary widely between the different
algorithms we use to solve different problems.
• However, no matter how we characterize these states,
what they ultimately come down to is that they indicate
what symbols to write based on what symbols there are.
• Hence, all we should be able to do is to be able to
discriminate between different states, but what we call
them is completely irrelevant.
• Moreover, although an algorithm can have any number
of stages defined, since we want an answer after a finite
number of steps, there can only be a finite number of
such states.
• One could also try and argue that we are cognitively only
able to discriminate between, or even simply define, a
finite number of states since our memory is limited.
Thus, again, there can only be a finite number of states.
The Scope and Limits of
Effective Computation IV
• Next, Turing reasoned that while one can write
as many symbols as one wants at any location
on the paper, one can only write one symbol at a
a time, and symbols have a discrete location on
the paper. Therefore, at any point in time the
number of symbols on the paper is finite, hence
we can number them, and hence we should be
able to do whatever we did before by writing the
symbols in one big long string of symbols,
possibly using other symbols to indicate
relationships between the original symbols, and
adding symbols to the left or right as needed.
The Scope and Limits of
Effective Computation V
• Moreover, to get to some location (whether to
read or write a symbol), we just need to be able
to go back and forth, one symbol at a time, along
this one big symbol string. We can add a few
states to indicate that we are in the process of
doing so, so this should pose no restrictions on
what we would be able to do.
• Finally, while the marks can be arbitrary, they
can only have a finite size, and hence there can
only be finitely many symbols, or else there
would have to be two symbols that are so much
alike that we can no longer perceptually
discriminate between them.
The Scope and Limits of
Effective Computation VI
• Turing thus obtained the following basic
components of effective computation:
– A finite set of states
– A finite set of symbols
– One big symbol string that can be added to on either
end
– An ability to move along this symbol string (to go left
or right)
– An ability to read a symbol
– An ability to write a symbol
Turing Machines Demo
Representations, Efficiency, and
Computational Power
• In our example, we used a ‘unary’ representation of
numbers (i.e. the number four was represented as
‘1111’), but we could also have used some other
representation, such as ‘4’ or ‘IV’.
• The choice in representation obviously has an effect on
the nature of the program that is needed to do the ‘right’
thing (now you need rules for encountering a ‘4’ instead
of a ‘I’).
• Depending on the problem, some representations lead to
computations that are simpler (I always wondered how
the Romans did long division!) and more efficient than
others.
• However, if one is able to eventually get the answer
using some representational scheme then, no matter
how inefficient that representational scheme is, the
computational ‘power’ of that scheme is equal to that of
some other scheme that gets the same answer.
0’s and 1’s
• As it turns out, we can always use a string of bits, i.e. a
string of 0’s and 1’s to represent objects, without losing
any computational power.
• This is indeed how the modern ‘digital computer’ does
things. That is, at the machine level, it’s all 0’s and 1’s.
• Of course, the 0’s and 1’s are just abstractions here; they
refer to some kind of physical dichotomy, e.g. hole in
punch card or not, voltage high or low, quantum spin up
or down, penny on piece of toilet paper or not, etc.
• The fact that one can decide to use anything to
represent anything is exactly why there can be
mechanical computers, electronic computers, biochemical computers, DNA computers, optical computers,
quantum computers and, as demonstrated by the Turing
machine, computers made of one (very) big roll of toilet
paper and (a lot of) pennies!
The Church-Turing Thesis
• Many definitions have been proposed to capture the
notion of an ‘effective computation’ other than TuringMachines.
• It turns out that all proposed definitions up to this date
are equivalent in the sense that whatever one
computational method is able to compute, any other
method can compute as well.
• The Church-Turing thesis states that Turing-machines
capture the notion of effective computation: whatever is
effectively computable, Turing-machines can compute.
• The Church-Turing thesis shows the amazing
computational power of Turing machines: Turing
machines can compute what this very laptop computes,
and they can do so using only 0’s and 1’s!
The Turing-Limit and
Hyper-Computation
• Still, there are certain problems that are not Turingcomputable, and therefore (by Church’s Thesis) probably
not effectively computable either.
– Example: The Halting Problem: deciding whether some Turingmachine will or will not halt for some input.
• This means that there is a non-trivial limit (called the
Turing-limit) to what can be effectively computed.
• There have been mathematical models of computation
proposed that go beyond the Turing-Limit. This kind of
computation is called hyper-computation.
• Hyper-computations are not effective computations as
they appeal to infinity in some way or other: infinitary
precision, infinitary time, etc.
• An interesting question is whether hyper-computation
can be physically implemented. Maybe certain aspects
of human cognition rely on hyper-computation?
Universal Turing Machines
• One of Turing’s great achievements was his
finding that one can make a Universal Turing
Machine, which is a Turing Machine U that can
simulate the behavior of any Turing Machine M
by giving a description of that machine M and
the input I that M would work on to machine U.
• This led to the notion of stored programs
(programs as part of the data), and thus to
programmable, all-purpose, computers.
• Thus, what was used as part of a proof of a
rather abstract mathematical result turned out to
revolutionize our lives!
Computationalism
• Computationalism is the view that cognition is
computation (i.e. computers can think, have beliefs, be
intelligent, self-conscious, etc.)
• The idea behind Computationalism is that the brain is,
like a computer, an information-processing device: it
takes information from the environment (perception), it
stores that information (memory/knowledge), infers new
information from existing information (reasoning,
anticipation), and makes decisions based on this
information (action).
• Computationalism fits well with the finding that one can
obtain powerful information-processing capacities using
very simple resources: Early views on the brain
supposed that neurons firing or not would constitute 0’s
and 1’s
Minimal Computationalism
• Notice that computationalism does not make any
claims regarding the computational architecture
underlying cognition.
• Thus, computationalism is compatible with:
– Logicism: the view that cognition is best studied
through formal logic-based representations and
manipulations
– Computerism: the view that our cognitive architecture
is like the Von Neumann architecture of a modern
computer
– Connectionism: the view that cognition is best studied
through neural networks
– Hyper-computation: computations that go beyond the
Turing-limit
Computing Things
• The symbols that computations manipulate are
representations of things.
• By manipulating those representations we come
to know something about the things that those
representations represent. Thus, things become
computable: ‘I can compute the ratio of 2
numbers’.
• I can use a computer to help me compute things.
• But now we have 2 notions of computation:
– Syntactic Computation: A computer going through a
process of symbol manipulation
– Semantic computation: Representing something by
some symbol string, manipulating that symbol string
into a new symbol string, and interpreting that new
symbol string into something meaningful.
Syntactic and Semantic
Computation
2,2
f
0011011000
4
001111000
M
Syntactic Computation
Semantic Computation
Meaning, Intentionality, and
The Chinese Room
• Semantic computation involves syntactic computation
and an interpreter to give some meaning to the syntactic
computation. But, how can syntactic computations by
themselves introduce any meaning?
• This relates to a distinction discussed by Dennett
between original (intrinsic) and derived intentionality: it
seems that the syntactic computation can only have
derived intentionality thanks to the interpreter (who has
intrinsic intentionality).
• John Searle uses exactly this line of reasoning in his
Chinese Room thought experiment to argue that
computations alone cannot lead to any intentionality, and
hence that computationalism is false.
• In other words, computations can lead to all kinds of
interesting functionality and behavior, but it isn’t
intelligence!
The Turing Test
Machine
Interrogator
Human
“I believe that in about fifty years’ time it will
be possible to programme computers, with a
storage capacity of about 109, to make them play
the imitation game so well that an average
interrogator will not have more than 70 per cent
chance of making the right identification after
5 minutes of questioning”
-Alan Turing (1950)
Turing’s Argument for AI
Premise 1: Machines can pass the Turing Test
Premise 2: Anything that passes the Turing Test
is intelligent
Conclusion: Machines can be intelligent
Can Machines pass the Turing
Test?
• Turing thinks that machines will at some point in
the future be able to pass the Turing Test,
because Turing thinks that passing the test
requires nothing more than some kind of
information processing ability, which is exactly
what computers do.
• In fact, Turing points to his Universal Machine to
demonstrate that a single, fairly simple, machine
can do all kinds of information-processing.
A Puzzle
• But wait, can’t we then just argue as follows:
– Intelligence requires nothing more than some kind of
information processing ability,
– Computers can have this information processing
ability
– Therefore, computers can be intelligent.
• Indeed, this is exactly how proponents of AI
make the argument today.
• So why didn’t Turing make this very argument?
Why bring in the game?
A Second Puzzle
• Also, why the strange set-up of the TuringTest? Why did Turing ‘pit’ a machine
against a human in some kind of contest?
Why not have the interrogator simply
interact with a machine and judge whether
or not the machine is intelligent based on
those interactions?
The Super-Simplified Turing Test
Interrogator
Machine
Answer: Bias
• The mere knowledge that we are dealing
with a machine will bias our judgment as
to whether that machine can think or not,
as we may bring certain preconceptions
about machines to the table.
• Moreover, knowing that we are dealing
with a machine will most likely lead us to
raise the bar for intelligence: it can’t write a
sonnet? Ha, I knew it!
• By shielding the interrogator from the
interrogated, such a bias and bar-raising is
eliminated in the Turing-Test.
The Simplified Turing Test
Interrogator
Machine or Human
Level the Playing Field
• Since we know we might be dealing with a
machine, we still raise the bar for the entity
on the other side being intelligent.
• Through his set-up of the test, Turing
made sure that the bar for being intelligent
wouldn’t be raised any higher for
machines than we do for fellow humans.
Is the Imitation Game a Test?
• Still, we are left with the first puzzle: if Turing wanted to
argue that machines can be intelligent, why bring up any
test at all?
• I believe that Turing never intended the test as part of
any argument for machine intelligence.
• Instead, Turing used the test to make us aware of the
point that we should level the playing field as far as
attributing intelligence goes: if we attribute intelligence to
humans based on their behavior then, just to be fair, we
should do it for machines as well.
• Thus, the convoluted set-up wasn’t merely a practical
consideration to eliminate bias: it was the whole point!
• In fact, Turing hardly uses the word ‘test’ in his original
article, and instead talks about it as the Imitation Game.
• Thus, Turing doesn’t say make any claims about
intelligence, but rather how we use and apply the word.
In Turing’s Words
The original question, “Can machines think?”, I believe
to be too meaningless to deserve discussion. Nevertheless
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.
-Alan Turing (1950)