1. Turing Machines and Computationalism
Minds & Machines
Fall 2005

2. Overview From Materialism to Computationalism Computations
Behaviorism
Identity Theory
Functionalism
Computationalism
Computations
What is a Computation?
Effective Computations
Turing Machines
Church-Turing Thesis
Universal Machines
The Brain as a Computer
Some Objections to Computationalism
Simulations
Semantics

3. Radical (Philosophical) Behaviorism
Mental properties are behavioral dispositions.
To be intelligent is to behave intelligent.
Mental properties reduce to properties between a biological agent and its environment. Whatever goes on inside the agent obviously contributes to the agent having those properties, but doesn’t constitute them.
Compare: a car’s speed, traction, and maneuverability.
Problem: What about our ‘inner mental life’; our thoughts, feelings, sensations, etc.?!

4. Methodological (Psychological) Behaviorism
There are inner mental states, but they can be studied in terms of behavioral dispositions.
Problems:
Not all mental states are caused by stimuli
Not all mental states cause observable behavior
For organisms with a sufficiently complex ‘inner mental life’, we’re trying to grasp the nature of a many-dimensional landscape by studying its projection onto a landscape with far less dimensions. Hence, it is possible for two agents with different mental states to have the same behavioral dispositions.
But most importantly: What are inner mental states?!

5. Identity Theory Mental states are physical states of the brain.
To have a belief X is to have a certain brain state.
Problems:
Carbon Chauvinism: Why does it have to be a carbon-based configuration of neurons? Why not using other elements or other material?
Also, what if you put the brain in a completely different kind of body or environment? In other words, isn’t the ‘meaning’ of brain states in part derived from the role that they have in the overall causal system?

6. Functionalism
Mental states of an agent can be defined relative to an abstract causal system as implemented by that agent’s sensory apparatus, motor control, and mediating mechanisms.
Functionalism can be seen as a kind of compromise between behaviorism and identity theory:
like behaviorism (and unlike identity theory) the emphasis is on the functionality of things, but
like identity theory (and unlike behaviorism) we are going to look what goes on inside of us

7. Multiple Realizability
Functionalism allows for completely different kinds of beings to be intelligent, as the relevant abstract causal/functional organization can be implemented in various ways.
Can computers be such beings?

8. Computationalism
Cognition can be defined in terms of information-processing:
Perception is taking in information from the environment
Memory/Beliefs/Knowledge is storing information
Reasoning is inferring new information from existing information
Planning is using information to make decisions
Etc.
Information-processing can be done through computations
Therefore, cognition is computation.
But, there are many kinds of computation. What are the computations that underlie computation? What kinds of computations are there anyway? In fact, what exactly is computation?

9. Formal Logic The housemaid or the butler did it
If the housemaid did it, the alarm
would have gone off 1.
H  B A.
The alarm did not go off 2.
HA A. 3. ~A A.
… therefore … 4. ~H
2, 3 MT 5. B
1, 4 DS
The butler did it!

10. 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

11. 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

12. 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.
Notice that according to this definition, humans can be computers too in the sense that they can follow that systematic procedure. That is, when we do long division on paper, we are computing, and thereby would be a computer.
Indeed, some 60 years ago, a ‘computer’ or ‘computist’ was understood to be a human being!
It was only by mechanizing these computations that we obtained computers as we now know them.

13. 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.

14. 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).

15. 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.

16. 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.

17. The Scope and Limits of Effective Computation V
Moreover, to get to some location in this string (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.

18. 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

19. Turing Machines (Demo)

20. 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, general-purpose, computers.

21. The Church-Turing Thesis
Many definitions have been proposed to capture the notion of an ‘effective computation’ other than Turing-Machines.
It turns out that all proposed definitions are equivalent in the sense that whatever one is able to compute using one computational method, one is able to compute with any of these other methods 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. For example, Turing machines can compute what this very laptop computes.

22. The Turing Limit
Still, there are certain problems that are not Turing-computable, and therefore (by the Church-Turing Thesis) not effectively computable either.
Example: The Halting Problem: deciding whether some Turing-machine 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.

23. Hypercomputation
There have been mathematical models of computation proposed that go beyond the Turing Limit. This kind of computation is called hyper-computation.
Hypercomputations attain their power through an appeal to infinity in some way or other: using infinitary precision, having an infinite amount of time, doing an infinite number of things in a finite amount of time, etc.
Hyper-computations are not effective computations, as we humans cannot follow algorithms that require us to measure things with infinite precision or to consciously perform an infinite number of actions.
Still, an interesting question is whether hyper-computation can be physically implemented. Indeed, maybe certain aspects of (unconscious) human cognition rely on hyper-computation?
That would certainly be a blow to cognitive science and AI, that have traditionally relied on computations below the Turing Limit.

24. The Impossibility of Zeus Machines
One thing is certain: we don’t implement a Zeus Machine, because Zeus machines are simply logically incoherent!
Remember, a Zeus machine is supposed to sequentially perform an infinite number of operations in a finite amount of time by performing each operation in half the time as it took the previous operation. And, as calculus shows, the time it would take to perform an infinite number of operations in this manner would be twice the time of the very first operation.
However, what this equation really shows is that if you add up an infinite number of terms, the sum converges to a finite amount. This is a far cry from showing that one could actually go through a process of summing up all these terms, one by one, and at any point be done with it. In fact, reaching the end of such an infinite series is by definition impossible!!

25. Zeno’s Paradox
The observation that one cannot finish a process of sequentially going through infinite sequence is of course at the heart of Zeno’s paradox: how can we ever move from A to B, if we have to sequentially go through an infinite number of points between A and B?
The answer is again: we can’t!
OK, but we obviously do move through space and time.
What’s up?
Maybe space and time are discrete, and so there is always only a finite number of points to go through.
Maybe motion isn’t the same as sequentially going through a number of points in space and time.

26. Symbols and Representations
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’.
It doesn’t matter what symbols we use! We can use ‘4’ to represent the number four, but we can also use ‘IV’ or or ‘5’ or ‘Bram’
Representations do have an effect on the nature of the program that is needed to do the ‘right’ thing (now you need rules for a ‘4’ instead of a ‘IV’), and also on the simplicity of the program (I have always wondered how the Romans did long division!).

27. 0’s and 1’s
An important result from computability theory is that all effective computations can be performed through the manipulation of bitstrings (strings of 0’s and 1’s) alone.
This is indeed how the modern ‘digital computer’ does things. That is, at the machine level, it’s all 0’s and 1’s.
Microsoft is actively working to patent 0 and 1. Smuckers has a patent on peanut butter and jelly sandwiches. One of the previous two claims is true.
The 0’s and 1’s are just abstractions though; they need to be physically implemented.
Thus, you need 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.

28. Implementations
A physical system implements a computer program if and only if that system implements a certain topology.
This topology is highly abstract:
Moving parts around is ok, as long as they’re still connected the same way
Stretching parts is ok, as long as they’re still connected the same way
Replacing parts is ok, as long as the new part has the same I/O properties as the original
This is exactly why there can be mechanical computers, electronic computers, bio-chemical computers, DNA computers, optical computers, and quantum computers!

29. Computationalism and the Brain
The brain seems to fit well with the thesis of computationalism:
The brain is quite unlike any other organ; organs like the heart, lungs, liver, etc. all do something very much physical: they collect, filter, release, and pump around physical material. The brain, however, seems to be just sitting there; Aristotle thought the brain was some kind of temperature regulator. Now we know that the brain takes in signals, and sends out signals through the nervous system. Thus, the brain does seem like an information-processor: a computer. So, if the mind is seen as dependent on the brain, then the mind is realized by a computer.
Also, we just saw that one can obtain powerful information-processing capacities using very simple resources. Indeed, early views on the brain supposed that neurons firing or not would constitute 0’s and 1’s.

30. Objection to Computationalism: Simulations
A computer simulation of a hurricane is just that: a simulation. It isn’t a real hurricane!
Similarly, simulating what a brain is doing is just that: a simulation of a brain, and not a real brain.

31. Response to the Simulation Objection
Well, there are two notions of ‘simulation’:
‘Computation’: a computer can simulate a hurricane in that we are able to use a computer to compute what the states are that a hurricane goes through. These states are described or displayed in some way or other, but there is no mapping between the states that that computer goes through and that the hurricane goes through. Similarly, computing the actions of a brain does indeed merely give us a description of the brain’s functioning.
‘Emulation’: However simulations in the above sense have nothing to do with the claim of computationalism which is about computations that do have the same functional organization as the brain, i.e. that emulate the brain!

32. Objection: How can you get Semantics from Syntax?
Ah, but similarly you have been using the word ‘computation’ in two different ways:
Syntactic computation: the manipulation of symbols in accordance to some algorithm
Semantic computation: the use of a syntactic computation in order to figure something out
So, syntactic computations can only become meaningful information-processing (semantic computation) if it is interpreted by some cognitive agent. In short, then, how do you get semantics from syntax?

33. Syntactic and Semantic Computationf x y ?? I O M
Syntactic Computation
Semantic Computation

34. Response
Well, good question. No one really knows how semantics, understanding, and intentionality comes into play.
Indeed, if an agent’s cognition comes about through semantic computation, then one needs to postulate a cognitive agent to perform the interpretation, which seems to lead to an infinite regress, so this is a real problem.