Turing Machines andComputationalism

Minds & Machines

Fall 2007

Overview

• From Materialism to Computationalism– Behaviorism– Identity Theory– Functionalism– Computationalism

• Computations– What is a Computation?– Effective Computations– Turing Machines– Church-Turing Thesis– Universal Machines– The Brain as a Computer

Behaviorism

• Mental properties are behavioral dispositions.• To be intelligent is to behave a certain way

– Do well on tests– Be able to solve problems– Give correct answers to questions– Deal with new situations– Etc.

• Compare: A car’s speed. – Being fast is behavioral property.

• Problem: What about our ‘inner mental life’; our thoughts, feelings, sensations, etc.?!

Mind-Brain Identity Theory

• Mental states are brain states.• 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?

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

Multiple Realizability

• Functionalism allows for completely different kinds of entities to be intelligent, as the relevant abstract causal/functional organization can be implemented in various ways.

• Can computers be such entities?

Functionalism, Chairs, and Computers

• We can be functionalist about chairs:– What makes a chair a chair is not what it is

made out of (indeed, you can have wooden, plastic, or metal chairs), but that you can sit in it, i.e. its functionality

• But, there is no way that we can program a computer so that it becomes a chair– ‘chairhood’ is not a functionality that can be

implemented by computer program.

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.

Computationalism and the Brain

• Notice that the argument on the previous slide is a purely conceptual one in that it is not based on any empirical evidence

• However, the nature of the brain provides several further, empirical reasons to support the thesis of computationalism– We will see a number of those in the rest of the

presentation

Computationalism and the Brain, Part I

• The brain fits with computationalism:– The brain is unlike any other organ; the heart, lungs,

liver, etc. all do something very much physical: they collect, filter, pump, etc. It’s all very physical.

– The brain, however, is quite different: Its function seems to be to take in signals, and send out signals, in communication with the nervous system.

– But, as such, the brain would be an information-processor: a computer.

– Indeed, we know that the nature of the mind changes when the brain changes.

A Broad Thesis

• Computationalism simply states that cognition is some form of computation.

• But, there are many kinds of computation.• What are the kinds of computations that

underlie cognition?• What kind of computer is the brain?

• Indeed, what exactly is computation?

Formal Logic

H B

HA

~A

~H

B

2, 3 MT

A.

A.

A.

5.

4.

3.

2.

1.

1, 4 DS

The housemaid or the butler did it

If the housemaid did it, the alarm would have gone off

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.• But again, 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.

• 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.

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

• The first thing Turing points out is that for any effective computation, we write down certain symbols depending on 2 things:– What symbols are already there– What ‘stage’ in the process we’re in

• For example, take multiplication. We’re going through several stages when doing this: sometimes we multiply two digits, sometimes we add a bunch of digits, etc.

The Scope and Limits of Effective Computation III

• What we are trying to achieve during each stage of some computation varies widely between the different algorithms we use to solve the different problems.

• However, no matter how we characterize these states, what they all 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 discriminate between a bunch of different 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, 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 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.

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)

Turing-Machines and Computationalism

• The claim of computationalism is not that our mind is implemented by a Turing-machine– Again, Turing-computation is only one form of

computation

• However, the theory of Turing-machines and Turing-computability can be used to argue for computationalism by pointing out certain features of the brain– Universal Turing Machines– 0’s and 1’s

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 your laptop computes.

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 the modern, programmable, general-purpose, computer.

Computationalism and the Brain, Part II

• There are reasons to believe that our brain is ‘programmable’ too:– The brain can create or erase neural connections– More or less neural resources can be devoted to certain tasks – In case of neural defects, other areas of the brain can take

over

• In short, the brain is highly ‘plastic’, and it is presumably as a result of this that we learn to become more adept at tasks, or learn to do completely new tasks.

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 ‘glfop&^Q^GH)!#@’ 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!).

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.

• You do need lots of these 0’s and 1’s!• But this is exactly how the modern ‘digital

computer’ does things. That is, at the machine level, it’s all 0’s and 1’s.

Physical Dichotomies

• 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.

Causal Topology

• A physical system implements a computer program if and only if that system implements a certain causal topology.

• This topology is highly abstract. As long as you keep the functionality of the parts, and the connections between the parts, the same, you can:– Move parts– Stretch parts– Replace parts

• This is why there can be mechanical computers, electronic computers, DNA computers, optical computers, and quantum computers!

Computationalism and the Brain, Part III

• Again, the brain fits with these results:– One can obtain powerful information-

processing capacities using very simple resources, as long as you have lots of them

– Well, our brain has 1011 neurons, and 1014 neural connections

– Early views on the brain supposed that neurons firing or not would constitute 0’s and 1’s.

Summary

• Two independent argument for computationalism:– One conceptual: cognition is information-

processing, and that’s exactly what computers do– One empirical: the mind seems dependent on the

brain, where the brain seems to be:• an information-processing device• programmable• exploiting large numbers of simple devices that can

support complex information-processing capacities