Computation and representation

Joe Lau

Overview of lecture

• What is computation?

• Brief history

• Computational explanations in cognitive science

• Levels of description

What is computation?

• A computational process = a formal operation on representation.

• Representation– A meaningful symbol

– “makes certain information explicit”

• Formal– Described by precise rule

– Can be carried out without knowing the meaning of the symbols

Initial comments

• No restriction on what the symbols represent.

• The definition of computation is independent of the material basis of the process.– The physical constitution of the process does

not matter.

• X can be simulated on a computer X carries out computations.

An example

• Consider this process :– input : a finite string of symbols containing

only symbols from “0123456789” – output : the same string with an additional

“0” added to the end• e.g. “1237” => “12370”

– meaning : the symbols are decimal numerals representing numbers

– process computes x10.

Another example

• Input : names of people

• Output : “yes”, “no”

• Process : looks up the input in a book. Output “yes” if there is a matching entry, “no” if not.

Brief history

• Mathematical theory– Alan Turing and others (1930s)

• Precursors– Concept of algorithm : 12C Islamic

mathematician Al’ Khowarizmi– Reasoning as symbol manipulation : 17C

Thomas Hobbes – Analytic engine : 19C Charles Babbage

The computational approach in cognitive science

• Assumption : computational processes are necessary for explaining mind and behavior

• Reason : – perceptual and cognitive processes involve

information processing– info. processing requires computations.– So perception and cognition requires

computations.

Comment on argument

• This is an empirical argument.– The assumptions could turn out to be wrong.– Perhaps it is possible to do information

processing without computations.– But no plausible alternative proposals so far.

• Having a mind might require more than computations.

Examples of computational theories in cognitive science

• Computational theories are prevalent in all areas of cognitive science, e.g.– Perception– Mental imagery– Reasoning– Language– etc.

Mental ImageryAre these the same objects?

Syntax

• “The VC told the wardens to stop drinking at midnight.”– stop [drinking at midnight]– [stop drinking] at midnight.

Styles of computation

• Parallel versus serial architecture

• Analog vs. discrete representation

• Electronic vs. other (e.g. chemical) medium

• Classical vs. quantum computation

Describing a computational system

• Three levels of description (David Marr) :– level of computation : what is computed and

why– level of algorithm : the procedure and

representations used– Level of implementation : the physical

hardware

• Higher level is independent of the lower one.

Illustration

• What is computed : x10

• Two different algorithms :– Add “0” to the end of the decimal numeral.– Add the number to itself ten times.

Churchland’s criticisms of Marr

• Criticism #1 : There are more than three levels.

• Criticism #2 : The higher levels are not independent of the lower levels.

Criticism #1

Reply to criticism #1

• Marr does not have to say that there are exactly three levels.

• Those are three kinds of levels.

Criticism #2

• Some AI people think that independence of levels means that they can understand intelligence without studying neurophysiology.

• Churchland– This is not possible.– The higher level is not independent of the

lower levels.

Reply to criticism #2

• Distinguish between conceptual and epistemic independence– Might not be epistemically independent :

To discover which algorithm is used one might have to know the hardware.

– But can still be conceptually independent : The algorithm can be defined and described independently of the implementation.

Summary

• What is computation?

• Why use computational explanations?

• Three levels of describing a computational system

• Remaining issues :– Theoretical objections to approach.– Further explanation of the concept of

computation and representation.