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Computation and representation Joe Lau

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Overview of lecture What is computation? Brief history Computational explanations in cognitive science Levels of description

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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

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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.

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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.

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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.

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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

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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.

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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.

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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.

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Mental Imagery Are these the same objects?

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

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Styles of computation Parallel versus serial architecture Analog vs. discrete representation Electronic vs. other (e.g. chemical) medium Classical vs. quantum computation

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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.

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Illustration What is computed : x10 Two different algorithms : –Add “0” to the end of the decimal numeral. –Add the number to itself ten times.

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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.

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Criticism #1

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Reply to criticism #1 Marr does not have to say that there are exactly three levels. Those are three kinds of levels.

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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.

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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.

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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.

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1Computer Sciences Department. Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER Reference 3Computer Sciences Department.

1Computer Sciences Department. Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER Reference 3Computer Sciences Department.

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