Cognitive processing as constraint satisfaction

Today’s goals

• One example of how “neutrally-inspired” mechanisms can address problems faced by symbolic approaches (in memory)

• One example of how models can be used formally as proofs of general principles.

• Introduction to HW1 models.

Aristotle: Memory is like a library.

Everyone else for next 2300 years: Yup.

Cognitive neuroscience: LOL, wut?

How like a library?

Ways memory is not like a library

• Content-addressable

• Generalizable

• Graceful degradation

• Graded, uncertain

• Supported by neural systems

Demo

Ways memory is not like a library

• Content-addressable

• Generalizable

• Graceful degradation

• Graded, uncertain

• Supported by neural systems

It does something, but how do you know it is doing the “right” thing?

• Suppose all weights are symmetrical• Weights represent “beliefs” about the

mutual compatibility of two features• If w > 0, “If A is on B should be on

and vice versa”• If w = 0, “A has no implications for B• If w < 0, “If A is on B should be off

and vice versa”• So for any pair of features you can

measure how well their activations “fit” the belief about their compatibility as:

G_ij = a_i * a_j * w_ij

G = How “good” is the pattern of activation, given knowledge of the mutual constraints between features?

To get a measure of how well the pattern across the whole network satisfies the “beliefs” coded in the weights you can sum this over all pairs of units…

Patterns that “fit” the constraints built into the weights and the external input will produce a large value in G…

An activation function that increases toward its maximum when net input is positive and decreases it toward zero when net input is negative is guaranteed to increase G for the whole network!

In other words, a local computation (should I raise or lower my behavior?) leads to system-wide improvement in “fitting” the constraints.

Schemata as constraint satisfaction

What do you know about restaurants?

• You eat there• May be a waiter• Order food• Sit down• Obtain food• Eat food• Pay for food• Leave tip

• Order food• At counter• From waiter

• Sit down• Choose seat• Shown to seat

• At table• At counter

• Obtain food• At counter• From waiter

• Eat food

• Pay for food• Before getting it• After getting it

• Use cash• Use credit

• Leave tip

Variables

Values

SchemaContains variables

Codes relations between variables

Contains constants

Variables

Specify different possible “fillers”

Might specify default values

Desiderata

• Should capture structure– Information about common elements,

variables, distributions of values for variables

• Should capture sub-structure– The “slot” in a given schema might be filled by

another more specific schema– E.G. “Paying”

• Should be flexible and applicable to new situations.

Problems• Not actually very flexible

– Eating in a dark restaurant?– Seems like you need to store all possible structures, implausible.– How can structured representations be adapted “on the fly”?

• Overly structured– Values of “fillers” for some variables depend on how others have been filled.– Not clear how to combine different structured representations– Relational information in schema depends on values of variables contained

• E.g. Order information of “Sitting” versus “Ordering” depends on value of variable “Obtain food”

• Acquisition– How do you learn these structures in the first place?

Similar issues arise in:

• Grammar

• Event representation

• Spatial representation

• Transitive inference

• Logic / reasoning

• Analogy

Schematic knowledge as constraint satisfaction

CeilingWallsDoor

WindowVery-large

LargeMedium

SmallVery-small

DeskTelephone

BedTypewriterBookshelf

CarpetBooks

Desk-chairClock

PictureFloor-lamp

SofaEasy-chairCoffee-cup

AshtrayFireplace

DrapesStove

SinkRefrigerator

ToasterCupboardCoffeepot

DresserTV

BathtubToiletScale

CoathangerComputer

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SofaEasy-chairCoffee-cup

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DrapesStove

SinkRefrigerator

ToasterCupboardCoffeepot

DresserTV

BathtubToiletScale

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“Variables” coded in weight structure…

Function (e.g., making inferences) achieved by activation updating / hill-climbing…

Different “schemas” correspond to different localmaxima in G….

Goodness landscape determines how discrete /combinable different schemas are…

“Schemas” adapt themselves to the situation specifiedby the input…

“Coalitions” of connected units can form “sub-schemas”to provide for different levels of structure…

Homework 1 begins this week!

20s 30s 40s Pusher Burglar Bookie

Jet Shark JH HS Col.

20s 30s 40s Pusher Burglar Bookie

Jet Shark JH HS Col.

Art-in Ken-in