Interaction between Variables is the Missing Factor in Cognitive Complexity

Graeme S. HalfordUniversity of Queensland

and

Glenda Andrews

Griffith University

The Relational Complexity Metric proposed by Halford, Wilson, & Phillips

(Behav & Brain Sciences, 1998, 21(6), 803-846)

Complexity of a cognitive

process is defined by the

number of variables that must

be related in a single cognitive

representation

Number of related variables corresponds to number of slots or arity of relations

A binary relation has two slots:

e.g. Larger-than(_____, _____)

Each slot can be filled in a variety of ways:

Larger-than(elephant, mouse)

Larger-than(mountain, molehill)

Larger-than(ocean-liner, rowing-boat)

Complexity of relations can be defined by the number of slots

Unary relations have one slot: e.g. class membership, as in dog(Fido)

Binary relations have two slots: e.g. larger(elephant, mouse)

Ternary relations: e.g. addition(2,3,5)

Quaternary relations: e.g. proportion(2,3,6,9)

Because each slot can be filled in a variety of ways, a slot corresponds to a variable or dimension, thus:

a unary relation is a set of points on one dimension

a binary relation is a set of points in two-dimensional space

. . . and so on . . .

an N-ary relation is a set of points in N-dimensional space

More complex relations impose

higher processing loads, in both

children and adults

The complexity of relations that can

be represented increases with age

Normative data available suggests that: unary relations can be processed at a median

age of one year

binary relations can be processed at a median age of two years

ternary relations can be processed at a median age of five years

quaternary relations can be processed at a median age of eleven years

Strategies for reducing cognitive complexity include:

Segmentation of task into components that do not overload capacity to process information in parallel

Conceptual chunking which is equivalent to collapsing variables: e.g. velocity = distance/time can be recoded to a binding between a variable and a constant (speed = 80 kph)

By devising strategies to reduce processing load, human beings can work within the limits of their processing capacity. Note that:

a strategy can usually be found to reduce the complexity of cognitive representations

human proficiency makes it difficult to analyse complexity effects, as the more complex a task is the more important strategies become

we need to define the conditions in which complexity effects can be observed

Complexity effects can

only be assessed

where chunking and

segmentation are

inhibited.

There is a major constraint on conceptual

chunking, because chunked relations become

inaccessible (e.g. if we think of velocity as a

single variable, we cannot determine what

happens to velocity if we travel the same

distance in half the time).

Complexity analyses exploit limits on chunking

and segmentation.

Variables cannot be chunked or segmented

where an interaction between them must be

processed.

This yields a principle on which rules

for complexity analysis are based:

Variables can be chunked or

segmented only if relations

between them do not need to be

processed

It follows that those tasks that

impose high processing loads

are those where chunking and

segmentation are constrained.

yellowblue

purple

red

blue green

green

red

green

red

blue

green

blue

red

Premises

Binary Ternary

Transitive Inference Task

Transitivity Transitive reasoning requires that the relation

GREEN above RED and RED above BLUE be

integrated to form an ordered triple, GREEN

above RED above BLUE.

GREEN above BLUE can be deduced from this.

Premise integration is ternary relational because

premise elements must be assigned to three slots.

There is a constraint on segmentation because:

because both premises must be considered in the same decision

green

red

Top

Middle

Bottom red

blue

green

red

Top

Middle

Bottom

Children’s performance: transitive inference

83.393.3 95.8 100 96.7

6.4

46.7

66.7 71.4

86.7

0

20

40

60

80

100

120

4 5 6 7 8

Age (years)

Percent children

succeeding

BinaryTernary

The transitive inference task, along with class

inclusion and a number of other Piagetian tasks, has been difficult for young children, though the causes of this have been highly controversial.

After allowances are made for task variables, there is still a source of difficulty that needs to

be explained.

We propose that relational complexity is the missing factor in the difficulty of these tasks, for children and adults (Halford, Wilson, & Phillips, 1998).

Class Inclusion

In the set {4 green circles, 3 yellow circles} green things and yellow things are included in circles.

This is a ternary relation between three classes; green, yellow, circles.

CIRCLES

GREEN CIRCLES YELLOW CIRCLES

There are also three binary relations:

green to circles, yellow to circles, green is the complement of yellow within

this set of elements

No one binary relation is sufficient

for understanding inclusion

The inclusion hierarchy cannot be decomposed into a set of binary relations without losing the essence of the concept.

The processing load is due to the need to allocate classes to all three slots in the same decision

To determine that circles are superordinate we must consider relations between circles, green elements and yellow elements.

Circles are not inherently superordinate. The class of circles is the superordinate because it

includes at least two subclasses. Similarly, green is a subordinate class because it is

included in circles, and because there is at least one other subordinate class of circles.

Conceptual chunking can be illustrated by considering a class:

circles, with subclasses: green, yellow/blue/orange.

yellow/blue/orange are chunked into the single class nongreen circles

CIRCLES

GREEN CIRCLES YELLOW/BLUE/ORANGE (NONGREEN) CIRCLES

So why not chunk green, yellow, blue and orange,

and thereby reduce the concept to a binary relation?

If we do we lose the inclusion hierarchy

At least three classes are needed to represent an inclusion hierarchy and it cannot be reduced to less than a ternary relation.

Transitivity and class inclusion are superficially different, yet both entail ternary relations:

B

>

Transitivity

CA

> >

Complement-of

Included-inIncluded-in

Circles

ClassInclusion

NongreenCircles

GreenCircles

Concept of mindWhiteBird

BlueBird

Blue Filter

In one version of the appearance-reality task, children are asked what colour the bird is really (white), and what colour does it appear when viewed through the filter (blue).

Children below about 4-5 years tend to answer that the bird is white and looks white, or that it is blue and looks blue.

The essential problem is that the relation between a property of an object and the person’s percept, is modulated by a third variable, the viewing condition

The concept of mind task is complex because it entails relating three variables

Thus it is the ternary relation:

Rappear-reality(object attribute,condition,percept)

COM is predicted by other ternary tasks

A B C

Tower of Hanoi

Goal To move all discs from peg A to peg C, without: • moving more than one disc at a time • placing a larger disc on a smaller disc

Complexity in the Tower of Hanoi

depends on the levels of embedding of the goal hierarchy

goal hierarchy metric can be subsumed under the relational complexity metric

moves with more subgoals entail relations with more dimensions of complexity

A B C

Consider a 2-disc problem

Main goal: Shift disc 2 to peg CSubgoal: Shift disc 1 to peg B

12

Shift is a relation, so shifting disc 2 to

peg C can be expressed as:

shift(2,C)

The goal hierarchy can be expressed as

the higher order relation:

Prior(shift(2,C),shift(1,B))A B C

12

BA

1

C

2

There are four roles to be filled.

The task is prima facie 4-dimensional

Consider a 3-disc problem

Main goal: Shift disc 3 to peg CSubgoal 1: Shift disc 2 to peg BSubgoal 2: Shift disc 1 to peg C

Prior(shift(3,C),Prior(shift(2,B),shift(1,C)))

Thus there are now 6 roles so the task is prima facie 6 dimensional

A B C

3

21

Conceptual chunking and segmentation can be used to reduce complexity

A B C

321

The first representation of the 3-disc puzzle can be simplified by chunking discs 1 and 2 into a “pyramid”:

Prior(shift(3, C),shift (1/2,B/C)).A B C

3

1/2

In considering the next step, 1/2 and B/C can be

unchunked, yielding: (Prior(shift(2,B),shift(1,C))

A B C

23 1

A B C

23 1

Thus conceptual chunking and segmentation enable the task to be divided into two 4 dimensional subtasks

We estimate that humans are limited to processing approximately 4 dimensions in parallel

This implies that humans would normally process no more than one goal and one subgoal in a single move

This is consistent with protocol information (VanLehn, 1991, Appendix, pp. 42-47).

Dimensional change card sort task Setting condition (S1 or S2) indicates whether to

sort by color or shape

Antecedent condition (A1 or A2) that assigns attributes (colors or shapes) to categories

The structure of the task can be expressed as: S1 A1 C1 S1 A2 C2

S2 A1 C2 S2 A2 C1

Processing load depends on whether the hierarchy can be decomposed:

No interaction so S1 and S2 subtasks can be processed independently

Interaction between S1/S2 and A1/A2 constrains decomposition

S1

A1 A2

C1 C2

S2

A1 A2

C3 C4

S1

A1 A2

C1 C2

S2

A1 A2

C2 C1

Weight and distance discriminations in the balance scale

Binary relational:

• Discrimination of weights with distance constant

• Discrimination of distances with weight constant

In conflict items, both weight and distance varied, and items were of three kinds:

weight dominant

distance dominant

balance (neither weight nor

distance dominant)

Experimental findings

Experiment 1: 2-year-old children succeeded on non-conflict weight and distance problems

Experiment 2: As for Experiment 1. Performance on conflict items did not exceed chance

Experiment 3: 3- to 4-year-olds succeeded on all except conflict balance problems, while 5- to 6-year-olds succeeded on all problem types

Pairwise Correlations and Descriptive Statistics for Balance Scale, Transitivity, and Class Inclusion Tasks and Age in Experiment 3

Balance Scale

Balance Scale 1.00

Transitivity

Transitivity .60** 1.00

Age (months)

Class Inclusion .58** .61** 1.00

Age .64** .72** 1.00.74**

Mean 1.15 1.21 59.947.11

SD 0.34 0.40 13.353.68

N 104 101 104101

Class Inclusion

Cross domain correspondences

Age groups

Task Domain

Transitivity

Hierarchical Classification

Class Inclusion

Cardinality

Sentence comprehension

3,4

11

37

15

20

24

5

54

35

39

60

57

6

71

65

67

79

52

7,8

80

61

90

85

57

Percentages of Children in each Age Group with Significantly Above-chance Performance on Binary-and Ternary-relation Items by Task Domain (Experiments 1 and 2 combined)

All tasks loaded on a single factor which

accounted for • 43% of the variance (Experiment 1) • 55% of the variance (Experiment 2)

Factor scores were correlated with • age (r = .80) • fluid intelligence (r = .79) • working memory (r = .66)

Correspondence across domains for tests at the same rank was observed.

Item Characteristic Curves

Person location

0

0.5

1

-3 -2 -1 0 1 2 3

TI2 TI3 HC2 HC3 CI2 CI3 CC2 CC3 HYP3

Conclusions

The relational complexity metric can account for many previously unexplained difficulties that children have with well-known tasks in numerous domains

Complexity of a cognitive process is defined by the relation that must be represented to perform the process. Complexity analyses are based on

principles that apply across domains.

Complexity of a cognitive process can be reduced by conceptual chunking and/or segmentation, subject to the constraint that variables cannot be chunked or segmented if relations between them must be used in making the current decision.

Effective relational complexity of a cognitive process is the minimum dimensionality to which a relation can be reduced without loss of

information.

Relational complexity analyses can be applied to tasks that entail both serial and parallel processing, including tasks with a hierarchical structure. Task complexity is defined as the effective relational complexity of the most complex process entailed in the task.

Those tasks that prove consistently difficult, for both children and adults, are those in which variables interact so that they have to be considered in a single decision, and segmentation or chunking

are constrained.

Now, a reflection:

It is not possible to determine the precocity of a cognitive process unless we can assess its complexity relative to other cognitive tasks. Some precocious performances may just be simpler performances.