Interaction between Variables is
the Missing Factor in Cognitive
Complexity
Graeme S. Halford
University 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 twodimensional space

. . . and so on . . .

an N-ary relation is a set of points in Ndimensional 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.
Transitive Inference Task
Premises
blue
red
yellow
green
purple
blue
green
red
green
green
red
blue
Binary
red
blue
Ternary
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
Top
Middle
Bottom
Top
green
red
green
Middle
red
Bottom
red
blue
Children’s performance:
transitive inference
120
100
95.8
93.3
100
96.7
83.3
86.7
80
Percent
children 60
succeeding
40
71.4
66.7
Binary
Ternary
46.7
20
6.4
0
4
5
6
Age (years)
7
8

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:
Transitivity
>
A
Class
Inclusion
B
>
>
C
Circles
Included-in
Green
Circles
Included-in
Complement-of
Nongreen
Circles
Concept of mind
White
Bird
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.
Blue
Bird
Blue Filter

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
Tower of Hanoi
A
B
C
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
Consider a 2-disc problem
A
B
C
1
2
Main goal: Shift disc 2 to peg C
Subgoal:
Shift disc 1 to peg B
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))
There are four roles to
be filled.
A
B
2
1
A
C
B
C
1
2
The task is prima facie
4-dimensional
Consider a 3-disc problem
A
B
C
1
2
3
Main goal: Shift disc 3 to peg C
Subgoal 1: Shift disc 2 to peg B
Subgoal 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
Conceptual chunking and segmentation can
be used to reduce complexity
B
A
C
1
2
3
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
1/2
3
B
C
In considering the next step, 1/2 and B/C can be
unchunked, yielding:
(Prior(shift(2,B),shift(1,C))
A
B
2
3
C
1
A
B
C
3
2
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:
S1
S2
S1
S2
A1
A2
A1
A2
A1
A2
A1
A2
C1
C2
C2
C1
C1
C2
C3
C4
Interaction between S1/S2 and
A1/A2 constrains decomposition
No interaction so S1 and S2
subtasks can be processed
independently
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 6year-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 Transitivity
Balance Scale
Class Inclusion
Age (months)
1.00
Transitivity
.60**
1.00
Class Inclusion
.58**
.61**
1.00
Age
.64**
.72**
.74**
1.00
Mean
1.15
1.21
7.11
59.94
SD
0.34
0.40
3.68
13.35
N
104
101
101
104
Cross domain correspondences
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)
Age groups
Task Domain
3,4
5
6
7,8
Transitivity
11
54
71
80
Hierarchical
Classification
37
35
65
61
Class Inclusion
15
39
67
90
Cardinality
20
60
79
85
Sentence
comprehension
24
57
52
57

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
1
0.5
0
-3
-2
-1
0
1
2
3
CC2
CC3
Person location
TI2
TI3
HC2
HC3
CI2
CI3
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.