Interaction between Variables is the Missing Factor in Cognitive Complexity
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Interaction between Variables is the Missing Factor in Cognitive Complexity
Graeme S. HalfordUniversity of Queensland
and
Glenda Andrews
Griffith University
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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
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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)
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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)
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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
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More complex relations impose
higher processing loads, in both
children and adults
The complexity of relations that can
be represented increases with age
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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
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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)
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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
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Complexity effects can
only be assessed
where chunking and
segmentation are
inhibited.
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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.
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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
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It follows that those tasks that
impose high processing loads
are those where chunking and
segmentation are constrained.
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yellowblue
purple
red
blue green
green
red
green
red
blue
green
blue
red
Premises
Binary Ternary
Transitive Inference Task
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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.
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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
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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
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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.
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We propose that relational complexity is the missing factor in the difficulty of these tasks, for children and adults (Halford, Wilson, & Phillips, 1998).
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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.
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CIRCLES
GREEN CIRCLES YELLOW CIRCLES
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There are also three binary relations:
green to circles, yellow to circles, green is the complement of yellow within
this set of elements
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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.
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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.
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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
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CIRCLES
GREEN CIRCLES YELLOW/BLUE/ORANGE (NONGREEN) CIRCLES
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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.
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Transitivity and class inclusion are superficially different, yet both entail ternary relations:
B
>
Transitivity
CA
> >
Complement-of
Included-inIncluded-in
Circles
ClassInclusion
NongreenCircles
GreenCircles
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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.
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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
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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
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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
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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)
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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
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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
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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
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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
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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).
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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
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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
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Weight and distance discriminations in the balance scale
Binary relational:
• Discrimination of weights with distance constant
• Discrimination of distances with weight constant
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In conflict items, both weight and distance varied, and items were of three kinds:
weight dominant
distance dominant
balance (neither weight nor
distance dominant)
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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
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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
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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)
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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.
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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
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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.
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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.
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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.
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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.