Interaction between Variables is the Missing Factor in Cognitive Complexity

1. Interaction between Variables is the Missing Factor in Cognitive Complexity Graeme S. HalfordUniversity of Queensland and Glenda Andrews Griffith University

2. The Relational Complexity Metricproposed 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

3. 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)

4. 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)

5. 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

6. More complex relations impose higher processing loads, in both children and adults The complexity of relations that can be represented increases with age

7. Normative data available suggests that: • unaryrelations can be processed at a median age of oneyear • binaryrelations can be processed at a median age of twoyears • ternary relations can be processed at a median age of five years • quaternary relations can be processed at a median age of eleven years

8. Strategies for reducing cognitive complexity include: • Segmentationof task into components that do not overload capacity to process information in parallel • Conceptual chunkingwhich 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)

9. 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

10. Complexity effects can only be assessed where chunking and segmentation are inhibited.

11. 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.

12. 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

13. It follows that those tasks that impose high processing loads are those where chunking and segmentation are constrained.

14. red green blue red Transitive Inference Task Premises blue yellow purple green green green red red blue blue Binary Ternary

15. Transitivity • Transitive reasoning requires that the relation GREENaboveRED and REDaboveBLUE be integrated to form an ordered triple, GREENabove REDaboveBLUE. • GREENaboveBLUE can be deduced from this. • Premise integration is ternary relational because premise elements must be assigned to three slots.

16. red blue Top Top green green Middle Middle red red Bottom Bottom There is a constraint on segmentation because: • because both premises must be considered in the same decision

17. Children’s performance: transitive inference

18. 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.

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

20. 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.

21. CIRCLES GREEN CIRCLES YELLOW CIRCLES

22. There are also three binary relations: • green to circles, • yellow to circles, • green is the complement of yellow within this set of elements

23. 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.

24. 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.

25. 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

26. CIRCLES GREEN CIRCLES YELLOW/BLUE/ORANGE (NONGREEN) CIRCLES

27. 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.

28. Transitivity and class inclusion are superficially different, yet both entail ternary relations: Transitivity B > > A C > Class Inclusion Circles Included-in Included-in Green Circles Nongreen Circles Complement-of

29. Concept of mind White 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. Bird Blue Bird Blue Filter

30. 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

31. B A 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

32. 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

33. Consider a 2-disc problem B A 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)

34. B C A 1 1 2 A B C 2 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. The task is prima facie 4-dimensional

35. B A C 1 2 3 Consider a 3-disc problem 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

36. B C A 1 2 3 B C A 1/2 3 Conceptual chunking and segmentation can be used to reduce complexity 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)).

37. 1 1 B C A B C A 2 2 3 3 In considering the next step, 1/2 and B/C can be unchunked, yielding: (Prior(shift(2,B),shift(1,C))

38. 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).

39. 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  C2S2 A2  C1

40. S1 S2 S1 S2 A1 A2 A1 A2 A1 A2 A1 A2 C1 C2 C3 C4 C1 C2 C2 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

41. Weight and distance discriminations in the balance scale Binary relational: • Discrimination of weights with distance constant • Discrimination of distances with weight constant

42. In conflict items, both weight and distance varied, and items were of three kinds: • weight dominant • distance dominant • balance (neither weight nor distance dominant)

43. 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

44. Class Inclusion Balance Scale Transitivity Age (months) Balance Scale 1.00 Transitivity .60** 1.00 .58** Class Inclusion .61** 1.00 .64** Age .72** .74** 1.00 Mean 1.15 1.21 7.11 59.94 SD 0.34 0.40 3.68 13.35 N 101 101 104 104 Pairwise Correlations and Descriptive Statistics for Balance Scale, Transitivity, and Class Inclusion Tasks and Age in Experiment 3

45. 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 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)

46. 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.

47. Item Characteristic Curves 1 0.5 0 -3 -2 -1 0 1 2 3 Person location TI2 TI3 HC2 HC3 CI2 CI3 CC2 CC3 HYP3

48. 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.

49. 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.

50. 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.