A Connectionist Model of the Development of Transitivity

A modular connectionist model covers all six established phenomena in transitivity development in children and predicts a new effect. In contrast, a symbolic-rule hypothesis based on logic captures none of these effects and is directly contradicted by one of them. In the model a constraintsatisfaction network generates a response based on input from a feed-forward comparison module and the particular question asked. Cycles to saturate the response module implement response times. Psychology of Transitivity Piaget and his colleagues (Inhelder & Piaget, 1964; Piaget, 1969) designed the transitivity problem to assess the development of children’s logical-inference abilities. This problem often employs sticks (or times) of different length, as shown in Figure 1. Given, for example, that a child learns that stick 2 is longer than stick 1, and that stick 3 is longer than stick 2, can the child infer that stick 3 must be longer than stick 1? This is not a perceptual problem in that the child only identifies a stick by its unique color, never seeing the actual stick lengths. Piaget’s evidence suggested that correct untrained inferences, such as comparing sticks 1 and 3, did not emerge until around seven years of age, thus providing an index of the child’s entry into the stage of concrete operations. Figure 1: A six-stick version of the transitivity task. Piaget’s long-dominant view of the transitivity task as being solved by logic was ultimately contradicted in an experiment measuring the time it took for people of different ages to make various inferences (Trabasso, Riley, & Wilson, 1975). Using a six-item version of the task like that in Figure 1, Trabasso et al. trained 6-year-olds, 9-yearolds, and university students on all adjacent pairs of sticks and then asked about all possible pairs of sticks, varying the question between Which stick is longer? and Which stick is shorter? Five different effects were reported. 1. A serial-position effect: learning the adjacent pairs near the ends of the array before the pairs near the middle. 2. A distance effect: faster inferences about pairs that are farther apart in length than for pairs close together in length. 3. An anchor effect: faster inferences about pairs involving an end anchor (sticks 1 or 6) than for pairs not involving an end anchor. 4. A congruity effect: faster inferences when the term used in the question (e.g., longer) is compatible with an end anchor (e.g., the longest stick) in the pair being compared than when the question term (e.g., longer) is incompatible with an end anchor in the pair being compared (e.g., the shortest stick). 5. An age effect: older participants learned the adjacent pairs faster and made inference comparisons faster and more accurately than did younger participants. 6. Other experiments with different comparison tasks found that the distance effect diminished with increasing age (Duncan & McFarland, 1980; Sekuler & Mierkiewicz, 1977). The first four of these effects have been replicated in a wide range of tasks involving symbolic comparisons along a dimension, e.g., numerical comparisons (Banks, 1977; Duncan & McFarland, 1980; Leth-Steenson & Marley, 2000; Sekuler & Mierkiewicz, 1977). The distance effect was particularly damaging to Piaget’s logical-inference interpretation because it is precisely opposite to what Piaget would presumably predict. Assuming that each inference takes some constant time, Piaget would have to predict that the more inferences required to make a comparison, the longer the comparison would take. For example, comparing sticks 2 and 3 requires no inference at all because participants are trained on such adjacent pairs. In contrast, comparing sticks 2 and 4 requires a single inference from two premises (S2 < S3 and S3 < S4, therefore S2 < S4). And comparing sticks 2 and 5 requires two inferences (the previous inference plus this one: S2 < S4 and S4 < S5, therefore S2 < S5). The larger the split (or difference) between sticks, the more inferences would be required. The splits are conventionally termed 1, 2, and 3 in these three comparisons, respectively. 1 2 3 4 5 6 Because of the distance effect in their response-time data, Trabasso et al. concluded that people don’t use logical inference per se to solve this task. They argued instead that participants construct a spatial image of the sticks while being trained on adjacent pairs and then consult this image when asked to make another comparison. The farther the sticks are apart within this spatial image, the easier it is to make a correct comparison. Despite a recent resurgence of interest in studying and modeling transitivity, there has been no computational model that covers all six of these effects. It is, in fact, computationally unclear how the brain might construct and consult spatial images in this way. The purpose of the present work is to build such a model with cascade-correlation (CC), a neural-network learning algorithm that has been used to simulate many other phenomena in cognitive development (Shultz, 2003). Another reason to use CC is that it searches in topology space, building the network, as well as in weight space. Like other feed-forward neural algorithms, CC produces responses in more or less constant time, and thus is not naturally suitable for covering response-time effects. To add this capability, we used a modular system of two networks, which we call constraint-satisfaction cascade-correlation (CSCC). A CC network learned to judge the relative lengths of adjacent sticks and a constraint-satisfaction (CS) network used that information plus information contained in the question to generate a response. The number of update cycles that the response module required to settle into a steady state was taken as an index of response time.