Algebraic Learning and Neural Network Models for Transitive and Non-transitive Responding
Transitive inference is a kind of deductive reasoning. Given the premises ''Anna is taller than Paul'' and ''Paul is taller than Mary'', adults and older children easily conclude that ''Anna is taller than Mary''. However, a related transitive responding ability has also been demonstrated in younger children and some animals with non-verbal tasks. For this, the premise statements are converted into an operant discrimination task. The subjects are offered the stimulus pairs A+ B-, B+ C-, C+ D-, D+ Ewhere + signifies that the choice of the relevant stimulus is rewarded and- indicates that its choice is penalised. When the subjects responded correctly to these premises, unreinforced tests with the conclusion stimulus pair BD were conducted. If they preferentially chose B, they were said to respond transitively. Algebraic conditioning models have been shown to be capable of reproducing such transitive behaviour. We describe a particularly simple algebraic model based on instrumental conditioning and then develop a neural network that yields transitive responding based on similar principles as the model. A variant of the model also incorporates a value transfer mechanism based on a classical conditioning process that appears to contribute occasionally to the itemordering underlying transitivity. Some humans, however, exhibit good premise pair performance but poor conclusion test performance. We consider model and network modifications that can account for this behaviour. A variant called the epsilonkappa model is shown to yield graded degrees of transitive responding with conclusion pairs while maintaining good performance on premise pairs.