This paper discusses learning spaces in the sense of Eppstein et al. (2008) to show that: (1) a learning space need not to have a base; (2) an essentially finite learning space need not to be well-graded; (3) the positive content family of a closed rooted medium need not to be a knowledge structure, and so it need not to be a learning space. These results disprove three assertions in Eppstein et al. (2008).

A variety of different recognition-memory models make different psychological assumptions, but similar predictions about ROC curves in old–new recognition-memory experiments. Some models assume that recognition responses are produced by a unitary process and other models assume they are a binary mixture of two qualitatively different types of responses. This note shows that despite their similar ROC predictions, the binary-mixture models make some striking predictions that the unitary models do not make. Specifically, in any experiment that includes conditions in which the mixture probability varies but the component distributions do not, the binary-mixture models predict that all response time probability density functions must intersect at the same time point (if they intersect at all). Similarly, they also all predict that if the ROC curves intersect, they must also all intersect at the same point.

In procedural knowledge space theory (PKST), a “problem space” is a formal representation of the knowledge that is needed for solving all of the problems of a certain type. The competence state of a real problem solver is a subset of the problem space which satisfies a specific condition, named the “sub-path assumption”. There could exist specific “symmetries” in a problem space that make certain parts of it “equivalent” up to those symmetries. Whenever an equivalence relation is introduced for elements in a problem space, the question almost naturally arises whether the collection of the induced equivalence classes forms, itself, a problem space. This is the main question addressed in the present article, which is restated as the problem of defining a homomorphism of one problem space into another problem space. Two types of homomorphisms are examined, which are named the “strong” and the “weak homomorphism”. The former corresponds to the usual notion of “operation preserving mapping”. The latter preserves operations in only one direction. Two algorithms are developed for testing the existence of homomorphisms between problem spaces. The notions and algorithms are illustrated in a series of three examples in which quite well-known neuro-psychological and cognitive tests are employed.

The Inhibited Elements Model (Brandon et al., 2000; Wagner and Brandon, 2000) has been proposed as an elemental model that may reproduce the configural model proposed by Pearce (1987, 1994), and although its mathematical formalization has been recently improved by Thorwart and Lachnit (2020), whether it actually reproduces Pearce’s model has remained an open question. In this work I further develop the mathematical formalization of the Inhibited Elements Model by casting it within the formalism proposed by Ghirlanda (2015, 2018). In doing so I will derive the conditions under which the Inhibited Elements Model reproduces Pearce’s model, showing that when all stimuli are assumed of the same “salience” these models coincide only when the application is restricted to compounds that either contain each other or have no common elements. Finally, the mathematical formalization developed here will be applied to the analytic comparison of the Inhibited Elements Model, Rescorla and Wagner’s model (Rescorla and Wagner, 1972; Wagner and Rescorla, 1972), and Pearce’s model in the context of several learning phenomena.

Within an Ames room, the perceived size of objects, such as people, changes dynamically when the objects move about within the room. The shape of the Ames room is not actually rectangular but it is perceived to be rectangular. Unfortunately, the geometrical properties of the Ames room have often been misunderstood, and rooms that have different shapes are also referred to as “Ames rooms” in many articles. In this study, the geometrical properties of the original Ames rooms constructed by Adelbert Ames, Jr. were analyzed and the generalization of the Ames room was discussed. We found that these original Ames rooms are 3D-to-3D perspective transformations of rectangular illusory rooms. Based on this analysis, we also developed a computational model that can construct a generalized Ames room that has a hexahedral shape with some free parameters that quantitatively control (i) the size and aspect-ratio of a rectangular illusory room, (ii) the amount of distortion of the Ames room from a rectangular room, and (iii) the viewpoint of an observer. This model was implemented as a computational program so that an Ames room can be constructed in a VR space. Note that the transformations of the Ames rooms can be applied to an arbitrary 3D scene and that they can be regarded as members of a subset of 3D-to-3D perspective transformations. Any perspective transformation in this subset distorts the 3D scene in such a way that the retinal image of the distorted scene, when seen from a specific viewpoint, is identical to the retinal image of the initial scene, when seen from a specific viewpoint. These generalizations allow us to control the conditions of an Ames room systematically with more flexibility when we study this illusion.

Self-control is core to human well-being. However, the lack of a well-specified, computationally tractable framework related to self-control makes it difficult to clarify underlying mechanisms, interpret relevant empirical phenomena, or develop interventions helpful in promoting self-control. To help address this gap, we invite consideration of the Comparison with Goal States Model (CGSM) for self-control. The CGSM amplifies activations related to available options whose representations are similar to representations of relevant goals and diminishes activations related to available options whose representations are dissimilar to representations of relevant goals. For example, influenced by healthy eating goals, the CGSM would amplify activations related to an apple and diminish activations related to a cookie, leading to an eventual preference for the apple, even though the cookie might be initially preferred. The CGSM successfully explicates observations related to reaction time in food choice, dynamics reflected in mouse-tracking trajectories, and showcases a mechanism by which hyperbolic discount curves in temporal discounting contexts might emerge. We use the CGSM to propose theoretical constraints on the nature of self-control and describe how multiple strategies have the potential to promote self-control.

We introduce a new framework for measuring the dynamics of category learning using Coupled Hidden Markov Models (CHMMs). The key assumptions of the framework are that people maintain a latent assignment of every stimulus to a category, and that they can update the assignments for all stimuli whenever they encounter any stimulus. These assumptions contrast with many existing accounts of category learning, which either do not allow for what is learned about one stimulus to influence the category association of others, or allow only for indirect influence. The CHMM framework allows tailored models to be developed for specific category learning tasks, taking as input the stimulus sequence and category responses people make, and producing as output inferences about the underlying dynamics of category assignments and the mechanics of the response processes. We demonstrate the framework by applying it to a categorization task considered by Lee and Navarro (2002), showing how the model measures the change in participants’ latent category assignments as they learn the category structure. We conclude by discussing potential applications of the CHMM framework to category learning situations involving prior knowledge, changing category structures, and category learning tasks that involve the consideration of multiple stimuli at one time.

The priority heuristic is a lexicographic semi-order for choosing between gambles. It has merits such as predicting, out-of-sample, people's majority choice more accurately than benchmarks such as prospect theory, having been axiomatized, and logically implying major violations of expected utility theory. The heuristic has shortcomings too, such as failing to account for individual differences and intricate choice patterns, and predicting less accurately than various model ensembles and neural networks in some environments. This note focuses on an important purported shortcoming of the heuristic, that it cannot produce valuations of gambles. I point out that the certainty equivalent of a gamble for the priority heuristic is known and suggest that this fact can be used to enhance the scope of the heuristic. Indeed, by making simple auxiliary assumptions and calculations, I demonstrate that the priority heuristic can explain the Saint Petersburg paradox and the equity premium puzzle, and to do so arguably more parsimoniously and plausibly than standard approaches.

Two theories of current interest and of mathematical and computational substance concerning knowledge assessment in education are discussed. These are the theory of knowledge structures and the theory of Bayesian networks as specifically related to educational assessment. In four separate sections, the two theories are compared by considering the sets of variables involved in their models, the set-theoretical and relational constructs defined on those variables, the probabilistic assumptions and properties, and the problems addressed by the theories in constructing their models. For the comparison, a common-base system of symbols and terms is adopted, which overcomes the peculiarities of expression in the corresponding streams of literature. This system gives us a better recognition of the similarities and differences between the two paradigms, and a precise appreciation of their arguments and abilities.