Tight Subdesigns of the Higman-Sims Design

The Higman-Sims design is an incidence structure of 176 points and 176 blocks of cardinality 50 with every two blocks meeting in 14 points. The automorphism group of this design is the Higman-Sims simple group. We demonstrate that the point set and the block set of the Higman-Sims design can be partitioned into subsets X1, X2,...,X11 and B1, B2,...,B11, respectively, so that the substructures (Xi, Bi), i = 1, 2,...,11, are isomorphic symmetric (16, 6, 2)-designs

Selective Influences, Mental Architectures, and Contextuality

Given a system with say two external factors α and β and two random outputs A and B in response to the external factors. α forms the context of A and β forms the context of B. When the marginal distribution of A is not affected by the change of β and the marginal distribution of B is not affected by the change of α, we say marginal selectivity present in the system. Can we say there is no context effect then? Our answer is “not yet for interdependent A and B”. If in addition, one can find a hidden variable R, so that A can be written as a function of α and R, and B can be written as a function of β and R, selective influences are established (Dzhafarov, 2003) and one speaks of “no context effect”. Perceptual separability understands if different stimulus attributes are perceived, evaluated, and responded in a separable fashion. Selective influences provide a new definition of perceptual separability. To realize the approach, we developed psychophysical matching experiments in which the responses A and B were extracted from an observer’s choice of a stimulus that was adjusted to match the fixed stimulus with attributes α and β. We used α and β (also A and B) as simple geometric properties of dots or lines. α and β are considered perceptually separable if selective influences of α and β on A and B are established. A mental architecture is a hypothetical network of underlying cognitive processes when a subject is performing a task. It is usually assumed that the durations of processes involved in the network are selectively influenced by different experimental factors. Usually the overall duration is observable but the duration components are not. One way to characterize different types of mental architectures, e.g. the parallel vs. the serial is to compute the interaction contrast of the distribution functions of the overall durations (Townsend & Nozawa, 1995). Note that for any given value of R, the duration components and the overall duration become deterministic quantities (Zhang & Dzhafarov, 2015). Consequently, one can easily compute the interaction contrast as the probabilistic problem is reduced to simple numerical combinatorics. Our work provides a simpler method than the previously used ones to investigate theories of mental architectures. In the behavioral systems, it is very likely that marginal selectivity is absent in a system. According to the contextuality-by-default theory (Dzhafarov & Kujala, 2014a, 2014b, 2014c; Dzhafarov, Kujala, & Cervantes, 2016; Dzhafarov, Kujala, & Larsson, 2015; Kujala & Dzhafarov, 2015, 2016; Kujala, Dzhafarov, & Larsson, 2015), if the covariance between A and B can be entirely attributed to α and β, and a hidden variable R, the system is not contextual. Otherwise it is contextual. Note that when marginal selectivity is present in a system, the framework of contextuality-by-default reduces to the framework of selective influences. Contextuality is tested for cyclic systems of ranks N = 4, 6, 8 using the psychophysical matching data