Geometrical organization of solutions to random linear Boolean equations

The random XORSAT problem deals with large random linear systems of Boolean variables. The difficulty of such problems is controlled by the ratio of number of equations to number of variables. It is known that in some range of values of this parameter, the space of solutions breaks into many disconnected clusters. Here we study precisely the corresponding geometrical organization. In particular, the distribution of distances between these clusters is computed by the cavity method. This allows to study the `x-satisfiability' threshold, the critical density of equations where there exist two solutions at a given distance.Comment: 20 page

Flexible and context-dependent categorization by eighteen-month-olds

One hundred 18‐month‐olds were tested using sequential touching and following 4 different priming contexts using sets of toys that could be simultaneously categorized at either the basic or global level. An exact expression of the expected mean sequence length for arbitrary categories was derived as a function of the number of touches made, and a finite mixture model analytic method was also used to explore individual variability in categorization. Toddlers could categorize flexibly and spontaneously selected the level of categorization as a function of the prior prime. Perceptual Variability emerged as a predictor of the level at which infants subsequently categorized. The infants were also able to classify objects as members of both basic‐ and global‐level categories simultaneously