  We present a general theory of series-parallel mental architectures with
selectively influenced stochastically non-independent components. A mental
architecture is a hypothetical network of processes aimed at performing a task,
of which we only observe the overall time it takes under variable parameters of
the task. It is usually assumed that the network contains several processes
selectively influenced by different experimental factors, and then the question
is asked as to how these processes are arranged within the network, e.g.,
whether they are concurrent or sequential. One way of doing this is to consider
the distribution functions for the overall processing time and compute certain
linear combinations thereof (interaction contrasts). The theory of selective
influences in psychology can be viewed as a special application of the
interdisciplinary theory of (non)contextuality having its origins and main
applications in quantum theory. In particular, lack of contextuality is
equivalent to the existence of a "hidden" random entity of which all the random
variables in play are functions. Consequently, for any given value of this
common random entity, the processing times and their compositions (minima,
maxima, or sums) become deterministic quantities. These quantities, in turn,
can be treated as random variables with (shifted) Heaviside distribution
functions, for which one can easily compute various linear combinations across
different treatments, including interaction contrasts. This mathematical fact
leads to a simple method, more general than the previously used ones, to
investigate and characterize the interaction contrast for different types of
series-parallel architectures.
