Bidirectional Grammar and Bidirectional Optimization
Reinhard Blutner, Anatoli Strigin
Abstract:
Recently, Gerd Niestegge developed a new approach to quantum mechanics
via conditional probabilities developing the well-known proposal to
consider the Lüders-von Neumann measurement as a non-classical
extension of probability conditionalization. I will apply his powerful
and rigorous approach to the treatment of concepts using a geometrical
model of meaning. In this model, instances are treated as vectors of a
Hilbert space H . In the present approach there are at least two
possibilities to form categories. The first possibility sees
categories as a mixture of its instances (described by a density
matrix). In the simplest case we get the classical probability theory
including the Bayesian formula. The second possibility sees categories
formed by a distinctive prototype which is the superposition of the
(weighted) instances. The construction of prototypes can be seen as
transferring a mixed quantum state into a pure quantum state freezing
the probabilistic characteristics of the superposed instances into the
structure of the formed prototype. Closely related to the idea of
forming concepts by prototypes is the existence of interference
effects. Such inference effects are typically found in macroscopic
quantum systems and I will discuss them in connection with several
puzzles of bounded rationality. The present approach nicely
generalizes earlier proposals made by authors such as Diederik Aerts,
Andrei Khrennikov, Ricardo Franco, and Jerome Busemeyer. Concluding, I
will suggest that an active dialogue between cognitive approaches to
logic and semantics and the modern approach of quantum information
science is mandatory.