Types and forgetfulness in categorical linguistics and quantum mechanics

The role of types in categorical models of meaning is investigated. A general scheme for how typed models of meaning may be used to compare sentences, regardless of their grammatical structure is described, and a toy example is used as an illustration. Taking as a starting point the question of whether the evaluation of such a type system 'loses information', we consider the parametrized typing associated with connectives from this viewpoint. The answer to this question implies that, within full categorical models of meaning, the objects associated with types must exhibit a simple but subtle categorical property known as self-similarity. We investigate the category theory behind this, with explicit reference to typed systems, and their monoidal closed structure. We then demonstrate close connections between such self-similar structures and dagger Frobenius algebras. In particular, we demonstrate that the categorical structures implied by the polymorphically typed connectives give rise to a (lax unitless) form of the special forms of Frobenius algebras known as classical structures, used heavily in abstract categorical approaches to quantum mechanics.Comment: 37 pages, 4 figure

Arrows of times, non-integer operators, self-similar structures, zeta functions and Riemann hypothesis: A synthetic categorical approach

© 2017 L & H Scientific Publishing, LLC. The authors have previously reported the existence of a morphism be- tween the Riemann zeta function and the "Cole and Cole" canonical transfer functions observed in dielectric relaxation, electrochemistry, mechanics and electromagnetism. The link with self-similar struc- tures has been addressed for a long time and likewise the discovered of the incompleteness which may be attached to any dynamics con- trolled by non-integer derivative operators. Furthermore it was al- ready shown that the Riemann Hypothesis can be associated with a transition of an order parameter given by the geometric phase at- tached to the fractional operators. The aim of this note is to show that all these properties have a generic basis in category theory. The highlighting of the incompleteness of non-integer operators considered as critical by some authors is relevant, but the use of the morphism with zeta function reduces the operational impact of this issue with- out limited its epistemological consequences

Two-way automata and transducers with planar behaviours are aperiodic

We consider a notion of planarity for two-way finite automata and transducers, inspired by Temperley-Lieb monoids of planar diagrams. We show that this restriction captures star-free languages and first-order transductions.Comment: 18 pages, DMTCS submissio