Optimal modeling for complex system design

The article begins with a brief introduction to the theory describing optimal data compression systems and their performance. A brief outline is then given of a representative algorithm that employs these lessons for optimal data compression system design. The implications of rate-distortion theory for practical data compression system design is then described, followed by a description of the tensions between theoretical optimality and system practicality and a discussion of common tools used in current algorithms to resolve these tensions. Next, the generalization of rate-distortion principles to the design of optimal collections of models is presented. The discussion focuses initially on data compression systems, but later widens to describe how rate-distortion theory principles generalize to model design for a wide variety of modeling applications. The article ends with a discussion of the performance benefits to be achieved using the multiple-model design algorithms

A Relational Formulation of the Theory of Types

This paper developes a relational---as opposed to a functional---theory of types. The theory is based on Hilbert and Bernays' eta operator plus the identity symbol, from which Church's lambda and the other usual operators are then defined. The logic is intended for use in the semantics of natural language

Switching internal times and a new perspective on the 'wave function of the universe'

Despite its importance in general relativity, a quantum notion of general covariance has not yet been established in quantum gravity and cosmology, where, given the a priori absence of coordinates, it is necessary to replace classical frames with dynamical quantum reference systems. As such, quantum general covariance bears on the ability to consistently switch between the descriptions of the same physics relative to arbitrary choices of quantum reference system. Recently, a systematic approach for such switches has been developed (arXiv:1809.00556, 1809.05093, 1810.04153). It links the descriptions relative to different choices of quantum reference system, identified as the correspondingly reduced quantum theories, via the reference-system-neutral Dirac quantization, in analogy to coordinate changes on a manifold. In this work, we apply this method to a simple cosmological model to demonstrate how to consistently switch between different internal time choices in quantum cosmology. We substantiate the argument that the conjunction of Dirac and reduced quantized versions of the theory defines a complete relational quantum theory that not only admits a quantum general covariance, but, we argue, also suggests a new perspective on the 'wave function of the universe'. It assumes the role of a perspective-neutral global state, without immediate physical interpretation, that, however, encodes all the descriptions of the universe relative to all possible choices of reference system at once and constitutes the crucial link between these internal perspectives. While, for simplicity, we use the Wheeler-DeWitt formulation, the method and arguments might be also adaptable to loop quantum cosmology.Comment: 14+7 pages. Invited contribution to the special issue "Progress in Group Field Theory and Related Quantum Gravity Formalisms", Eds. S. Carrozza, S. Gielen and D. Oriti. Minor clarifications, updated references, matches published versio

Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church's intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms of Church's intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin's intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi

Minisuperspaces: Observables and Quantization

A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e.\ phase space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through ``deparametrization'' and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory {\it inspite of the fact that the evolution is implemented by a 1-parameter family of unitary transformations}. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to quantum theory following an independent avenue. The two quantum theories --based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables-- are compared and shown to be equivalent.Comment: 34 page

Geometries, Non-Geometries, and Fluxes

Using F-theory/heterotic duality, we describe a framework for analyzing non-geometric T2-fibered heterotic compactifications to six- and four-dimensions. Our results suggest that among T2-fibered heterotic string vacua, the non-geometric compactifications are just as typical as the geometric ones. We also construct four-dimensional solutions which have novel type IIB and M-theory dual descriptions. These duals are non-geometric with three- and four-form fluxes not of (2,1) or (2,2) Hodge type, respectively, and yet preserve at least N=1 supersymmetry.Comment: 68 pages, 1 figure. v2: added references, minor changes. v3: minor change