Results 1 
6 of
6
Big toy models: Representing physical systems as Chu spaces. Synthese, 2011. Online First, April 2011. Available as arXiv:0910.2393
 m,n 33 S. Abramsky. Relational Hidden Variables and NonLocality. Studia Logica
, 2012
"... We pursue a modeloriented rather than axiomatic approach to the foundations of Quantum Mechanics, with the idea that new models can often suggest new axioms. This approach has often been fruitful in Logic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, we ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
We pursue a modeloriented rather than axiomatic approach to the foundations of Quantum Mechanics, with the idea that new models can often suggest new axioms. This approach has often been fruitful in Logic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, we aim for a ‘big toy model’, in which both quantum and classical systems can be faithfully represented — as well as, possibly, more exotic kinds of systems. To this end, we show how Chu spaces can be used to represent physical systems of various kinds. In particular, we show how quantum systems can be represented as Chu spaces over the unit interval in such a way that the Chu morphisms correspond exactly to the physically meaningful symmetries of the systems — the unitaries and antiunitaries. In this way we obtain a full and faithful functor from the groupoid of Hilbert spaces and their symmetries to Chu spaces. We also consider whether it is possible to use a finite value set rather than the unit interval; we show that three values suffice, while the two standard possibilistic reductions to two values both fail to preserve fullness. 1
Completeness of the finitary Moss logic
 In Areces and Goldblatt [3
"... abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type. ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type.
Proof systems for the coalgebraic cover modality
 Same volume. Clemens Kupke, Alexander Kurz and Yde Venema
, 2008
"... abstract. We investigate an alternative presentation of classical and positive modal logic where the coalgebraic cover modality is taken as primitive. For each logic, we present a sound and complete Hilbertstyle axiomatization. Moreover, we give a twosided sound and complete sequent calculus for t ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
abstract. We investigate an alternative presentation of classical and positive modal logic where the coalgebraic cover modality is taken as primitive. For each logic, we present a sound and complete Hilbertstyle axiomatization. Moreover, we give a twosided sound and complete sequent calculus for the negationfree language, and for the language with negation we provide a onesided sequent calculus which is sound, complete and cutfree.
Completeness for the coalgebraic cover modality
 Logical Methods in Computer Science
"... We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor T: Set → Set, extends that of classical propositional logic with the socalled coalgebraic cover modality ∇T. The semantics ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor T: Set → Set, extends that of classical propositional logic with the socalled coalgebraic cover modality ∇T. The semantics of ∇T is defined in terms of a categorically defined relation lifting operation T. As the main contributions of our paper we introduce a derivation system M, and prove that M provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, and we employ Pattinson’s stratification method, showing that our derivation system can be stratified in ω many layers, corresponding to the modal depth of the formulas involved. In the proof of our main result we identify some new concepts and obtain some auxiliary results of independent interest. We survey properties of the notion T of relation lifting, induced by an arbitrary but fixed set functor T. We introduce a category Pres of Boolean algebra presentations, and establish an adjunction between Pres and the category BA of Boolean algebras. Given the fact that our derivation system M involves only formulas of depth one, it can be encoded as a functor
Coalgebras, Chu Spaces, and Representations of Physical Systems
, 2009
"... We revisit our earlier work on the representation of quantum systems as Chu spaces, and investigate the use of coalgebra as an alternative framework. On the one hand, coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimu ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
We revisit our earlier work on the representation of quantum systems as Chu spaces, and investigate the use of coalgebra as an alternative framework. On the one hand, coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimulation and coalgebraic logic. However, the standard coalgebraic framework does not accommodate contravariance, and is too rigid to allow physical symmetries to be represented. We introduce a fibrational structure on coalgebras in which contravariance is represented by indexing. We use this structure to give a universal semantics for quantum systems based on a final coalgebra construction. We characterize equality in this semantics as projective equivalence. We also define an analogous indexed structure for Chu spaces, and use this to obtain a novel categorical description of the category of Chu spaces. We use the indexed structures of Chu spaces and coalgebras over a common base to define a truncation functor from coalgebras to Chu spaces. This truncation functor is used to lift the full and faithful representation of the groupoid of physical symmetries on Hilbert spaces into Chu spaces, obtained in our previous work, to the coalgebraic semantics.
GENERALIZED POWERLOCALES VIA RELATION LIFTING
"... Abstract. This paper introduces an endofunctor VT ..."
(Show Context)