53 research outputs found
Resource modalities in game semantics
The description of resources in game semantics has never achieved the
simplicity and precision of linear logic, because of a misleading conception:
the belief that linear logic is more primitive than game semantics. We advocate
instead the contrary: that game semantics is conceptually more primitive than
linear logic. Starting from this revised point of view, we design a categorical
model of resources in game semantics, and construct an arena game model where
the usual notion of bracketing is extended to multi- bracketing in order to
capture various resource policies: linear, affine and exponential
A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine
Combining insights from the study of type refinement systems and of monoidal
closed chiralities, we show how to reconstruct Lawvere's hyperdoctrine of
presheaves using a full and faithful embedding into a monoidal closed
bifibration living now over the compact closed category of small categories and
distributors. Besides revealing dualities which are not immediately apparent in
the traditional presentation of the presheaf hyperdoctrine, this reconstruction
leads us to an axiomatic treatment of directed equality predicates (modelled by
hom presheaves), realizing a vision initially set out by Lawvere (1970). It
also leads to a simple calculus of string diagrams (representing presheaves)
that is highly reminiscent of C. S. Peirce's existential graphs for predicate
logic, refining an earlier interpretation of existential graphs in terms of
Boolean hyperdoctrines by Brady and Trimble. Finally, we illustrate how this
work extends to a bifibrational setting a number of fundamental ideas of linear
logic.Comment: Identical to the final version of the paper as appears in proceedings
of LICS 2016, formatted for on-screen readin
Topological Quantum Programming in TED-K
While the realization of scalable quantum computation will arguably require
topological stabilization and, with it, topological-hardware-aware quantum
programming and topological-quantum circuit verification, the proper
combination of these strategies into dedicated topological quantum programming
languages has not yet received attention. Here we describe a fundamental and
natural scheme that we are developing, for typed functional (hence verifiable)
topological quantum programming which is topological-hardware aware -- in that
it natively reflects the universal fine technical detail of topological q-bits,
namely of symmetry-protected (or enhanced) topologically ordered Laughlin-type
anyon ground states in topological phases of quantum materials.
What makes this work is: (1) our recent result that wavefunctions of
realistic and technologically viable anyon species -- namely of su(2)-anyons
such as the popular Majorana/Ising anyons but also of computationally universal
Fibonacci anyons -- are reflected in the twisted equivariant differential (TED)
K-cohomology of configuration spaces of codimension=2 nodal defects in the host
material's crystallographic orbifold; (2) combined with our earlier observation
that such TED generalized cohomology theories on orbifolds interpret
intuitionistically-dependent linear data types in cohesive homotopy type theory
(HoTT), supporting a powerful modern form of modal quantum logic.
In this short note we give an exposition of the basic ideas, a quick review
of the underlying results and a brief indication of the basic language
constructs for anyon braiding via TED-K in cohesive HoTT. The language system
is under development at the "Center for Quantum and Topological Systems" at the
Research Institute of NYU, Abu Dhabi.Comment: 8 pages, 1 figure; extended abstract for contribution to: PlanQC2022
https://icfp22.sigplan.org/home/planqc-202
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