23,375 research outputs found
Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism
This essay examines the philosophical significance of -logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of -logical validity can then be countenanced within a coalgebraic logic, and -logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of -logical validity correspond to those of second-order logical consequence, -logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets
How to quantify coherence: Distinguishing speakable and unspeakable notions
Quantum coherence is a critical resource for many operational tasks.
Understanding how to quantify and manipulate it also promises to have
applications for a diverse set of problems in theoretical physics. For certain
applications, however, one requires coherence between the eigenspaces of
specific physical observables, such as energy, angular momentum, or photon
number, and it makes a difference which eigenspaces appear in the
superposition. For others, there is a preferred set of subspaces relative to
which coherence is deemed a resource, but it is irrelevant which of the
subspaces appear in the superposition. We term these two types of coherence
unspeakable and speakable respectively. We argue that a useful approach to
quantifying and characterizing unspeakable coherence is provided by the
resource theory of asymmetry when the symmetry group is a group of
translations, and we translate a number of prior results on asymmetry into the
language of coherence. We also highlight some of the applications of this
approach, for instance, in the context of quantum metrology, quantum speed
limits, quantum thermodynamics, and NMR. The question of how best to treat
speakable coherence as a resource is also considered. We review a popular
approach in terms of operations that preserve the set of incoherent states,
propose an alternative approach in terms of operations that are covariant under
dephasing, and we outline the challenge of providing a physical justification
for either approach. Finally, we note some mathematical connections that hold
among the different approaches to quantifying coherence.Comment: A non-technical summary of the results and applications is provided
in the first section. V5 close to the published version. Typos correcte
On palimpsests in neural memory: an information theory viewpoint
The finite capacity of neural memory and the
reconsolidation phenomenon suggest it is important to be able
to update stored information as in a palimpsest, where new
information overwrites old information. Moreover, changing
information in memory is metabolically costly. In this paper, we
suggest that information-theoretic approaches may inform the
fundamental limits in constructing such a memory system. In
particular, we define malleable coding, that considers not only
representation length but also ease of representation update,
thereby encouraging some form of recycling to convert an old
codeword into a new one. Malleability cost is the difficulty of
synchronizing compressed versions, and malleable codes are of
particular interest when representing information and modifying
the representation are both expensive. We examine the tradeoff
between compression efficiency and malleability cost, under a
malleability metric defined with respect to a string edit distance.
This introduces a metric topology to the compressed domain. We
characterize the exact set of achievable rates and malleability as
the solution of a subgraph isomorphism problem. This is all done
within the optimization approach to biology framework.Accepted manuscrip
Fragility of the Commons under Prospect-Theoretic Risk Attitudes
We study a common-pool resource game where the resource experiences failure
with a probability that grows with the aggregate investment in the resource. To
capture decision making under such uncertainty, we model each player's risk
preference according to the value function from prospect theory. We show the
existence and uniqueness of a pure Nash equilibrium when the players have
heterogeneous risk preferences and under certain assumptions on the rate of
return and failure probability of the resource. Greater competition, vis-a-vis
the number of players, increases the failure probability at the Nash
equilibrium; we quantify this effect by obtaining bounds on the ratio of the
failure probability at the Nash equilibrium to the failure probability under
investment by a single user. We further show that heterogeneity in attitudes
towards loss aversion leads to higher failure probability of the resource at
the equilibrium.Comment: Accepted for publication in Games and Economic Behavior, 201
- …