893 research outputs found
Computability and Algorithmic Complexity in Economics
This is an outline of the origins and development of the way computability theory and algorithmic complexity theory were incorporated into economic and finance theories. We try to place, in the context of the development of computable economics, some of the classics of the subject as well as those that have, from time to time, been credited with having contributed to the advancement of the field. Speculative thoughts on where the frontiers of computable economics are, and how to move towards them, conclude the paper. In a precise sense - both historically and analytically - it would not be an exaggeration to claim that both the origins of computable economics and its frontiers are defined by two classics, both by Banach and Mazur: that one page masterpiece by Banach and Mazur ([5]), built on the foundations of Turing’s own classic, and the unpublished Mazur conjecture of 1928, and its unpublished proof by Banach ([38], ch. 6 & [68], ch. 1, #6). For the undisputed original classic of computable economics is RabinÃs effectivization of the Gale-Stewart game ([42];[16]); the frontiers, as I see them, are defined by recursive analysis and constructive mathematics, underpinning computability over the computable and constructive reals and providing computable foundations for the economist’s Marshallian penchant for curve-sketching ([9]; [19]; and, in general, the contents of Theoretical Computer Science, Vol. 219, Issue 1-2). The former work has its roots in the Banach-Mazur game (cf. [38], especially p.30), at least in one reading of it; the latter in ([5]), as well as other, earlier, contributions, not least by Brouwer.
Honest elementary degrees and degrees of relative provability without the cupping property
An element a of a lattice cups to an element b>ab>a if there is a c<bc<b such that a∪c=ba∪c=b. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if b is a sufficiently large honest elementary degree, then b has the anti-cupping property, which means that there is an a with 0<Ea<Eb0<Ea<Eb that does not cup to b. For comparison, we also modify a result of Cai to show, in several versions of the degrees of relative provability that are closely related to the honest elementary degrees, that in fact all non-zero degrees have the anti-cupping property, not just sufficiently large degrees
Sumcheck-based delegation of quantum computing to rational server
Delegated quantum computing enables a client with a weak computational power
to delegate quantum computing to a remote quantum server in such a way that the
integrity of the server is efficiently verified by the client. Recently, a new
model of delegated quantum computing has been proposed, namely, rational
delegated quantum computing. In this model, after the client interacts with the
server, the client pays a reward to the server. The rational server sends
messages that maximize the expected value of the reward. It is known that the
classical client can delegate universal quantum computing to the rational
quantum server in one round. In this paper, we propose novel one-round rational
delegated quantum computing protocols by generalizing the classical rational
sumcheck protocol. The construction of the previous rational protocols depends
on gate sets, while our sumcheck technique can be easily realized with any
local gate set. Furthermore, as with the previous protocols, our reward
function satisfies natural requirements. We also discuss the reward gap. Simply
speaking, the reward gap is a minimum loss on the expected value of the
server's reward incurred by the server's behavior that makes the client accept
an incorrect answer. Although our sumcheck-based protocols have only
exponentially small reward gaps as with the previous protocols, we show that a
constant reward gap can be achieved if two non-communicating but entangled
rational servers are allowed. We also discuss that a single rational server is
sufficient under the (widely-believed) assumption that the learning-with-errors
problem is hard for polynomial-time quantum computing. Apart from these
results, we show, under a certain condition, the equivalence between
and delegated quantum computing protocols. Based on this
equivalence, we give a reward-gap amplification method.Comment: 28 pages, 1 figure, Because of the character limitation, the abstract
was shortened compared with the PDF fil
Plato's cave and differential forms
In the 1970s and again in the 1990s, Gromov gave a number of theorems and
conjectures motivated by the notion that the real homotopy theory of compact
manifolds and simplicial complexes influences the geometry of maps between
them. The main technical result of this paper supports this intuition: we show
that maps of differential algebras are closely shadowed, in a technical sense,
by maps between the corresponding spaces. As a concrete application, we prove
the following conjecture of Gromov: if and are finite complexes with
simply connected, then there are constants and such that
any two homotopic -Lipschitz maps have a -Lipschitz homotopy (and
if one of the maps is a constant, can be taken to be .) We hope that it
will lead more generally to a better understanding of the space of maps from
to in this setting.Comment: 39 pages, 1 figure; comments welcome! This is the final version to be
published in Geometry & Topolog
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