146 research outputs found
Generalizations of the Recursion Theorem
We consider two generalizations of the recursion theorem, namely Visser's ADN
theorem and Arslanov's completeness criterion, and we prove a joint
generalization of these theorems
Predicativity and parametric polymorphism of Brouwerian implication
A common objection to the definition of intuitionistic implication in the
Proof Interpretation is that it is impredicative. I discuss the history of that
objection, argue that in Brouwer's writings predicativity of implication is
ensured through parametric polymorphism of functions on species, and compare
this construal with the alternative approaches to predicative implication of
Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
The Modal Logics of Kripke-Feferman Truth
We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results
Anticoncentration theorems for schemes showing a quantum speedup
One of the main milestones in quantum information science is to realise
quantum devices that exhibit an exponential computational advantage over
classical ones without being universal quantum computers, a state of affairs
dubbed quantum speedup, or sometimes "quantum computational supremacy". The
known schemes heavily rely on mathematical assumptions that are plausible but
unproven, prominently results on anticoncentration of random prescriptions. In
this work, we aim at closing the gap by proving two anticoncentration theorems
and accompanying hardness results, one for circuit-based schemes, the other for
quantum quench-type schemes for quantum simulations. Compared to the few other
known such results, these results give rise to a number of comparably simple,
physically meaningful and resource-economical schemes showing a quantum speedup
in one and two spatial dimensions. At the heart of the analysis are tools of
unitary designs and random circuits that allow us to conclude that universal
random circuits anticoncentrate as well as an embedding of known circuit-based
schemes in a 2D translation-invariant architecture.Comment: 12+2 pages, added applications sectio
Random walks in Euclidean space
Consider a sequence of independent random isometries of Euclidean space with
a previously fixed probability law. Apply these isometries successively to the
origin and consider the sequence of random points that we obtain this way. We
prove a local limit theorem under a suitable moment condition and a necessary
non-degeneracy condition. Under stronger hypothesis, we prove a limit theorem
on a wide range of scales: between e^(-cl^(1/4)) and l^(1/2), where l is the
number of steps.Comment: 62 pages, 1 figure, revision based on referee's report, proofs and
results unchange
The Modal Logics of Kripke-Feferman Truth
We determine the modal logic of fixed-point models of truth and their
axiomatizations by Solomon Feferman via Solovay-style completeness results.
Given a fixed-point model , or an axiomatization thereof, we
find a modal logic such that a modal sentence is a theorem of
if and only if the sentence obtained by translating the modal
operator with the truth predicate is true in or a theorem of
under all such translations. To this end, we introduce a novel version of
possible worlds semantics featuring both classical and nonclassical worlds and
establish the completeness of a family of non-congruent modal logics whose
internal logic is subclassical with respect to this semantics
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