2,456 research outputs found
A Systematic Approach to Canonicity in the Classical Sequent Calculus
International audienceThe sequent calculus is often criticized for requiring proofs to contain large amounts of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps---such as instantiating a block of quantifiers---by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proof-nets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical first-order logic. The essential element of our approach is the use of a multi-focused sequent calculus as the means of abstracting away the details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that make the foci as parallel as possible are canonical. Moreover, such proofs are isomorphic to expansion proofs---a well known, minimalistic, and parallel generalization of Herbrand disjunctions---for classical first-order logic. This technique is a systematic way to recover the desired essence of any sequent proof without abandoning the sequent calculus
Permutability in proof terms for intuitionistic sequent calculus with cuts
This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.Partially financed by FCT through project UID/MAT/00013/2013, and by COST action CA15123 EUTYPES. The first and the last authors were partially financed by Fundação para
a Ciência e a Tecnologia (FCT) through project UID/MAT/00013/2013. The first author got
financial support by the COST action CA15123 EUTYPES.info:eu-repo/semantics/publishedVersio
Multidimensional hyperbolic billiards
The theory of planar hyperbolic billiards is already quite well developed by
having also achieved spectacular successes. In addition there also exists an
excellent monograph by Chernov and Markarian on the topic. In contrast, apart
from a series of works culminating in Sim\'anyi's remarkable result on the
ergodicity of hard ball systems and other sporadic successes, the theory of
hyperbolic billiards in dimension 3 or more is much less understood. The goal
of this work is to survey the key results of their theory and highlight some
central problems which deserve particular attention and efforts
The mirror conjecture for minuscule flag varieties
We prove Rietsch's mirror conjecture that the Dubrovin quantum connection for
minuscule flag varieties is isomorphic to the character D-module of the
Berenstein-Kazhdan geometric crystal. The idea is to recognize the quantum
connection as Galois and the geometric crystal as automorphic. We reveal
surprising relations with the works of Frenkel-Gross, Heinloth-Ng\^o-Yun and
Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results
where Hecke eigensheaves are determined by their local ramification. As
corollaries we obtain combinatorial identities for counts of rational curves
and the Peterson variety presentation of the small quantum cohomology ring
Gabor Frames for Quasicrystals, -theory, and Twisted Gap Labeling
We study the connection between Gabor frames for quasicrystals, the topology
of the hull of a quasicrystal and the -theory of the twisted
groupoid -algebra arising from a quasicrystal. In
particular, we construct a finitely generated projective module
\mathcal{H}_\L over related to time-frequency analysis,
and any multiwindow Gabor frame for can be used to construct an
idempotent in representing \mathcal{H}_\L in
We show for lattice subsets in dimension two, this
element corresponds to the Bott element in allowing
us to prove a twisted version of Bellissard's gap labeling theorem
Fr\'echet Modules and Descent
We study several aspects of the study of Ind-Banach modules over Banach rings
thereby synthesizing some aspects of homological algebra and functional
analysis. This includes a study of nuclear modules and of modules which are
flat with respect to the projective tensor product. We also study metrizable
and Fr\'{e}chet Ind-Banach modules. We give explicit descriptions of projective
limits of Banach rings as ind-objects. We study exactness properties of
projective tensor product with respect to kernels and countable products. As
applications, we describe a theory of quasi-coherent modules in Banach
algebraic geometry. We prove descent theorems for quasi-coherent modules in
various analytic and arithmetic contexts.Comment: improved versio
Characterization of strong normalizability for a sequent lambda calculus with co-control
We study strong normalization in a lambda calculus of proof-terms
with co-control for the intuitionistic sequent calculus. In this sequent
lambda calculus, the management of formulas on the left hand
side of typing judgements is “dual" to the management of formulas
on the right hand side of the typing judgements in Parigot’s lambdamu
calculus - that is why our system has first-class “co-control".
The characterization of strong normalization is by means of intersection
types, and is obtained by analyzing the relationship with
another sequent lambda calculus, without co-control, for which a
characterization of strong normalizability has been obtained before.
The comparison of the two formulations of the sequent calculus,
with or without co-control, is of independent interest. Finally, since
it is known how to obtain bidirectional natural deduction systems
isomorphic to these sequent calculi, characterizations are obtained
of the strongly normalizing proof-terms of such natural deduction
systems.The authors would like to thank the anonymous
referees for their valuable comments and helpful suggestions.
This work was partly supported by FCT—Fundação para a Ciência
e a Tecnologia, within the project UID-MAT-00013/2013; by
COST Action CA15123 - The European research network on types
for programming and verification (EUTypes) via STSM; and by the
Ministry of Education, Science and Technological Development,
Serbia, under the projects ON174026 and III44006.info:eu-repo/semantics/publishedVersio
Estimation under group actions: recovering orbits from invariants
Motivated by geometric problems in signal processing, computer vision, and
structural biology, we study a class of orbit recovery problems where we
observe very noisy copies of an unknown signal, each acted upon by a random
element of some group (such as Z/p or SO(3)). The goal is to recover the orbit
of the signal under the group action in the high-noise regime. This generalizes
problems of interest such as multi-reference alignment (MRA) and the
reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain
matching lower and upper bounds on the sample complexity of these problems in
high generality, showing that the statistical difficulty is intricately
determined by the invariant theory of the underlying symmetry group.
In particular, we determine that for cryo-EM with noise variance
and uniform viewing directions, the number of samples required scales as
. We match this bound with a novel algorithm for ab initio
reconstruction in cryo-EM, based on invariant features of degree at most 3. We
further discuss how to recover multiple molecular structures from heterogeneous
cryo-EM samples.Comment: 54 pages. This version contains a number of new result
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