776 research outputs found

    Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations

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    Given a closed, oriented Lagrangian submanifold LL in a Liouville domain M‟\overline{M}, one can define a Maurer-Cartan element with respect to a certain L∞L_\infty-structure on the string homology H^∗S1(LL;R)\widehat{H}_\ast^{S^1}(\mathcal{L}L;\mathbb{R}), completed with respect to the action filtration. When the first Gutt-Hutchings capacity of M‟\overline{M} is finite, and LL is a K(π,1)K(\pi,1) space, it leads to interesting geometric implications. In particular, we show that LL bounds a non-constant pseudoholomorphic disc of Maslov index 2. This confirms a general form of Audin's conjecture and generalizes the works of Fukaya and Irie in the case of Cn\mathbb{C}^n to a wide class of Liouville manifolds. In particular, when dim⁥R(M‟)=6\dim_\mathbb{R}(\overline{M})=6, every closed, orientable, prime Lagrangian 3-manifold L⊂M‟L\subset\overline{M} is diffeomorphic to a spherical space form, or S1×ΣgS^1\times\Sigma_g, where ÎŁg\Sigma_g is a closed oriented surface.Comment: 75 pages, 6 figures. v2: slightly simplified the argument, references adde

    Categorical Coherence from Term Rewriting Systems

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    International audienceThe celebrated Squier theorem allows to prove coherence properties of algebraic structures, such as MacLane’s coherence theorem for monoidal categories, based on rewriting techniques. We are interested here in extending the theory and associated tools simultaneously in two directions. Firstly, we want to take in account situations where coherence is partial, in the sense that it only applies for a subset of structural morphisms (for instance, in the case of the coherence theorem for symmetric monoidal categories, we do not want to strictify the symmetry). Secondly, we are interested in structures where variables can be duplicated or erased. We develop theorems and rewriting techniques in order to achieve this, first in the setting of abstract rewriting systems, and then extend them to term rewriting systems, suitably generalized in order to take coherence in account. As an illustration of our results, we explain how to recover the coherence theorems for monoidal and symmetric monoidal categories

    A Total Break of the Scrap Digital Signature Scheme

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    Recently a completely new post-quantum digital signature scheme was proposed using the so called ``scrap automorphisms\u27\u27. The structure is inherently multivariate, but differs significantly from most of the multivariate literature in that it relies on sparsity and rings containing zero divisors. In this article, we derive a complete and total break of Scrap, performing a key recovery in not much more time than verifying a signature. We also generalize the result, breaking unrealistic instances of the scheme for which there is no particularly efficient signing algorithm and key sizes are unmanageable

    Multi-graded Featherweight Java

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    Resource-aware type systems statically approximate not only the expected result type of a program, but also the way external resources are used, e.g., how many times the value of a variable is needed. We extend the type system of Featherweight Java to be resource-aware, parametrically on an arbitrary grade algebra modeling a specific usage of resources. We prove that this type system is sound with respect to a resource-aware version of reduction, that is, a well-typed program has a reduction sequence which does not get stuck due to resource consumption. Moreover, we show that the available grades can be heterogeneous, that is, obtained by combining grades of different kinds, via a minimal collection of homomorphisms from one kind to another. Finally, we show how grade algebras and homomorphisms can be specified as Java classes, so that grade annotations in types can be written in the language itself

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Normal Form Bisimulations By Value

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    Normal form bisimilarities are a natural form of program equivalence resting on open terms, first introduced by Sangiorgi in call-by-name. The literature contains a normal form bisimilarity for Plotkin's call-by-value λ\lambda-calculus, Lassen's \emph{enf bisimilarity}, which validates all of Moggi's monadic laws and can be extended to validate η\eta. It does not validate, however, other relevant principles, such as the identification of meaningless terms -- validated instead by Sangiorgi's bisimilarity -- or the commutation of \letexps. These shortcomings are due to issues with open terms of Plotkin's calculus. We introduce a new call-by-value normal form bisimilarity, deemed \emph{net bisimilarity}, closer in spirit to Sangiorgi's and satisfying the additional principles. We develop it on top of an existing formalism designed for dealing with open terms in call-by-value. It turns out that enf and net bisimilarities are \emph{incomparable}, as net bisimilarity does not validate Moggi's laws nor η\eta. Moreover, there is no easy way to merge them. To better understand the situation, we provide an analysis of the rich range of possible call-by-value normal form bisimilarities, relating them to Ehrhard's relational model.Comment: Rewritten version (deleted toy similarity and explained proof method on naive similarity) -- Submitted to POPL2

    A survey of local-global methods for Hilbert's Tenth Problem

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    Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide correctly, for each f∈Z[X1,
,Xn]f\in\mathbb{Z}[X_{1},\dots,X_{n}], whether the diophantine equation f(X1,...,Xn)=0f(X_{1},...,X_{n})=0 has a solution in R. The celebrated `Davis-Putnam-Robinson-Matiyasevich theorem' shows that {\bf H10} for Z\mathbb{Z} is unsolvable, i.e.~there is no such algorithm. Since then, Hilbert's Tenth Problem has been studied in a wide range of rings and fields. Most importantly, for {number fields and in particular for Q\mathbb{Q}}, H10 is still an unsolved problem. Recent work of Eisentr\"ager, Poonen, Koenigsmann, Park, Dittmann, Daans, and others, has dramatically pushed forward what is known in this area, and has made essential use of local-global principles for quadratic forms, and for central simple algebras. We give a concise survey and introduction to this particular rich area of interaction between logic and number theory, without assuming a detailed background of either subject. We also sketch two further directions of future research, one inspired by model theory and one by arithmetic geometry

    A Complete Finite Equational Axiomatisation of the Fracterm Calculus for Common Meadows

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    We analyse abstract data types that model numerical structures with a concept of error. Specifically, we focus on arithmetic data types that contain an error flag ⊄\bot whose main purpose is to always return a value for division. To rings and fields we add a division operator x/yx/y and study a class of algebras called \textit{common meadows} wherein x/0=⊄x/0 = \bot. The set of equations true in all common meadows is named the \textit{fracterm calculus of common meadows}. We give a finite equational axiomatisation of the fracterm calculus of common meadows and prove that it is complete and that the fracterm calculus is decidable

    On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras

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    For a fixed finite algebra ?, we consider the decision problem SysTerm(?): does a given system of term equations have a solution in ?? This is equivalent to a constraint satisfaction problem (CSP) for a relational structure whose relations are the graphs of the basic operations of ?. From the complexity dichotomy for CSP over fixed finite templates due to Bulatov [Bulatov, 2017] and Zhuk [Zhuk, 2017], it follows that SysTerm(?) for a finite algebra ? is in P if ? has a not necessarily idempotent Taylor polymorphism and is NP-complete otherwise. More explicitly, we show that for a finite algebra ? in a congruence modular variety (e.g. for a quasigroup), SysTerm(?) is in P if the core of ? is abelian and is NP-complete otherwise. Given ? by the graphs of its basic operations, we show that this condition for tractability can be decided in quasi-polynomial time
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