IA Scholar Query: Essential and density topologies on s2-continuous posets.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSun, 12 Jun 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Divergences on Monads for Relational Program Logics
https://scholar.archive.org/work/y2jo6bw6yvg4fh2yginhsbuvae
Several relational program logics have been introduced for integrating reasoning about relational properties of programs and measurement of quantitative difference between computational effects. Towards a general framework for such logics, in this paper, we formalize quantitative difference between computational effects as divergence on monad, then develop a relational program logic acRL that supports generic computational effects and divergences on them. To give a categorical semantics of acRL supporting divergences, we give a method to obtain graded strong relational liftings from divergences on monads. We derive two instantiations of acRL for the verification of 1) various differential privacy of higher-order functional probabilistic programs and 2) difference of distribution of costs between higher-order functional programs with probabilistic choice and cost counting operations.Tetsuya Sato, Shin-ya Katsumatawork_y2jo6bw6yvg4fh2yginhsbuvaeSun, 12 Jun 2022 00:00:00 GMTStatic Analysis of Probabilistic Programs: An Algebraic Approach
https://scholar.archive.org/work/zlv4j3kxgndxrcvg7g3ukozhbu
Probabilistic programs are programs that can draw random samples from probability distributions and involve random control flows. They are becoming increasingly popular and have been applied in many areas such as algorithm design, cryptographic protocols, uncertainty modeling, and statistical inference. Formal reasoning about probabilistic programs comes with unique challenges, because it is usually not tractable to obtain the exact result distributions of probabilistic programs. This thesis focuses on an algebraic approach for static analysis of probabilistic programs. The thesis first provides a brief background on measure theory and introduces an imperative arithmetic probabilistic programming language Appl with a novel hyper-graph program model. Second, the thesis presents an algebraic denotational semantics for Appl that can be instantiated with different models of nondeterminism. The thesis also develops a new model of nondeterminism that involves nondeterminacy among state transformers and presents a domain-theoretic characterization of the new model. Based on the algebraic denotational semantics, the thesis proposes a general algebraic framework PMAF for designing, implementing, and proving the correctness of static analyses of probabilistic programs. The thesis also includes a concrete static analysis—central-moment analysis for cost accumulators in probabilistic programs—and elaborates implementation strategies to improve the usability and efficiency of the analysis. There is a gap between the general PMAF framework and the central-moment analysis, in the sense that the former is based on abstraction and iterative approximation, but the latter is based on constraint solving. The thesis provides some preliminary results on bridging the gap, via the development of novel regular hyper-path expressions, which finitely represent possibly-infinite hyperpaths on control-flow hyper-graphs of probabilistic programs without nondeterminism, and DMKAT algebraic structures, which can be used to interpret regular hyp [...]Di Wangwork_zlv4j3kxgndxrcvg7g3ukozhbuMon, 06 Jun 2022 00:00:00 GMTMatching minors in bipartite graphs
https://scholar.archive.org/work/ulk6jjkjnzh2hnh5ex5nky27a4
In this thesis we adapt fundamental parts of the Graph Minors series of Robertson and Seymour for the study of matching minors and investigate a connection to the study of directed graphs. We develope matching theoretic to established results of graph minor theory: We characterise the existence of a cross over a conformal cycle by means of a topological property. Furthermore, we develope a theory for perfect matching width, a width parameter for graphs with perfect matchings introduced by Norin. here we show that the disjoint alternating paths problem can be solved in polynomial time on graphs of bounded width. Moreover, we show that every bipartite graph with high perfect matching width must contain a large grid as a matching minor. Finally, we prove an analogue of the we known Flat Wall theorem and provide a qualitative description of all bipartite graphs which exclude a fixed matching minor.Sebastian Wiederrecht, Technische Universität Berlin, Stephan Kreutzerwork_ulk6jjkjnzh2hnh5ex5nky27a4Tue, 19 Apr 2022 00:00:00 GMTWhole-grain Petri nets and processes
https://scholar.archive.org/work/sv5ygs34tvewrmyq7c6usjl5qu
We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in the style of Meseguer and Montanari, in terms of free coloured props, and allows the following unification: for P a Petri net, the Segal space of P-processes is shown to be the free coloured prop-in-groupoids on P. There is also an unfolding semantics à la Winskel, which bypasses the classical symmetry problems: with the new formalism, every Petri net admits a universal unfolding, which in turn has associated an event structure and a Scott domain. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native. (Collective-token semantics emerges from rather drastic quotient constructions à la Best-Devillers, involving taking π_0 of the groupoids of states.)Joachim Kockwork_sv5ygs34tvewrmyq7c6usjl5quFri, 08 Apr 2022 00:00:00 GMTContextuality in foundations and quantum computation
https://scholar.archive.org/work/zbbhqhiodfaq7k6cs6kntxs5ha
Contextuality is a key concept in quantum theory. We reveal just how important it is by demonstrating that quantum theory builds on contextuality in a fundamental way: a number of key theorems in quantum foundations can be given a unifi ed presentation in the topos approach to quantum theory, which is based on contextuality as the common underlying principle. We review existing results and complement them by providing contextual reformulations for Stinespring's and Bell's theorem. Both have a number of consequences that go far beyond the evident confirmation of the unifying character of contextuality in quantum theory. Complete positivity of quantum channels is already encoded in contexts, nonlocality arises from a notion of composition of contexts, and quantum states can be singled out among more general non-signalling correlations over the composite context structure by a notion of time orientation in subsystems, thus solving a much discussed open problem in quantum information theory. We also discuss nonlocal correlations under the generalisation to orthomodular lattices and provide generalised Bell inequalities in this setting. The dominant role of contextuality in quantum foundations further supports a recent hypothesis in quantum computation, which identifi es contextuality as the resource for the supposed quantum advantage over classical computers. In particular, within the architecture of measurement-based quantum computation, the resource character of nonlocality and contextuality exhibits rather clearly. We study contextuality in this framework and generalise the strong link between contextuality and computation observed in the qubit case to qudit systems. More precisely, we provide new proofs of contextuality as well as a universal implementation of computation in this setting, while emphasising the crucial role played by phase relations between measurement eigenstates. Finally, we suggest a fine-grained measure for contextuality in the form of the number of qubits required for implementation in the no [...]Markus Frembs, Terence Rudolph, Engineering And Physical Sciences Research Councilwork_zbbhqhiodfaq7k6cs6kntxs5haWed, 23 Mar 2022 00:00:00 GMTRegular matrices of unbounded linear operators
https://scholar.archive.org/work/rtrckz6grjhfdmpj3sv5zf3v2m
Let X,Y be Banach spaces, and fix a linear operator T ∈ℒ(X,Y), and ideals ℐ, 𝒥 on ω. We obtain Silverman–Toeplitz type theorems on matrices A=(A_n,k: n,k ∈ω) of linear operators in ℒ(X,Y), so that 𝒥-lim Ax=T(ℐ-lim x) for every X-valued sequence x=(x_0,x_1,...) which is ℐ-convergent [and bounded]. This allows us to establish the relationship between the classical Silverman–Toeplitz characterization of regular matrices and its multidimensional analogue for double sequences, its variant for matrices of linear operators, and the recent version (for the scalar case) in the context of ideal convergence. As byproducts, we obtain characterizations of several matrix classes and a generalization of the classical Hahn–Schur theorem. In the proofs we will use an ideal version of the Banach–Steinhaus theorem which has been recently obtained by De Bondt and Vernaeve in [J. Math. Anal. Appl. 495 (2021)].Paolo Leonettiwork_rtrckz6grjhfdmpj3sv5zf3v2mMon, 31 Jan 2022 00:00:00 GMTQuantalic spectra of semirings
https://scholar.archive.org/work/47mfn2tc4nbeviqhglafi7vnji
Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a commutative localic semiring R (the pointfree analogue of a topological semiring), we define a spectrum of R which generalises the Stone spectrum of a distributive lattice, the Zariski spectrum of a commutative ring, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann-Lawson spectrum of a continuous frame. We then provide an explicit construction of this spectrum under conditions on R which are satisfied by our main examples. Our results are constructively valid and hence admit interpretation in any elementary topos with natural number object. For this reason the spectrum we construct should actually be a locale instead of a topological space. A simple modification to our construction gives rise to a quantic spectrum in the form of a commutative quantale. Such a quantale contains 'differential' information in addition to the purely topological information of the localic spectrum. In the case of a discrete ring, our construction produces the quantale of ideals. This prompts us to study the quantale of ideals in more detail. We discuss some results from abstract ideal theory in the setting of quantales and provide a tentative definition for what it might mean for a quantale to be nonsingular by analogy to commutative ring theory.Graham Manuellwork_47mfn2tc4nbeviqhglafi7vnjiMon, 17 Jan 2022 00:00:00 GMTResource Theories as Quantale Modules
https://scholar.archive.org/work/kmtyeaior5amlodmsxck7y7l3q
We aim to counter the tendency for specialization in science by advancing a language that can facilitate the translation of ideas and methods between disparate contexts. The focus is on questions of "resource-theoretic nature". In a resource theory, one identifies resources and allowed manipulations that can be used to transform them. Some of the main questions are: How to optimize resources? What are the trade-offs between them? Can a given resource be converted to another one via the allowed manipulations? Because of their ubiquity, methods used to answer them in one context can be used to tackle corresponding questions in new contexts. The translation occurs in two stages. Firstly, methods are generalized to the abstract language. Then, one can determine whether potentially novel contexts can accommodate them. We focus on the first stage, by introducing two variants of an abstract framework in which existing and yet unidentified resource theories can be represented. Using these, the task of generalizing concrete methods is tackled in chapter 4 by studying the ways in which meaningful measures of resources may be constructed. One construction expresses a notion of cost (or yield) of a resource. Among other applications, it may be used to extend measures from a subset of resources to a larger domain. Another construction allows the translation of resource measures from one resource theory to another. Special cases include resource robustness and weight measures, as well as relative entropy based measures quantifying minimal distinguishability from freely available resources. We instantiate some of these ideas in a resource theory of distinguishability in chapter 5. It describes the utility of systems with probabilistic behavior for the task of distinguishing between hypotheses, which said behavior may depend on.Tomáš Gondawork_kmtyeaior5amlodmsxck7y7l3qSat, 04 Dec 2021 00:00:00 GMTThe Markov-quantile process attached to a family of marginals
https://scholar.archive.org/work/cdotmy4gfbf3njcq5l2bqz4a6q
Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L'accès aux articles de la revue « Journal de l'École polytechnique -Mathématiques » (http://jep.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l'édition scientifique ouverte www.centre-mersenne.orgCharles Boubel, Nicolas Juilletwork_cdotmy4gfbf3njcq5l2bqz4a6qWed, 10 Nov 2021 00:00:00 GMTEfficient Sampling and Structure Learning of Bayesian Networks
https://scholar.archive.org/work/ejgznqpopjezxl5jkkjmiuiasu
Bayesian networks are probabilistic graphical models widely employed to understand dependencies in high dimensional data, and even to facilitate causal discovery. Learning the underlying network structure, which is encoded as a directed acyclic graph (DAG) is highly challenging mainly due to the vast number of possible networks in combination with the acyclicity constraint. Efforts have focussed on two fronts: constraint-based methods that perform conditional independence tests to exclude edges and score and search approaches which explore the DAG space with greedy or MCMC schemes. Here we synthesise these two fields in a novel hybrid method which reduces the complexity of MCMC approaches to that of a constraint-based method. Individual steps in the MCMC scheme only require simple table lookups so that very long chains can be efficiently obtained. Furthermore, the scheme includes an iterative procedure to correct for errors from the conditional independence tests. The algorithm offers markedly superior performance to alternatives, particularly because DAGs can also be sampled from the posterior distribution, enabling full Bayesian model averaging for much larger Bayesian networks.Jack Kuipers, Polina Suter, Giusi Moffawork_ejgznqpopjezxl5jkkjmiuiasuTue, 09 Nov 2021 00:00:00 GMTThe local and global versions of the Whittaker category
https://scholar.archive.org/work/fwp6wsrmzfal5k5g5jtb3s4khu
Given a category C acted on by the loop group G((t)), we define its Whittaker model Whit(C) as C^N((t)),χ, where χis a non-degenerate character. We study the properties of this construction. When C is the category of sheaves on the quotient of G((t)) by a congruence subgroup, we find a "finite-dimensional" model for Whit(C); the corresponding geometric object is Drinfeld's compactification, denoted Bun_N (with poles and level structure).Dennis Gaitsgorywork_fwp6wsrmzfal5k5g5jtb3s4khuSat, 30 Oct 2021 00:00:00 GMT