1,733 research outputs found

    The Diophantine problem in Chevalley groups

    Full text link
    In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R)G_\pi (\Phi,R), where Φ\Phi is an indecomposable root system of rank >1> 1, RR is an arbitrary commutative ring with 11. We establish a variant of double centralizer theorem for elementary unipotents xα(1)x_\alpha(1). This theorem is valid for arbitrary commutative rings with 11. The result is principle to show that any one-parametric subgroup XαX_\alpha, αΦ\alpha \in \Phi, is Diophantine in GG. Then we prove that the Diophantine problem in Gπ(Φ,R)G_\pi (\Phi,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in RR. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.Comment: 44 page

    Entanglement of Sections: The pushout of entangled and parameterized quantum information

    Full text link
    Recently Freedman & Hastings asked for a mathematical theory that would unify quantum entanglement/tensor-structure with parameterized/bundle-structure via their amalgamation (a hypothetical pushout) along bare quantum (information) theory. As a proposed answer to this question, we first make precise a form of the relevant pushout diagram in monoidal category theory. Then we prove that the pushout produces what is known as the *external* tensor product on vector bundles/K-classes, or rather on flat such bundles (flat K-theory), i.e., those equipped with monodromy encoding topological Berry phases. The bulk of our result is a further homotopy-theoretic enhancement of the situation to the "derived category" (infinity-category) of flat infinity-vector bundles ("infinity-local systems") equipped with the "derived functor" of the external tensor product. Concretely, we present an integral model category of simplicial functors into simplicial K-chain complexes which conveniently presents the infinity-category of parameterized HK-module spectra over varying base spaces and is equipped with homotopically well-behaved external tensor product structure. In concluding we indicate how this model category serves as categorical semantics for the linear-multiplicative fragment of Linear Homotopy Type Theory (LHoTT), which is thus exhibited as a universal quantum programming language. This is the context in which we recently showed that topological anyonic braid quantum gates are native objects in LHoTT.Comment: 71 pages, various figure

    Revisiting Language Support for Generic Programming: When Genericity Is a Core Design Goal

    Get PDF
    ContextGeneric programming, as defined by Stepanov, is a methodology for writing efficient and reusable algorithms by considering only the required properties of their underlying data types and operations. Generic programming has proven to be an effective means of constructing libraries of reusable software components in languages that support it. Generics-related language design choices play a major role in how conducive generic programming is in practice.InquirySeveral mainstream programming languages (e.g. Java and C++) were first created without generics; features to support generic programming were added later, gradually. Much of the existing literature on supporting generic programming focuses thus on retrofitting generic programming into existing languages and identifying related implementation challenges. Is the programming experience significantly better, or different when programming with a language designed for generic programming without limitations from prior language design choices?ApproachWe examine Magnolia, a language designed to embody generic programming. Magnolia is representative of an approach to language design rooted in algebraic specifications. We repeat a well-known experiment, where we put Magnolia’s generic programming facilities under scrutiny by implementing a subset of the Boost Graph Library, and reflect on our development experience.KnowledgeWe discover that the idioms identified as key features for supporting Stepanov-style generic programming in the previous studies and work on the topic do not tell a full story. We clarify which of them are more of a means to an end, rather than fundamental features for supporting generic programming. Based on the development experience with Magnolia, we identify variadics as an additional key feature for generic programming and point out limitations and challenges of genericity by property.GroundingOur work uses a well-known framework for evaluating the generic programming facilities of a language from the literature to evaluate the algebraic approach through Magnolia, and we draw comparisons with well-known programming languages.ImportanceThis work gives a fresh perspective on generic programming, and clarifies what are fundamental language properties and their trade-offs when considering supporting Stepanov-style generic programming. The understanding of how to set the ground for generic programming will inform future language design.</p

    Filtered colimit elimination from Birkhoff's variety theorem

    Full text link
    Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon "filtered colimit elimination," and study a sufficient condition for it. We show that if a locally finitely presentable category A\mathscr{A} satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to A\mathscr{A}.Comment: 22 page

    Categorical structures for deduction

    Get PDF
    We begin by introducing categorized judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and first-order logic as special kinds of categorized judgemental theories. We believe our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. For first-order logic we offer a deep analysis of structural rules, describing some of their properties, and putting them into context. We then put one of the main constructions introduced, namely that of categorized judgemental dependent type theories, to the test: we frame it in the general context of categorical models for dependent types, describe a few examples, study its properties, and use it to model subtyping and as a tool to prove intrinsic properties hidden in other models. Somehow orthogonally, then, we show a different side as to how categories can help the study of deductive systems: we transport a known model from set-based categories to enriched categories, and use the information naturally encoded into it to describe a theory of fuzzy types. We recover structural rules, observe new phenomena, and study different possible enrichments and their interpretation. We open the discussion to include different takes on the topic of definitional equality

    Canonical Algebraic Generators in Automata Learning

    Get PDF
    Many methods for the verification of complex computer systems require the existence of a tractable mathematical abstraction of the system, often in the form of an automaton. In reality, however, such a model is hard to come up with, in particular manually. Automata learning is a technique that can automatically infer an automaton model from a system -- by observing its behaviour. The majority of automata learning algorithms is based on the so-called L* algorithm. The acceptor learned by L* has an important property: it is canonical, in the sense that, it is, up to isomorphism, the unique deterministic finite automaton of minimal size accepting a given regular language. Establishing a similar result for other classes of acceptors, often with side-effects, is of great practical importance. Non-deterministic finite automata, for instance, can be exponentially more succinct than deterministic ones, allowing verification to scale. Unfortunately, identifying a canonical size-minimal non-deterministic acceptor of a given regular language is in general not possible: it can happen that a regular language is accepted by two non-isomorphic non-deterministic finite automata of minimal size. In particular, it thus is unclear which one of the automata should be targeted by a learning algorithm. In this thesis, we further explore the issue and identify (sub-)classes of acceptors that admit canonical size-minimal representatives. In more detail, the contributions of this thesis are three-fold. First, we expand the automata (learning) theory of Guarded Kleene Algebra with Tests (GKAT), an efficiently decidable logic expressive enough to model simple imperative programs. In particular, we present GL*, an algorithm that learns the unique size-minimal GKAT automaton for a given deterministic language, and prove that GL* is more efficient than an existing variation of L*. We implement both algorithms in OCaml, and compare them on example programs. Second, we present a category-theoretical framework based on generators, bialgebras, and distributive laws, which identifies, for a wide class of automata with side-effects in a monad, canonical target models for automata learning. Apart from recovering examples from the literature, we discover a new canonical acceptor of regular languages, and present a unifying minimality result. Finally, we show that the construction underlying our framework is an instance of a more general theory. First, we see that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on a category of subobjects with respect to an epi-mono factorisation system. Second, we explore the abstract theory of generators and bases for algebras over a monad: we discuss bases for bialgebras, the product of bases, generalise the representation theory of linear maps, and compare our ideas to a coalgebra-based approach

    Asynchronous Algorithmic Alignment with Cocycles

    Full text link
    State-of-the-art neural algorithmic reasoners make use of message passing in graph neural networks (GNNs). But typical GNNs blur the distinction between the definition and invocation of the message function, forcing a node to send messages to its neighbours at every layer, synchronously. When applying GNNs to learn to execute dynamic programming algorithms, however, on most steps only a handful of the nodes would have meaningful updates to send. One, hence, runs the risk of inefficiencies by sending too much irrelevant data across the graph -- with many intermediate GNN steps having to learn identity functions. In this work, we explicitly separate the concepts of node state update and message function invocation. With this separation, we obtain a mathematical formulation that allows us to reason about asynchronous computation in both algorithms and neural networks

    Functorial String Diagrams for Reverse-Mode Automatic Differentiation

    Get PDF
    We formulate a reverse-mode automatic differentiation (RAD) algorithm for (applied) simply typed lambda calculus in the style of Pearlmutter and Siskind [Barak A. Pearlmutter and Jeffrey Mark Siskind, 2008], using the graphical formalism of string diagrams. Thanks to string diagram rewriting, we are able to formally prove for the first time the soundness of such an algorithm. Our approach requires developing a calculus of string diagrams with hierarchical features in the spirit of functorial boxes, in order to model closed monoidal (and cartesian closed) structure. To give an efficient yet principled implementation of the RAD algorithm, we use foliations of our hierarchical string diagrams

    The representation theory of the increasing monoid

    Get PDF
    We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras
    corecore