2,987 research outputs found
Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets
We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
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Reconciling Shannon and Scott with a Lattice of Computable Information
This paper proposes a reconciliation of two different theories of information. The first, originally proposed in a lesser-known work by Claude Shannon (some five years after the publication of his celebrated quantitative theory of communication), describes how the information content of channels can be described qualitatively, but still abstractly, in terms of information elements, where information elements can be viewed as equivalence relations over the data source domain. Shannon showed that these elements have a partial ordering, expressing when one information element is more informative than another, and that these partially ordered information elements form a complete lattice. In the context of security and information flow this structure has been independently rediscovered several times, and used as a foundation for understanding and reasoning about information flow. The second theory of information is Dana Scott\u27s domain theory, a mathematical framework for giving meaning to programs as continuous functions over a particular topology. Scott\u27s partial ordering also represents when one element is more informative than another, but in the sense of computational progress - i.e. when one element is a more defined or evolved version of another. To give a satisfactory account of information flow in computer programs it is necessary to consider both theories together, in order to understand not only what information is conveyed by a program (viewed as a channel, \ue0 la Shannon) but also how the precision with which that information can be observed is determined by the definedness of its encoding (\ue0 la Scott). To this end we show how these theories can be fruitfully combined, by defining the Lattice of Computable Information (LoCI), a lattice of preorders rather than equivalence relations. LoCI retains the rich lattice structure of Shannon\u27s theory, filters out elements that do not make computational sense, and refines the remaining information elements to reflect how Scott\u27s ordering captures possible varieties in the way that information is presented. We show how the new theory facilitates the first general definition of termination-insensitive information flow properties, a weakened form of information flow property commonly targeted by static program analyses
Every topos has an optimal noetherian form
The search, of almost a century long, for a unified axiomatic framework for
establishing homomorphism theorems of classical algebra (such as Noether
isomorphism theorems and homological diagram lemmas) has led to the notion of a
`noetherian form', which is a generalization of an abelian category suitable to
encompass categories of non-abelian algebraic structures (such as non-abelian
groups, or rings with identity, or cocommutative Hopf algebras over any field,
and many others). In this paper, we show that, surprisingly, even the category
of sets, and more generally, any topos, fits under the framework of a
noetherian form. Moreover, we give an intrinsic characterization of such
noetherian form and show that it is very closely related to the known
noetherian form of a semi-abelian category. In fact, we show that for a pointed
category having finite products and sums, the existence of the type of
noetherian form that any topos possesses is equivalent to the category being
semi-abelian (this result is unexpected since only trivial toposes can be
semi-abelian). We also show that these noetherian forms are optimal, in a
suitable sense.Comment: 66 pages, submitted for publicatio
Infinite Permutation Groups and the Origin of Quantum Mechanics
We propose an interpretation for the meets and joins in the lattice of
experimental propositions of a physical theory, answering a question of
Birkhoff and von Neumann in [1]. When the lattice is atomistic, it is
isomorphic to the lattice of definably closed sets of a finitary relational
structure in First Order Logic. In terms of mapping experimental propositions
to subsets of the atomic phase space, the meet corresponds to set intersection,
while the join is the definable closure of set union. The relational structure
is defined by the action of the lattice automorphism group on the atomic layer.
Examining this correspondence between physical theories and infinite group
actions, we show that the automorphism group must belong to a family of
permutation groups known as geometric Jordan groups. We then use the
classification theorem for Jordan groups to argue that the combined
requirements of probability and atomicism leave uncountably infinite Steiner
2-systems (of which projective spaces are standard examples) as the sole class
of options for generating the lattice of particle Quantum Mechanics.Comment: 23 page
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