Types, rings, and games

Algebraic equations on complex numbers and functional equations on generating functions are often used to solve combinatorial problems. But the introduction of common arithmetic operators such as subtraction and division always causes panic in the world of objects which are generated from constants by applying products and coproducts. Over the years, researchers have been endeavouring to interpretate some absurd calculations on objects which lead to meaningful combinatorial results. This thesis investigates connections between algebraic equations on complex numbers and isomorphisms of recursively defined objects. We are attempting to work out conditions under which isomorphisms between recursively defined objects can be decided by equalities between polynomials on multi-variables with integers as coefficients

Types, rings, and games

Algebraic equations on complex numbers and functional equations on generating functions are often used to solve combinatorial problems. But the introduction of common arithmetic operators such as subtraction and division always causes panic in the world of objects which are generated from constants by applying products and coproducts. Over the years, researchers have been endeavouring to interpretate some absurd calculations on objects which lead to meaningful combinatorial results. This thesis investigates connections between algebraic equations on complex numbers and isomorphisms of recursively defined objects. We are attempting to work out conditions under which isomorphisms between recursively defined objects can be decided by equalities between polynomials on multi-variables with integers as coefficients

LOGICAL COMBINATORS FOR SYSTEM CONFIGURATION

System configuration describes the construction of complex engineering systems from their component parts. The configuration language is at a meta-Ievel to a specification language and expresses the horizontal structuring of specifications and modules by extension and parameterization; it also expresses the implementation, of both specifications and modules during the development of a software system. The logic chosen for system configuration is many-sorted first-order logic which possesses the Craig interpolation property. Configuration is expressed precisely within the logical framework by the operation of combinators on recursively defined configured objects of sorts in the set {specification, module}; each configured object is a named theory presentation. Properties of commutativity between the combinators are illustrated by equivalent paths in the three-dimensional development space for configuration. The actual building of configured objects is expressed by constructing diagrams within a categorical workspace that is based on the structure of a KZ-doctrine

Algorithmic information and incompressibility of families of multidimensional networks

This article presents a theoretical investigation of string-based generalized representations of families of finite networks in a multidimensional space. First, we study the recursive labeling of networks with (finite) arbitrary node dimensions (or aspects), such as time instants or layers. In particular, we study these networks that are formalized in the form of multiaspect graphs. We show that, unlike classical graphs, the algorithmic information of a multidimensional network is not in general dominated by the algorithmic information of the binary sequence that determines the presence or absence of edges. This universal algorithmic approach sets limitations and conditions for irreducible information content analysis in comparing networks with a large number of dimensions, such as multilayer networks. Nevertheless, we show that there are particular cases of infinite nesting families of finite multidimensional networks with a unified recursive labeling such that each member of these families is incompressible. From these results, we study network topological properties and equivalences in irreducible information content of multidimensional networks in comparison to their isomorphic classical graph.Comment: Extended preprint version of the pape

Why Numbers Are Sets

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's. I give a detailed mathematical demonstration that 0 is {} and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals

Neural Task Programming: Learning to Generalize Across Hierarchical Tasks

In this work, we propose a novel robot learning framework called Neural Task Programming (NTP), which bridges the idea of few-shot learning from demonstration and neural program induction. NTP takes as input a task specification (e.g., video demonstration of a task) and recursively decomposes it into finer sub-task specifications. These specifications are fed to a hierarchical neural program, where bottom-level programs are callable subroutines that interact with the environment. We validate our method in three robot manipulation tasks. NTP achieves strong generalization across sequential tasks that exhibit hierarchal and compositional structures. The experimental results show that NTP learns to generalize well to- wards unseen tasks with increasing lengths, variable topologies, and changing objectives.Comment: ICRA 201

Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$. The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even complexes of coherent sheaves) with all groups \mExt^q for $q \geq 2$ vanishing, then $X$ also admits a tilting sheaf (tilting bundle, or tilting complex, respectively) obtained as a universal extension of this exceptional sequence.Comment: 15 page