Algebraic structures of tropical mathematics

Tropical mathematics often is defined over an ordered cancellative monoid \tM, usually taken to be (\RR, +) or (\QQ, +). Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid \tM is replaced by R: = L\times \tM, where $L$ is a given indexing semiring (not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is trivial, i.e, $L=\{1\}$, $R$ is the usual bipotent max-plus algebra. When $L=\{1,\infty\}$ we recover the "standard" supertropical structure with its "ghost" layer. When L = \NN we can describe multiple roots of polynomials via a "layering function" $s: R \to L$. Likewise, one can define the layering $s: R^{(n)} \to L^{(n)}$ componentwise; vectors $v_1, \dots, v_m$ are called tropically dependent if each component of some nontrivial linear combination \sum \a_i v_i is a ghost, for "tangible" \a_i \in R. Then an $n\times n$ matrix has tropically dependent rows iff its permanent is a ghost. We explain how supertropical algebras, and more generally layered algebras, provide a robust algebraic foundation for tropical linear algebra, in which many classical tools are available. In the process, we provide some new results concerning the rank of d-independent sets (such as the fact that they are semi-additive),put them in the context of supertropical bilinear forms, and lay the matrix theory in the framework of identities of semirings.Comment: 19 page

An oak is an oak, or not? Understanding and dealing with confusion and disagreement in biological classification

Human interaction with the living world, in science and beyond, always involves classification. While it has been a long-standing scientific goal to produce a single all-purpose taxonomy of life to cater for this need, classificatory practice is often subject to confusion and disagreement, and many philosophers have advocated forms of classificatory pluralism. This entails that multiple classifications should be allowed to coexist, and that whichever classification is best, is context-dependent. In this paper, we discuss some practical consequences of classificatory pluralism, in particular with regard to how one is supposed to find the best classification for a given context. We do so by means of a case study concerning oaks, in particular the pedunculate oak (Quercus robur L.) and the sessile oak (Quercus petraea (Matt.) Liebl.), two important putative species that present several classificatory challenges; and by applying one recent philosophical framework conceptualizing classification, the so-called Grounded Functionality Account (GFA) of (natural) kinds. We show how the GFA elucidates several issues related to oak classification and gives directions to optimize classificatory practices, and discuss some implications for scientific taxonomy

Designing role-based view for object-relational databases

In a federated database system, a view mechanism is crucial since it is used to define exportable subsets of data ; to perform a virtual restructuring d ataset; and to construct the integrated schema. The view service in federated databa se systems must be capable of retaining as much semantic information as possible. The object-oriented ( 0 - 0 ) model was considered the suitable canonical data model since it meets the original criteria for canonical model selection. However, with the emergence of stronger object-relational (0 -R ) model, the re is a clear argument for using an 0 - R canonical model in the federation. Hence, research should now focus on th e development of semantically powerful view mechanism for th e newer model. Meanwhile, the availability of real 0 -R technologies offers researchers the opportunity to develop different forms of view mechanisms. The concept of roles has been widely studied in 0 - 0 modelling and development. The role model represents some characteristics that the traditional 0-0 model lacked, such as object migration, multiple occurrences and context-dependent access. While many forms of 0-0 views were designed for the 0-0 canonical model, one option was to extend the 0-0 model to incorporate a role model. In a role model, the real entity is modelled in the form of a role rather than an object. An object represents the permanent properties of an entity is a root object; and an object represents the temporary properties of an entity is a role object. The contribution of this research is to design a view system that employees the concept of roles for the 0 -R canonical model in a federated database system. In this thesis, an examination of the current 0 -R metamodel is provided first in order to provide an environment for recognising the roleview metadata and measuring the view performance; then a Roleview Definition Language (RDL) is introduced, along with the semantics for defining virtual classes and generating virtua l extents; finally, a working prototype is provided to prove th e role-based view system is implementable and the syntax is semantically correct

A dependent nominal type theory

Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of beta-eta-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles

Statistical dependency parsing of Turkish

This paper presents results from the first statistical dependency parser for Turkish. Turkish is a free-constituent order language with complex agglutinative inflectional and derivational morphology and presents interesting challenges for statistical parsing, as in general, dependency relations are between “portions” of words called inflectional groups. We have explored statistical models that use different representational units for parsing. We have used the Turkish Dependency Treebank to train and test our parser but have limited this initial exploration to that subset of the treebank sentences with only left-to-right non-crossing dependency links. Our results indicate that the best accuracy in terms of the dependency relations between inflectional groups is obtained when we use inflectional groups as units in parsing, and when contexts around the dependent are employed

Classification systems offer a microcosm of issues in conceptual processing: A commentary on Kemmerer (2016)

This is a commentary on Kemmerer (2016), Categories of Object Concepts Across Languages and Brains: The Relevance of Nominal Classification Systems to Cognitive Neuroscience, DOI: 10.1080/23273798.2016.1198819

On Equivalence and Canonical Forms in the LF Type Theory

Decidability of definitional equality and conversion of terms into canonical form play a central role in the meta-theory of a type-theoretic logical framework. Most studies of definitional equality are based on a confluent, strongly-normalizing notion of reduction. Coquand has considered a different approach, directly proving the correctness of a practical equivalance algorithm based on the shape of terms. Neither approach appears to scale well to richer languages with unit types or subtyping, and neither directly addresses the problem of conversion to canonical. In this paper we present a new, type-directed equivalence algorithm for the LF type theory that overcomes the weaknesses of previous approaches. The algorithm is practical, scales to richer languages, and yields a new notion of canonical form sufficient for adequate encodings of logical systems. The algorithm is proved complete by a Kripke-style logical relations argument similar to that suggested by Coquand. Crucially, both the algorithm itself and the logical relations rely only on the shapes of types, ignoring dependencies on terms.Comment: 41 page

Explicit Substitutions for Contextual Type Theory

In this paper, we present an explicit substitution calculus which distinguishes between ordinary bound variables and meta-variables. Its typing discipline is derived from contextual modal type theory. We first present a dependently typed lambda calculus with explicit substitutions for ordinary variables and explicit meta-substitutions for meta-variables. We then present a weak head normalization procedure which performs both substitutions lazily and in a single pass thereby combining substitution walks for the two different classes of variables. Finally, we describe a bidirectional type checking algorithm which uses weak head normalization and prove soundness.Comment: In Proceedings LFMTP 2010, arXiv:1009.218