45 research outputs found
Relational lattices via duality
The natural join and the inner union combine in different ways tables of a
relational database. Tropashko [18] observed that these two operations are the
meet and join in a class of lattices-called the relational lattices- and
proposed lattice theory as an alternative algebraic approach to databases.
Aiming at query optimization, Litak et al. [12] initiated the study of the
equational theory of these lattices. We carry on with this project, making use
of the duality theory developed in [16]. The contributions of this paper are as
follows. Let A be a set of column's names and D be a set of cell values; we
characterize the dual space of the relational lattice R(D, A) by means of a
generalized ultrametric space, whose elements are the functions from A to D,
with the P (A)-valued distance being the Hamming one but lifted to subsets of
A. We use the dual space to present an equational axiomatization of these
lattices that reflects the combinatorial properties of these generalized
ultrametric spaces: symmetry and pairwise completeness. Finally, we argue that
these equations correspond to combinatorial properties of the dual spaces of
lattices, in a technical sense analogous of correspondence theory in modal
logic. In particular, this leads to an exact characterization of the finite
lattices satisfying these equations.Comment: Coalgebraic Methods in Computer Science 2016, Apr 2016, Eindhoven,
Netherland
Positive definite metric spaces
Magnitude is a numerical invariant of finite metric spaces, recently
introduced by T. Leinster, which is analogous in precise senses to the
cardinality of finite sets or the Euler characteristic of topological spaces.
It has been extended to infinite metric spaces in several a priori distinct
ways. This paper develops the theory of a class of metric spaces, positive
definite metric spaces, for which magnitude is more tractable than in general.
Positive definiteness is a generalization of the classical property of negative
type for a metric space, which is known to hold for many interesting classes of
spaces. It is proved that all the proposed definitions of magnitude coincide
for compact positive definite metric spaces and further results are proved
about the behavior of magnitude as a function of such spaces. Finally, some
facts about the magnitude of compact subsets of l_p^n for p \le 2 are proved,
generalizing results of Leinster for p=1,2, using properties of these spaces
which are somewhat stronger than positive definiteness.Comment: v5: Corrected some misstatements in the last few paragraphs. Updated
reference
The logic of the future in quantum theory
According to quantum mechanics, statements about the future made by sentient beings like us are, in general, neither true nor false; they must satisfy a many-valued logic. I propose that the truth value of such a statement should be identified with the probability that the event it describes will occur. After reviewing the history of related ideas in logic, I argue that it gives an understanding of probability which is particularly satisfactory for use in quantum mechanics. I construct a lattice of future-tense propositions, with truth values in the interval , and derive logical properties of these truth values given by the usual quantum-mechanical formula for the probability of a history
Topological analysis of representations
International audienceComputable analysis is the theoretical study of the abilities of algorithms to process infinite objects. The algorithms abilities depend on the way these objects are presented to them. We survey recent results on the problem of identifying the properties of objects that are decidable or semidecidable, for several concrete classes of objects and representations of them. Topology is at the core of this study, as the decidable and semidecidable properties are closely related to the open sets induced by the representation
Categorial Compositionality III: F-(co)algebras and the Systematicity of Recursive Capacities in Human Cognition
Human cognitive capacity includes recursively definable concepts, which are prevalent in domains involving lists, numbers, and languages. Cognitive science currently lacks a satisfactory explanation for the systematic nature of such capacities (i.e., why the capacity for some recursive cognitive abilities–e.g., finding the smallest number in a list–implies the capacity for certain others–finding the largest number, given knowledge of number order). The category-theoretic constructs of initial F-algebra, catamorphism, and their duals, final coalgebra and anamorphism provide a formal, systematic treatment of recursion in computer science. Here, we use this formalism to explain the systematicity of recursive cognitive capacities without ad hoc assumptions (i.e., to the same explanatory standard used in our account of systematicity for non-recursive capacities). The presence of an initial algebra/final coalgebra explains systematicity because all recursive cognitive capacities, in the domain of interest, factor through (are composed of) the same component process. Moreover, this factorization is unique, hence no further (ad hoc) assumptions are required to establish the intrinsic connection between members of a group of systematically-related capacities. This formulation also provides a new perspective on the relationship between recursive cognitive capacities. In particular, the link between number and language does not depend on recursion, as such, but on the underlying functor on which the group of recursive capacities is based. Thus, many species (and infants) can employ recursive processes without having a full-blown capacity for number and language
A 2-categorical extension of Etingof–Kazhdan quantisation
Let k be a field of characteristic zero. Etingof and Kazhdan constructed a
quantisation U_h(b) of any Lie bialgebra b over k, which depends on the choice
of an associator Phi. They prove moreover that this quantisation is functorial
in b. Remarkably, the quantum group U_h(b) is endowed with a Tannakian
equivalence F_b from the braided tensor category of Drinfeld-Yetter modules
over b, with deformed associativity constraints given by Phi, to that of
Drinfeld-Yetter modules over U_h(b). In this paper, we prove that the
equivalence F_b is functorial in b.Comment: Small revisions in Sections 2 and 6. An appendix added on the
equivalence between admissible Drinfeld-Yetter modules over a QUE and modules
over its quantum double. To appear in Selecta Math. 71 page
Categorial Compositionality: A Category Theory Explanation for the Systematicity of Human Cognition
Classical and Connectionist theories of cognitive architecture seek to explain systematicity (i.e., the property of human cognition whereby cognitive capacity comes in groups of related behaviours) as a consequence of syntactically and functionally compositional representations, respectively. However, both theories depend on ad hoc assumptions to exclude specific instances of these forms of compositionality (e.g. grammars, networks) that do not account for systematicity. By analogy with the Ptolemaic (i.e. geocentric) theory of planetary motion, although either theory can be made to be consistent with the data, both nonetheless fail to fully explain it. Category theory, a branch of mathematics, provides an alternative explanation based on the formal concept of adjunction, which relates a pair of structure-preserving maps, called functors. A functor generalizes the notion of a map between representational states to include a map between state transformations (or processes). In a formal sense, systematicity is a necessary consequence of a higher-order theory of cognitive architecture, in contrast to the first-order theories derived from Classicism or Connectionism. Category theory offers a re-conceptualization for cognitive science, analogous to the one that Copernicus provided for astronomy, where representational states are no longer the center of the cognitive universe—replaced by the relationships between the maps that transform them
Category Theoretic Analysis of Hierarchical Protein Materials and Social Networks
Materials in biology span all the scales from Angstroms to meters and typically consist of complex hierarchical assemblies of simple building blocks. Here we describe an application of category theory to describe structural and resulting functional properties of biological protein materials by developing so-called ologs. An olog is like a “concept web” or “semantic network” except that it follows a rigorous mathematical formulation based on category theory. This key difference ensures that an olog is unambiguous, highly adaptable to evolution and change, and suitable for sharing concepts with other olog. We consider simple cases of beta-helical and amyloid-like protein filaments subjected to axial extension and develop an olog representation of their structural and resulting mechanical properties. We also construct a representation of a social network in which people send text-messages to their nearest neighbors and act as a team to perform a task. We show that the olog for the protein and the olog for the social network feature identical category-theoretic representations, and we proceed to precisely explicate the analogy or isomorphism between them. The examples presented here demonstrate that the intrinsic nature of a complex system, which in particular includes a precise relationship between structure and function at different hierarchical levels, can be effectively represented by an olog. This, in turn, allows for comparative studies between disparate materials or fields of application, and results in novel approaches to derive functionality in the design of de novo hierarchical systems. We discuss opportunities and challenges associated with the description of complex biological materials by using ologs as a powerful tool for analysis and design in the context of materiomics, and we present the potential impact of this approach for engineering, life sciences, and medicine.Presidential Early Career Award for Scientists and Engineers (N000141010562)United States. Army Research Office. Multidisciplinary University Research Initiative (W911NF0910541)United States. Office of Naval Research (grant N000141010841)Massachusetts Institute of Technology. Dept. of MathematicsStudienstiftung des deutschen VolkesClark BarwickJacob Luri