A sheaf-theoretic approach to pattern matching and related problems

AbstractWe present a general theory of pattern matching by adopting an extensional, geometric view of patterns. Representing the geometry of the pattern via a Grothendieck topology, the extension of the matching relation for a constant target and varying pattern forms a sheaf. We derive a generalized version of the Knuth-Morris-Pratt string-matching algorithm by gradually converting this extensional description into an intensional description, i.e., an algorithm. The generality of this approach is illustrated by briefly considering other applications: Earley's algorithm for parsing, Waltz filtering for scene analysis, matching modulo commutativity, and the n-queens problem

Pattern matching : a sheaf-theoretic approach

A general theory of pattern matching is presented by adopting an extensional, geometric view of patterns. The extension of the matching relation consists of the occurrences of all possible patterns in a particular target. The geometry of the pattern describes the structure of the pattern and the spatial relationships among parts of the pattern. The extension and the geometry, when combined, produce a structure called a sheaf. Sheaf theory is a well developed branch of mathematics which studies the global consequences of locally defined properties. For pattern matching, an occurrence of a pattern, a global property of the pattern, is obtained by gluing together occurrences of parts of the pattern, which are locally defined properties.A sheaf-theoretic view of pattern rnatching provides a uniforrn treatrnent of pattern matching on any kind of data structure-strings, trees, graphs, hypergraphs, and so on. Such a parametric description is achieved by using the language of category theory, a highly abstract description of commonly occurring structures and relationships in mathematics.A generalized version of the Knuth-Morris-Pratt pattern matching algorithm is derived by gradually converting the extensional description of pattern rnatching as a sheaf into an intensional description. The algorithm results from a synergy of four very general program synthesis/transformation techniques: (1) Divide and conquer: exploit the sheaf condition; assemble a full match by gluing together partial matches; (2) Finite differencing: collect and update partial matches incrementally while traversing the target; (3) Backtracking: instead of saving all partial matches, save just one; when this partial match cannot be extended, fail back to another; (4) Partial evaluation: precompute pattern-based (and therefore constant) computations.The derivation is carried out in a general frarnework using Grothendieck topologies. By appropriately instantiating the underlying data structures and topologies, the sarne scheme results in matching algorithms for patterns with variables and with multiple patterns. Slight variations of the derivation result in Earley's algorithm for context-free parsing, and Waltz filtering, a relaxation algorithm for providing 3-D interpretations to 2-D irnages.Other applications of a geometric view of patterns are briefly considered: rewrites, parallel algorithms, induction and computability

Cluster varieties from Legendrian knots

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.Comment: 47 page

Sheaf Theory through Examples

An approachable introduction to elementary sheaf theory and its applications beyond pure math. Sheaves are mathematical constructions concerned with passages from local properties to global ones. They have played a fundamental role in the development of many areas of modern mathematics, yet the broad conceptual power of sheaf theory and its wide applicability to areas beyond pure math have only recently begun to be appreciated. Taking an applied category theory perspective, Sheaf Theory through Examples provides an approachable introduction to elementary sheaf theory and examines applications including n-colorings of graphs, satellite data, chess problems, Bayesian networks, self-similar groups, musical performance, complexes, and much more. With an emphasis on developing the theory via a wealth of well-motivated and vividly illustrated examples, Sheaf Theory through Examples supplements the formal development of concepts with philosophical reflections on topology, category theory, and sheaf theory, alongside a selection of advanced topics and examples that illustrate ideas like cellular sheaf cohomology, toposes, and geometric morphisms. Sheaf Theory through Examples seeks to bridge the powerful results of sheaf theory as used by mathematicians and real-world applications, while also supplementing the technical matters with a unique philosophical perspective attuned to the broader development of ideas

Sheaf Theory as a Foundation for Heterogeneous Data Fusion

A major impediment to scientific progress in many fields is the inability to make sense of the huge amounts of data that have been collected via experiment or computer simulation. This dissertation provides tools to visualize, represent, and analyze the collection of sensors and data all at once in a single combinatorial geometric object. Encoding and translating heterogeneous data into common language are modeled by supporting objects. In this methodology, the behavior of the system based on the detection of noise in the system, possible failure in data exchange and recognition of the redundant or complimentary sensors are studied via some related geometric objects. Applications of the constructed methodology are described by two case studies: one from wildfire threat monitoring and the other from air traffic monitoring. Both cases are distributed (spatial and temporal) information systems. The systems deal with temporal and spatial fusion of heterogeneous data obtained from multiple sources, where the schema, availability and quality vary. The behavior of both systems is explained thoroughly in terms of the detection of the failure in the systems and the recognition of the redundant and complimentary sensors. A comparison between the methodology in this dissertation and the alternative methods is described to further verify the validity of the sheaf theory method. It is seen that the method has less computational complexity in both space and time

Loops and Knots as Topoi of Substance. Spinoza Revisited

The relationship between modern philosophy and physics is discussed. It is shown that the latter develops some need for a modernized metaphysics which shows up as an ultima philosophia of considerable heuristic value, rather than as the prima philosophia in the Aristotelian sense as it had been intended, in the first place. It is shown then, that it is the philosophy of Spinoza in fact, that can still serve as a paradigm for such an approach. In particular, Spinoza's concept of infinite substance is compared with the philosophical implications of the foundational aspects of modern physical theory. Various connotations of sub-stance are discussed within pre-geometric theories, especially with a view to the role of spin networks within quantum gravity. It is found to be useful to intro-duce a separation into physics then, so as to differ between foundational and empirical theories, respectively. This leads to a straightforward connection bet-ween foundational theories and speculative philosophy on the one hand, and between empirical theories and sceptical philosophy on the other. This might help in the end, to clarify some recent problems, such as the absence of time and causality at a fundamental level. It is implied that recent results relating to topos theory might open the way towards eventually deriving logic from physics, and also towards a possible transition from logic to hermeneutic.Comment: 42 page

The Sen Limit

F-theory compactifications on elliptic Calabi-Yau manifolds may be related to IIb compactifications by taking a certain limit in complex structure moduli space, introduced by A. Sen. The limit has been characterized on the basis of SL(2,Z) monodromies of the elliptic fibration. Instead, we introduce a stable version of the Sen limit. In this picture the elliptic Calabi-Yau splits into two pieces, a P^1-bundle and a conic bundle, and the intersection yields the IIb space-time. We get a precise match between F-theory and perturbative type IIb. The correspondence is holographic, in the sense that physical quantities seemingly spread in the bulk of the F-theory Calabi-Yau may be rewritten as expressions on the log boundary. Smoothing the F-theory Calabi-Yau corresponds to summing up the D(-1)-instanton corrections to the IIb theory.Comment: 41 pp, 1 figure, LaTe