203 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Data analysis with merge trees
Today’s data are increasingly complex and classical statistical techniques need growingly more refined mathematical tools to be able to model and investigate them. Paradigmatic situations are represented by data which need to be considered up to some kind of trans- formation and all those circumstances in which the analyst finds himself in the need of defining a general concept of shape. Topological Data Analysis (TDA) is a field which is fundamentally contributing to such challenges by extracting topological information from data with a plethora of interpretable and computationally accessible pipelines. We con- tribute to this field by developing a series of novel tools, techniques and applications to work with a particular topological summary called merge tree. To analyze sets of merge trees we introduce a novel metric structure along with an algorithm to compute it, define a framework to compare different functions defined on merge trees and investigate the metric space obtained with the aforementioned metric. Different geometric and topolog- ical properties of the space of merge trees are established, with the aim of obtaining a deeper understanding of such trees. To showcase the effectiveness of the proposed metric, we develop an application in the field of Functional Data Analysis, working with functions up to homeomorphic reparametrization, and in the field of radiomics, where each patient is represented via a clustering dendrogram
Polynomial time multiplication and normal forms in free bands
We present efficient computational solutions to the problems of checking
equality, performing multiplication, and computing minimal representatives of
elements of free bands. A band is any semigroup satisfying the identity and the free band is the free object in the
variety of -generated bands. Radoszewski and Rytter developed a linear time
algorithm for checking whether two words represent the same element of a free
band. In this paper we describe an alternate linear time algorithm for checking
the same problem. The algorithm we present utilises a representation of words
as synchronous deterministic transducers that lend themselves to efficient
(quadratic in the size of the alphabet) multiplication in the free band. This
representation also provides a means of finding the short-lex least word
representing a given free band element with quadratic complexity.Comment: 32 pages, 13 figures (amended to improve intro, and fix some minor
typos
Rethinking inconsistent mathematics
This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the rolee of logic and ethics in the practice of mathematics
Syntax-semantics interface: an algebraic model
We extend our formulation of Merge and Minimalism in terms of Hopf algebras
to an algebraic model of a syntactic-semantic interface. We show that methods
adopted in the formulation of renormalization (extraction of meaningful
physical values) in theoretical physics are relevant to describe the extraction
of meaning from syntactic expressions. We show how this formulation relates to
computational models of semantics and we answer some recent controversies about
implications for generative linguistics of the current functioning of large
language models.Comment: LaTeX, 75 pages, 19 figure
A rigorous treatment of Meek's method for single transferable vote with formal proofs of key properties
This thesis presents a mechanised formalisation of key concepts and properties of Meek's method of Single Transferable Vote (STV). This method is currently in use in a number of local elections in New Zealand, the Royal Statistical Society, and even the Stack Exchange network. Using a formal approach, we show that the iterative solution to the surplus transfer round of Meek's method converges to a unique and valid solution, and connect a functional implementation of its key components to a more abstract and generalised proof.
Along the way, we consider and address issues present in existing pen-and-paper proofs, and discuss a general representation of strict ballots suitable for the proof patterns encountered in our formal development and for the implementation of Meek's method.
We believe that this work pushes the boundaries of interactive theorem proving for the formal verification of voting algorithms, and offers multiple promising avenues for further work on formally verifying the correctness and termination of STV methods in Isabelle/HOL
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Unbounded loops in quantum programs: categories and weak while loops
Control flow of quantum programs is often divided into two different classes: classical and quantum. Quantum programs with classical control flow have their conditional branching determined by the classical outcome of measurements, and these collapse quantum data. Conversely, quantum control flow is coherent, i.e. it does not perturb quantum data; quantum walk-based algorithms are practical examples where coherent quantum feedback plays a major role. This dissertation has two main contributions: (i) a categorical study of coherent quantum iteration and (ii) the introduction of weak while loops.
(i) The objective is to endow categories of quantum processes with a traced monoidal structure capable of modelling iterative quantum loops. To this end, the trace of a morphism is calculated via the execution formula, which adds up the contribution of all possible paths of the control flow. Haghverdi's unique decomposition categories are generalised to admit additive inverses and equipped with convergence criteria using basic topology. In this setting, it is possible to prove the validity of the execution formula as a categorical trace on certain categories of quantum processes. Among these there are categories of quantum processes over finite dimensional Hilbert spaces (as previously shown by Bartha), but also certain categories of quantum processes over infinite dimensional Hilbert spaces, such as a category of time-shift invariant quantum processes over discrete time.
(ii) A weak while loop is a classical control flow primitive that offers a trade-off between the collapse caused on each iteration and the amount of information gained. The trade-off may be adjusted by tuning a parameter and, in certain situations, it is possible to set its value so that we may control the algorithm without sacrificing its quantum speed-up. As an example, it is shown that Grover's search problem can be implemented using a weak while loop, maintaining the same time complexity as the standard Grover's algorithm (as previously shown by Mizel). In a more general setting, sufficient conditions are provided that let us determine, with arbitrarily high probability, a worst-case estimate of the number of iterations the loop will run for
Conjunctive Queries for Logic-Based Information Extraction
This thesis offers two logic-based approaches to conjunctive queries in the
context of information extraction. The first and main approach is the
introduction of conjunctive query fragments of the logics FC and FC[REG],
denoted as FC-CQ and FC[REG]-CQ respectively. FC is a first-order logic based
on word equations, where the semantics are defined by limiting the universe to
the factors of some finite input word. FC[REG] is FC extended with regular
constraints. The second approach is to consider the dynamic complexity of FC.Comment: Based on the author's PhD thesis and contains work from two
conference publications (arXiv:2104.04758, arXiv:1909.10869) which are joint
work with Dominik D. Freydenberge
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