The Wonder of Colors and the Principle of Ariadne

The Principle of Ariadne, formulated in 1988 ago by Walter Carnielli and Carlos Di Prisco and later published in 1993, is an infinitary principle that is independent of the Axiom of Choice in ZF, although it can be consistently added to the remaining ZF axioms. The present paper surveys, and motivates, the foundational importance of the Principle of Ariadne and proposes the Ariadne Game, showing that the Principle of Ariadne, corresponds precisely to a winning strategy for the Ariadne Game. Some relations to other alternative. set-theoretical principles are also briefly discussed

POLYNOMIAL RING CALCULUS FOR MODAL LOGICS: A NEW SEMANTICS AND PROOF METHOD FOR MODALITIES

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra-Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics.41150170Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fonds National de la Recherche LuxembourgFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [2004/14107-2, 05/04123-3]CNPq [300702/2005-1

Pattern Functional Dependencies for Data Cleaning

Patterns (or regex-based expressions) are widely used to constrain the format of a domain (or a column), e.g., a Year column should contain only four digits, and thus a value like "1980-" might be a typo. Moreover, integrity constraints (ICs) defined over multiple columns, such as (conditional) functional dependencies and denial constraints, e.g., a ZIP code uniquely determines a city in the UK, have been widely used in data cleaning. However, a promising, but not yet explored, direction is to combine regex- and IC-based theories to capture data dependencies involving partial attribute values. For example, in an employee ID such as"F-9-107", "F" is sufficient to determine the finance department. Inspired by the above observation, we propose a novel class of ICs, called pattern functional dependencies (PFDs), to model fine-grained data dependencies gleaned from partial attribute values. These dependencies cannot be modeled using traditional ICs, such as (conditional) functional dependencies, which work on entire attribute values. We also present a set of axioms for the inference of PFDs, analogous to Armstrong's axioms for FDs, and study the complexity of consistency and implication analysis of PFDs. Moreover, we devise an effective algorithm to automatically discover PFDs even in the presence of errors in the data. Our extensive experiments on 15 real-world datasets show that our approach can effectively discover valid and useful PFDs over dirty data, which can then be used to detect data errors that are hard to capture by other types of ICs

Rejection in Łukasiewicz's and Słupecki's Sense

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic

A coalgebraic perspective on logical interpretations

In Computer Science stepwise refinement of algebraic specifications is a well-known formal methodology for rigorous program development. This paper illustrates how techniques from Algebraic Logic, in particular that of interpretation, understood as a multifunction that preserves and reflects logical consequence, capture a number of relevant transformations in the context of software design, reuse, and adaptation, difficult to deal with in classical approaches. Examples include data encapsulation and the decomposition of operations into atomic transactions. But if interpretations open such a new research avenue in program refinement, (conceptual) tools are needed to reason about them. In this line, the paper’s main contribution is a study of the correspondence between logical interpretations and morphisms of a particular kind of coalgebras. This opens way to the use of coalgebraic constructions, such as simulation and bisimulation, in the study of interpretations between (abstract) logics.Fundação para a Ciência e a Tecnologia (FCT