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Concept# Logic

Summary

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.
Logic studies arguments, which consist of a set of premises together with a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" to the conclusion "I don't have to work". Premises and conclusions express propositions

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We present an algorithm to solve XPath decision problems under regular tree type constraints and show its use to statically type-check XPath queries. To this end, we prove the decidability of a logic with converse for finite ordered trees whose time complexity is a simple exponential of the size of a formula. The logic corresponds to the alternation free modal mu-calculus without greatest fixpoint, restricted to finite trees, and where formulas are cycle-free.

2007This thesis presents Fuzzy CoCo, a novel approach for system design, conducive to explaining human decisions. Based on fuzzy logic and coevolutionary computation, Fuzzy CoCo is a methodology for constructing systems able to accurately predict the outcome of a human decision-making process, while providing an understandable explanation of the underlying reasoning. Fuzzy logic provides a formal framework for constructing systems exhibiting both good numeric performance (precision) and linguistic representation (interpretability). From a numeric point of view, fuzzy systems exhibit nonlinear behavior and can handle imprecise and incomplete information. Linguistically, they represent knowledge in the form of rules, a natural way for explaining decision processes. Fuzzy modeling —meaning the construction of fuzzy systems— is an arduous task, demanding the identification of many parameters. This thesis analyses the fuzzy-modeling problem and different approaches to coping with it, focusing on evolutionary fuzzy modeling —the design of fuzzy inference systems using evolutionary algorithms— which constitutes the methodological base of my approach. In order to promote this analysis the parameters of a fuzzy system are classified into four categories: logic, structural, connective, and operational. The central contribution of this work is the use of an advanced evolutionary technique —cooperative coevolution— for dealing with the simultaneous design of connective and operational parameters. Cooperative coevolutionary fuzzy modeling succeeds in overcoming several limitations exhibited by other standard evolutionary approaches: stagnation, convergence to local optima, and computational costliness. Designing interpretable systems is a prime goal of my approach, which I study thoroughly herein. Based on a set of semantic and syntactic criteria, regarding the definition of linguistic concepts and their causal connections, I propose a number of strategies for producing more interpretable fuzzy systems. These strategies are implemented in Fuzzy CoCo, resulting in a modeling methodology providing high numeric precision, while incurring as little a loss of interpretability as possible. After testing Fuzzy CoCo on a benchmark problem —Fisher's Iris data— I successfully apply the algorithm to model the decision processes involved in two breast-cancer diagnostic problems: the WBCD problem and the Catalonia mammography interpretation problem. For the WBCD problem, Fuzzy CoCo produces systems both of high performance and high interpretability, comparable (if not better) than the best systems demonstrated to date. For the Catalonia problem, an evolved high-performance system was embedded within a web-based tool —called COBRA— for aiding radiologists in mammography interpretation. Several aspects of Fuzzy CoCo are thoroughly analyzed to provide a deeper understanding of the method. These analyses show the consistency of the results. They also help derive a stepwise guide to applying Fuzzy CoCo, and a set of qualitative relationships between some of its parameters that facilitate setting up the algorithm. Finally, this work proposes and explores preliminarily two extensions to the method: Island Fuzzy CoCo and Incremental Fuzzy CoCo, which together with the original CoCo constitute a family of coevolutionary fuzzy modeling techniques. The aim of these extensions is to guide the choice of an adequate number of rules for a given problem. While Island Fuzzy CoCo performs an extended search over different problem sizes, Incremental Fuzzy CoCo bases its search power on a mechanism of incremental evolution.

This thesis proposes to address the well-know database integration problem with a new method that combines functionality from database conceptual modeling techniques with functionality from logic-based reasoners. We elaborate on a hybrid - modeling+validation - integration approach for spatio-temporal information integration on the schema level. The modeling part of our methodology is supported by the spatio-temporal conceptual model MADS, whereas the validation part of the integration process is delegated to the description logics validation services. We therefore adhere to the principle that, rather than extending either formalism to try to cover all desirable functionality, a hybrid system, where the database component and the logic component would cooperate, each one performing the tasks for which it is best suited, is a viable solution for semantically rich information management. First, we develop a MADS-based flexible integration approach where the integrated schema designer has several viable ways to construct a final integrated schema. For different related schema elements we provide the designer with four general policies and with a set of structural solutions or structural patterns within each policy. To always guarantee an integrated solution, we provide for a preservation policy with multi-representation structural pattern. To state the inter-schema mappings, we elaborate on a correspondence language with explicit spatial and temporal operators. Thus, our correspondence language has three facets: structural, spatial, and temporal, allowing to relate the thematic representation as well as the spatial and temporal features. With the inter-schema mappings, the designer can state correspondences between related populations, and define the conditions that rule the matching at the instance level. These matching rules can then be used in query rewriting procedures or to match the instances within the data integration process. We associate a set of putative structural patterns to each type of population correspondence, providing a designer with a patterns' selection for flexible integrated schema construction. Second, we enhance our integration method by employing validation services of the description logic formalism. It is not guaranteed that the designer can state all the inter-schema mappings manually, and that they are all correct. We add the validation phase to ensure validity and completeness of the inter-schema mappings set. Inter-schema mappings cannot be validated autonomously, i.e., they are validated against the data model and the schemas they link. Thus, to implement our validation approach, we translate the data model, the source schemas and the inter-schema mappings into a description logic formalism, preserving the spatial and temporal semantics of the MADS data model. Thus, our modeling approach in description logic insures that the model designer will correctly define spatial and temporal schema elements and inter-schema mappings. The added value of the complete translation (i.e., including the data model and the source schemas) is that we validate not only the inter-schema mappings, but also the compliance of the source schemas to the data model, and infer implicit relationships within them. As the result of the validation procedure, the schema designer obtains the complete and valid set of inter-schema mappings and a set of valid (flexible) schematic patterns to apply to construct an integrated schema that meets application requirements. To further our work, we model a framework in which a schema designer is able to follow our integration method and realize the schema integration task in an assisted way. We design two models, UML and SEAM models, of a system that provides for integration functionalities. The models describe a framework where several tools are employed together, each involved in the service it is best suited for. We define the functionalities and the cooperation between the composing elements of the framework and detail the logics of the integration process in an UML activity diagram and in a SEAM operation model.

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