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Default Reasoning via Topology and Mathematical Analysis: A Preliminary Report

Koutras, Costas D. and Liaskos, Konstantinos and Moyzes, Christos and Rantsoudis, Christos Default Reasoning via Topology and Mathematical Analysis: A Preliminary Report. (2018) In: 16th International Conference on Principles of Knowledge Representation and Reasoning (KR 2018), 30 October 2018 - 2 November 2018 (Tempe, Arizona, United States).

(Document in English)

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Official URL: https://aaai.org/ocs/index.php/KR/KR18/paper/view/18017


A default consequence relation α|~β (if α, then normally β) can be naturally interpreted via a `most' generalized quantifier: α|~β is valid iff in `most' α-worlds, β is also true. We define various semantic incarnations of this principle which attempt to make the set of (α ∧ β)-worlds `large' and the set of (α ∧ ¬ β)-worlds `small'. The straightforward implementation of this idea on finite sets is via `clear majority'. We proceed to examine different `majority' interpretations of normality which are defined upon notions of classical mathematics which formalize aspects of `size'. We define default consequence using the notion of asymptotic density from analytic number theory. Asymptotic density provides a way to measure the size of integer sequences in a way much more fine-grained and accurate than set cardinality. Further on, in a topological setting, we identify `large' sets with dense sets and `negligibly small' sets with nowhere dense sets. Finally, we define default consequence via the concept of measure, classically developed in mathematical analysis for capturing `size' through a generalization of the notions of length, area and volume. The logics defined via asymptotic density and measure are weaker than the KLM system P, the so-called `conservative core' of nonmonotonic reasoning, and they resemble to probabilistic consequence. Topology goes a longer way towards system P but it misses Cautious Monotony (CM) and AND. Our results show that a `size'-oriented interpretation of default reasoning is context-sensitive and in `most' cases it departs from the preferential approach.

Item Type:Conference or Workshop Item (Paper)
HAL Id:hal-03623002
Audience (conference):International conference proceedings
Uncontrolled Keywords:
Institution:Other partners > American University of the Middle East - AUM (KUWAIT)
French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UT3 (FRANCE)
Université de Toulouse > Université Toulouse - Jean Jaurès - UT2J (FRANCE)
Université de Toulouse > Université Toulouse 1 Capitole - UT1 (FRANCE)
Other partners > University of Liverpool (UNITED KINGDOM)
Laboratory name:
Deposited On:15 May 2019 08:42

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