Developing understanding of triangle

As children develop concepts of shape they move from a visual understanding to a property based approach to classification. In this study two cohorts, one a longitudinal study from grade 1 to 4 and the other a sample across a school from pre-school to grade 8, were asked to identify triangles. The resulting data shows errors of inclusion are greater than errors of exclusion and suggests an order in which particular properties are attended to as children learn

Proof and Proving in Mathematics Education

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Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

Knowledge Representation, Reasoning and Learning for Non-Extractive Reading Comprehension

abstract: While in recent years deep learning (DL) based approaches have been the popular approach in developing end-to-end question answering (QA) systems, such systems lack several desired properties, such as the ability to do sophisticated reasoning with knowledge, the ability to learn using less resources and interpretability. In this thesis, I explore solutions that aim to address these drawbacks. Towards this goal, I work with a specific family of reading comprehension tasks, normally referred to as the Non-Extractive Reading Comprehension (NRC), where the given passage does not contain enough information and to correctly answer sophisticated reasoning and ``additional knowledge" is required. I have organized the NRC tasks into three categories. Here I present my solutions to the first two categories and some preliminary results on the third category. Category 1 NRC tasks refer to the scenarios where the required ``additional knowledge" is missing but there exists a decent natural language parser. For these tasks, I learn the missing ``additional knowledge" with the help of the parser and a novel inductive logic programming. The learned knowledge is then used to answer new questions. Experiments on three NRC tasks show that this approach along with providing an interpretable solution achieves better or comparable accuracy to that of the state-of-the-art DL based approaches. The category 2 NRC tasks refer to the alternate scenario where the ``additional knowledge" is available but no natural language parser works well for the sentences of the target domain. To deal with these tasks, I present a novel hybrid reasoning approach which combines symbolic and natural language inference (neural reasoning) and ultimately allows symbolic modules to reason over raw text without requiring any translation. Experiments on two NRC tasks shows its effectiveness. The category 3 neither provide the ``missing knowledge" and nor a good parser. This thesis does not provide an interpretable solution for this category but some preliminary results and analysis of a pure DL based approach. Nonetheless, the thesis shows beyond the world of pure DL based approaches, there are tools that can offer interpretable solutions for challenging tasks without using much resource and possibly with better accuracy.Dissertation/ThesisDoctoral Dissertation Computer Science 201

Negotiating with a logical-linguistic protocol in a dialogical framework

This book is the result of years of reflection. Some time ago, while working in commodities, the author felt how difficult it was to decide the order in which to use arguments during a negotiation process. What would happen if we translated the arguments into cards and played them according to the rules of the Bridge game? The results were impressive. There was potential for improvement in the negotiation process. The investigation went deeper, exploring players, cards, deals and the information concealed in the players´ announcements, in the cards and in the deals. This new angle brought the research to NeuroLinguistic Patterns and cryptic languages, such as Russian Cards. In the following pages, the author shares her discovery of a new application for Logical Dialogues: Negotiations, tackled from basic linguistic structures placed under a dialogue form as a cognitive system which ‘understands’ natural language, with the aim to solve conflicts and even to serve peace

International Workshop on Description Logics : Bonn, May 28/29, 1994

This collection of papers forms the permanent record of the 1994 Description Logic Workshop, that was held at the Gustav Stresemann Institut in Bonn, Germany on 28 and 29 May 1994, immediately after the Fourth International Conference on Principles of Knowledge Representation and Reasoning. The workshop was set up to be as informal as possible, so this collection cannot hope to capture the discussions associated with the workshop. However, we hope that it will serve to remind participants of their discussion at the workshop, and provide non-participants with indications of the topics that were discussed at the workshop. The workshop consisted of seven regular sessions and one panel session. Each regular session had about four short presentations on a single theme, but also had considerable time reserved for discussion. The themes of the sessions were Foundations of Description Logics, Architecture of Description Logics and Description Logic Systems, Language Extensions, Expanding Description Logics, General Applications of Description Logics, Natural Language Applications of Description Logics, Connections between Description Logics and Databases, and the Future of Description Logics and Description Logic Systems. The session on Foundations of Description Logics concentrated on computational properties of description logics, correspondences between description logics and other formalisms, and on semantics of description logics, Similarly, there is discussion on how to develop tractable desription logics, for some notion of tractable, and whether it is useful to worry about achieving tractability at all. Several of the participants argued in favour of a very expressive description logic. This obviously precludes tractability or even decidability of complete reasoning. Klaus Schild proposed that for some purposes one could employ "model checking" (i .e., a closed world assumption) instead of "theorem proving," and has shown that this is still tractable for very large languages. Maurizio Lenzerini's opinion was that it is important to have decidable languages. Tractability cannot be achieved in several application areas because there one needs very expressive constructs: e.g., axioms, complex role constructors, and cycles with fixed-point semantics. For Bob MacGregor, not even decidability is an issue since he claims that Loom's incomplete reasoner is sufficient for his applications. The discussion addressed the question of whether there is still need for foundations, and whether the work on foundation done until now really solved the problems that the designers of early DL systems had. Both questions were mostly answered in the affirmative, with the caveat that new research on foundations should make sure that it is concerned with "real" problems, and not just generates new problems. In the session on Architecture of Description Logics and Description Logic Systems the participants considered different ways of putting together description logics and description logic systems. One way of doing this is to have a different kind of inference strategy for description logics, such as one based on intuitionistic logics or one based directly on rules of inference-thus allowing variant systems. Another way of modifying description logic systems is to divide them up in different ways, such as making a terminology consist of a schema portion and a view portion. Some discussion in this session concerned whether architectures should be influenced by application areas, or even by particular applications. There was considerable discussion at the workshop on how Description Logics should be extended or expanded to make them more useful. There are several methods to do this. The first is to extend the language of descriptions, e.g ., to represent n-ary relations, temporal information, or whole-part relationships, all of which were discussed at the workshop. The second is to add in another kind of reasoning, such as default reasoning, while still keeping the general framework of description logic reasoning. The third is to incorporate descriptions or description-like constructs in a larger reasoner, such as a first order reasoner. This was the approach taken in OMEGA and is the approach being taken in the Loom project. There have been many extensions of the first two kinds proposed for description logics, including several presented at the workshop. One quest ion discussed at the workshop was whether these extensions fit in well with the philosophy of description logic. Another question was whether the presence of many proposals for extensions means that description logics are easy to expand, or that description logics are inadequate representation formalisms? The general consensus was that description logics adequately capture a certain kind of core reasoning and that they lend themselves to incorporation with other kinds of reasoning. Care must be taken, however, to keep the extended versions true to the goals of description logics. The sessions on Applications of Description Logics had presentations on applications of description logics in various areas, including configuration, tutoring, natural language processing, and domain modeling. Most of these applications are research applications, funded by government research programs. There was discussion of what is needed to have more fielded applications of description logics. The session on Connections between Description Logics and Databases considered three kinds of connections between Description Logics and Databases: 1. using Description Logics for expressing database schemas, including local schemas, integrated schemas, and views, integrity constraints, and queries; 2. using Description Logic reasoning for various database-related reasoning, including schema integration and validation, and query optimization, and query validation and organization; and 3. making Description Logic reasoners more like Database Mangagement Systems via optimization. All three of these connections are being actively investigated by the description logic community. The panel session on the Future of Description Logics and Description Logic Systems discussed where the future of description logics will lie. There seems to be a consensus that description logics must forge tighter connections with other formalisms, such as databases or object-oriented systems. In this way, perhaps, description logics will find more real applications