Modular Design Patterns for Hybrid Learning and Reasoning Systems: a taxonomy, patterns and use cases

The unification of statistical (data-driven) and symbolic (knowledge-driven) methods is widely recognised as one of the key challenges of modern AI. Recent years have seen large number of publications on such hybrid neuro-symbolic AI systems. That rapidly growing literature is highly diverse and mostly empirical, and is lacking a unifying view of the large variety of these hybrid systems. In this paper we analyse a large body of recent literature and we propose a set of modular design patterns for such hybrid, neuro-symbolic systems. We are able to describe the architecture of a very large number of hybrid systems by composing only a small set of elementary patterns as building blocks. The main contributions of this paper are: 1) a taxonomically organised vocabulary to describe both processes and data structures used in hybrid systems; 2) a set of 15+ design patterns for hybrid AI systems, organised in a set of elementary patterns and a set of compositional patterns; 3) an application of these design patterns in two realistic use-cases for hybrid AI systems. Our patterns reveal similarities between systems that were not recognised until now. Finally, our design patterns extend and refine Kautz' earlier attempt at categorising neuro-symbolic architectures.Comment: 20 pages, 22 figures, accepted for publication in the International Journal of Applied Intelligenc

Categorical Ontology I - Existence

The present paper is the first piece of a series whose aim is to develop an approach to ontology and metaontology through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of \clE and thus of \Omega_\clE yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of techniques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges' famous paradoxes of Tlön's ``nine copper coins'', and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Categorical Ontology I - Existence

The present paper approaches ontology and metaontology through mathematics, and more precisely through category theory. We exploit the theory of elementary toposes to claim that a satisfying “theory of existence”, and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of E and thus of Ω E yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of tech- niques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges’ famous paradoxes of Tlön’s “nine copper coins”, and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Instructional strategies in explicating the discovery function of proof for lower secondary school students

In this paper, we report on the analysis of teaching episodes selected from our pedagogical and cognitive research on geometry teaching that illustrate how carefully-chosen instructional strategies can guide Grade 8 students to see and appreciate the discovery function of proof in geometr

Emerging trends proceedings of the 17th International Conference on Theorem Proving in Higher Order Logics: TPHOLs 2004

technical reportThis volume constitutes the proceedings of the Emerging Trends track of the 17th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2004) held September 14-17, 2004 in Park City, Utah, USA. The TPHOLs conference covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification. There were 42 papers submitted to TPHOLs 2004 in the full research cate- gory, each of which was refereed by at least 3 reviewers selected by the program committee. Of these submissions, 21 were accepted for presentation at the con- ference and publication in volume 3223 of Springer?s Lecture Notes in Computer Science series. In keeping with longstanding tradition, TPHOLs 2004 also offered a venue for the presentation of work in progress, where researchers invite discussion by means of a brief introductory talk and then discuss their work at a poster session. The work-in-progress papers are held in this volume, which is published as a 2004 technical report of the School of Computing at the University of Utah

Categorical Ontology I - Existence

The present paper is the first piece of a series whose aim is to develop an approach to ontology and metaontology through category theory. We exploit the theory of elementary toposes to claim that a satisfying ``theory of existence'', and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of \clE and thus of \Omega_\clE yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of techniques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges' famous paradoxes of Tlön's ``nine copper coins'', and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works

Categorical Ontology I - Existence

The present paper approaches ontology and metaontology through mathematics, and more precisely through category theory. We exploit the theory of elementary toposes to claim that a satisfying “theory of existence”, and more at large ontology itself, can both be obtained through category theory. In this perspective, an ontology is a mathematical object: it is a category, the universe of discourse in which our mathematics (intended at large, as a theory of knowledge) can be deployed. The internal language that all categories possess prescribes the modes of existence for the objects of a fixed ontology/category. This approach resembles, but is more general than, fuzzy logics, as most choices of E and thus of Ω E yield nonclassical, many-valued logics. Framed this way, ontology suddenly becomes more mathematical: a solid corpus of tech- niques can be used to backup philosophical intuition with a useful, modular language, suitable for a practical foundation. As both a test-bench for our theory, and a literary divertissement, we propose a possible category-theoretic solution of Borges’ famous paradoxes of Tlön’s “nine copper coins”, and of other seemingly paradoxical construction in his literary work. We then delve into the topic with some vistas on our future works