Perspectives of Neuro--Symbolic Integration -- Extended Abstract --

There is an obvious tension between symbolic and subsymbolic theories, because both show complementary strengths and weaknesses in corresponding applications and underlying methodologies. The resulting gap in the foundations and the applicability of these approaches is theoretically unsatisfactory and practically undesirable. We sketch a theory that bridges this gap between symbolic and subsymbolic approaches by the introduction of a Topos-based semi-symbolic level used for coding logical first-order expressions in a homogeneous framework. This semi-symbolic level can be used for neural learning of logical first-order theories. Besides a presentation of the general idea of the framework, we sketch some challenges and important open problems for future research with respect to the presented approach and the field of neuro-symbolic integration, in general

Dimensions of Neural-symbolic Integration - A Structured Survey

Research on integrated neural-symbolic systems has made significant progress in the recent past. In particular the understanding of ways to deal with symbolic knowledge within connectionist systems (also called artificial neural networks) has reached a critical mass which enables the community to strive for applicable implementations and use cases. Recent work has covered a great variety of logics used in artificial intelligence and provides a multitude of techniques for dealing with them within the context of artificial neural networks. We present a comprehensive survey of the field of neural-symbolic integration, including a new classification of system according to their architectures and abilities.Comment: 28 page

Neural-Symbolic Learning and Reasoning: Contributions and Challenges

The goal of neural-symbolic computation is to integrate robust connectionist learning and sound symbolic reasoning. With the recent advances in connectionist learning, in particular deep neural networks, forms of representation learning have emerged. However, such representations have not become useful for reasoning. Results from neural-symbolic computation have shown to offer powerful alternatives for knowledge representation, learning and reasoning in neural computation. This paper recalls the main contributions and discusses key challenges for neural-symbolic integration which have been identified at a recent Dagstuhl seminar

Topos and Stacks of Deep Neural Networks

Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Giraud's stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannon's entropy (P.Baudot and D.B. 2015). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Loef) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendieck's derivators

Some resonances between Eastern thought and Integral Biomathics in the framework of the WLIMES formalism for modelling living systems

Forty-two years ago, Capra published “The Tao of Physics” (Capra, 1975). In this book (page 17) he writes: “The exploration of the atomic and subatomic world in the twentieth century has …. necessitated a radical revision of many of our basic concepts” and that, unlike ‘classical’ physics, the sub-atomic and quantum “modern physics” shows resonances with Eastern thoughts and “leads us to a view of the world which is very similar to the views held by mystics of all ages and traditions.“ This article stresses an analogous situation in biology with respect to a new theoretical approach for studying living systems, Integral Biomathics (IB), which also exhibits some resonances with Eastern thought. Stepping on earlier research in cybernetics1 and theoretical biology,2 IB has been developed since 2011 by over 100 scientists from a number of disciplines who have been exploring a substantial set of theoretical frameworks. From that effort, the need for a robust core model utilizing advanced mathematics and computation adequate for understanding the behavior of organisms as dynamic wholes was identified. At this end, the authors of this article have proposed WLIMES (Ehresmann and Simeonov, 2012), a formal theory for modeling living systems integrating both the Memory Evolutive Systems (Ehresmann and Vanbremeersch, 2007) and the Wandering Logic Intelligence (Simeonov, 2002b). Its principles will be recalled here with respect to their resonances to Eastern thought

Topological Foundations of Cognitive Science

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki

A hierarchy of languages, logics, and mathematical theories

We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal. We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence