Ulf Krumnack

Syntactic principles of heuristic-driven theory projection

Analogy making is a central construct in human cognition and plays an important role to explain cognitive abilities. While various psychologically or neurally inspired theories for analogical reasoning have been proposed, there is a lack of a logical foundation for analogical reasoning in artificial intelligence and cognitive science. We aim to close this gap and propose heuristic-driven theory projection (HDTP), a mathematically sound framework for analogy making. HDTP represents knowledge about the source and the target domain as first-order logic theories and compares them for structural commonalities using anti-unification. The paper provides an overview of the syntactic principles of HDTP, explains all phases of analogy making at a formal level, and illustrates these phases with examples.

Restricted higher-order anti-unification for analogy making

Anti-unification has often be used as a tool for analogy making. But while first-order anti-unification is too simple for many applications, general higher-order anti-unification is too complex and leads into theoretical difficulties. In this paper we present a restricted framework for higher-order substitutions and show that anti-unification is well-defined in this setting. A complexity measure for generalizations can be introduced in a quite natural way, which allows for selecting preferred generalizations. An algorithm for computing such generalizations is presented and the utility of complexity for anti-unifying sets of terms is discussed by an extended example.

A computational account of conceptual blending in basic mathematics

We present an account of a process by which different conceptualisations of number can be blended together to form new conceptualisations via recognition of common features, and judicious combination of their distinctive features. The accounts of number are based on Lakoff and Nunezs cognitively-based grounding metaphors for arithmetic. The approach incorporates elements of analogical inference into a generalised framework of conceptual blending, using some ideas from the work of Goguen. The ideas are worked out using Heuristic-Driven Theory Projection (HDTP, a method based on higher-order anti-unification). HDTP provides generalisations between domains, giving a crucial step in the process of finding commonalities between theories. In addition to generalisations, HDTP can also transfer concepts from one domain to another, allowing the construction of new conceptual blends. Alongside the methods by which conceptual blends may be constructed, we provide heuristics to guide this process.

Heuristic-Driven Theory Projection: An Overview

This chapter provides a concise overview of Heuristic-Driven Theory Projection (HDTP), a powerful framework for computing analogies. The chapter attempts to illuminate HDTP from several different perspectives. On the one hand, the syntactic basis of HDTP is formally specified, in particular, restricted higher-order anti-unification together with a complexity measure is described as the core process to compute a generalization given two input domains (source and target). On the other hand, the substitution-governed alignment and mapping process is analyzed together with the transfer of knowledge from source to target in order to induce hypotheses on the target domain. Additionally, this chapter presents some core ideas concerning the semantics of HDTP as well as the algorithm that computes analogies given two input domains. Finally, some further remarks describe the different (but important) roles heuristics play in this framework.

Algorithmic Aspects of Theory Blending

In Cognitive Science, conceptual blending has been proposed as an important cognitive mechanism that facilitates the creation of new concepts and ideas by constrained combination of available knowledge. It thereby provides a possible theoretical foundation for modeling high-level cognitive faculties such as the ability to understand, learn, and create new concepts and theories. This paper describes a logic-based framework which allows a formal treatment of theory blending, discusses algorithmic aspects of blending within the framework, and provides an illustrating worked out example from mathematics.

Re-representation in a Logic-Based Model for Analogy Making

Analogical reasoning plays an important role for cognitively demanding tasks. A major challenge in computing analogies concerns the problem of adapting the representation of the domains in a way that the analogous structures become obvious, i.e. finding and, in certain circumstances, generating appropriate representations that allow for computing an analogical relation. We propose to resolve this re-representation problem of analogy making in a logical framework based on the anti-unification of logical theories. The approach is exemplified using examples from qualitative reasoning (naive physics) and mathematics.

Theory Blending as a Framework for Creativity in Systems for General Intelligence

Being creative is a central property of humans in solving problems, adapting to new states of affairs, applying successful strategies in previously unseen situations, or coming up with new conceptualizations. General intelligent systems should have the potential to realize such forms of creativity to a certain extent. We think that creativity and productivity issues can be best addressed by taking cognitive mechanisms into account, such as analogy making, concept blending, computing generalizations and the like.

Structure-sensitive learning of text types

In this paper, we discuss the structure based classification of documents based on their logical document structure, i.e., their DOM trees.We describe a method using predefined structural features and also four tree kernels suitable for such structures. We evaluate the methods experimentally on a corpus containing the DOM trees of newspaper articles, and on the well-known SUSANNE corpus. We will demonstrate that, for the two corpora, many text types can be learned based on structural features only.

What Is a Derived Signature Morphism

The notion of signature morphism is basic to the theory of institutions. It provides a powerful primitive for the study of specifications, their modularity and their relations in an abstract setting. The notion of derived signature morphism generalises signature morphisms to more complex constructions, where symbols may be mapped not only to symbols, but to arbitrary terms. The purpose of this work is to study derived signature morphisms in an institution-independent way. We will recall and generalize two known approaches to derived signature morphisms, introduce a third one, and discuss their pros and cons. We especially study the existence of colimits of derived signature morphisms. The motivation is to give an independent semantics to the notion of derived signature morphism, query and substitution in the context of the Distributed Ontology, Modeling and Specification Language DOL.

Refinements of restricted higher-order anti-unification for heuristic-driven theory projection

Empirical research supports the belief that structural commonalities between two domains are the main guidance for the construction of analogies. Restricted higher-order anti-unification has been shown suitable to find structural commonalties and generate mappings between domains in the symbolic analogy model Heuristic-Driven Theory Projection (HDTP). This paper will describe how to enforce and integrate restrictions on mappings between symbols from a many-to-many up to a one-to-one symbol correspondence. We will also discuss how sorts together with sortal ontologies can be incorporated into anti-unification within HDTP and thereby restrict possible mappings between domains.