Blending in the Hub

Conceptual blending has been employed very successfully to understand the process of concept invention, studied particularly within cognitive psychology and linguistics. However, despite this influential research, within computational creativity little effort has been devoted to fully formalise these ideas and to make them amenable to computational techniques. We here present the basic formalisation of conceptual blending, as sketched by the late Joseph Goguen, and show how the Distributed Ontology Language DOL can be used to declaratively specify blending diagrams. Moreover, we discuss in detail how the workflow and creative act of generating and evaluating a new, blended concept can be managed and computationally supported within Ontohub, a DOL-enabled theory repository with support for a large number of logical languages and formal linking constructs. Concept Invention via Blending In the general methodology of conceptual blending introduced by Fauconnier and Turner (2003), the blending of two thematically rather different conceptual spaces yields a new conceptual space with emergent structure, selectively combining parts of the given spaces whilst respecting common structural properties.1 The ‘imaginative’ aspect of blending is summarised as follows in Turner (2007): [. . . ] the two inputs have different (and often clashing) organising frames, and the blend has an organising frame that receives projections from each of those organising frames. The blend also has emergent structure on its own that cannot be found in any of the inputs. Sharp differences between the organising frames of the inputs offer the possibility of rich clashes. Far from blocking the construction of the network, such clashes offer challenges to the imagination. The resulting blends can turn out to be highly imaginative. A classic example for this is the blending of the concepts house and boat, yielding as most straightforward blends the The usage of the term ‘conceptual space’ in blending theory is not to be confused with the usage established by Gärdenfors (2000). concepts of a houseboat and a boathouse, but also an amphibious vehicle (Goguen and Harrell, 2009). In the almost unlimited space of possibilities for combining existing ontologies to create new ontologies with emergent structure, conceptual blending can be built on to provide a structural and logic-based approach to ‘creative’ ontological engineering. This endeavour primarily raises the following two challenges: (1) when combining the terminologies of two ontologies, the shared semantic structure is of particular importance to steer possible combinations. This shared semantic structure leads to the notion of base ontology, which is closely related to the notion of ‘tertium comparationis’ found in the classic rhetoric and poetic theories, but also in more recent cognitive theories of metaphor (see, e.g., Jaszczolt (2003)); (2) having established a shared semantic structure, there is typically still a huge number of possibilities that can capitalise on this information in the combination process: here, structural optimality principles as well as ontology evaluation techniques take on a central role in selecting interesting blends. We believe that the principles governing ontological blending are quite distinct from the rather informal principles employed in blending phenomena in language or poetry, or the rather strict principles ruling blending in mathematics, in particular in the way formal inconsistencies are dealt with. For instance, whilst blending in poetry might be particularly inventive or imaginative when the structure of the basic categories found in the input spaces is almost completely ignored, and whilst the opposite, i.e., rather strict adherence to sort structure, is important in areas such as mathematics in order to generate meaningful blends2, ontological blending is situated somewhere in the middle: re-arrangement and new combination of basic categories can be rather interesting, but has to be finely controlled through corresponding interfaces, often regulated by or related to choices found in foundational or upper ontologies. For instance when creating the theory of transfinite cardinals by blending the perfective aspect of counting up to any fixed finite number with the imperfective aspect of ‘endless counting’ (Núñez, 2005). The core contributions of the paper can be summarised as follows.3 We: • sketch the logical analysis of conceptual blending in terms of blending diagrams and colimits, as originally proposed by Joseph Goguen, and give an abstract definition of ontological blendoids capturing the basic intuitions of conceptual blending in the ontological setting; • provide a formal language for declaratively specifying blending diagrams by employing the OWL4 fragment of the distributed ontology language DOL for blending. This provides a structured approach to ontology languages and combines the simplicity and good tool support for OWL with the more complex blending facilities of OBJ3 (Goguen and Malcolm, 1996) or Haskell (Kuhn, 2002); • discuss the capabilities of the Ontohub/Hets ecosystem with regard to collaboratively managing, creating, and evaluating blended concepts and theories; this includes an investigation of the evaluation problem in blending, together with a discussion of structural optimality principles and current automated reasoning support. We close with a detailed discussion of open problems and future work. Blending Computationalised Goguen has created the field of algebraic semiotics which logically formalises the structural aspects of semiotic signs, sign systems, and their mappings (Goguen, 1999). In Goguen and Harrell (2009), algebraic semiotics has been applied to user interface design and blending. Algebraic semiotics does not claim to provide a comprehensive formal theory of blending—indeed, Goguen and Harrell admit that many aspects of blending, in particular concerning the meaning of the involved notions, as well as the optimality principles for blending, cannot be captured formally. However, the structural aspects can be formalised and provide insights into the space of possible blends. Goguen defines semiotic systems to be algebraic theories that can be formulated by using the algebraic specification language OBJ (Goguen and Malcolm, 1996). Moreover, a special case of a semiotic system is a conceptual space: it consists only of constants and relations, one sort, and axioms that define that certain relations hold on certain instances. As we focus on standard ontology languages, namely OWL and first-order logic, we here replace the logical language OBJ. As structural aspects in the ontology language are necessary for blending, we augment these languages with structuring mechanisms known from algebraic specification theory (Kutz et al., 2008). This allows to translate most parts of Goguen’s theory to these ontology languages. Goguen’s main insight has been that semiotic systems and conceptual spaces can be related via morphisms, and that blending is comparable to colimit computation, a construction that abstracts the operation of disjoint unions modulo This paper elaborates on ideas first introduced in Hois et al. (2010); detailed technical definitions are given in Kutz et al. (2012). With ‘OWL’ we refer to OWL 2 DL, see http://www.w3. org/TR/owl2-overview/ base morphisms O1 O2 B Base Ontology Blendoid Input 1 Input 2 blendoid morphisms Figure 1: The basic integration network for blending: concepts in the base ontology are first refined to concepts in the input ontologies and then selectively blended into the blendoid. the identification of certain parts, explained in more detail below. In particular, the blending of two concepts is often a pushout (also called a blendoid in this context). Some basic definitions:5 Non-logical symbols are grouped into signatures, which for our purposes can be regarded as collections of kinded symbols (e.g. concept names, relation names). Signature morphisms are maps between signatures that preserve (at least) kinds of symbols (i.e. map concept names to concept names, relations to relations, etc.). A theory or ontology pairs a signature with a set of sentences over that signature, and an theory morphism (or interpretation) between two theories is just a signature morphism between the underlying signatures that preserves logical consequence, that is, ρ : T1 → T2 is a theory morphism if T2 |= ρ(T1), i.e. all the translations of sentences of T1 along ρ follow from T2. This construction is completely logic independent. Signature/theory morphisms are an essential ingredient for describing conceptual blending in a logical way. We now give a general definition of ontological blending capturing the basic intuition that a blend of input ontologies shall partially preserve the structure imposed by base ontologies, but otherwise be an almost arbitrary extension or fragment of the disjoint union of the input ontologies with appropriately identified base space terms. For the following definition, which we first introduced in Kutz et al. (2012), a diagram consists of a set of ontologies and a set of morphisms between them. The colimit of a diagram is similar to a disjoint union of its ontologies, with some identifications of shared parts as specified by the morphisms in the diagram. We refrain from presenting the category-theoretic definition here (which can be found in Note that these definitions apply to OWL, but also to many other logics. Indeed, they apply to any logic formalised as an institution (Goguen and Burstall, 1992). Adámek, Herrlich, and Strecker (1990)), but explain the colimit operation using the examples below. Definition 1 (Ontological Base Diagram) An ontological base diagram is a diagram D for which the minimal nodes (Bi)i∈Dmin⊆|D| are called base ontologies, the maximal nodes (Ij)j∈Dmax⊆|D| called input ontologies, and where the theory morphisms μij : Bi → Ij are called the base morphis