A Modular Consistency Proof for DOLCE

We propose a novel technique for proving the consistency of large, complex and heterogeneous theories for which ‘standard’ automated reasoning methods are considered insufficient. In particular, we exemplify the applicability of the method by establishing the consistency of the foundational ontology DOLCE, a large, first-order ontology. The approach we advocate constructs a global model for a theory, in our case DOLCE, built from smaller models of subtheories together with amalgamability properties between such models. The proof proceeds by (i) hand-crafting a so-called architectural specification of DOLCE which reflects the way models of the theory can be built, (ii) an automated verification of the amalgamability conditions, and (iii) a (partially automated) series of relative consistency proofs. Introduction The field of formal ontology may be subdivided into the study of domain ontologies, devoted to specific application areas, and foundational ontologies, axiomatising fundamental and domain-independent concepts. Foundational ontologies, such as SUMO (Niles and Pease 2001), DOLCE (Gangemi et al. 2002), GFO (Heller and Herre 2004), and BFO (Grenon, Smith, and Goldberg 2004), are typically specified in some variant of first-order logic, and their firstorder theories tend to be rather large. DOLCE, for instance, consists of a few hundred axioms, and SUMO of several thousand. Automated and semi-automated theorem proving systems have successfully been applied to reasoning about foundational ontologies. In particular, using automated provers, a number of inconsistencies in SUMO have been found (Voronkov 2006; Horrocks and Voronkov 2006), and SUMO has been corrected accordingly. The problem of proving the consistency of ontologies, however, is much harder in general. In the literature, two main approaches for proving consistency are described: model finders and relative consistency proofs. There are several model finders for first-order logic available. Some of them search for finite models by a translation to propositional logic (and then using SAT solvers) (e.g. Isabelle-refute (Weber 2005)), some Copyright c © 2011, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. of them use more advanced methods like the model evolution calculus (e.g. Darwin (Baumgartner and Tinelli 2003; Baumgartner, Fuchs, and Tinelli 2004)), or resolution via detecting a saturated set of clauses (e.g. SPASS (Weidenbach et al. 2002)). However, these techniques currently only suffice to find models for relatively small first-order theories—they do not scale to DOLCE, let alone SUMO. In fact, the difficulties already arise for the rather small sub-theories ‘classical extensional parthood’ (CEP) and ‘constitution’ (CON) of DOLCE. CEP is a theory of mereology, and it is straightforward to see that finite models for it can be obtained by powersets of finite sets, where the empty set has to be excluded. The singleton sets are then just the atoms of the mereology. The above first-order model finders could not find models with more that four atoms for these theories. Moreover, several weeks of computation time did not suffice to find a model for the whole of DOLCE. An alternative way of proving consistency is to use a relative consistency proof, that is, to provide a theory interpretation into some other theory that is known (or assumed) to be consistent. An obvious disadvantage of this approach is that it not only requires the manual construction of such a theory interpretation, but that such an interpretation will also typically be rather large and complex. In this work, we propose to construct models not in a monolithic, but in a structured way. We employ a set of operations for model construction that have been introduced in the context of software specification under the name of architectural specifications. These allow for decomposing the task of constructing a model for a (large) theory into smaller subtasks. These subtasks include: (a) automatically finding (or manually constructing) models for (relatively) small theories, (b) proving the conservativity of theory extensions, which can be done performing (local) relative consistency proofs, and (c) establishing amalgamability between already constructed models (or model classes). Relative Consistency Proofs For the purposes of this paper we shall identify a (first-order) ontology with a theory in first-order logic, namely a signature (set of non-logical symbols) and a set of axioms. We will say that a theory is consistent (=satisfiable) if it has a model; by completeness this is equivalent to formal consistency, which means that no contradiction can be derived. When we are unable to directly establish that a certain theory, say T , is consistent, we can instead show that it is consistent provided some other theory T ′ is. The general method behind this is as follows: T ′ is extended conservatively with new definitions (call the resulting theory T ′′), and then T is interpreted in T ′′, via a theory morphism (interpretation of theories) σ : T → T ′′. Now if T ′ is consistent, it has a model. Since T ′′ is a conservative extension, it has a model, too, and this model can be reduced (via σ) to a model of T . Hence, altogether, consistency of T ′ implies that of T . Let us make the notion of ‘conservative extension with new definitions’ a bit more precise. A conservative extension of a theory T is a theory extension ι : T → T ′ such that for any model M of T , there is a ι-expansion of M to a T ′-model M ′, i.e. such that the reduct M |ι of M ′ via ι is again M . If the ι-expansion is in fact always unique, then the theory extension is called definitional. We can summarise the above as follows: diagrammatically, we can represent relative consistency proofs in the form of conservativity triangles as shown in Fig. 1, i.e. we are given theories T, T ′, T ′′, and signature morphisms σ : T → T ′′, ι1 : T ′ → T ′′, and ι2 : T ′ → T such that the following triangle commutes: