A description logic framework for commonsense conceptual combination integrating typicality, probabilities and cognitive heuristics

ABSTRACT

We propose a nonmonotonic Description Logic of typicality able to account for the phenomenon of the combination of prototypical concepts. The proposed logic relies on the logic of typicality $\mathcal{A}\mathcal{L}\mathcal{C}+{\mathbf{T}}_{\mathbf{R}}$, whose semantics is based on the notion of rational closure, as well as on the distributed semantics of probabilistic Description Logics, and is equipped with a cognitive heuristic used by humans for concept composition.

We first extend the logic of typicality $\mathcal{A}\mathcal{L}\mathcal{C}+{\mathbf{T}}_{\mathbf{R}}$ by typicality inclusions of the form $p::\mathbf{T}(C)⊑D$, whose intuitive meaning is that ‘we believe with degree $p$ about the fact that typical Cs are Ds’. As in the distributed semantics, we define different scenarios containing only some typicality inclusions, each one having a suitable probability. We then exploit such scenarios in order to ascribe typical properties to a concept $C$ obtained as the combination of two prototypical concepts. We also show that reasoning in the proposed Description Logic is ExpTime-complete as for the underlying standard Description Logic $\mathcal{A}\mathcal{L}\mathcal{C}$.

KEYWORDS:

• Nonmonotonic reasoning
• description logics
• common-sense reasoning
• cognitive modelling

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Thus, providing additional arguments in favour of the theoretical positions according to which it is possible to reconcile compositionality and prototypes (Hampton, 2000; Prinz, 2012).

2. We want to stress that, as in any probabilistic formal framework, probabilities are assumed to come from an application domain. This is true also for other frameworks such as, for example, fuzzy logics or probabilistic extensions of logic programs. In this paper, we focus on the proposal of the formalism itself, therefore the machinery for obtaining probabilities from a dataset of the application domain is out of the scope.

3. In Theorem 10 in (Giordano et al., 2015) the authors have shown that for any consistent KB there exists a finite minimal canonical model of KB.

4. It is worth-noting that a general framework for the automatic identification of a HEAD/MODIFIER combination is currently not available in literature. In this work, we will take for granted that some methods for the correct identification of these pairs exist and we will focus on the reasoning part.

5. The reason why we only allow typicality inclusions equipped with probabilities $p>0.5$ is detailed in the following.

6. In the sense that prototypes are usually intended as statistically representative members of a category (e.g. the prototype of $Bird$ in, let’s say, Europe is different with respect to the prototype of $Bird$ in New Zealand, i.e. a kiwi which do not fly, since the types of birds encountered by these two populations are statistically different and therefore the learned typical members differ as well). This assumption is also reflected in computational frameworks of cognitive semantics where prototypes for conceptual representations are naturally calculated/learned as centroids in vector space models of meaning (Gärdenfors, 2004, 2014).

7. It is worth noticing that in our logic, the uncertain/graded component of typicality is captured by the ranked semantics underlying the operator $T$ in the logic $\mathcal{A}\mathcal{L}\mathcal{C}+{\mathbf{T}}_{\mathbf{R}}$. On the other hand, the epistemic uncertainty is modelled by the interpretation of probabilities of the DISPONTE semantics.

8. It could be possible to consider an alternative semantics whose models are equipped with multiple preference relations, whence with multiple typicality operators. In this case, it should be possible to distinguish different aspects of exceptionality, however, the approach based on a single preference relation in Giordano et al. (2015) ensures good computational properties (reasoning in the resulting nonmonotonic logic $\mathcal{A}\mathcal{L}\mathcal{C}+{\mathbf{T}}_{\mathbf{R}}$ has the same complexity of the standard $\mathcal{A}\mathcal{L}\mathcal{C}$), whereas adopting multiple preference relations could lead to higher complexities.

9. It is worth noticing that here the degree $q$ does not play any role. Indeed, a typicality inclusion $\mathbf{T}(C)⊑D$ holds in a model only if it satisfies the semantic condition of the underlying DL of typicality, i.e. minimal (typical) elements of $C$ are elements of $D$. The degree of belief $q$ will have a crucial role in the application of the distributed semantics, allowing the definition of scenarios as well as the computation of their probabilities.

10. This choice is motivated by the challenges provided by the task of commonsense conceptual combination itself: in order to generate plausible novel compounds it is necessary to maintain a certain level of ‘surprise’ in the combination, since obvious inheritance of attributes does not have any explanatory power for human-like and human-level concept combination (Hampton, 1987). For this reason, both scenarios inheriting all the properties of the two concepts and all the properties of the HEAD are discarded. In this respect, the typicality-based inheritance procedure of our logic falls within the so-called functional compositionality, as introduced in Pelletier (2017).

11. The inconsistency arises when the knowledge base is extended by two contradicting properties for the combined concept, for instance the knowledge base extended by both $\mathbf{T}(Pet\sqcap Fish)⊑Warm$ and $\mathbf{T}(Pet\sqcap Fish)⊑\mathrm{\neg }Warm$ would be consistent only if there are no pet fishes.

12. It is worth noticing that the logic $T^{CL}$ would select the same scenario also in the more challenging situation in which degrees of properties of the HEAD are strictly lower, for instance in case inclusion $1$ would be replaced in $\mathcal{T}$ by $0.7::\mathbf{T}(Fish)⊑\mathrm{\neg }Affectionate$.

13. As we will see in the next sections, in application domains in which ${\mathbf{T}}^{\mathrm{C}\mathrm{L}}$ is exploited, e.g. in the field of computational creativity, where it is not always easy to define which is the ‘correct’ combination, this situation is not uncommon. As a consequence, in such cases, the final decision about what selection represents the most appropriate combination, is left to the human decision-makers.

14. The attempt of modelling a reasoning error could seem prima facie, counterintuitive in an AI setting. However, it is worth-noting that this type of reasoning corresponds to a very powerful evolutionary heuristics developed by humans and strongly relying on common-sense knowledge. The use of typical knowledge in cognitive tasks, in fact, has to do with the constraints that concern every finite agent that has a limited access to the knowledge relevant for a given task. Consider, for example the following variant of the Linda problem. Let us suppose that a certain individual Pluto is described as follows. He weighs about 250 kg, and he is approximately two metres tall. His body is covered with a thick, dark fur, he has a large mouth with robust teeth and paws with long claws. He roars and growls. Now, given this information, we have to evaluate the plausibility of the two following alternatives: a) Pippo is a mammal; b) Pippo is a mammal, and he is wild and dangerous. Which is the ‘correct’ answer? According to the dictates of the normative theory of probability, it is surely a). But if you encounter Pippo in the wilderness, it would probably be best to run.

15. https://tvtropes.org.

16. Technically this kind of combination is called conceptual blending and is slightly different from a classical combination since the generated concept is an entirely new one and is not a subset of classes generating it, see (Nagai & Taura, 2006) for more details on the subtle differences between the two tasks.

17. In this example, we consider the first $AntiHero$-revised knowledge base obtained in Section “Anti Hero”.

18. Of course, the number of properties can be considered as a parameter through which it is possible to play with the mechanisms underlying the logic ${\mathbf{T}}^{\mathrm{C}\mathrm{L}}$.