Introduction

Considering logic-based ontology representation languages, in Description Logics (DLs) many proposals for defining defeasibility and typicality have been formalised: as a matter of fact, most of them emerge from existing approaches in non-monotonic logics, as in [1,2]. On the other hand, little attention has been devoted to study the capability of these approaches in capturing the interpretation of typicality and exceptions from the point of view of formal ontology and cognitive aspects. Consequently, the philosophical and cognitive assumptions of this kind of reasoning are often overlooked and need a committed discussion in order to understand the capabilities of the current approaches.

Considering this, we recently initiated this discussion with an analysis of generics [3], sentences reporting a regularity regarding particular facts that can be generalised but tolerate exceptions. Our analysis (presented in [4]) highlighted three desiderata for non-monotonic reasoning: D1. Exceptionality: generics and non-monotonic reasoning both admit exceptions and much of the effort in the research has been dedicated to explain and model how exceptions can be tolerated. We think that another important aspect that should be considered is why something is an exception, i.e. how to include in the formal representation also the justification or explanation of why an instance is considered exceptional or not.

D2. Gradability: normality is a graded notion in the case of typicality, for example, instead of typical individuals and atypical ones with respect to some concept, we have more or less typical individuals. For instance, it would not be possible to divide wolves between typical wolves and atypical ones in absolute terms, but there would be wolves that are more or less typical according to the specific features of each individual.

D3. Content sensitivity: non-monotonic reasoning cannot be modelled by using only an extensional approach. This means that we cannot rely on pure extensional semantics, i.e. seeing the relation among concepts only in the light of relationships between sets. We need to take into account the semantics of the concepts involved in a broader sense, for example by relying on notions like typicality and saliency. The intuition here is that to explain why an individual is exceptional, for example, one would need some insights into the meaning (or, the content) of the statements of which the individual is an exception.

According to these desiderata, in this paper we sketch a new formal account for non-monotonic reasoning in DLs based on a graded reading of typicality. Intuitively, in the case of a conflict between two facts about an individual, we can decide which one should be accepted according to how much the individual in question is typical w.r.t. such facts. For example: we know that dogs are trusted, whereas wolves are not; we know also that Balto is a wolfdog hybrid; we can ask ourselves now, should we infer that Balto is trusted or not? In our approach, we want to use the additional information we have about Balto being a dog and Balto being a wolf to see if he is a more typical instance of a dog or wolf and, according to this, infer if he is trusted or not. More specifically, our approach is based on two main elements: prototype definitions and a typicality score. Prototype definitions are inspired by the prototype theory of concepts [5] and its representation based on the tooth operator as introduced, for example, in [6]. According to the endorsers of the prototype theory about concepts, being a member of a concept does not mean to satisfy a precise definition, but rather to satisfy enough features or constituents of that concept [7]. The second key element is the typicality score for individuals: this is calculated by inspecting to what extent the individual satisfies the features of the prototype. The aim of the score is to measure how typical the individual is with respect to the prototype considered: in case of a conflict on prototype-related properties, the score provides a preference determining which conclusion should prevail for that specific individual.

We remark that the current presentation of the formalisation is still an initial proposal and includes some constraints to simplify its exposition: some of the possible refinements and extensions are briefly discussed in the conclusions.

DLs with Weighted Prototypes

On the base of the idea above, we distinguish two parts in our knowledge bases: the actual DL knowledge base, which represents the knowledge of interest and can contain defeasible axioms and information about features of individuals, and a separate set containing prototype definitions. In the following we sketch a proposal for a syntax and semantics of such enriched KBs.

Syntax

The following definitions are independent from the DL language used for representing the main knowledge base: we consider a fixed concept language ℒ Σ (such as for example 𝒜ℒ𝒞) based on a DL signature Σ with disjoint and non-empty sets NC of concept names, NR of role names, and NI of individual names. Furthermore, we identify a subset of the concept names as denoting prototype names by assuming a subset NP ⊆ NC and a set of feature names NF ⊆ NC with NP ∩ NF = ∅.

Definition 1 (Features).

A basic feature is a concept name 𝐶 ∈ NF. A general feature is a complex concept in language ℒ Σ using only basic features as concept names.

For simplicity, we call general concepts the concepts composed only of concepts in NC∖NP∪NF.

The features associated with prototypes together with the degree of their importance are given in prototype definitions. 1 In particular, to allow for a direct comparison across prototype scores, we here constrain the weights of features to be in the [0, 1] interval and to add up to 1, i.e. prototypes are positive and normalised.

Definition 2 (Positive normalised prototype definition).

Let 𝑃 ∈ NP be a prototype name, let 𝐶 1 , . . . , 𝐶 𝑚 be general features of ℒ Σ and let 𝑤 = (𝑤 1 , . . . , 𝑤 𝑚 ) ∈ R 𝑚 be a weight vector, where for every 𝑖 ∈ {1, . . . , 𝑚} we have 𝑤 𝑖 ∈ (0, 1] and ∑︀ 𝑖∈{1,...,𝑚} 𝑤 𝑖 = 1. Then, the expression

𝑃 (𝐶 1 : 𝑤 1 , . . . , 𝐶 𝑚 : 𝑤 𝑚 )

is called a prototype definition for 𝑃 .

In the knowledge part of the KB, we can use prototype names in DL axioms to describe properties of the members of such classes. Here we consider the case in which prototype names are only used as primitive concepts on the left hand side of concept inclusions.

In particular, we call a concept inclusion of the type 𝑃 ⊑ 𝐷 a prototype axiom if 𝑃 ∈ NP and 𝐷 is a (possibly general) concept of ℒ Σ . Intuitively, these axioms are not absolute and can be "overridden" by prototype instances (cf. defeasible axioms in [8]), also depending on the "degree of membership" of the individual to the given prototype (i.e., the satisfaction of its features). Prototype axioms can be seen as corresponding to generic sentences since they express generalisations that admits exceptions. Such exceptions can thus override the truth of a prototype axiom for that specific individual.

As noted above, we consider knowledge bases which can contain prototype axioms and which are enriched with an accessory KB, the PBox 𝒫 providing prototype definitions.

Definition 3 (Prototyped Knowledge Base, PKB).

A prototyped knowledge base, PKB for short, in language ℒ Σ is a triple K = ⟨𝒯 , 𝒜, 𝒫⟩ where: Note that a PKB ⟨𝒯 , 𝒜, ∅⟩ can be seen as a standard DL knowledge base.

-𝒯 = 𝑇 𝑃 ⊎ 𝑇 𝐶 is

Example 1. We can now represent the example described in the introduction as a prototyped knowledge base 𝒦 = ⟨𝒯 , 𝒜, 𝒫⟩ as follows: Below we will construct a semantics for this kind of PKB which will entail and justify the conclusion that balto is a trusted dog which is a wolf, without being inconsistent. Note that in the case of the instances pluto and alberto no contradiction arises, thus we want that the axioms in 𝒯 are applied to them normally. ◇

𝒯 = 𝑇 𝑃 = { Dog ⊑ Trusted,

Semantics

The semantics of PKBs is based on standard interpretations for the underlying DL ℒ Σ . However, we need to introduce additional semantic structure to manage exceptions to prototype axioms, exploiting the prototype definition expressions in 𝒫.

Definition 4 (PKB interpretations). A PKB interpretation is a description logic interpretationℐ = ⟨∆ ℐ , • ℐ ⟩ for signature Σ with a non-empty domain, ∆ ℐ , 𝑎 ℐ ∈ ∆ ℐ for every 𝑎 ∈ NI, 𝐴 ℐ ⊆ ∆ ℐ for every 𝐴 ∈ NC, 𝑅 ℐ ⊆ ∆ ℐ × ∆ ℐ

for every 𝑅 ∈ NR, and where the extension of complex concepts is defined recursively as usual for language ℒ Σ .

Note that we are not giving a DL interpretation to the prototype definition expressions in 𝒫.

We consider the notion of axiom instantiation and clashing assumptions as defined in [8]. Given an axiom 𝛼 ∈ ℒ Σ with FO-translation ∀x.𝜑 𝛼 (x), the instantiation of 𝛼 with a tuple e of individuals in NI, written 𝛼(e), is the specialisation of 𝛼 to e, i.e., 𝜑 𝛼 (e), depending on the type of 𝛼.

Definition 5 (Clashing assumptions and clashing sets).

A clashing assumption is a pair ⟨𝛼, e⟩ such that 𝛼(e) is an axiom instantiation of 𝛼, and 𝛼 ∈ 𝑇 𝑃 is a prototype axiom.

A clashing set for ⟨𝛼, e⟩ is a satisfiable set 𝑆 of ABox assertions s.t. 𝑆 ∪ {𝛼(e)} is unsatisfiable.

Intuitively, a clashing assumption ⟨𝑃 ⊑ 𝐷, 𝑒⟩ states that we assume that 𝑒 is an exception to the prototype axiom 𝑃 ⊑ 𝐷 in a given PKB interpretation. Then, the fact that a clashing set 𝑆 for ⟨𝑃 ⊑ 𝐷, 𝑒⟩ is verified by such an interpretation gives a "justification" of the validity of the assumption of overriding. This intuition is reflected in the definition of models: we first extend PKB interpretations with a set of clashing assumptions.

Definition 6 (CAS-interpretation).

A CAS-interpretation is a structure ℐ CAS = ⟨ℐ, 𝜒⟩ where ℐ is a PKB interpretation and 𝜒 is a set of clashing assumptions.

Then, CAS-models for a PKB K are CAS-interpretations that verify "strict" axioms in 𝑇 𝐶 and defeasibly apply prototype axioms in 𝑇 𝑃 (excluding the exceptional instances in 𝜒).

Definition 7 (CAS-model). Given a PKB K, a CAS-interpretationℐ CAS = ⟨ℐ, 𝜒⟩ is a CAS- model for K (denoted ℐ CAS |= K),ℐ ′ CAS = ⟨ℐ ′ , 𝜒⟩ of K that is NI-congruent with ℐ CAS , it holds that ℐ ′ |= 𝑆 ⟨𝛼,e⟩ . A CAS model ℐ CAS of a PKB K is justified, if every ⟨𝛼, e⟩ ∈ 𝜒 is justified in K.

We define the consequence from justified CAS-models:

K |= 𝐽𝐶𝐴𝑆 𝛼 if ℐ CAS |= 𝛼 for every justified CAS-model ℐ CAS of K.

The main intuition of prototype definitions is that each member of a prototype is associated with a score which denotes the "degree of typicality" of the instance with respect to the concept described by the prototype. As in [6], such a degree is computed from the prototype features that are satisfied by the instances and their score. Ideally, the prototype score of an individual allows us to determine a preference over models: axioms on prototypes with higher score are preferred to the ones on lower scoring prototypes. Formally, a simple score function can be defined as follows:

Definition 9 (Prototype score). Given a prototype definition 𝑃 (𝐶 1 : 𝑤 1 , ..., 𝐶 𝑚 : 𝑤 𝑚 ), we define the score function score 𝑃 : NI → [0, 1] for prototype 𝑃 as:

score 𝑃 (𝑎) = ∑︁ K |= 𝐽𝐶𝐴𝑆 𝐶 𝑖 (𝑎)

𝑤 𝑖

The scoring function can then be used to define preferences over models: in particular, we want to prefer justified CAS models where the exceptions appear on elements of the less scoring prototypes. This can be encoded as follows:

Definition 10 (Preference SP). 𝜒 1 > 𝜒 2 if, for every ⟨𝑃 ⊑ 𝐷, 𝑒⟩ ∈ 𝜒 1 ∖ 𝜒 2 such that there exists a ⟨𝑄 ⊑ 𝐸, 𝑒⟩ ∈ 𝜒 2 ∖ 𝜒 1 , it holds that score 𝑃 K (𝑒) < score 𝑄 K (𝑒).

Given two CAS-interpretations

ℐ 1 CAS = ⟨ℐ 1 , 𝜒 1 ⟩ and ℐ 2 CAS = ⟨ℐ 2 , 𝜒 2 ⟩, we say that ℐ 1 CAS is preferred to ℐ 2 CAS (denoted ℐ 1 CAS > ℐ 2 CAS ) if 𝜒 1 > 𝜒 2 .

Finally, we define the notion of PKB model as a minimal justified model for the PKB.

Definition 11 (PKB model). An interpretationℐ is a PKB model of K (denoted, ℐ |= K) if -K has some justified CAS model ℐ CAS = ⟨ℐ, 𝜒⟩.

-there exists no justified ℐ ′ CAS = ⟨ℐ ′ , 𝜒 ′ ⟩ that is preferred to ℐ CAS .

The consequence from PKB models of K (denoted K |= 𝛼) allows us to use the degree of typicality of instances to verify which of the conflicting prototype axioms should apply.

Example 2. Considering the PKB reported in the example above, assume to have two PKB interpretations ℐ 1 and ℐ 2 associated respectively with the following two sets of clashing assumptions 𝜒 1 = {⟨Wolf ⊑ ¬Trusted, balto⟩} and 𝜒 2 = {⟨Dog ⊑ Trusted, balto⟩}.

We have now two CAS-interpretations corresponding to ⟨ℐ Moreover, we can note that for pluto and alberto we can standardly infer Trusted(pluto) and ¬Trusted(alberto). The reason is that the clashing assumptions are referred to specific individuals, and since there are no contradicting assertions for pluto and alberto, there are no clashing sets that justify their assumptions as exceptions. Therefore, axioms in 𝒯 apply to them standardly. ◇

Discussion and Conclusions

We presented an initial formalisation for a non-monotonic extension of DLs with the aim of satisfying three desiderata extracted from a critical discussion on generics and the prototype theory about concepts. We note that our formalism meets the desiderata: (D1). the formalisation is based on the idea that we need to justify an exception to an axiom by looking at how typical it is: in other words, we use typicality to decide with respect to which of the conflicting axioms (which correspond to generics) the individual is an exception; (D2). we are using a graded notion of typicality: we do not simply have typical and atypical individuals, but we compute a score which is comparable across prototypes; (D3). the notion of typicality that we introduce is not extensional: by using the scores to represent it, we are relying on a characteristic which goes beyond an extensional set-theoretic treatment.

In future work, we want to extend the cognitive and ontological study of exceptions also by comparing it with other accounts for typicality and defeasibility in DLs. Regarding our formalisation, we need to explore and refine the formal consequences of our approach in greater detail. In particular, we need to discuss what are the best options to compute the scores in order to have a balanced score for every prototype and how to extend this computation to roles, possibly following some of the ideas outlined in [9,10], where novel tooth-operators for role-successor counting are studied. The preference relation can also be refined: for example, comparisons on clashing assumptions can be restricted to the axioms that are actually incompatible. We need also to understand better how to allow for more interaction between concepts used for prototypes and features, for example by allowing nested definitions of prototypes, use prototype concepts as features, and compute scores with defeasible features.

Finally, we need an extensive comparison with related works. On the one hand, we will confront our approach to existing formalisms for defeasible reasoning in DLs like [11,12]. Of particular interest for this purpose are formalisms using weights and having a multi-preferential relation over the individuals with respect to the concepts they are instances of, as, for instance, [13,14]. On the other hand, we will analyse also works that share our approach to taking into account, in a central way, the results coming from cognitive science and philosophy for developing formal systems in the field of knowledge representation and in particular using the language of DLs. Examples of such works, particularly interested in the notion of typicality, are [15,16].