Preferential Temporal Description Logics
with Typicality and Weighted Knowledge Bases
Mario Alviano1 , Laura Giordano2,* and Daniele Theseider Dupré2
1
    DEMACS, University of Calabria, Via Bucci 30/B, 87036 Rende (CS), Italy
2
    DISIT, University of Piemonte Orientale, Viale Michel 11, 1512 Alessandria, Italy


                                         Abstract
                                         In this paper we define an extension of a temporal description logic with a typicality operator, to allow for
                                         defeasible reasoning in a preferential temporal description logic. We show that a preferential extension of
                                         LTL𝒜ℒ𝒞 with typicality can be polynomially encoded into LTL𝒜ℒ𝒞 , and the approach allows borrowing
                                         some decidability and complexity results. We consider as well a multi-preferential temporal semantic for
                                         temporal weighted knowledge bases with typicality.




1. Introduction
Preferential extensions of Description Logics (DLs) allow reasoning with exceptions through
the identification of prototypical properties of individuals or classes of individuals. Defeasible
inclusions are allowed in the knowledge base, to model typical, defeasible, non-strict properties
of individuals. Their semantics extends DL semantics with a preference relation among domain
individuals, along the lines of the preferential semantics introduced by Kraus, Lehmann and
Magidor [1, 2] (KLM for short). Preferential extensions and rational extensions of the description
logic 𝒜ℒ𝒞 [3] have been studied [4, 5], and several different closure constructions have been
developed [6, 7, 8, 9, 10, 11], inspired by Lehmann and Magidor’s rational closure [2] and
Lehmann’s lexicographic closure [12]. More recently, multi-preferential extensions of DLs have
been developed, by allowing multiple preference relations with respect to different concepts
[13, 14, 15], as the semantic for ranked and for weighted knowledge bases with typicality.
   Temporal extensions of Description Logics are very well-studied in DLs literature, see the
survey on temporal DLs and their complexity and decidability [16]. While preferential extensions
of LTL with defeasible temporal operators have been recently studied [17, 18, 19] to enrich
temporal formalisms with non-monotonic reasoning features, preferential extensions (and, more
specifically, typicality based extensions) of temporal DLs have not been considered so far, up to

CILC’23: 38th Italian Conference on Computational Logic, June 21–23, 2023, Udine, Italy
*
 Corresponding author.
" mario.alviano@unical.it (M. Alviano); laura.giordano@uniupo.it (L. Giordano); dtd@uniupo.it (D. Theseider
Dupré)
~ https://alviano.net/ (M. Alviano); https://people.unipmn.it/laura.giordano/ (L. Giordano);
https://people.unipmn.it/dtd/ (D. Theseider Dupré)
 0000-0002-2052-2063 (M. Alviano); 0000-0001-9445-7770 (L. Giordano); 0000-0001-6798-4380 (D. Theseider
Dupré)
                                       © 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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    Workshop
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                  ISSN 1613-0073
                                       CEUR Workshop Proceedings (CEUR-WS.org)
our knowledge.
   To fill this gap, in this paper we develop a preferential extension of Temporal DLs, based on
the approach proposed in [5] to define a description logic with typicality. More specifically, we
build over a temporal extension of 𝒜ℒ𝒞, LTL𝒜ℒ𝒞 [16], based on Linear Time Temporal Logic
(LTL), and develop its extension with typicality.
   Generalizing the approach in [5], we define a preferential temporal description logic LTL𝒜ℒ𝒞
with typicality, LTLT   𝒜ℒ𝒞 , by adding to the language a typicality operator T that selects the
most typical instances of a concept 𝐶. The resulting temporal DL with typicality allows for
representing temporal properties of concepts which admit exceptions, e.g., for instance that,
normally, professors teach at least a course until the end of the semester, although exceptions are
permitted.
   We show that the preferential extension of LTL𝒜ℒ𝒞 with typicality can be polynomially
encoded into LTL𝒜ℒ𝒞 , and this approach allows borrowing decidability and complexity results
from LTL𝒜ℒ𝒞 . We also consider a multi-preferential extension of LTL𝒜ℒ𝒞 , by allowing a concept-
wise preferential semantics where different preferences are associated to different concepts. The
encoding also applies to this case. We discuss possible extensions of the closure constructions
for weighted knowledge bases [14, 20] to the temporal case. It allows for a finer grained
representation of the plausibility of prototypical properties of a concept, including temporal
properties, by assigning weights to the different typicality properties.


2. The Description Logic 𝒜ℒ𝒞
In this section we recall the syntax and semantics of the description logic 𝒜ℒ𝒞 [3] and of its
temporal extension LTL𝒜ℒ𝒞 [16].

2.1. 𝒜ℒ𝒞
Let 𝑁𝐶 be a set of concept names, 𝑁𝑅 a set of role names and 𝑁𝐼 a set of individual names. The
set of 𝒜ℒ𝒞 concepts (or, simply, concepts) can be defined inductively as follows:
    • 𝐴 ∈ 𝑁𝐶 , ⊤ and ⊥ are concepts;
    • if 𝐶 and 𝐷 are concepts, and 𝑟 ∈ 𝑁𝑅 , then 𝐶 ⊓ 𝐷, 𝐶 ⊔ 𝐷, ¬𝐶, ∀𝑟.𝐶, ∃𝑟.𝐶 are concepts.
An 𝒜ℒ𝒞 knowledge base (KB) 𝐾 is a pair (𝒯 , 𝒜), where 𝒯 is a TBox and 𝒜 is an ABox. The
TBox 𝒯 is a set of concept inclusions (or subsumptions) 𝐶 ⊑ 𝐷, where 𝐶, 𝐷 are concepts. The
ABox 𝒜 is a set of assertions of the form 𝐶(𝑎) and 𝑟(𝑎, 𝑏) where 𝐶 is a concept, 𝑎 and 𝑏 are
individual names in 𝑁𝐼 and 𝑟 a role name in 𝑁𝑅 .
   An 𝒜ℒ𝒞 interpretation is defined as a pair 𝐼 = ⟨Δ, ·𝐼 ⟩ where: Δ is a domain — a set whose
elements are denoted by 𝑥, 𝑦, 𝑧, . . . , and ·𝐼 is an extension function that maps each concept name
𝐶 ∈ 𝑁𝐶 to a set 𝐶 𝐼 ⊆ Δ, each role name 𝑟 ∈ 𝑁𝑅 to a binary relation 𝑟𝐼 ⊆ Δ × Δ, and each
individual name 𝑎 ∈ 𝑁𝐼 to an element 𝑎𝐼 ∈ Δ. It is extended to complex concepts:
   ⊤𝐼 = Δ,      ⊥𝐼 = ∅,         (¬𝐶)𝐼 = Δ∖𝐶 𝐼 ,
   (∃𝑟.𝐶) = {𝑥 ∈ Δ | ∃𝑦.(𝑥, 𝑦) ∈ 𝑟𝐼 and 𝑦 ∈ 𝐶 𝐼 },
         𝐼                                                     (𝐶 ⊓ 𝐷)𝐼 = 𝐶 𝐼 ∩ 𝐷𝐼 ,
   (∀𝑟.𝐶)𝐼 = {𝑥 ∈ Δ | ∀𝑦.(𝑥, 𝑦) ∈ 𝑟𝐼 ⇒ 𝑦 ∈ 𝐶 𝐼 },               (𝐶 ⊔ 𝐷)𝐼 = 𝐶 𝐼 ∪ 𝐷𝐼 .
The notions of satisfiability of a KB in an interpretation and entailment are defined as follows:
Definition 1 (Satisfiability and entailment). Given an 𝒜ℒ𝒞 interpretation 𝐼 = ⟨Δ, ·𝐼 ⟩:
  - 𝐼 satisfies an inclusion 𝐶 ⊑ 𝐷 if 𝐶 𝐼 ⊆ 𝐷𝐼 ;
  - 𝐼 satisfies an assertion 𝐶(𝑎) (resp., 𝑟(𝑎, 𝑏)) if 𝑎𝐼 ∈ 𝐶 𝐼 (resp., (𝑎𝐼 , 𝑏𝐼 ) ∈ 𝑟𝐼 ).
Given a KB 𝐾 = (𝒯 , 𝒜), an interpretation 𝐼 satisfies 𝒯 (resp. 𝒜) if 𝐼 satisfies all inclusions in
𝒯 (resp. all assertions in 𝒜); 𝐼 is a model of 𝐾 if 𝐼 satisfies 𝒯 and 𝒜.
  A concept inclusion 𝐹 = 𝐶 ⊑ 𝐷 (resp., an assertion 𝐶(𝑎), 𝑟(𝑎, 𝑏)), is entailed by 𝐾, written
𝐾 |= 𝐹 , if for all models 𝐼 =⟨Δ, ·𝐼 ⟩ of 𝐾, 𝐼 satisfies 𝐹 .

   Given a knowledge base 𝐾, the subsumption problem is the problem of deciding whether an
inclusion 𝐶 ⊑ 𝐷 is entailed by 𝐾. The satisfiability problem is the problem of deciding whether
a knowlwdge base 𝐾 has a model. The concept satisfiability problem is the problem of deciding,
for a concept 𝐶, whether 𝐶 is consistent with 𝐾 (i.e., whether there exists a model 𝐼 of 𝐾, such
that 𝐶 𝐼 ̸= ∅).


3. The Temporal Description Logic LTL𝒜ℒ𝒞
The concepts of the temporal description logic LTL𝒜ℒ𝒞 can be formed from standard constructors
using the temporal operators ○ (next), 𝒰 (until), ◇ (eventually) and □ (always) of linear time
temporal logic (LTL). The set of temporally extended concepts is as follows:

       𝐶 ::= 𝐴 | ⊤ | ⊥ | 𝐶 ⊓ 𝐷 | 𝐶 ⊔ 𝐷 | ¬𝐶 | ∀𝑟.𝐶 | ∃𝑟.𝐶 | ○𝐶 | 𝐶𝒰𝐷 | ◇𝐶 | □𝐶

where 𝐴 ∈ 𝑁𝐶 , and 𝐶 and 𝐷 are temporally extended concepts.
   A temporal interpretation for LTL𝒜ℒ𝒞 is a pair ℐ = (Δℐ , ·ℐ ), where Δℐ is a nonempty
domain; ·ℐ is an extension function that maps each concept name 𝐶 ∈ 𝑁𝐶 to a set 𝐶 ℐ ⊆ N × Δℐ ,
each role name 𝑟 ∈ 𝑁𝑅 to a relation 𝑟ℐ ⊆ N × Δℐ × Δℐ , and each individual name 𝑎 ∈ 𝑁𝐼 to
an element 𝑎ℐ ∈ Δℐ . Following [16] we assume individual names to be rigid, i.e., having the
same interpretation at any time point. In a pair (𝑛, 𝑑) ∈ N × Δℐ , 𝑛 represents a time point and
𝑑 a domain element; (𝑛, 𝑑) ∈ 𝐶 ℐ means that 𝑑 is an instance of concept 𝐶 at time point 𝑛, and
similarly for (𝑛, 𝑑1 , 𝑑2 ) ∈ 𝑟ℐ . Function ·ℐ is extended to complex concepts as follows:

      ⊤ℐ = N × Δℐ         ⊥ℐ = ∅       (¬𝐶)ℐ = (N × Δℐ )∖𝐶 ℐ
      (𝐶 ⊓ 𝐷)ℐ = 𝐶 ℐ ∩ 𝐷ℐ           (𝐶 ⊔ 𝐷)ℐ = 𝐶 ℐ ∪ 𝐷ℐ
      (∃𝑟.𝐶)ℐ = {(𝑛, 𝑥) ∈ N × Δℐ | ∃𝑦.(𝑛, 𝑥, 𝑦) ∈ 𝑟ℐ and (𝑛, 𝑦) ∈ 𝐶 ℐ }
      (∀𝑟.𝐶)ℐ = {(𝑛, 𝑥) ∈ N × Δℐ | ∀𝑦.(𝑛, 𝑥, 𝑦) ∈ 𝑟ℐ ⇒ (𝑛, 𝑦) ∈ 𝐶 ℐ }
      (○𝐶)ℐ = {(𝑛, 𝑥) ∈ N × Δℐ | (𝑛 + 1, 𝑥) ∈ 𝐶 ℐ }
      (◇𝐶)ℐ = {(𝑛, 𝑥) ∈ N × Δℐ | ∃𝑚 ≥ 𝑛 such that (𝑚, 𝑥) ∈ 𝐶 ℐ }
      (□𝐶)ℐ = {(𝑛, 𝑥) ∈ N × Δℐ | ∀𝑚 ≥ 𝑛, (𝑚, 𝑥) ∈ 𝐶 ℐ }
      (𝐶𝒰𝐷)ℐ = {(𝑛, 𝑥) ∈ N × Δℐ | ∃𝑚 ≥ 𝑛 s.t. (𝑚, 𝑥) ∈ 𝐷ℐ
                                       and (𝑘, 𝑥) ∈ 𝐶 ℐ , ∀𝑘 (𝑛 ≤ 𝑘 < 𝑚)}
While the definition above assumes a constant domain (i.e., that the domain elements are the
same at all time points), in the following we will also consider the case with expanding domains,
when there is a sequence of increasing domains Δℐ0 ⊆ Δℐ1 ⊆ . . ., one for each time point.
   Let a TBox 𝒯 be a set of concept inclusions 𝐶 ⊑ 𝐷, where 𝐶, 𝐷 are temporally extended
concepts, as above. It has been proven that concept satisfiability in LTL𝒜ℒ𝒞 w.r.t. TBoxes is
E XP T IME-complete, both with expanding domains [21] and with constant domains [16].
   The complexity of other cases and, specifically, the cases of temporal ABoxes [22] and temporal
TBoxes (which allow temporal operators over concept inclusions), have as well been studied in
the literature, and we refer to [16] for a discussion of the result and algorithms for satisfiability
checking.
   In the next section we develop a preferential extension of LTL𝒜ℒ𝒞 . For simplicity, we focus
on the case of non-temporal ABox and TBox, i.e., with the TBox containing a set of concept
inclusions 𝐶 ⊑ 𝐷, where 𝐶, 𝐷 are temporally extended concepts, but without temporal operator
applied to the concept inclusions themselves.


4. LTLT
      𝒜ℒ𝒞 : A Preferential Extension of LTL𝒜ℒ𝒞 with Typicality

In this section we define an extension of the temporal description logic LTL𝒜ℒ𝒞 allowing
typicality concepts of the form T(𝐶), where 𝐶 is a LTL𝒜ℒ𝒞 concept. The instances of T(𝐶) are
intended to be the typical instances of a concept 𝐶. Following [5], we call T a typicality operator.
The concept T(𝐶) can be used on the left hand side of concept inclusions to express defeasible
properties of a concept 𝐶 of the form T(𝐶) ⊑ 𝐷, meaning that the typical instances of concept
𝐶 are also instances of concept 𝐷 (normally, 𝐶’s are 𝐷’s). We can therefore distinguish between
properties that hold for all instances of 𝐶, expressed by strict inclusions (𝐶 ⊑ 𝐷), and those
that only hold for the typical instances of 𝐶, expressed by typicality or defeasible inclusions
(T(𝐶) ⊑ 𝐷).
   Unlike [5, 9], where a typicality operator was introduced for 𝒜ℒ𝒞, here we do not require that
the typicality operator only occurs on the left hand side of concept inclusions, and this choice
is in agreement with [23, 24]. As usual, we assume that the typicality operator T cannot be
nested. Extended concepts can be built by combining the concept constructors in LTL𝒜ℒ𝒞 with
the typicality operator. They can freely occur in concept inclusions, such as, for instance, in the
following ones:

                           T(Professor ) ⊑ (∃teaches.Course)𝒰Semester _End
              ∃lives_in.Town ⊓ Young ⊑ T(◇∃granted .Loan)

The first inclusion means that normally professors teach at least a course until the end of the
semester (but exceptions are allowed). The second one means that persons living in town and
being young are typical in the set of individuals eventually being granted a loan.
   We define a preferential extension, LTLT 𝒜ℒ𝒞 , of LTL𝒜ℒ𝒞 . As for the preferential extension of
the logic 𝒜ℒ𝒞 [5], we define the semantics of LTLT  𝒜ℒ𝒞 in terms of preferential models, extending
ordinary models of LTLT  𝒜ℒ𝒞  with a preference relation < on the domain, whose intuitive meaning
is to compare the “typicality” of domain elements, that is to say, 𝑥 < 𝑦 means that domain element
𝑥 is more typical than 𝑦. The typical instances of an (extended) concept 𝐶 (the instances of
T(𝐶)) are the instances 𝑥 of 𝐶 that are minimal with respect to the preference relation < (i.e., no
other instances of 𝐶 are preferred to 𝑥).
  In the following, we will consider a collection of preference relations <𝑛 , one for each time
point 𝑛. They will be defined as the projections of a relation < over the single time points.

Definition 2 (Preferential temporal interpretations for LTLT            T
                                                           𝒜ℒ𝒞 ). An LTL𝒜ℒ𝒞 interpretation is a
                   ℐ      ℐ
structure ℳ = (Δ , <, · ) where:

    • (Δℐ , ·ℐ ) is a temporal interpretation as for LTL𝒜ℒ𝒞 , as introduced in Section 3, but the
      interpretation function ·ℐ is extended to typicality concepts (see below);
    • the relation < ⊆ N × Δℐ × Δℐ associates to each time point 𝑛 a preference <𝑛 over the
      domain Δℐ such that, for all 𝑛 ∈ N, <𝑛 = {(𝑎, 𝑏) | (𝑛, 𝑎, 𝑏) ∈ <} and relation <𝑛 is an
      irreflexive, transitive and well-founded relation over Δℐ ;
    • the interpretation of typicality concepts T(𝐶) is defined as follows:

                          (T(𝐶))ℐ = {(𝑛, 𝑑) | 𝑑 ∈ Min <𝑛 (𝐶𝑛ℐ ), for 𝑛 ∈ N}

      where 𝐶𝑛ℐ = {𝑑 | (𝑛, 𝑑) ∈ 𝐶 ℐ } are the instances of 𝐶 at time point 𝑛, and Min <𝑛 (𝑆) =
      {𝑢 : 𝑢 ∈ 𝑆 and ∄𝑧 ∈ 𝑆 s.t. 𝑧 <𝑛 𝑢}.

Furthermore, we say that relation <𝑛 is well-founded if, for all 𝑆 ⊆ Δℐ , for all 𝑥 ∈ 𝑆, either
𝑥 ∈ Min <𝑛 (𝑆) or ∃𝑦 ∈ Min <𝑛 (𝑆) such that 𝑦 <𝑛 𝑥.

   For each timepoint 𝑛, relation <𝑛 has the properties of preference relation in KLM preferential
interpretations [1, 2]. When modularity also holds for <𝑛 (i.e., for all 𝑥, 𝑦, 𝑧 ∈ Δℐ , 𝑥 <𝑛 𝑦
implies (𝑥 <𝑛 𝑧 or 𝑧 <𝑛 𝑦)), <𝑛 has the properties of preference relation in rational (or ranked)
KLM interpretations [2]. In the following, however, we will not restrict to modular relations <𝑛 .
   The relation < can be regarded as a function associating to each time point 𝑛 a preference
relation <𝑛 over Δℐ , i.e., <𝑛 ⊆ Δℐ × Δℐ . At each time point 𝑛, the typicality concept T(𝐶) is
interpreted as the set of maximally preferred 𝐶-elements, according to the preference relation <𝑛
for time point 𝑛.
   As for the temporal language LTL𝒜ℒ𝒞 [16], although in this section we have used a con-
stant domain Δℐ in a preferential temporal interpretation, expanding domains could have been
considered as well, by letting a domain Δℐ𝑛 , for each time point 𝑛.
   The notions of satisfiability and model of a knowledge base can be easily extended to LTLT 𝒜ℒ𝒞
with non-temporal ABox and TBox. As 𝒜 is a non-temporal ABox, the assertions in 𝒜 are
evaluated at time point 0. On the other hand, all inclusions in the (non-temporal) TBox 𝒯 have to
be satisfied at all time points.

Definition 3 (Satisfiability in LTLT                        T                           ℐ    ℐ
                                        𝒜ℒ𝒞 ). Given an LTL𝒜ℒ𝒞 interpretation ℳ = ⟨Δ , <, · ⟩, ℳ
                                                ℐ     ℐ
satisfies a concept inclusion 𝐶 ⊑ 𝐷 iff 𝐶 ⊆ 𝐷 ; ℳ satisfies an assertion 𝐶(𝑎) (resp., 𝑟(𝑎, 𝑏))
iff (0, 𝑎ℐ ) ∈ 𝐶 ℐ (resp., (0, 𝑎ℐ , 𝑏ℐ ) ∈ 𝑟ℐ ).
Given an LTLT    𝒜ℒ𝒞 knowledge base 𝐾 = (𝒯 , 𝒜), the interpretation ℳ is a model of 𝐾 if
ℳ satisfies all concept inclusions in 𝒯 and all assertions in 𝒜. An LTLT         𝒜ℒ𝒞 knowledge base
𝐾 = (𝒯 , 𝒜) is satisfiable in LTLT          if a  model ℳ = ⟨Δ ℐ , <, ·ℐ ⟩ of 𝐾 exists.
                                        𝒜ℒ𝒞
    The fact that each irreflexive and transitive relation <𝑛 on Δ is well-founded guarantees that,
for any <𝑛 , there are no infinite descending chains of elements of Δℐ .
    At any time point 𝑛 there is a possibly different relation <𝑛 , which allows to identify the
typical instances of a concept 𝐶 at any time point 𝑛. As observed in [5] for 𝒜ℒ𝒞 with typicality,
the meaning of T can be split into two parts: for any element 𝑥 ∈ Δℐ , 𝑥 ∈ (T(𝐶))𝐼 when (i)
𝑥 ∈ 𝐶 𝐼 , and (ii) there is no 𝑦 ∈ 𝐶 𝐼 such that 𝑦 < 𝑥 (note that, for 𝒜ℒ𝒞 with typicality, there is a
single preference relation < on the domain Δℐ ). In order to isolate the second part of the meaning
of T, one can introduce a Gödel-Löb modality (for which we use the symbol □< , while □ is used
for the temporal operator always), and interpret the preference relation < as the inverse of the
accessibility relation of this modality. Well-foundedness of < ensures that typical elements of 𝐶 𝐼
exist whenever 𝐶 𝐼 ̸= ∅, by avoiding infinitely descending chains of elements. The interpretation
of □< in ℳ is as follows: (□< 𝐶)𝐼 = {𝑥 ∈ Δℐ | for every 𝑦 ∈ Δℐ , if 𝑦 < 𝑥 then 𝑦 ∈ 𝐶 𝐼 }. For
the case of 𝒜ℒ𝒞 with typicality, it has been proven that 𝑥 is a typical instance of 𝐶 if and only if
it is an instance of 𝐶 and □< ¬𝐶, that is: given an interpretation ℳ, a concept 𝐶 and an element
𝑥 ∈ Δ, 𝑥 ∈ (T(𝐶))𝐼 iff 𝑥 ∈ (𝐶 ⊓ □< ¬𝐶)𝐼 [5].
    This modal interpretation of the typicality operator T in terms of a Gödel-Löb modality □<
has been used to define an encoding of 𝒮ℛ𝒪ℐ𝒬𝑃 T into 𝒮ℛ𝒪ℐ𝒬 [23] as well as for encoding a
preferential extension of 𝒮ℋℐ𝒬 into 𝒮ℋℐ𝒬, by introducing a new role 𝑃< in the DL language
to represent the preference relation. In the next section, we will extend this encoding to the
temporal case for 𝒜ℒ𝒞.


5. Encoding of LTLT
                  𝒜ℒ𝒞 in LTL𝒜ℒ𝒞

In this section we show that reasoning in LTLT  𝒜ℒ𝒞 can be reduced polynomially to reasoning in
LTL𝒜ℒ𝒞 . The idea, as reported above, is to define an encoding of the typicality concept in the
temporal description logic, by interpreting T(𝐶) as a formula 𝐴 ⊓ □< ¬𝐴, where the accessibility
relation of the modality □< is the inverse of the preference relation.
   The interpretation of T(𝐶) at a time point 𝑛 is to be evaluated based on the preference relation
< at time point 𝑛, i.e., based on <𝑛 . We represent the preference relation < in a preferential
temporal interpretation ℳ (see Definition 2) by introducing a new role 𝑃< in the language. Also,
we represent a concept T(𝐶) with the concept 𝐶 ⊓ □¬𝐶 , where □¬𝐶 is a new concept name
which is intended to capture the meaning of formula □< ¬𝐶 (dropping the < to make notation
lighter). Finally, we will introduce additional concept inclusion axioms to capture the interplay
between role 𝑃< and the new concepts □¬𝐶 , as well as to enforce the properties of the preference
relations <𝑛 .
   Let 𝐾 = (𝒯 , 𝒜) be a LTLT      𝒜ℒ𝒞 knowledge base and let 𝑁𝐶 , 𝑁𝑅 , 𝑁𝐼 be the set of con-
cept names, role names and individual names in the language of 𝐾. We define the encoding
𝐾 ′ = (𝒯 ′ , 𝒜′ ) of 𝐾 in LTL𝒜ℒ𝒞 over the concept names, role names and individual names in
𝑁𝐶′ , 𝑁𝑅′ , 𝑁𝐼′ , as follows.
   The language of 𝐾 ′ contains all the individual names, concept names and role names in
the language of 𝐾 (i.e., 𝑁𝐶 ⊆ 𝑁𝐶′ , 𝑁𝑅 ⊆ 𝑁𝑅′ , 𝑁𝐼 ⊆ 𝑁𝐼′ ). For each T(𝐴) occurring in 𝐾
(where 𝐴 is any, possibly complex, temporally extended concept), we introduce in 𝑁𝐶′ a new
atomic concept □¬𝐴 and, for each inclusion 𝐶 ⊑ 𝐷 ∈ 𝒯 , we introduce in 𝒯 ′ the inclusion
𝐶 ′ ⊑ 𝐷′ , where 𝐶 ′ and 𝐷′ are obtained from 𝐶 and 𝐷, respectively, by replacing the occurrence
of any concept T(𝐴) with the concept 𝐴 ⊓ □¬𝐴 . Note that concept □¬𝐴 may have a different
interpretation at each time point.
   As mentioned above, to capture the properties of the □< modality, a new role name 𝑃< is
introduced to represent the relation < in preferential models, and the following concept inclusion
axioms are introduced in 𝒯 ′ , for all concepts 𝐴 such that T(𝐴) occurs in 𝒯 :
                                      □¬𝐴 ⊑ ∀𝑃< .(¬𝐴 ⊓ □¬𝐴 )                                        (1)

                                      ¬□¬𝐴 ⊑ ∃𝑃< .(𝐴 ⊓ □¬𝐴 )                                        (2)
The first inclusion accounts for the transitivity of the preference relations <𝑛 .        The second
inclusion accounts for the smoothness (see [2]) of the preference relations <𝑛 , i.e., the fact that if
an element is not a typical 𝐴 element at a time point 𝑛, then there must be a typical 𝐴 element
preferred to it according to <𝑛 . The property holds for a well-founded relation <𝑛 .
  We also define ABox 𝒜′ by replacing each occurrence of the concept T(𝐴) in any individual
assertions 𝐶(𝑑) in 𝒜, with the concept 𝐴⊓□¬𝐴 , and by including in 𝒜′ all the resulting assertions.
All the assertions of the form 𝑅(𝑎, 𝑏) ∈ 𝒜 are included unaltered in 𝒜′ .
Proposition 1. For a temporal knowledge base 𝐾 = (𝒯 , 𝒜) in LTLT                 ′
                                                                     𝒜ℒ𝒞 , let 𝐾 be the encoding
of 𝐾 in LTL𝒜ℒ𝒞 . It holds that 𝐾 is satisfiable in LTL𝒜ℒ𝒞 iff 𝐾 ′ is satisfiable in LTL𝒜ℒ𝒞 .
                                                      T


  As it is clear, the encoding above is polynomial in the size of the knowledge base 𝐾 and, more
precisely, if |𝐾| is the size of 𝐾, the size of 𝐾 ′ is 𝑂(|𝐾|).
  As a consequence of Proposition 1, the decidability and complexity results that have been
proven to hold for the temporal description logic LTL𝒜ℒ𝒞 also extend to the preferential temporal
description logic LTLT  𝒜ℒ𝒞 . Note that our encoding does not depend on assumptions on constant
domains, and it works as well for expanding domains.
  In particular, for non-temporal TBoxes 𝒯 , that is, a set of concept inclusions 𝐶 ⊑ 𝐷, where
𝐶, 𝐷 are LTLT   𝒜ℒ𝒞 concepts, the following holds as a consequence of the encoding above and of
the results for LTL𝒜ℒ𝒞 with expanding domains and with constant domains [21, 16].
Corollary 1. Concept satisfiability in LTLT
                                          𝒜ℒ𝒞 w.r.t. TBoxes is E XP T IME -complete, both with
expanding domains and with constant domains.
  Note that this encoding which exploits 𝒜ℒ𝒞 constructs can as well be adopted for more
expressive logics, although for expressive DLs alternative encodings might be viable. The
preferential extension LTLT   𝒜ℒ𝒞 and its encoding in LTL𝒜ℒ𝒞 can as well be considered for
knowledge bases with temporal TBoxes and temporal ABoxes, with minor modifications of the
proof of Proposition 1. While we leave the detailed treatment of these cases for future work, in
the next sections, we move to consider a multi-preferential semantics for temporal 𝒜ℒ𝒞 with
typicality, as well as possible closure constructions for these extension.


6. A Multi-preferential Temporal Extension of 𝒜ℒ𝒞
Following [13, 14, 20], we can consider a multi-preferential extension of temporal 𝒜ℒ𝒞 with
                                       T,𝑚
typicality LTLT
              𝒜ℒ𝒞 . Let us call it 𝐿𝑇 𝐿𝒜ℒ𝒞 , by associating a preference relation <𝐶𝑖 with each
concept 𝐶𝑖 in a set of distinguished concepts 𝒞 = {𝐶1 , . . . , 𝐶𝑘 }. The underlying idea is that the
distinguished concepts 𝐶𝑖 represent the aspects with respect to which domain individuals are
compared. For instance, Tom may be more typical than Bob as a student (tom <S bob), but less
typical as an employed student (bob <ES tom).
   In the temporal case, this means that, at each time point 𝑛, there are different preference
relations <𝑛𝐶1 , . . . , <𝑛𝐶𝑘 one for each 𝐶𝑖 ∈ 𝒞. Let us assume, for the moment, that the typicality
operator only applies to the distinguished concepts 𝐶𝑖 . The notion of multi-preferential temporal
interpretation for 𝐿𝑇 𝐿T,𝑚  𝒜ℒ𝒞 is defined as follows:

Definition 4 (Multi-preferential temporal interpretations for 𝐿𝑇 𝐿T,𝑚
                                                                  𝒜ℒ𝒞 ). Let 𝒞 = {𝐶1 , . . . , 𝐶𝑘 }
                                               T,𝑚
be a set of distinguished concepts. An 𝐿𝑇 𝐿𝒜ℒ𝒞 interpretation over 𝒞 is a structure ℳ =
(Δℐ , <𝐶1 , . . . , <𝐶𝑘 , ·ℐ ) where:
    • Δℐ is a nonempty domain;
    • for each 𝐶𝑖 ∈ 𝒞, relation <𝐶𝑖 ⊆ N × Δℐ × Δℐ associates to each time point 𝑛 a preference
      <𝑛𝐶𝑖 over the domain Δℐ such that, for all 𝑛 ∈ N, <𝑛𝐶𝑖 = {(𝑎, 𝑏) | (𝑛, 𝑎, 𝑏) ∈<𝐶𝑖 } and
      relation <𝑛𝐶𝑖 is an irreflexive, transitive and well-founded relation over Δℐ ;
    • The interpretation function ·ℐ , introduced in Section 3 for temporally extended goals, is
      extended to typicality concepts T(𝐶𝑖 ) as follows:

                         (T(𝐶𝑖 ))ℐ = {(𝑛, 𝑑) | 𝑑 ∈ Min <𝑛𝐶 (𝐶𝑖,𝑛
                                                             ℐ
                                                                 ), for 𝑛 ∈ N}
                                                             𝑖

             ℐ = {𝑑 | (𝑛, 𝑑) ∈ 𝐶 ℐ } are the instances of 𝐶 at time point 𝑛.
      where 𝐶𝑖,𝑛                𝑖                          𝑖

   Let us define an 𝐿𝑇 𝐿T,𝑚                                             T
                          𝒜ℒ𝒞 knowledge base 𝐾 = ⟨𝒯 , 𝒜⟩ as an LTL𝒜ℒ𝒞 knowledge base in
which only typicality concepts of the form T(𝐶𝑖 ) may occur. An encoding in LTL𝒜ℒ𝒞 of the
different preference relations associated to concepts can be defined in a similar way as for the
single preference relation <, but requires to introduce a new role 𝑃<𝐶𝑖 , for each distinguished
concept 𝐶𝑖 ∈ 𝒞, as well as a new concept name □¬𝐶𝑖 to encode a typicality concept T(𝐶𝑖 )
occurring in the TBox with concept 𝐶𝑖 ⊓ □¬𝐶𝑖 in LTL𝒜ℒ𝒞 . The two axioms, (1) and (2) need as
well to be introduced for all 𝐶𝑖 ∈ 𝒞.
   The result that concept satisfiability in LTLT𝒜ℒ𝒞 w.r.t. TBoxes is E XP T IME -complete also
extends to the multi-preferential temporal 𝒜ℒ𝒞 under the 𝐿𝑇 𝐿T,𝑚
                                                               𝒜ℒ𝒞 semantics.


6.1. Global Preference
Note that, given the preferences <𝑛𝐶𝑖 for the distinguished concepts, one can interpret T(𝐶𝑖 ) at
time point 𝑛 as the set of minimal 𝐶𝑖 elements w.r.t. <𝑛𝐶𝑖 . However, to provide an interpretation of
the typicality concept T(𝐶) for an arbitrary 𝐶 (such as, for instance, T(Employee ⊓ Student)),
one would need to define a preference relation with respect to 𝐶 or, in alternative, a notion
of global preference relation. Many notions of preference combination have been considered
and studied in the literature [25, 26]. Following [13], a notion of global preference < can, for
instance, be defined by exploiting a modified Pareto combination of the preference relations
<𝐶1 , . . . , <𝐶𝑘 , which takes into account the specificity relation ≻ among concepts, e.g., that
concept PhDStudent is more specific than concept Student (PhDStudent ≻ Student), and its
properties override the properties of Student, when conflicting. The global preference relation
<𝑛 at time point 𝑛 can be defined from <𝑛𝐶1 , . . . , <𝑛𝐶𝑘 as follows:
              𝑥 <𝑛 𝑦 iff (𝑖) 𝑥 <𝑛𝐶𝑖 𝑦, for some 𝐶𝑖 ∈ 𝒞, and
                         (𝑖𝑖) for all 𝐶𝑗 ∈ 𝒞, 𝑥 ≤𝑛𝐶𝑗 𝑦 or ∃𝐶ℎ (𝐶ℎ ≻ 𝐶𝑗 and 𝑥 <𝑛𝐶ℎ 𝑦).
We interpret T(𝐶), for an arbitrary concept 𝐶, at time point 𝑛, as the set of minimal 𝐶-elements
with respect to <𝑛 , i.e., (T(𝐶))ℐ = {(𝑛, 𝑑) | 𝑑 ∈ Min <𝑛 (𝐶𝑛ℐ ), for 𝑛 ∈ N}, where 𝐶𝑛ℐ = {𝑑 |
(𝑛, 𝑑) ∈ 𝐶 ℐ } are the instances of concept 𝐶 at time point 𝑛. This leads to the definition of a
concept-wise multi-preferential temporal interpretation (cw𝑚 -interpretation) for 𝐿𝑇 𝐿T,𝑚    𝒜ℒ𝒞 as
                                          𝐼                                  𝐼            T,𝑚
a a tuple ℳ = ⟨Δ, <𝐶1 , . . . , <𝐶𝑘 , <, · ⟩, where ⟨Δ, <𝐶1 , . . . , <𝐶𝑘 , · ⟩ is an 𝐿𝑇 𝐿𝒜ℒ𝒞 multi-
preferential temporal interpretation (see Definition 4), and <⊆ N × Δ × Δ is a global preference
relation, defined from the relations <𝑛 ⊆ Δℐ × Δℐ as follows: (𝑛, 𝑎, 𝑏) ∈< iff (𝑎, 𝑏) ∈<𝑛 .


7. Temporal Weighted KBs and a Closure Construction
Preferential logics provide a rather weak notion non-monotonic inference and can be used as the
basis for stronger notions of entailment, based on canonical minimal models and closure construc-
tions [2, 12]. Similar constructions have been developed for preferential DLs [6, 9, 10, 13]. In this
section we show that these constructions can be extended to the preferential temporal description
logic considered above. In the following, we define a notion of temporal weighted 𝐿𝑇 𝐿T,𝑚       𝒜ℒ𝒞
knowledge base by allowing for weighted defeasible inclusions for the distinguished concepts, as
done in [14, 20] for the (non-temporal) DLs with typicality. The idea is to allow for the definition
of prototypical properties of a class, with (positive or negative) weights representing the degree
of plausibility/implausibility of each property, including as well the temporal dimension.
   A temporal weighted 𝐿𝑇 𝐿T,𝑚   𝒜ℒ𝒞 knowledge base 𝐾 over 𝒞 is a tuple ⟨𝒯 , 𝒯𝐶1 , . . . , 𝒯𝐶𝑘 , 𝒜⟩,
where 𝒯 is a TBox in 𝐿𝑇 𝐿𝒜ℒ𝒞 , 𝒜 is an ABox in 𝐿𝑇 𝐿T,𝑚
                             T,𝑚
                                                          𝒜ℒ𝒞 and, for each 𝐶𝑖 ∈ 𝒞, 𝒯𝐶𝑖 is a set of
weighted defeasible inclusions, of the form T(𝐶𝑖 ) ⊑ 𝐷ℎ , having weight 𝑤ℎ𝑖 , a real number.
   Consider, for instance, the weighted knowledge base 𝐾 = ⟨𝒯𝑠𝑡𝑟𝑖𝑐𝑡 , 𝒯𝐸𝑚𝑝 , 𝒯𝑆𝑡𝑢𝑑𝑒𝑛𝑡 , 𝒜⟩, over
the set of distinguished concepts 𝒞 = {Emp, Student, Professor }, with empty ABox, and with
𝒯 containing the inclusions:
           Emp ⊑ Adult               Adult ⊑ ∃has_SSN .⊤            PhdStudent ⊑ Student
           T(𝑃 𝑟𝑜𝑓 𝑒𝑠𝑠𝑜𝑟) ⊑ (∃𝑇 𝑒𝑎𝑐ℎ𝑒𝑠.𝐶𝑜𝑢𝑟𝑠𝑒)𝒰𝑆𝑒𝑚𝑒𝑠𝑡𝑒𝑟_𝐸𝑛𝑑
where, for instance, 𝒯𝐸𝑚𝑝 contains the weighted defeasible inclusions:
   (𝑑1 ) T(Emp) ⊑ Young, - 50                (𝑑2 ) T(Emp) ⊑ ∃has_boss.Emp, 100
   (𝑑3 ) T(Emp) ⊑ ∃has_classes.⊤, -70;
and 𝒯𝑆𝑡𝑢𝑑𝑒𝑛𝑡 contains the defeasible inclusions:
   (𝑑4 ) T(Student) ⊑ Young, 90              (𝑑5 ) T(Student) ⊑ ∃has_classes.⊤, 80
   (𝑑6 ) T(Student) ⊑ ∃hasScholarship.⊤, -30
   (𝑑7 ) T(𝑆𝑡𝑢𝑑𝑒𝑛𝑡) ⊑ ◇(𝑃 𝑟𝑜𝑚𝑜𝑡𝑒𝑑 ⊔ 𝑅𝑒𝑗𝑒𝑐𝑡𝑒𝑑), 100
The meaning is that, while an employee normally has a boss, he is not likely to be young or have
classes. Furthermore, between the two defeasible inclusions (𝑑1 ) and (𝑑3 ), the second one is
considered to be less plausible than the first one. Negative weights represent penalties. Given
two employees Tom and Bob such that, at time point 𝑛, Tom is not young, has no boss and has
classes, while Bob is not young, has a boss and has no classes, considering the weights above, we
will regard Bob as being more typical than Tom as an employee at 𝑛.
   Given a temporal interpretation ℐ = (Δℐ , ·ℐ ) for LTL𝒜ℒ𝒞 , we say that 𝑥 ∈ Δℐ satisfies
T(𝐶𝑖 ) ⊑ 𝐷 in ℐ at time point 𝑛, if (𝑛, 𝑥) ̸∈ 𝐶𝑖ℐ or (𝑛, 𝑥) ∈ 𝐷ℐ (otherwise 𝑥 violates T(𝐶𝑖 ) ⊑ 𝐷
in ℐ at time point 𝑛). For a concept 𝐶𝑖 ∈ 𝒞 and a domain element 𝑥 ∈ Δℐ , the weight 𝑊𝑖,𝑛  ℐ (𝑥) of

𝑥 w.r.t. 𝐶𝑖 in ℐ at time point 𝑛, is defined considering the inclusions (T(𝐶𝑖 ) ⊑ 𝐷ℎ , 𝑤ℎ𝑖 ) ∈ 𝒯𝐶𝑖 ,
as follows:                           {︃ ∑︀
                                                         𝑖              ℐ
                                                               if 𝑥 ∈ 𝐶𝑖,𝑛
                            ℐ                        ℐ 𝑤ℎ
                                            ℎ:(𝑛,𝑥)∈𝐷ℎ
                         𝑊𝑖,𝑛 (𝑥) =                                                             (3)
                                         −∞                    otherwise
where −∞ is added at the bottom of real values. Informally, given an interpretation ℐ, for
(𝑛, 𝑥) ∈ 𝐶𝑖ℐ , the weight 𝑊𝑖,𝑛 ℐ (𝑥) of 𝑥 wrt 𝐶 at time point 𝑛 is the sum of the weights of all
                                                 𝑖
defeasible inclusions for 𝐶𝑖 satisfied by 𝑥 in ℐ at time point 𝑛. The more plausible are the satisfied
inclusions, the higher is the weight of 𝑥. The lowest weight, −∞, is given to all domain elements
which are not instances of 𝐶𝑖 at time point 𝑛.
   Based on this notion of weight of a domain element wrt a concept, a preference relation <𝑛𝐶𝑖
can be built from a given interpretation ℐ and a weighted knowledge base 𝐾. At time point 𝑛,
an element 𝑥 is preferred to element 𝑦 wrt 𝐶𝑖 if the sum of the weights of the defaults in 𝒯𝐶𝑖
satisfied by 𝑥 at 𝑛 is higher than the sum of the weights of defaults in 𝒯𝐶𝑖 satisfied by 𝑦 at 𝑛: for
𝑥, 𝑦 ∈ Δ,
                                               ℐ          ℐ
                                𝑥 <𝑛𝐶𝑖 𝑦 iff 𝑊𝑖,𝑛 (𝑥) > 𝑊𝑖,𝑛 (𝑦)                                  (4)
Note that <𝑛𝐶𝑗 is a strict modular and well-founded partial order, and all 𝐶𝑖 -elements are preferred
wrt <𝐶𝑖 to the domain elements which are not instances of 𝐶𝑖 . The higher is the weight of an
element wrt 𝐶𝑖 (at 𝑛) the more preferred is the element w.r.t. 𝐶𝑖 at time point 𝑛. In the example
above, 𝑊𝑖,𝑛ℐ (𝑏𝑜𝑏) = 30 > 𝑊 ℐ (𝑡𝑜𝑚) = −70 (for 𝐶 = Emp) and, hence, bob <
                               𝑖,𝑛                    𝑖                               Emp tom, i.e.,
Bob is more typical than Tom as an employee.
   Let us define a concept-wise multi-preferential temporal semantics (cw𝑚 temporal semantics)
for a weighted knowledge base.
Definition 5. A concept-wise multi-preferential temporal model (cw𝑚 -model) of a weighted
𝐿𝑇 𝐿T,𝑚
     𝒜ℒ𝒞 knowledge base 𝐾 = ⟨𝒯 , 𝒯𝐶1 , . . . , 𝒯𝐶𝑘 , 𝒜⟩ over 𝒞 is a concept-wise multi-preferential
interpretation ℳ = ⟨Δℐ , <𝐶1 , . . . , <𝐶𝑘 , <, ·ℐ ⟩, such that: for all 𝑗 = 1, . . . , 𝑘,
                             <𝐶𝑗 = {(𝑛, 𝑥, 𝑦) : 𝑛 ∈ N and 𝑥 <𝑛𝐶𝑗 𝑦},

where each <𝑛𝐶𝑗 is defined from 𝒯𝐶𝑗 and ⟨Δℐ , ·ℐ ⟩, according to condition (4); < is the resulting
global preference relation, as defined in Section 6.1; and ⟨Δℐ , <, ·ℐ ⟩ satisfies 𝒯 and 𝒜 according
to satisfiability in Definition 3.
  Based on the notion of cw𝑚 -model of a KB, the notions of concept-wise entailment (or cw𝑚 -
entailment) and canonical cw𝑚 -entailment can be defined in a natural way for weigthed KBs in
𝐿𝑇 𝐿T,𝑚
     𝒜ℒ𝒞 , as in the non-temporal case [20].
  Let us restrict consideration to canonical models, i.e., models which are large enough to
contain all the relevant domain elements (see [13]). Let Conc 𝐾 be the set of all non-temporal
concepts 𝐶 occurring in 𝐾 plus their complements ¬𝐶.
Definition 6. Given a ranked knowledge base 𝐾 = ⟨𝒯 , 𝒯𝐶1 , . . . , 𝒯𝐶𝑘 , 𝒜⟩ a model ℳ =
⟨Δℐ , <𝐶1 , . . . , <𝐶𝑘 , <, ·ℐ ⟩ of 𝐾 is canonical for 𝐾 if, for any set of concepts {𝐷1 , . . . , 𝐷𝑚 } ⊆
Conc 𝐾 such that 𝐷1 ⊓ . . . ⊓ 𝐷𝑚 is satisfiable with respect to ⟨𝒯 , 𝒜⟩, it holds that for all time
points 𝑛, there exists a domain element 𝑥 ∈ Δℐ𝑛 such that (𝑛, 𝑥) ∈ 𝐷𝑖ℐ for all 𝑖 = 1, . . . , 𝑚.
   The idea is that, in a canonical model for 𝐾, any conjunction of concepts occurring in 𝐾, or
their complements, when consistent with the TBox 𝒯 and the ABox 𝒜 of 𝐾, must have some
instance in the domain at each time point 𝑛. Existence of canonical interpretations has been
proven in the non-temporal case for knowledge bases which are consistent under the preferential
(or ranked) semantics for typicality [9]. A similar construction can be developed for the temporal
case, exploiting the fact that, in the case we have considered (that of KBs with non-temporal
TBoxes and non-temporal ABoxes), the interaction between the temporal component and the DL
component of the temporal DL is rather limited (see [16]).
Definition 7 (cw𝑚 -entailment [14]). An inclusion T(𝐶) ⊑ 𝐷 is cw𝑚 -entailed from a weighted
knowledge base 𝐾 if it is satisfied in all canonical cw𝑚 -models ℳ of 𝐾.
   The study of the properties of this semantic, such as the KLM properties [2], which have been
studied for description logics with typicality in the non-temporal case, will be considered for
future work, as well as the development of alternative semantic constructions.


8. Conclusions
In this paper we have developed a preferential temporal description logics with typicality LTLT𝒜ℒ𝒞 .
The monotonic logic LTLT      𝒜ℒ𝒞  can  be further extended   to define a semantics   for  weighted
knowledge bases, by introducing multiple preferences. The paper discusses these extensions,
showing that the concept-wise multi-preferential semantic in [13] adapts smoothly to the temporal
case.
   On a different route, a preferential LTL with defeasible temporal operators has been studied in
[18, 19]. The decidability of meaningful fragments of the logic has been proven, and tableaux
based proof methods for such fragments have been developed [17, 19]. Instead, our approach does
not consider defeasible temporal operators (nor preferences over time points), but it combines
standard LTL operators with the typicality operator in a temporal 𝒜ℒ𝒞 (where preferences are
over the domain elements).
   A different approach for combining defeasibility in temporal DL formalism has been proposed
in [27], by combining a temporal action logic [28] for reasoning about actions (whose semantics
is based on a notion of temporal answer set) and an ℰℒ⊥ ontology. The approach provides a
polynomial encoding of an action theory extended with an ℰℒ⊥ knowledge base in normal form,
into the language of the temporal action logic. The temporal action logic studied in [28] is based
on an extension of LTL, called Dynamic Linear Time Temporal Logic (DLTL) introduced in [29],
which allows for complex actions. The proof methods for this action logic are based on ASP
encodings of bounded model checking [28, 30], and can then be exploited for reasoning about
actions in an extended action theory. Defeasibility in [27], and in the related work on reasoning
about actions in Description Logics [31, 32, 33] (often not based on temporal logics), is concerned
with the non-monotonicity of the frame problem and, in the literature, different solutions are
explored. Our paper, instead, aims at representing temporal properties of concepts which admit
exceptions, through a notion of typicality, and is not specifically intended for reasoning about
actions.
   The encoding of LTLT   𝒜ℒ𝒞 into LTL𝒜ℒ𝒞 provides decidability and complexity results for the
                       T
monotonic logic LTL𝒜ℒ𝒞 for free. For the multi-preferential case, proof methods for defeasible
temporal reasoning with weighted knowledge bases have to be investigated, possibly for fragments
of 𝐿𝑇 𝐿T,𝑚
         𝒜ℒ𝒞 . This will be subject of future work.


Acknowledgments
We thank the anonymous referees for their helpful comments. This work was partially sup-
ported by GNCS-INdAM. Mario Alviano was partially supported by Italian Ministry of Research
(MUR) under PNRR project FAIR “Future AI Research”, CUP H23C22000860006, under PNRR
project Tech4You “Technologies for climate change adaptation and quality of life improvement”,
CUP H23C22000370006, and under PNRR project SERICS “SEcurity and RIghts in the Cy-
berSpace”, CUP H73C22000880001; by Italian Ministry of Health (MSAL) under POS project
RADIOAMICA, CUP H53C22000650006; by the LAIA lab (part of the SILA labs).


References
 [1] S. Kraus, D. Lehmann, M. Magidor, Nonmonotonic reasoning, preferential models and
     cumulative logics, Artificial Intelligence 44 (1990) 167–207.
 [2] D. Lehmann, M. Magidor, What does a conditional knowledge base entail?, Artificial
     Intelligence 55 (1992) 1–60.
 [3] F. Baader, D. Calvanese, D. McGuinness, D. Nardi, P. Patel-Schneider, The Description
     Logic Handbook - Theory, Implementation, and Applications, Cambridge, 2007.
 [4] K. Britz, J. Heidema, T. Meyer, Semantic preferential subsumption, in: G. Brewka, J. Lang
     (Eds.), KR 2008, AAAI Press, Sidney, Australia, 2008, pp. 476–484.
 [5] L. Giordano, V. Gliozzi, N. Olivetti, G. L. Pozzato, ALC+T: a preferential extension of
     Description Logics, Fundamenta Informaticae 96 (2009) 1–32.
 [6] G. Casini, U. Straccia, Rational Closure for Defeasible Description Logics, in: T. Janhunen,
     I. Niemelä (Eds.), JELIA 2010, volume 6341 of LNCS, Springer, Helsinki, 2010, pp. 77–90.
 [7] G. Casini, U. Straccia, Defeasible inheritance-based description logics, Journal of Artificial
     Intelligence Research (JAIR) 48 (2013) 415–473.
 [8] G. Casini, T. Meyer, I. J. Varzinczak, , K. Moodley, Nonmonotonic Reasoning in Description
     Logics: Rational Closure for the ABox, in: 26th International Workshop on Description
     Logics (DL 2013), volume 1014 of CEUR Workshop Proceedings, 2013, pp. 600–615.
 [9] L. Giordano, V. Gliozzi, N. Olivetti, G. L. Pozzato, Semantic characterization of rational
     closure: From propositional logic to description logics, Art. Int. 226 (2015) 1–33.
[10] K. Britz, G. Casini, T. Meyer, K. Moodley, U. Sattler, I. Varzinczak, Principles of klm-style
     defeasible description logics, ACM Trans. Comput. Log. 22 (2021) 1:1–1:46.
[11] L. Giordano, V. Gliozzi, A reconstruction of multipreference closure, Artif. Intell. 290
     (2021).
[12] D. J. Lehmann, Another perspective on default reasoning, Ann. Math. Artif. Intell. 15
     (1995) 61–82.
[13] L. Giordano, D. Theseider Dupré, An ASP approach for reasoning in a concept-aware
     multipreferential lightweight DL, TPLP 10(5) (2020) 751–766.
[14] L. Giordano, D. Theseider Dupré, Weighted defeasible knowledge bases and a multiprefer-
     ence semantics for a deep neural network model, in: Proc. JELIA 2021, May 17-20, volume
     12678 of LNCS, Springer, 2021, pp. 225–242.
[15] L. Giordano, D. Theseider Dupré, An ASP approach for reasoning on neural networks under
     a finitely many-valued semantics for weighted conditional knowledge bases, Theory Pract.
     Log. Program. 22 (2022) 589–605. URL: https://doi.org/10.1017/S1471068422000163.
[16] C. Lutz, F. Wolter, M. Zakharyaschev, Temporal description logics: A survey, in: TIME,
     2008, pp. 3–14.
[17] A. Chafik, F. C. Alili, J. Condotta, I. Varzinczak, A one-pass tree-shaped tableau for
     defeasible LTL, in: TIME 2021, September 27-29, 2021, Klagenfurt, Austria, volume 206
     of LIPIcs, 2021.
[18] A. Chafik, F. C. Alili, J. Condotta, I. Varzinczak, On the decidability of a fragment of
     preferential LTL, in: TIME 2020, September 23-25, 2020, Bozen-Bolzano, Italy, volume
     178 of LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
[19] A. Chafik, Defeasible temporal logics for the specification and verification of exception-
     tolerant systems, PhD Thesis, Artois University, 2022.
[20] L. Giordano, D. Theseider Dupré, Weighted conditional EL⊥ knowledge bases with integer
     weights: an ASP approach, in: Proc. 37th Int. Conf. on Logic Programming, ICLP 2021
     (Technical Communications), Sept. 20-27, 2021, volume 345 of EPTCS, 2021, pp. 70–76.
[21] K. Schild, Combining terminological logics with tense logic, in: EPIA, 1993, pp. 105–120.
[22] F. Baader, S. Ghilardi, C. Lutz, LTL over description logic axioms, in: Proceedings of the
     21st International Workshop on Description Logics (DL2008), Dresden, Germany, May
     13-16, 2008, volume 353 of CEUR Workshop Proc., CEUR-WS.org, 2008.
[23] L. Giordano, V. Gliozzi, Encoding a preferential extension of the description logic SROIQ
     into SROIQ, in: Proc. ISMIS 2015, volume 9384 of LNCS, Springer, 2015, pp. 248–258.
[24] R. Booth, G. Casini, T. Meyer, I. Varzinczak, On rational entailment for propositional
     typicality logic, Artif. Intell. 277 (2019).
[25] G. Brewka, A rank based description language for qualitative preferences, in: 6th Europ.
     Conf. on Artificial Intelligence, ECAI’2004, Valencia, Spain, August 22-27, 2004, 2004, pp.
     303–307.
[26] M. Huelsman, M. Truszczynski, Relating preference languages by their expressive power,
     in: Proceedings of the Thirty-Fifth International Florida Artificial Intelligence Research
     Society Conference, FLAIRS 2022, Hutchinson Island, Jensen Beach, Florida, USA, May
     15-18, 2022, 2022.
[27] L. Giordano, A. Martelli, D. Theseider Dupré, Reasoning about actions with ℰℒ ontologies
     and temporal answer sets for DLTL, in: Logic Programming and Nonmonotonic Reasoning
     - 16th International Conference, LPNMR, volume 13416 of Lecture Notes in Computer
     Science, Springer, 2022, pp. 231–244.
[28] L. Giordano, A. Martelli, D. Theseider Dupré, Reasoning about actions with temporal
     answer sets, Theory and Practice of Logic Programming 13 (2013) 201–225.
[29] J. Henriksen, P. Thiagarajan, Dynamic Linear Time Temporal Logic, Annals of Pure and
     Applied logic 96 (1999) 187–207.
[30] L. Giordano, A. Martelli, D. Theseider Dupré, Achieving completeness in the verification
     of action theories by bounded model checking in ASP, J. Log. Comp. 25 (2015) 1307–30.
[31] F. Baader, C. Lutz, M. Milicic, U. Sattler, F. Wolter, Integrating description logics and
     action formalisms: First results, in: Proc. AAAI 2005, 2005, pp. 572–577.
[32] H. Liu, C. Lutz, M. Milicic, F. Wolter, Reasoning about actions using description logics
     with general tboxes, in: Proc. JELIA 2006, Liverpool, UK, 2006, pp. 266–279.
[33] F. Baader, M. Lippmann, H. Liu, Using causal relationships to deal with the ramification
     problem in action formalisms based on description logics, in: LPAR-17, 2010, pp. 82–96.