=Paper= {{Paper |id=Vol-3733/paper16 |storemode=property |title=Many-valued Temporal Weighted Knowledge Bases with Typicality for Explainability |pdfUrl=https://ceur-ws.org/Vol-3733/paper16.pdf |volume=Vol-3733 |authors=Mario Alviano,Marco Botta,Roberto Esposito,Laura Giordano,Daniele Theseider Dupré |dblpUrl=https://dblp.org/rec/conf/cilc/AlvianoBE0D24 }} ==Many-valued Temporal Weighted Knowledge Bases with Typicality for Explainability== https://ceur-ws.org/Vol-3733/paper16.pdf
                         Many-valued Temporal Weighted Knowledge Bases with
                         Typicality for Explainability
                         Mario Alviano1,* , Marco Botta2,* , Roberto Esposito2,* , Laura Giordano3,* and
                         Daniele Theseider Dupré3,*
                         1
                           DEMACS, University of Calabria, Via Bucci 30/B, 87036 Rende (CS), Italy
                         2
                           Dipartimento di Informatica, Università di Torino, Corso Svizzera 185, 10149 Torino, Italy
                         3
                           DISIT, University of Piemonte Orientale, Viale Michel 11, 15121 Alessandria, Italy


                                      Abstract
                                      In this paper, we develop a many-valued semantics for the description logic 𝐿𝑇 𝐿𝒜ℒ𝒞 , a temporal extension of
                                      description logic 𝒜ℒ𝒞, based on Linear-time Temporal Logic (LTL). We add a typicality operator to represent
                                      defeasible properties, and discuss the use of the (many-valued) temporal conditional logic and of weighted KBs
                                      for explaining the dynamic behaviour of a network.

                                      Keywords
                                      Preferential Logics, Temporal Logics, Many-valued Description Logics, Explainability




                         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, 6], and several different closure constructions have been developed [7, 8, 9, 10, 11, 12], inspired by
                         Lehmann and Magidor’s rational closure [2] and Lehmann’s lexicographic closure [13]. More recently,
                         multi-preferential extensions of DLs have been developed, by allowing multiple preference relations
                         with respect to different concepts [14, 15, 16], as the semantic for ranked and weighted knowledge
                         bases with typicality.
                            LTL extensions of Description Logics are very well-studied in DLs literature, and we refer to [17, 18]
                         for surveys on temporal DLs and their complexity and decidability. While preferential extensions of
                         LTL with defeasible temporal operators have been recently studied [19, 20, 21] to enrich temporal
                         formalisms with non-monotonic reasoning features, a preferential extension of a temporal DL has been
                         proposed in [22], based on the approach proposed in [5] to define a description logic with typicality.
                         More specifically, in [22] we build over a temporal extension of 𝒜ℒ𝒞, LTL𝒜ℒ𝒞 [17], based on Linear
                         Time Temporal Logic (LTL), to develop a temporal 𝒜ℒ𝒞 with typicality, LTLT          𝒜ℒ𝒞 . Generalizing the
                         approach in [5], a typicality operator T (that selects the most typical instances of a concept) is added to
                         LTL𝒜ℒ𝒞 to represent temporal properties of concepts which admit exceptions.
                            It is proven that the preferential extension of LTLT𝒜ℒ𝒞 can be polynomially encoded into LTL𝒜ℒ𝒞 , and
                         this approach allows borrowing decidability and complexity results from LTL𝒜ℒ𝒞 . A similar encoding

                          CILC 2024: 39th Italian Conference on Computational Logic, June 26-28, 2024, Rome, Italy
                         *
                           Corresponding author.
                          $ mario.alviano@unical.it (M. Alviano); marco.botta@unito.it (M. Botta); roberto.esposito@unito.it (R. Esposito);
                          laura.giordano@uniupo.it (L. Giordano); dtd@uniupo.it (D. Theseider Dupré)
                           https://alviano.net/ (M. Alviano); http://informatica.unito.it/persone/marco.botta/ (M. Botta);
                          http://informatica.unito.it/persone/roberto.esposito (R. Esposito); 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)
                                   © 2024 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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can be given for a multi-preferential extension of LTLT 𝒜ℒ𝒞 , by allowing a concept-wise preferential
semantics, where different preferences are associated to different concepts.
  In this paper, we aim at developing a many-valued extension of LTL𝒜ℒ𝒞 with typicality, which
makes it possible to represent a concept inclusions such as

                          ∃lives_in.Town ⊓ Young ⊑ T(♢Granted _Loan),

(meaning that who lives in town and is young, normally is eventually granted a loan), where the
interpretation of some concepts (e.g., Young) may be non-crisp.
   In the paper we first recall fuzzy extensions of 𝒜ℒ𝒞 and temporal extensions of 𝒜ℒ𝒞. Then, we
develop a many-valued extension of LTL𝒜ℒ𝒞 , by building on many-valued DLs, which are widely
studied in the literature, both for the fuzzy case [23, 24, 25, 26, 27] and for the finitely-valued case
[28, 29, 30, 31]. Then we add a typicality operator to the language of the many-valued LTL𝒜ℒ𝒞 , to get
a many-valued temporal extension of 𝒜ℒ𝒞 with typicality.
   We discuss extensions of the closure constructions for weighted knowledge bases with typicality
[15, 32, 33] to the temporal case. This 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. We discuss how the preferential temporal logic can be used to provide a logical
interpretation of the transient behaviour of (recurrent) neural networks.


2. Fuzzy 𝒜ℒ𝒞
Fuzzy description logics have been widely studied in the literature for representing vagueness in DLs
[23, 24, 25, 26, 27], based on the idea that concepts and roles can be interpreted as fuzzy sets. Formulas
in Mathematical Fuzzy Logic [34] have a degree of truth in an interpretation rather than being true or
false; similarly, axioms in a fuzzy DL have a degree of truth, usually in the interval [0, 1]. The finitely
many-valued case is also well studied for DLs [28, 29, 30, 31]. We first recall the semantics of a fuzzy
extension of 𝒜ℒ𝒞, following [25]; then we will consider the finitely-valued case.
   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:
      (i) 𝐴 ∈ 𝑁𝐶 , ⊤ and ⊥ are concepts;
      (ii) if 𝐶 and 𝐷 are concepts, then 𝑟 ∈ 𝑁𝑅 , then 𝐶 ⊓ 𝐷, 𝐶 ⊔ 𝐷, ¬𝐶, ∀𝑟.𝐶, ∃𝑟.𝐶 are
      concepts.
A fuzzy interpretation for 𝒜ℒ𝒞 is a pair 𝐼 = ⟨∆, ·𝐼 ⟩ where: ∆ is a non-empty domain and ·𝐼 is fuzzy
interpretation function that assigns to each concept name 𝐴 ∈ 𝑁𝐶 a function 𝐴𝐼 : ∆ → [0, 1], to each
role name 𝑟 ∈ 𝑁𝑅 a function 𝑟𝐼 : ∆ × ∆ → [0, 1], and to each individual name 𝑎 ∈ 𝑁𝐼 an element
𝑎𝐼 ∈ ∆. A domain element 𝑥 ∈ ∆ belongs to the extension of 𝐴 to some degree in [0, 1], i.e., 𝐴𝐼 is a
fuzzy set.
   The interpretation function ·𝐼 is extended to complex concepts as follows:
     ⊤𝐼 (𝑥) = 1,            ⊥𝐼 (𝑥) = 0,
           𝐼
     (¬𝐶) (𝑥) = ⊖𝐶 (𝑥),𝐼

     (𝐶 ⊓ 𝐷)𝐼 (𝑥) = 𝐶 𝐼 (𝑥) ⊗ 𝐷𝐼 (𝑥),
     (𝐶 ⊔ 𝐷)𝐼 (𝑥) = 𝐶 𝐼 (𝑥) ⊕ 𝐷𝐼 (𝑥),
     (∃𝑟.𝐶)𝐼 (𝑥) = sup𝑦∈Δ 𝑟𝐼 (𝑥, 𝑦) ⊗ 𝐶 𝐼 (𝑦),
     (∀𝑟.𝐶)𝐼 (𝑥) = inf 𝑦∈Δ 𝑟𝐼 (𝑥, 𝑦) ▷ 𝐶 𝐼 (𝑦),
where 𝑥 ∈ ∆, and ⊗, ⊕, ▷ and ⊖ are arbitrary but fixed t-norm, s-norm, implication function, and
negation function, chosen among the combination functions of some fuzzy logic. In particular, in
Gödel logic 𝑎 ⊗ 𝑏 = 𝑚𝑖𝑛{𝑎, 𝑏}, 𝑎 ⊕ 𝑏 = 𝑚𝑎𝑥{𝑎, 𝑏}, 𝑎 ▷ 𝑏 = 1 if 𝑎 ≤ 𝑏 and 𝑏 otherwise; ⊖𝑎 = 1 if
𝑎 = 0 and 0 otherwise. In Łukasiewicz logic, 𝑎 ⊗ 𝑏 = 𝑚𝑎𝑥{𝑎 + 𝑏 − 1, 0}, 𝑎 ⊕ 𝑏 = 𝑚𝑖𝑛{𝑎 + 𝑏, 1},
𝑎 ▷ 𝑏 = 𝑚𝑖𝑛{1 − 𝑎 + 𝑏, 1} and ⊖𝑎 = 1 − 𝑎. Following [25], we will not commit to a specific choice of
combination functions,
  A fuzzy 𝒜ℒ𝒞 knowledge base 𝐾 is a pair (𝒯 , 𝒜) where 𝒯 is a fuzzy TBox and 𝒜 a fuzzy ABox. A
fuzzy TBox is a set of fuzzy concept inclusions of the form 𝐶 ⊑ 𝐷 𝜃 𝑛, where 𝐶 ⊑ 𝐷 is an 𝒜ℒ𝒞 concept
inclusion axiom, 𝜃 ∈ {≥, ≤, >, <} and 𝑛 ∈ [0, 1]. A fuzzy ABox 𝒜 is a set of fuzzy assertions of the
form 𝐶(𝑎)𝜃𝑛 or 𝑟(𝑎, 𝑏)𝜃𝑛, where 𝐶 is an 𝒜ℒ𝒞 concept, 𝑟 ∈ 𝑁𝑅 , 𝑎, 𝑏 ∈ 𝑁𝐼 , 𝜃 ∈ {≥, ≤, >, <} and
𝑛 ∈ [0, 1]. Following Bobillo and Straccia [27], we assume that fuzzy interpretations are witnessed, i.e.,
the sup and inf are attained at some point of the involved domain. The interpretation function ·𝐼 is also
extended to axioms as follows:

                    (𝐶 ⊑ 𝐷)𝐼 = inf 𝑥∈Δ𝐼 𝐶 𝐼 (𝑥) ▷ 𝐷𝐼 (𝑥)           (𝐶(𝑎))𝐼 = 𝐶 𝐼 (𝑎𝐼 )

Definition 1 (Satisfiability and entailment for ℒ𝒞 𝑛 knowledge bases). Let 𝐾 = (𝒯 , 𝒜) be a
weighted ℒ𝒞 𝑛 knowledge base, and 𝐼 be an interpretation. The satisfiability relation |= is defined as
follows:

    • 𝐼 |= 𝐶 ⊑ 𝐷 𝜃𝛼 if (𝐶 ⊑ 𝐷)𝐼 𝜃𝛼;
    • 𝐼 |= 𝐶(𝑎) 𝜃𝛼 if 𝐶 𝐼 (𝑎𝐼 ) 𝜃𝛼;
    • 𝐼 |= 𝑟(𝑎, 𝑏) 𝜃 𝑛 if 𝑟𝐼 (𝑎𝐼 , 𝑏𝐼 )𝜃 𝑛.
    • for a set 𝑆 of axioms, 𝐼 |= 𝑆 if 𝐼 |= 𝐸 for all 𝐸 ∈ 𝑆;
    • 𝐼 |= 𝐾 if 𝐼 |= 𝒯 and 𝐼 |= 𝒜.

If 𝐼 |= Γ, we say that 𝐼 satisfies Γ or that 𝐼 is a model of Γ (for Γ being an axiom, a set of axioms, or a
KB). An axiom 𝐸 is entailed by 𝐾, written 𝐾 |= 𝐸, if 𝐼 |= 𝐸 holds for all models 𝐼 of 𝐾.

   For the finitely many-valued case, we assume the truth space to be 𝒞𝑛 = {0, 𝑛1 , . . . , 𝑛−1  𝑛
                                                                                             𝑛 , 𝑛 }, for an
integer 𝑛 ≥ 1 [28, 29, 30]. In the following, we will use 𝒜ℒ𝒞 𝑛 to refer to a finitely-valued extension
of 𝒜ℒ𝒞 interpreted over the truth space 𝒞𝑛 , without committing to a specific choice of combination
functions.


3. The temporal Description Logic 𝐿𝑇 𝐿𝒜ℒ𝒞
The temporal Description Logic 𝐿𝑇 𝐿𝒜ℒ𝒞 is a temporal extension of 𝒜ℒ𝒞 based on linear time temporal
logic (LTL) The concepts of 𝐿𝑇 𝐿𝒜ℒ𝒞 can be formed by adding to the constructors of 𝒜ℒ𝒞 the temporal
operators ○ (next), 𝒰 (until), ♢ (eventually) and □ (always) of LTL. Temporal extensions of Description
Logics are very well-studied in the literature; see, for instance, the survey on temporal DLs and their
complexity and decidability by Lutz et al. [17].
   The set of temporally extended concepts is the following:

           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 𝐿𝑇 𝐿𝒜ℒ𝒞 is a pair ℐ = (∆ℐ , ·ℐ ), where ∆ℐ is a nonempty domain; ·ℐ is
an extension function that maps each concept name 𝐶 ∈ 𝑁𝐶 to a set 𝐶 ℐ ⊆ N × ∆ℐ , each role name
𝑟 ∈ 𝑁𝑅 to a relation 𝑟ℐ ⊆ N × ∆ℐ × ∆ℐ , and each individual name 𝑎 ∈ 𝑁𝐼 to an element 𝑎ℐ ∈ ∆ℐ .
Following [17] we assume individual names to be rigid, i.e., having the same interpretation at any time
point. In a pair (𝑛, 𝑑) ∈ N × ∆ℐ , 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 × ∆ℐ          ⊥ℐ = ∅       (¬𝐶)ℐ = (N × ∆ℐ )∖𝐶 ℐ
      (𝐶 ⊓ 𝐷)ℐ = 𝐶 ℐ ∩ 𝐷ℐ           (𝐶 ⊔ 𝐷)ℐ = 𝐶 ℐ ∪ 𝐷ℐ
      (∃𝑟.𝐶)ℐ = {(𝑛, 𝑥) ∈ N × ∆ℐ | ∃𝑦.(𝑛, 𝑥, 𝑦) ∈ 𝑟ℐ and (𝑛, 𝑦) ∈ 𝐶 ℐ }
      (∀𝑟.𝐶)ℐ = {(𝑛, 𝑥) ∈ N × ∆ℐ | ∀𝑦.(𝑛, 𝑥, 𝑦) ∈ 𝑟ℐ ⇒ (𝑛, 𝑦) ∈ 𝐶 ℐ }
      (○𝐶)ℐ = {(𝑛, 𝑥) ∈ N × ∆ℐ | (𝑛 + 1, 𝑥) ∈ 𝐶 ℐ }
      (♢𝐶)ℐ = {(𝑛, 𝑥) ∈ N × ∆ℐ | ∃𝑚 ≥ 𝑛 such that (𝑚, 𝑥) ∈ 𝐶 ℐ }
      (□𝐶)ℐ = {(𝑛, 𝑥) ∈ N × ∆ℐ | ∀𝑚 ≥ 𝑛, (𝑚, 𝑥) ∈ 𝐶 ℐ }
      (𝐶𝒰𝐷)ℐ = {(𝑛, 𝑥) ∈ N × ∆ℐ | ∃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 ∆ℐ0 ⊆ ∆ℐ1 ⊆ . . ., one for each time point.
   For simplicity, in the following we will focus on the case of non-temporal TBox, i.e., to a 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.
   The notions of satisfiability and model of a knowledge base can be easily extended to LTLT     𝒜ℒ𝒞 with
non-temporal TBox. All inclusions in the (non-temporal) TBox 𝒯 are regarded as global temporal
constraints, and have to be satisfied at all time points, i.a., a concept inclusion 𝐶 ⊑ 𝐷 is satisfied in an
interpretation ℐ if 𝐶 ℐ ⊆ 𝐷ℐ .
   It has been proven that, for non-temporal TBoxes, concept satisfiability in LTL𝒜ℒ𝒞 w.r.t. non-
temporal TBoxes is ExpTime-complete, both with expanding domains [35] and with constant domains
[17]. The complexity of other cases and, specifically, the cases of temporal ABoxes [36] and temporal
TBoxes (which allow temporal operators over concept inclusions), have as well been studied in the
literature, and we refer to [17] for a discussion of the result and algorithms for satisfiability checking.
   In [22] we have shown that, in the two-valued case, a typicality operator can be added to LTL𝒜ℒ𝒞 and
that a preferential extension of LTL𝒜ℒ𝒞 with typicality can be polynomially encoded into LTL𝒜ℒ𝒞 . The
encoding allows borrowing some decidability and complexity results from 𝐿𝑇 𝐿𝒜ℒ𝒞 to its preferential
version with typicality.
   In the following section, we first develop a many-valued semantics for LTL𝒜ℒ𝒞 and, then, we define
the typicality operator. Finally, we extend the notion of weighted KBs to the temporal, many-valued
case.


4. A many-valued semantics for 𝐿𝑇 𝐿𝒜ℒ𝒞
Let us now move to the many-valued case. To define a temporal extension of 𝐿𝑇 𝐿𝒜ℒ𝒞 with typicality,
we develop a many-valued semantics for 𝐿𝑇 𝐿𝒜ℒ𝒞 , by interpreting, at each time point, all concepts and
role names over a truth degree set 𝒮 equipped with a preorder relation ≤𝒮 , a bottom element 0𝒮 , and a
top element 1𝒮 . We denote by <𝒮 and ∼𝒮 the related strict preference relation and equivalence relation.
In the following we will assume 𝒮 to be the unit interval [0, 1] or the finite set 𝒞𝑛 , for an integer 𝑛 ≥ 1,
and that ⊗, ⊕, ▷ and ⊖ are a t-norm, an s-norm, an implication function, and a negation function
in some well known system of many-valued logic. In particular, in the following we will restrict to
continuous t-norms.
   A many-valued temporal interpretations for 𝐿𝑇 𝐿𝒜ℒ𝒞 is a pair ℐ = (∆ℐ , ·ℐ ), where ∆ℐ is a non-
empty domain; ·ℐ is an interpretation function that maps each concept name 𝐴 ∈ 𝑁𝐶 to a function
𝐴ℐ : N × ∆ℐ → 𝒮, each role name 𝑟 ∈ 𝑁𝑅 to a function 𝑟ℐ : N × ∆ℐ × ∆ℐ → 𝒮, and each individual
name 𝑎 ∈ 𝑁𝐼 to an element 𝑎ℐ ∈ ∆ℐ . Again, in the following definition we assume individual names
to be rigid, i.e., having the same interpretation at any time point 𝑛. Given a time point 𝑛 ∈ N and a
domain element 𝑑 ∈ ∆ℐ , the interpretation 𝐴ℐ of a concept name 𝐴 assigns to the pair (𝑛, 𝑑) a value
𝐴ℐ (𝑛, 𝑑) ∈ 𝒮 representing the degree of membership of 𝑑 in concept 𝐴 at time point 𝑛; and similarly
for roles.
   The interpretation function ·𝐼 is extended to complex concepts as follows (where, for the semantics
of the temporal operators, we adapt a formulation from [37]):
      ⊥ℐ (𝑛, 𝑥) = 0, ⊤ℐ (𝑛, 𝑥) = 1
      (¬𝐶)ℐ (𝑛, 𝑥) = ⊖𝐶 ℐ (𝑛, 𝑥)
      (𝐶 ⊓ 𝐷)ℐ (𝑛, 𝑥) = 𝐶 ℐ (𝑛, 𝑥) ⊗ 𝐷ℐ (𝑛, 𝑥)
      (𝐶 ⊔ 𝐷)ℐ (𝑛, 𝑥) = 𝐶 ℐ (𝑛, 𝑥) ⊕ 𝐷ℐ (𝑛, 𝑥)
      (∃𝑟.𝐶)ℐ (𝑛, 𝑥) = 𝑠𝑢𝑝𝑦∈Δ 𝑟ℐ (𝑛, 𝑥, 𝑦) ⊗ 𝐶 ℐ (𝑛, 𝑦)
      (∀𝑟.𝐶)ℐ (𝑛, 𝑥) = 𝑖𝑛𝑓𝑦∈Δ 𝑟ℐ (𝑛, 𝑥, 𝑦) ▷ 𝐶 ℐ (𝑛, 𝑦)
      (○𝐶)ℐ (𝑛, 𝑥) = 𝐶 ℐ (𝑛 + 1, 𝑥)
      (♢𝐶)ℐ (𝑛, 𝑥) = 𝑚≥𝑛 𝐶 ℐ (𝑚, 𝑥)
                    ⨁︀

      (□𝐶)ℐ (𝑛, 𝑥) = 𝑚≥𝑛 𝐶 ℐ (𝑚, 𝑥)
                    ⨂︀

      (𝐶𝒰𝐷)ℐ (𝑛, 𝑥) = 𝑚≥𝑛 (𝐷ℐ (𝑚, 𝑥) ⊗ 𝑚−1   ℐ
                      ⨁︀              ⨂︀
                                        𝑘=𝑛 𝐶 (𝑘, 𝑥))

  The semantics of ♢, □ and 𝒰 requires a passage to the limit. Following [37], one can introduce a
bounded version for ♢, □ and 𝒰, by adding new temporal operators ♢𝑡 (eventually in the next 𝑡 time
points), □𝑡 (always within 𝑡 time points) and 𝒰𝑡 , with the interpretation:

      (♢𝑡 𝐶)ℐ (𝑛, 𝑥) = 𝑛+𝑡      ℐ
                       ⨁︀
                          𝑚=𝑛 𝐶 (𝑚, 𝑥)
      (□𝑡 𝐶)ℐ (𝑛, 𝑥) = 𝑛+𝑡      ℐ
                       ⨂︀
                         𝑚=𝑛 𝐶 (𝑚, 𝑥)
      (𝐶𝒰𝑡 𝐷)ℐ (𝑛, 𝑥) = 𝑛+𝑡
                        ⨁︀         ℐ
                                              ⨂︀𝑚−1 ℐ
                           𝑚=𝑛 (𝐷 (𝑚, 𝑥) ⊗        𝑘=𝑛 𝐶 (𝑘, 𝑥))

so that (♢𝐶)ℐ (𝑛, 𝑥) = 𝑙𝑖𝑚𝑡→+∞ (♢𝑡 𝐶)ℐ (𝑛, 𝑥) and (□𝐶)ℐ (𝑛, 𝑥) = 𝑙𝑖𝑚𝑡→+∞ (□𝑡 𝐶)ℐ (𝑛, 𝑥) and
(𝐶𝒰𝐷)ℐ (𝑛, 𝑥) = 𝑙𝑖𝑚𝑡→+∞ (𝐶𝒰𝑡 𝐷)ℐ (𝑛, 𝑥). The existence of the limits is ensured by the fact that
(♢𝐶)ℐ (𝑛, 𝑥) and (𝐶𝒰𝐷)ℐ (𝑛, 𝑥) are increasing in 𝑛, while (□𝐶)ℐ (𝑛, 𝑥) is decreasing in 𝑛.
   Note that, here, we have not considered the additional temporal operators (“soon”, “almost always”,
etc.) introduced by Frigeri et al. [37] for representing vagueness in the temporal dimension. As a
consequence, for the case 𝒮 = [0, 1], the semantics above is an extension to 𝒜ℒ𝒞 of the FLTL (Fuzzy
Linear-time Temporal Logic) semantics by Lamine and Kabanza [38].

Proposition 1. For all concepts 𝐶 and 𝐷, and for all time points 𝑛, the following properties hold:
  (♢𝐶)ℐ (𝑛, 𝑥) = 𝐶 𝐼 (𝑛, 𝑥) ⊕ (♢𝐶)ℐ (𝑛 + 1, 𝑥)
  (□𝐶)ℐ (𝑛, 𝑥) = 𝐶 𝐼 (𝑛, 𝑥) ⊗ (□𝐶)ℐ (𝑛 + 1, 𝑥)
  (𝐶𝒰𝐷)ℐ (𝑛, 𝑥) = 𝐷𝐼 (𝑛, 𝑥) ⊕ (𝐶 𝐼 (𝑛, 𝑥) ⊗ (𝐶𝒰𝐷)ℐ (𝑛 + 1, 𝑥))

Note that, although in this section we have considered a constant domain ∆ℐ , for a many-valued prefer-
ential temporal interpretation ℐ, expanding domains could have been considered as well, considering a
domain ∆ℐ𝑛 for each time point 𝑛, with condition ∆ℐ0 ⊆ ∆ℐ1 ⊆ . . ., as for LTL𝒜ℒ𝒞 in the the two-valued
case [17].
   As in [22], for simplicity, we consider knowledge bases with non-temporal TBox and ABox, where a
non-temporal TBox 𝒯 is a set of concept inclusions 𝐶 ⊑ 𝐷, where (as in the two-valued case) 𝐶, 𝐷
are temporally extended concepts, but no temporal operator is applied in front of concept inclusions
themselves. The notions of satisfiability and model of a knowledge base can be easily generalized to a
many-valued LTL𝒜ℒ𝒞 knowledge base 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, concept inclusions in the
(non-temporal) TBox 𝒯 are evaluated by considering all time points 𝑛.
   Given a many-valued temporal interpretation ℐ = ⟨∆ℐ , ·ℐ ⟩, the interpretation function ·𝐼 is extended
to inclusion axioms as follows:

                         (𝐶 ⊑ 𝐷)𝐼 = inf 𝑥∈Δ𝐼 ,𝑛∈N (𝐶 𝐼 (𝑛, 𝑥) ▷ 𝐷𝐼 (𝑛, 𝑥))

  Let 𝐾 be an LTL𝒜ℒ𝒞 knowledge base 𝐾 = (𝒯 , 𝒜) with non-temporal ABox and TBox.
Definition 2 (Satisfiability in many-valued LTL𝒜ℒ𝒞 ). Given a many-valued temporal interpretation
for ℐ = ⟨∆ℐ , ·ℐ ⟩, satisfiability of an axiom in ℐ is defined as follows:
    • ℐ |= 𝐶 ⊑ 𝐷 𝜃𝛼 if (𝐶 ⊑ 𝐷)ℐ 𝜃𝛼;
    • ℐ |= 𝐶(𝑎) 𝜃𝛼 if 𝐶 ℐ (0, 𝑎ℐ ) 𝜃𝛼;
    • ℐ |= 𝑟(𝑎, 𝑏) 𝜃 𝛼 if 𝑟ℐ (0, 𝑎ℐ , 𝑏ℐ )𝜃 𝛼.
The interpretation ℐ is a model of 𝐾 = (𝒯 , 𝒜) if ℐ satisfies all concept inclusions in 𝒯 and all assertions in
𝒜. A knowledge base 𝐾 = (𝒯 , 𝒜) is satisfiable in the many-valued extension of LTL𝒜ℒ𝒞 if a many-valued
temporal model ℐ = ⟨∆ℐ , ·ℐ ⟩ of 𝐾 exists.


5. A many-valued 𝐿𝑇 𝐿𝒜ℒ𝒞 withTypicality
As in the two-valued case [22], the language of a many-valued 𝐿𝑇 𝐿𝒜ℒ𝒞 can be extended with typicality
concepts of the form T(𝐶) representing the set of typical instances of concept 𝐶. The typicality operator
T may occur both in concepts of TBox and ABox, but it cannot be nested. Unlike [5, 10], 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 of the form T(𝐶) ⊑ 𝐷, and this choice is in agreement
with [39, 40]. 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, the following ones (adapted from [22]):

                                     T(Professor ) ⊑ (∃teaches.Course)𝒰Retired
                       ∃lives_in.Town ⊓ Young ⊑ T(♢Granted _Loan)

Note that, while the semantics in [22] was two-valued, in this example, the interpretation of some
concepts (e.g., Young and Granted _Loan) may have a non-crisp value in [0, 1]. Indeed, being young is
a fuzzy concept and Granted _Loan may have a degree of truth, for the different domain individuals
(depending, e.g., on the outcome of some classifier on input exemplars).
   From the semantic side, in the many valued case, the degree of membership of domain individuals
in concept 𝐶 at the different time points 𝑛 induces a preference relation ≺𝑛𝐶 over the domain. Such
preference relations are used to define the typical 𝐶-elements at the different time points.
   Given a temporal interpretation ℐ = ⟨∆ℐ , ·ℐ ⟩ over a truth degree set 𝒮, a preference relation ≺𝑛𝐶 on
∆ℐ can be associated to any concept 𝐶 and time point 𝑛 ∈ N, based on the many valued interpretation
of concepts in ℐ and on the the strict partial order <𝒮 : for all 𝑥, 𝑦 ∈ ∆ℐ ,
                               𝑥 ≺𝑛𝐶 𝑦 if and only if 𝐶 ℐ (𝑛, 𝑦) <𝒮 𝐶 ℐ (𝑛, 𝑥),
where 𝑥 ≺𝑛𝐶 𝑦 means that 𝑥 is preferred to 𝑦 wrt 𝐶 at time point 𝑛.
  The many-valued temporal semantics introduced in the previous section easily extends to the language
with typicality. Note that this semantics is inherently multi-preferential.
  We regard typical 𝐶-elements (at time point 𝑛) as the domain elements 𝑥 which are preferred with
respect to ≺𝑛𝐶 among all domain elements (and such that 𝐶 ℐ (𝑥) ̸= 0). The interpretation of typicality
concepts T(𝐶) can be defined as follows:
Definition 3. Given an interpretation ℐ = ⟨∆ℐ , ·ℐ ⟩, for all 𝑛 ∈ N, 𝑥 ∈ ∆ℐ , (T(𝐶))ℐ (𝑛, 𝑥) = 𝐶 ℐ (𝑥),
if there is no 𝑦 ∈ ∆ℐ such that 𝑦 ≺𝑛𝐶 𝑥; (T(𝐶))ℐ (𝑛, 𝑥) = 0, otherwise.
When (T(𝐶))ℐ (𝑥) > 0, 𝑥 is said to be a typical 𝐶-element in ℐ. Note that, when ≤𝒮 is a total preorder
(as it is in the cases 𝒮 = [0, 1] and 𝒮 = 𝒞𝑛 ), relation ≺𝑛𝐶 is an irreflexive, transitive and modular relation
over ∆ℐ , like ranked preference relations in KLM-style ranked interpretations by Lehmann and Magidor
[2]. For finitely-many truth values, ≺𝑛𝐶 is also well-founded.
   For 𝐿𝑇 𝐿𝒜ℒ𝒞 with typicality, the notion of satisfiability of an axiom in a multi-preferential temporal
interpretation ℐ and the notion of model of a KB, are the ones given in Definition 2 (again for non-
temporal KBs).
  In the following, we will denote with LTL𝒜ℒ𝒞 𝑛 T the many-valued extension of 𝐿𝑇 𝐿𝒜ℒ𝒞 with
typicality, with truth degree set 𝒮 = 𝒞𝑛 , for 𝑛 ≥ 1, and with 𝐿𝑇 𝐿𝒜ℒ𝒞 F T the fuzzy extension of
𝐿𝑇 𝐿𝒜ℒ𝒞 with typicality (where 𝒮 = [0, 1]).


6. Weighted temporal knowledge bases
Besides a set of strict concept inclusions in the TBox, weighted KBs also allow a set of typicality inclusions
(or defeasible inclusions), each one with a weight. Weighted typicality inclusions for a concept 𝐶𝑖 have
the form (T(𝐶𝑖 ) ⊑ 𝐷𝑗 , 𝑤𝑖𝑗 ), and describe the prototypical properties of 𝐶𝑖 -elements (where 𝐷𝑗 is a
concept, and the weight 𝑤𝑖𝑗 is a real number). A concept 𝐶𝑖 for which weighted typicality inclusions
are provided is said to be a distinguished concept.
   A weighted ℒ𝒞 𝑛 T knowledge base is a tuple ⟨𝒯 , 𝒟, 𝒜⟩, where the (strict) TBox 𝒯 is a set of concept
inclusions, the defeasible TBox 𝒟 is a set of weighted typicality inclusions for the distinguished concepts
𝐶𝑖 , and 𝒜 is a set of assertions.
   Consider the weighted 𝒜ℒ𝒞T knowledge base 𝐾 = ⟨𝒯 , 𝒟, 𝒜⟩, over the set of distin-
guished concepts {Student, Employee, Person, . . .}, with 𝒯 containing, for instance, the inclusion
Student ⊑ Person ≥ 1 .
   The set 𝒟 of weighted typicality inclusions may contain, e.g., the following inclusions, describing
the prototypical properties of concept Student:
     (T(Student) ⊑ Has_Classes, +50),
     (T(Student) ⊑ Active,+35) ,
     (T(Student) ⊑ ∃has_Boss.⊤, -70),
That is, a student normally has classes and is active, but she usually does not have a boss (negative
weight). Accordingly, a student having classes, but not a boss, is more typical than an active student
having classes and a boss. In the two valued case, one can evaluate how typical are two domain
individuals mary and tom as students, by considering their weight with respect of concept Student,
i.e., by summing the (positive or negative) weights of the defeasible inclusions satisfied by mary and
tom, and comparing them. The higher the weight the more typical is the individual. In the many-value
case, in defining the weight of a domain element 𝑥 with respect to a distinguished concept 𝐶𝑖 , we have
to consider that, in an interpretation ℐ, at time point 𝑛, element 𝑥 may belong to other concepts to
some degree (e.g., at time point 𝑛, mary may be active with degree 0.8, i.e., Active ℐ (𝑛, mary) = 0.8).
   Given a many-valued temporal interpretation ℐ = ⟨∆ℐ , ·ℐ ⟩, the weight of 𝑥 ∈ ∆ℐ with respect to a
distinguished concept 𝐶𝑖 at time point 𝑛 is given by
                                  ℐ (𝑥) =                               ℐ
                                            ∑︀
                               𝑊𝑖,𝑛            (T(𝐶𝑖 )⊑𝐷𝑗 ,𝑤𝑖𝑗 )∈𝒟 𝑤𝑖𝑗 𝐷𝑗 (𝑛, 𝑥).

                                       ℐ (𝑥), the more typical is 𝑥 as an instance of 𝐶 ), at time point
Intuitively, the higher the value of 𝑊𝑖,𝑛                                                 𝑖
𝑛 (considering the defeasible properties of 𝐶𝑖 ). Here, the membership degree 𝐷𝑗ℐ (𝑛, 𝑥) of 𝑥 in each
concept 𝐷𝑗 at time point 𝑛 is considered.
   The notions of faithful, coherent and 𝜙-coherent semantics introduced for many-valued weighted
KBs [41, 15, 16] can be smoothly extended to the temporal case. Generalizing from the non-temporal
case, we expect the membership degree of a domain element 𝑥 in a concept 𝐶𝑖 at a time point 𝑛 to be
in agreement with the weight of 𝑥 with respect to concept 𝐶𝑖 , at the same time point 𝑛. We consider
some different agreement conditions at time point 𝑛, as follows.
   A many-valued temporal interpretation ℐ = ⟨∆ℐ , ·ℐ ⟩ is faithful at 𝑛 if, for all 𝑥, 𝑦 ∈ ∆ℐ ,
                                                 ℐ          ℐ
                                    𝑥 ≺𝑛𝐶𝑖 𝑦 ⇒ 𝑊𝑖,𝑛 (𝑥) > 𝑊𝑖,𝑛 (𝑦)

The interpretation ℐ is coherent at 𝑛 if, for all 𝑥, 𝑦 ∈ ∆ℐ ,
                                                   ℐ          ℐ
                                    𝑥 ≺𝑛𝐶𝑖 𝑦 iff 𝑊𝑖,𝑛 (𝑥) > 𝑊𝑖,𝑛 (𝑦)

Given a collection of monotonically non-decreasing functions 𝜙𝑖 : R → 𝒮, one for each concept 𝐶𝑖 ∈ 𝒞:
  - the interpretation ℐ is 𝜙-coherent at 𝑛 if, for all 𝑥 ∈ ∆ℐ ,

                                         𝐶𝑖ℐ (𝑛, 𝑥) = 𝜙𝑖 (𝑊𝑖,𝑛
                                                            ℐ
                                                               (𝑥))

  - the interpretation ℐ is transient 𝜙-coherent at 𝑛 if, for all 𝑥 ∈ ∆ℐ ,

                                       𝐶𝑖ℐ (𝑛 + 1, 𝑥) = 𝜙𝑖 (𝑊𝑖,𝑛
                                                              ℐ
                                                                 (𝑥))

   It is easy to see that a many-valued temporal interpretation ℐ = ⟨∆ℐ , ·ℐ ⟩ determines, at each
                                                                              𝑛    𝑛             𝑛
time point 𝑛, a (non-temporal) many-valued interpretation 𝐽 𝑛 = ⟨∆𝐽 , ·𝐽 ⟩, where ∆𝐽 = ∆ℐ , (for
                   𝑛
𝐴 ∈ 𝑁𝐶 ), and 𝑟𝐽 (𝑥, 𝑦) = 𝑟ℐ (𝑛, 𝑥, 𝑦) (for 𝑟 ∈ 𝑁𝑅 ). Letting the interpretation of typicality in 𝐽 𝑛
exploit the preference relations ≺𝑛𝐶𝑖 for each 𝐶𝑖 (see Section 5), i.e., the preference relation induced
by the many-valued interpretation of concept 𝐶𝑖 in 𝐽 𝑛 , a many-valued temporal interpretation ℐ
can be regarded as a sequence 𝐽 0 , 𝐽 1 , 𝐽 2 , . . . of many-valued preferential interpretations, as the ones
considered in [33]. At each single time point the KLM properties of preferential consequence relation
are then be expected to hold.
   When considering the single time point 𝑛, the condition that the interpretation ℐ is coherent (resp.,
faithful, 𝜙-coherent) at 𝑛, means that the preferential interpretation 𝐽 𝑛 is coherent (resp., faithful,
𝜙-coherent) according to their definition in [33]. Different notions of agreement at different time points
can then be combined to give rise to different semantics of a temporal weighted KB, and different
notions of entailment (based on different closure constructions).


7. Temporal weighted KBs and the transient behaviour of a neural
   network
In [33] it has been shown that many-valued Weighted KBs with typicality can provide a logical inter-
pretation to some neural network model. Specifically, the 𝜙-coherent semantics allows to capture the
stationary states of multilayer networks as well as of networks with cyclic dependencies. In this section,
we are interested in the transient behavior of a network.
   Let us first recall from [42] the model of a neuron as an information-processing unit in ∑︀  an (artificial)
neural network. A neuron 𝑘 can be described by the following pair of equations: 𝑢𝑘 = 𝑛𝑗=1 𝑤𝑘𝑗 𝑥𝑗 ,
and 𝑦𝑘 = 𝜙(𝑢𝑘 + 𝑏𝑘 ), where 𝑥1 , . . . , 𝑥𝑛 are the input signals, 𝑤𝐾1 , . . . , 𝑤𝑘𝑛 are synaptic weights; 𝑏𝑘
is the bias, 𝜙 an activation function, and 𝑦𝑘 is the output signal of unit∑︀     𝑘. By adding a new synapse
with input 𝑥0 = +1 and synaptic weight 𝑤𝑘0 = 𝑏𝑘 , one can write: 𝑢𝑘 = 𝑛𝑗=0 𝑤𝑘𝑗 𝑥𝑗 , and 𝑦𝑘 = 𝜙(𝑢𝑘 ),
where 𝑢𝑘 is called the induced local field of the neuron. The neuron can be represented as a directed
graph, where the input signals 𝑥1 , . . . , 𝑥𝑛 and the output signal 𝑦𝑘 of neuron 𝑘 are nodes of the graph.
An edge from 𝑥𝑗 to 𝑦𝑘 , labelled 𝑤𝑘𝑗 , means that 𝑥𝑗 is an input signal of neuron 𝑘 with synaptic weight
𝑤𝑘𝑗 .
   A neural network can then be seen as “a directed graph consisting of nodes with interconnecting
synaptic and activation links" [42]: nodes in the graph are the neurons (the processing units) and the
weight 𝑤𝑖𝑗 on the edge from node 𝑗 to node 𝑖 represents “the strength of the connection [..] by which
unit 𝑗 transmits information to unit 𝑖" [43]. Source nodes (i.e., nodes without incoming edges) produce
the input signals to the graph. Neural network models are classified by their synaptic connection
topology. In a feedforward network the architectural graph is acyclic, while in a recurrent network it
contains cycles. In a recurrent network at least one feedback exists, so that “the output of a node in
the system influences in part the input applied to that particular element" [42]. A time delay may be
associated to feedback connections.
   Let us consider a trained network 𝒩 . We do not put restrictions on the topology the network.
Following the approach in [33], 𝒩 can be mapped into a (non-temporal) weighted conditional knowledge
base 𝐾 𝒩 [15, 33], by regarding the units in the network as concept names and the synaptic connections
between units as weighted inclusions.
   If 𝐶𝑘 is the concept name associated to unit 𝑘 and 𝐶𝑗1 , . . . , 𝐶𝑗𝑚 are the concept names associated
to units 𝑗1 , . . . , 𝑗𝑚 , whose output signals are the input signals for unit 𝑘, with synaptic weights
𝑤𝑘,𝑗1 , . . . , 𝑤𝑘,𝑗𝑚 , then unit 𝑘 can be associated a set 𝒯𝐶𝑘 of weighted typicality inclusions: T(𝐶𝑘 ) ⊑ 𝐶𝑗1
with 𝑤𝑘,𝑗1 , . . . , T(𝐶𝑘 ) ⊑ 𝐶𝑗𝑚 with 𝑤𝑘,𝑗𝑚 .
   It has been proven that the input-output behavior of a multilayer network 𝒩 can be captured by a
preferential interpretation 𝐼𝒩      Δ built over a set of input stimuli ∆ (e.g., the test set), through a simple

construction, which exploits the activity level of units for the input stimuli.
   A logical characterization of a trained multi-layer network 𝒩 is established [33] by proving that the
preferential interpretation 𝐼𝒩     Δ , describing the network behavior over a set ∆ of input stimuli, is indeed

a 𝜙-coherent model of the weighted knowledge base 𝐾 𝒩 and, vice-versa, that any 𝜙-coherent model of
the knowledge base 𝐾 𝒩 captures the behavior of the network over some set ∆ of input stimuli. Also
in the case the network is not feedforward, the 𝜙-coherent semantics allows the stationary states of the
network 𝒩 to be captured.
   This approach allows for the verification of conditional properties of the network (of the form
T(𝐶) ⊏ 𝐷 ≥ 𝜃) by model checking over the preferential interpretation 𝐼𝒩              Δ , or by using entailment
                                                𝒩
from the conditional knowledge base 𝐾 (e.g., in an ASP encoding of a finitely-valued semantics[32]).
Both the model checking and entailment approach have been used in the verification of properties of
feedforward neural networks for the recognition of basic emotions.
   In the temporal case, when we consider a temporal preferential model ℐ of the weighted knowledge
base 𝐾 𝒩 , we may represent different states of the network at different time points.
   When ℐ is 𝜙-coherent at time point 𝑛, the condition (stated above) that, for all 𝑥 ∈ ∆ℐ ,
                                                            ∑︁
                                          𝐶𝑖ℐ (𝑛, 𝑥) = 𝜙𝑖 (    𝑤𝑖ℎ 𝐷ℎℐ (𝑛, 𝑥))
                                                        ℎ

imposes that the (non-temporal) interpretation 𝐽 𝑛 at time point 𝑛 represents a stationary                state of
network 𝒩 . In such a case, 𝜙𝑖 plays the role of the activation function, and the sum ℎ 𝑤𝑖ℎ 𝐷ℎℐ (𝑛, 𝑥)
                                                                                                 ∑︀
plays the role of the induced local field.
   However, the temporal formalism also allows to capture the dynamic behavior of the network beyond
stationary states, and this is especially interesting when the network 𝒩 is recurrent. In this case, the
knowledge base 𝐾 𝒩 contains cyclic dependencies in DBox.
   By imposing the condition that ℐ is a transient 𝜙-coherent interpretation at all time points 𝑛, one can
enforce that the interpretations 𝐽 0 , 𝐽 1 , 𝐽 2 , . . . at successive time points describe the dynamic evolution
of the activity of units in the network (where the activity of each unit at time point 𝑛 + 1 depends
on the activity of incoming units at time point 𝑛). The temporal formalism provides a semantics for
capturing the trajectories of the network state. Alternatively, time delayed feedback connections can be
easily captured by temporal operators in 𝐾 𝒩 .
   Once a trained neural network has been represented as a weighted defeasible knowledge base 𝐾 𝒩 ,
entailment allows for temporal properties to be proved over the runs representing the evolution of
the network, an approach which may be computationally quite costly, depending on the size of the
neural network and on the length of the runs. The non-temporal case is already challenging, and
we refer to complexity results and to an experimentation of some different ASP based encodings of
defeasible entailment for the verification of properties of a neural network, both in the feedforward
case and in the cyclic case [33, 44]. The model checking approach, on the other hand, does not require
to consider in the model ℐ𝒩 Δ the activity of all units, but only of the units involved in the properties to

be verified. Similarly, not all time points need to be considered, but only those corresponding to the
states of interest.
   An interesting direction for future work, is an extension to the temporal case of the model-checking
approach developed in Datalog [45, 33] for the verification of conditional properties of a network, for
post-hoc verification.
8. Conclusions
In this paper, we develop a many-valued, temporal description logic with typicality, extending 𝐿𝑇 𝐿𝒜ℒ𝒞
to deal with defeasible reasoning. Our extension of LTL𝒜ℒ𝒞 builds, on the one hand, on fuzzy and
many-valued DLs, and, on the other hand, on preferential DLs with typicality. We have first developed
a many-valued semantics for LTL𝒜ℒ𝒞 , and then added to the language a typicality operator, based on a
(multi-) preferential semantics. Finally, we have defined an extension of weighted knowledge bases
with typicality to the temporal many-valued case, for representing prototypical properties of different
classes in the temporal case.
   On a different route, a preferential LTL with defeasible temporal operators has been studied in [20, 21],
where the decidability of meaningful fragments of the logic is proven, and tableaux based proof methods
for such fragments is developed [19, 21]. Our approach does not consider defeasible temporal operators
nor preferences over time points, but combines standard LTL operators with the typicality operator in
a many-valued temporal 𝒜ℒ𝒞. Preferences are over domain elements, but they change over time.
   In previous work, we have developed a preferential temporal description logics with typicality
LTLT                                         T
     𝒜ℒ𝒞 [22]. The monotonic logic LTL𝒜ℒ𝒞 is further extended with multiple preferences. Such
extensions show that the concept-wise multi-preferential semantic in [14] adapts smoothly to the
temporal case. In the two-valued case, the semantics for rank and weighted 𝒜ℒ𝒞 knowledge bases has
been defined based on semantic closure constructions [14, 15], developed in the spirit of Lehmann’s
lexicographic closure [13], Kern-Isberner’s c-representations [46, 47] and Weydert’s algebraic semi-
qualitative approach [48], Casini and Straccia’s fuzzy rational closure [49], but allowing for multiple
preferences defining a ranking on individuals for each concept. In this paper, we have considered the
temporal many-valued case and developed a semantics for weighted knowledge bases that deals with
different agreement conditions at the different time points, leading to different closure constructions for
the temporal conditional logics.
   Much work has been recently devoted to the combination of neural networks and symbolic reasoning
[50, 51, 52]. While conditional weighted KBs have been shown to capture (in the many-valued case) the
stationary states of a neural network (or its finite approximation) [15, 33], and allow for combining
empirical knowledge with elicited knowledge for reasoning and for post-hoc verification, adding a
temporal dimension opens to the possibility of verifying properties concerning the dynamic behaviour
of the network, based on a model checking approach or an entailment based approach.
   A different approach for dealing with defeasibility in temporal DL formalism has been proposed in
[53], by combining a (dynamic) temporal action logic [54] for reasoning about actions (whose semantics
is based on a notion of temporal answer set) and an ℰℒ⊥ ontology. The temporal action logic allows for
complex actions, and the proof methods are based on ASP encodings of bounded model checking [54].
   Extending the above mentioned ASP encodings to deal with model checking in temporal preferential
interpretations is a direction of future work. Future work also includes studying the decidability for
fragments of the logic and exploiting the formalism for explainability.


Acknowledgement
We thank the anonymous referees for their helpful suggestions. This work was partially supported by
GNCS-INdAM project 2024 “LCXAI: Logica Computazionale per eXplainable Artificial Intelligence”.
Mario Alviano was partially supported by Italian Ministry of University and Research (MUR) under PRIN
project PRODE “Probabilistic declarative process mining”, CUP H53D23003420006 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 CyberSpace”, CUP H73C22000880001; by Italian Ministry of
Health (MSAL) under POS projects CAL.HUB.RIA (CUP H53C22000800006) and RADIOAMICA (CUP
H53C22000650006); by Italian Ministry of Enterprises and Made in Italy under project STROKE 5.0
(CUP B29J23000430005); and by the LAIA lab (part of the SILA labs).
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