In this paper we propose a many-valued temporal conditional logic, and exploit it in the verification of properties of an argumentation graph, in a gradual semantics. We start from a many-valued logic with typicality, and extend it with the temporal operators of the Linear Time Temporal Logic (LTL), thus providing a formalism which is able to capture the dynamics of a system, trough strict and defeasible temporal properties. We then consider an instantiation of the formalism for gradual argumentation.
Preferential approaches to commonsense reasoning [
This paper deals with the propositional setting, and aims at extending the many-valued conditional
logic with typicality developed in [
Preferential extensions of LTL with defeasible temporal operators have been recently studied [28, 29, 30] to enrich temporal formalisms with non-monotonic reasoning features, by considering defeasible versions of the LTL operators. Our approach, instead, consists in adding the standard LTL operators to a (many-valued) conditional logic with typicality, an approach similar to the temporal preferential extension considered for Description Logics (DLs) in [31], where the logic LTLALC [32], extending ALC with LTL operators, has been further extended with a typicality operator, to develop a (two-valued) temporal ALC with typicality, called LTLATLC.
As in the Propositional Typicality Logic by Booth et al. [33] (and in the DLs with typicality [34]) the
conditionals are formalized based on material implication (resp., concept inclusion) plus the typicality
operator T. The typicality operator allows for the definition of conditional implications T(α ) → β ,
meaning that “normally if α holds, β holds". They correspond to conditional implications α |∼ β in
KLM logics [
The preferential semantics of the logic exploits multiple preference relations <α with respect to diferent formulas α , following the approach developed for ranked and weighted KBs in description logics, based on a multi-preferential semantics [35, 36] and for the propositional calculus in [37], where preference are allowed with respect to diferent aspects.
The schedule of the paper is the following. Section 2 develops a many-valued preferential logic with
typicality, which is inspired to [
In this section, we recall a many-valued preferential logic with typicality developed in [
Let L be a propositional many-valued logic, whose formulas are built from a set P rop of propositional variables using the logical connectives ∧, ∨, ¬ and →, as usual. We assume that ⊥ (representing falsity) and ⊤ (representing truth) are formulas of L. We consider a many-valued semantics for formulas, over a truth degree set D, equipped with a preorder relation ≤ D, a bottom element 0D, and a top element 1D. We denote by <D and ∼ D the related strict preference relation and equivalence relation (often, we will omit explicitly referring to D, and simply write ≤ <, ∼ , 0 and 1).
Let ⊗ , ⊕ , ⊖ and be the truth degree functions in D for the connectives ∧, ∨, ¬ and → (respectively).
When D is [
We further extend the language of L by adding a typicality operator, as introduced by Booth et
al. [33] for the propositional calculus, and by Giordano et al. for preferential description logics [39].
Intuitively, “a sentence of the form T(α ) is understood to refer to the typical situations in which α
holds" [33]. The typicality operator allows the formulation of conditional implications (or defeasible
implications) of the form T(α ) → β whose meaning is that “normally, if α then β ”, or “in the typical
situations when α holds, β also holds”. They correspond to conditional implications α |∼ β of KLM
preferential logics [
The interpretation of a typicality formula T(α ) is defined with respect to a preferential interpretation.
The KLM preferential semantics [
Here we consider a many-valued multi-preferential semantics. The propositions at each world w ∈ W have a value in D and multiple preference relations <A⊆ W × W are associated to formulas A of L.
Multi-preferential semantics have previously been proposed by Gill in [40], used in defining refinements of the rational closure construction [37], and for description logics, in ranked defeasible KBs [41] and, in the many-valued case, in weighted conditional KBs [21, 22]. The semantics below exploits a set of preference relations <Ai associated to formulas Ai of L 2.
A (multi-)preferential interpretation is a triple M = ⟨W , {<Ai }, v⟩ where: • W is a non-empty set of worlds; • each <Ai ⊆ W × W is an irreflexive and transitive relation on W ; • v : W × Prop →− D is a valuation function, assigning a truth value in D to any propositional variable in each world w ∈ W .
The valuation v is inductively extended to all formulas in LT as follows: v(w, ⊥) = 0D v(w, A ∧ B) = v(w, A) ⊗ v(w, B) v(w, A → B) = v(w, A) v(w, B) v(w, ⊤) = 1D
v(w, A ∨ B) = v(w, A) ⊕ v(w, B) v(w, ¬A) = ⊖ v(w, A) and the interpretation of a typicality formula T(A) in M, at a world w, is defined as: v(w, T(A)) = v(w, A) if ∄w ′ ∈ W s.t. w ′ <A w 0D otherwise When v(w, T(A)) ̸= 0D, w is a typical/normal A-world in M. Note that we do not assume wellfoundedness of <A.
A ranked interpretation is a (multi-)preferential interpretation M = ⟨W , {<Ai }, v⟩ for which each preference relation <Ai is modular, i.e., for all x, y, z, if x <Ai y then x <Ai z or z <Ai y.
In general, some conditions may be needed to enforce an agreement between the truth values of a formula Ai at the diferent worlds in M and preference relation <Ai among them. The preferences <Ai might have been determined by some closure construction, such as those exploiting the ranks or weights of conditionals, as in [41, 21]. Similar conditions, called coherence, faithfulness and φ-coherence conditions, have for instance been introduced in the multi-preferential semantics for DLs with typicality in [21, 22]. 2If we limit our consideration to finite KBs, we can restrict our attention to interpretations with a finite set of preference relations <Ai , one for each formula Ai such that T(Ai) occurs in the KB or in the query.
We call a (multi-)preferential interpretation M = ⟨W, {<Ai }, v⟩ coherent if, for all w, w′ ∈ W, and preference relation <Ai , v(w, Ai) > v(w′, Ai) ⇐⇒ w <Ai w′ that is, the ordering among Ai valuations in w and w′ is justified by the preference relation <Ai ; and viceversa. A weaker condition is faithfulness. A (multi-)preferential interpretation M = ⟨W, {<Ai }, v⟩ is faithful if, for all w, w′ ∈ W, and preference relation <Ai ,
v(w, Ai) > v(w′, Ai) ⇒ w <Ai w′
Clearly, a preferential interpretation M might be coherent with respect to a preference relation <Ai , while being only faithful with respect to another one <Aj .
Let us now define the satisfiability of a graded implication with form A → B ≥ l (or A → B ≤ u) with respect to an interpretation M, where l and u are constants corresponding to truth values in D and A and B are formulas of LT.
We can first define the truth degree of an implication A → B in M as follows: Definition 2. Given a preferential interpretation M = ⟨W, {<Ai }, v⟩, the truth degree of an implication A → B wrt. M is defined as:
(A → B)M = infw∈W (v(w, A) v(w, B)).
The satisfiability of a graded implication in a preferential interpretation M is evaluated globally to the preferential interpretation M.
Definition 3. A preferential interpretation M = ⟨W, {<Ai }, v⟩, satisfies a graded implication A → B ≥ l (written M |= A → B ≥ l) if (A → B)M ≥ l. Similarly, I satisfies a graded implication A → B ≤ u (written M |= A → B ≤ u) if (A → B)M ≤ u.
Let a knowledge base K be a set of graded implications. A model of K is an interpretation M which satisfies all the graded implications in K. Given a knowledge base K, we say that K entails a graded implication A → B ≥ l if A → B ≥ l is satisfied in all the models of K (and similarly for a graded implication A → B ≤ l). In the following, we will refer to the entailment of graded implications A → B ≥ 1 as 1-entailment.
It can be proven that the the usual KLM preferential semantics [
In this section we extend the language of the logic LT with the temporal operators ⃝ (next), U (until), 3 (eventually) and 2 (always) of Linear Time Temporal Logic (LTL) [42].
We extend the language of graded implications, by allowing temporal and typicality operators to occur in a graded implication A → B ≥ l (or A → B ≥ l) in A and in B, with the only restriction that T should not be nested. For instance, lives_in_town ∧ young → T(3granted _loan) ≥ 0.8 3T(granted _loan) → lives_in_town ∧ young ≥ 0.8. are graded implications. We define the semantics of the logic in agreement with the fuzzy LTL semantics by Frigeri et al. [43].
A temporal (multi-)preferential interpretation is a triple I = ⟨W, {<nAi }n∈N, v⟩ where: • W is a non-empty set of worlds; • each <nAi ⊆ W × W is an irreflexive and transitive relation on W; • v : N × W × Prop →− D is a valuation function assigning, at each time point n, a truth value to any propositional variable in each world w ∈ W.
When there is no w′ ∈ W s.t. w′ <nA w, we say that w is a normal situation for A at timepoint n.
In a preferential interpretation I = ⟨W, {<nAi }n∈N, v⟩, the valuation v(n, w, A) of a formula A, in world w, at time point n ∈ N, can be defined inductively as follows: v(n, w, ⊥) = 0D v(n, w, ⊤) = 1D v(n, w, A ∧ B) = v(n, w, A) ⊗ v(n, w, B) v(n, w, A ∨ B) = v(n, w, A) ⊕ v(n, w, B) v(n, w, T(A)) = v0D(n, w, A) oifth∄ewr w′∈iseW s.t. w ′ <nA w v(n, w, ⃝ A) = v(n + 1, w, A) v(n, w, 3A) = Lm≥ n v(m, w, A) v(n, w, AU B) = Lm≥ n(v(m, w, B)⊗
Nkm=−n1 v(k, w, A))
v(n, w, ¬A) = ⊖ v(n, w, A) v(n, w, 2A) = Nm≥ n v(m, w, A) so that v(n, w, 3tA) = Lnm+=tn v(m, w, A) v(n, w, AUtB) = Lnm+=tn(v(m, w, B)⊗ v(n, w, 3A) = limt→+∞ v(n, w, 3tA) v(n, w, 2A) = limt→+∞ v(n, w, 2tA) v(n, w, AU B) = limt→+∞v(n, w, AUtB) The semantics of 3, 2 and U requires a passage to the limit. Following [43], we introduce a bounded version for 3, 2 and U , by adding new temporal operators 3t (eventually in the next t time points), 2t (always within t time points) and Ut, with the interpretation: v(n, w, 2tA) = Nnm+=tn v(m, w, A)
Nkm=−n1 v(k, w, A)) The existence of the limits is ensured by the fact that v(n, w, 3tA) and v(n, w, AUtB) are increasing in t, while v(n, w, 2tA) is decreasing in t (assuming the monotonicity properties of t-norms and t-conorms for ⊗ and ⊕ ) [43].
The notions of coherence and faithfulness can be extended to temporal multi-preferential interpretation. E.g., a multi-preferential interpretation is coherent if, for all w, w′ ∈ W, n ∈ N and preference relation <Ai , v(n, w, Ai) > v(n, w′, Ai) ⇐⇒ w <nAi w′ (and similarly for faithfulness).
Note that, here, we have not considered the additional temporal operators (“soon”, “almost always”,
etc.) introduced by Frigeri et al. [43] for representing vagueness in the temporal dimension (which
can be considered for future work). As a consequence, for the case D = [
Proposition 1. For any formulas A and B, time point n and world w, the following holds: v(n, w, 3A) = v(n, w, A) ⊕ v(n + 1, w, 3A) v(n, w, 2A) = v(n, w, A) ⊗ v(n + 1, w, 2A) v(n, w, AU B) = v(n, w, B)⊕ (v(n, w, A) ⊗ v(n + 1, w, AU B))
It can be seen that a temporal many-valued interpretation I = ⟨W, {<nAi }n∈N, v⟩ can be regarded as a sequence of (non-temporal) preferential interpretations M0, M1, M2, . . . over the same set of worlds W, where each Mn is defined as follows: Mn = ⟨W, {<nAi }, vn⟩, where w <nAi w′ holds in Mn if w <nAi w′ holds in I, for all w, w′ ∈ W; and vn(w, A) = v(n, w, A), for all w ∈ W and propositional variable A.
Definition 5. Given a temporal preferential interpretation I = ⟨W, {<nAi }n∈N, v⟩ the truth degree of an implication A → B in I at time point n is defined as:
(A → B)I,n = infw∈W (v(n, w, A) v(n, w, B)).
Let us now define the satisfiability of a graded implication in a preferential interpretation I =
⟨W, {<nAi }n∈N, v⟩. Rather than regarding graded implications as global constraints, that have to hold
at all time points, we can allow for boolean combination of graded implications (as in [
α ::= A → B ≥ l | A → B ≥ l | α ∧ β | ¬α | ⃝ α | 3α | 2α | α U β , where α and β stand for temporal graded formulas. Note that temporal operators may occur both within graded implications (A → B ≥ l) and in front of them, and of their boolean combinations.
An example of temporal graded formula is the following conjunction: 2(T(professor ) → teaches U retired ≥ 0.7) ∧
(lives_in_town ∧ young → T(3granted _loan) ≥ 0.8) where the graded implication in the first conjunct is prefixed by a 2 operator, while the second one is not.
We will evaluate the satisfiability of a temporal graded formula at the initial time point 0 of a temporal preferential interpretation I. Let us first define the interpretation of temporal graded formulas at a time point n of a temporal interpretation I as follows:
I, n |= A → B ≥ l if (A → B)I,n ≥ l I, n |= A → B ≤ l if (A → B)I,n ≤ l I, n |= α ∧ β if I, n |= α and I, n |= β I, n |= ¬α if I, n ̸|= α I, n |= ⃝ α if I, n + 1 |= α I, n |= 3α if exists m ≥ n such that I, m |= α I, n |= 2α if for all m ≥ n, I, m |= α
I, n |= α U β if exists m ≥ n such that I, m |= β and, for all n ≤ k < m, I, k |= α Let us define the notions of satisfiability and entailment.
Definition 6 (Satisfiability and entailment) . A temporal graded formula α is satisfied in a temporal preferential interpretation I = ⟨W, {<nAi }n∈N, v⟩ if I, 0 |= α .
A preferential interpretation I = ⟨W, {<nAi }n∈N, v⟩ is a model of a temporal conditional knowledge base K, if I satisfies all the temporal graded formulas in K.
A temporal conditional knowledge base K entails a temporal graded formula α if α is satisfied in all the models I of K.
Observe that any graded implication A → B ≥ l is either satisfied or not at a time point n of a temporal interpretation I, i.e., either I, n |= A → B ≥ l or I, n ̸|= A → B ≥ l (and similarly for the graded implications A → B ≤ l). Hence, the interpretation above of temporal graded formulas in I at a time point n is two-valued (although it builds over the degree of an implication A → B in I at time point n, which has a truth value (A → B)I,n in D, see Definition 5).
Note that, in the temporal graded formula given above, the graded implication in the first conjunct (T(professor ) → teaches U retired ≥ 0.7) is required to hold at all the time points of the interpretation I (as it is prefixed by 2), while the second conjunct (lives_in_town ∧ young → T(3granted _loan) ≥ 0.8) has to hold only at time point 0.
Decidability and complexity of the diferent decision problems (the satisfiability, the model checking and entailment problems) have to be studied for this temporal many-valued conditional logic, for diferent choices of D and of combination functions. Satisfiability is decidable in the two-valued case, when we restrict to preference relations <Ai with respect to a finite number of formulas (for instance, by restricting to the formulas occurring in a finite KB, and to the respective preferences). Under such conditions, and assuming that all the temporal graded formulas in a KB are prefixed by the 2 operator, the propositional temporal logic with typicality introduced above can be regarded as a special case of LT LALC with typicality, which has been shown to be decidable in the two-valued case and for a finite number of preference relations [31].
As in the two-valued non-temporal case, the notion of preferential entailment considered in the previous
section is rather weak. For the KLM logics, diferent closure constructions have been proposed to
strengthen entailment by restricting to a subset of the preferential models of a conditional knowledge
base K. Let us just mention, the rational closure [
In the following we consider a construction that has been proposed for weighted knowledge bases in
defeasible description logics, where defeasible implications have a weight. We reformulate the semantics
of weighted KBs in [21, 22] in the propositional context, for the temporal case, by assuming that D is
the unit interval [
A weighted KB is a set of weighted typicality implication of the form (T(Ai) → Bj , wij ), where Ai and Bj are propositions, and the weight wij is a real number, representing the plausibility or implausibility of the conditional implication. For instance, for a proposition student , we may have a set of weighted defeasible implications: (T(student ) → has_Classes, +50) (T(student ) → 3holds_Degree,+30) (T(student ) → has_Boss, -40) that represent prototypical properties of students, i.e., that a student normally has classes and will eventually reach the degree, and she usually does not have a boss (negative weight). Accordingly, a student having classes, but not a boss, is more typical than a student having classes and a boss. Similarly, one may introduce a set of weighted conditionals for other formulas, e.g., for employee.
Based on the set of weighted conditionals for a formula Ai, one can constrain the preferences between worlds according to <Ai . For instance, consider an interpretation I = ⟨W, {<nAi }n∈N, v⟩ in which a world w describes a student (v(0, w, student ) = 1) that in the initial state has classes (v(0, w, has_Classes) = 1) but not a boss (v(0, w, has_Boss) = 0), and that at time point 8 will reach the degree (v(8, w, hold _Degree) = 1) ; while world w′ describes a student (v(0, w′, student ) = 1) that in the initial state has classes (v(0, w′, has_Classes) = 1) and has a boss (v(0, w′, has_Boss) = 1), and will reach the degree at time point 7 (v(7, w′, hold _Degree) = 1.
The idea is that the preference relation <s0tudent in I should consider the situation described at w at time point 0, more normal than the situation described by w′ (i.e., w <s0tudent w′), as the sum of the weights of the defeasible implications satisfied by world w at time point 0 (50 + 30 = 80) is greater than the sum of the weights of the defeasible implications satisfied by world w′ (50 + 30 − 40 = 40) at time point 0.
We have to further consider that the propositions may be non-crisp, e.g., v(0, w, has_Classes) = 0.7, and this has some impact on the degree to which a conditional implication (e.g., T(student ) → has_Classes), is satisfied.
Given a weighted knowledge base K, we call distinguished propositions those propositions Ai such that at least a weighted defeasible implications of the form (T(Ai) → Bj , wij ) occurs in K.
Let K be a temporal weighted KB. Given a many-valued temporal interpretation I = ⟨W, {<nAi }n∈N, v⟩, the weight of a world x ∈ W with respect to a distinguished proposition Ai at time point n is given by
WAIi,n(x) = P(T(Ai)→Bj,wij)∈K wij · v(n, x, Bj ).
Intuitively, the higher the value of WiI,n(x), the more normal is the state of afairs x, at time point n, concerning the properties of A in K. This constrains the preference relation <Ai in I. We extend the coherent and faithful multi-preferential semantics for weighted knowledge bases to the temporal case: Definition 7. A many-valued temporal preferential interpretation I = ⟨W, {<nAi }n∈N, v⟩ satisfies a weighted KB K if, for all distinguished formulas Ai and time points n, it holds that: x <nAi y ⇐⇒
WiI,n(x) > WiI,n(y)
The condition in Definition 7, together with the coherence (faithfulness) condition introduced in Section 2, guarantees that the many-valued interpretation I agrees with the weighted inclusions in K, at each time point n.
A weighted (defeasible) knowledge base KD can coexist with a strict knowledge base KS (i.e., a set of temporal graded formulas). This is in agreement with the usual approach in defeasible DLs, which distinguishes between a strict TBox and a defeasible TBox.
In previous sections, we have developed a many-valued, temporal logic with typicality, extending with
LT L operators the many-valued conditional logic with typicality proposed in [
The idea in [
For the definition of an argumentation graph, let us adapt the notion of edge-weighted QBAF by Potyka [24] to a generic domain D. A (weighted) argumentation graph is a quadruple G = ⟨A, R, σ 0, π ⟩, where A is a set of arguments, R ⊆ A × A a set of edges, σ 0 : A → D assigns a base score of arguments, and π : R → R is a weight function assigning a positive or negative weight to edges (when the graph is not weighted, we assume function π only distinguishes between attacks and supports).
A many-valued labelling σ of G over D is a function σ : A → D, which assigns to each argument
an acceptability degree (or a strength) in the domain of argument valuation D (for Ai ∈ A, σ (Ai) is
the acceptability degree of argument Ai in σ ). Whatever gradual semantics S is considered for the
argumentation graph G (see, e.g., [
If we consider all arguments Ai ∈ A as propositional variables (P rop = A), each many-valued labelling σ can be regarded as a world wσ ∈ W in a many-valued preferential interpretation MG = ⟨W, {<Ai }, v⟩, such that v(wσ , Ai) = σ (Ai).
More precisely, in [
The approach can be extended to the temporal case, based on the temporal many-valued logic with typicality developed in Section 3. It can allow to reason about the dynamics of an argumentation graph, for instance, when the weights of edges might change in time, e.g. when learning the weights. Indeed, a multilayer neural network can be regarded as a weighted knowledge base [26, 22], but also as a weighted argumentation graph [24, 25], based on the strong relationships of the two formalisms [23]. As another example, the structure of an argumentation graph can be updated through the interaction of diferent agents in time, such as in [ 27] via Argumentative Exchanges, giving rise to a sequence of argumentation graphs Gn (over the same set of arguments A).
When the argumentation graph changes dynamically, a temporal many-valued interpretation I = ⟨W, {<nAi }n∈N, v⟩ can be seen as a sequence of (non-temporal) preferential interpretations M0, M1, M2, . . . (as in Section 3), where each Mn = ⟨W, {<nAi }, vn⟩ is constructed from the labellings Σ nS of the argumentation graph Gn at time point n, as for MG above. Temporal graded formulas over arguments, e.g., 2(T(A1 ) → A2U A3 ∨ A3) ≥ 0.7, can then be verified over I.
As mentioned above, this verification approach has been studied, for the non-temporal case, in the verification of properties of argumentation graphs under the φ-coherent gradual semantics [20], and an ASP approach has been developed for the verification of graded conditional implications over arguments and over boolean combination of arguments. Extending the ASP approach to deal with the temporal case, for specific fragments of the language, is a direction for future work.
The paper proposes a framework in which diferent (many-valued) preferential logics with typicality can be captured, together with their temporal extensions, with LTL operators. The interpretation of the typicality operator is based on a multi-preferential semantics, and an extension of weighted conditional knowledge bases to the temporal (many-valued) case is proposed.
The approach is parametric with respect to the choice of a specific many-valued logic (and their combination functions), but also with respect to the definition of the preference relations <Ai , which might exploit diferent closure constructions, among the many studied in the literature, in the spirit of Lehmann’s lexicographic closure [45]. The two-valued case, with a single preference relation can as well be regarded as a special cases of this preferential temporal formalism.
On a diferent route, a preferential logics with defeasible LTL operators has been studied in [ 29, 46]. The decidability of diferent fragments of the logic has been proven, and tableaux based proof methods for such fragments have been developed [28, 46]. 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 logic. In our approach, preferences between worlds change over time.
Much work has been recently devoted to the combination of neural networks and symbolic reasoning [47, 48, 49]. 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) [ 21, 22, 23], 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 behavior of a network, based on a model checking approach or an entailment based approach.
Extending the above mentioned ASP encodings to deal with 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, and for reasoning about the dynamics of gradual argumentation graphs in gradual semantics.
We thank the anonymous referees for their helpful comments. This research was partially supported by INDAM-GNCS. 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). semantics, in: 5th Workshop on Advances in Argumentation in Artif. Intell., 2021, Milan, Italy, Nov. 29, volume 3086 of CEUR Workshop Proc., 2021. [20] M. Alviano, L. Giordano, D. Theseider Dupré, Preferential reasoning with typicality in ASP over weighted argumentation graphs in a gradual semantics, in: ASPOCP 2023, Proc. ICLP Workshops 2023, CEUR Workshop Proceedings 3437, 2023. [21] L. Giordano, D. Theseider Dupré, Weighted defeasible knowledge bases and a multipreference semantics for a deep neural network model, in: Proc. JELIA 2021, May 17-20, volume 12678 of LNCS, Springer, 2021, pp. 225–242. [22] M. Alviano, F. Bartoli, M. Botta, R. Esposito, L. Giordano, D. Theseider Dupré, A preferential interpretation of multilayer perceptrons in a conditional logic with typicality, Int. Journal of Approximate Reasoning 164 (2024). [23] M. Alviano, L. Giordano, D. Theseider Dupré, Weighted knowledge bases with typicality and defeasible reasoning in a gradual argumentation semantics, Intelligenza Artificiale 18 (2024). [24] N. Potyka, Interpreting neural networks as quantitative argumentation frameworks, in: ThirtyFifth AAAI Conference on Artificial Intelligence, AAAI 2021, February 2-9, 2021, AAAI Press, 2021, pp. 6463–6470. [25] A. S. d’Avila Garcez, K. Broda, D. M. Gabbay, Symbolic knowledge extraction from trained neural networks: A sound approach, Artif. Intell. 125 (2001) 155–207. [26] L. Giordano, From weighted conditionals of multilayer perceptrons to a gradual argumentation semantics, in: 5th Workshop on Advances in Argumentation in Artif. Intell., volume 3086 of CEUR Workshop Proceedings, 2021. [27] A. Rago, H. Li, F. Toni, Interactive explanations by conflict resolution via argumentative exchanges, in: Proc. 20th Int. Conf. on Principles of Knowledge Representation and Reasoning, 2023, pp. 582–592. [28] 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. [29] 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. [30] A. Chafik, Defeasible temporal logics for the specification and verification of exception-tolerant systems, PhD Thesis, Artois University, 2022. [31] M. Alviano, L. Giordano, D. Theseider Dupré, Preferential temporal description logics with typicality and weighted knowledge bases, in: Proc. 38th Italian Conf. on Computational Logic, Udine, Italy, June 21-23, 2023, volume 3428 of CEUR Workshop Proceedings, 2023. [32] C. Lutz, F. Wolter, M. Zakharyaschev, Temporal description logics: A survey, in: TIME, 2008, pp.
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