In this paper we develop a preferential interpretation of gradual argumentation and propose an approach for temporal conditional reasoning about argumentation graphs. The approach exploits a two-valued temporal conditional logic with typicality, combining a preferential logic with Linear Time Temporal Logic (LTL). It introduces a dynamic dimension to conditional reasoning in gradual argumentation, enabling the verification of conditional properties across time, such as trends in the evolution of argument strength.
1. Conceptual simplification: By moving from a many-valued to a two-valued framework, we retain expressive power while simplifying the formal semantics and reasoning mechanisms. 2. Temporal extension: We introduce a dynamic dimension to conditional reasoning in gradual argumentation, enabling the verification of conditional properties across time, such as trends in the evolution of argument strength.
The proposed logic can be used to analyze time-dependent properties of argumentation graphs, including iterative updates of strength functions, and is also applicable to weighted knowledge bases and neural networks interpreted as weighted argumentation frameworks. This opens up new opportunities for explainability, temporal verification, and symbolic-neural integration.
In more detail, our approach combines preferential approaches to commonsense reasoning [14, 15, 16, 17, 18, 19, 20, 21, 22] with the Linear Time Temporal Logic (LTL) [23]. Preferential extensions of LTL with defeasible temporal operators have been recently studied to enrich temporal formalisms with non-monotonic reasoning features, by considering defeasible versions of the LTL operators [24, 25, 26]. In this regard, we follow a diferent route, adding the standard LTL operators to a conditional logic with typicality, an approach similar to the one pursued for Description Logics (DLs), through an extension of the temporal description logic LTLℒ [27] with the typicality operator [28]. As in the Propositional Typicality Logic (PTL) by Booth et al. [29] (and in the DLs with typicality [30]) the conditionals will be formalized based on material implication plus a typicality operator T. The typicality operator allows for the definition of conditional implications T( ) → , meaning that “normally if holds, holds". They correspond to conditional implications |∼ in KLM logics [17, 19]. In the paper, we will consider a multi-preferential logic, where preferences are associated to aspects (and to arguments).
The structure of the paper is as follows. Section 2 provides the necessary background on gradual argumentation and two-valued multi-preferential conditional logic. In Section 3, we show how such logic can be instantiated for reasoning on argumentation graphs. Section 4 introduces the temporal extension of the logic, and Section 5 describes how it can be instantiated to model temporal aspects of gradual argumentation. Section 6 concludes and discusses future directions.
In this section we provide preliminaries on gradual argumentation semantics and on a two-valued multi-preferential semantics for conditionals.
In this following, we shortly recap gradual argumentation semantics, following Baroni, Rago and Toni [6, 7], and consider a specific semantic from [31, 32, 13].
As in the Quantitative Bipolar Argumentation Framework (QBAF) by Baroni et al. [6, 7], we let the domain of argument interpretation be a set , equipped with a preorder relation ≤ . In the literature, this assumption is considered general enough to include the domain of argument valuations in most gradual argumentation semantics [3, 33, 4, 34, 5, 6, 8, 11]. We do not assume that contains a minimum element and a maximum element. If they exist, we denote them by 0 and 1 (or simply 0 and 1), respectively. If not, we will add the two elements 0 and 1 at the bottom and top of the values in , respectively. For the definition of an argumentation graph, we consider the definition of edge-weighted QBAF by Potyka [11], for a generic domain .
We let a weighted argumentation graph to be a quadruple = ⟨, ℛ, 0, ⟩, where is a set of arguments, ℛ ⊆ × a set of edges, the base score function 0 : → assigns a base score to arguments, and : ℛ → R is a weight function assigning a positive or negative weight to edges. An example of argumentation graph is in Figure 1, where a base score is not given.
A ir (, ) ∈ ℛ is regarded as a support of argument to argument when the weight (, ) is positive and as an attack of argument to argument when (, ) is negative. argument valuation .
The properties of edge-weighted argumentation graphs with weights in the interval [
Whatever semantics is considered for an argumentation graph , we will assume that the semantics identifies a set Σ of many-valued labellings (also called strength functions, or weightings) of the graph over a domain of argument valuation . A many-valued labelling of over is a total function : → , which assigns to each argument an acceptability degree (or a strength) in the domain of
When, in the following, we want to consider a set of possible values for the initial score of Σ 0 = { 01, 02, . . . , 0} is a finite set of possible initial scores . arguments, we will represent the argumentation graph as a triple = (, ℛ, Σ 0, ), where
argument has neither supports nor attacks.
Let us recall the definition of the -coherent semantics of an argumentation graph [13]. We let
be the interval [
For a weighted graph = ⟨, ℛ, 0, ⟩ and a many-valued labelling , the weight on is defined as a partial function : → R, assigning a positive or negative support (relative to labelling () =
∑︁ ∈− () ( , ) ( ) () is left undefined when R− (Ai ) = ∅. : R → , a -coherent many-valued labelling of is defined as follows:
Given a weighted argumentation graph = ⟨, ℛ, 0, ⟩ and a non-decreasing function () = ︂{ ( ()) 0() otherwise for all Ai ∈ s.t. R− (Ai ) ̸= ∅ initial score in Σ 0.
The semantics is a perceptron-like semantics which has been inspired by some preferential semantics for weighted KBs developed in [35, 10], and has some relations with the semantics proposed by Potyka [11] for interpreting neural networks (see [13] for comparisons). In this perceptron like view, the argumentation graph plays the role of a (possibly recurrent) multilayer network, where arguments correspond to units, edges (with their weights) correspond to synaptic connections between units, () corresponds to the induced local field of unit , and corresponds to the activation function. The acceptability degree of an argument in a labelling corresponds to the activity of unit in a stationary state of the network. We refer to [9, 13] for further details, including properties of the semantics and comparisons with other argumentation semantics.
We denote by Σ the set of all the -coherent many-valued labelling of = ⟨, ℛ, ⟩, for all the possible choices of the initial score; by Σ 0 the set of the -coherent many-valued labelling for the initial score 0, and by Σ Σ0 the set of the -coherent many-valued labelling for all the values of Example 1. In the -coherent semantics for the weighted argumentation graph in Figure 1, in the ifnitely-valued case with
= , for = 5, with being the approximation in of the logistic function, Σ contains 36 many-valued -coherent labellings, while, for = 9, Σ contains 100 -coherent labellings. In this case, there is a labelling for each combination of values of the base score for 2 and 6. For instance,
= (3/5, 0, 3/5, 3/5, 1/5, 0) (meaning that (1) = 3/5, (2) = 0, and so on) is a many-valued -coherent labelling of for = 5. (1) (2)
Above, the notion of -coherent many-valued labelling of is defined through a set of equations, as in Gabbay’s equational approach to argumentation networks [36]. A definition of the -coherent semantics can also be given, in the style of the gradual semantics in the framework by Amgoud and Doder [8]. We refer to [13] for details and for a discussion of the properties of the semantics. In particular, one cannot assume that, for an initial score 0, there is a unique strength function (a unique -coherent labelling).
In the following we recall the multi-preferential semantics from [37], and slightly extend it.
As mentioned in the introduction, the conditional logic extends a propositional language with a typicality operator T, following the approach used in the description logic ℒ + T [38] as well as in the Propositional Typicality Logic (PTL) [29]. Intuitively, “a sentence of the form T( ) is understood to refer to the typical situations in which holds" [29]. As in PTL [29], the typicality operator cannot be nested. When an implication has the form T( ) → , it is called a defeasible implication, whose meaning is that “normally, if then ”. An implication → is called strict, if it does not contain occurrences of the typicality operator.
The KLM preferential semantics [17, 19, 16] exploits a single preference relation between worlds: a set of worlds , with their valuation and a preference relation < among worlds (where < ′ means that world is more normal than world ′). A conditional |∼ is satisfied in a KLM preferential interpretation, if holds in all the most normal worlds satisfying , i.e., in all <-minimal worlds satisfying . Here, instead, we consider a multi-preferential semantics, where preference relations are associated with distinguished propositional formulas 1, . . . , (called distinguished propositions in the following). In the semantics, a preference relation will be associated with each distinguished proposition , where < ′ means that world is less atypical than world ′ concerning aspect/property (e.g., <student ′ means that describes a less atypical situation for a student than ′).
In the following we will consider finite KBs over a set Prop of propositional variables, and a finite set of distinguished propositions 1, . . . , (propositional formulas over Prop). Preferential interpretations are equipped with a set of worlds and a finite set of preference relations <1 , . . . , < , where, for each distinguished proposition , < is a strict partial order on the set of worlds . For the moment, we assume that, in any typicality formula T(), is a distinguished proposition.
A (multi-)preferential interpretation is a triple ℳ = ⟨ , {< }, ⟩ where: ∙ is a non-empty set of worlds; ∙ each < ⊆ × is an irreflexive and transitive relation on ; ∙ : →− 2Prop is a valuation function, assigning to each world a set of propositional variables in Prop.
A ranked interpretation is a (multi-)preferential interpretation ℳ = ⟨ , {< }, ⟩ for which all preference relations < are modular, that is: for all , , , if < then < or < . A relation < is well-founded if it does not allow for infinitely descending chains of worlds 0, 1, 2, . . ., with 1 < 0, 2 < 1, etc. The valuation is inductively extended to all formulae: ℳ, |= ⊤
ℳ, ̸|= ⊥ ℳ, |= if ∈ (), for all ∈ Prop ℳ, |= ∧ if ℳ, |= ∨ if ℳ, |= ¬ if ℳ, |= → if ℳ, |= T() if ℳ, |= and ℳ, |= ℳ, |= or ℳ, |= ℳ, ̸|=
ℳ, |= implies ℳ, |= ℳ, |= and ∄w ′ ∈ s.t. w ′ <Ai w and ℳ, ′ |= .
Whether T() is satisfied at a world or not also depends on the other worlds of the interpretation ℳ. Restricting our consideration to modular interpretations, leads to the notions of satisfiability and validity of a formula in the ranked (or rational) multi-preferential semantics. Diferently from [ 37] (and from KLM semantics [17, 19]), here we do not assume well-foundedness of the preference relations.
An implication of the form T() → , with in ℒ, corresponds to a conditional |∼ in KLM logics [17]. It can be easily proven that, when all the preference relations < coincide with a single well-founded preference relation <, a multi-preferential interpretation ℳ corresponds to a KLM preferential interpretation, and a defeasible implication T() → (with and in ℒ) has the semantics of a KLM conditional |∼ . The multi-preferential semantics is a generalization of the KLM preferential semantics.
Given a preferential interpretation ℳ, a formula is satisfied in ℳ if ℳ, |= for some world ∈ . A formula is valid in ℳ (written ℳ |= ) if ℳ, |= , for all the worlds ∈ . A formula is valid if is valid in all the preferential interpretations ℳ.
Let a knowledge base be a set of (strict or defeasible) implications. A preferential model of is a multi-preferential interpretation ℳ such that ℳ |= → , for all implications → in . Given a knowledge base , we say that an implication → is preferentially entailed from if ℳ |= → holds, for all preferential models ℳ of . We say that → is rationally entailed from if ℳ |= → holds, for all ranked models ℳ of .
It is well known that preferential entailment and rational entailment are weak. As with the rational closure [19] and the lexicographic closure [39] for KML conditionals, also in the multi-preferential case one can strengthen entailment by restricting to specific preferential models, based on some closure constructions, which allow to define the preference relations < from a knowledge base , e.g., by exploiting the ranks or weights of conditional implications, when available [40, 41, 10].
In [
As we have seen, the semantics of can then be regarded, abstractly, as a pair (, Σ ): a domain of argument valuation and a set of labellings Σ over the domain.
If we consider the set of arguments as propositional variables, each labelling can be regarded as a world ∈ in a many-valued preferential interpretation ℳ which contains a preference relation < , for each argument (here we are assuming that the single arguments 1, . . . , correspond to the distinguished propositions).
More precisely, a gradual semantics (, Σ ) of an argumentation graph can be associated with a preferential interpretation ℳ = ⟨, {<1 , . . . , < }, ⟩, defined by letting: - = { | ∈ Σ } - for all the arguments ∈ , and a threshold value ∈ : ( , ) = ︂{ 0 1 if (Ai ) ≤ t otherwise (3) - for all the arguments ∈ , and worlds , ′ ∈ :
< ′ if () > ′()
The choice of the threshold depends on the semantics and on the domain. For instance, for the domain
= [
Example 2. For instance, referring to the argumentation graph in Example 1, the labelling (strength function) = (3/5, 0, 3/5, 3/5, 1/5, 0) gives rise to a world in the preferential interpretation ℳ of the graph. Assuming a threshold = 2/5, we have ( , 1) = ( , 3) = ( , 4) = 1 (as (1) > , (3) > and (4) > ), while ( , 2) = ( , 5) = ( , 6) = 0.
Furthermore, for a labelling ′ ∈ Σ such that ′(3) = 4/5, we will have: ′ <3 , as ′(3) > (3). That is, labelling ′ represents a more typical situation in which argument 3 holds, with respect to labelling .
Once a preferential interpretation ℳ of an argumentation graph with respect to a gradual semantics , has been constructed, such interpretation can be used in the verification of strict and conditional graded implications (by checking their validity in the model ℳ), for explanation, e.g., by validating conditional relations between arguments.
For instance, given a weighted argumentation graph describing the rules for assigning loans, e.g., involving the arguments living _in_town, young and granted _loan, and a gradual semantics , one may want to verify the property
T(granted _loan) → living _in_town ∧ young (normally the loan is granted to people living in town and being young) or the property living _in_town ∧ young → T(granted _loan) (living in town and being young implies that normally the loan is granted). The implications above can be checked for validity over the preferential interpretation ℳ, constructed from the set of labellings Σ of the graph in the semantics .
Let us continue the example concerning the argumentation graph in Figure 1, under the -coherent argumentation semantics.
Example 3. As mentioned before, for the weighted argumentation graph in Figure 1, in the -coherent argumentation semantics there are 36 labellings in case of a domain with = 5. Since 2 supports 3, which in turn attacks 5, some relation can be expected between 2 and 5. Assuming a threshold = 2/5, the following conditional implication turns out to be valid in the interpretation ℳ: T(2) → ¬5
T(1 ∨ 2) → ¬5 that is, in the situations (labellings) which maximize the acceptability of argument 2, argument ¬5 holds. The corresponding strict implication 2 → ¬5 does not hold.
In this example, we may wonder whether model ℳ also validates the implication: It turns out, however, that the last conditional implication is outside the language we have defined.
So far we have assumed that the distinguished propositions correspond to single arguments. We lift this condition and allow typicality formulas T(), with a boolean combination of arguments, to include typicality formulas as T(1 ∨ 2) in the example above, and also the typicality formula in the conditional implication:
T(living _in_town ∧ young ) → granted _loan To deal with these conditionals, we have to extend preferential interpretations by allowing preference relations < associated with boolean combinations of arguments , and generalizing the semantic condition of typicality formulas (in Definition 2) to any boolean combination of arguments , as follows: ℳ, |= T() if
ℳ, |= and ∄w ′ ∈ s.t. w ′ <A w and ℳ, ′ |= For instance, for evaluating T(1 ∨ 2) → ¬5, we need preference relation <1∨2 .
The definition of of the preference relation < for a boolean combination of arguments can rely on the strength of the atomic arguments in the gradual semantics. The strength of boolean arguments in a labelling can be defined inductively from the strength of the atomic arguments in the gradual semantics by exploiting suitable truth degree functions ⊗ , ⊕ , ⊖ in , as follows: (¬) = ⊖ () ( ∧ ) = () ⊗ () ( ∨ ) = () ⊕ () where and are boolean combinations of arguments.
When is [
Then, the preference relation < can be defined as:
< ′ if () > ′() for all worlds , ′ ∈ .
We can reconsider the previous example, based on this generalization of the preferential semantics. Example 4. Based on the choice of truth degree functions above, one can prove that the conditional implication T(1 ∨ 2) → ¬5 is valid in the preferential interpretation of the argumentation graph in Figure 1 under the -coherent semantics, while the strict implication 1 ∨ 2 → ¬5 is not valid.
Note that the idea of interpreting the strength function as a valuation in a many-valued logic, was
previously used in many-valued preferential interpretations [
In this section, we extend the two-valued conditional logic in Section 2.3 with the operators of the Linear Time Temporal Logic (LTL) [23].
Compared with the preferential semantics above, the semantics of T also considers the temporal dimension, through a set of time points in N. The valuation function assigns, at each time point ∈ N, a truth value to each propositional variable in a world ∈ ; the preference relations < (with respect to each distinguished proposition ) are relative to time points. Evolution in time may change the valuation of propositions at the worlds, and it may also change the preference relations between worlds ( might represent a typical situation for a student at time point 0, but not at time point 50).
A temporal (multi-)preferential interpretation (or T interpretation) is a triple ℐ =
⟨, {< }∈N, ⟩ where: • is a non-empty set of worlds; • for each and ∈ N, < ⊆ × is an irreflexive and transitive relation on ; • : N × →− 2Prop is a valuation function assigning, at each time point , a set of propositional variables in Prop to each world ∈ .
For ∈ and ∈ N, (, ) is the set of the propositional variables which are true in world at time point . If there is no ′ ∈ s.t. ′ < , we say that is a normal situation for at time point .
Given an T interpretation ℐ = ⟨, {< }∈N, ⟩, we define inductively the truth value of a formula in a world at time point (written ℐ, , |= ), as follows: ℐ, , |= ⊤
ℐ, , ̸|= ⊥ ℐ, , |= if ∈ (, ), for all ∈ Prop ℐ, , |= ∧ if ℐ, , |= and ℐ, , |= ℐ, , |= ∨ if ℐ, , |= or ℐ, , |= ℐ, , |= ¬ if ℐ, , ̸|= ℐ, , |= → if ℐ, , |= implies ℐ, , |= ℐ, , |= if ℐ, + 1, |= ℐ, , |= ◇ if there is an ≥ such that ℐ, , |= ℐ, , |= □ if for all ≥ , ℐ, , |= ℐ, , |= if there is an ≥ such that ℐ, , |= and,
for all such that ≤ < , ℐ, , |= ℐ, , |= T() if ℐ, , |= and ∄w ′ ∈ s.t. w ′ <nAi w and ℐ, , ′ |= . Note that whether a world represents a typical situation for at a time point depends on the preference between worlds at time point .
A temporal conditional KB is a set of T formulas. We evaluate the satisfiability of a temporal graded formula at the initial time point 0 of a temporal preferential interpretation ℐ. Definition 4. An T formula is satisfied in a temporal preferential interpretation ℐ = ⟨, {< }∈N, ⟩ if ℐ, 0, |= , for some world ∈ . An T formula is valid in a temporal preferential interpretation ℐ = ⟨, {< }∈N, ⟩ if ℐ, 0, |= , for all worlds ∈ . An T formula is valid, if is valid in all temporal preferential interpretations ℐ. An T formula is satisfiable , if is satisfied in some temporal preferential interpretation ℐ.
It can be shown that the problem of deciding the satisfiability of an T formula can be polynomially reduced to the problem of deciding the satisfiability of a concept in the description ltohgeictypicalTitℒyoipnetrraotdour.cedinT[28],hwashbicehenexptreonvdesnthtoe bteemdpecoirdaalbdleeswcrhipetnioanfinliotegiscetofweℒll-f[o2u7n]dweidth ℒ preference relations <1 , . . . , < is considered, and concept inclusions are regarded as global temporal constraints. In turn, the decidability of concept satisfiability in T relies on the result that concept satisfiability for LTLℒ w.r.t. TBoxes is in ExpTime (and, actually,iℒtis ExpTime-complete), both with expanding domains [43] and with constant domains [27].
As for the non-temporal case, we aim at instantiating the two-valued temporal conditional logic introduced in the previous section to the gradual argumentation setting, to make it suitable for capturing the dynamics of strength functions in time.
In Amgoud and Doder’s framework of gradual semantics [8], it is proven that a uniform iterative way of calculating strengths of arguments can be applied to any semantics based on well-defined evaluation methods, for which convergence is guaranteed.
In the following, we consider an iterative formulation of the -coherent semantics, and describe a possible construction of a temporal interpretation for it. In particular, starting from an initial score 0, one can iteratively define a sequence of labellings 0, 1, 2, ..., as follows: () = ︂{ 0() ( − 1 ()) if R− (Ai ) = ∅ if R− (Ai ) ̸= ∅ (4) In the general case, for the -coherent semantics one cannot guarantee that the sequence 0, 1, 2, ..., starting from an initial score 0, converges to some -coherent labelling, unless the argumentation graph is acyclic. Convergence conditions for edge-weighted QBAFs have been studied by Potyka, both in the discrete and in the continuous case [11].
Although the sequence of labellings 0, 1, 2, ..., may not converge, one may be interested in verifying temporal properties over the sequence of labellings (or a finite stretch of it). More generally, one may be interested in considering a finite set Σ 0 of possible initial score functions { 01, 02, . . . , 0}. The associated sequences 0 , 1 , 2 , ..., one for each initial score function 0 , determine a set of runs, from which a temporal preferential interpretation ℐ of the argumentation graph can be constructed.
In the sequence of labellings 0 , 1 , 2 , ... obtained from each initial score function 0 ∈ Σ 0, the labelling is the one obtained from 0 at iteration .
Let the set of arguments be the set of propositional variables of the temporal conditional logic. We can build a temporal preferential interpretation ℐ = ⟨, {< }∈N, ⟩ of an argumentation graph , from a set of initial score functions Σ 0, by introducing a world ∈ for each initial score function 0 in Σ 0. The valuation of propositions in a world at time point , will be determined by the strength () of the argument in the labelling .
More precisely we define a temporal preferential interpretation ℐ = ⟨, {< }∈N, ⟩ of an argumentation graph , with respect to Σ 0, by letting: - = { | 0 ∈ Σ 0} - for all the arguments ∈ , and a threshold value ∈ : - for all the arguments ∈ , and worlds ℎ, ∈ :
∈ (, ) if () > ; < ℎ if () > ℎ().
Note that a temporal many-valued interpretation ℐ = ⟨, {< }∈N, ⟩ can be seen as a sequence of (non-temporal) preferential interpretations ℳ0, ℳ1, ℳ2, . . ., where each ℳ = ⟨, {< }, ⟩ is constructed from all the labelling Σ of the argumentation graph at the iteration (one for each initial score function in Σ 0), as for ℳ above.
Once a temporal preferential interpretation ℐ has been constructed, the validity of temporal conditional formulas over arguments, such as □(T(A1 ) → 2 A3 ∨ 3), can be verified over the constructed preferential interpretation ℐ. In the loan example, for instance, one may want to check whether normally, young people leaving in town are eventually granted a loan, i.e., T(living _in_town ∧ young ) → ◇granted _loan.
For boolean combinations of arguments in typicality formulas, the non-temporal solution from Section 3.1 extends naturally to the temporal case. Moreover, the approach described here is not limited to the -coherent semantics, but can also be adapted to other gradual argumentation semantics with iterative formulations, such as those in [8, 11].
In [44], we introduced a many-valued temporal logic with typicality by extending the many-valued
conditional logic of [
A particularly promising application lies in the analysis of neural networks. Indeed, a multilayer neural network can be interpreted as a weighted knowledge base [9, 10] or as a weighted argumentation graph [12, 11], given the strong semantic relationships also explored in [13]. In the many-valued setting, conditional weighted KBs have already been shown to capture the stationary states of such networks (or suitable approximations thereof) [35, 10, 13], supporting post-hoc symbolic reasoning and verification. The addition of a temporal dimension, as proposed here, opens the way to verifying properties of dynamic behavior — such as how activations evolve, how information propagates, or how explanatory patterns shift over time. From a logical perspective, our approach ofers an alternative to other defeasible temporal logics that enrich temporal operators directly, such as those studied in [25, 45]. While those frameworks include tableaux methods and address decidability of fragments with defeasible temporal connectives [24], our logic maintains standard operators and encodes defeasibility entirely through temporal evolution of preferential structures (namely, by allowing preference relations among worlds to change over time). This yields a clean separation between temporal and nonmonotonic aspects, which may facilitate integration with other symbolic reasoning tools. Finally, our work contributes to neuro-symbolic integration [46, 47, 48] by providing a principled framework that links typicality-based conditional logic with neural models in a temporally dynamic and semantically transparent way.
Directions for further research include: developing ASP encodings for the temporal logic proposed here; studying the decidability and complexity of fragments of the logic; investigating convergence and stability in iterative dynamics; applying the framework to explainability of evolving argumentation graphs and time-dependent neural-symbolic systems.
This research was partially supported by INDAM-GNCS. Mario Alviano was partially supported by the 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 the Italian Ministry of Health (MSAL) under POS projects CAL.HUB.RIA (CUP H53C22000800006) and RADIOAMICA (CUP H53C22000650006); by the Italian Ministry of Enterprises and Made in Italy under project STROKE 5.0 (CUP B29J23000430005); under PN RIC project ASVIN “Assistente Virtuale Intelligente di Negozio” (CUP B29J24000200005); and by the LAIA lab (part of the SILA labs). Mario Alviano is member of Gruppo Nazionale Calcolo Scientifico-Istituto Nazionale di Alta Matematica (GNCS-INdAM).
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