In this paper, we develop a many-valued semantics for the description logic ℒ, a temporal extension of description logic ℒ, 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. 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 ℒ [3] have been studied [4, 5, 6], and several diferent 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 diferent 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 ℒ, LTLℒ [17], based on Linear Time Temporal Logic (LTL), to develop a temporal ℒ with typicality, LTLTℒ . Generalizing the approach in [5], a typicality operator T (that selects the most typical instances of a concept) is added to LTLℒ to represent temporal properties of concepts which admit exceptions. It is proven that the preferential extension of LTLTℒ can be polynomially encoded into LTLℒ , and this approach allows borrowing decidability and complexity results from LTLℒ . A similar encoding
can be given for a multi-preferential extension of LTLTℒ, by allowing a concept-wise preferential semantics, where diferent preferences are associated to diferent concepts.
In this paper, we aim at developing a many-valued extension of LTLℒ 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 ℒ and temporal extensions of ℒ. Then, we
develop a many-valued extension of LTLℒ, by building on many-valued DLs, which are widely
studied in the literature, both for the fuzzy case [
We discuss extensions of the closure constructions for weighted knowledge bases with typicality
[
Let be a set of concept names, a set of role names and a set of individual names. The set of ℒ concepts (or, simply, concepts) can be defined inductively as follows: (i) ∈ , ⊤ and ⊥ are concepts; (ii) if and are concepts, then ∈ , then ⊓ , ⊔ , ¬, ∀., ∃. are concepts.
A fuzzy interpretation for ℒ is a pair = ⟨∆ , · ⟩ where: ∆ is a non-empty domain and · is fuzzy
interpretation function that assigns to each concept name ∈ a function : ∆ → [
The interpretation function · is extended to complex concepts as follows: ⊤ () = 1, ⊥ () = 0, (¬) () = ⊖ (), ( ⊓ ) () = () ⊗ (), ( ⊔ ) () = () ⊕ (), (∃.) () = sup∈Δ (, ) ⊗ (), (∀.) () = inf∈Δ (, ) ▷ (), where ∈ ∆ , 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 ⊗ = {, }, ⊕ = {, }, ▷ = 1 if ≤ and otherwise; ⊖ = 1 if = 0 and 0 otherwise. In Łukasiewicz logic, ⊗ = { + − 1, 0}, ⊕ = { + , 1}, ▷ = {1 − + , 1} and ⊖ = 1 − . Following [25], we will not commit to a specific choice of combination functions,
A fuzzy ℒ knowledge base is a pair ( , ) where is a fuzzy TBox and a fuzzy ABox. A
fuzzy TBox is a set of fuzzy concept inclusions of the form ⊑ , where ⊑ is an ℒ concept
inclusion axiom, ∈ {≥ , ≤ , >, <} and ∈ [
If |= Γ , we say that satisfies Γ or that is a model of Γ (for Γ being an axiom, a set of axioms, or a KB). An axiom is entailed by , written |= , if |= holds for all models of .
For the finitely many-valued case, we assume the truth space to be = {0, 1 , . . . , − 1 , }, for an integer ≥ 1 [28, 29, 30]. In the following, we will use ℒ to refer to a finitely-valued extension of ℒ interpreted over the truth space , without committing to a specific choice of combination functions.
ℒ
The temporal Description Logic ℒ is a temporal extension of ℒ based on linear time temporal
logic (LTL) The concepts of ℒ can be formed by adding to the constructors of ℒ the temporal
operators ○ (next), (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. [
The set of temporally extended concepts is the following:
::= | ⊤ | ⊥ | ⊓ | ⊔ | ¬ | ∃. | ∀. | ○ | | ♢ | □ where ∈ , and and are temporally extended concepts.
A temporal interpretation for ℒ is a pair ℐ = (∆ ℐ , · ℐ ), where ∆ ℐ is a nonempty domain; · ℐ is
an extension function that maps each concept name ∈ to a set ℐ ⊆ N × ∆ ℐ , each role name
∈ to a relation ℐ ⊆ N × ∆ ℐ × ∆ ℐ , and each individual name ∈ to an element ℐ ∈ ∆ ℐ .
Following [
(¬)ℐ = (N × ∆ ℐ )∖ℐ ( ⊔ )ℐ = ℐ ∪ ℐ (∃.)ℐ = {(, ) ∈ N × ∆ ℐ | ∃.(, , ) ∈ ℐ and (, ) ∈ ℐ } (∀.)ℐ = {(, ) ∈ N × ∆ ℐ | ∀.(, , ) ∈ ℐ ⇒ (, ) ∈ ℐ } (○ )ℐ = {(, ) ∈ N × ∆ ℐ | ( + 1, ) ∈ ℐ } (♢ )ℐ = {(, ) ∈ N × ∆ ℐ | ∃ ≥ such that (, ) ∈ ℐ } (□ )ℐ = {(, ) ∈ N × ∆ ℐ | ∀ ≥ , (, ) ∈ ℐ } ( )ℐ = {(, ) ∈ N × ∆ ℐ | ∃ ≥ s.t. (, ) ∈ ℐ
and (, ) ∈ ℐ , ∀ ( ≤ < )} 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 ∆ 0ℐ ⊆ ∆ 1ℐ ⊆ . . ., 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 ⊑ , where , 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 LTLTℒ with non-temporal TBox. All inclusions in the (non-temporal) TBox are regarded as global temporal constraints, and have to be satisfied at all time points, i.a., a concept inclusion ⊑ is satisfied in an interpretation ℐ if ℐ ⊆ ℐ .
It has been proven that, for non-temporal TBoxes, concept satisfiability in LTLℒ w.r.t.
nontemporal TBoxes is ExpTime-complete, both with expanding domains [35] and with constant domains
[
In [
In the following section, we first develop a many-valued semantics for LTLℒ and, then, we define the typicality operator. Finally, we extend the notion of weighted KBs to the temporal, many-valued case.
ℒ
Let us now move to the many-valued case. To define a temporal extension of ℒ with typicality,
we develop a many-valued semantics for ℒ, by interpreting, at each time point, all concepts and
role names over a truth degree set equipped with a preorder relation ≤ , a bottom element 0 , and a
top element 1 . We denote by < and ∼ the related strict preference relation and equivalence relation.
In the following we will assume to be the unit interval [
A many-valued temporal interpretations for ℒ is a pair ℐ = (∆ ℐ , · ℐ ), where ∆ ℐ is a nonempty domain; · ℐ is an interpretation function that maps each concept name ∈ to a function ℐ : N × ∆ ℐ → , each role name ∈ to a function ℐ : N × ∆ ℐ × ∆ ℐ → , and each individual name ∈ to an element ℐ ∈ ∆ ℐ . Again, in the following definition we assume individual names to be rigid, i.e., having the same interpretation at any time point . Given a time point ∈ N and a domain element ∈ ∆ ℐ , the interpretation ℐ of a concept name assigns to the pair (, ) a value ℐ (, ) ∈ representing the degree of membership of in concept at time point ; and similarly for roles.
The interpretation function · is extended to complex concepts as follows (where, for the semantics of the temporal operators, we adapt a formulation from [37]): ⊥ℐ (, ) = 0, ⊤ℐ (, ) = 1 (¬)ℐ (, ) = ⊖ ℐ (, ) ( ⊓ )ℐ (, ) = ℐ (, ) ⊗ ℐ (, ) ( ⊔ )ℐ (, ) = ℐ (, ) ⊕ ℐ (, ) (∃.)ℐ (, ) = ∈Δ ℐ (, , ) ⊗ ℐ (, ) (∀.)ℐ (, ) = ∈Δ ℐ (, , ) ▷ ℐ (, ) (○ )ℐ (, ) = ℐ ( + 1, ) (♢ )ℐ (, ) = ⨁︀≥ ℐ (, ) (□ )ℐ (, ) = ⨂︀≥ ℐ (, ) ( )ℐ (, ) = ⨁︀≥ (ℐ (, ) ⊗
⨂︀=−1 ℐ (, ))
The semantics of ♢ , □ and requires a passage to the limit. Following [37], one can introduce a bounded version for ♢ , □ and , by adding new temporal operators ♢ (eventually in the next time points), □ (always within time points) and , with the interpretation: (♢ )ℐ (, ) = ⨁︀+= ℐ (, ) (□ )ℐ (, ) = ⨂︀+= ℐ (, ) ()ℐ (, ) = ⨁︀+=(ℐ (, ) ⊗
⨂︀=−1 ℐ (, )) so that (♢ )ℐ (, ) = →+∞(♢ )ℐ (, ) and (□ )ℐ (, ) = →+∞(□ )ℐ (, ) and ( )ℐ (, ) = →+∞()ℐ (, ). The existence of the limits is ensured by the fact that (♢ )ℐ (, ) and ( )ℐ (, ) are increasing in , while (□ )ℐ (, ) is decreasing in .
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 = [
Proposition 1. For all concepts and , and for all time points , the following properties hold:
(♢ )ℐ (, ) = (, ) ⊕ (♢ )ℐ ( + 1, )
(□ )ℐ (, ) = (, ) ⊗ (□ )ℐ ( + 1, )
( )ℐ (, ) = (, ) ⊕ ( (, ) ⊗ ( )ℐ ( + 1, ))
Note that, although in this section we have considered a constant domain ∆ ℐ , for a many-valued
preferential temporal interpretation ℐ, expanding domains could have been considered as well, considering a
domain ∆ ℐ for each time point , with condition ∆ 0ℐ ⊆ ∆ 1ℐ ⊆ . . ., as for LTLℒ in the the two-valued
case [
As in [
Given a many-valued temporal interpretation ℐ = ⟨∆ ℐ , · ℐ ⟩, the interpretation function · is extended to inclusion axioms as follows:
( ⊑ ) = inf ∈Δ ,∈N( (, ) ▷ (, ))
Let be an LTLℒ knowledge base = ( , ) with non-temporal ABox and TBox. Definition 2 (Satisfiability in many-valued LTLℒ). Given a many-valued temporal interpretation for ℐ = ⟨∆ ℐ , · ℐ ⟩, satisfiability of an axiom in ℐ is defined as follows: • ℐ |= ⊑ • ℐ |= () • ℐ |= (, )
if ( ⊑ )ℐ ; if ℐ (0, ℐ ) ;
if ℐ (0, ℐ , ℐ ) .
The interpretation ℐ is a model of = ( , ) if ℐ satisfies all concept inclusions in and all assertions in . A knowledge base = ( , ) is satisfiable in the many-valued extension of LTLℒ if a many-valued temporal model ℐ = ⟨∆ ℐ , · ℐ ⟩ of exists. 5. A many-valued
withTypicality
ℒ
As in the two-valued case [
T(Professor ) ⊑ (∃teaches.Course) Retired
∃lives_in.Town ⊓ Young ⊑ T(♢ Granted _Loan)
Note that, while the semantics in [
From the semantic side, in the many valued case, the degree of membership of domain individuals in concept at the diferent time points induces a preference relation ≺ over the domain. Such preference relations are used to define the typical -elements at the diferent time points.
Given a temporal interpretation ℐ = ⟨∆ ℐ , · ℐ ⟩ over a truth degree set , a preference relation ≺ on ∆ ℐ can be associated to any concept and time point ∈ N, based on the many valued interpretation of concepts in ℐ and on the the strict partial order < : for all , ∈ ∆ ℐ ,
≺ if and only if ℐ (, ) < ℐ (, ), where ≺ means that is preferred to wrt at time point .
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 -elements (at time point ) as the domain elements which are preferred with respect to ≺ among all domain elements (and such that ℐ () ̸= 0). The interpretation of typicality concepts T() can be defined as follows: Definition 3. Given an interpretation ℐ = ⟨∆ ℐ , · ℐ ⟩, for all ∈ N, ∈ ∆ ℐ , (T())ℐ (, ) = ℐ (), if there is no ∈ ∆ ℐ such that ≺ ; (T())ℐ (, ) = 0, otherwise.
When (T())ℐ () > 0, is said to be a typical -element in ℐ. Note that, when ≤ is a total preorder
(as it is in the cases = [
For ℒ 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 nontemporal KBs).
In the following, we will denote with LTLℒT the many-valued extension of ℒ with
typicality, with truth degree set = , for ≥ 1, and with ℒFT the fuzzy extension of
ℒ with typicality (where = [
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 have the form (T() ⊑ , ), and describe the prototypical properties of -elements (where is a concept, and the weight is a real number). A concept for which weighted typicality inclusions are provided is said to be a distinguished concept.
A weighted ℒT knowledge base is a tuple ⟨ , , ⟩, where the (strict) TBox is a set of concept inclusions, the defeasible TBox is a set of weighted typicality inclusions for the distinguished concepts , and is a set of assertions.
Consider the weighted ℒT knowledge base = ⟨ , , ⟩, over the set of distinguished concepts {Student , Employee, Person, . . .}, with containing, for instance, the inclusion Student ⊑ Person ≥ 1 .
The set 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 with respect to a distinguished concept , we have to consider that, in an interpretation ℐ, at time point , element may belong to other concepts to some degree (e.g., at time point , mary may be active with degree 0.8, i.e., Activeℐ (, mary ) = 0.8).
Given a many-valued temporal interpretation ℐ = ⟨∆ ℐ , · ℐ ⟩, the weight of ∈ ∆ ℐ with respect to a distinguished concept at time point is given by
ℐ,() = ∑︀(T()⊑,)∈ ℐ (, ).
Intuitively, the higher the value of ℐ,(), the more typical is as an instance of ), at time point (considering the defeasible properties of ). Here, the membership degree ℐ (, ) of in each concept at time point is considered.
The notions of faithful, coherent and -coherent semantics introduced for many-valued weighted
KBs [
A many-valued temporal interpretation ℐ = ⟨∆ ℐ , · ℐ ⟩ is faithful at if, for all , ∈ ∆ ℐ , The interpretation ℐ is coherent at if, for all , ∈ ∆ ℐ , ≺ ⇒
ℐ,() > ℐ,() ≺ if
ℐ,() > ℐ,() Given a collection of monotonically non-decreasing functions : R → , one for each concept ∈ : - the interpretation ℐ is -coherent at if, for all ∈ ∆ ℐ , - the interpretation ℐ is transient -coherent at if, for all ∈ ∆ ℐ ,
ℐ (, ) = (ℐ,()) ℐ ( + 1, ) = (ℐ,())
It is easy to see that a many-valued temporal interpretation ℐ = ⟨∆ ℐ , · ℐ ⟩ determines, at each time point , a (non-temporal) many-valued interpretation = ⟨∆ , · ⟩, where ∆ = ∆ ℐ , (for ∈ ), and (, ) = ℐ (, , ) (for ∈ ). Letting the interpretation of typicality in exploit the preference relations ≺ for each (see Section 5), i.e., the preference relation induced by the many-valued interpretation of concept in , a many-valued temporal interpretation ℐ can be regarded as a sequence 0, 1, 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 , the condition that the interpretation ℐ is coherent (resp., faithful, -coherent) at , means that the preferential interpretation is coherent (resp., faithful, -coherent) according to their definition in [ 33]. Diferent notions of agreement at diferent time points can then be combined to give rise to diferent semantics of a temporal weighted KB, and diferent notions of entailment (based on diferent closure constructions).
In [33] it has been shown that many-valued Weighted KBs with typicality can provide a logical interpretation to some neural network model. Specifically, the -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 can be described by the following pair of equations: = ∑︀=1 , and = ( + ), where 1, . . . , are the input signals, 1, . . . , are synaptic weights; is the bias, an activation function, and is the output signal of unit . By adding a new synapse with input 0 = +1 and synaptic weight 0 = , one can write: = ∑︀=0 , and = (), where is called the induced local field of the neuron. The neuron can be represented as a directed graph, where the input signals 1, . . . , and the output signal of neuron are nodes of the graph. An edge from to , labelled , means that is an input signal of neuron with synaptic weight .
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 on the edge from node to node represents “the strength of the connection [..] by which unit transmits information to unit " [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 . We do not put restrictions on the topology the network.
Following the approach in [33], can be mapped into a (non-temporal) weighted conditional knowledge
base [
If is the concept name associated to unit and 1 , . . . , are the concept names associated to units 1, . . . , , whose output signals are the input signals for unit , with synaptic weights ,1 , . . . , , , then unit can be associated a set of weighted typicality inclusions: T() ⊑ 1 with ,1 , . . . , T() ⊑ with , .
It has been proven that the input-output behavior of a multilayer network can be captured by a preferential interpretation 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 is established [33] by proving that the preferential interpretation Δ , describing the network behavior over a set ∆ of input stimuli, is indeed a -coherent model of the weighted knowledge base and, vice-versa, that any -coherent model of the knowledge base captures the behavior of the network over some set ∆ of input stimuli. Also in the case the network is not feedforward, the -coherent semantics allows the stationary states of the network to be captured.
This approach allows for the verification of conditional properties of the network (of the form T() ⊏ ≥ ) by model checking over the preferential interpretation Δ, or by using entailment from the conditional knowledge base (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 , we may represent diferent states of the network at diferent time points.
When ℐ is -coherent at time point , the condition (stated above) that, for all ∈ ∆ ℐ , ℐ (, ) = (∑︁ ℎ ℎℐ (, ))
ℎ imposes that the (non-temporal) interpretation at time point represents a stationary state of network . In such a case, plays the role of the activation function, and the sum ∑︀ℎ ℎ ℎℐ (, ) 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 is recurrent. In this case, the knowledge base contains cyclic dependencies in DBox.
By imposing the condition that ℐ is a transient -coherent interpretation at all time points , one can enforce that the interpretations 0, 1, 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 + 1 depends on the activity of incoming units at time point ). 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 .
Once a trained neural network has been represented as a weighted defeasible knowledge base , 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 diferent 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 Δ the activity of all units, but only of the units involved in the properties to to consider in the model ℐ 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.
In this paper, we develop a many-valued, temporal description logic with typicality, extending ℒ to deal with defeasible reasoning. Our extension of LTLℒ 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 LTLℒ, 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 diferent classes in the temporal case.
On a diferent route, a preferential LTL with defeasible temporal operators has been studied in [
In previous work, we have developed a preferential temporal description logics with typicality
LTLTℒ [
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) [
A diferent 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 ℰ ℒ⊥ 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.
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). 2005, Crete, May 29 - June 1, volume 3532 of LNCS, Springer, 2005, pp. 167–181. [24] G. Stoilos, G. B. Stamou, V. Tzouvaras, J. Z. Pan, I. Horrocks, Fuzzy OWL: uncertainty and the semantic web, in: OWLED*05 Workshop, volume 188 of CEUR Workshop Proc., 2005. [25] T. Lukasiewicz, U. Straccia, Description logic programs under probabilistic uncertainty and fuzzy vagueness, Int. J. Approx. Reason. 50 (2009) 837–853. [26] S. Borgwardt, F. Distel, R. Peñaloza, The limits of decidability in fuzzy description logics with general concept inclusions, Artif. Intell. 218 (2015) 23–55. [27] F. Bobillo, U. Straccia, Reasoning within fuzzy OWL 2 EL revisited, Fuzzy Sets Syst. 351 (2018) 1–40. [28] A. García-Cerdaña, E. Armengol, F. Esteva, Fuzzy description logics and t-norm based fuzzy logics,
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