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 for 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 [2, 3] (KLM for short). Multi-preferential extensions of DLs have been developed, to provide a semantics for ranked and weighted knowledge bases with typicality [4, 5, 6]. Temporal extensions of Description Logics are very well-studied in DLs literature [7, 8]. Preferential extensions of Linear Time Temporal Logic (LTL) with defeasible temporal operators have been recently studied [9, 10] to enrich temporal formalisms with non-monotonic reasoning features. On a diferent route, a preferential extension of the temporal description logic LTLTℒ has been proposed in [11], extending LTLℒ [7] with a typicality operator T, which selects the most typical instances of a concept, to represent defeasible temporal properties of concepts, i.e., temporal properties 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 short paper, an abridged version of [12], we describe a many-valued extension of LTLℒ with typicality, making it possible to represent concept inclusions such as
∃lives_in.Town ⊓ Young ⊑ T(♢ Granted _Loan), (meaning that people living in town and being young, normally are eventually granted a loan), where the interpretation of some concepts (such as, Young ) may be non-crisp.
This many-valued temporal extension of ℒ builds on many-valued DLs, which are widely studied in the literature, both for the fuzzy case [13, 14, 15] and for the finitely-valued case [ 16, 17]. We then add a typicality operator, to get a many-valued temporal extension of ℒ with typicality.
We briefly discuss the definition of a closure construction 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) is defined inductively from concept names and the ⊤ and ⊥ concepts, using intersection ⊓ , union ⊔ , negation ¬, as well as universal and existential restrictions ∀., ∃..
A fuzzy interpretation , given a non-empty domain Δ, assigns to each individual name ∈ an
element ∈ Δ; to each concept name ∈ a function : Δ → [
⊥ () = 0 ( ⊓ ) () = () ⊗ () (∃.) () = sup∈Δ (, ) ⊗ () (¬) () = ⊖ () ( ⊔ ) () = () ⊕ () (∀.) () = inf∈Δ (, ) ▷ () where ∈ Δ, 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 [15], we do not commit to a specific choice of combination functions,
A fuzzy ℒ knowledge base is a pair ( , ) where is a fuzzy TBox and is a fuzzy ABox.
A fuzzy TBox is a set of fuzzy concept inclusions of the form ⊑ , where ⊑ is an ℒ
concept inclusion axiom, ∈ {≥ , ≤ , >, <} and ∈ [
This allows defining the satisfiability of fuzzy concept inclusions, |= ⊑ if ( ⊑ ) ; while, for fuzzy assertions, |= () if ( ) , and |= (, ) if ( , ) . 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), meaning that satisfies all the axioms in Γ.
For the finitely many-valued case, we assume the truth space to be 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 assume to be the unit interval [
ℒ
Temporal extensions of DLs, their complexity and decidability are very well-studied in the literature
(see, e.g., [
While we refer to [
A many-valued temporal interpretations for ℒ is a pair ℐ = (Δℐ , · ℐ ), where Δℐ is a
nonempty domain; · ℐ is an interpretation function that maps each concept name ∈ to a function
ℐ : N × Δℐ → , each role name ∈ to a function ℐ : N × Δℐ × Δℐ → , and each individual
name ∈ to an element ℐ ∈ Δℐ . For simplicity, following [
⊤ℐ (, ) = 1 ( ⊓ )ℐ (, ) = ℐ (, ) ⊗ ℐ (, ) (∃.)ℐ (, ) = ∈Δ ℐ (, , ) ⊗ ℐ (, ) (∀.)ℐ (, ) = ∈Δ ℐ (, , ) ▷ ℐ (, ) ( )ℐ (, ) = ⨁︀≥ (ℐ (, ) ⊗ ⨂︀=−1 ℐ (, )) (¬)ℐ (, ) = ⊖ ℐ (, ) ( ⊔ )ℐ (, ) = ℐ (, ) ⊕ ℐ (, ) (○ )ℐ (, ) = ℐ ( + 1, ) (♢ )ℐ (, ) = ⨁︀≥ ℐ (, ) (□ )ℐ (, ) = ⨂︀≥ ℐ (, ) The semantics of ♢ , □ and requires a passage to the limit. Following [22], bounded versions for ♢ , □ and can be introduced, using additional 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 .
Here, we have not considered the additional temporal operators (“soon”, “almost always”, etc.)
introduced by Frigeri et al. [22] for representing vagueness in the temporal dimension. As a consequence,
for the case = [
(□ )ℐ (, ) = (, )⊗ (□ )ℐ (+1, )
( )ℐ (, ) = (, ) ⊕ ( (, ) ⊗ ( )ℐ ( + 1, ))
Although we have considered a constant domain Δℐ , for a many-valued preferential temporal
interpretation ℐ, expanding domains could be considered, as for LTLℒ in the two-valued case [
For simplicity, we consider knowledge bases with non-temporal TBox and ABox, where a nontemporal TBox is a set of concept inclusions ⊑ , where , are temporally extended concepts, and 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 LTLℒ knowledge base with non-temporal ABox and TBox. The assertions in a non-temporal ABox are evaluated at time point 0. Concept inclusions in the non-temporal TBox are evaluated by considering all time points .
Given a many-valued temporal interpretation ℐ = ⟨Δℐ , · 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. Given a many-valued temporal interpretation for ℐ = ⟨Δℐ , · ℐ ⟩, satisfiability of an axiom in ℐ is defined as: • 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 = ⟨Δℐ , · ℐ ⟩ of exists. 4. A many-valued
with Typicality ℒ As in the two-valued case [11], the language of a many-valued ℒ can be extended with typicality concepts of the form T() representing the set of typical instances of concept . The typicality operator T may occur both in concepts of TBox and ABox, but it cannot be nested. Extended concepts can be built by combining the concept constructors in LTLℒ with the typicality operator, by allowing T() as a concept. They can freely occur in concept inclusions as in:
T(Professor ) ⊑ (∃teaches.Course) Retired ∃lives_in.Town ⊓ Young ⊑ T(♢ Granted _Loan)
Inclusions of the form T() ⊑ correspond to conditionals |∼ in KLM preferential logics [
Given a temporal interpretation ℐ = ⟨Δℐ , · ℐ ⟩ over a truth degree set , a preference relation ≺ on Δℐ is induced by the many valued interpretation of in ℐ, at time point , as follows: for all , ∈ Δℐ , ≺ if and only if ℐ (, ) < ℐ (, ), where ≺ means that is preferred to with respect to at time point .
The many-valued temporal semantics introduced in the previous section easily extends to the language with typicality (see below). 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 ). Note that this semantics is inherently multi-preferential. The interpretation of typicality concepts T() can be defined as follows: Definition 1. Given an interpretation ℐ = ⟨Δℐ , · ℐ ⟩, for all ∈ N, ∈ Δℐ , (T())ℐ (, ) = ℐ (, ), if there is no ∈ Δℐ 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 as in Section 3.
In the following, we 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). The concepts for which weighted typicality inclusions are provided are called distinguished concepts.
A weighted temporal knowledge base is a tuple ⟨ , , ⟩, where the (strict) TBox is a set of strict inclusions, the defeasible TBox is a set of weighted typicality inclusions, and is a set of assertions.
Consider the weighted LTLℒT knowledge base = ⟨ , , ⟩, over the set of distinguished concepts {Student , Employee, Person, . . .}, with containing, for instance, the inclusion Student ⊑ Person ≥ 1 . and containing the following weighted typicality 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 to 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).
The many-valued temporal interpretation ℐ = ⟨Δℐ , · ℐ ⟩ with = [
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.
For LTLℒT and ℒFT, the notions of faithful, coherent and -coherent semantics
introduced for many-valued weighted KBs in [
A many-valued temporal interpretation ℐ can be regarded as a sequence 0, 1, 2, . . . of manyvalued preferential interpretations (as those considered in [19]), for each time point. 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 particular, a notion of transient -coherence at (i.e., for all ∈ Δℐ , ℐ ( + 1, ) = (ℐ,())) is introduced in [12] to provide a logical characterization of the transient behavior of a recurrent multilayer network.
In [19] 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 subsection, we are interested in the transient behavior of a network.
Let us consider a trained network . We do not put restrictions on the topology the network.
Following the approach in [19], can be mapped into a (non-temporal) weighted conditional knowledge
base [
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 Δ (e.g., the test set), through a simple construction, which exploits the activity level of units for the input stimuli.
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 for finitely-valued semantics [ 18]). 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 [24, 19].
When we consider a temporal preferential model ℐ of the weighted knowledge base , we can represent diferent states of the network at diferent time points. When ℐ is -coherent at time point , the coherence condition above 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.
The temporal formalism also allows to capture the dynamic behavior of the network beyond stationary states. When the network is recurrent, 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, as well as time delayed feedback connections.
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 concepts in the temporal case.
On a diferent route, preferential extensions of LTL with defeasible temporal operators have been
recently studied by Chafik et al. [
Much work has been recently devoted to the combination of neural networks and symbolic reasoning
[25, 26, 27]. 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) [
An interesting direction for future work, is an extension to the temporal case of the model-checking approach developed in Datalog [24, 19] for the verification of conditional properties of a network, for post-hoc verification.
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). [11] M. Alviano, L. Giordano, D. Theseider Dupré, Preferential temporal description logics with typicality and weighted knowledge bases, in: Proc. 38th Italian Conference on Computational Logic, Udine, Italy, June 21-23, volume 3428 of CEUR Workshop Proc., 2023. [12] M. Alviano, M. Botta, R. Esposito, L. Giordano, D. Theseider Dupré, Many-valued temporal weighted knowledge bases with typicality for explainability, in: Proc. 39th Italian Conference on Computational Logic, Rome, Italy, June 26-28, 2024, volume 3733 of CEUR Workshop Proc., 2024. [13] U. Straccia, Towards a fuzzy description logic for the semantic web (preliminary report), in: ESWC 2005, Crete, May 29 - June 1, volume 3532 of LNCS, Springer, 2005, pp. 167–181. [14] 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. [15] T. Lukasiewicz, U. Straccia, Description logic programs under probabilistic uncertainty and fuzzy vagueness, Int. J. Approx. Reason. 50 (2009) 837–853. [16] A. García-Cerdaña, E. Armengol, F. Esteva, Fuzzy description logics and t-norm based fuzzy logics,
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