In this paper we introduce a temporal, conditional logic with typicality, which combines a multi-preferential conditional logic, allowing for defeasible reasoning, with the temporal modalities of Linear Time Temporal Logic. The combination provides a formalism which is capable of capturing the dynamics of a system, trough its strict and defeasible temporal properties. The paper also studies weighted temporal conditional knowledge bases for strengthening preferential entailment.
and weighted knowledge bases (KBs) for Description Logics [20, 21]. The idea is that a world may represent a more typical situation describing a student, compared to ′ (w <stud w ′) but, vice-versa, world ′ may represent a more typical situation describing an employee, compared to (w ′ <emp w ). Under this respect, the semantic we consider is a generalization of the KLM preferential semantics, which exploits a single preference relation on worlds. Decidability of the satisfiability problem is proven.
The paper provides a construction for strengthening preferential entailment in the temporal conditional logic, based on weighted temporal KBs, in which conditional implications are associated a weight representing its plausibility or implausibility. For instance, for the proposition student , we may have a set of weighted temporal conditionals Kstudent = {(T(student ) → has_Classes, +50), (T(student ) → ◇holds_Degree, +30), (T(student ) → has_Boss, − 40) }, that describes prototypical properties of students, i.e., that a student normally has classes and will eventually get a degree, but she usually does not have a boss (negative weight). Accordingly, a student having classes, but not a boss, is regarded as being more typical than a student having classes and a boss.
An approach for combining preferences is also presented, defining the preference relations for boolean combinations of distinguished propositions (e.g., student ∧ employee, if student and employee are distinguished propositions). These preference relations are defined by combining the weights of the distinguished propositions at each time point, by generalizing an approach recently proposed for propositional conditional logic (and Conditional ASP) [22].
The schedule of the paper is the following. Section 2 develops a many-valued preferential logic with typicality. Section 2.2 extends the logic with LTL modalities to develop a temporal conditional logic. In Section 3 weighted temporal conditional KBs are introduced, and a construction for combining preferences is developed. Section 4 concludes the paper.
In this section we define a two-valued preferential logic with typicality, which generalizes Kraus
Lehmann and Magidor’s preferential semantics [
We consider a propositional language , whose formulae are built from a set of propositional variables using the boolean connectives ∧, ∨, ¬ and → of propositional logic. We assume that ⊥ (representing falsity) and ⊤ (representing truth) are formulae of .
A propositional language with a typicality operator is introduced following the approach used in the description logic ℒ + T [23] as well as in the Propositional Typicality Logic (PTL) [17]. Let T be the language with typicality. Intuitively, “a sentence of the form T( ) is understood to refer to the typical situations in which holds" [17]. As in PTL [17], the typicality operator cannot be nested. An implication of the form T( ) → is called a defeasible implication, meaning that “normally, if then ”. An implication → is called strict, if it does not contain occurrences of the typicality operator.
The KLM preferential semantics [
In the following, we shortly recall the multi-preferential semantics from [22]. We consider finite KBs, and a finite set of distinguished propositions 1, . . . , . 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.
Definition 1.
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, |= and ℳ, |= ℳ, |= or ℳ, |= ℳ, |= ¬ if
ℳ, ̸|= ℳ, |= → if ℳ, |= T() if
ℳ, |= implies ℳ, |= ℳ, |= and ∄w ′ ∈ s.t. w ′ <Ai w and ℳ, ′ |= .
Whether T() is satisfied at a world 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. Unlike [22] here we do not assume well-foundedness of the preference relations.
When an implication has the form T() → , with in ℒ, it stands for a conditional |∼
in KLM logics [
It is well known that preferential entailment and rational entailment are weak. As with rational closure
[
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 ) 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).
Definition 2. ⟨ , {< }∈N, ⟩ where:
A temporal (multi-)preferential interpretation (or T interpretation) is a triple ℐ = • is a non-empty set of worlds; • each < ⊆ × 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 formula at the initial time point 0 of a temporal preferential interpretation ℐ, as in LTL. Definition 3 (Satisfiability and entailment) . A T formula is satisfied in a temporal preferential interpretation ℐ = ⟨, {< }∈N, ⟩ if ℐ, 0, |= for some world ∈ . A T formula is valid in the temporal preferential interpretation ℐ = ⟨, {< }∈N, ⟩ (written ℐ |= ) if ℐ, 0, |= , for all worlds ∈ . A preferential interpretation ℐ = ⟨, {< }∈N, ⟩ is a model of a temporal conditional knowledge base , if ℐ |= holds, for all the formulas in .
A temporal conditional knowledge base preferentially (rationally) entails a formula if is satisfied in all the (ranked) models ℐ of .
It can be shown that the problem of deciding the satisfiability of a T formula can be polynomially reduced to the problem of deciding the satisfiability of a concept in the description logic Tℒ introduced in [15], which extends the temporal description logic ℒ [14] with the typicality operator. Tℒ has been proven to be decidable when a finite set of well-founded preference relations <1 , . . . , < is considered, and concept inclusions are regarded as global temporal scaotnissfiatrbailiintytsf.oTrheLTdeLcidℒabwili.rt.yt. oTfBcooxnecseipstinsaEtxispfiaTbiimliety(ainnd, actuaTllℒy,itreisliEesxpoTnimthee-coremsuplltetteh),abtoctohnwceiptht expanding domains [26] and with constant domains [14].
As for KLM logics, the notion of preferential and rational entailment of temporal conditionals is weak. Preferential entailment can be strengthened by restricting to a subset of the temporal preferential models of a conditional knowledge base .
In the next section we will consider weighted conditionals and extend to the temporal case the construction for reasoning with weighted conditional knowledge bases developed in [22]. In a companion paper [27], dealing with a deontic temporal conditional logic, a construction exploiting temporal deontic conditionals with ranks is developed.
A weighted temporal conditional knowledge base includes weighted temporal conditionals of the form (T() → , ), where and are LTL formulas, and the weight is a real number, representing the plausibility or implausibility of the conditional implication. For instance, for the proposition student , we may have the set of weighted temporal conditionals Kstudent in Section 1, while for employee the set of conditionals Kemployee = {T(student ) → has_Boss, +40), (T(student ) → ◇get _wage,+50), (T(student ) → has_Classes, − 30)}.
A set of weighted conditionals is introduced for each distinguished formula . They coexist with a strict part of the knowledge base, , containing formulas with no occurrences of the typicality operator. A weighted temporal conditional knowledge base (with respect to the distinguished formulas 1, . . . , ) is a tuple = ⟨, 1 , . . . , ⟩, where is the strict part of the knowledge base, and 1 , . . . , are the sets of weighted conditionals (the defeasible part of the KB).
Given a temporal conditional interpretation ℐ = ⟨, {< }∈N, ⟩, the preference relation < associated with the distinguished proposition at time point is determined based on the weights of worlds at , depending on the weights of the conditionals in .
More precisely, we define for each world ∈ and time point , the weight (, ) of world at time point with respect to proposition , as follows: (, ) =
∑︁ :ℐ,,|=, ℎ, Informally, the weight (, ) of at wrt is the sum of the weights of all conditional implications T() → , for , such that , is satisfied by at time point in the interpretation ℐ. The more plausible are the conditional properties for satisfied in at , the higher is the weight of at time point with respect to . We let (, ) = 0, when = ∅ (there are no defeasible implications (T() → , , ℎ, ) for the distinguished proposition ). We can define a preorder relation ≤ associated with proposition at time point as follows: for all 1, 2 ∈ , 1 ≤ 2 if
(1, ) ≥ (2, ).
For instance, from the knowledge base containing the set of conditionals Kstudent and Kemployee , the preference relations <sntudent are determined as above, and we are able to conclude that T(student ) → has_Classes ∧ ◇holds_Degree, while we cannot conclude that T(student ) → has_Classes∧ has_Boss. (1) (2) (3) The strict partial order < induced by ≤ is defined as usual: 1 < 2 if 1 ≤ 2 and 2 ̸≤ 1.
Definition 4 (Model of a weighted temporal KB). A temporal multi-preferential interpretation ℐ = ⟨, {< }∈N, ⟩ is an T model of a weighted KB = ⟨, 1 , . . . , ⟩ if it is a model of Aafnodrm,fuolraa ll disisetnintagiuliesdhefrdofmormawuleaisghte, d<is Tthkenstorwicltepdagretibaalsoerder iifnduicsesdatbiysfietdheinparellorthdeer ≤ .T models of .
So far, we have only considered typicality formulas of the form T(), with a distinguished formula. To evaluate whether formulas like:
T(student ∧ employee) → has_Classes ∧ ◇holds_Degree are entailed from a weighted KB, we extend the formalism to deal with general typicality formulas T(), with a boolean combination of distinguished proposition. To deal with such formulas, in this section we extend to the temporal case the approach proposed in [22] for combining preferences in a Conditional ASP based on weighted knowledge bases.
First, for the language of T, which includes typicality formulas T() with a boolean combination of distinguished formulas, the truth of T() in a world at time point in an interpretation ℐ is defined, as for distinguished formulas (see Definition 2), as follows:
ℐ, , |= T() if ℐ, , |= and ∄w ′ ∈ s.t. w ′ <nA w and ℐ, , ′ |= .
For evaluating the truth of a typicality formula T() at time point , we need to define the preference relation < associated to the formula at time point . For instance, evaluating formula (3) above in an interpretation ℐ at timepoint 0, requires determining the most normal situations for an employed student at a time point 0. This requires the preference relation <s0tudent∧employee to be determined. Assume that the weighted knowledge base , includes the sets of conditionals Kstudent and Kemployee . Then the weight Wstudent∧employee (w , 0 ) of world at time point 0 with respect to proposition student ∧ employee can be computed from the weights Wstudent (w , 0 ) and Wemployee (w , 0 ). We define the preference relation < associated with a complex formula (a boolean combination of the ’s) at time point , by inductively extending to complex formulae the notion of the weight of a world , at time point .
Given a weighted temporal knowledge base , let Max and Min be, resp., the maximum and the minimum value of the weight (, ), for each distinguished proposition in , world and time point . These values can be determined from the weights of the conditionals in . We let: 1∧2 (, ) = (1 (, ), 2 (, )) 1∨2 (, ) = (1 (, ), 2 (, )) ¬ (, ) = − (, ) +
The preference relation < associated with a complex formula can be defined by exploiting the weight function , as for distinguished propositions: For all 1, 2 ∈ , we let: 1 < 2 if (1, ) > (2, ).
The preference relation < is modular. By generalizing the result in [22] to the temporal case, the following proposition can be proved.
Proposition 1. Let and be boolean combinations of the distinguished propositions 1, . . . , . If and are equivalent in propositional logic, then, for all worlds 12 ∈ , 1 < 2 if 1 < 2.
The notion of model of a weighted temporal KB, and the notion of entailment from a weighted knowledge base , can be suitably extended, by letting the preference relation < be defined as above, for any boolean combinations of the distinguished propositions 1, . . . , .
This logic can be used in conjunction with a formalism which generates a set of trajectories from a set of possible initial states (e.g., the runs of a business process). The verification of conditional properties over a set of runs is based on the preference relations computed from a weighted temporal KB.
The short paper presents a formalism which combines a preferential logic with typicality and the temporal logic LTL. The interpretation of the typicality operator is based on a multi-preferential semantics, and an extension of weighted conditional knowledge bases to the temporal case is proposed. The extension exploits the weights of conditionals for combining preferences between worlds with respect to diferent formulas.
On a diferent route, a preferential logic with defeasible LTL operators has been studied in [ 12, 28]. The decidability of diferent fragments of the logic has been proven, and tableaux based proof methods for such fragments have been developed [11, 28]. Our approach does not consider defeasible temporal operators nor preferences over time points, but it combines standard LTL operators with the typicality operator in a temporal logic.
In [16] we considered a many-valued temporal logic with typicality for the verification of temporal properties of gradual argumentation graphs in gradual argumentation semantics.
Future work includes: developing proof methods, e.g., based on ASP encodings, as done for the non-temporal case [29, 22]; exploiting the formalism for explainability and for defeasible reasoning about dynamic systems and, specifically, for reasoning about the dynamics of argumentation graphs in gradual semantics (see [30], but also to reason about obligations and permissions in a deontic temporal conditional logic [27].
While conditional weighted KBs have been shown to capture the stationary states of some neural networks (or their finite approximation) [ 25], and allow for combining empirical knowledge with elicited knowledge for post-hoc verification, adding a temporal dimension also opens to the possibility of verifying properties concerning the dynamic behavior of a network.
Acknowledgements: 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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