In this paper we propose a many-valued temporal conditional logic. We start from a many-valued logic with typicality, and extend it with the temporal operators of the Linear Time Temporal Logic (LTL), thus providing a formalism which is able to capture the dynamics of a system, trough strict and defeasible temporal properties. We consider the many-valued case, while the two-valued case can be regarded as a special case. In this short paper we report about our work on a temporal extension of a many-valued conditional logic, based on a preferential approach to commonsense reasoning [1, 2, 3, 4, 5, 6]. The paper develops a propositional many-valued temporal logic with typicality, by extending the many-valued conditional logic with typicality introduced in [7] with temporal operators from the Linear Time Temporal Logic (LTL). This allows considering the temporal dimension, when reasoning about the defeasible typicality properties of a system, for explanation. Preferential extensions of LTL with defeasible temporal operators have been recently studied [8, 9, 10] to enrich temporal formalisms with non-monotonic reasoning features, by considering defeasible versions of the LTL operators. Our approach, instead, adds the standard LTL operators to a (manyvalued) conditional logic with typicality, an approach similar to the preferential extension considered for Description Logics (DLs), where the logic LTLℒ [11], extending ℒ with LTL operators, has been further extended with a typicality operator, in the two-valued [12] and many-valued case [13] to allow for conditional reasoning. As in the Propositional Typicality Logic (PTL) by Booth et al. [14] (and in the DLs with typicality [15]) the conditionals are formalized based on material implication (resp., concept inclusions) plus the typicality operator T. Conditional implications T( ) → , meaning that “normally if holds, holds", corresponds to conditionals |∼ in KLM logics [4, 6]. In this paper, as in [7], we further consider a many-valued semantics, so that a formula is given a value in a truth degree set , and the two-valued case can be regarded as a special case, obtained for = {0, 1}. As the logic is many-valued, we consider graded conditionals of the form (T( ) → ) ≥ l , resp., (T( ) → ) ≤ l , meaning that “normally if holds, then holds, with degree at least (resp., at most) " (in the following, we will omit the parentheses in (T( ) → ) ≥ l , and simply write T( ) → ≥ l ). For instance, the formalism allows for representing graded implications as: living _in_Town ∧Young → T(◇Granted _Loan) ≥ , meaning that living in town and being young, implies that normally the
loan is eventually granted, and the implication has degree at least , where the interpretation of some propositions (e.g., Young ) may be non-crisp.
The preferential semantics of the logic exploits multiple preference relations < with respect to diferent formulas , following the multi-preferential semantics developed for ranked and weighted DL knowledge bases (KBs) [16, 17], as well as for propositional conditionals, based on preferences with respect to diferent aspects [ 18]. The semantics considered in this paper generalized the approach in [18], which specifically deals with a multi-preferential extensions of the rational closure semantics.
The schedule of the paper is the following. Section 2 develops a many-valued preferential logic with typicality. Section 3 extends such logic with LTL modalities to develop a temporal many-valued conditional logic, and temporal graded formulas. Section 4 concludes the paper. An extended version of the paper, also dealing with weighted knowledge bases and gradual argumentation, can be found in [19].
Let ℒ be a propositional many-valued logic, whose formulas are built from a set of propositional variables using the logical connectives ∧, ∨, ¬ and →, as usual. We assume that ⊥ and ⊤are formulas of ℒ. We consider a many-valued semantics for formulas, 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 (often we will omit explicitly referring to ).
Let ⊗ , ⊕ , ⊖ and ▷ be the truth degree functions in for the connectives ∧, ∨, ¬ and → (respectively).
When is [
We further extend the language of ℒ by adding a typicality operator as introduced by Booth et al. [14] for propositional calculus, and by Giordano et al. for preferential description logics [22]. Intuitively, “a sentence of the form T() is understood to refer to the typical situations in which holds" [14]. The typicality operator allows the formulation of conditional implications (or defeasible implications) of the form T() → whose meaning is that “normally, if then ”, or “in the typical situations when holds, also holds”. As in PTL [14], the typicality operator cannot be nested. When and do not contain occurrences of the typicality operator, an implication → is called strict. We call ℒT the language obtained by extending ℒ with a unary typicality operator T.
The interpretation of a typicality formula T() is defined with respect to a preferential interpretation.
The KLM preferential semantics [
Here we consider a many-valued multi-preferential semantics for conditionals. The propositions at each world ∈ have a value in and multiple preference relations < ⊆ × are associated to formulas of ℒ. Multi-preferential semantics have been previously considered for defining refinements of the rational closure construction [23, 18], as well as for defeasible DLs, both in the two-valued and in the many-valued case, for ranked KBs [24, 25, 26].
A (multi-)preferential interpretation is a triple ℳ = ⟨, {< }, ⟩ where: • is a non-empty set of worlds; • each < ⊆ × is a strict partial order relation on ; • : × Prop →− is a valuation function, assigning a truth value in to each propositional variable at each world ∈ .
The valuation is inductively extended to all formulas in ℒT: (, ⊥) = 0
(, ¬) = ⊖ (, ) (, T()) = ︂{ (, ⊤) = 1
if ∄w ′ ∈ s.t. w ′ <A w
(, ∧) = (, )⊗ (, )
(, )
0
(, → ) = (, ) ▷ (, )
(, ∨) = (, )⊕ (, )
in ℳ is defined as follows:
semantics [
When (, T()) ̸= 0, is a typical/normal -world in ℳ. Note that, diferently from the KLM Let us define the satisfiability in
ℳ of a graded implication, with form → ≥ or → ≤ , where and are constants corresponding to truth values in and and are formulas of ℒT.
Given a preferential interpretation ℳ = ⟨, {< }, ⟩, the truth degree of an implication → ( → )ℳ = ∈ ((, ) ▷ (, )).
The satisfiability of a graded implication is evaluated globally to the preferential interpretation ℳ. ≥ (written ℳ |= → ≥ ) if ( → )ℳ ≥ . Similarly, for → ≤ .
A preferential interpretation ℳ = ⟨, {< }, ⟩, satisfies a graded implication → In general, some conditions may be needed to enforce an agreement between the truth values of a formula at the diferent worlds in
ℳ and the preference relations < among them. The preferences < might have been determined by some closure construction, such as those exploiting the ranks or weights of conditionals in [24, 25]. Similar conditions, called coherence, faithfulness and -coherence conditions have, for instance, been introduced in the multi-preferential semantics for DLs with typicality in [25, 26]. preference relation < ,
A (multi-)preferential interpretation ℳ
= ⟨, {< }, ⟩ is coherent if, for all , ′ ∈ , and (, ) > (′, ) ⇐⇒ < ′ that is, the ordering among the values of in and ′ is justified by the preference relation <; and vice-versa. A weaker condition is faithfulness, only requires that (, ) > (′, ) ⇒ < ′.
Clearly, a preferential interpretation ℳ might be coherent with respect to a preference relation < , while being only faithful with respect to another < .
We let a knowledge base be a set of graded implications. A model of is an interpretation ℳ which satisfies all the graded implications in . Given a knowledge base , we say that entails a graded implication → ≥ if → ≥ is satisfied in all the models of (and similarly for → ≤ ). In the following, we will refer to the entailment of a graded implication → ≥ 1 as well-founded.
The KLM properties of a preferential consequence relation can be reformulated in the many-valued setting, and it can be proven that, for the choice of combination functions as in Gödel logic, they hold for 1-entailment, under the assumptions that ℳ is coherent and the preference relations < are
Note that KLM preferential interpretations, with a single well-founded preference relation, can be
regarded as a special case of multi-preferential interpretations. It can be proven that any KLM preferential
interpretation [
In this section we extend the language of the logic ℒ ◇ (eventually) and □ (always) of Linear Time Temporal Logic (LTL) [27].
T with the temporal operators ○ (next), (until),
First, we allow temporal operators and typicality operators to occur in a graded implication → ≥ (or → ≥ ) in and in , with the only restriction that T should not be nested. For instance, lives_in_town ∧ young →
T(◇granted _loan) ≥ 0.8 and ◇T(granted _loan) → lives_in_town ∧ young ≥ 0.8. are graded implication. Then, we will allow for combining graded implications.
The semantics of the many-valued temporal logic with typicality is defined in agreement with the semantics by Frigeri et al. [28].
A temporal (multi-)preferential interpretation is a triple ℐ = ⟨ , {< }∈N, ⟩ where: • is a non-empty set of worlds; • each < ⊆ × • : N × ×
Prop →−
is partial order on ; any propositional variable in each world ∈ .
is a valuation function assigning, at each time point, a truth value to When there is no ′ ∈ s.t. ′ < , we say that is a normal situation for at timepoint .
In a preferential interpretation ℐ = ⟨ , {< }∈N, ⟩, the valuation (, , ) of a formula , in world at time point ∈ N, can be defined inductively as follows: (, , ⊥) = 0 (, , ⊤) = 1
(, , ∧ ) = (, , ) ⊗ (, , ) (, , T()) = 0
︂{ (, , ) if ∄w ′ ∈ s.t. w ′ <nA w (, , ) = ⨁︀ (, , ○ ) = ( + 1, , ) (, , ◇) = ⨁︀≥ (, , ) ≥ ((, , )⊗
= ⨂︀− 1 (, , )) (, , □) = ⨂︀≥ (, , ) (, , ¬) = ⊖ (, , ) (, , ∨ ) = (, , ) ⊕ (, , ) (always within time points) and , with the interpretation: The semantics of ◇, □ and requires a passage to the limit. Following [28], we introduce a bounded version for ◇, □ and , by adding new temporal operators ◇ (eventually in the next time points), □ so that (, , ◇) = ⨁︀+= (, , ) (, , ) = ⨁︀+=((, , )⊗ (, , ◇) = →+∞ (, , ◇) (, , ) = →+∞(, , ).
=
⨂︀− 1 (, , ))
(, , □) = ⨂︀+= (, , )
(, , □) = →+∞ (, , □)
and (, , □) is decreasing in (assuming usual properties of t-norms and t-conorms for ⊗
Existence of the limits is ensured by the fact that (, , ◇) and (, , ) are increasing in ,
and ⊕ ).
to the semantics of FLTL (Fuzzy Linear-time Temporal Logic) by Lamine and Kabanza [29], i.e.,
As a consequence, for the case = [
Given a temporal preferential interpretation ℐ = ⟨ , {< }∈N, ⟩ the truth degree of an implication → in ℐ at time point is: ( → )ℐ, = ∈ ((, , ) ▷ (, , )). , ′ ∈ ; and (, ) = (, , ), for all ∈ .
Note that a temporal many-valued interpretation ℐ = ⟨ , {< }∈N, ⟩ can be regarded as a sequence of (non-temporal) preferential interpretations ℳ0 , ℳ1, ℳ2, . . . where each ℳ is defined as follows: ℳ = ⟨ , {< }, ⟩, where < ′ holds in ℳ if < ′ holds in ℐ, for all
In the temporal case, rather than regarding graded implications as global constraints, that have to hold at all the time points, we allow for boolean combination of graded implications (as done in [7]) and also for the temporal operators to occur in front of the graded implications and of their boolean combinations. We call such formulas temporal graded formulas. A temporal graded formula is defined as follows: ::= → ≥ | → ≥ | ∧ | ¬ | ○ | ◇ | □ | , example of temporal graded formula is the following conjunction: where and stand for temporal graded formulas. Note that temporal operators may occur both within graded implications ( → ≥ ) and in front of them, and of their boolean combinations. An □(T(professor ) → ℎ retired ≥
0.7) ∧ (lives_in_town ∧ young → T(◇granted _loan) ≥ 0.8) where the graded implication in the first conjunct is prefixed by a □ operator, while the second one is not.
A temporal conditional KB is a set of temporal graded formulas. We evaluate the satisfiability of a temporal graded formula at the initial time point 0 of a temporal preferential interpretation ℐ, essentially, as in LTL. Observe that any graded implication → ≥ is either satisfied or not at a time point of a temporal interpretation ℐ, i.e., either ℐ, |= → ≥ or ℐ, ̸|= → ≥ (and similarly for the graded implications with ≤ ). Hence, the interpretation above of temporal graded formulas in ℐ at a time point is two-valued (although it builds over the degree of an implication → in ℐ at time point , which has a truth value ( → )ℐ, in ). We refer to [19] for the detailed definition of satisfiability of a temporal graded formula at time point , and define the notions of satisfiability and entailment as follows.
Definition 5 (Satisfiability and entailment) . A temporal graded formula is satisfied in a temporal preferential interpretation ℐ = ⟨, {< }∈N, ⟩ if ℐ, 0 |= .
A preferential interpretation ℐ = ⟨, {< }∈N, ⟩ is a model of a temporal conditional knowledge base , if ℐ satisfies all the temporal graded formulas in .
A temporal conditional knowledge base entails a temporal graded formula if is satisfied in all the models ℐ of .
Note that, in the temporal graded formula given above, the graded implication in the first conjunct (T(professor ) → ℎ retired ≥ 0.7) is required to hold at all the time points of the interpretation ℐ (as it is prefixed by □), while the second conjunct (lives_in_town ∧ young → T(◇granted _loan) ≥ 0.8) has to hold only at time point 0.
Decidability and complexity of the diferent decision problems (the satisfiability, the model checking and entailment problems) have to be studied for this temporal many-valued conditional logic, for diferent choices of and of combination functions. In the two-valued case, a related formalism which extends the temporal description logic ℒ [11] with the typicality operator, has been shown to be decidable when only a finite set of preference relations < is considered [12], and concept inclusions are regarded as global temporal constraints.
As in the two-valued non-temporal case, the notion of preferential entailment considered in this section is rather weak. For the KLM logics, some diferent closure constructions have been proposed to strengthen entailment by restricting to a subset of the preferential models of a conditional knowledge base . Let us just mention, the rational closure [6], the lexicographic closure [30], and the MP-closure [18].
In this direction, we consider weighted temporal KBs, which allow defeasible implications with a weight, following an approach first proposed for weighted KBs in defeasible DLs [ 26, 13]. A weighted KB is a set of weighted typicality implication of the form (T() → , ), where and are propositions, and the weight is a real number, representing the plausibility or implausibility of the conditional implication. For instance, for a proposition student , we may have a set of weighted defeasible implications: (T(student ) → has_Classes, +50), (T(student ) → ◇holds_Degree,+30) , (T(student ) → has_Boss, -40), that represent prototypical properties of students, i.e., that a student normally has classes and will eventually reach the degree, but she usually does not have a boss (negative weight). Accordingly, a student having classes, but not a boss, is more typical than a student having classes and a boss.
A weighted (defeasible) knowledge base can coexist with a strict knowledge base (i.e., a set of graded implications), as usual in defeasible DLs. We refer to the extended version [19] for a semantics of weighted temporal KBs and for an instantiation of the approach for gradual argumentation.
The paper proposes a framework in which diferent (many-valued) preferential logics with typicality can be captured, together with their temporal extensions, with the operators from 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 (many-valued) case is suggested. In [19] we also consider an instantiation of the formalism to the verification of temporal properties of gradual argumentation graphs, an approach which extends the (multi-)preferential (typicality-based) approach for the verification of conditional properties of argumentation graphs in gradual argumentation semantics proposed in [7].
On a diferent route, in the two-valued case, a preferential logics with defeasible LTL operators
has been studied in [9, 31]. The decidability of diferent fragments of the logic has been proven, and
tableaux based proof methods for such fragments have been developed [8, 31]. Our approach does
not consider defeasible temporal operators nor preferences over time points, but combines standard
LTL operators with the typicality operator in a many-valued temporal logic. We have not considered
the additional temporal operators (“soon”, “almost always”, etc.) introduced by Frigeri et al. [28]
for representing vagueness in the temporal dimension, they can be considered for future work. The
quantitative value of satisfaction of extended LTL formulas is also considered in [32]. Our approach,
besides being many-valued, exploit a typicality operator, which allows for conditional implications and
makes the logic non-monotonic [
Future work also includes studying the decidability for fragments of the logic, developing proof methods, as in the non-temporal case [20, 33], exploiting the formalism for explainability, and for reasoning about the dynamics of argumentation graphs in a gradual semantics.
While conditional weighted KBs have been shown to capture the stationary states of some neural networks (or their finite approximation) [ 25, 26], and allow for combining empirical knowledge with elicited knowledge for post-hoc verification, adding a temporal dimension opens to the possibility of verifying properties concerning the dynamic behavior of a network.
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). [6] D. Lehmann, M. Magidor, What does a conditional knowledge base entail?, Artificial Intelligence 55 (1992) 1–60. [7] M. Alviano, L. Giordano, D. Theseider Dupré, Typicality, conditionals and a probabilistic semantics for gradual argumentation, in: Proc. 21st International Workshop on Non-Monotonic Reasoning (NMR 2023), Rhodes, Greece, September 2-4, 2023, volume 3464 of CEUR Workshop Proceedings, CEUR-WS.org, 2023, pp. 4–13. [8] A. Chafik, F. C. Alili, J. Condotta, I. Varzinczak, A one-pass tree-shaped tableau for defeasible LTL, in: TIME 2021, September 27-29, 2021, Klagenfurt, Austria, volume 206 of LIPIcs, 2021. [9] A. Chafik, F. C. Alili, J. Condotta, I. Varzinczak, On the decidability of a fragment of preferential LTL, in: TIME 2020, September 23-25, 2020, Bozen-Bolzano, Italy, volume 178 of LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. [10] A. Chafik, F. C. Alili, J. Condotta, I. Varzinczak, Defeasible linear temporal logic, J. Appl. Non
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