In this paper we introduce a temporal, deontic, conditional logic with typicality, T. It combines a multipreferential conditional logic, which can be used for defeasible reasoning, with a temporal and deontic logic. The combination provides a formalism which is able to capture the dynamics of a system, through its strict and defeasible temporal properties, and also to reason about obligations and permissions. Temporal ranked knowledge bases are introduced for strengthening preferential entailment.
In this paper we aim at introducing a temporal, deontic, conditional logic with typicality, called T,
based on a preferential approach to commonsense reasoning [
Preferential approaches to commonsense reasoning have their roots in conditional logics [
Extending the conditional logic with the temporal and deontic operators allows considering the temporal dimension when reasoning about the defeasible properties of a system, e.g., to represent statements such as “normally students will get a degree", and also to capture rules describing obligations admitting exceptions (for instance the rule that normally people have the obligation to pay taxes within a deadline, but, in case of violation, they have the obligation to pay with fine).
The formalism can be exploited for explanation, and for reasoning about fulfillment of obligations,
in the verification of the compliance of a process (like a business process) to norms. It is well known
from the literature that “many normative rules allow for exceptions. Without defeasibility, it would be
impossible to distinguish exceptions from violations" [
In this regard, in this work we aim at a conditional extension of a fragment of the Deontic Dynamic
Linear Time Temporal Logic (Deontic DLTL) studied in [
Preferential extensions of LTL with defeasible temporal operators have been recently studied [
As in the Propositional Typicality Logic (PTL) by Booth et al. [18] (and in the DLs with typicality [19]) the conditionals are formalized based on material implication (resp., concept inclusions) plus the typicality operator T. A conditional statement “normally if holds, holds" is represented by a conditional implication T( ) → , meaning that “in the typical situations in which holds, also holds”. For instance, in this temporal formalism, the conditional implication: T(Student ) → ◇get _degree means that, normally, students will eventually get a degree (although not all students will).
When and are formulas of the propositional calculus, an implication T( ) → is intended to
correspond to the conditional |∼ in KLM logics [
After extending the conditional logic with typicality with deontic and temporal modalities, the paper provides a decidability result for the deontic temporal conditional logic T. Then, for strengthening preferential entailment, it introduces ranked knowledge bases (i.e., knowledge bases in which conditional implications are associated with a rank) and develops an approach for reasoning from a ranked knowledge base in the temporal setting, based on a closure construction in the style of the lexicographic closure [25] for KLM logics, by exploiting the strategy # from Brewka’s framework for qualitative preferences [26].
The schedule of the paper is the following. Section 2 introduces the two-valued preferential logic with typicality following [24]. Section 3 extends such a logic with the LTL modalities and with the deontic modalities to develop a temporal deontic conditional logic, and it proves the decidability of the logic. Section 4 focuses on ranked KBs and discusses a closure construction for the temporal case. Section 5 concludes the paper.
In this section, we recall the definition of a two-valued preferential logic with typicality from [ 24] and
slightly generalize it. The preferential semantics of the logic generalizes Kraus Lehmann and Magidor’s
(KLM) preferential semantics [
We consider a propositional language , whose formulae are built from a set Prop of propositional variables using the boolean connectives ∧, ∨, ¬ and → of propositional logic. We assume that ⊥ (representing falsity) and ⊤ (representing truth) are formulae of .
A typicality operator is introduced following the approach used in the description logic ℒ + T [27] as well as in the Propositional Typicality Logic (PTL), by Booth et al. [18]. We 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" [18]. As in PTL [18], the typicality operator cannot be nested. In an implication → , and may contain occurrences of the typicality operator. When T does not occur in nor , the implication → is called strict. When an implication has the form T( ) → , it is called a defeasible implication, whose meaning is that “normally, if then ”, and corresponds to KLM conditional |∼ .
The KLM preferential semantics [
Here, instead, we consider a multi-preferential semantics, where preference relations are associated with distinguished propositional formulas 1, . . . , (called distinguished propositions in the following). The idea is that how much a situation (a world) is normal (or less atypical) with respect to another one, depends on the aspects considered for comparison. The semantics introduced below exploits a set of preference relations < , each associated to a distinguished proposition , where < ′ means that world is less atypical than world ′ concerning aspect . As mentioned above, for two worlds and ′ in , it may be the case that <student ′, but ′ <employee .
In the following, we limit our consideration to finite KBs, and restrict our attention to a finite set of distinguished propositions 1, . . . , . Preferential interpretations are equipped with a finite set of preference relations <1 , . . . , < , one for each distinguished proposition . For each , we let < ⊆ × be a strict partial order on the set of worlds . So far we assume that, in any typicality formula T(), is a distinguished proposition, but we will lift this restriction in Section 4.2.
A (multi-)preferential interpretation is a triple ℳ = ⟨ , {< }, ⟩ where: ∙ is a non-empty set of worlds; ∙ for each , < ⊆ × is an irreflexive and transitive relation on ; ∙ : →− 2Prop is a valuation function, assigning to each world ∈ a set of propositional variables in Prop (the variables which are true in ).
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 < .
The valuation is inductively extended to all formulae of T: 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.
Let [[]]ℳ be the set of all the worlds in ℳ satisfying a formula (i.e., [[]]ℳ = { ∈ : ℳ, |= }) and let < () be the set of <-minimal worlds in , for any set of worlds ⊆ , and strict partial order <, that is: < () = { ∈ | there is no ′ ∈ , such that ′ < }. For a well-founded preference relation <, one can reformulate the semantic condition for the typicality operator as follows:
ℳ, |= T() if ∈ < ([[]]ℳ).
In this work, we do not assume that all the preference relations < are well-founded, so that the
semantics above is more general than the one in [24], and than the usual KLM semantics [
A formula is satisfiable in the multi-preferential semantics if there exist a multi-preferential interpretation ℳ = ⟨ , {< }, ⟩ and a world ∈ such that ℳ, |= . A formula is valid in an interpretation ℳ (written ℳ |= ) if, for all worlds ∈ , ℳ, |= . A formula is valid in the multi-preferential semantics (simply, A is valid) if is valid in any multi-preferential 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.
When an implication has the form T() → , with in , it corresponds to a conditional
|∼ in KLM logics [
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 [
In this section we further extend the language T with the temporal operators (next), (until), ◇
(eventually) and □ (always) of Linear Time Temporal Logic (LTL) [
We also introduce in the language a deontic operator O, which is read “it is obligatory that", and the
possibility operator P (“it is permitted that") with a semantics as in Standard Deontic Logic [
The syntax for temporal deontic conditional formulas, shortly, T formulas, is the following: ::= | ∧ | ∨ | ¬ | | | ◇ | □ | O() | P() | T() where and are well-formed formulas and is a distinguished formula.
Combining the modalities allows formulating conditions on the evolution of a system, taking into consideration the obligations of agents, their fulfillment or violation. For instance, normally, when receiving an invoice, one has the obligation to pay within a deadline (1)
T(_) → O(¬1 ) and the violation to pay within deadline 1 generates an obligation to pay with a fine within a new deadline 2:
O(¬1 )) ∧ 1 ∧ ¬ → O(¬2 _ ) The implication above is strict, but it might as well be formulated as a defeasible implication. Other examples of T formulae are:
□(T(professor ) → ℎ retired ) lives _in_town ∧ young → T(◇granted _loan)
Compared with the preferential semantics in the previous section, the semantics of a temporal deontic logic with typicality has also to consider 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). At each time point an accessibility relation is used for evaluating obligations at time point . ℐ = ⟨ , {< }∈N, {}∈N, ⟩ where:
A temporal deontic (multi-)preferential interpretation (or T interpretation) is a triple • 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 ∈ N, ⊆ × is a serial accessibility relation.
For ∈ and ∈ N, (, ) is the set of the propositional variables which are true in world at time point . When there is no ′ ∈ s.t. ′ < , we say that is a normal situation for at time point . is the accessibility relation of the deontic modality O at time point . We have assumed that, for all time points ∈ N, is serial, that is: for all ∈ , there is a ′ ∈ such that (, ′) ∈ . This is the usual assumption in SDL. O() is true in a world at time point if is true in all the worlds which are ideal with respect to at time point (i.e., in all the worlds ′ such that (, ′) ∈ ).
Given an T interpretation ℐ = ⟨ , {< }∈N, {}∈N, ⟩, we define inductively the truth 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, , |= O() if for all ′ ∈ , such that (, ′) ∈ , ℐ, , ′ |= ℐ, , |= T() if ℐ, , |= and ∄w ′ ∈ s.t. w ′ <nAi w and ℐ, , ′ |= .
Note that a temporal interpretation ℐ = ⟨, {< }∈N, {}∈N, ⟩ can be regarded as a sequence of (non-temporal) deontic preferential interpretations ℳ0, ℳ1, ℳ2, . . . where each ℳ is defined as follows: ℳ = ⟨, {< }, , ⟩, where < ′ holds in ℳ if < ′ holds in ℐ, for all , ′ ∈ ; () = (, ), for all ∈ , and in ℳ is the accessibility relation at time point in ℐ.
A temporal deontic conditional KB is a set of T formulas. We evaluate the satisfiability of a temporal formula in a temporal preferential interpretation ℐ, by verifying its truth at the initial time point 0 of the interpretation ℐ. .
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 (written ℐ |= ) if ℐ, 0, |= , for all worlds ∈ .
A preferential interpretation ℐ = ⟨, {< }∈N, ⟩ is a model of a temporal deontic conditional knowledge base , if ℐ |= holds, for all the formulas in .
A temporal deontic conditional knowledge base entails a formula if ℐ |= for all the models ℐ of
The temporal deontic logic with typicality introduced in this section can be proven to be decidable. when
the preference relations < are well-founded. The problem of deciding the satisfiability of a T
formula can be polynomially reduced to deciding the satisfiability of a concept in the description
logic Tℒ introduced in [
Concept satisfiability in ℒTℒ can be polynomially reduced to concept satisfiability in ℒ,
when a finite set of well-founded preference relations <1 , . . . , < is considered, and concept
inclusions are regarded as global temporal constraints [
To provide a sketch of the decidability result for T, let us introduce in the language of the description logic Tℒ a concept name , for each proposition ∈ Prop, and a role , associated to the deontic operator O. Given an T formula , we define the concept associated to in the description logic Tℒ, as follows (by induction on the structure of the formula): = , if ∈ Prop ¬ = ¬. ∧ = ⊓ ∨ = ⊔ O() = ∀. T() = T( ) = □ = □ ◇ = ◇ P() = ∃. In order to enforce seriality for the deontic modality O, we let the concept inclusion ⊤ ⊑ ∃.⊤ belong to the TBox . As assumed before, such an inclusion is global and holds for all individuals at any time point. Note that the typicality operator can be used in Tℒ, so that any formula T() can be mapped to a concept T(). The encoding above of a formula into a concept is clearly polynomial in the size of the formula , and the TBox only contains one axiom.
It can be proven that a T formula is satisfiable if and only if the concept is satisfiable in the
description logic Tℒ w.r.t. TBox . We omit the proof and refer to [
Proposition 1. The satisfiability of an T formula is in ExpTime, both with expanding domains and with constant domains.
As for KLM logics, the notion of preferential entailment considered in this section is rather weak.
For 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 [
In particular, here we will focus on ranked knowledge bases, that were first explored by Brewka in his framework for qualitative preferences [29, 26], where basic preference relations ≥ are associated with diferent ranked knowledge bases , and new preference relations are defined by combining the basic preference relations. More precisely, we allow for ranked temporal conditional KBs, in which conditional implications have a rank, a natural number. Each distinguished formula is associated with a set of conditionals, which are used for defining the preorders ≤ , as well as the strict preference relations < (the associated strict partial orders). An approach using ranks was adopted in [22] in a conditional extension of a lightweight description logic, and in [21] in a framework for modular, multi-concept preferential semantics based on the lexicographic closure (where ranks are determined by the rational closure construction). Here we extend the construction to the temporal case, and we exploit user defined ranks for the conditionals, based on the strategy # from Brewka’s framework for qualitative preferences [26], extended to the temporal case.
Let us introduce the ranked conditional KBs for the temporal case, through an example. Let be a set of conditional implications, with their ranks, for the distinguished formula student . They describe the typical properties of students. The higher is the rank, the higher is the priority of the conditional formula: 1: T(student ) → O(have_Classes) , 0 2: T(student ) → ◇get _Degree, 0 3: T(student ) → ¬has_Boss, 0 4: T(student ∧ employee) → has_Boss, 1 Normally, a student has the obligation to have classes, she will eventually get a degree, and she does not have a boss; while a student who is also an employee has normally a boss. The rules above are intended to define a preference relation <. In particular, worlds violating conditionals with lower ranks are preferred with respect to worlds violating conditionals with higher ranks. We use the same relation, concerning defaults, as in Lehmann’s lexicographic closure [25], but here we assume that the ranks of the conditionals are given. For instance, a world describing a student having classes and not having a boss, and that will eventually get a degree, is more typical than a world ′ describing a student and employee having classes and a boss, and that will eventually get the degree. In particular, ′ violates the conditional 3, while does not violate any conditional (the formal definition is given below).
In a ranked conditional knowledge base, the sets of ranked conditionals 1 , . . . , associated with the distinguished formulas 1, . . . , , coexist with a strict part of the knowledge base , i.e., a set of formulas which do not contain the typicality operator. For instance, by the strict implication student → O(paid _taxes (get _degree ∨ withdraw )) in , all students have the obligation to pay the enrollment taxes until they either get the degree or withdraw, with no exceptions. A ranked conditional knowledge base is represented by a tuple = ⟨ , 1 , . . . , ⟩, with the strict part, and 1 , . . . , (the defeasible part of the KB) are sets of ranked conditionals. Each is a set of pairs (T() → , ) associating defeasible implications with a rank. Note that is not required to coincide with . As in the example above we allow any formula in a ranked conditional implication (T() → , ).
Given a ranked conditional knowledge base = ⟨ , 1 , . . . , ⟩, and a preferential temporal interpretation ℐ, one can define a preorder ≤ for each distinguished proposition at time point , by considering the rank of each defeasible implication (T() → , ) in .
(, ) be the set of typicality inclusions in with rank , being satisfied by world at We let time point in a temporal interpretation ℐ, that is:
(, ) = {T() → | (T() → , ) ∈ and ℐ, , ̸|= or ℐ, , |= }.
We define the preference relations ≤ as follows: Definition 4. Given a ranked conditional knowledge base = ⟨ , 1 , . . . , ⟩, and an T interpretation ℐ = ⟨, {< }∈N, {}∈N, ⟩, for all worlds 1, 2 ∈ , we let 1 ≤ 2 if either | (1, )| = | (2, )|, for all , or ∃ such that | (1, )| > | (2, )| and, ∀ℎ > , |ℎ (1, )| = |ℎ (2, )|.
Informally, the preference relation ≤ gives higher preference to worlds violating a smaller number of conditional implications with higher rank for at time point . It corresponds to the strategy # in Brewka’s framework for qualitative preferences [26], transferred to the temporal case. The preorder ≤ is total. The strict preference relation < is defined as usual from the preorder relation as: < ′ if ≤ ′ and ′ ̸≤ .
Definition 5. An T interpretation ℐ = ⟨, {< }∈N, {}∈N, ⟩, is a model of the ranked knowledge base = ⟨ , 1 , . . . , ⟩, if ℐ is a model of according to Definition 3 and, for each distinguished proposition , the preference relation ≤ is defined according to Definition 4.
A formula is entailed from a ranked T knowledge base = ⟨ , 1 , . . . , ⟩ if ℐ |= for all the models ℐ of the ranked knowledge base .
Let us continue with our example. Whether an obligation is fulfilled or not at a world, depends on the trajectory starting at that world. One can represent a situation in which an obligation, normally, has to be fulfilled, but it may have exceptions. For instance, the obligation O(_ (_ ∨ ℎ)) should by normally fulfilled but, when it is not, there is a new obligation to pay with a ifne within (e.g.) the next May (a contrary to duty obligation). Also, the obligation is normally canceled if there is a tax amnesty. This policy can be encoded by a set of defeasible implications (for sake of conciseness, we will use O(1) as a short name for the formula O(_ (_ ∨ ℎ))): 7: T(O(o1 ) ∧ tax _amnesty )→ X ¬O(o1 ), 2 4: T(O(o1 )) → paid _taxes (got _degree ∨ withdrawn), 0 5: T(O(o1 ) ∧ paid _taxes ∧ ¬got _degree ∧ ¬withdraw ) → X O(o1 ), 0 6: T(O(o1 ) ∧ ¬paid _taxes ∧ ¬got _degree ∧ ¬withdrawn)
→ X O(¬may pay _fine) ∧ X ¬O(o1 ), 1 By conditional 5, the obligation O(1) normally persists to the next state if it has not been violated (paid _taxes holds) and it is not already fulfilled (by reaching a degree or withdrawing). Conditional 6, with rank 1, states that, in a typical situation in which obligation O(1) is violated (i.e., O(1) holds, but taxes have not been paid, and the student neither has received a degree nor has withdrawn), in the next state a new obligation to pay with fine within May is added, and O(1) is canceled. By 7 the obligation O(1) is normally canceled if there is a tax amnesty (conditional with rank 2).
In this example, we may assume that the conditionals concerning the payment of taxes for students belong as well to the set of ranked conditionals . Based on the semantics above, this set of ranked conditionals induces a preference relation <, which enable us to prove, for instance the property that normally students will pay taxes until they get a degree or withdraw:
T(Student ) → paid _taxes (got _degree ∨ withdrawn) (i.e., that the obligation for students to pay taxes is normally fulfilled).
In the general case, we may want to verify conditional properties of the form T() → , where is not a distinguished proposition. For instance, we may want to check whether the conditional
T(Student ∧ Employee) → paid _taxes (got _degree ∨ withdrawn) is entailed from a ranked KB also in case Student ∧ Employee is not a distinguished proposition. When general typicality formulas of the form T() are admitted (provided does not contain the typicality operator), we also needs to generalize the semantic condition for evaluating the typicality operator T().
The semantic condition in Definition 2 can be extended to all typicality formulas, as follows: ℐ, , |= T() if ℐ, , |= and ∄w ′ ∈ s.t. w ′ <nA w and ℐ, , ′ |= .
This requires to provide a definition of the preference relation < also for the cases when is not a distinguished formula. New preference relations can be obtained by combining the preference relations <1 , . . . , < , based on Brewka’s framework for preference combination [26].
In Brewka’s framework, qualitative preferences [29, 26] are defined, starting from basic preference descriptions to define preorders on models (propositional interpretations). More precisely, a logical preference description language, LPD, is introduced, to combine basic preference descriptions 1 and 2 into complex ones. If 1 and 2 are preference descriptions in LPD, also 1 ∧ 2, 1 ∨ 2, ¬1 and 1 > 2 are preference descriptions in LPD, where: 1 ∧ 2 is defined as the set theoretic intersection of relations 1 and 2; 1 ∨ 2 is defined as the transitive closure of the set theoretic union of 1 and 2; ¬1 is defined as the inverse of relation 1, and 1 > 2 is intended to express preferences among expressions (in particular, preference 1 has higher priority with respect to preference 2; we refer to [26] for details).
In our context, the preference relations ≤ 1 , . . . , ≤ , associated with the distinguished propositions 1, . . . , play the role of basic preference descriptions to be combined, based on the framework above. Given the preferences ≤ and ≤ , we let: ≤ ∧ = (≤ ∩ ≤ ), ≤ ∨ = (≤ ∪ ≤ )+, and ≤ ¬ = (≤ )− 1, where (≤ )− 1 is the inverse of preference relation ≤ , (≤ ∪ ≤ )+ is the transitive closure of the set theoretic union of ≤ and ≤ , and (≤ ∩ ≤ ) is the set theoretic intersection of ≤ and ≤ .
Note that, when ≤ and ≤ are total preorders, ≤ ¬ and ≤ ¬ are total preorders; and the union ≤ ∪ ≤ is transitive, and is as well a total preorder (in this case, applying the transitive closure is not needed). Note also that ≤ ∧ is a preorder, but not necessarily total. When ≤ and ≤ are ranked preference relations, it holds that: ≤ ∧ ′ if ≤ ′ and ′ ≤ ′ ≤ ∨ ′ if ≤ ′ or ′ ≤ ′ ≤ ¬ ′ if ′ ≤ As observed by Brewka, the preference description ¬(1 ∨ 2) is diferent from ¬1 ∧ ¬2 and, in our context, the preference relation ≤ ¬(1∨2) difers from ≤ ¬1∧¬2 . To avoid the problem that equivalent formulas may be associated to diferent preference relations, we let the relation ≤ associated to a boolean formula be the preference relation associated to its conjunctive normal form (CNF). The preference relation associated to a formula in CNF can be computed from the basic ranked preferences, taking their inverse (still total preorders), then computing the preferences of the disjunctions as unions of total preorders, and finally, computing the intersections of the preferences for the disjuncts.
Once a preference relation ≤ has been associated to each formula , the induced strict partial order < can be used in the evaluation of the typicality formula T() while computing entailment.
This approach for dealing with ranked KBs provides an example of the constructions which can
be adopted for reasoning in the temporal deontic conditional logic with ranked knowledge bases.
Alternative constructions could be used for ranked knowledge bases, including, for instance, exploiting
diferent lexicographic orders, while approaches, based on weighted knowledge bases can also be
considered (as done for temporal defeasible Description Logics [
The paper proposes a temporal, deontic, conditional logic with typicality, T, based on a preferential semantics and exploiting the operators of LTL and the deontic operators of SDL. The interpretation of the typicality operator is based on a multi-preferential semantics, and exploits an extension of ranked conditional knowledge bases to the temporal deontic case.
Our starting point for defining T is a multi-preferential logic with typicality, a logic which allows for conditional reasoning based on a multi-preferential semantics. It is defined along the lines of preferential logics with typicality, such as the description logic ℒ + T [27] and the Propositional Typicality Logic (PTL) [18] and, more precisely, along the lines of multi-preferential logics with typicality, which have been used for conditional reasoning about multilayer networks [23] and about gradual argumentation semantics [30, 31], and have been recently exploited in a conditional extension of Answer Set Programming (Conditional ASP) [24]. The paper borrows and extends the propositional multi-preferential semantics in [24].
Future work includes studying diferent closure constructions, including constructions based on weighted KBs for temporal conditionals, as well as considering extensions of Answer Set Programming with temporal conditionals and providing reasoning tools.
On a diferent route, in the two-valued case, a preferential logics with defeasible LTL operators has
been studied in [
Related work also include the many-valued conditional logic for the verification of temporal properties of gradual argumentation graphs under a gradual argumentation semantics developed in [33, 34]. In this paper we have considered a two-valued temporal conditional logic, also including the deontic operators and we have proved a decidability result. We have as well developed a closure construction based on ranked knowledge bases, and their combination.
The Temporal Modal Defeasible Logic [
A Dynamic Deontic and Temporal Logic has been proposed by Dignum and Kuiper [37] to reason
about obligations and deadlines. In particular, they provide a formalization of achievement obligations
as obligations with an until formula as argument. This idea has been later exploited in a simple Deontic
Dynamic Temporal Logic (DDLTL) [
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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