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A temporal modal defeasible logic for formalizing social commitments in dialogue and argumentation models

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Abstract

In this paper, we extend a temporal defeasible logic with a modal operator Committed to formalize commitments that agents undertake as a consequence of communicative actions (speech acts) during dialogues. We represent commitments as modal sentences. The defeasible dual of the modal operator Committed is a modal operator called Exempted. The logical setting makes the social-commitment based semantics of speech acts verifiable and practical; it is possible to detect if, and when, a commitment is violated and/or complied with. One of the main advantages of the proposed system is that it allows for capturing the nonmonotonic behavior of the commitments induced by the relevant speech acts.

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Correspondence to Nadim Obeid.

Appendix A: Time theory based on points and intervals (PI)

Appendix A: Time theory based on points and intervals (PI)

Let ti, tj, tk, tr, tm ,tn ∈ I and tp, tp1 ∈ P. Let → be the implication of classical logic and A ⇔ B iff (A → B) ∧ (B → A).

A time structure is a tuple, MT = 〈P, I, <P , Meets, In 〉 where

  1. (1)

    P and I are non-empty sets of points and intervals respectively,

  2. (2)

    <P is a precedence relation on points of time. <P has the following properties:

    1. (P1)

      (tp1 <P tp2) ∧ (tp2 <P tp3) → tp1 <P tp3 (Transitivity)

    2. (P2)

      ¬ (tp1 <P tp1) (Irreflexivity)

    3. (P3)

      (tp1 <P tp2) ∨ (tp1 = tp2) ∨ (tp2 <P tp1) (linearity)

    4. (P4)

      (∀t p) (∃ tp1)(tp <P tp1) (U-Unboundedness)

    5. (P5)

      (∀t p) (∃ tp1)(tp1 <P tp) (L-Unboundedness)

    6. (P6)

      (∀t p1, tp2)(tp1 <P tp2)(∃ t p3)((tp1 <P tp3) ∧ (tp3 <P tp2)) (Density)

    (P4) (resp. P5) states that for any time point tp, there exists a point tp1 that comes after it, U-Unboundedness (resp. before it, L-Unboundedness).

  1. (3)

    Meets is axiomatized [1] as follows:

    1. (I1)

      (∀ ti, tj)(∃ tk) (Meets(ti, tk) ∧ Meets(tk, tj) →

      (∀ tr) (Meets(ti, tr) ≡ Meets(tj, tr))

    1. (I2)

      (∀ ti, tj)(∃ tk) (Meets(tk, ti) ∧ Meets(tk, tj) →

      (∀ tr) (Meets(tr, ti) ≡ Meets(tr, tj))

    1. (I3)

      (∀ ti, tj, tk, tr)(Meets(ti, tj) ∧ Meets(tk, tr) →

      Meets(ti, tr) XOR

      (∃ tm)(Meets(ti, tm) ∧ Meets(tm,tr) XOR

      (∃ tn)(Meets(tk, tn) ∧ Meets(tn, tj)

    1. (I4)

      (∀ ti)((∃ tj, tk)(Meets(tj, ti) ∧ Meets(ti, tk))

    1. (I5)

      (∀ ti, tj (Meets(ti, tj) → (∃ tk = ti + t j,)(∃ tm,tn)(Meets(tm, ti) ∧

      Meets(ti, tj) ∧ Meets(tj, tn) ∧

      Meets(tm, tk) ∧ Meets(tk, tn))

    where XOR denotes exclusive OR. (I1) and (I2) state that every interval has a unique start point and a unique end point. (I3) defines all the possible relations between any two meeting places. (I4) states that every interval has one interval that precedes and an interval that succeeds it. tk = ti + t j is only definable if Meets(i, j) holds and k contains exactly ti, tj and their meeting points tp, i.e., tk = ti ∪{tp} ∪t j. (I5) states that for any two adjacent intervals ti and tj, there exists an interval tk such that tk = ti + t j.

  1. (4)

    In is a point-interval relation that is governed by the following axiom:

(PI1):

(∀t i)(∃ t p1, tp2) (In(tp1, ti) ∧ In(tp2, ti) ∧ (tp1≠ tp2) ∧ (tp1<P tp2)

We may add the following definition:

Definition A.1

Let t ∈P ∪I.

Duration (t) = 0 iff t ∈P and

Duration(t) >0 iff t ∈I.

Given the above set of axioms we may define other interval-interval relations. It is well known that there are 13 differentbinary relations between intervals on a linear order (and quite a few more on a partial ordering) as shown in Fig. 1.

Fig. 1
figure 1

Binary relations between intervals

We may also define point-interval relations. Let tp,tp1,tp2 ∈P and t,t1 ∈I.Begin(tp,t) states thattpis the lower limit(beginning) of t. End(tp,t)states that t p is the Upperlimit (end) of t. Begin(tp,t)and End(tp,t)can be defined as:

(Def1):

Begin(tp,t)iff (∀t p1)[(In(tp1,t) → tpPtp1) ∧

(∀t p2)if (tp2≠tpand (In(tp1, t) → tp2 < Ptp1)then tp2 < Ptp].

(Def2):

End(tp,t)iff (∀t p1)[(In(tp1,t) → tp1 < Ptp) ∧

(∀t p2)if (tp2≠tpand (In(tp1, t) → tp1 < Ptp2)then tp < Ptp2].

From these definitions, we may derive the following axioms:

(PI2):

(∀t) (∀t p) (∀t p1)(Begin(tp,t) ∧End(tp1,t) → tp<P tp11)

(PI3):

(∀t)(∃ tp)(∃ tp1)(Begin(tp,t) ∧End(tp1,t))

(PI4):

(∀t)(Begin(tp,t) ∧Begin(tp1,t)) → tp = tp1

(PI5):

(∀t)(End(tp,t) ∧End(tp1,t)) → tp = tp1

(PI6):

(∀t) (∀t1)(Begin(tp,t) ∧End(tp1,t) ∧Begin(tp, t1) ∧End(tp1, t1)) → t = t1.

(Def3):

Before(tp,t) iff tp <P tp1 where Begin(tp1, t).

(Def4):

After(tp,t) iff tp2 <P tp where End(tp2, t).

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Moubaiddin, A., Salah, I. & Obeid, N. A temporal modal defeasible logic for formalizing social commitments in dialogue and argumentation models. Appl Intell 48, 608–627 (2018). https://doi.org/10.1007/s10489-017-0983-3

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