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A temporal defeasible logic for handling access control policies


Access control policies are specified within systems to ensure confidentiality of their information. Available knowledge about policies is usually incomplete and uncertain. An essential goal in reasoning is to reach conclusions which can be justified. However, since justification does not necessarily guarantee truth, the best we can do is to derive “plausible/ tentative” conclusions from partial and conflicting information. Policies are typically expressed as rules that could be complex and include timing constraints. Complex sets of access policies can contain conflicts e.g., a rule allows access while another rule prevents it. In this paper, we aim at providing a formalism for specifying authorization policies of a dynamic system. We present a temporal defeasible logic (TDL) which allows us to specify temporal policies and to handle conflicts. It can be shown that the proposed model is a generalization of the role-based access control model.

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Correspondence to Khair Eddin Sabri.

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

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

Let i, j, k, r, m ,n ∈ I and p, p 1 ∈ P. Let → be the implication of classical logic and A \(\leftrightarrow \) B iff (A → B)&(B → A).

A time structure is a tuple, M T=〈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)

      (p 1 <P p 2) & (p 2 <P p 3) → p 1 <P p 3 (Transitivity)

    2. (P2)

      \(\lnot \) (p 1 <P p 1) (Irreflexivity)

    3. (P3)

      (p 1 <P p 2) ∨ (p 1= p 2) ∨ (p 2 <P p 1) (linearity)

    4. (P4)

      (∀p) (∃ p 1)(p <P p 1) (U-Unboundedness)

    5. (P5)

      (∀p) (∃ p 1)(p 1 <P p) (L-Unboundedness)

    6. (P6)

      (∀p 1, p 2)(p 1 <P p 2)(∃p 3)(p 1 <P p 3 & p 3 <P p 2) (Density)

(P4) (resp. P5) states that for any time point p, there exists a point p 1 that comes after it, U-Unboundedness (resp. before it, L-Unboundedness).

  1. (3)

    Meets is axiomatized [1] as follows:

  1. (I1)

    (∀ i, j) (∃ k) (Meets(i, k) & Meets(j, k) → (∀ r) (Meets(i, r) ≡ Meets(j, r))

  2. (I2)

    (∀ i, j) (∃ k) (Meets(k, i) & Meets(k, j) → (∀ r) (Meets(r, i) ≡ Meets(r, j))

  3. (I3)

    (∀ i, j, k, r)(Meets(i, j) & Meets(k, r) → Meets(i, r) XOR

    (∃ m)(Meets(i, m) & Meets(m,r) XOR

    (∃ n)(Meets(k, n) & Meets(n, j)

  4. (I4)

    (∀ i)((∃ j, k)(Meets(j, i) & Meets(i, k))

  5. (I5)

    (∀ i, j)(Meets(i, j) → (∃ k = i + j,)(∃ m,n)(Meets(m, i) & Meets(i, j) &Meets(j, n) & Meets(m, k) & Meets(k, n))

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. k = i + j is only definable if Meets(i, j) holds and k contains exactly i, j and their meeting points p, i.e., k = i\(\cup \){p} \(\cup \)j. (I5) states that for any two adjacent intervals i and j, there exists an interval k such that k = i + j.

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

(PI 1):

(∀i)(∃p 1, p 2) (In(p 1, i) & In(p 2, i) & (p 1 ≠ p 2) & (p 1 <P p 2))

We may add the following definition:

Definition A.1

Let t ∈ P\(\cup \)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 different binary relations between intervals on a linear order (and quite a few more on a partial ordering) as shown in Fig. 2.

Fig. 2
figure 2

Binary relations between intervals

We may also define point-interval relations. Let p, p 1, p 2 ∈ P and t, t 1 ∈ I. Begin(p,t) states that p is the lower limit (beginning) of t. End(p,t) states that p is the Upper limit (end) of t. Begin(p,t) and End(p,t) can be defined as:


Begin(p,t) iff (∀p 1)[(In(p 1, t) → p ≤ P p 1) and

(∀p 2) if (p 2 ≠ p and (In(p 1 , t) → p 2 <P p 1) then p 2 <P p].(Def2) End(p,t) iff (∀p 1)[(In(p 1, t) → p 1 <P p) and

(∀p 2):

if (p 2 ≠ p and (In(p 1 , t) → p 1 <P p 2) then p <P p2].

From these definitions, we may derive the following axioms:

(PI 2):

(∀t) (∀p) (∀p 1)(Begin(p,t)&End(p 1,t) → p <P p 1)

(PI 3):

(∀t)(∃p)(∃p 1)(Begin(p,t)&End(p 1,t))

(PI 4):

(∀t)(Begin(p,t)&Begin(p 1,t)) → p = p1

(PI 5):

(∀t)(End(p,t)&End(p 1,t)) → p = p1

(PI 6):

(∀t) (∀t1)(Begin(p,t)&End(p 1,t)&Begin(p, t1)&End (p 1, t 1)) → t = t 1.


Before(p,t) iff p <P p 1 where Begin(p 1, t).


After(p,t) iff p 2 <P p where End(p 2, t).

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Sabri, K.E., Obeid, N. A temporal defeasible logic for handling access control policies. Appl Intell 44, 30–42 (2016).

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