An Integer 0-1 Linear Programming Approach for Computing Inconsistency Degree in Product-Based Possibilistic DL-Lite

Abstract

This paper considers the problem of computing inconsistency degree of uncertain knowledge bases expressed in product-based possibilistic DL-Lite, which is an extension of DL-Lite to deal with uncertainty in the product-based possibility theory framework. Indeed, computing the inconsistency degree is at the heart of any query answering process in such knowledge bases. Unlike previous work where uncertainty is only considered at the ABox level, in the present work both ABox and TBox may be uncertain. We discuss the new form of conflicts and how to obtain them by a generalized negative closure procedure. Then, we model the inconsistency degree computation as an integer 0-1 linear programming problem and we show the efficiency of this choice by a comparison with two other solutions, using the weighted Max-SAT and the approximate greedy algorithm for the weighted set cover problem, respectively.

Appendix: Proofs

Proof

of Proposition 1.

In Proposition 1, ${\mathcal {K}}_1$ represents a Pb-$\pi$-DL-Lite KB obtained from ${\mathcal {K}}$ by adding the assumption that B(a) (resp. R(a, b)) is surely false. Note that item 2. follows immediately from item 1. of Proposition 1 and Eq. 5. Hence, it is enough to show that item 1. of Proposition 1 holds. So, let us show that $Inc({\mathcal {K}}_1)= N_{{\mathcal {K}}}(B(a))$ (a similar reasoning is valid for $Inc({\mathcal {K}}_1)= N_{{\mathcal {K}}}(R(a,b))$).

Let I be a DL-lite interpretation. Since ${\mathcal {K}}_1$ is composed of ${\mathcal {K}}\cup \{(D \sqsubseteq \lnot B, 1), (D(a), 1)\}$, we have two cases:

• If $I\,\,\models \,\,D(a)$ and $I \,\models \, D \sqsubseteq \lnot B$ (Hence $I \nvDash B(a)$) then $\pi _{{\mathcal {K}}_1}(I) = \pi _{{\mathcal {K}}}(I)$.

• If $I \nvDash D(a)$ or $I \nvDash D \sqsubseteq \lnot B$ then $\pi _{{\mathcal {K}}_1}(I) = 0$.

Hence, we have: $\max _{I\in \varOmega } (\pi _{{\mathcal {K}}_1}(I)) = \max _{I\in \varOmega , I \nvDash B(a)} (\pi _{{\mathcal {K}}}(I))$. It follows from Eqs. 3 and 4 that $Inc({\mathcal {K}}_1) = 1 - \max _{I\in \varOmega , I \nvDash B(a)} (\pi _{{\mathcal {K}}}(I)) = N_{{\mathcal {K}}}(B(a))$    $\blacksquare$

Proof

of Proposition 2.

Let K be a Pb-$\pi$-DL-Lite KB, $LP_{{\mathcal {K}}}$ be the corresponding integer linear program defined as in Definition 4. We consider that the formulas (axioms and assertions) of ${\mathcal {K}}$ are indexed from 1 to $n = |{\mathcal {T}}\cup {\mathcal {A}}|$. The formula $\langle \varPhi _i, \alpha _i \rangle$ in ${\mathcal {K}}$ corresponds to the variable of index i in $LP_{{\mathcal {K}}}$.

$\Rightarrow$) Suppose that $Inc({\mathcal {K}}) = \lambda$. Then by Definition 3, $\lambda = 1 - \max _{I\in \varOmega } (\pi _{{\mathcal {K}}}(\textit{I}))$. Let $I_1$ be an interpretation such that $\pi _{{\mathcal {K}}}(I_1) = \max _{I\in \varOmega } (\pi _{{\mathcal {K}}}(\textit{I}))$.

Let J be the set of indices of formulas falsified by $I_1$, i.e., . By Definition 2, $\pi _{{\mathcal {K}}}(I_1) = \prod _{i \in J} (1-\alpha _i)$ and hence $\lambda = 1- \pi _{{\mathcal {K}}}(I_1)$.

Let $Y=(y_1, \dots , y_n)$ be a binary vector such that $y_i = 1$ if $i\in J$ and $y_i=0$ otherwise. Let us show that Y is an optimal solution of $LP_{{\mathcal {K}}}$ whose objective function is $F(\lambda )$.

Since for all $i\notin J$, $I_1\,\models \, \langle \varPhi _i, \alpha _i \rangle$, it is clear that the set $S=\{\langle \varPhi _i, \alpha _i \rangle \in {\mathcal {T}}\cup {\mathcal {A}}~|~i\notin J\}$ is consistent and hence does not contain any supported conflict. This means that for every supported conflict ${\mathcal {C}}_j$ there is a formula $\langle \varPhi _i,\alpha _i\rangle \in {\mathcal {C}}_j$ such that $i\in J$, i.e., $y_i = 1$. It follows that the corresponding constraint $C_j$ is satisfied by Y.

The objective function of Y is: $\sum _{i=1}^{n} F(\alpha _i).y_i$ = $\sum _{i\in J} F(\alpha _i)$ = $\sum _{i\in J} -ln(1-\alpha _i)$ = $-ln(\prod _{i\in J} (1-\alpha _i))$ = $F(1-\prod _{i\in J} (1-\alpha _i))$ = $F(1 - \pi _{{\mathcal {K}}}(I_1))$ = $F(\lambda )$.

Now, to show that Y is an optimal solution, suppose for the sake of contradiction that there is a solution $Z=(z_1, \dots , z_n)$ of $LP_{{\mathcal {K}}}$ such that $\sum _{i=1}^{n} F(\alpha _i).z_i < \sum _{i=1}^{n} F(\alpha _i).y_i$. Let $J'$ be the set of indices of variables put to 1 in Z: $J'=\{i~|~z_i = 1\}$ and let $I_2$ be an interpretation that satisfies all the formulas $\langle \varPhi _i, \alpha _i \rangle$ where $i\notin J'$. Such an interpretation exists because the set of formulas $H=\{\langle \varPhi _i,\alpha _i \rangle ~|~i\notin J'\}$ is consistent. Indeed if we suppose that this is not the case, it follows that there is a supported conflict ${\mathcal {C}}_j\subseteq H$ i.e., for all $\langle \varPhi _i,\alpha _i\rangle \in {\mathcal {C}}_j$ we have $z_i = 0$. This means that the constraint $C_j$ is not satisfied by Z which contradicts the fact that Z is a solution of $LP_{{\mathcal {K}}}$. The possibility degree of $I_2$ is given by: $\pi _{{\mathcal {K}}}(I_2)=\prod _{i\in J'} (1-\alpha _i)$. Now, it holds that:

$\sum _{i=1}^{n} F(\alpha _i).z_i < \sum _{i=1}^{n} F(\alpha _i).y_i$ $\Leftrightarrow$ $\sum _{i\in J'} F(\alpha _i) < \sum _{i\in J} F(\alpha _i)$

$\Leftrightarrow$ $\sum _{i\in J'} -ln(1-\alpha _i) < \sum _{i\in J} -ln(1-\alpha _i)$

$\Leftrightarrow$ $-ln(\prod _{i\in J'}(1-\alpha _i)) < -ln(\prod _{i\in J}(1-\alpha _i))$

$\Leftrightarrow$ $ln(\prod _{i\in J'}(1-\alpha _i)) > ln(\prod _{i\in J}(1-\alpha _i))$

$\Leftrightarrow$ $\prod _{i\in J'}(1-\alpha _i) > \prod _{i\in J}(1-\alpha _i)$ $\Leftrightarrow$ $\pi _{{\mathcal {K}}} (I_2) > \pi _{{\mathcal {K}}} (I_1)$.

This contradicts the fact that $\lambda$ is the inconsistency degree of ${\mathcal {K}}$.

$\Leftarrow$) Let $Y=(y_1,\dots ,y_n)$ be an optimal solution of $LP_{{\mathcal {K}}}$ whose objective function value is $F(\lambda ) = \sum _{i=1}^{n} F(\alpha _i).y_i$. Let $J = \{i~|~y_i = 1\}$. It is easy to check that: $\lambda = 1 - \prod _{i\in J} (1-\alpha _i)$.

Let $I_1$ be an interpretation that satisfies all the formulas $\langle \varPhi _i, \alpha _i \rangle$ where $i\notin J$. Such an interpretation exists because the set of formulas $H=\{\langle \varPhi _i,\alpha _i \rangle ~|~i\notin J\}$ is consistent. Indeed if we suppose that this is not the case, it follows that there is a supported conflict ${\mathcal {C}}_j\subseteq H$ i.e., for all $\langle \varPhi _i,\alpha _i\rangle \in {\mathcal {C}}_j$ we have $y_i = 0$. This means that the constraint $C_j$ is not satisfied by Y which contradicts the fact that Y is a solution of $LP_{{\mathcal {K}}}$. The possibility degree of $I_1$ is given by $\pi _{{\mathcal {K}}}(I_1)=\prod _{i\in J} (1-\alpha _i)$ and hence, $\lambda = 1-\pi _{{\mathcal {K}}}(I_1)$. To show that $\lambda$ is the inconsistency degree of ${\mathcal {K}}$ it suffices to show that $\pi _{{\mathcal {K}}}(I_1) = \max _{I\in \varOmega } (\pi _{{\mathcal {K}}}(I))$.

Suppose for the sake of contradiction that there is an interpretation $I_2$ such that $\pi _{{\mathcal {K}}}(I_2) > \pi _{{\mathcal {K}}}(I_1)$. Let $J'$ be the set of indices of formulas falsified by $I_2$: and let $Z=(z_1,\dots ,z_n)$ be a binary vector such that $z_i = 1$ if $i\in J'$ and $z_i=0$ otherwise.

Let us show that Z satisfies all the constraints of $LP_{{\mathcal {K}}}$. Since for all $i\notin J'$, $I_2\,\models \, \langle \varPhi _i, \alpha _i \rangle$, it is clear that the set $S=\{\langle \varPhi _i, \alpha _i \rangle \in {\mathcal {T}}\cup {\mathcal {A}}~|~i\notin J'\}$ is consistent, i.e., does not contain any supported conflict. This means that for every supported conflict ${\mathcal {C}}_j$ there is a formula $\langle \varPhi _i,\alpha _i\rangle \in {\mathcal {C}}_j$ such that $i\in J$, i.e., $z_i = 1$. It follows that the corresponding constraint $C_j$ is satisfied by Y. Now, it holds that:

$\pi _{{\mathcal {K}}} (I_2) > \pi _{{\mathcal {K}}} (I_1)$ $\Leftrightarrow$ $\prod _{i\in J'}(1-\alpha _i) > \prod _{i\in J}(1-\alpha _i)$

$\Leftrightarrow$ $ln(\prod _{i\in J'}(1-\alpha _i)) > ln(\prod _{i\in J}(1-\alpha _i))$ $\Leftrightarrow$ $\sum _{i\in J'} ln(1-\alpha _i) > \sum _{i\in J} ln(1-\alpha _i)$

$\Leftrightarrow$ $-\sum _{i\in J'} ln(1-\alpha _i) < -\sum _{i\in J} ln(1-\alpha _i)$

$\Leftrightarrow$ $\sum _{i\in J'} -ln(1-\alpha _i) < \sum _{i\in J} -ln(1-\alpha _i)$

$\Leftrightarrow$ $\sum _{i\in J'} F(\alpha _i) < \sum _{i\in J} F(\alpha _i)$ $\Leftrightarrow$ $\sum _{i= 1}^{n} F(\alpha _i).z_i < \sum _{i= 1}^{n} F(\alpha _i).y_i$.

But this contradicts the fact that Y is an optimal solution of $LP_{{\mathcal {K}}}$    $\blacksquare$