A Bird’s-Eye View of Forgetting in Answer-Set Programming

Abstract

Forgetting is an operation that allows the removal, from a knowledge base, of middle variables no longer deemed relevant, while preserving all relationships (direct and indirect) between the remaining variables. When investigated in the context of Answer-Set Programming, many different approaches to forgetting have been proposed, following different intuitions, and obeying different sets of properties.

This talk will present a bird’s-eye view of the complex landscape composed of the properties and operators of forgetting defined over the years in the context of Answer-Set Programming, zooming in on recent findings triggered by the formulation of the so-called strong persistence, a property based on the strong equivalence between an answer-set program and the result of forgetting modulo the forgotten atoms, which seems to best encode the requirements of the forgetting operation.

The author was partially supported by FCT UID/CEC/04516/2013. This paper is substantially based on [1, 2].

A Answer-Set Programming

We assume a propositional signature $\mathcal {A}$, a finite set of propositional atoms⁵. An (extended) logic program P over $\mathcal {A}$ is a finite set of (extended) rules of the form

$$\begin{aligned} a_1 \vee \ldots \vee a_k&\leftarrow b_1,..., b_l, not\,c_{1},..., not\,c_m, not\,not\,d_1,..., not\,not\,d_n \; , \end{aligned}$$

(1)

where all $a_1,\ldots ,a_k,b_1,\ldots , b_l,c_{1},\ldots , c_m$, and $d_{1},\ldots , d_n$ are atoms of $\mathcal {A}$.⁶ Such rules r are also commonly written in a more succinct way as

$$\begin{aligned} A \leftarrow B, not\,C, not\,not\,D \; , \end{aligned}$$

(2)

where we have $A = \{a_1,\ldots ,a_k\}$, $B =\{b_1,\ldots , b_l\}$, $C =\{c_{1},\ldots , c_m\}$, $D =\{d_{1},\ldots , d_n\}$, and we will use both forms interchangeably. By $\mathcal {A}(P)$ we denote the set of atoms appearing in P. This class of logic programs, $\mathcal {C}_{e}$, includes a number of special kinds of rules r: if $n=0$, then we call r disjunctive; if, in addition, $k\le 1$, then r is normal; if on top of that $m=0$, then we call r Horn, and fact if also $l=0$. The classes of disjunctive, normal and Horn programs, $\mathcal {C}_{d}$, $\mathcal {C}_{n}$, and $\mathcal {C}_{H}$, are defined resp. as a finite set of disjunctive, normal, and Horn rules. We also call extended rules with $k\le 1$ non-disjunctive, thus admitting a non-standard class $\mathcal {C}_{nd}$, called non-disjunctive programs, different from normal programs. Given a program P and a set I of atoms, the reduct $P^I$ is defined as $P^I = \{ A \leftarrow B : r \text { of the form (2) in } P, C \cap I=\emptyset , D \subseteq I\}$.

An HT-interpretation is a pair $\langle X,Y\rangle$ s.t. $X\subseteq Y \subseteq \mathcal {A}$. Given a program P, an HT-interpretation $\langle X,Y\rangle$ is an HT-model of P if $Y\,\models \,P$ and $X\,\models \,P^{Y}$, where $\models$ denotes the standard consequence relation for classical logic. We admit that the set of HT-models of a program P are restricted to $\mathcal {A}(P)$ even if $\mathcal {A}(P)\subset \mathcal {A}$. We denote by $\mathcal {HT}(P)$ the set of all HT-models of P. A set of atoms Y is an answer set of P if $\langle Y,Y\rangle \in \mathcal {HT}(P)$, but there is no $X\subset Y$ such that $\langle X,Y\rangle \in \mathcal {HT}(P)$. The set of all answer sets of P is denoted by $\mathcal {AS}(P)$. We say that two programs $P_1, P_2$ are equivalent if $\mathcal {AS}(P_1)=\mathcal {AS}(P_2)$ and strongly equivalent, denoted by $P_1\equiv P_2$, if $\mathcal {AS}(P_1\cup R)=\mathcal {AS}(P_2\cup R)$ for any $R\in \mathcal {C}_{e}$. It is well-known that $P_1\equiv P_2$ exactly when $\mathcal {HT}(P_1)=\mathcal {HT}(P_2)$ [45]. We say that $P'$ is an HT-consequence of P, denoted by $P\,\models _\mathsf{HT}P'$, whenever $\mathcal {HT}(P)\subseteq \mathcal {HT}(P')$. The V -exclusion of a set of answer sets (a set of HT-interpretations) $\mathcal {M}$, denoted $\mathcal {M}_{\parallel V}$, is ${\{X\backslash V\mid X\in \mathcal {M}\}}$ (${\{\langle X\backslash V,Y\setminus V\rangle \mid \langle X,Y\rangle \in \mathcal {M}\}}$). Finally, given two sets of atoms $X,X'\subseteq \mathcal {A}$, we write $X\sim _V X'$ whenever $X \backslash V=X' \backslash V$.