hex-Programs with Existential Quantification

Abstract

hex-programs extend ASP by external sources. In this paper, we present domain-specific existential quantifiers on top of hex-programs, i.e., ASP programs with external access which may introduce new values that also show up in the answer sets. Pure logical existential quantification corresponds to a specific instance of our approach. Programs with existential quantifiers may have infinite groundings in general, but for specific reasoning tasks a finite subset of the grounding can suffice. We introduce a generalized grounding algorithm for such problems, which exploits domain-specific termination criteria in order to generate a finite grounding for bounded model generation. As an application we consider query answering over existential rules. In contrast to other approaches, several extensions can be naturally integrated into our approach. We further show how terms with function symbols can be handled by hex-programs, which in fact can be seen as a specific form of existential quantification.

This research has been supported by the Austrian Science Fund (FWF) project P20840, P20841, P24090, and by the Vienna Science and Technology Fund (WWTF) project ICT08-020.

Appendix: Term Bounding Function \(b_{ synsem }\)

The TBF $b_{ synsem }$ [6] builds on the positive attribute dependency graph $G_A(\varPi )$, whose nodes are the attributes of $\varPi$ and whose edges model the information flow between them. E.g., if for rule $r$ we have $p(\mathbf{X}) \,{\in }\, H(r)$ and $q(\mathbf{Y}) \,{\in }\, B^{+}(r)$ such that $X_i \,{=}\, Y_j$ for some $X_i\,{\in }\, \mathbf{X}$ and $Y_j\,{\in }\, \mathbf{Y}$, then we have a flow from $q{\upharpoonright }_{}j$ to $p{\upharpoonright }_{}i$. A cycle $K$ in $G_A(\varPi )$ is benign wrt. a set of safe attributes $S$, if there exists a well-ordering $\le _{\mathcal {C}}$ of $\mathcal {C}$, such that for every $\&{g}[\mathbf{X}]_r{\upharpoonright }_{\textsc {o}}j \not \in S$ in the cycle, $f_{\text { } \& g}(\mathbf {A},x_1, \ldots , x_m,t_1, \ldots t_n) = 0$ whenever

• some $x_i$ for $1 \,{\le }\, i \,{\le }\, m$ is a predicate parameter, $\&{g}[\mathbf{X}]_r{\upharpoonright }_{\textsc {i}}i\,{\not \in }\,S$ is in $K$, and we have $(s_1, \ldots , s_{ ar (x_i)})\,{\in }\, ext (\mathbf {A}, x_i)$, and $t_j \not \le _{\mathcal {C}} s_k$ for some $1 \le k \le ar (x_i)$; or

• for some $1\le i\le m$, $type (\&{g},i)=\mathtt {const}$, $\&{g}[\mathbf{X}]_r{\upharpoonright }_{\textsc {i}}i \not \in S$ is in $K$, and $t_j \not \le _{\mathcal {C}} x_i$.

A cycle in $G_A(\varPi )$ is called malign wrt. $S$ if it is not benign. Then $b_{ synsem }$ is as follows.

Definition 5

(Syntactic and Semantic Term Bounding Function). We define the TBF $b_{ synsem }(\varPi , r, S, B)$ such that $t \in b_{ synsem }(\varPi , r, S, B)$ iff

• (i) $t$ is a constant in $r$; or

• (ii) there is an ordinary atom $q(s_1, \ldots , s_{ ar (q)}) \in B^{+}(r)$ such that $t = s_j$, for some $1 \le j \le ar (q)$ and $q{\upharpoonright }_{}j \in S$; or

• (iii) for some external atom $\&{g}[\mathbf{X}](\mathbf{Y}) \in B^{+}(r)$, we have that $t = Y_i$ for some $Y_i\in \mathbf{Y}$, and for each $X_i \in \mathbf{X}$, $X_i \in B$, if $\tau (\&{g}, i) = \mathbf {const},$ and $X_i{\upharpoonright }_{}1, \ldots , X_i{\upharpoonright }_{} ar (X_i) \in S$ if $\tau (\&{g}, i) = \mathbf {pred}$; or

• (iv) $t$ is captured by some attribute $\alpha$ in $B^{+}(r)$ that is not reachable from malign cycles in $G_A(\varPi )$ wrt. $S$, i.e., if $\alpha \,{=}\,p{\upharpoonright }_{}i$ then $t\,{=}t_i$ for some $p(t_1,\ldots ,t_\ell ) \,{\in }\, B^{+}(r)$, and if $\alpha \,{=}\, \&{g}[\mathbf{X}]_r{\upharpoonright }_{T}i$ then $t \,{=}\,X^T_i$ for some $\&{g}[\mathbf{X}^{\textsc {i}}](\mathbf{X}^{\textsc {o}})$ ${\in }\, B^{+}(r)$ where the input and output vectors are $\mathbf{X}^T{=}X_1^T,\ldots ,X^T_{ ar _{}(\&{g})}$; or

• (v) $t \,{=}\, Y_i$ for some $\&{g}[\mathbf{X}](\mathbf{Y})\,{\in }\, B^{+}(r)$, where $\{y_i \mid \mathbf{x} \in (\mathcal {P} \cup \mathcal {C})^{ ar _{\textsc {i}}(\&{g})}, \mathbf{y} \in \mathcal {C}^{ ar _{\textsc {o}}(\&{g})},$ $f_{\text { } \& g}(\mathbf {A},\mathbf{x},\mathbf{y})\,{=}\, 1\}$ is finite for all assignments $\mathbf {A}$.

• (vi) $t \,{\in }\, \mathbf{X}$ for some $\&{g}[\mathbf{X}](\mathbf{Y}) \,{\in }\, B^{+}(r)$, where $U \,{\in }\, B$ for every $U\,{\in }\, \mathbf{Y}$ and $\{\mathbf{x} \mid \mathbf{x} \,{\in }\, (\mathcal {P} {\cup }\mathcal {C})^{ ar _{\textsc {i}}(\&{g})},$ $f_{\text { } \& g}(\mathbf {A},\mathbf{x},\mathbf{y}) = 1\}$ is finite for every $\mathbf {A}$ and $\mathbf{y} \in \mathcal {C}^{ ar _{\textsc {o}}(\&{g})}$.