Polynomial Approximation to Well-Founded Semantics for Logic Programs with Generalized Atoms: Case Studies

Md. Solimul Chowdhury; Fangfang Liu; Wu Chen; Arash Karimi; Jia-Huai You

doi:10.1007/978-3-319-17822-6_16

LOPSTR 2014: Logic-Based Program Synthesis and Transformation pp 279-296 | Cite as

Polynomial Approximation to Well-Founded Semantics for Logic Programs with Generalized Atoms: Case Studies

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Authors and affiliations

Md. Solimul ChowdhuryEmail author
Fangfang Liu
Wu Chen
Arash Karimi
Jia-Huai You

Conference paper

First Online: 23 April 2015

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8981)

Abstract

The well-founded semantics of normal logic programs has two main utilities, one being an efficiently computable semantics with a unique intended model, and the other serving as polynomial time constraint propagation for the computation of answer sets of the same program. When logic programs are generalized to support constraints of various kinds, the semantics is no longer tractable, which makes the second utility doubtful. This paper considers the possibility of tractable but incomplete methods, which in general may miss information in the computed result, but never generates wrong conclusions. For this goal, we first formulate a well-founded semantics for logic programs with generalized atoms, which generalizes logic programs with arbitrary aggregates/constraints/dl-atoms. As a case study, we show that the method of removing non-monotone dl-atoms for the well-founded semantics by Eiter et al. actually falls into this category. We also present a case study for logic programs with standard aggregates.

Keywords

Polynomial approximation Well-founded semantics Generalized atoms

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A Appendix: Proof of Theorem 1

Proof

We prove the claim by induction on the construction of $\textit{lfp}(W_{\pi (\textit{KB})})$ and $\textit{lfp}(W_{\textit{KB}})$.

(a)

Base: $W^0_{\pi (\textit{KB})}|_\tau = W^0_\textit{KB}= \emptyset $.
(b)

Step: Assume, for all $k \ge 0$, $W^k_{\pi (\textit{KB})}|_\tau \subseteq W^k_\textit{KB}$, and prove $W^{k+1}_{\pi (\textit{KB})}|_\tau \subseteq W^{k+1}_\textit{KB}$. By definition, we know

$$\begin{aligned}&W^{k+1}_{\pi (\textit{KB})} |_\tau = T_{\pi (\textit{KB})} (W^k_{\pi (\textit{KB})}) |_\tau \cup \lnot U_{\pi (\textit{KB})}(W^k_{\pi (\textit{KB})})|_\tau \end{aligned}$$

(3)

$$\begin{aligned}&W^{k+1}_\textit{KB}= T_\textit{KB}(W^k_\textit{KB}) \cup \lnot . U_\textit{KB}(W^k_\textit{KB}) \end{aligned}$$

(4)

To prove $W^{k+1}_{\pi (\textit{KB})}|_\tau \subseteq W^{k+1}_\textit{KB}$, it is sufficient to prove both of

$$\begin{aligned}&T_{\pi (\textit{KB})} (W^k_{\pi (\textit{KB})}) |_\tau \subseteq T_\textit{KB}(W^k_\textit{KB})\end{aligned}$$

(5)

$$\begin{aligned}&U_{\pi (\textit{KB})}(W^k_{\pi (\textit{KB})})|_\tau \subseteq U_\textit{KB}(W^k_\textit{KB}) \end{aligned}$$

(6)

Below, let us assume that at most one dl-atom may appear in a rule. The proof can be generalized to arbitrary rules by the same argument, for the transformation of one dl-atom at a time.

(i)

We first prove (5). Let $a \in T_{\pi (\textit{KB})} (W^k_{\pi (\textit{KB})}) |_\tau $. By definition, $\exists r' \in \pi (P)$ such that $B(r')$ is persistently true under $W^k_{\pi (\textit{KB})}$. WLOG, assume for some $r \in P$, $\pi (r) = \{r',r''\}$, as illustrated in (7) below, in which a dl-atom appears positively, which is replaced by rules in (7) of $\pi (r)$.
$$\begin{aligned}&r:~ a \leftarrow \ldots , DL [ \ldots , S_j {\cap \!\!\!\!-} p_j , \ldots ; Q](e), \ldots \end{aligned}$$

(7)

$$\begin{aligned}&r':a \leftarrow \ldots , DL [ \ldots , S_j {\cup \!\!\!\!-} \bar{p_j}, \ldots ; Q](\mathbf{e}), \ldots \nonumber \\&r'':\bar{p_j} (\mathbf{e}) \leftarrow \lnot DL [S^{\prime }_j \uplus {p_j}; S^{\prime }](\mathbf{e}) \end{aligned}$$

(8)
Let $D$ denote the dl-atom in (7), and $D'$ the corresponding dl-atom in (7). If the operator ${\cap \!\!\!\!-}$ does not occur in $D$, then trivially $B(r)$ is persistently true under $W^k_\textit{KB}$, and thus $a \in T_\textit{KB}(W^k_\textit{KB})$.

Otherwise, since $B(r')$ is persistently true under $W^k_{\pi (\textit{KB})}$, we have $D'$ is persistently true under $W^k_{\pi (\textit{KB})}$, and by the fact that $D'$ is monotone, we have $W^k_{\pi (\textit{KB})} \models _L D'$. The atom $\bar{p_j} (\mathbf{e})$ may or may not play a role in the entailment $L\cup \bigcup _{i=1}^m D'_i \models Q(\mathbf{e})$ (cf. Definition (6)). The proof is trivial if it does not. Otherwise, $\bar{p_j} (\mathbf{e})$ is well-founded already w.r.t. $W^k_{\pi (\textit{KB})}$, and by the last rule in (7), $p_j(\mathbf{e})$ is unfounded w.r.t. $W^{k'}_{\pi (\textit{KB})}$, for some $k' < k$. By induction hypothesis, we know $W^{k'}_{\pi (\textit{KB})} |_\tau \subseteq W^{k'}_\textit{KB}$, thus $p_j(\mathbf{e})$ is unfounded w.r.t. $W^k_\textit{KB}$. It follows $D$ is persistently true under $W^k_\textit{KB}$, and by the assumption that $D$ is the only dl-atom in (7) and that $B(r')$ is persistently true under $W^k_{\pi (\textit{KB})}$, we have $B(r)$ is persistently true under $W^k_\textit{KB}$. Thus, $a \in T_\textit{KB}(W^k_\textit{KB})$.

If $D$ appears negatively in rule body, the proof is similar because, given a partial interpretation $S$, $\lnot D$ is persistently true under $S$ iff $D$ is persistently false under $S$, and we just need to swap well-founded and unfounded in the argument above.
(ii)

To prove (6), assume that $a \in U_{\pi (\textit{KB})} (W^k_{\pi (\textit{KB})}) |_\tau $ and we show $a \in U_\textit{KB}(W^k_\textit{KB})$. Consider the case of (7). WLOG, assume that $r$ is the only rule in $P$ with $a$ in the head. If the fact $a \in U_{\pi (\textit{KB})} (W^k_{\pi (\textit{KB})})$ is independent of $\bar{p_j}(\mathbf{e})$, the proof is trivial. Otherwise, that $a \in U_{\pi (\textit{KB})} (W^k_{\pi (\textit{KB})})$ is because $D'$ is persistently false under $\lnot . U' \cup W^k_{\pi (\textit{KB})}$, where $U'$ is the greatest unfounded set relative to $W^k_{\pi (\textit{KB})}$. This implies that $\bar{p_j}(\mathbf{e})$ must be unfounded w.r.t. $W^k_{\pi (\textit{KB})}$, and it follows that $p_j(\mathbf{e})$ is well-founded already w.r.t. $W^{k'}_{\pi (\textit{KB})}$, for some $k' < k$. Then, by induction hypothesis, we have that $p_j(\mathbf{e})$ is well-founded w.r.t. $W^k_\textit{KB}$, which implies that $B(r)$ is persistently false under $\lnot . U \cup W^k_\textit{KB}$, where $U$ is the greatest unfounded set w.r.t. $W^k_\textit{KB}$. It follows $a \in U_\textit{KB}(W^k_\textit{KB})$. The proof for the case where a dl-atom appears negatively in rule body is similar.

Hence, the proof is completed. $\square $

References

1.
Alviano, M., Calimeri, F., Faber, W., Leone, N., Perri, S.: Unfounded sets and well-founded semantics of answer set programs with aggregates. J. Artif. Intell. Res. 42, 487–527 (2011)MATHMathSciNetGoogle Scholar
2.
Alviano, M., Faber, W.: The complexity boundary of answer set programming with generalized atoms under the FLP semantics. In: Cabalar, P., Son, T.C. (eds.) LPNMR 2013. LNCS, vol. 8148, pp. 67–72. Springer, Heidelberg (2013) CrossRefGoogle Scholar
3.
Artale, A., Calvanese, D., Kontchakov, R., Zakharyaschev, M.: The DL-Lite family and relations. J. Artif. Intell. Res. 36, 1–69 (2009)MATHMathSciNetGoogle Scholar
4.
Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, Cambridge (2003)Google Scholar
5.
Baral, C.: Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, New York (2003) CrossRefMATHGoogle Scholar
6.
Bordeaux, L., Hamadi, Y., Zhang, L.: Propositional satisfiability and constraint programming: a comparative survey. ACM Comput. Surv. 38(4), 1–54 (2006)CrossRefGoogle Scholar
7.
Eiter, T., Fink, M., Krennwallner, T., Redl, C., Schüller, P.: Efficient hex-program evaluation based on unfounded sets. J. Artif. Intell. Res. (JAIR) 49, 269–321 (2014)MATHGoogle Scholar
8.
Eiter, T., Ianni, G., Lukasiewicz, T., Schindlauer, R., Tompits, H.: Combining answer set programming with description logics for the semantic web. Artif. Intell. 172(12–13), 1495–1539 (2008)CrossRefMATHMathSciNetGoogle Scholar
9.
Eiter, T., Ianni, G., Schindlauer, R., Tompits, H.: A uniform integration of higher-order reasoning and external evaluations in answer-set programming. In: IJCAI, pp. 90–96 (2005)Google Scholar
10.
Eiter, T., Lukasiewicz, T., Ianni, G., Schindlauer, R.: Well-founded semantics for description logic programs in the semantic web. ACM Trans. Comput. Log. 12(2) (2011), Article 3Google Scholar
11.
Faber, W.: Unfounded sets for disjunctive logic programs with arbitrary aggregates. In: Baral, C., Greco, G., Leone, N., Terracina, G. (eds.) LPNMR 2005. LNCS (LNAI), vol. 3662, pp. 40–52. Springer, Heidelberg (2005) CrossRefGoogle Scholar
12.
Faber, W., Pfeifer, G., Leone, N., Dell’Armi, T., Ielpa, G.: Design and implementation of aggregate functions in the DLV system. TPLP 8(5–6), 545–580 (2008)MATHMathSciNetGoogle Scholar
13.
Knorr, M., Júlio Alferes, J., Hitzler, P.: Local closed world reasoning with description logics under the well-founded semantics. Artif. Intell. 175(9–10), 1528–1554 (2011)CrossRefMATHGoogle Scholar
14.
Liu, G., Goebel, R., Janhunen, T., Niemelä, I., You, J.-H.: Strong equivalence of logic programs with abstract constraint atoms. In: Delgrande, J.P., Faber, W. (eds.) LPNMR 2011. LNCS, vol. 6645, pp. 161–173. Springer, Heidelberg (2011) CrossRefGoogle Scholar
15.
Liu, G., You, J.-H.: Lparse programs revisited: semantics and representation of aggregates. In: Garcia de la Banda, M., Pontelli, E. (eds.) ICLP 2008. LNCS, vol. 5366, pp. 347–361. Springer, Heidelberg (2008) CrossRefGoogle Scholar
16.
Liu, L., Pontelli, E., Cao Son, T., Truszczyński, M.: Logic programs with abstract constraint atoms. Artif. Intell. 174, 295–315 (2010)CrossRefMATHGoogle Scholar
17.
Marek, V.W., Truszczynski, M: Logic programs with abstract constraint atoms. In: Proceedings of AAAI-04, pp. 86–91 (2004)Google Scholar
18.
Pelov, N., Denecker, M., Bruynooghe, M.: Well-founded and stable semantics of logic programs with aggregates. Theory Pract. Log. Program. 7, 301–353 (2007)CrossRefMATHMathSciNetGoogle Scholar
19.
Rossi, F., Van Beek, P., Walsh, T. (eds.): Global constraints. Handbook of Constraint Programming. Elsevier, New York (2006)Google Scholar
20.
Simons, P., Niemel$\ddot{a}$, I., Soininen, T.: Extending and implementing the stable model semantics. Artif. Intell. 138(1–2), 181–234 (2002)Google Scholar
21.
Cao Son, T., Pontelli, E.: A constructive semantic characterization of aggregates in answer set programming. TPLP 7(3), 355–375 (2007)MATHGoogle Scholar
22.
Pontelli, E., Huy Tu, P.: Answer sets for logic programs with arbitrary abstract constraint atoms. J. Artif. Intell. Res. 29, 353–389 (2007)MATHGoogle Scholar
23.
Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5(2), 285–309 (1955)CrossRefMATHMathSciNetGoogle Scholar
24.
Van Gelder, A., Ross, K.A., Schlipf, J.S.: The well-founded semantics for general logic programs. J. ACM 38(3), 620–650 (1991)MATHGoogle Scholar
25.
Wang, Y., Lin, F., Zhang, M., You, J.-H.: A well-founded semantics for basic logic programs with arbitrary abstract constraint atoms. In: AAAI (2012)Google Scholar

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Authors and Affiliations

Md. Solimul Chowdhury
- 1
Email author
Fangfang Liu
- 2
Wu Chen
- 3
Arash Karimi
- 1
Jia-Huai You
- 1

1.Department of Computing ScienceUniversity of AlbertaEdmontonCanada
2.College of Computer and Information ScienceShanghai UniversityBaoshanChina
3.College of Computer and Information ScienceSouthwest UniversityChongqingChina

Polynomial Approximation to Well-Founded Semantics for Logic Programs with Generalized Atoms: Case Studies

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