Abstract
In this paper, we introduce novel algorithms to solve projected answer set counting 
. #PAs asks to count the number of answer sets with respect to a given set of projection atoms, where multiple answer sets that are identical when restricted to the projection atoms count as only one projected answer set. Our algorithms exploit small treewidth of the primal graph of the input instance by dynamic programming (DP).
We establish a new algorithm for head-cycle-free (HCF) programs and lift very recent results from projected model counting to #PAs when the input is restricted to HCF programs. Further, we show how established DP algorithms for tight, normal, and disjunctive answer set programs can be extended to solve #PAs. Our algorithms run in polynomial time while requiring double exponential time in the treewidth for tight, normal, and HCF programs, and triple exponential time for disjunctive programs.
Finally, we take the exponential time hypothesis (ETH) into account and establish lower bounds of bounded treewidth algorithms for #PAs. Under ETH, one cannot significantly improve our obtained worst-case runtimes.
This work extends an abstract [11] explaining only concepts, and a preliminary workshop paper, and has been supported by Austrian Science Fund (FWF): Y698 and DFG: HO 1294/11-1. Hecher is also affiliated with University of Potsdam, Germany.
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Notes
- 1.
Let \(G=(V,E)\) be a digraph and \(W \subseteq V\). Then, a cycle in G is a W-cycle if it contains all vertices from W.
- 2.
Proofs marked with “\(\star \)” are in extended version at: https://tinyurl.com/y6gkrblc.
- 3.
\(\text {Post-order}(T,n)\) provides the sequence of nodes for tree T rooted at n.
- 4.
Note that in Listing 1, \(\mathbb {A}\) takes in addition as input set P and table \(\iota _t\), used later. Later, P represents the projection atoms and \(\iota _t\) is a table at t from an earlier traversal.
- 5.
Later we use (among others) \(\mathtt {PCNT}_{{\mathbb {PHC}}}\) where \(\mathbb {A}={\mathbb {PHC}} \).
- 6.
Table \(\nu (t)\) contains rows obtained by recursively following origins of \(\tau (n)\) for root n.
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Fichte, J.K., Hecher, M. (2019). Treewidth and Counting Projected Answer Sets. In: Balduccini, M., Lierler, Y., Woltran, S. (eds) Logic Programming and Nonmonotonic Reasoning. LPNMR 2019. Lecture Notes in Computer Science(), vol 11481. Springer, Cham. https://doi.org/10.1007/978-3-030-20528-7_9
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