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**Published in:** Dagstuhl Seminar Proceedings, Volume 5171, Nonmonotonic Reasoning, Answer Set Programming and Constraints (2005)

We discuss proof schemes, a kind of context-dependent proofs for logic
programs. We show usefullness of these constructs both in the context of
normal logic programs and their generalizations due to Niemela and
collaborators. As an application we show the following result. For every
cardinality-constraint logic program P there is a logic program PÃ‚Â´ with the
same heads, but with bodies consisting of atoms and negated atoms such
that P and PÃ‚Â´ have same stable models. It is worth noting that another
proof of same result can be obtained from the results by Lifschitz and
collaborators.

Victor W. Marek and Jeffrey B. Remmel. Normal Form Theorem for Logic Programs with Cardinality Constraints. In Nonmonotonic Reasoning, Answer Set Programming and Constraints. Dagstuhl Seminar Proceedings, Volume 5171, pp. 1-34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{marek_et_al:DagSemProc.05171.5, author = {Marek, Victor W. and Remmel, Jeffrey B.}, title = {{Normal Form Theorem for Logic Programs with Cardinality Constraints}}, booktitle = {Nonmonotonic Reasoning, Answer Set Programming and Constraints}, pages = {1--34}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {5171}, editor = {Gerhard Brewka and Ilkka Niemel\"{a} and Torsten Schaub and Miroslaw Truszczynski}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.05171.5}, URN = {urn:nbn:de:0030-drops-2598}, doi = {10.4230/DagSemProc.05171.5}, annote = {Keywords: Proof scheme, cardinality constraints} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 5171, Nonmonotonic Reasoning, Answer Set Programming and Constraints (2005)

We propose a set of desiderata for extensions of Answer Set Programming to capture domains where the objects of interest are infinite sets and yet we can still process ASP programs effectively. We propose two different schemes to do this. One is to extend cardinality type constraints to set constraints which involve codes for finite, recursive and recursively enumerable sets. A second scheme to modify logic programming to reason about sets directly. In this setting, we can also augment logic programming with certain
monotone inductive operators so that we can reason about families of sets which have structure such a closed sets of a topological space or
subspaces of a vector space. We observe that under such conditions, the classic Gelfond-Lifschitz construction generalizes to at least two different notions of stable models.

Jeffrey B. Remmel and Victor W. Marek. Set Based Logic Programming. In Nonmonotonic Reasoning, Answer Set Programming and Constraints. Dagstuhl Seminar Proceedings, Volume 5171, pp. 1-26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{remmel_et_al:DagSemProc.05171.8, author = {Remmel, Jeffrey B. and Marek, Victor W.}, title = {{Set Based Logic Programming}}, booktitle = {Nonmonotonic Reasoning, Answer Set Programming and Constraints}, pages = {1--26}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {5171}, editor = {Gerhard Brewka and Ilkka Niemel\"{a} and Torsten Schaub and Miroslaw Truszczynski}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.05171.8}, URN = {urn:nbn:de:0030-drops-2667}, doi = {10.4230/DagSemProc.05171.8}, annote = {Keywords: ASP, codes for infinite sets, stable model generalizations} }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Victor Marek, Anil Nerode, and Peter H. Schmitt. Non-Classical Logics in Computer Science (Dagstuhl Seminar 9338). Dagstuhl Seminar Report 73, pp. 1-36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1993)

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@TechReport{marek_et_al:DagSemRep.73, author = {Marek, Victor and Nerode, Anil and Schmitt, Peter H.}, title = {{Non-Classical Logics in Computer Science (Dagstuhl Seminar 9338)}}, pages = {1--36}, ISSN = {1619-0203}, year = {1993}, type = {Dagstuhl Seminar Report}, number = {73}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemRep.73}, URN = {urn:nbn:de:0030-drops-149616}, doi = {10.4230/DagSemRep.73}, }

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