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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

The article investigates the relation between three well-known hypotheses.
- H_{union}: the union of disjoint ≤^p_m-complete sets for NP is ≤^p_m-complete
- H_{opps}: there exist optimal propositional proof systems
- H_{cpair}: there exist ≤^{pp}_m-complete disjoint NP-pairs The following results are obtained:
- The hypotheses are pairwise independent under relativizable proofs, except for the known implication H_{opps} ⇒ H_{cpair}.
- An answer to Pudlák’s question for an oracle relative to which ¬H_{cpair}, ¬H_{opps}, and UP has ≤^p_m-complete sets.
- The converse of Köbler, Messner, and Torán’s implication NEE ∩ TALLY ⊆ coNEE ⇒ H_{opps} fails relative to an oracle, where NEE =^{df} NTIME(2^O(2ⁿ)).
- New characterizations of H_{union} and two variants in terms of coNP-completeness and p-producibility of the set of hard formulas of propositional proof systems.

Titus Dose and Christian Glaßer. NP-Completeness, Proof Systems, and Disjoint NP-Pairs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dose_et_al:LIPIcs.STACS.2020.9, author = {Dose, Titus and Gla{\ss}er, Christian}, title = {{NP-Completeness, Proof Systems, and Disjoint NP-Pairs}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {9:1--9:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.9}, URN = {urn:nbn:de:0030-drops-118707}, doi = {10.4230/LIPIcs.STACS.2020.9}, annote = {Keywords: NP-complete, propositional proof system, disjoint NP-pair, oracle} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Pudlák [P. Pudlák, 2017] lists several major conjectures from the field of proof complexity and asks for oracles that separate corresponding relativized conjectures. Among these conjectures are:
- DisjNP: The class of all disjoint NP-pairs has no many-one complete elements.
- SAT: NP contains no many-one complete sets that have P-optimal proof systems.
- UP: UP has no many-one complete problems.
- NP cap coNP: NP cap coNP has no many-one complete problems.
As one answer to this question, we construct an oracle relative to which DisjNP, neg SAT, UP, and NP cap coNP hold, i.e., there is no relativizable proof for the implication DisjNP wedge UP wedge NP cap coNP ==> SAT. In particular, regarding the conjectures by Pudlák this extends a result by Khaniki [Khaniki, 2019]. Since Khaniki [Khaniki, 2019] constructs an oracle showing that the implication SAT ==> DisjNP has no relativizable proof, we obtain that the conjectures DisjNP and SAT are independent in relativized worlds, i.e., none of the implications DisjNP ==> SAT and SAT ==> DisjNP can be proven relativizably.

Titus Dose. P-Optimal Proof Systems for Each NP-Set but no Complete Disjoint NP-Pairs Relative to an Oracle. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dose:LIPIcs.MFCS.2019.47, author = {Dose, Titus}, title = {{P-Optimal Proof Systems for Each NP-Set but no Complete Disjoint NP-Pairs Relative to an Oracle}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {47:1--47:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.47}, URN = {urn:nbn:de:0030-drops-109918}, doi = {10.4230/LIPIcs.MFCS.2019.47}, annote = {Keywords: NP-complete, proof systems, disjoint NP-pair, oracle, UP} }

Document

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We investigate the computational complexity of balance problems for {-,*}-circuits computing finite sets of natural numbers. These problems naturally build on problems for integer expressions and integer circuits studied by Stockmeyer and Meyer (1973), McKenzie and Wagner (2007), and Glaßer et al. (2010).
Our work shows that the balance problem for {-,*}-circuits is undecidable which is the first natural problem for integer circuits or related constraint satisfaction problems that admits only one arithmetic operation and is proven to be undecidable.
Starting from this result we precisely characterize the complexity of balance problems for proper subsets of {-,*}. These problems turn out to be complete for one of the classes L, NL, and NP.

Titus Dose. Balance Problems for Integer Circuits. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dose:LIPIcs.MFCS.2018.5, author = {Dose, Titus}, title = {{Balance Problems for Integer Circuits}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.5}, URN = {urn:nbn:de:0030-drops-95874}, doi = {10.4230/LIPIcs.MFCS.2018.5}, annote = {Keywords: computational complexity, integer expressions, integer circuits} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, addition, and multiplication is allowed turns out to be equivalent to the complement of polynomial identity testing (PIT).
Our results imply the following improvements and insights on problems studied in earlier papers. We improve the bounds for the membership problem MC(\cup,\cap,¯,+,×) studied by McKenzie and Wagner 2007 and for the equivalence problem EQ(\cup,\cap,¯,+,×) studied by Glaßer et al. 2010. Moreover, it turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a major open problem in algebraic computing complexity:
1. membership problem MC(\cap,+,×) studied by McKenzie and Wagner 2007
2. integer membership problems MC_Z(+,×), MC_Z(\cap,+,×) studied by Travers 2006
3. equivalence problem EQ(+,×) studied by Glaßer et al. 2010

Dominik Barth, Moritz Beck, Titus Dose, Christian Glaßer, Larissa Michler, and Marc Technau. Emptiness Problems for Integer Circuits. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{barth_et_al:LIPIcs.MFCS.2017.33, author = {Barth, Dominik and Beck, Moritz and Dose, Titus and Gla{\ss}er, Christian and Michler, Larissa and Technau, Marc}, title = {{Emptiness Problems for Integer Circuits}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.33}, URN = {urn:nbn:de:0030-drops-80641}, doi = {10.4230/LIPIcs.MFCS.2017.33}, annote = {Keywords: computational complexity, integer expressions, integer circuits, polynomial identity testing} }

Document

**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We study the computational complexity of constraint satisfaction problems that are based on integer expressions and algebraic circuits. On input of a finite set of variables and a finite set of constraints the question is whether the variables can be mapped onto finite subsets of N (resp., finite intervals over N) such that all constraints are satisfied. According to the operations allowed in the constraints, the complexity varies over a wide range of complexity classes such as L, P, NP, PSPACE, NEXP, and even Sigma_1, the class of c.e. languages.

Titus Dose. Complexity of Constraint Satisfaction Problems over Finite Subsets of Natural Numbers. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dose:LIPIcs.MFCS.2016.32, author = {Dose, Titus}, title = {{Complexity of Constraint Satisfaction Problems over Finite Subsets of Natural Numbers}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {32:1--32:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.32}, URN = {urn:nbn:de:0030-drops-64461}, doi = {10.4230/LIPIcs.MFCS.2016.32}, annote = {Keywords: computational complexity, constraint satisfaction problems, integer expressions and circuits} }

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