4 Search Results for "Beck, Moritz"

Document
DOI: 10.4230/LIPIcs.MFCS.2025.18
Supercritical Size-Width Tree-Like Resolution Trade-Offs for Graph Isomorphism

Authors: Christoph Berkholz, Moritz Lichter, and Harry Vinall-Smeeth

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We study the refutation complexity of graph isomorphism in the tree-like resolution calculus. Torán and Wörz [Jacobo Torán and Florian Wörz, 2023] showed that there is a resolution refutation of narrow width k for two graphs if and only if they can be distinguished in (k+1)-variable first-order logic (FO^{k+1}). While DAG-like narrow width k resolution refutations have size at most n^k, tree-like refutations may be much larger. We show that there are graphs of order n, whose isomorphism can be refuted in narrow width k but only in tree-like size 2^{Ω(n^{k/2})}. This is a supercritical trade-off where bounding one parameter (the narrow width) causes the other parameter (the size) to grow above its worst case. The size lower bound is super-exponential in the formula size and improves a related supercritical trade-off by Razborov [Alexander A. Razborov, 2016]. To prove our result, we develop a new variant of the k-pebble EF-game for FO^k to reason about tree-like refutation size in a similar way as the Prover-Delayer games in proof complexity. We analyze this game on the compressed CFI graphs introduced by Grohe, Lichter, Neuen, and Schweitzer [Martin Grohe et al., 2023]. Using a recent improved robust compressed CFI construction of de Rezende, Fleming, Janett, Nordström, and Pang [Susanna F. de Rezende et al., 2024], we obtain a similar bound for width k (instead of the stronger but less common narrow width) and make the result more robust.
Cite as

Christoph Berkholz, Moritz Lichter, and Harry Vinall-Smeeth. Supercritical Size-Width Tree-Like Resolution Trade-Offs for Graph Isomorphism. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 18:1-18:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{berkholz_et_al:LIPIcs.MFCS.2025.18,
  author =	{Berkholz, Christoph and Lichter, Moritz and Vinall-Smeeth, Harry},
  title =	{{Supercritical Size-Width Tree-Like Resolution Trade-Offs for Graph Isomorphism}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{18:1--18:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.18},
  URN =		{urn:nbn:de:0030-drops-241253},
  doi =		{10.4230/LIPIcs.MFCS.2025.18},
  annote =	{Keywords: Proof complexity, Resolution, Width, Tree-like size, Supercritical trade-off, Lower bound, Finite model theory, CFI graphs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.24
Super-Critical Trade-Offs in Resolution over Parities via Lifting

Authors: Arkadev Chattopadhyay and Pavel Dvořák

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
Razborov [Alexander A. Razborov, 2016] exhibited the following surprisingly strong trade-off phenomenon in propositional proof complexity: for a parameter k = k(n), there exists k-CNF formulas over n variables, having resolution refutations of O(k) width, but every tree-like refutation of width n^{1-ε}/k needs size exp(n^Ω(k)). We extend this result to tree-like Resolution over parities, commonly denoted by Res(⊕), with parameters essentially unchanged. To obtain our result, we extend the lifting theorem of Chattopadhyay, Mande, Sanyal and Sherif [Arkadev Chattopadhyay et al., 2023] to handle tree-like affine DAGs. We introduce additional ideas from linear algebra to handle forget nodes along long paths.
Cite as

Arkadev Chattopadhyay and Pavel Dvořák. Super-Critical Trade-Offs in Resolution over Parities via Lifting. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2025.24,
  author =	{Chattopadhyay, Arkadev and Dvo\v{r}\'{a}k, Pavel},
  title =	{{Super-Critical Trade-Offs in Resolution over Parities via Lifting}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{24:1--24:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.24},
  URN =		{urn:nbn:de:0030-drops-237186},
  doi =		{10.4230/LIPIcs.CCC.2025.24},
  annote =	{Keywords: Proof complexity, Lifting, Resolution over parities}
}
Document
DOI: 10.4230/LIPIcs.ECOOP.2025.22
Quantifying Cache Side-Channel Leakage by Refining Set-Based Abstractions

Authors: Jacqueline L. Mitchell and Chao Wang

Published in: LIPIcs, Volume 333, 39th European Conference on Object-Oriented Programming (ECOOP 2025)

Abstract
We propose an improved abstract interpretation based method for quantifying cache side-channel leakage by addressing two key components of precision loss in existing set-based cache abstractions. Our method targets two key sources of imprecision: (1) imprecision in the abstract transfer function used to update the abstract cache state when interpreting a memory access and (2) imprecision due to the incompleteness of the set-based domain. At the center of our method are two key improvements: (1) the introduction of a new transfer function for updating the abstract cache state which carefully leverages information in the abstract state to prevent the spurious aging of memory blocks and (2) a refinement of the set-based domain based on the finite powerset construction. We show that both the new abstract transformer and the domain refinement enjoy certain enhanced precision properties. We have implemented the method and compared it against the state-of-the-art technique on a suite of benchmark programs implementing both sorting algorithms and cryptographic algorithms. The experimental results show that our method is effective in improving the precision of cache side-channel leakage quantification.
Cite as

Jacqueline L. Mitchell and Chao Wang. Quantifying Cache Side-Channel Leakage by Refining Set-Based Abstractions. In 39th European Conference on Object-Oriented Programming (ECOOP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 333, pp. 22:1-22:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mitchell_et_al:LIPIcs.ECOOP.2025.22,
  author =	{Mitchell, Jacqueline L. and Wang, Chao},
  title =	{{Quantifying Cache Side-Channel Leakage by Refining Set-Based Abstractions}},
  booktitle =	{39th European Conference on Object-Oriented Programming (ECOOP 2025)},
  pages =	{22:1--22:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-373-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{333},
  editor =	{Aldrich, Jonathan and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2025.22},
  URN =		{urn:nbn:de:0030-drops-233140},
  doi =		{10.4230/LIPIcs.ECOOP.2025.22},
  annote =	{Keywords: Abstract interpretation, side-channel, leakage quantification, cache}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2017.33
Emptiness Problems for Integer Circuits

Authors: Dominik Barth, Moritz Beck, Titus Dose, Christian Glaßer, Larissa Michler, and Marc Technau

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

Abstract
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
Cite as

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.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}
}
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