3 Search Results for "Aichinger, Erhard"

Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.161
Nonuniform Deterministic Finite Automata over Finite Algebraic Structures

Authors: Paweł M. Idziak, Piotr Kawałek, and Jacek Krzaczkowski

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Nonuniform deterministic finite automata (NUDFA) over monoids were invented by Barrington in [Barrington, 1985] to study boundaries of nonuniform constant-memory computation. Later, results on these automata helped to identify interesting classes of groups for which equation satisfiability problem (PolSat) is solvable in (probabilistic) polynomial time [Mikael Goldmann and Alexander Russell, 2002; Idziak et al., 2022]. Based on these results, we present a full characterization of groups, for which the identity checking problem (called PolEqv) has a probabilistic polynomial-time algorithm. We also go beyond groups, and propose how to generalise the notion of NUDFA to arbitrary finite algebraic structures. We study satisfiability of these automata in this more general setting. As a consequence, we present a full description of finite algebras from congruence modular varieties for which testing circuit equivalence CEqv can be solved by a probabilistic polynomial-time procedure. In our proofs we use two computational complexity assumptions: randomized Expotential Time Hypothesis and Constant Degree Hypothesis.
Cite as

Paweł M. Idziak, Piotr Kawałek, and Jacek Krzaczkowski. Nonuniform Deterministic Finite Automata over Finite Algebraic Structures. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 161:1-161:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{idziak_et_al:LIPIcs.ICALP.2025.161,
  author =	{Idziak, Pawe{\l} M. and Kawa{\l}ek, Piotr and Krzaczkowski, Jacek},
  title =	{{Nonuniform Deterministic Finite Automata over Finite Algebraic Structures}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{161:1--161:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.161},
  URN =		{urn:nbn:de:0030-drops-235386},
  doi =		{10.4230/LIPIcs.ICALP.2025.161},
  annote =	{Keywords: program satisfiability, circuit equivalence, identity checking}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.4
The Complexity of Checking Quasi-Identities over Finite Algebras with a Mal'cev Term

Authors: Erhard Aichinger and Simon Grünbacher

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
We consider finite algebraic structures and ask whether every solution of a given system of equations satisfies some other equation. This can be formulated as checking the validity of certain first order formulae called quasi-identities. Checking the validity of quasi-identities is closely linked to solving systems of equations. For systems of equations over finite algebras with finitely many fundamental operations, a complete P/NPC dichotomy is known, while the situation appears to be more complicated for single equations. The complexity of checking the validity of a quasi-identity lies between the complexity of term equivalence (checking whether two terms induce the same function) and the complexity of solving systems of polynomial equations. We prove that for each finite algebra with a Mal'cev term and finitely many fundamental operations, checking the validity of quasi-identities is coNP-complete if the algebra is not abelian, and in P when the algebra is abelian.
Cite as

Erhard Aichinger and Simon Grünbacher. The Complexity of Checking Quasi-Identities over Finite Algebras with a Mal'cev Term. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{aichinger_et_al:LIPIcs.STACS.2023.4,
  author =	{Aichinger, Erhard and Gr\"{u}nbacher, Simon},
  title =	{{The Complexity of Checking Quasi-Identities over Finite Algebras with a Mal'cev Term}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{4:1--4:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.4},
  URN =		{urn:nbn:de:0030-drops-176560},
  doi =		{10.4230/LIPIcs.STACS.2023.4},
  annote =	{Keywords: quasi-identities, conditional identities, systems of equations}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.72
Solving Systems of Equations in Supernilpotent Algebras

Authors: Erhard Aichinger

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

Abstract
Recently, M. Kompatscher proved that for each finite supernilpotent algebra A in a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this algebra. Let mu be the maximal arity of the fundamental operations of A, and let d := |A|^{log_2 mu + log_2 |A| + 1}. Applying a method that G. Károlyi and C. Szabó had used to solve equations over finite nilpotent rings, we show that for A, there is c in N such that a solution of every system of s equations in n variables can be found by testing at most c n^{sd} (instead of all |A|^n possible) assignments to the variables. This also yields new information on some circuit satisfiability problems.
Cite as

Erhard Aichinger. Solving Systems of Equations in Supernilpotent Algebras. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 72:1-72:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{aichinger:LIPIcs.MFCS.2019.72,
  author =	{Aichinger, Erhard},
  title =	{{Solving Systems of Equations in Supernilpotent Algebras}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{72:1--72:9},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.72},
  URN =		{urn:nbn:de:0030-drops-110162},
  doi =		{10.4230/LIPIcs.MFCS.2019.72},
  annote =	{Keywords: Supernilpotent algebras, polynomial equations, polynomial mappings, circuit satisfiability}
}
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