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Documents authored by Mrkonjić, Lovro

Document
DOI: 10.4230/OASIcs.Tannen.3
Semiring Provenance in the Infinite

Authors: Sophie Brinke, Erich Grädel, Lovro Mrkonjić, and Matthias Naaf

Published in: OASIcs, Volume 119, The Provenance of Elegance in Computation - Essays Dedicated to Val Tannen (2024)

Abstract
Semiring provenance evaluates database queries or logical statements not just by true or false but by values in some commutative semiring. This permits to track which combinations of atomic facts are responsible for the truth of a statement, and to derive further information, for instance concerning costs, confidence scores, number of proof trees, or access levels to protected data. The focus of this approach, proposed and developed to a large extent by Val Tannen and his collaborators, has first been on (positive) database query languages, but has later been extended, again in collaboration with Val, to a systematic semiring semantics for first-order logic (and other logical systems), as well as to a method for the strategy analysis of games. So far, semiring provenance has been studied for finite structures. To extend the semiring provenance approach for first-order logic to infinite domains, the semirings need to be equipped with addition and multiplication operators over infinite collections of values. This needs solid algebraic foundations, and we study here the necessary and desirable properties of semirings with infinitary operations to provide a well-defined and informative provenance analysis over infinite domains. We show that, with suitable definitions for such infinitary semiring, large parts of the theory of semiring provenance can be succesfully generalised to infinite structures.
Cite as

Sophie Brinke, Erich Grädel, Lovro Mrkonjić, and Matthias Naaf. Semiring Provenance in the Infinite. In The Provenance of Elegance in Computation - Essays Dedicated to Val Tannen. Open Access Series in Informatics (OASIcs), Volume 119, pp. 3:1-3:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{brinke_et_al:OASIcs.Tannen.3,
  author =	{Brinke, Sophie and Gr\"{a}del, Erich and Mrkonji\'{c}, Lovro and Naaf, Matthias},
  title =	{{Semiring Provenance in the Infinite}},
  booktitle =	{The Provenance of Elegance in Computation - Essays Dedicated to Val Tannen},
  pages =	{3:1--3:26},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-320-1},
  ISSN =	{2190-6807},
  year =	{2024},
  volume =	{119},
  editor =	{Amarilli, Antoine and Deutsch, Alin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Tannen.3},
  URN =		{urn:nbn:de:0030-drops-200997},
  doi =		{10.4230/OASIcs.Tannen.3},
  annote =	{Keywords: Semiring semantics, first-order logic, semirings with infinitary operations}
}
Document
DOI: 10.4230/LIPIcs.CSL.2024.19
Ehrenfeucht-Fraïssé Games in Semiring Semantics

Authors: Sophie Brinke, Erich Grädel, and Lovro Mrkonjić

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abstract
Ehrenfeucht-Fraïssé games provide a fundamental method for proving elementary equivalence (and equivalence up to a certain quantifier rank) of relational structures. We investigate the soundness and completeness of this method in the more general context of semiring semantics. Motivated originally by provenance analysis of database queries, semiring semantics evaluates logical statements not just by true or false, but by values in some commutative semiring; this can provide much more detailed information, for instance concerning the combinations of atomic facts that imply the truth of a statement, or practical information about evaluation costs, confidence scores, access levels or the number of successful evaluation strategies. There is a wide variety of different semirings that are relevant for provenance analysis, and the applicability of classical logical methods in semiring semantics may strongly depend on the algebraic properties of the underlying semiring. While Ehrenfeucht-Fraïssé games are sound and complete for logical equivalences in classical semantics, and thus on the Boolean semiring, this is in general not the case for other semirings. We provide a detailed analysis of the soundness and completeness of model comparison games on specific semirings, not just for classical Ehrenfeucht-Fraïssé games but also for other variants based on bijections or counting. Finally we propose a new kind of games, called homomorphism games, based on the fact that there exist locally very different semiring interpretations that can be proved to be elementarily equivalent via separating sets of homomorphisms. We prove that these homomorphism games provide a sound and complete method for logical equivalences on finite lattice semirings.
Cite as

Sophie Brinke, Erich Grädel, and Lovro Mrkonjić. Ehrenfeucht-Fraïssé Games in Semiring Semantics. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{brinke_et_al:LIPIcs.CSL.2024.19,
  author =	{Brinke, Sophie and Gr\"{a}del, Erich and Mrkonji\'{c}, Lovro},
  title =	{{Ehrenfeucht-Fra\"{i}ss\'{e} Games in Semiring Semantics}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{19:1--19:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello 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.CSL.2024.19},
  URN =		{urn:nbn:de:0030-drops-196623},
  doi =		{10.4230/LIPIcs.CSL.2024.19},
  annote =	{Keywords: Semiring semantics, elementary equivalence, Ehrenfeucht-Fra\"{i}ss\'{e} games}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2021.133
Elementary Equivalence Versus Isomorphism in Semiring Semantics

Authors: Erich Grädel and Lovro Mrkonjić

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
We study the first-order axiomatisability of finite semiring interpretations or, equivalently, the question whether elementary equivalence and isomorphism coincide for valuations of atomic facts over a finite universe into a commutative semiring. Contrary to the classical case of Boolean semantics, where every finite structure is axiomatised up to isomorphism by a first-order sentence, the situation in semiring semantics is rather different, and depends on the underlying semiring. We prove that for a number of important semirings, including min-max semirings, and the semirings of positive Boolean expressions, there exist finite semiring interpretations that are elementarily equivalent but not isomorphic. The same is true for the polynomial semirings that are universal for the classes of absorptive, idempotent, and fully idempotent semirings, respectively. On the other side, we prove that for other, practically relevant, semirings such as the Viterby semiring 𝕍, the tropical semiring 𝕋, the natural semiring ℕ and the universal polynomial semiring ℕ[X], all finite semiring interpretations are first-order axiomatisable, and thus elementary equivalence implies isomorphism.
Cite as

Erich Grädel and Lovro Mrkonjić. Elementary Equivalence Versus Isomorphism in Semiring Semantics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 133:1-133:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gradel_et_al:LIPIcs.ICALP.2021.133,
  author =	{Gr\"{a}del, Erich and Mrkonji\'{c}, Lovro},
  title =	{{Elementary Equivalence Versus Isomorphism in Semiring Semantics}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{133:1--133:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.133},
  URN =		{urn:nbn:de:0030-drops-142022},
  doi =		{10.4230/LIPIcs.ICALP.2021.133},
  annote =	{Keywords: Semiring semantics, elementary equivalence, axiomatisability}
}
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