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**Published in:** LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

The number of quantifiers needed to express first-order (FO) properties is captured by two-player combinatorial games called multi-structural games. We analyze these games on binary strings with an ordering relation, using a technique we call parallel play, which significantly reduces the number of quantifiers needed in many cases. Ordered structures such as strings have historically been notoriously difficult to analyze in the context of these and similar games. Nevertheless, in this paper, we provide essentially tight bounds on the number of quantifiers needed to characterize different-sized subsets of strings. The results immediately give bounds on the number of quantifiers necessary to define several different classes of Boolean functions. One of our results is analogous to Lupanov’s upper bounds on circuit size and formula size in propositional logic: we show that every Boolean function on n-bit inputs can be defined by a FO sentence having (1+ε)n/log(n) + O(1) quantifiers, and that this is essentially tight. We reduce this number to (1 + ε)log(n) + O(1) when the Boolean function in question is sparse.

Marco Carmosino, Ronald Fagin, Neil Immerman, Phokion G. Kolaitis, Jonathan Lenchner, and Rik Sengupta. On the Number of Quantifiers Needed to Define Boolean Functions. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{carmosino_et_al:LIPIcs.MFCS.2024.34, author = {Carmosino, Marco and Fagin, Ronald and Immerman, Neil and Kolaitis, Phokion G. and Lenchner, Jonathan and Sengupta, Rik}, title = {{On the Number of Quantifiers Needed to Define Boolean Functions}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {34:1--34:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.34}, URN = {urn:nbn:de:0030-drops-205907}, doi = {10.4230/LIPIcs.MFCS.2024.34}, annote = {Keywords: logic, combinatorial games, Boolean functions, quantifier number} }

Document

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

In 1981, Neil Immerman described a two-player game, which he called the "separability game" [Neil Immerman, 1981], that captures the number of quantifiers needed to describe a property in first-order logic. Immerman’s paper laid the groundwork for studying the number of quantifiers needed to express properties in first-order logic, but the game seemed to be too complicated to study, and the arguments of the paper almost exclusively used quantifier rank as a lower bound on the total number of quantifiers. However, last year Fagin, Lenchner, Regan and Vyas [Fagin et al., 2021] rediscovered the game, provided some tools for analyzing them, and showed how to utilize them to characterize the number of quantifiers needed to express linear orders of different sizes. In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results. First we carefully distinguish minimum number of quantifiers from the more usual descriptive complexity measures, minimum quantifier rank and minimum number of variables. Then, for each positive integer k, we give an explicit example of a property of finite structures (in particular, of finite graphs) that can be expressed with a sentence of quantifier rank k, but where the same property needs 2^Ω(k²) quantifiers to be expressed. We next give the precise number of quantifiers needed to distinguish two rooted trees of different depths. Finally, we give a new upper bound on the number of quantifiers needed to express s-t connectivity, improving the previous known bound by a constant factor.

Ronald Fagin, Jonathan Lenchner, Nikhil Vyas, and Ryan Williams. On the Number of Quantifiers as a Complexity Measure. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fagin_et_al:LIPIcs.MFCS.2022.48, author = {Fagin, Ronald and Lenchner, Jonathan and Vyas, Nikhil and Williams, Ryan}, title = {{On the Number of Quantifiers as a Complexity Measure}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {48:1--48:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert 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.MFCS.2022.48}, URN = {urn:nbn:de:0030-drops-168460}, doi = {10.4230/LIPIcs.MFCS.2022.48}, annote = {Keywords: number of quantifiers, multi-structural games, complexity measure, s-t connectivity, trees, rooted trees} }

Document

**Published in:** LIPIcs, Volume 127, 22nd International Conference on Database Theory (ICDT 2019)

A document spanner models a program for Information Extraction (IE) as a function that takes as input a text document (string over a finite alphabet) and produces a relation of spans (intervals in the document) over a predefined schema. A well-studied language for expressing spanners is that of the regular spanners: relational algebra over regex formulas, which are regular expressions with capture variables. Equivalently, the regular spanners are the ones expressible in non-recursive Datalog over regex formulas (which extract relations that constitute the extensional database). This paper explores the expressive power of recursive Datalog over regex formulas. We show that such programs can express precisely the document spanners computable in polynomial time. We compare this expressiveness to known formalisms such as the closure of regex formulas under the relational algebra and string equality. Finally, we extend our study to a recently proposed framework that generalizes both the relational model and the document spanners.

Liat Peterfreund, Balder ten Cate, Ronald Fagin, and Benny Kimelfeld. Recursive Programs for Document Spanners. In 22nd International Conference on Database Theory (ICDT 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 127, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{peterfreund_et_al:LIPIcs.ICDT.2019.13, author = {Peterfreund, Liat and Cate, Balder ten and Fagin, Ronald and Kimelfeld, Benny}, title = {{Recursive Programs for Document Spanners}}, booktitle = {22nd International Conference on Database Theory (ICDT 2019)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-101-6}, ISSN = {1868-8969}, year = {2019}, volume = {127}, editor = {Barcelo, Pablo and Calautti, Marco}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2019.13}, URN = {urn:nbn:de:0030-drops-103155}, doi = {10.4230/LIPIcs.ICDT.2019.13}, annote = {Keywords: Information Extraction, Document Spanners, Polynomial Time, Recursion, Regular Expressions, Datalog} }

Document

**Published in:** LIPIcs, Volume 68, 20th International Conference on Database Theory (ICDT 2017)

We develop a unifying approach to declarative entity linking by introducing the notion of an entity linking framework and an accompanying notion of the certain links in such a framework. In an entity linking framework, logic-based constraints are used to express properties of the desired link relations in terms of source relations and, possibly, in terms of other link relations. The definition of the certain links in such a framework makes use of weighted repairs and consistent answers in inconsistent databases. We demonstrate the modeling capabilities of this approach by showing that numerous concrete entity linking scenarios can be cast as such entity linking frameworks for suitable choices of constraints and weights. By using the certain links as a measure of expressive power, we investigate the relative expressive power of several entity linking frameworks and obtain sharp comparisons. In particular, we show that we gain expressive power if we allow constraints that capture non-recursive collective entity resolution, where link relations may depend on other link relations (and not just on source relations). Moreover, we show that an increase in expressive power also takes place when we allow constraints that incorporate preferences as an additional mechanism for expressing "goodness" of links.

Douglas Burdick, Ronald Fagin, Phokion G. Kolaitis, Lucian Popa, and Wang-Chiew Tan. Expressive Power of Entity-Linking Frameworks. In 20th International Conference on Database Theory (ICDT 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 68, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{burdick_et_al:LIPIcs.ICDT.2017.10, author = {Burdick, Douglas and Fagin, Ronald and Kolaitis, Phokion G. and Popa, Lucian and Tan, Wang-Chiew}, title = {{Expressive Power of Entity-Linking Frameworks}}, booktitle = {20th International Conference on Database Theory (ICDT 2017)}, pages = {10:1--10:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-024-8}, ISSN = {1868-8969}, year = {2017}, volume = {68}, editor = {Benedikt, Michael and Orsi, Giorgio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2017.10}, URN = {urn:nbn:de:0030-drops-70554}, doi = {10.4230/LIPIcs.ICDT.2017.10}, annote = {Keywords: entity linking, entity resolution, constraints, repairs, certain links} }

Document

**Published in:** LIPIcs, Volume 31, 18th International Conference on Database Theory (ICDT 2015)

The aim of this paper is to introduce and develop a truly declarative framework for entity linking and, in particular, for entity resolution. As in some earlier approaches, our framework is based on the systematic use of constraints. However, the constraints we adopt are link-to-source constraints, unlike in earlier approaches where source-to-link constraints were used to dictate how to generate links. Our approach makes it possible to focus entirely on the intended properties of the outcome of entity linking, thus separating the constraints from any procedure of how to achieve that outcome. The core language consists of link-to-source constraints that specify the desired properties of a link relation in terms of source relations and built-in predicates such as similarity measures. A key feature of the link-to-source constraints is that they employ disjunction, which enables the declarative listing of all the reasons as to why two entities should be linked. We also consider extensions of the core language that capture collective entity resolution, by allowing inter-dependence between links.
We identify a class of "good" solutions for entity linking specifications, which we call maximum-value solutions and which capture the strength of a link by counting the reasons that justify it. We study natural algorithmic problems associated with these solutions, including the problem of enumerating the "good" solutions, and the problem of finding the certain links, which are the links that appear in every "good" solution. We show that these problems are tractable for the core language, but may become intractable once we allow inter-dependence between link relations. We also make some surprising connections between our declarative framework, which is deterministic, and probabilistic approaches such as ones based on Markov Logic Networks.

Douglas Burdick, Ronald Fagin, Phokion G. Kolaitis, Lucian Popa, and Wang-Chiew Tan. A Declarative Framework for Linking Entities. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 25-43, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{burdick_et_al:LIPIcs.ICDT.2015.25, author = {Burdick, Douglas and Fagin, Ronald and Kolaitis, Phokion G. and Popa, Lucian and Tan, Wang-Chiew}, title = {{A Declarative Framework for Linking Entities}}, booktitle = {18th International Conference on Database Theory (ICDT 2015)}, pages = {25--43}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-79-8}, ISSN = {1868-8969}, year = {2015}, volume = {31}, editor = {Arenas, Marcelo and Ugarte, Mart{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2015.25}, URN = {urn:nbn:de:0030-drops-49759}, doi = {10.4230/LIPIcs.ICDT.2015.25}, annote = {Keywords: entity linking, entity resolution, constraints, certain links} }

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