Document

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We study the separability problem for automatic relations (i.e., relations on finite words definable by synchronous automata) in terms of recognizable relations (i.e., finite unions of products of regular languages). This problem takes as input two automatic relations R and R', and asks if there exists a recognizable relation S that contains R and does not intersect R'. We show this problem to be undecidable when the number of products allowed in the recognizable relation is fixed. In particular, checking if there exists a recognizable relation S with at most k products of regular languages that separates R from R' is undecidable, for each fixed k ⩾ 2. Our proofs reveal tight connections, of independent interest, between the separability problem and the finite coloring problem for automatic graphs, where colors are regular languages.

Pablo Barceló, Diego Figueira, and Rémi Morvan. Separating Automatic Relations. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{barcelo_et_al:LIPIcs.MFCS.2023.17, author = {Barcel\'{o}, Pablo and Figueira, Diego and Morvan, R\'{e}mi}, title = {{Separating Automatic Relations}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.17}, URN = {urn:nbn:de:0030-drops-185514}, doi = {10.4230/LIPIcs.MFCS.2023.17}, annote = {Keywords: Automatic relations, recognizable relations, separability, finite colorability} }

Document

**Published in:** LIPIcs, Volume 255, 26th International Conference on Database Theory (ICDT 2023)

We show that the problem of whether a query is equivalent to a query of tree-width k is decidable, for the class of Unions of Conjunctive Regular Path Queries with two-way navigation (UC2RPQs). A previous result by Barceló, Romero, and Vardi [Pablo Barceló et al., 2016] has shown decidability for the case k = 1, and here we show that decidability in fact holds for any arbitrary k > 1. The algorithm is in 2ExpSpace, but for the restricted but practically relevant case where all regular expressions of the query are of the form a^* or (a_1 + ... + a_n) we show that the complexity of the problem drops to Π^p_2.
We also investigate the related problem of approximating a UC2RPQ by queries of small tree-width. We exhibit an algorithm which, for any fixed number k, builds the maximal under-approximation of tree-width k of a UC2RPQ. The maximal under-approximation of tree-width k of a query q is a query q' of tree-width k which is contained in q in a maximal and unique way, that is, such that for every query q'' of tree-width k, if q'' is contained in q then q'' is also contained in q'.

Diego Figueira and Rémi Morvan. Approximation and Semantic Tree-Width of Conjunctive Regular Path Queries. In 26th International Conference on Database Theory (ICDT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 255, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{figueira_et_al:LIPIcs.ICDT.2023.15, author = {Figueira, Diego and Morvan, R\'{e}mi}, title = {{Approximation and Semantic Tree-Width of Conjunctive Regular Path Queries}}, booktitle = {26th International Conference on Database Theory (ICDT 2023)}, pages = {15:1--15:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-270-9}, ISSN = {1868-8969}, year = {2023}, volume = {255}, editor = {Geerts, Floris and Vandevoort, Brecht}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2023.15}, URN = {urn:nbn:de:0030-drops-177575}, doi = {10.4230/LIPIcs.ICDT.2023.15}, annote = {Keywords: graph databases, conjunctive regular path queries, semantic optimization, tree-width, containment, approximation} }

Document

**Published in:** LIPIcs, Volume 255, 26th International Conference on Database Theory (ICDT 2023)

We consider the dichotomy conjecture for consistent query answering under primary key constraints. It states that, for every fixed Boolean conjunctive query q, testing whether q is certain (i.e. whether it evaluates to true over all repairs of a given inconsistent database) is either polynomial time or coNP-complete. This conjecture has been verified for self-join-free and path queries.
We propose a simple inflationary fixpoint algorithm for consistent query answering which, for a given database, naively computes a set Δ of subsets of database repairs with at most k facts, where k is the size of the query q. The algorithm runs in polynomial time and can be formally defined as:
1) Initialize Δ with all sets S of at most k facts such that S⊧ q.
2) Add any set S of at most k facts to Δ if there exists a block B (i.e., a maximal set of facts sharing the same key) such that for every fact a ∈ B there is a set S' ∈ Δ contained in S ∪ {a}. The algorithm answers "q is certain" iff Δ eventually contains the empty set. The algorithm correctly computes certainty when the query q falls in the polynomial time cases of the known dichotomies for self-join-free queries and path queries. For arbitrary Boolean conjunctive queries, the algorithm is an under-approximation: the query is guaranteed to be certain if the algorithm claims so. However, there are polynomial time certain queries (with self-joins) which are not identified as such by the algorithm.

Diego Figueira, Anantha Padmanabha, Luc Segoufin, and Cristina Sirangelo. A Simple Algorithm for Consistent Query Answering Under Primary Keys. In 26th International Conference on Database Theory (ICDT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 255, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{figueira_et_al:LIPIcs.ICDT.2023.24, author = {Figueira, Diego and Padmanabha, Anantha and Segoufin, Luc and Sirangelo, Cristina}, title = {{A Simple Algorithm for Consistent Query Answering Under Primary Keys}}, booktitle = {26th International Conference on Database Theory (ICDT 2023)}, pages = {24:1--24:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-270-9}, ISSN = {1868-8969}, year = {2023}, volume = {255}, editor = {Geerts, Floris and Vandevoort, Brecht}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2023.24}, URN = {urn:nbn:de:0030-drops-177663}, doi = {10.4230/LIPIcs.ICDT.2023.24}, annote = {Keywords: consistent query answering, primary keys, conjunctive queries} }

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**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We study languages of unambiguous VASS, that is, Vector Addition Systems with States, whose transitions read letters from a finite alphabet, and whose acceptance condition is defined by a set of final states (i.e., the coverability language). We show that the problem of universality for unambiguous VASS is ExpSpace-complete, in sheer contrast to Ackermann-completeness for arbitrary VASS, even in dimension 1. When the dimension d ∈ ℕ is fixed, the universality problem is PSpace-complete if d ≥ 2, and coNP-hard for 1-dimensional VASSes (also known as One Counter Nets).

Wojciech Czerwiński, Diego Figueira, and Piotr Hofman. Universality Problem for Unambiguous VASS. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{czerwinski_et_al:LIPIcs.CONCUR.2020.36, author = {Czerwi\'{n}ski, Wojciech and Figueira, Diego and Hofman, Piotr}, title = {{Universality Problem for Unambiguous VASS}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {36:1--36:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.36}, URN = {urn:nbn:de:0030-drops-128486}, doi = {10.4230/LIPIcs.CONCUR.2020.36}, annote = {Keywords: unambiguity, vector addition systems, universality problems} }

Document

**Published in:** LIPIcs, Volume 155, 23rd International Conference on Database Theory (ICDT 2020)

We study the containment problem for UC2RPQ, that is, two-way Regular Path Queries, closed under conjunction, projection and union. We show a dichotomy property between PSpace-c and ExpSpace-c based on a property on the underlying graph of queries. We show that for any class C of graphs, the containment problem for queries whose underlying graph is in C is in PSpace if and only if C has bounded bridgewidth. Bridgewidth is a graph measure we introduce to this end, defined as the maximum size of a minimal edge separator of a graph.

Diego Figueira. Containment of UC2RPQ: The Hard and Easy Cases. In 23rd International Conference on Database Theory (ICDT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 155, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{figueira:LIPIcs.ICDT.2020.9, author = {Figueira, Diego}, title = {{Containment of UC2RPQ: The Hard and Easy Cases}}, booktitle = {23rd International Conference on Database Theory (ICDT 2020)}, pages = {9:1--9:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-139-9}, ISSN = {1868-8969}, year = {2020}, volume = {155}, editor = {Lutz, Carsten and Jung, Jean Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2020.9}, URN = {urn:nbn:de:0030-drops-119330}, doi = {10.4230/LIPIcs.ICDT.2020.9}, annote = {Keywords: Regular Path Queries (RPQ), 2RPQ, CRPQ, C2RPQ, UC2RPQ, graph databases, containment, inclusion, equivalence, dichotomy, graph measure, bridge-width (bridgewidth), minimal edge separator, minimal cut-set, max-cut, tree-width (treewidth)} }

Document

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

The class of synchronous relations, also known as automatic or regular, is one of the most studied subclasses of rational relations. It enjoys many desirable closure properties and is known to be logically characterized: the synchronous relations are exactly those that are defined by a first-order formula on the structure of all finite words, with the prefix, equal-length and last-letter predicates. Here, we study the quantifier alternation hierarchy of this logic. We show that it collapses at level Sigma_3 and that all levels below admit decidable characterizations. Our results reveal the connections between this hierarchy and the well-known hierarchy of first-order defined languages of finite words.

Diego Figueira, Varun Ramanathan, and Pascal Weil. The Quantifier Alternation Hierarchy of Synchronous Relations. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{figueira_et_al:LIPIcs.MFCS.2019.29, author = {Figueira, Diego and Ramanathan, Varun and Weil, Pascal}, title = {{The Quantifier Alternation Hierarchy of Synchronous Relations}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {29:1--29:14}, 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.2019.29}, URN = {urn:nbn:de:0030-drops-109735}, doi = {10.4230/LIPIcs.MFCS.2019.29}, annote = {Keywords: synchronous relations, automatic relations, first-order logic, characterization, quantifier alternation} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We study the boundedness problem for unions of conjunctive regular path queries with inverses (UC2RPQs). This is the problem of, given a UC2RPQ, checking whether it is equivalent to a union of conjunctive queries (UCQ). We show the problem to be ExpSpace-complete, thus coinciding with the complexity of containment for UC2RPQs. As a corollary, when a UC2RPQ is bounded, it is equivalent to a UCQ of at most triple-exponential size, and in fact we show that this bound is optimal. We also study better behaved classes of UC2RPQs, namely acyclic UC2RPQs of bounded thickness, and strongly connected UCRPQs, whose boundedness problem is, respectively, PSpace-complete and Pi_2^P-complete. Most upper bounds exploit results on limitedness for distance automata, in particular extending the model with alternation and two-wayness, which may be of independent interest.

Pablo Barceló, Diego Figueira, and Miguel Romero. Boundedness of Conjunctive Regular Path Queries (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 104:1-104:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barcelo_et_al:LIPIcs.ICALP.2019.104, author = {Barcel\'{o}, Pablo and Figueira, Diego and Romero, Miguel}, title = {{Boundedness of Conjunctive Regular Path Queries}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {104:1--104:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.104}, URN = {urn:nbn:de:0030-drops-106803}, doi = {10.4230/LIPIcs.ICALP.2019.104}, annote = {Keywords: regular path queries, boundedness, limitedness, distance automata} }

Document

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

A standard approach to define k-ary word relations over a finite alphabet A is through k-tape finite state automata that recognize regular languages L over {1, ..., k} x A, where (i,a) is interpreted as reading letter a from tape i. Accordingly, a word w in L denotes the tuple (u_1, ..., u_k) in (A^*)^k in which u_i is the projection of w onto i-labelled letters. While this formalism defines the well-studied class of rational relations, enforcing restrictions on the reading regime from the tapes, which we call synchronization, yields various sub-classes of relations. Such synchronization restrictions are imposed through regular properties on the projection of the language L onto {1, ..., k}. In this way, for each regular language C subseteq {1, ..., k}^*, one obtains a class Rel({C}) of relations. Synchronous, Recognizable, and Length-preserving rational relations are all examples of classes that can be defined in this way.
We study basic properties of these classes of relations, in terms of closure under intersection, complement, concatenation, Kleene star and projection. We characterize the classes with each closure property. For the binary case (k=2) this yields effective procedures.

María Emilia Descotte, Diego Figueira, and Santiago Figueira. Closure Properties of Synchronized Relations. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{descotte_et_al:LIPIcs.STACS.2019.22, author = {Descotte, Mar{\'\i}a Emilia and Figueira, Diego and Figueira, Santiago}, title = {{Closure Properties of Synchronized Relations}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {22:1--22:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.22}, URN = {urn:nbn:de:0030-drops-102614}, doi = {10.4230/LIPIcs.STACS.2019.22}, annote = {Keywords: synchronized word relations, rational, closure, characterization, intersection, complement, Kleene star, concatenation} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

A natural approach to define binary word relations over a finite alphabet A is through two-tape finite state automata that recognize regular languages over {1, 2} x A, where (i,a) is interpreted as reading letter a from tape i. Accordingly, a word w in L denotes the pair (u_1,u_2) in A^* x A^* in which u_i is the projection of w onto i-labelled letters. While this formalism defines the well-studied class of Rational relations (a.k.a. non-deterministic finite state transducers), enforcing restrictions on the reading regime from the tapes, which we call synchronization, yields various sub-classes of relations. Such synchronization restrictions are imposed through regular properties on the projection of the language onto {1,2}. In this way, for each regular language C subseteq {1,2}^*, one obtains a class Rel({C}) of relations. Regular, Recognizable, and length-preserving rational relations are all examples of classes that can be defined in this way.
We study the problem of containment for synchronized classes of relations: given C,D subseteq {1,2}^*, is Rel({C}) subseteq Rel({D})? We show a characterization in terms of C and D which gives a decidability procedure to test for class inclusion. This also yields a procedure to re-synchronize languages from {1, 2} x A preserving the denoted relation whenever the inclusion holds.

María Emilia Descotte, Diego Figueira, and Gabriele Puppis. Resynchronizing Classes of Word Relations. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 123:1-123:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{descotte_et_al:LIPIcs.ICALP.2018.123, author = {Descotte, Mar{\'\i}a Emilia and Figueira, Diego and Puppis, Gabriele}, title = {{Resynchronizing Classes of Word Relations}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {123:1--123:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.123}, URN = {urn:nbn:de:0030-drops-91270}, doi = {10.4230/LIPIcs.ICALP.2018.123}, annote = {Keywords: synchronized word relations, containment, resynchronization} }

Document

**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Whether the reachability problem for branching vector addition systems, or equivalently the provability problem for multiplicative exponential linear logic, is decidable has been a long-standing open question. The one-dimensional case is a generalisation of the extensively studied one-counter nets, and it was recently established polynomial-time complete provided counter updates are given in unary. Our main contribution is to determine the complexity when the encoding is binary: polynomial-space complete.

Diego Figueira, Ranko Lazic, Jérôme Leroux, Filip Mazowiecki, and Grégoire Sutre. Polynomial-Space Completeness of Reachability for Succinct Branching VASS in Dimension One. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 119:1-119:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{figueira_et_al:LIPIcs.ICALP.2017.119, author = {Figueira, Diego and Lazic, Ranko and Leroux, J\'{e}r\^{o}me and Mazowiecki, Filip and Sutre, Gr\'{e}goire}, title = {{Polynomial-Space Completeness of Reachability for Succinct Branching VASS in Dimension One}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {119:1--119:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.119}, URN = {urn:nbn:de:0030-drops-74374}, doi = {10.4230/LIPIcs.ICALP.2017.119}, annote = {Keywords: branching vector addition systems, reachability problem} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

In the context of statistical databases, the release of accurate statistical information about the collected data often puts at risk the privacy of the individual contributors. The goal of differential privacy is to maximise the utility of a query while protecting the individual records in the database. A natural way to achieve differential privacy is to add statistical noise to the result of the query.
In this context, a mechanism for releasing statistical information is thus a trade-off between utility and privacy. In order to balance these two "conflicting" requirements, privacy preserving mechanisms calibrate the added noise to the so-called sensitivity of the query, and thus a precise estimate of the sensitivity of the query is necessary to determine the amplitude of the noise to be added.
In this paper, we initiate a systematic study of sensitivity of counting queries over relational databases. We first observe that the sensitivity of a Relational Algebra query with counting is not computable in general, and that while the sensitivity of Conjunctive Queries with counting is computable, it becomes unbounded as soon as the query includes a join. We then consider restricted classes of databases (databases with constraints), and study the problem of computing the sensitivity of a query given such constraints. We are able to establish bounds on the sensitivity of counting conjunctive queries over constrained databases. The kind of constraints studied here are: functional dependencies and cardinality dependencies. The latter is a natural generalisation of functional dependencies that allows us to provide tight bounds on the sensitivity of counting conjunctive queries.

Myrto Arapinis, Diego Figueira, and Marco Gaboardi. Sensitivity of Counting Queries. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 120:1-120:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arapinis_et_al:LIPIcs.ICALP.2016.120, author = {Arapinis, Myrto and Figueira, Diego and Gaboardi, Marco}, title = {{Sensitivity of Counting Queries}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {120:1--120:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.120}, URN = {urn:nbn:de:0030-drops-62552}, doi = {10.4230/LIPIcs.ICALP.2016.120}, annote = {Keywords: Differential privacy, sensitivity, relational algebra} }

Document

**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

While the theory of languages of words is very mature, our understanding of relations on words is still lagging behind. And yet such relations appear in many new applications such as verification of parameterized systems, querying graph-structured data, and information extraction, for instance. Classes of well-behaved relations typically used in such applications are obtained by adapting some of the equivalent definitions of regularity of words for relations, leading to non-equivalent notions of recognizable, regular, and rational relations.
The goal of this paper is to propose a systematic way of defining classes of relations on words, of which these three classes are just natural examples, and to demonstrate its advantages compared to some of the standard techniques for studying word relations. The key idea is that of a synchronization of a pair of words, which is a word over an extended alphabet. Using it, we define classes of relations via classes of regular languages over a fixed alphabet, just {1,2} for binary relations. We characterize some of the standard classes of relations on words via finiteness of parameters of synchronization languages, called shift, lag, and shiftlag. We describe these conditions in terms of the structure of cycles of graphs underlying automata, thereby showing their decidability. We show that for these classes there exist canonical synchronization languages, and every class of relations can be effectively re-synchronized using those canonical representatives. We also give sufficient conditions on synchronization languages, defined in terms of injectivity and surjectivity of their Parikh images, that guarantee closure under intersection and complement of the classes of relations they define.

Diego Figueira and Leonid Libkin. Synchronizing Relations on Words. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 518-529, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{figueira_et_al:LIPIcs.STACS.2014.518, author = {Figueira, Diego and Libkin, Leonid}, title = {{Synchronizing Relations on Words}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {518--529}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.518}, URN = {urn:nbn:de:0030-drops-44849}, doi = {10.4230/LIPIcs.STACS.2014.518}, annote = {Keywords: Word Relations, Regular, Rational, Recognizable} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We study the satisfiability problem for XPath over XML documents of bounded depth. We define two parameters, called match width and braid width, that assign a number to any class of documents. We show that for all k, satisfiability for XPath restricted to bounded depth documents with match width at most k is decidable; and that XPath is undecidable on any class of documents with unbounded braid width. We conjecture that these two parameters are equivalent, in the sense that a class of documents has bounded match width iff it has bounded braid width.

Vince Bárány, Mikolaj Bojanczyk, Diego Figueira, and Pawel Parys. Decidable classes of documents for XPath. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 99-111, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{barany_et_al:LIPIcs.FSTTCS.2012.99, author = {B\'{a}r\'{a}ny, Vince and Bojanczyk, Mikolaj and Figueira, Diego and Parys, Pawel}, title = {{Decidable classes of documents for XPath}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {99--111}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.99}, URN = {urn:nbn:de:0030-drops-38512}, doi = {10.4230/LIPIcs.FSTTCS.2012.99}, annote = {Keywords: XPath, XML, class automata, data trees, data words, satisfiability} }

Document

**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

A data tree is a tree whose every node carries a label from a finite alphabet and a datum from some infinite domain. We introduce a new model of automata over unranked data trees with a decidable emptiness problem. It is essentially a bottom-up alternating automaton with one register, enriched with epsilon-transitions that perform tests on the data values of the subtree. We show that it captures the expressive power of the vertical fragment of XPath -- containing the child, descendant, parent and ancestor axes -- obtaining thus a decision procedure for its satisfiability problem.

Diego Figueira and Luc Segoufin. Bottom-up automata on data trees and vertical XPath. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 93-104, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{figueira_et_al:LIPIcs.STACS.2011.93, author = {Figueira, Diego and Segoufin, Luc}, title = {{Bottom-up automata on data trees and vertical XPath}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {93--104}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.93}, URN = {urn:nbn:de:0030-drops-30029}, doi = {10.4230/LIPIcs.STACS.2011.93}, annote = {Keywords: Decidability, XPath, Data trees, Bottom-up tree automata} }

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