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

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

We present results concerning the expressiveness and decidability of a popular graph learning formalism, graph neural networks (GNNs), exploiting connections with logic. We use a family of recently-discovered decidable logics involving "Presburger quantifiers". We show how to use these logics to measure the expressiveness of classes of GNNs, in some cases getting exact correspondences between the expressiveness of logics and GNNs. We also employ the logics, and the techniques used to analyze them, to obtain decision procedures for verification problems over GNNs. We complement this with undecidability results for static analysis problems involving the logics, as well as for GNN verification problems.

Michael Benedikt, Chia-Hsuan Lu, Boris Motik, and Tony Tan. Decidability of Graph Neural Networks via Logical Characterizations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 127:1-127:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{benedikt_et_al:LIPIcs.ICALP.2024.127, author = {Benedikt, Michael and Lu, Chia-Hsuan and Motik, Boris and Tan, Tony}, title = {{Decidability of Graph Neural Networks via Logical Characterizations}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {127:1--127:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.127}, URN = {urn:nbn:de:0030-drops-202708}, doi = {10.4230/LIPIcs.ICALP.2024.127}, annote = {Keywords: Logic, Graph Neural Networks} }

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**Published in:** LIPIcs, Volume 271, 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)

Recently Dependency Quantified Boolean Formula (DQBF) has attracted a lot of attention in the SAT community. Intuitively, a DQBF is a natural extension of quantified boolean formula where for each existential variable, one can specify the set of universal variables it depends on. It has been observed that a DQBF with k existential variables - henceforth denoted by k-DQBF - is essentially a k-CNF formula in succinct representation. However, beside this and the fact that the satisfiability problem is NEXP-complete, not much is known about DQBF.
In this paper we take a closer look at k-DQBF and show that a number of well known classical results on k-SAT can indeed be lifted to k-DQBF, which shows a strong resemblance between k-SAT and k-DQBF. More precisely, we show the following.
a) The satisfiability problem for 2- and 3-DQBF is PSPACE- and NEXP-complete, respectively.
b) There is a parsimonious polynomial time reduction from arbitrary DQBF to 3-DQBF.
c) Many polynomial time projections from SAT to languages in NP can be lifted to polynomial time reductions from the satisfiability of DQBF to languages in NEXP.
d) Languages in the class NSPACE[s(n)] can be reduced to the satisfiability of 2-DQBF with O(s(n)) universal variables.
e) Languages in the class NTIME[t(n)] can be reduced to the satisfiability of 3-DQBF with O(log t(n)) universal variables.
The first result parallels the well known classical results that 2-SAT and 3-SAT are NL- and NP-complete, respectively.

Long-Hin Fung and Tony Tan. On the Complexity of k-DQBF. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fung_et_al:LIPIcs.SAT.2023.10, author = {Fung, Long-Hin and Tan, Tony}, title = {{On the Complexity of k-DQBF}}, booktitle = {26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-286-0}, ISSN = {1868-8969}, year = {2023}, volume = {271}, editor = {Mahajan, Meena and Slivovsky, Friedrich}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2023.10}, URN = {urn:nbn:de:0030-drops-184729}, doi = {10.4230/LIPIcs.SAT.2023.10}, annote = {Keywords: Dependency quantified boolean formulas, existential variables, complexity} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

During the last decades, a lot of effort was put into identifying decidable fragments of first-order logic. Such efforts gave birth, among the others, to the two-variable fragment and the guarded fragment, depending on the type of restriction imposed on formulae from the language. Despite the success of the mentioned logics in areas like formal verification and knowledge representation, such first-order fragments are too weak to express even the simplest statistical constraints, required for modelling of influence networks or in statistical reasoning.
In this work we investigate the extensions of these classical decidable logics with percentage quantifiers, specifying how frequently a formula is satisfied in the indented model. We show, surprisingly, that all the mentioned decidable fragments become undecidable under such extension, sharpening the existing results in the literature. Our negative results are supplemented by decidability of the two-variable guarded fragment with even more expressive counting, namely Presburger constraints. Our results can be applied to infer decidability of various modal and description logics, e.g. Presburger Modal Logics with Converse or ALCI, with expressive cardinality constraints.

Bartosz Bednarczyk, Maja Orłowska, Anna Pacanowska, and Tony Tan. On Classical Decidable Logics Extended with Percentage Quantifiers and Arithmetics. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bednarczyk_et_al:LIPIcs.FSTTCS.2021.36, author = {Bednarczyk, Bartosz and Or{\l}owska, Maja and Pacanowska, Anna and Tan, Tony}, title = {{On Classical Decidable Logics Extended with Percentage Quantifiers and Arithmetics}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {36:1--36:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.36}, URN = {urn:nbn:de:0030-drops-155478}, doi = {10.4230/LIPIcs.FSTTCS.2021.36}, annote = {Keywords: statistical reasoning, knowledge representation, satisfiability, fragments of first-order logic, guarded fragment, two-variable fragment, (un)decidability} }

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

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We consider the extension of FO² with quantifiers that state that the number of elements where a formula holds should belong to a given ultimately periodic set. We show that both satisfiability and finite satisfiability of the logic are decidable. We also show that the spectrum of any sentence is definable in Presburger arithmetic. In the process we present several refinements to the "biregular graph method". In this method, decidability issues concerning two-variable logics are reduced to questions about Presburger definability of integer vectors associated with partitioned graphs, where nodes in a partition satisfy certain constraints on their in- and out-degrees.

Michael Benedikt, Egor V. Kostylev, and Tony Tan. Two Variable Logic with Ultimately Periodic Counting. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 112:1-112:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{benedikt_et_al:LIPIcs.ICALP.2020.112, author = {Benedikt, Michael and Kostylev, Egor V. and Tan, Tony}, title = {{Two Variable Logic with Ultimately Periodic Counting}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {112:1--112:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.112}, URN = {urn:nbn:de:0030-drops-125197}, doi = {10.4230/LIPIcs.ICALP.2020.112}, annote = {Keywords: Presburger Arithmetic, Two-variable logic} }

Document

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

We introduce three formal models of distributed systems for query evaluation on massive databases: Distributed Streaming with Register Automata (DSAs), Distributed Streaming with Register Transducers (DSTs), and Distributed Streaming with Register Transducers and Joins (DSTJs). These models are based on the key-value paradigm where the input is transformed into a dataset of key-value pairs, and on each key a local computation is performed on the values associated with that key resulting in another set of key-value pairs. Computation proceeds in a constant number of rounds, where the result of the last round is the input to the next round, and transformation to key-value pairs is required to be generic. The difference between the three models is in the local computation part. In DSAs it is limited to making one pass over its input using a register automaton, while in DSTs it can make two passes: in the first pass it uses a finite-state automaton and in the second it uses a register transducer. The third model DSTJs is an extension of DSTs, where local computations are capable of constructing the Cartesian product of two sets. We obtain the following results: (1) DSAs can evaluate first-order queries over bounded degree databases; (2) DSTs can evaluate semijoin algebra queries over arbitrary databases; (3) DSTJs can evaluate the whole relational algebra over arbitrary databases; (4) DSTJs are strictly stronger than DSTs, which in turn, are strictly stronger than DSAs; (5) within DSAs, DSTs and DSTJs there is a strict hierarchy w.r.t. the number of rounds.

Frank Neven, Nicole Schweikardt, Frédéric Servais, and Tony Tan. Distributed Streaming with Finite Memory. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 324-341, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{neven_et_al:LIPIcs.ICDT.2015.324, author = {Neven, Frank and Schweikardt, Nicole and Servais, Fr\'{e}d\'{e}ric and Tan, Tony}, title = {{Distributed Streaming with Finite Memory}}, booktitle = {18th International Conference on Database Theory (ICDT 2015)}, pages = {324--341}, 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.2015.324}, URN = {urn:nbn:de:0030-drops-49939}, doi = {10.4230/LIPIcs.ICDT.2015.324}, annote = {Keywords: distributed systems, relational algebra, semijoin algebra, register automata, register transducers.} }

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