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Documents authored by Le, Xuan Bach

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Le, Xuan Bach

Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.19
Decidability and Complexity of Tree Share Formulas

Authors: Xuan Bach Le, Aquinas Hobor, and Anthony W. Lin

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
Fractional share models are used to reason about how multiple actors share ownership of resources. We examine the decidability and complexity of reasoning over the "tree share" model of Dockins et al. using first-order logic, or fragments thereof. We pinpoint a connection between the basic operations on trees union, intersection, and complement and countable atomless Boolean algebras, allowing us to obtain decidability with the precise complexity of both first-order and existential theories over the tree share model with the aforementioned operations. We establish a connection between the multiplication operation on trees and the theory of word equations, allowing us to derive the decidability of its existential theory and the undecidability of its full first-order theory. We prove that the full first-order theory over the model with both the Boolean operations and the restricted multiplication operation (with constants on the right hand side) is decidable via an embedding to tree-automatic structures.
Cite as

Xuan Bach Le, Aquinas Hobor, and Anthony W. Lin. Decidability and Complexity of Tree Share Formulas. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{le_et_al:LIPIcs.FSTTCS.2016.19,
  author =	{Le, Xuan Bach and Hobor, Aquinas and Lin, Anthony W.},
  title =	{{Decidability and Complexity of Tree Share Formulas}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{19:1--19:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.19},
  URN =		{urn:nbn:de:0030-drops-68544},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.19},
  annote =	{Keywords: Fractional Share Models, Resource Accounting, Countable Atomless Boolean Algebras, Word Equations, Tree Automatic Structures}
}

Le, Xuan-Bach

Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2019.103
Monadic Decomposability of Regular Relations (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors: Pablo Barceló, Chih-Duo Hong, Xuan-Bach Le, Anthony W. Lin, and Reino Niskanen

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

Abstract
Monadic decomposibility - the ability to determine whether a formula in a given logical theory can be decomposed into a boolean combination of monadic formulas - is a powerful tool for devising a decision procedure for a given logical theory. In this paper, we revisit a classical decision problem in automata theory: given a regular (a.k.a. synchronized rational) relation, determine whether it is recognizable, i.e., it has a monadic decomposition (that is, a representation as a boolean combination of cartesian products of regular languages). Regular relations are expressive formalisms which, using an appropriate string encoding, can capture relations definable in Presburger Arithmetic. In fact, their expressive power coincide with relations definable in a universal automatic structure; equivalently, those definable by finite set interpretations in WS1S (Weak Second Order Theory of One Successor). Determining whether a regular relation admits a recognizable relation was known to be decidable (and in exponential time for binary relations), but its precise complexity still hitherto remains open. Our main contribution is to fully settle the complexity of this decision problem by developing new techniques employing infinite Ramsey theory. The complexity for DFA (resp. NFA) representations of regular relations is shown to be NLOGSPACE-complete (resp. PSPACE-complete).
Cite as

Pablo Barceló, Chih-Duo Hong, Xuan-Bach Le, Anthony W. Lin, and Reino Niskanen. Monadic Decomposability of Regular Relations (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. 103:1-103:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{barcelo_et_al:LIPIcs.ICALP.2019.103,
  author =	{Barcel\'{o}, Pablo and Hong, Chih-Duo and Le, Xuan-Bach and Lin, Anthony W. and Niskanen, Reino},
  title =	{{Monadic Decomposability of Regular Relations}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{103:1--103:14},
  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.2019.103},
  URN =		{urn:nbn:de:0030-drops-106790},
  doi =		{10.4230/LIPIcs.ICALP.2019.103},
  annote =	{Keywords: Transducers, Automata, Synchronized Rational Relations, Ramsey Theory, Variable Independence, Automatic Structures}
}
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