DYN-FO AND REGULAR TREES
Keywords: monadic second-order logic; first-order logic; dynamic complexity class; Dyn-FO; logic for dynamic objects
Internship atLimos, Clermont-Ferrand, supervised by Mamadou M. Kant´e.
Dyn-FO. From the seminal paper of Courcelle [4] we know that every monadic second-order property can be solved in linear time in graph classes of bounded tree-width (provided a tree-decomposition of small width is given, but this can be done in linear time following the papers [1] and [2]).
This theorem can be extended to graph classes of bounded clique-width [6, 5]
(replace also linear time by cubic time). These theorems by Courcelle et al.
are based on an old theorem stating that monadic second-order property on trees can be solved in linear time on regular trees (regular in the sense of recognizable by tree-automata [3]), and the fact that graphs of bounded tree- width (or clique-width) are monadic second-order interpretations of regular trees (see [5, Chapters 1-7] for more details and applications).
Even if these theorems are interesting and enough general, their use in several areas such as database theory is limited due to the fact that these theorems deal mostly with static objects, while in databases we mainly deal with dynamic objects, ie we allow update queries such as additions, deletions and modifications of tuples. We can of course at each update recompute the decomposition, but this is counter-productive and moreover we do not know yet how to update a decomposition in terms of the sizes of the updates.
Instead, we would like to be able to update the answers as the database changes, the time complexity depending on the sizes of the changes and not on the entire size of the database. In [8] the authors address this problem and introduce the notion of Dyn-FO, which is the set of properties that can be maintained and queried in first-order logic. Examples of Dyn-FO properties are bipartiteness, computation of minimum spanning trees, undirected graph connectivity, etc. Notice that none of these examples is a static first-order property.
These last years, Schwentick et al. address the problem of identifying classes of properties that are Dyn-FO, and at the same time some lower bounds results stating that some properties are not in Dyn-LwithLa frag- ment of first-order logic. They prove in particular that regular languages are in Dyn-FO, meaning that every monadic second-order property on words is in Dyn-FO [7]. See [9, 10] for more examples.
The goal. We are interested in the extension of the results in words to graphs. For that, we will mimic the techniques by Courcelle et al. and will
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2 DYN-FO AND REGULAR TREES
be first interested in regular trees. The goal of the internship is to first read the papers [8] and [7]. In a second phase it would be desirable to prove or disprove the following conjecture.
Conjecture 1. Every monadic second-order property on trees is in Dyn-FO.
Of course, once this is done we would be interested in finding lower bounds or improving this theorem with some fragments of first-order logic. Notice that even if this is an exploratory subject, we are confident that the tech- niques in [7] could be adapted to prove Conjecture 1.
All materials will be provided and for further information please feel free to contact me [email protected]. In the bibliography section, you can find some references.
References
[1] Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth.SIAM J. Comput., 25(6):1305–1317, 1996.
[2] Hans L. Bodlaender, P˚al Grøn˚as Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michal Pilipczuk. Ano(ckn) 5-approximation algorithm for treewidth. InFOCS, pages 499–508. IEEE Computer Society, 2013.
[3] Hubert Comon, Max Dauchet, Remi Gilleron, Florent Jacquemard, Denis Lugiez, Christof L¨oding, Sophie Tison, and Marc Tommasi. Tree automata techniques and applications. available freely at http://tata.gforge.inria.fr/, 2007.
[4] Bruno Courcelle. The monadic second-order logic of graphs. i. recognizable sets of finite graphs.Inf. Comput., 85(1):12–75, 1990.
[5] Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic: a Language Theoretic Approach, volume 138 of Encyclopedia of Math- ematics and its Applications. Cambridge University Press, 2012. Available at www.labri.fr/perso/courcell/Book/TheBook.pdf.
[6] Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable op- timization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125–150, 2000.
[7] Wouter Gelade, Marcel Marquardt, and Thomas Schwentick. The dynamic com- plexity of formal languages. ACM Trans. Comput. Log., 13(3), 2012. Available at http://arxiv.org/abs/0812.1915.
[8] Sushant Patnaik and Neil Immerman. Dyn-fo: A parallel, dynamic complexity class.
Journal of Computer and System Sciences, 55:199–209, 1997.
[9] Thomas Zeume and Thomas Schwentick. On the quantifier-free dynamic complexity of reachability. In Krishnendu Chatterjee and Jir´ı Sgall, editors,MFCS, volume 8087 of LNCS, pages 837–848. Springer, 2013. Available at http://arxiv.org/abs/1306.3056.
[10] Thomas Zeume and Thomas Schwentick. Dynamic conjunctive queries. In Nicole Schweikardt, Vassilis Christophides, and Vincent Leroy, editors,Proc. 17th Interna- tional Conference on Database Theory (ICDT), Athens, Greece, March 24-28, 2014., pages 38–49. OpenProceedings.org, 2014.
