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Reachability is in DynFO [article]

Samir Datta, Raghav Kulkarni, Anish Mukherjee, Thomas Schwentick, Thomas Zeume
2017 arXiv   pre-print
This article confirms their conjecture that the Reachability query is in DynFO. As a byproduct it is shown that the rank of a matrix with small values can be maintained in DynFO(+,x).  ...  It is further shown that the (size of the) maximum matching of a graph can be maintained in non-uniform DynFO, another extension of DynFO, with non-uniform initialisation of the auxiliary relations.  ...  In this article, we confirm the conjecture of Patnaik and Immerman [34] that the reachability query for general directed graphs is indeed in DynFO. Theorem 1. Reachability is in DynFO.  ... 
arXiv:1502.07467v3 fatcat:tpwyiktbmjc4jh5j5zg7uzhsfy

Dynamic Complexity of Reachability: How Many Changes Can We Handle?

Samir Datta, Pankaj Kumar, Anish Mukherjee, Anuj Tawari, Nils Vortmeier, Thomas Zeume, Emanuela Merelli, Artur Czumaj, Anuj Dawar
2020 International Colloquium on Automata, Languages and Programming  
In 2015, it was shown that reachability for arbitrary directed graphs can be updated by first-order formulas after inserting or deleting single edges.  ...  Later, in 2018, this was extended for changes of size (log n)/(log log n), where n is the size of the graph. Changes of polylogarithmic size can be handled when also majority quantifiers may be used.  ...  In other words, the reachability query is contained in the dynamic complexity class DynFO [19] .  ... 
doi:10.4230/lipics.icalp.2020.122 dblp:conf/icalp/DattaK0TVZ20 fatcat:bcnuyzbx4fam3gnh2qkgalvp2e

Dynamic complexity of Reachability: How many changes can we handle? [article]

Samir Datta, Pankaj Kumar, Anish Mukherjee, Anuj Tawari, Nils Vortmeier, Thomas Zeume
2020 arXiv   pre-print
Later, in 2018, this was extended for changes of size log n/loglog n, where n is the size of the graph. Changes of polylogarithmic size can be handled when also majority quantifiers may be used.  ...  In 2015, it was shown that reachability for arbitrary directed graphs can be updated by first-order formulas after inserting or deleting single edges.  ...  Acknowledgements The first author would like to thank Chetan Gupta for finding a problem in a previous version of the proof of Theorem 15.  ... 
arXiv:2004.12739v1 fatcat:ml337i6yhbg6hpok3hbr3d7yym

Reachability and Distances under Multiple Changes

Samir Datta, Anish Mukherjee, Nils Vortmeier, Thomas Zeume, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
In other words, Reachability is in DynFO. In this article we extend the framework to changes of multiple edges at a time, and study the Reachability and Distance queries under these changes.  ...  Some preliminary results towards showing that distances are in DynFO are discussed.  ...  The result that Reachability is in DynFO does not imply NL ⊆ DynFO, as DynFO is only known to be closed under very weak reductions, called bounded first-order reductions, under which Reachability is not  ... 
doi:10.4230/lipics.icalp.2018.120 dblp:conf/icalp/Datta0VZ18 fatcat:bnp2mkzqengwjgxmrifytufxzq

The dynamic complexity of formal languages

Wouter Gelade, Marcel Marquardt, Thomas Schwentick
2012 ACM Transactions on Computational Logic  
Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF.  ...  It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation.  ...  Among others they showed that the dynamic complexity of the following problems is in DynFO: Reachability in undirected graphs, minimum spanning forests, multiplication, regular languages, the Dyck languages  ... 
doi:10.1145/2287718.2287719 fatcat:amvn2uah6fb7latv5cw4ycos7q

Dynamic Graph Queries

Pablo Muñoz, Nils Vortmeier, Thomas Zeume, Marc Herbstritt
2016 International Conference on Database Theory  
Further we study the closely related problems of maintaining distances in graphs and reachability in product graphs.  ...  Evaluating graph queries in this dynamic context is a challenging task, as those queries often combine first-order and navigational features.  ...  This query can easily be maintained in DynFO, as it is bounded first-order reducible to reachability.  ... 
doi:10.4230/lipics.icdt.2016.14 dblp:conf/icdt/MunozVZ16 fatcat:f4owlmo3kzhjxcqozvg645e5ja

On the Quantifier-Free Dynamic Complexity of Reachability [chapter]

Thomas Zeume, Thomas Schwentick
2013 Lecture Notes in Computer Science  
The dynamic complexity of the reachability query is studied in the dynamic complexity framework of Patnaik and Immerman, restricted to quantifier-free update formulas.  ...  It is shown that, with this restriction, the reachability query cannot be dynamically maintained, neither with binary auxiliary relations nor with unary auxiliary functions, and that ternary auxiliary  ...  All attempts to show that Reachability cannot be maintained in DynFO have failed.  ... 
doi:10.1007/978-3-642-40313-2_73 fatcat:iozpve6kdvhxvh65tvlbrwj6lq

Reachability and Distances under Multiple Changes [article]

Samir Datta, Anish Mukherjee, Nils Vortmeier, Thomas Zeume
2018 arXiv   pre-print
In other words, Reachability is in DynFO. In this article we extend the framework to changes of multiple edges at a time, and study the Reachability and Distance queries under these changes.  ...  Some preliminary results towards showing that distances are in DynFO are discussed.  ...  The result that Reachability is in DynFO does not imply NL ⊆ DynFO, as DynFO is only known to be closed under very weak reductions, called bounded first-order reductions, under which Reachability is not  ... 
arXiv:1804.08555v1 fatcat:eydrzofihbcrvlvqtqpjvu3ywa

Dynamic Complexity of the Dyck Reachability [chapter]

Patricia Bouyer, Vincent Jugé
2017 Lecture Notes in Computer Science  
A breakthrough was the proof that the Dyck reachability problem in acyclic directed graphs, which is a non-redundant LogCFL-complete problem, belongs to DynFO [16].  ...  In the case of a unary Dyck alphabet, the Dyck reachability problem lies in NL (non-deterministic logarithmic space), and even lies in DynFO in the case of undirected graphs; this contrasts with the case  ...  In fact, the 1-letter Dyck reachability problem (over directed graphs) is bfo + -reducible to the problem of computing distances in directed graphs, whose membership in DynFO or DynFO + is a long-standing  ... 
doi:10.1007/978-3-662-54458-7_16 fatcat:jrabog6lhnccviy7cal7h6yody

On the quantifier-free dynamic complexity of Reachability [article]

Thomas Zeume, Thomas Schwentick
2013 arXiv   pre-print
The dynamic complexity of the reachability query is studied in the dynamic complexity framework of Patnaik and Immerman, restricted to quantifier-free update formulas.  ...  Further inexpressibility results are given for the reachability query in a different setting as well as for a syntactical restriction of quantifier-free update formulas.  ...  All attempts to show that Reachability cannot be maintained in DynFO have failed.  ... 
arXiv:1306.3056v1 fatcat:jzqf7rodpzce5clsmqaxwvvbvm

Dynamic Graph Queries [article]

Pablo Muñoz and Nils Vortmeier and Thomas Zeume
2015 arXiv   pre-print
Further we study the closely related problems of maintaining distances in graphs and reachability in product graphs.  ...  Evaluating graph queries in this dynamic context is a challenging task, as those queries often combine first-order and navigational features.  ...  Yet as an immediate consequence of the dynamic first-order update program for reachability exhibited in [6] , all fixed regular path queries (and, since DynFO is closed under conjunctions, also conjunctions  ... 
arXiv:1512.05511v1 fatcat:3utoxdfn5bhqjdta254yv6orri

Dynamic Complexity of the Dyck Reachability [article]

Patricia Bouyer, Vincent Jugé
2017 arXiv   pre-print
Some of them are P-complete (under a strong notion of reduction) while the others lie either in DynFO or in NL.  ...  We study the dynamic complexity of Dyck reachability problems in directed and undirected graphs, where updates may add or delete edges.  ...  In fact, the 1-letter Dyck reachability problem (over directed graphs) is bfo + -reducible to the problem of computing distances in directed graphs, whose membership in DynFO or DynFO + is a long-standing  ... 
arXiv:1610.07499v2 fatcat:42ov2rdgbjhmjadh64fgv2mkoy

Dynamic Complexity of Planar 3-Connected Graph Isomorphism [chapter]

Jenish C. Mehta
2014 Lecture Notes in Computer Science  
In this work, we show that isomorphism of Planar 3-connected graphs can be decided in DynFO + (which is DynFO with some polynomial precomputation).  ...  In this setting, DynFO, or Dynamic First-Order, is one of the smallest and the most natural complexity class (since SQL queries can be expressed in First-Order Logic), and contains those problems whose  ...  In [13] , it was shown that Undirected Reachability (which is in the static class L), lies in the complexity class DynFO.  ... 
doi:10.1007/978-3-319-06686-8_21 fatcat:zlmya2t5dzakbg26b2xdar3asy

Dynamic Complexity of Group Problems [article]

Samir Datta and Asif Khan and Shivdutt Sharma and Yadu Vasudev and Shankar Ram Vasudevan
2022 arXiv   pre-print
Since then many natural problems related to reachability and matching have been placed in DynFO and related classes DMVZ18,DKMTVZ20,DTV21.  ...  "Is t a product of elements from S? ") using an FO query at every step. This places the dynamic CGM problem (for groups) when the ambient magma is specified via a table in DynFO.  ...  Vasudevan 23:7 Generalising the notion of ord(•) in a group to a magma we say that ord(g) is the length of the unique cycle reachable from g in X g .  ... 
arXiv:2206.00371v1 fatcat:3swsktm7jfhhhdy33bhp24iodi

The dynamic descriptive complexity of k -clique

Thomas Zeume
2017 Information and Computation  
In this work the dynamic descriptive complexity of the kclique query is studied in a framework introduced by Patnaik and Immerman.  ...  k − 2 (even in the presence of unary auxiliary functions).  ...  In [16] it was shown that reachability and 3-clique cannot be maintained in the binary quantifier-free fragment of DynFO.  ... 
doi:10.1016/j.ic.2017.04.005 fatcat:sgkimqrc2fa2ppnmgsfx7domyu
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