Abstract
Motivated by distributed data processing applications, we introduce a class of labeled directed acyclic graphs constructed using sequential and parallel composition operations, and study automata and logics over them. We show that deterministic and non-deterministic acceptors over such graphs have the same expressive power, which can be equivalently characterized by Monadic Second-Order logic and the graded µ-calculus. We establish closure under composition operations and decision procedures for membership, emptiness, and inclusion. A key feature of our graphs, called synchronized series-parallel graphs (SSPG), is that parallel composition introduces a synchronization edge from the newly introduced source vertex to the sink. The transfer of information enabled by such edges is crucial to the determinization construction, which would not be possible for the traditional definition of series-parallel graphs.
SSPGs allow both ordered ranked parallelism and unordered unranked parallelism. The latter feature means that in the corresponding automata, the transition function needs to account for an arbitrary number of predecessors by counting each type of state only up to a specified constant, thus leading to a notion of counting complexity that is distinct from the classical notion of state complexity. The determinization construction translates a nondeterministic automaton with n states and k counting complexity to a deterministic automaton with 2n2 states and kn counting complexity, and both these bounds are shown to be tight. Furthermore, for nondeterministic automata a bound of 2 on counting complexity suffices without loss of expressiveness.
- Rajeev Alur, Phillip Hilliard, Zachary G Ives, Konstantinos Kallas, Konstantinos Mamouras, Filip Niksic, Caleb Stanford, Val Tannen, and Anton Xue. 2021. Synchronization schemas. In Proceedings of the 40th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems. 1–18. https://doi.org/10.1145/3452021.3458317
Google Scholar
Digital Library
- Rajeev Alur and Parthasarathy Madhusudan. 2004. Visibly pushdown languages. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing. 202–211. https://doi.org/10.1145/1007352.1007390
Google Scholar
Digital Library
- Rajeev Alur and P. Madhusudan. 2009. Adding Nesting Structure to Words. J. ACM, 56, 3 (2009), Article 16, may, 43 pages. issn:0004-5411 https://doi.org/10.1145/1516512.1516518
Google Scholar
Digital Library
- Rajeev Alur and Pavol Černý. 2011. Streaming Transducers for Algorithmic Verification of Single-Pass List-Processing Programs. In Proceedings of the 38th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL ’11). Association for Computing Machinery, New York, NY, USA. 599–610. isbn:9781450304900 https://doi.org/10.1145/1926385.1926454
Google Scholar
Digital Library
- Roberto Amadini. 2021. A survey on string constraint solving. ACM Computing Surveys (CSUR), 55, 1 (2021), 1–38. https://doi.org/10.1145/3484198
Google Scholar
Digital Library
- Arvind Arasu, Shivnath Babu, and Jennifer Widom. 2003. CQL: A language for continuous queries over streams and relations. In International Workshop on Database Programming Languages. 1–19. https://doi.org/10.1007/978-3-540-24607-7_1
Google Scholar
Cross Ref
- Arvind Arasu, Shivnath Babu, and Jennifer Widom. 2006. The CQL continuous query language: semantic foundations and query execution. The VLDB Journal, 15, 2 (2006), 121–142. https://doi.org/10.1007/s00778-004-0147-z
Google Scholar
Digital Library
- Michael A Arbib and Yehoshafat Give’on. 1968. Algebra automata I: Parallel programming as a prolegomena to the categorical approach. Information and Control, 12, 4 (1968), 331–345. https://doi.org/10.1016/S0019-9958(68)90374-4
Google Scholar
Cross Ref
- Brian Babcock, Shivnath Babu, Mayur Datar, Rajeev Motwani, and Jennifer Widom. 2002. Models and issues in data stream systems. In Proceedings of the twenty-first ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems. 1–16. https://doi.org/10.1145/543613.543615
Google Scholar
Digital Library
- Brenda S Baker. 1978. Tree transducers and tree languages. Information and Control, 37, 3 (1978), 241–266. https://doi.org/10.1016/S0019-9958(78)90538-7
Google Scholar
Cross Ref
- Everardo Bárcenas, Edgard Benítez-Guerrero, and Jesús Lavalle. 2015. On the Model Checking of the Graded μ -calculus on Trees. In Mexican International Conference on Artificial Intelligence. 178–189. https://doi.org/10.1007/978-3-319-27060-9_14
Google Scholar
Cross Ref
- Hans Bekić. 1984. Definable operations in general algebras, and the theory of automata and flowcharts. Springer Berlin Heidelberg, Berlin, Heidelberg. 30–55. isbn:978-3-540-38933-0 https://doi.org/10.1007/BFb0048939
Google Scholar
Cross Ref
- Henrik Björklund and Thomas Schwentick. 2010. On notions of regularity for data languages. Theoretical Computer Science, 411, 4-5 (2010), 702–715. https://doi.org/10.1007/978-3-540-74240-1_9
Google Scholar
Cross Ref
- Mikoł aj Bojańczyk, Anca Muscholl, Thomas Schwentick, Luc Segoufin, and Claire David. 2006. Two-variable logic on words with data. In 21st Annual IEEE Symposium on Logic in Computer Science (LICS’06). 7–16. https://doi.org/10.1109/LICS.2006.51
Google Scholar
Digital Library
- Symeon Bozapalidis and Antonios Kalampakas. 2008. Graph automata. Theoretical Computer Science, 393, 1-3 (2008), 147–165. https://doi.org/j.tcs.2007.11.022
Google Scholar
Digital Library
- Julian C. Bradfield and Igor Walukiewicz. 2018. The mu-calculus and Model Checking. In Handbook of Model Checking, Edmund M. Clarke, Thomas A. Henzinger, Helmut Veith, and Roderick Bloem (Eds.). Springer, 871–919. https://doi.org/10.1007/978-3-319-10575-8_26
Google Scholar
Cross Ref
- Julien Carme, Joachim Niehren, and Marc Tommasi. 2004. Querying unranked trees with stepwise tree automata. In International Conference on Rewriting Techniques and Applications. 105–118. https://doi.org/10.1007/978-3-540-25979-4_8
Google Scholar
Cross Ref
- Carl Chapman and Kathryn T Stolee. 2016. Exploring regular expression usage and context in Python. In Proceedings of the 25th International Symposium on Software Testing and Analysis. 282–293. https://doi.org/10.1145/2931037.2931073
Google Scholar
Digital Library
- Hubert Comon, Max Dauchet, Rémi Gilleron, Florent Jacquemard, Denis Lugiez, Christof Löding, Sophie Tison, and Marc Tommasi. 2008. Tree Automata Techniques and Applications. https://hal.inria.fr/hal-03367725
Google Scholar
- Bruno Courcelle. 1989. On recognizable sets and tree automata. In Algebraic Techniques. Elsevier, 93–126. https://doi.org/10.1016/B978-0-12-046370-1.50009-7
Google Scholar
Cross Ref
- Bruno Courcelle. 1990. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation, 85, 1 (1990), 12–75. issn:0890-5401 https://doi.org/10.1016/0890-5401(90)90043-H
Google Scholar
Digital Library
- Julien Cristau, Christof Löding, and Wolfgang Thomas. 2005. Deterministic automata on unranked trees. In International Symposium on Fundamentals of Computation Theory. 68–79. https://doi.org/10.1007/11537311_7
Google Scholar
Digital Library
- Silvano Dal Zilio and Denis Lugiez. 2003. XML schema, tree logic and sheaves automata. In International Conference on Rewriting Techniques and Applications. 246–263. https://doi.org/10.1007/s00200-006-0016-7
Google Scholar
Cross Ref
- Loris D’Antoni and Margus Veanes. 2021. Automata modulo theories. Commun. ACM, 64, 5 (2021), 86–95. https://doi.org/10.1145/3419404
Google Scholar
Digital Library
- Stéphane Demri and Ranko Lazić. 2009. LTL with the freeze quantifier and register automata. ACM Transactions on Computational Logic (TOCL), 10, 3 (2009), 1–30. https://doi.org/10.1145/1507244.1507246
Google Scholar
Digital Library
- Volker Diekert and Yves Métivier. 1997. Partial commutation and traces. In Handbook of formal languages. Springer, 457–533. https://doi.org/10.1007/978-3-642-59126-6_8
Google Scholar
Cross Ref
- Volker Diekert and Grzegorz Rozenberg. 1995. The book of traces. World scientific. https://doi.org/10.1142/2563
Google Scholar
Cross Ref
- Rayna Dimitrova and Rupak Majumdar. 2018. Reachability Analysis of Reversal-Bounded Automata on Series—Parallel Graphs. Acta Inf., 55, 2 (2018), mar, 153–189. issn:0001-5903 https://doi.org/10.1007/s00236-016-0290-1
Google Scholar
Digital Library
- John Doner. 1970. Tree acceptors and some of their applications. J. Comput. System Sci., 4, 5 (1970), 406–451. https://doi.org/10.1016/S0022-0000(70)80041-1
Google Scholar
Digital Library
- Frank Drewes, Hans-Joerg Kreowski, and Annegret Habel. 1997. Hyperedge replacement graph grammars. 95–162. isbn:ISBN:98-102288-48 https://doi.org/10.1142/9789812384720_0002
Google Scholar
Cross Ref
- Loris D’Antoni and Margus Veanes. 2017. The power of symbolic automata and transducers. In International Conference on Computer Aided Verification. 47–67. https://doi.org/10.1007/978-3-319-63387-9_3
Google Scholar
Cross Ref
- Joost Engelfriet. 2015. Tree Automata and Tree Grammars. CoRR, abs/1510.02036 (2015), 80 pages. arXiv:1510.02036. arxiv:1510.02036
Google Scholar
- Ferenc Gécseg and Magnus Steinby. 1997. Tree languages. In Handbook of formal languages. Springer, 1–68. https://doi.org/10.1007/978-3-642-59126-6_1
Google Scholar
Cross Ref
- Buğra Gedik. 2014. Partitioning functions for stateful data parallelism in stream processing. The VLDB Journal, 23, 4 (2014), 517–539. https://doi.org/10.1007/s00778-013-0335-9
Google Scholar
Digital Library
- Sumit Gulwani. 2011. Automating string processing in spreadsheets using input-output examples. ACM Sigplan Notices, 46, 1 (2011), 317–330. https://doi.org/10.1145/1926385.1926423
Google Scholar
Digital Library
- Hossein Hojjat, Philipp Rümmer, and Ali Shamakhi. 2019. On strings in software model checking. In Asian Symposium on Programming Languages and Systems. 19–30. https://doi.org/10.1007/978-3-030-34175-6_2
Google Scholar
Cross Ref
- John E. Hopcroft and Richard M. Karp. 1973. An n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs. SIAM J. Comput., 2, 4 (1973), 225–231. https://doi.org/10.1137/0202019 arxiv:https://doi.org/10.1137/0202019.
Google Scholar
Digital Library
- D. Janin and G. Lenzi. 2001. Relating levels of the mu-calculus hierarchy and levels of the monadic hierarchy. In Proceedings 16th Annual IEEE Symposium on Logic in Computer Science. 347–356. https://doi.org/10.1109/LICS.2001.932510
Google Scholar
Cross Ref
- David Janin and Igor Walukiewicz. 1996. On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. In CONCUR ’96: Concurrency Theory, Ugo Montanari and Vladimiro Sassone (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg. 263–277. isbn:978-3-540-70625-0 https://doi.org/10.1007/3-540-61604-7_60
Google Scholar
Cross Ref
- A. B. Kahn. 1962. Topological Sorting of Large Networks. Commun. ACM, 5, 11 (1962), nov, 558–562. issn:0001-0782 https://doi.org/10.1145/368996.369025
Google Scholar
Digital Library
- Konstantinos Kallas, Filip Niksic, Caleb Stanford, and Rajeev Alur. 2022. Stream processing with dependency-guided synchronization. In Proceedings of the 27th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming. 1–16. https://doi.org/10.1145/3503221.3508413
Google Scholar
Digital Library
- Tsutomu Kamimura and Giora Slutzki. 1981. Parallel and two-way automata on directed ordered acyclic graphs. Information and Control, 49, 1 (1981), 10–51. https://doi.org/10.1016/S0019-9958(81)90438-1
Google Scholar
Cross Ref
- Tsutomu Kamimura and Giora Slutzki. 1981. Transductions of dags and trees. Mathematical systems theory, 15, 1 (1981), 225–249. https://doi.org/10.1007/BF01786981
Google Scholar
Cross Ref
- Michael Kaminski and Nissim Francez. 1994. Finite-memory automata. Theoretical Computer Science, 134, 2 (1994), 329–363. https://doi.org/10.1016/0304-3975(94)90242-9
Google Scholar
Digital Library
- Dexter Kozen. 1983. Results on the propositional μ -calculus. Theoretical Computer Science, 27, 3 (1983), 333–354. issn:0304-3975 https://doi.org/10.1016/0304-3975(82)90125-6 Special Issue Ninth International Colloquium on Automata, Languages and Programming (ICALP) Aarhus, Summer 1982.
Google Scholar
Cross Ref
- Orna Kupferman, Ulrike Sattler, and Moshe Y Vardi. 2002. The complexity of the graded μ -calculus. In International Conference on Automated Deduction. 423–437. https://doi.org/10.1007/3-540-45620-1_34
Google Scholar
Cross Ref
- Dietrich Kuske. 2000. Infinite series-parallel posets: logic and languages. In International Colloquium on Automata, Languages, and Programming. 648–662. https://doi.org/10.1007/3-540-45022-X_55
Google Scholar
Cross Ref
- Kamal Lodaya and Pascal Weil. 1998. Series-parallel posets: algebra, automata and languages. In Annual Symposium on Theoretical Aspects of Computer Science. 555–565. https://doi.org/10.1007/BFb0028590
Google Scholar
Cross Ref
- Kamal Lodaya and Pascal Weil. 2000. Series–parallel languages and the bounded-width property. Theoretical Computer Science, 237, 1-2 (2000), 347–380. https://doi.org/10.1016/S0304-3975(00)00031-1
Google Scholar
Digital Library
- Denis Lugiez. 2005. Multitree automata that count. Theoretical Computer Science, 333, 1-2 (2005), 225–263. https://doi.org/j.tcs.2004.10.023
Google Scholar
Digital Library
- Konstantinos Mamouras, Caleb Stanford, Rajeev Alur, Zachary G Ives, and Val Tannen. 2019. Data-trace types for distributed stream processing systems. In Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation. 670–685. https://doi.org/10.1145/3314221.3314580
Google Scholar
Digital Library
- Antoni Mazurkiewicz. 1986. Trace theory. In Advanced course on Petri nets. 278–324. https://doi.org/10.1007/3-540-17906-2_30
Google Scholar
Cross Ref
- Derek G Murray, Frank McSherry, Rebecca Isaacs, Michael Isard, Paul Barham, and Martín Abadi. 2013. Naiad: a timely dataflow system. In Proceedings of the Twenty-Fourth ACM Symposium on Operating Systems Principles. 439–455. https://doi.org/10.1145/2517349.2522738
Google Scholar
Digital Library
- Derek G Murray, Frank McSherry, Michael Isard, Rebecca Isaacs, Paul Barham, and Martin Abadi. 2016. Incremental, iterative data processing with timely dataflow. Commun. ACM, 59, 10 (2016), 75–83. https://doi.org/10.1145/2983551
Google Scholar
Digital Library
- Frank Neven, Thomas Schwentick, and Victor Vianu. 2004. Finite state machines for strings over infinite alphabets. ACM Transactions on Computational Logic (TOCL), 5, 3 (2004), 403–435. https://doi.org/10.1145/1013560.1013562
Google Scholar
Digital Library
- Joachim Niehren and Andreas Podelski. 1993. Feature automata and recognizable sets of feature trees. In Colloquium on Trees in Algebra and Programming. 356–375. https://doi.org/10.1007/3-540-56610-4_76
Google Scholar
Cross Ref
- Fabian Reiter. 2015. Distributed graph automata. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science. 192–201. https://doi.org/10.1109/LICS.2015.27
Google Scholar
Digital Library
- Patrick M Rondon, Ming Kawaguci, and Ranjit Jhala. 2008. Liquid types. In Proceedings of the 29th ACM SIGPLAN Conference on Programming Language Design and Implementation. 159–169. https://doi.org/10.1145/1375581.1375602
Google Scholar
Digital Library
- Luc Segoufin. 2006. Automata and logics for words and trees over an infinite alphabet. In International Workshop on Computer Science Logic. 41–57. https://doi.org/10.1007/11874683_3
Google Scholar
Digital Library
- Ambuj Shatdal and Jeffrey F Naughton. 1995. Adaptive parallel aggregation algorithms. Acm Sigmod Record, 24, 2 (1995), 104–114. https://doi.org/10.1145/223784.223801
Google Scholar
Digital Library
- Yael Shemesh and Nissim Francez. 1994. Finite-state unification automata and relational languages. Information and Computation, 114, 2 (1994), 192–213. https://doi.org/10.1006/inco.1994.1085
Google Scholar
Digital Library
- James W. Thatcher and Jesse B. Wright. 1968. Generalized finite automata theory with an application to a decision problem of second-order logic. Mathematical systems theory, 2, 1 (1968), 57–81. https://doi.org/10.1007/BF01691346
Google Scholar
Cross Ref
- Wolfgang Thomas. 1990. Infinite trees and automaton definable relations over ω -words. In Annual Symposium on Theoretical Aspects of Computer Science. 263–277. https://doi.org/10.1016/0304-3975(92)90090-3
Google Scholar
Digital Library
- Wolfgang Thomas. 1997. Elements of an automata theory over partial orders. In Partial order methods in verification. 29, American Mathematical Society, 25–40. https://doi.org/10.1090/dimacs/029/02
Google Scholar
Cross Ref
- Peter A. Tucker, David Maier, Tim Sheard, and Leonidas Fegaras. 2003. Exploiting punctuation semantics in continuous data streams. IEEE Transactions on Knowledge and Data Engineering, 15, 3 (2003), 555–568. https://doi.org/10.1109/TKDE.2003.1198390
Google Scholar
Digital Library
- Igor Walukiewicz. 1996. Monadic second order logic on tree-like structures. In STACS 96, Claude Puech and Rüdiger Reischuk (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg. 399–413. isbn:978-3-540-49723-3
Google Scholar
Index Terms
A Robust Theory of Series Parallel Graphs
Recommendations
The circular chromatic number of series-parallel graphs
In this article, we consider the circular chromatic number χc(G) of series-parallel graphs G. It is well known that series-parallel graphs have chromatic number at most 3. Hence, their circular chromatic numbers are at most 3. If a series-parallel graph ...
Small Drawings of Outerplanar Graphs, Series-Parallel Graphs, and Other Planar Graphs
In this paper, we study small planar drawings of planar graphs. For arbitrary planar graphs, ź(n2) is the established upper and lower bound on the worst-case area. A long-standing open problem is to determine for what graphs a smaller area can be ...
Equistable series-parallel graphs
Special issue on stability in graphs and related topicsA graph is called equistable when there is a non-negative weight function on its vertices such that a set S of vertices has total weight 1 if and only if S is maximal stable. We characterize those series-parallel graphs that are equistable, generalizing ...






Comments