ABSTRACT
We introduce a novel variant of BSS machines called Separate Branching BSS machines (S-BSS in short) and develop a Fagin-type logical characterisation for languages decidable in nondeterministic polynomial time by S-BSS machines. We show that NP on S-BSS machines is strictly included in NP on BSS machines and that every NP language on S-BSS machines is a countable disjoint union of closed sets in the usual topology of Rn. Moreover, we establish that on Boolean inputs NP on S-BSS machines without real constants characterises a natural fragment of the complexity class ∃R (a class of problems polynomial time reducible to the true existential theory of the reals) and hence lies between NP and PSPACE. Finally we apply our results to determine the data complexity of probabilistic independence logic.
- Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. 2018. The art gallery problem is ∃R-complete. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018. 65--73. https://doi.org/10.1145/3188745.3188868Google ScholarDigital Library
- Michael Benedikt, Martin Grohe, Leonid Libkin, and Luc Segoufin. 2003. Reachability and connectivity queries in constraint databases. J. Comput. System Sci. 66, 1 (2003), 169--206. https://doi.org/10.1016/S0022-0000(02)00034-X Special Issue on PODS 2000.Google ScholarDigital Library
- Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale. 1997. Complexity and Real Computation. Springer-Verlag, Berlin, Heidelberg.Google Scholar
- Lenore Blum, Mike Shub, and Steve Smale. 1989. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.) 21, 1 (07 1989), 1--46. https://projecteuclid.org:443/euclid.bams/1183555121Google Scholar
- Peter Bürgisser and Felipe Cucker. 2006. Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets. J. Complexity 22, 2 (2006), 147--191. https://doi.org/10.1016/j.jco.2005.11.001Google ScholarDigital Library
- John F. Canny. 1988. Some Algebraic and Geometric Computations in PSPACE. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA. 460--467. https://doi.org/10.1145/62212.62257Google ScholarDigital Library
- Jukka Corander, Antti Hyttinen, Juha Kontinen, Johan Pensar, and Jouko Väänänen. 2019. A logical approach to context-specific independence. Ann. Pure Appl. Logic 170, 9 (2019), 975--992. https: //doi.org/10.1016/j.apal.2019.04.004Google ScholarCross Ref
- Felipe Cucker and Klaus Meer. 1999. Logics Which Capture Complexity Classes Over The Reals. J. Symb. Log. 64, 1 (1999), 363--390. https: //doi.org/10.2307/2586770Google ScholarCross Ref
- Arnaud Durand, Miika Hannula, Juha Kontinen, Arne Meier, and Jonni Virtema. 2018. Approximation and dependence via multi-team semantics. Ann. Math. Artif. Intell. 83, 3-4 (2018), 297--320. https://doi.org/10.1007/s10472-017-9568-4Google ScholarDigital Library
- Arnaud Durand, Miika Hannula, Juha Kontinen, Arne Meier, and Jonni Virtema. 2018. Probabilistic Team Semantics. In Foundations of Information and Knowledge Systems - 10th International Symposium, FoIKS 2018, Budapest, Hungary, May 14-18, 2018, Proceedings. 186--206. https://doi.org/10.1007/978-3-319-90050-6_11Google Scholar
- Pietro Galliani. 2008. Game Values and Equilibria for Undetermined Sentences of Dependence Logic. (2008). MSc Thesis. ILLC Publications, MoL-2008-08.Google Scholar
- Pietro Galliani and Lauri Hella. 2013. Inclusion Logic and Fixed Point Logic. In Computer Science Logic 2013 (CSL 2013) (Leibniz International Proceedings in Informatics (LIPIcs)), Simona Ronchi Della Rocca (Ed.), Vol. 23. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 281--295. https://doi.org/10.4230/LIPIcs.CSL.2013.281Google Scholar
- Erich Grädel and Yuri Gurevich. 1998. Metafinite Model Theory. Inf. Comput. 140, 1 (1998), 26--81. https://doi.org/10.1006/inco.1997.2675Google ScholarDigital Library
- Erich Grädel and Stephan Kreutzer. 1999. Descriptive Complexity Theory for Constraint Databases. In Computer Science Logic, 13th International Workshop, CSL '99, 8th Annual Conference of the EACSL, Madrid, Spain, September 20-25, 1999, Proceedings. 67--81. https://doi.org/10.1007/3-540-48168-0_6Google Scholar
- Erich Grädel and Klaus Meer. 1995. Descriptive complexity theory over the real numbers. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, 29 May-1 June 1995, Las Vegas, Nevada, USA. 315--324. https://doi.org/10.1145/225058.225151Google ScholarDigital Library
- Miika Hannula, Åsa Hirvonen, Juha Kontinen, Vadim Kulikov, and Jonni Virtema. 2019. Facets of Distribution Identities in Probabilistic Team Semantics. In JELIA (Lecture Notes in Computer Science), Vol. 11468. Springer, 304--320.Google Scholar
- Miika Hannula and Juha Kontinen. 2016. A finite axiomatization of conditional independence and inclusion dependencies. Inf. Comput. 249 (2016), 121--137. https://doi.org/10.1016/j.ic.2016.04.001Google ScholarDigital Library
- Uffe Flarup Hansen and Klaus Meer. 2006. Two logical hierarchies of optimization problems over the real numbers. Math. Log. Q. 52, 1 (2006), 37--50. https://doi.org/10.1002/malq.200510021Google ScholarCross Ref
- Tapani Hyttinen, Gianluca Paolini, and Jouko Väänänen. 2017. A Logic for Arguing About Probabilities in Measure Teams. Arch. Math. Logic 56, 5-6 (2017), 475--489. https://doi.org/10.1007/s00153-017-0535-xGoogle ScholarDigital Library
- Paris C. Kanellakis, Gabriel M. Kuper, and Peter Z. Revesz. 1995. Constraint Query Languages. J. Comput. Syst. Sci. 51, 1 (1995), 26--52. https://doi.org/10.1006/jcss.1995.1051Google ScholarDigital Library
- Pascal Koiran. 1994. Computing over the Reals with Addition and Order. Theor. Comput. Sci. 133, 1 (1994), 35--47. https://doi.org/10.1016/0304-3975(93)00063-BGoogle ScholarDigital Library
- Andreas Krebs, Arne Meier, Jonni Virtema, and Martin Zimmermann. 2018. Team Semantics for the Specification and Verification of Hyperproperties. In MFCS (LIPIcs), Vol. 117. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 10:1--10:16.Google Scholar
- Stephan Kreutzer. 2000. Fixed-Point Query Languages for Linear Constraint Databases. In Proceedings of the Nineteenth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, May 15-17, 2000, Dallas, Texas, USA. 116--125. https://doi.org/10.1145/335168.335214Google ScholarDigital Library
- Klaus Meer. 2000. Counting problems over the reals. Theor. Comput. Sci. 242, 1-2 (2000), 41--58. https://doi.org/10.1016/S0304-3975(98)00190-XGoogle ScholarDigital Library
- Marcus Schaefer. 2009. Complexity of Some Geometric and Topological Problems. In Graph Drawing, 17th International Symposium, GD 2009, Chicago, IL, USA, September 22-25, 2009. Revised Papers. 334--344. https://doi.org/10.1007/978-3-642-11805-0_32Google ScholarDigital Library
- Marcus Schaefer and Daniel Stefankovic. 2017. Fixed Points, Nash Equilibria, and the Existential Theory of the Reals. Theory Comput. Syst. 60, 2 (2017), 172--193. https://doi.org/10.1007/s00224-015-9662-0Google ScholarDigital Library
- Jouko Väänänen. 2007. Dependence Logic. Cambridge University Press.Google Scholar
- S. Willard. 2004. General Topology. Dover Publications. https://books.google.co.jp/books?id=-o8xJQ7Ag2cCGoogle Scholar
Index Terms
- Descriptive complexity of real computation and probabilistic independence logic
Recommendations
Fixed-Parameter Tractability, Definability, and Model-Checking
In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory.
We propose to consider parameterized model-checking problems for various fragments of first-order logic as ...
The Descriptive Complexity Approach to LOGCFL
Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal ...
Hierarchies in Dependence Logic
We study fragments D(k∀) and D(k-dep) of dependence logic defined either by restricting the number k of universal quantifiers or the width of dependence atoms in formulas. We find the sublogics of existential second-order logic corresponding to these ...
Comments