Tight Bounds of Circuits for Sum-Product Queries
Article No.: 87, Pages 1 - 20
Abstract
In this paper, we ask the following question: given a Boolean Conjunctive Query (CQ), what is the smallest circuit that computes the provenance polynomial of the query over a given semiring? We answer this question by giving upper and lower bounds. Notably, it is shown that any circuit F that computes a CQ over the tropical semiring must have size log |F| ≥ (1-ε) · da-entw for any ε >0, where da-entw is the degree-aware entropic width of the query. We show a circuit construction that matches this bound when the semiring is idempotent. The techniques we use combine several central notions in database theory: provenance polynomials, tree decompositions, and disjunctive Datalog programs. We extend our results to lower and upper bounds for formulas (i.e., circuits where each gate has outdegree one), and to bounds for non-Boolean CQs.
References
[1]
Antoine Amarilli, Pierre Bourhis, and Pierre Senellart. 2015. Provenance Circuits for Trees and Treelike Instances. In ICALP (2) (Lecture Notes in Computer Science, Vol. 9135). Springer, 56--68.
[2]
Antoine Amarilli and Benny Kimelfeld. 2022. Uniform Reliability of Self-Join-Free Conjunctive Queries. Log. Methods Comput. Sci. 18, 4 (2022).
[3]
Albert Atserias, Martin Grohe, and Dániel Marx. 2008. Size Bounds and Query Plans for Relational Joins. In FOCS. IEEE Computer Society, 739--748.
[4]
Walter Baur and Volker Strassen. 1983. The Complexity of Partial Derivatives. Theor. Comput. Sci. 22 (1983), 317--330.
[5]
Markus Bläser. 2013. Fast Matrix Multiplication. Theory Comput. 5 (2013), 1--60.
[6]
Karl Bringmann, Nofar Carmeli, and Stefan Mengel. 2022. Tight Fine-Grained Bounds for Direct Access on Join Queries. In PODS. ACM, 427--436.
[7]
Nofar Carmeli and Luc Segoufin. 2023. Conjunctive Queries With Self-Joins, Towards a Fine-Grained Enumeration Complexity Analysis. In PODS. ACM, 277--289.
[8]
Terence H. Chan and Raymond W. Yeung. 2002. On a relation between information inequalities and group theory. IEEE Trans. Inf. Theory 48, 7 (2002), 1992--1995.
[9]
Daniel Deutch, Tova Milo, Sudeepa Roy, and Val Tannen. 2014. Circuits for Datalog Provenance. In ICDT. OpenProceedings. org, 201--212.
[10]
Tomasz Gogacz and Szymon Torunczyk. 2017. Entropy Bounds for Conjunctive Queries with Functional Dependencies. In ICDT (LIPIcs, Vol. 68). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 15:1--15:17.
[11]
Todd J. Green, Gregory Karvounarakis, and Val Tannen. 2007. Provenance semirings. In PODS. ACM, 31--40.
[12]
Stasys Jukna. 2015. Lower Bounds for Tropical Circuits and Dynamic Programs. Theory Comput. Syst. 57, 1 (2015), 160--194.
[13]
Ahmet Kara, Hung Q. Ngo, Milos Nikolic, Dan Olteanu, and Haozhe Zhang. 2020. Maintaining Triangle Queries under Updates. ACM Trans. Database Syst. 45, 3 (2020), 11:1--11:46.
[14]
Aziz Amezian El Khalfioui and Jef Wijsen. 2023. Consistent Query Answering for Primary Keys and Conjunctive Queries with Counting. In ICDT (LIPIcs, Vol. 255). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 23:1--23:19.
[15]
Mahmoud Abo Khamis, Phokion G. Kolaitis, Hung Q. Ngo, and Dan Suciu. 2021. Bag Query Containment and Information Theory. ACM Trans. Database Syst. 46, 3 (2021), 12:1--12:39.
[16]
Mahmoud Abo Khamis, Hung Q. Ngo, XuanLong Nguyen, Dan Olteanu, and Maximilian Schleich. 2018. In-Database Learning with Sparse Tensors. In PODS. ACM, 325--340.
[17]
Mahmoud Abo Khamis, Hung Q. Ngo, XuanLong Nguyen, Dan Olteanu, and Maximilian Schleich. 2020. Learning Models over Relational Data Using Sparse Tensors and Functional Dependencies. ACM Trans. Database Syst. 45, 2 (2020), 7:1--7:66.
[18]
Mahmoud Abo Khamis, Hung Q. Ngo, and Atri Rudra. 2016. FAQ: Questions Asked Frequently. In PODS. ACM, 13--28.
[19]
Mahmoud Abo Khamis, Hung Q. Ngo, and Dan Suciu. 2023. What do Shannon-type Inequalities, Submodular Width, and Disjunctive Datalog have to do with one another? arXiv:1612.02503
[20]
Balagopal Komarath, Anurag Pandey, and Chengot Sankaramenon Rahul. 2022. Monotone Arithmetic Complexity of Graph Homomorphism Polynomials. In ICALP (LIPIcs, Vol. 229). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 83:1--83:20.
[21]
Edward A. Lee and Alberto L. Sangiovanni-Vincentelli. 1998. A framework for comparing models of computation. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 17, 12 (1998), 1217--1229.
[22]
Dan Olteanu. 2020. The Relational Data Borg is Learning. Proc. VLDB Endow. 13, 12 (2020), 3502--3515.
[23]
Dan Olteanu and Jiewen Huang. 2008. Using OBDDs for Efficient Query Evaluation on Probabilistic Databases. In SUM (Lecture Notes in Computer Science, Vol. 5291). Springer, 326--340.
[24]
Dan Olteanu and Jakub Závodný. 2015. Size Bounds for Factorised Representations of Query Results. ACM Trans. Database Syst. 40, 1 (2015), 2:1--2:44.
[25]
Neil Robertson and Paul D. Seymour. 1984. Graph minors. III. Planar tree-width. J. Comb. Theory, Ser. B 36, 1 (1984), 49--64.
[26]
John E. Savage. 1998. Models of computation - exploring the power of computing. Addison-Wesley.
[27]
Claus-Peter Schnorr. 1976. A Lower Bound on the Number of Additions in Monotone Computations. Theor. Comput. Sci. 2, 3 (1976), 305--315.
[28]
Amir Shaikhha, Maximilian Schleich, and Dan Olteanu. 2021. An Intermediate Representation for Hybrid Database and Machine Learning Workloads. Proc. VLDB Endow. 14, 12 (2021), 2831--2834.
[29]
Yilei Wang and Ke Yi. 2022. Query Evaluation by Circuits. In PODS. ACM, 67--78.
Index Terms
- Tight Bounds of Circuits for Sum-Product Queries
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May 2024
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Association for Computing Machinery
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Published: 14 May 2024
Published in PACMMOD Volume 2, Issue 2
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