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Classification of Non-Redundancy of Boolean Predicates of Arity 4

Authors Joshua Brakensiek , Venkatesan Guruswami , Aaron Putterman

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ACM Subject Classification
  • Theory of computation → Constraint and logic programming
Keywords
  • constraint satisfaction problem
  • redundancy

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Abstract

Given a constraint satisfaction problem (CSP) predicate P ⊆ D^r, the non-redundancy (NRD) of P is the maximum-sized instance on n variables such that for every clause of the instance, there is an assignment which satisfies all clauses but that one. The study of NRD for various CSPs is an active area of research which combines ideas from extremal combinatorics, logic, lattice theory, and other techniques. Complete classifications are known in the cases r = 2 and (|D| = 2, r = 3). 
In this paper, we give a near-complete classification of the case (|D| = 2, r = 4). Of the 400 distinct non-trivial Boolean predicates of arity 4, we implement an algorithmic procedure which perfectly classifies 397 of them. Of the remaining three, we solve two by reducing to extremal combinatorics problems - leaving the last one as an open question. Along the way, we identify the first Boolean predicate whose non-redundancy asymptotics are non-polynomial.

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Joshua Brakensiek, Venkatesan Guruswami, and Aaron Putterman. Classification of Non-Redundancy of Boolean Predicates of Arity 4. In 32nd International Conference on Principles and Practice of Constraint Programming (CP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 379, pp. 8:1-8:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CP.2026.8

Author Details

Joshua Brakensiek
  • Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA
Venkatesan Guruswami
  • Simons Institute for the Theory of Computing and the University of California, Berkeley, Berkeley, CA, USA
Aaron Putterman
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Funding

  • Brakensiek, Joshua: Supported in part by the Simons Investigator award of Venkatesan Guruswami and NSF grants CCF-2211972 and DMS-2503280.
  • Guruswami, Venkatesan: Supported in part by a Simons Investigator award and NSF award CCF-2211972.
  • Putterman, Aaron: Supported in part by the Simons Investigator Awards of Madhu Sudan and Salil Vadhan and AFOSR award FA9550-25-1-0112.

Acknowledgements

We thank Santhoshini Velusamy and Amatya Sharma for valuable conversations which indirectly led to fixing some typos in Appendix A.

Supplementary Materials

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