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#SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank

Authors Nutan Limaye , Adarsh Srinivasan , Srikanth Srinivasan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • probabilistic polynomials
  • probabilistic rank
  • circuit satisfiability
  • circuit lower bounds
  • polynomial method
  • threshold circuits

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Abstract

There is a large body of work that shows how to leverage lower bound techniques for circuit classes to obtain satisfiability algorithms that run in better than brute-force time [Ramamohan Paturi et al., 1997; Ryan Williams, 2014]. For circuits with threshold gates, there are several such algorithms based on either  
- Probabilistic Representations by low-degree polynomials, which allow for the use of fast polynomial evaluation algorithms, or 
- Low rank, which allows for an efficient reduction to rectangular matrix multiplication.  In this paper, we use a related notion of probabilistic rank to obtain satisfiability algorithms for circuit classes contained in ACC⁰∘3-PTF, i.e. constant-depth circuits with modular counting gates and a single layer of degree-3 polynomial threshold functions. 
Even for the special case of a single 3-PTF, it is not clear how to use either of the above two strategies to get a non-trivial satisfiability algorithm. The best known algorithm in this case previously was based on memoization and yields worse guarantees than our algorithm.

Cite As Get BibTex

Nutan Limaye, Adarsh Srinivasan, and Srikanth Srinivasan. #SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 67:1-67:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.67

Author Details

Nutan Limaye
  • {IT} {University} of {Copenhagen}, Denmark
Adarsh Srinivasan
  • Rutgers University, Piscataway, NJ, USA
Srikanth Srinivasan
  • {University} of {Copenhagen}, Denmark

Funding

  • Limaye, Nutan: Received funding from the Independent Research Fund Denmark (grant agreement No. 10.46540/3103-00116B) and is also supported by the Basic Algorithms Research Copenhagen (BARC), funded by VILLUM Foundation Grants 16582 and 54451, and Digital Research Centre Denmark, project P40.
  • Srinivasan, Adarsh: Supported by the National Science Foundation under Grants CCF-2313372 and CCF-2443697. Part of this work was done during a visit to ITU Copenhagen and BARC funded by Basic Algorithms Research Copenhagen(BARC), supported by VILLUM Foundation Grants 16582 and 54451, and while at INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria. This work was partially funded from the Ministry of Education and Science of Bulgaria (support for INSAIT, part of the Bulgarian National Roadmap for Research Infrastructure).
  • Srinivasan, Srikanth: Funded by the European Research Council (ERC) under grant agreement no. 101125652 (ALBA).

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