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Alternation Depth of Threshold Decision Lists

Authors Vladimir Podolskii , Morgan Prior

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LIPIcs.ICALP.2026.148.pdf
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ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Communication complexity
Keywords
  • linear decision lists
  • threshold decision lists
  • rectangle decision lists
  • threshold circuits

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Abstract

Linear decision lists are a computational model for Boolean functions. A linear decision list is built from a sequence of linear threshold function queries which are evaluated one by one: if a query returns true, the list outputs the value of the function, and if the answer is false, the process continues to the next query. The size of a linear decision list is the number of queries in it.
Linear decision lists form a natural and nontrivial subclass of depth-2 threshold circuits, the class of circuits that currently marks the frontier of explicit circuit lower bounds. Although some techniques for proving lower bounds against linear decision lists exist, they are quite limited, leaving important open problems unresolved. Moreover, for the related model of exact linear decision lists, no strong lower bounds are known.
We initiate the study of alternation depth of decision lists with linear threshold queries. The alternation depth is defined as the number of alternations in the sequence of output values of the decision list. We show that linear decision lists, both with bounded and unbounded weights in the threshold queries, form fine hierarchies with respect to alternation depth. A similar hierarchy exists for rectangle decision lists, the model closely related to communication complexity with NP oracles. We prove strong separations within these hierarchies and between them.
Next, we give a superpolynomial lower bound for an explicit function for exact linear decision lists of depth below n/log n. Such lower bounds were not previously known and do not follow directly from existing methods. We also establish a fine depth hierarchy for exact linear decision lists.
To prove these hierarchy separations, we use an iterative technique combined with existing techniques such as fooling sets and the analysis of blocky matrices. For the lower bound on exact linear decision lists, we combine the discrepancy method with an iterative analysis of blocky matrices.

Cite As Get BibTex

Vladimir Podolskii and Morgan Prior. Alternation Depth of Threshold Decision Lists. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 148:1-148:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.148

Author Details

Vladimir Podolskii
  • Tufts University, Medford, MA, USA
Morgan Prior
  • Tufts University, Medford, MA, USA

Acknowledgements

We would like to thank anonymous reviewers for numerous helpful comments that significantly improved the quality of the paper.

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