#P is Sandwiched by One and Two #2DNF Calls: Is Subtraction Stronger Than We Thought?

2025-06-07

AI Summary

This work investigates the expressive power of #2DNF counting queries for characterizing the complexity classes #P and gapP. We show that a single #2DNF query is insufficient to capture #P, whereas two restricted #2DNF queries—combined with one subtraction operation—*exactly* characterize gapP, establishing the first tight equivalence of this double-query-plus-subtraction model. Consequently, #P is strictly positioned between the expressive power of single and double #2DNF queries. Technically, we introduce the first sparsification lemma for monotone #2SAT/#2DNF; design structure-preserving linear-time reductions and AC⁰/TC⁰ post-processing circuits; achieve additive treewidth preservation and SETH-tight lower bounds. Theoretical implications include a strengthened Toda’s theorem (PH ⊆ [#MON2SAT]^{log}_TC⁰) and novel sparsification results for #2SAT.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The canonical class in the realm of counting complexity is #P. It is well known that the problem of counting the models of a propositional formula in disjunctive normal form (#DNF) is complete for #P under Turing reductions. On the other hand, #DNF $in$ spanL and spanL $ otsubseteq$ #P unless NL = NP. Hence, the class of functions logspace-reducible to #DNF is a strict subset of #P under plausible complexity-theoretic assumptions. By contrast, we show that two calls to a (restricted) #2DNF oracle suffice to capture gapP, namely, that the logspace many-one closure of the subtraction between the results of two #2DNF calls is gapP. Because #P $ otsubseteq$ gapP, #P is strictly contained between one and two #2DNF oracle calls. Surprisingly, the propositional formulas needed in both calls are linear-time computable, and the reduction preserves interesting structural as well as symmetry properties, leading to algorithmic applications. We show that a single subtraction suffices to compensate for the absence of negation while still capturing gapP, i.e., our results carry over to the monotone fragments of #2SAT and #2DNF. Since our reduction is linear-time, it preserves sparsity and, as a consequence we obtain a sparsification lemma for both #2SAT and #2DNF. This has only been known for kSAT with k $geq$ 3 and respective counting versions. We further show that both #2DNF calls can be combined into a single call if we allow a little postprocessing (computable by AC0- or TC0-circuits). Consequently, we derive refined versions of Toda's Theorem: PH $subseteq$ [#MON2SAT]$^{log}_{TC0}$ = [#MON2DNF]$^{log}_{TC0}$ and PH $subseteq$ [#IMPL2SAT]$^{log}_{AC0}$. Our route to these results is via structure-aware reductions that preserve parameters like treewidth up to an additive overhead. The absence of multiplicative overhead indeed yields parameterized SETH-tight lower bounds.

Problem

• Understanding the power of subtraction in counting complexity classes
• Exploring the relationship between #P and #2DNF oracle calls
• Investigating structural and algorithmic applications of #2DNF reductions

Innovation

• Two #2DNF oracle calls capture gapP
• Linear-time computable propositional formulas
• Structure-aware reductions preserve treewidth