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arXiv:2610.31638

No Perfect Randomized Composition

Alex Meiburg

Published Oct 22, 2026• 0 citations
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Abstract

Does computing a composition of total Boolean functions require the product of the randomized query complexities of its factors? We give a negative answer. There are families of total functions $f,g$ such that $R(f\circ g) =Θ\!\left(\frac1{\log R(g)}\right)R(f)R(g)$ This result requires first separating linearized query complexity $\LR$, as well as noisy randomized query complexity $noisyR$, from standard randomized query complexity $R$. Both constructions essentially come from totalizing existing partial function separations, albeit via different techniques. By adapting these techniques, we also construct total functions separating every adjacent pair in the chain from sabotage $RS$, max-conflict $\barχ$, and linearised query complexity $LR$. We show that AND-of-ORs separates conflict $χ$ from max-conflict $\barχ$ complexity and makes conflict and sabotage incomparable. Finally, randomized complexity can be smaller than its composition limit $R^*$ by the maximal factor $Θ(\log R)$. The total--total composition separation is also formalized in Lean.

DOI
10.48550/arxiv.2610.31638

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Does computing a composition of total Boolean functions require the product of the randomized query complexities of its factors? We give a negative answer. There are families of total functions f,gf,gf,g such that R(f∘g)=Θ ⁣(1log⁡R(g))R(f)R(g)R(f\circ g) =Θ\!\left(\frac1{\log R(g)}\right)R(f)R(g)R(f∘g)=Θ(logR(g)1​)R(f)R(g) This result requires first separating linearized query complexity \LR\LR\LR, as well as noisy randomized query complexity noisyRnoisyRnoisyR, from standard randomized query complexity RRR. Both constructions essentially come from totalizing existing partial function separations, albeit via different techniques. By adapting these techniques, we also construct total functions separating every adjacent pair in the chain from sabotage RSRSRS, max-conflict χˉ\barχχˉ​, and linearised query complexity LRLRLR. We show that AND-of-ORs separates conflict χχχ from max-conflict χˉ\barχχˉ​ complexity and makes conflict and sabotage incomparable. Finally, randomized complexity can be smaller than its composition limit R∗R^*R∗ by the maximal factor Θ(log⁡R)Θ(\log R)Θ(logR). The total--total composition separation is also formalized in Lean.
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Let Rϵ\mathsf {R}_\epsilonRϵ​ denote the randomized query complexity for error probability ε, and R:=R1/3\mathsf {R}:=\mathsf {R}_{1/3}R:=R1/3​ . In this work, we investigate whether a perfect composition theorem R(f∘gn)=Ω(R(f)⋅R(g))\mathsf {R}(f \circ g^n)=\Omega (\mathsf {R}(f)\cdot \mathsf {R}(g))R(f∘gn)=Ω(R(f)⋅R(g)) holds for a relation f⊆{0,1}n×Sf \subseteq \lbrace 0,1\rbrace ^n \times \mathcal {S}f⊆{0,1}n×S and a total inner function g:{0,1}m→{0,1}g:\lbrace 0,1\rbrace ^m \rightarrow \lbrace 0,1\rbraceg:{0,1}m→{0,1} . Ben-David and Kothari (ICALP 2016, TOC 2018) showed that R(f∘gn)=Ω(R(f)⋅RS(g))\mathsf {R}(f \circ g^n)=\Omega (\mathsf {R}(f)\cdot \mathsf {RS}(g))R(f∘gn)=Ω(R(f)⋅RS(g)) , where RS\mathsf {RS}RS is the sabotage complexity, a measure that they defined. Their result holds even when f is any relation and g is any partial function. Ben-David and Kothari asked in their article whether Sabotage complexity RS\mathsf {RS}RS is asymptotically equivalent to R\mathsf {R}R for total Boolean functions. In this article, we show that RS\mathsf {RS}RS is asymptotically at least the maximum distributional query complexity for any product distribution (i.e., probability distribution where the individual bits are independently distributed). In light of the minimax theorem, which states that R\mathsf {R}R is the maximum distributional query complexity for any distribution, our result makes progress toward answering the composition question. We prove our result by introducing a novel complexity measure called Rprod\mathsf {R}_{}^{\mathsf {prod}}Rprod​ , and proving it to be equivalent to RS\mathsf {RS}RS up to a logarithmic factor. How tight is our bound? We show that our bound is polynomially loose for the complete binary NAND tree function, for which RS\mathsf {RS}RS is polynomially larger than the distributional query complexity for any product distribution. Our result also confirms that RS\mathsf {RS}RS and R\mathsf {R}R are asymptotically equivalent for this function. We thus also show a polynomial separation between R\mathsf {R}R and the maximum distributional query complexity for any product distribution which, to our knowledge, was not known prior to our work.
We study the composition question for bounded-error randomized query complexity: Is R(f o g) = Omega(R(f) R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Theta(log R(g)), in between f and g allows us to prove R(f o h o g) = Omega(R(f) R(h) R(g)). We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, RS(f o g) >= RS(f) RS(g), and a composition theorem with randomized query complexity, R(f o g) = Omega(R(f) RS(g)). It is also a quadratically tight lower bound for total functions and can be quadratically superior to the partition bound, the best known general lower bound for randomized query complexity. Using this technique we also show implications for lifting theorems in communication complexity. We show that a general lifting theorem for zero-error randomized protocols implies a general lifting theorem for bounded-error protocols.
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Let Rϵ(⋅)R_\epsilon(\cdot)Rϵ​(⋅) stand for the bounded-error randomized query complexity with error ϵ>0\epsilon > 0ϵ>0. For any relation f⊆{0,1}n×Sf \subseteq \{0,1\}^n \times Sf⊆{0,1}n×S and partial Boolean function g⊆{0,1}m×{0,1}g \subseteq \{0,1\}^m \times \{0,1\}g⊆{0,1}m×{0,1}, we show that R1/3(f∘gn)∈Ω(R4/9(f)⋅R1/3(g))R_{1/3}(f \circ g^n) \in \Omega(R_{4/9}(f) \cdot \sqrt{R_{1/3}(g)})R1/3​(f∘gn)∈Ω(R4/9​(f)⋅R1/3​(g)​), where f∘gn⊆({0,1}m)n×Sf \circ g^n \subseteq (\{0,1\}^m)^n \times Sf∘gn⊆({0,1}m)n×S is the composition of fff and ggg. We give an example of a relation fff and partial Boolean function ggg for which this lower bound is tight. We prove our composition theorem by introducing a new complexity measure, the max conflict complexity χˉ(g)\bar \chi(g)χˉ​(g) of a partial Boolean function ggg. We show χˉ(g)∈Ω(R1/3(g))\bar \chi(g) \in \Omega(\sqrt{R_{1/3}(g)})χˉ​(g)∈Ω(R1/3​(g)​) for any (partial) function ggg and R1/3(f∘gn)∈Ω(R4/9(f)⋅χˉ(g))R_{1/3}(f \circ g^n) \in \Omega(R_{4/9}(f) \cdot \bar \chi(g))R1/3​(f∘gn)∈Ω(R4/9​(f)⋅χˉ​(g)); these two bounds imply our composition result. We further show that χˉ(g)\bar \chi(g)χˉ​(g) is always at least as large as the sabotage complexity of ggg, introduced by Ben-David and Kothari.