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.
Let
Rϵ denote the randomized query complexity for error probability ε, and
R:=R1/3 . In this work, we investigate whether a perfect composition theorem
R(f∘gn)=Ω(R(f)⋅R(g)) holds for a relation
f⊆{0,1}n×S and a total inner function
g:{0,1}m→{0,1} . Ben-David and Kothari (ICALP 2016, TOC 2018) showed that
R(f∘gn)=Ω(R(f)⋅RS(g)) , where
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 is asymptotically equivalent to
R for total Boolean functions. In this article, we show that
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 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 , and proving it to be equivalent to
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 is polynomially larger than the distributional query complexity for any product distribution. Our result also confirms that
RS and
R are asymptotically equivalent for this function. We thus also show a polynomial separation between
R and the maximum distributional query complexity for any product distribution which, to our knowledge, was not known prior to our work.
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∘g)=Θ(logR(g)1)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
χˉ, and linearised query complexity
LR. We show that AND-of-ORs separates conflict
χ from max-conflict
χˉ complexity and makes conflict and sabotage incomparable. Finally, randomized complexity can be smaller than its composition limit
R∗ by the maximal factor
Θ(logR). The total--total composition separation is also formalized in Lean.
We prove two new results about the randomized query complexity of composed functions. First, we show that the randomized composition conjecture is false: there are families of partial Boolean functions f and g such that R(f°g) ≪ R(f)R(g). In fact, we show that the left hand side can be polynomially smaller than the right hand side (though in our construction, both sides are polylogarithmic in the input size of f). Second, we show that for all f and g, R(f°g) = Ω(noisyR(f) R(g)), where noisyR(f) is a measure describing the cost of computing f on noisy oracle inputs. We show that this composition theorem is the strongest possible of its type: for any measure M(·) satisfying R(f°g)=Ω(M(f)R(g)) for all f and g, it must hold that noisyR(f)=Ω(M(f)) for all f. We also give a clean characterization of the measure noisyR(f): it satisfies noisyR(f)=Θ(R(f°GapMajn)/R(GapMajn)), where n is the input size of f and GapMajn is the √n-gap majority function on n bits.
For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)}) , where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g. The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n). The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g.