For any Boolean function f:{0,1}ⁿ → {0,1} with a complexity measure having value k ≪ n, is it possible to restrict the function f to Θ(k) variables while keeping the complexity preserved at Θ(k)? Instantiation of this question for the measure of circuit complexity of the Boolean function was shown to be related to circuit lower bounds (Buresh-Oppenheim and Santhanam, 2006). Variants of the above question were also shown to have connections to the log-rank conjecture in communication complexity (Hrubeš, 2024) and lower bounds in proof complexity (Razborov, 2016). In the context of communication and query complexity, this question was recently studied by Göös, Newman, Riazanov and Sokolov (2024). They showed, among other results, that query complexity cannot be condensed losslessly. In this work, we show that there exists a Boolean function f such that any restriction of f to O(ℳ(f)) variables has ℳ(⋅)-complexity at most Õ(ℳ(f)^{2/3}), where ℳ is one of block sensitivity (bs), fractional block sensitivity (fbs), certificate complexity (𝖢), deterministic query complexity (𝖣), zero-error randomized query complexity (𝖱₀), and AND (and OR)-decision tree query complexity. This improves upon the results of Göös, Newman, Riazanov, and Sokolov (2024) for 𝖣 and 𝖱₀, and in particular answers their open question about the condensation of block sensitivity. We complement the negative results on lossless condensation with positive results about lossy condensation. In particular, we show that for every Boolean function f there exists a restriction of f to O(ℳ(f)) variables such that its ℳ(⋅)-complexity is at least Ω(ℳ(f)^{1/2}), where ℳ ∈ {bs,fbs,𝖢,UC_{min},UC₁,UC,𝖣,deg̃,λ}. In addition, we show lossy condensation for randomized and quantum query complexity with a slightly smaller exponent.

Given Boolean functions f, g : 𝔽₂ⁿ → {-1,+1}, we say they are linearly isomorphic if there exists A ∈ GL_n(𝔽₂) such that f(x) = g(Ax) for all x. We study this problem in the tolerant property testing framework under the known-unknown model, where g is given explicitly and f is accessible only via oracle queries, meaning the algorithm may adaptively request the value of f(x) for inputs x ∈ 𝔽₂ⁿ of its choice. Given parameters ε ≥ 0 and ω > 0, the goal is to distinguish whether there exists A ∈ GL_n(𝔽₂) such that the normalized Hamming distance between f and g(Ax) is at most ε, or whether for every A ∈ GL_n(𝔽₂) the distance is at least ε+ω. Our main result is a tolerant tester making Õ ((m/ω) ⁴) queries to f, where m is an upper bound on the spectral norm of g, improving the previous Õ ((m/ω) ^{24}) bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of Ω(m²) for constant ω (for example, ω = 1/4), improving the prior Ω(log m) lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana-McFarland functions from symmetric-key cryptography.

International audience

Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e. functions obtained by composing a base function with itself a number of times. Let h^d denote the standard d-fold composition of the base function h. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: - The outer function f:{0,1}ⁿ → {0,1} is a recursive function of the form h^d, with h being any base function and d = Ω(log log n). - The inner function is a recursive function of the form h^d, with h being any constant arity base function (other than AND and OR) and d = Ω(log log n), where n is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be efficiently eliminated if the inner or outer function is a recursive function.

For any Boolean functions f and g, the question whether R(f∘g) = Θ̃(R(f) ⋅ R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg̃(f∘g) = Θ̃(deg̃(f)⋅deg̃(g)). These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg̃ compose. A recent landmark result (Ben-David and Blais, 2020) showed that R(f∘g) = Ω(noisyR(f)⋅ R(g)). This implies that composition holds whenever noisyR(f) = Θ̃(R(f)). We show two results: 1. When R(f) = Θ(n), then noisyR(f) = Θ(R(f)). In other words, composition holds whenever the randomized query complexity of the outer function is full. 2. If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg̃(f∘g) = Ω(M(f) ⋅ deg̃(g)) (for some non-trivial complexity measure M(⋅)) was known to the best of our knowledge. We prove that deg̃(f∘g) = Ω̃(√{bs(f)} ⋅ deg̃(g)), where bs(f) is the block sensitivity of f. This implies that deg̃ composes when deg̃(f) is asymptotically equal to √{bs(f)}. It is already known that both R and deg̃ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.

The role of symmetry in Boolean functions f:{0, 1}ⁿ → {0, 1} has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of 𝖲_n, is an important class of functions in the study of Boolean functions. A function f:{0, 1}ⁿ → {0, 1} is called transitive (or weakly-symmetric) if there exists a transitive group 𝖦 of 𝖲_n such that f is invariant under the action of 𝖦. In other words, the value of the function remains unchanged even after the input bits of f are moved around according to some permutation σ ∈ 𝖦. Understanding various complexity measures of transitive functions has been a rich area of research for the past few decades. This work studies transitive functions in light of several combinatorial measures. The question that we try to address in this paper is what are the maximum separations between various pairs of combinatorial measures for transitive functions. Such study for general Boolean functions has been going on for many years. Aaronson et al. (STOC, 2021) have nicely compiled the current best-known results for general Boolean functions. But before this paper, no such systematic study had been done on the case of transitive functions. Separations between a pair of combinatorial measures are shown by constructing interesting functions that demonstrate the separation. Over the past three decades, various interesting classes of functions have been designed for this purpose. In this context, one of the celebrated classes of functions is the "pointer functions". Ambainis et al. (JACM, 2017) constructed several functions, which are modifications of the pointer function in Göös et al. (SICOMP, 2018 / FOCS, 2015), to demonstrate the separation between various pairs of measures. In the last few years, pointer functions have been used to show separation between various other pairs of measures (Eg: Mukhopadhyay et al. (FSTTCS, 2015), Ben-David et al. (ITCS, 2017), Göös et al. (ToCT, 2018 / ICALP, 2017)). However, the pointer functions themselves are not transitive. Based on the various kinds of pointer functions, we construct new transitive functions, which we use to demonstrate similar separations between various pairs of combinatorial measures as demonstrated by the original pointer functions. Our construction of transitive functions depends crucially on the construction of particular classes of transitive groups whose actions, though involved, help to preserve certain structural features of the input strings. The transitive groups we construct may be of independent interest in other areas of mathematics and theoretical computer science. We summarize the current knowledge of relations between various combinatorial measures of transitive functions in a table similar to the table compiled by Aaronson et al. (STOC, 2021) for general functions.