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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.