# Composition limits and separating examples for some boolean function complexity measures

## Abstract

Block sensitivity (bs(f)), certificate complexity (C(f)) and fractional certificate complexity (C*(f)) are three fundamental combinatorial measures of complexity of a boolean function f. It has long been known that bs(f) ≤ C*(f) ≤ C(f) = O(bs(f)2). We provide an infinite family of examples for which C(f) grows quadratically in C*(f) (and also bs(f)) giving optimal separations between these measures. Previously the biggest separation known was $$C(f) = C*(f)^{\log _{4,5} 5}$$. We also give a family of examples for which C*(f)= Ω (bs(f)3/2).

These examples are obtained by composing boolean functions in various ways. Here the composition fog of f with g is obtained by substituting for each variable of f a copy of g on disjoint sets of variables. To construct and analyse these examples we systematically investigate the behaviour under function composition of these measures and also the sensitivity measure s(f). The measures s(f), C(f) and C*(f) behave nicely under composition: they are submultiplicative (where measure m is submultiplicative if m(fog) ≤ m(f)m(g)) with equality holding under some fairly general conditions. The measure bs(f) is qualitatively different: it is not submultiplicative. This qualitative difference was not noticed in the previous literature and we correct some errors that appeared in previous papers. We define the composition limit of a measure m at function f, m lim(f) to be the limit as k grows of m(f (k))1/k, where f (k) is the iterated composition of f with itself k-times. For any function f we show that bs lim(f) = (C*)lim(f) and characterize s lim(f); (C*)lim(f), and C lim(f) in terms of the largest eigenvalue of a certain set of 2×2 matrices associated with f.

This is a preview of subscription content, access via your institution.

## References

1. [1]

S. Aaronson: Quantum certificate complexity, J. Comput. Syst. Sci. 74 (2008), 313–322.

2. [2]

H. Buhrman and R. de Wolf: Complexity measures and decision tree complexity: a survey, Theor. Comput. Sci. 288 (2002), 21–43.

3. [3]

S. Bublitz, U. Schurfeld and I. Wegener: Properties of complexity measures for PRAMs and WRAMs, Theor. Comput. Sci. 48 (1986), 53–73.

4. [4]

P. Hatami, R. Kulkarni and D. Pankratov: Variations on the Sensitivity Con-jecture, Number 4 in Graduate Surveys, Theory of Computing Library, 2011.

5. [5]

C. D. Meyer: Matrix analysis and applied linear algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2000.

6. [6]

G. Midrijanis: Exact quantum query complexity for total boolean functions, arXiv preprint quant-ph/0403168, 2004.

7. [7]

N. Nisan: CREW PRAMs and Decision Trees, SIAM J. Comput. 20 (1991), 999–1007.

8. [8]

E. R. Scheinerman, D. H. Ullman and C. Berge: Fractional Graph Theory: A Rational Approach to the Theory of Graphs, Dover Books on Mathematics Series, Dover Publications, 2011.

9. [9]

A. Tal: Properties and applications of boolean function composition, Electronic Col-loquium on Computational Complexity (ECCC) 19 2012.

10. [10]

A. Tal: Properties and applications of boolean function composition, in: ITCS (2013), 441–454.

## Author information

Authors

### Corresponding author

Correspondence to Justin Gilmer.

Supported by NSF grant CCF 083727.

Supported by NSF grants CCF-083727 and CCF-1218711.

Work partially done as a Postdoctoral researcher at DIMACS, Rutgers University.

## Rights and permissions

Reprints and Permissions

Gilmer, J., Saks, M. & Srinivasan, S. Composition limits and separating examples for some boolean function complexity measures. Combinatorica 36, 265–311 (2016). https://doi.org/10.1007/s00493-014-3189-x