Boolean Functions, Invariance Groups, and Parallel Complexity

Complexity of Boolean Functions

s of Presentation: Paul Beame: Time-Space Tradeoffs and Multiparty Communication Complexity 9 Beate Bollig: Exponential Lower Bounds for Integer Multiplication Using Universal Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Philipp Wölfel: On k-wise l-mixed Boolean Functions . . . . . . . . . . . . . . 10 Elizabeta Okol’nishnikova: On One Lower Bound for Branching Progra...

The complexity of Boolean functions

Compositional complexity of Boolean functions

We define two measures, y and c, of complexity for Boolean functions. These measures are related to issues of functional decomposition which (for continuous functions) were studied by Arnol’d, Kolmogorov, VituSkin and others in connection with Hilbert’s 13th Problem. ‘This perspective was first applied to Boolean functions in (11. Our complexity measures differ from those which were considered ...

Symmetry groups of boolean functions

We prove that every abelian permutation group, but known exceptions, is the symmetry group of a boolean function. This solves the problem posed in the book by Clote and Kranakis. In fact, our result is proved for a larger class of groups, namely, for all groups contained in direct sums of regular groups. We investigate the problem of representability of permutation groups by the symmetry groups...

Relationships between Boolean Functions and Symmetric Groups

We study the relations between boolean functions and symmetric groups. We consider elements of a symmetric group as variable transformation operators for boolean functions. Boolean function may be xed or permuted by these operators. We give some properties relating the symmetric group Sn and boolean functions on Vn.