Stasys Jukna

Boolean function complexity

Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterloo that have been discovered over the past several decades, right up to results from the last year or two. Many open problems, marked as Research Problems, are mentioned along the way. The problems are mainly of combinatorial flavor but their solutions could have great consequences in circuit complexity and computer science. The book will be of interest to graduate students and researchers in the fields of computer science and discrete mathematics.

Entropy of contact circuits and lower bounds on their complexity

Abstract A method for obtaining lower bounds on the contact circuit complexity of explicitly defined Boolean functions is given. It appears as one of possible concretizations of a more general “convolutional” approach to the lower bounds problem worked out by the author in 1984 [12]. The method is based on an appropriate notion of “inner information” or “entropy” of finite objects (circuits, Boolean functions, etc.). Lower bounds on the complexity are obtained by means of entropy-preserving embeddings of circuits into the more restricted ones. This allows to prove in a uniform and easy way that contact circuits, which are local in a sense that the function computed by a subcircuit weakly depends on the whole circuit, require 2 Ω(√n) or even 2 Ω( n log n ) contacts to compute some explicitly defined n -argument Boolean functions from NP.

On Graph Complexity

By the complexity of a graph we mean the minimum number of union and intersection operations needed to obtain the whole set of its edges starting from stars. This measure of graphs is related to the circuit complexity of boolean functions.We prove some lower bounds on the complexity of explicitly given graphs. This yields some new lower bounds for boolean functions, as well as new proofs of some known lower bounds in the graph-theoretic framework. We also formulate several combinatorial problems whose solution would have intriguing consequences in computational complexity.

Neither reading few bits twice nor reading illegally helps much

We first consider the so-called (1, +s)-branching programs in which along every consistent path at most s variables are tested more than once. We prove that any such program computing a characteristic function of a linear code C has size at least 2Ω(min/s d1, d2s), where d1 and d2 are the minimal distances of C and its dual C⊥. We apply this criterion to explicit linear codes and obtain a super-polynomial lower bound for s = o(nlogn). Then we introduce a natural generalization of read-k-times and (1, +s)-branching programs that we call semantic branching programs. These programs correspond to corrupting Turing machines which, unlike eraser machines, are allowed to read input bits even illegally, i.e. in excess of their quota on multiple readings, but in that case they receive in response an unpredictably corrupted value. We generalize the above-mentioned bound to the semantic case, and also prove exponential lower bounds for semantic read-once nondeterministic branching programs.

Some bounds on multiparty communication complexity of pointer jumping

Abstract. We introduce the model of conservative one-way multiparty complexity and prove lower and upper bounds on the complexity of pointer jumping.¶ The pointer jumping function takes as its input a directed layered graph with a starting node and k layers of n nodes, and a single edge from each node to one node from the next layer. The output is the node reached by following k edges from the starting node. In a conservative protocol, the ith player can see only the node reached by following the first i–1 edges and the edges on the jth layer for each j > i. This is a restriction of the general model where the ith player sees edges of all layers except for the ith one. In a one-way protocol, each player communicates only once in a prescribed order: first Player 1 writes a message on the blackboard, then Player 2, etc., until the last player gives the answer. The cost is the total number of bits written on the blackboard.¶Our main results are the following bounds on k-party conservative one-way communication complexity of pointer jumping with k layers:¶ (1) A lower bound of order

On P versus NP CO-NP for decision trees and read-once branching programs

\Omega(n/k^2)

Note: On covering graphs by complete bipartite subgraphs

for any

The Effect of Null-Chains on the Complexity of Contact Schemes

k = O(n^{1/3-\varepsilon})

Top-down lower bounds for depth 3 circuits

.¶(2) Matching upper and lower bounds of order

Top-down lower bounds for depth-three circuits

\Theta(n\,{\rm log}^{(k-1)}n)