5 research outputs found
Quasipolynomial simulation of DNNF by a non-determinstic read-once branching program
We prove that dnnfs can be simulated by Non-deterministic Read-Once Branching Programs (nrobps) of quasi-polynomial size. As a result, all the exponential lower bounds for nrobps immediately apply for dnnfs
Notes on Boolean Read-k and Multilinear Circuits
A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f
is a read-k circuit if the polynomial produced (purely syntactically) by the
arithmetic (+,x) version of the circuit has the property that for every prime
implicant of f, the polynomial contains at least one monomial with the same set
of variables, each appearing with degree at most k. Every monotone circuit is a
read-k circuit for some k. We show that already read-1 (OR,AND) circuits are
not weaker than monotone arithmetic constant-free (+,x) circuits computing
multilinear polynomials, are not weaker than non-monotone multilinear
(OR,AND,NOT) circuits computing monotone Boolean functions, and have the same
power as tropical (min,+) circuits solving combinatorial minimization problems.
Finally, we show that read-2 (OR,AND) circuits can be exponentially smaller
than read-1 (OR,AND) circuits.Comment: A throughout revised version. To appear in Discrete Applied
Mathematic
On the Incompressibility of Monotone DNFs
We prove optimal lower bounds for multilinear circuits and for monotone circuits with bounded depth. These lower bounds state that, in order to compute certain functions, these circuits need exactly as many OR gates as the respective DNFs. The proofs exploit a property of the functions that is based solely on prime implicant structure. Due to this feature, the lower bounds proved also hold for approximations of the considered functions that are similar to slice functions. Known lower bound arguments cannot handle these kinds of approximations. In order to show limitations of our approach, we prove that cliques of size n − 1 can be detected in a graph with n vertices by monotone formulas with O (log n) OR gates. Our lower bound for multilinear circuits improves a lower bound due to Borodin, Razborov and Smolensky for nondeterministic read-once branching programs computing the clique function.