39,548 research outputs found
A Lower Bound of Conditional Branches for Boolean Satisfiability on Post Machines
We establish a lower bound of conditional branches for deciding the
satisfiability of the conjunction of any two Boolean formulas from a set called
a full representation of Boolean functions of variables - a set containing
a Boolean formula to represent each Boolean function of variables. The
contradiction proof first assumes that there exists a Post machine (Post's
Formulation 1) that correctly decides the satisfiability of the conjunction of
any two Boolean formulas from such a set by following an execution path that
includes fewer than conditional branches. By using multiple runs of this
Post machine, with one run for each Boolean function of variables, the
proof derives a contradiction by showing that this Post machine is unable to
correctly decide the satisfiability of the conjunction of at least one pair of
Boolean formulas from a full representation of -variable Boolean functions
if the machine executes fewer than conditional branches. This lower bound
of conditional branches holds for any full representation of Boolean
functions of variables, even if a full representation consists solely of
minimized Boolean formulas derived by a Boolean minimization method. We discuss
why the lower bound fails to hold for satisfiability of certain restricted
formulas, such as 2CNF satisfiability, XOR-SAT, and HORN-SAT. We also relate
the lower bound to 3CNF satisfiability. The lower bound does not depend on
sequentiality of access to the boxes in the symbol space and will hold even if
a machine is capable of non-sequential access.Comment: This article draws heavily from arXiv:1406.597
Minimization for Generalized Boolean Formulas
The minimization problem for propositional formulas is an important
optimization problem in the second level of the polynomial hierarchy. In
general, the problem is Sigma-2-complete under Turing reductions, but
restricted versions are tractable. We study the complexity of minimization for
formulas in two established frameworks for restricted propositional logic: The
Post framework allowing arbitrarily nested formulas over a set of Boolean
connectors, and the constraint setting, allowing generalizations of CNF
formulas. In the Post case, we obtain a dichotomy result: Minimization is
solvable in polynomial time or coNP-hard. This result also applies to Boolean
circuits. For CNF formulas, we obtain new minimization algorithms for a large
class of formulas, and give strong evidence that we have covered all
polynomial-time cases
Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas
We show lower bounds of and on the
randomized and quantum communication complexity, respectively, of all
-variable read-once Boolean formulas. Our results complement the recent
lower bound of by Leonardos and Saks and
by Jayram, Kopparty and Raghavendra for
randomized communication complexity of read-once Boolean formulas with depth
. We obtain our result by "embedding" either the Disjointness problem or its
complement in any given read-once Boolean formula.Comment: 5 page
Stratification and enumeration of Boolean functions by canalizing depth
Boolean network models have gained popularity in computational systems
biology over the last dozen years. Many of these networks use canalizing
Boolean functions, which has led to increased interest in the study of these
functions. The canalizing depth of a function describes how many canalizing
variables can be recursively picked off, until a non-canalizing function
remains. In this paper, we show how every Boolean function has a unique
algebraic form involving extended monomial layers and a well-defined core
polynomial. This generalizes recent work on the algebraic structure of nested
canalizing functions, and it yields a stratification of all Boolean functions
by their canalizing depth. As a result, we obtain closed formulas for the
number of n-variable Boolean functions with depth k, which simultaneously
generalizes enumeration formulas for canalizing, and nested canalizing
functions
- …