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

Bounds on the Average Sensitivity of Nested Canalizing Functions

Nested canalizing Boolean (NCF) functions play an important role in biological motivated regulative networks and in signal processing, in particular describing stack filters. It has been conjectured that NCFs have a stabilizing effect on the network dynamics. It is well known that the average sensitivity plays a central role for the stability of (random) Boolean networks. Here we provide a tight upper bound on the average sensitivity for NCFs as a function of the number of relevant input variables. As conjectured in literature this bound is smaller than 4/3 This shows that a large number of functions appearing in biological networks belong to a class that has very low average sensitivity, which is even close to a tight lower bound.Comment: revised submission to PLOS ON

Truth Table Minimization of Computational Models

Complexity theory offers a variety of concise computational models for computing boolean functions - branching programs, circuits, decision trees and ordered binary decision diagrams to name a few. A natural question that arises in this context with respect to any such model is this: Given a function f:{0,1}^n \to {0,1}, can we compute the optimal complexity of computing f in the computational model in question? (according to some desirable measure). A critical issue regarding this question is how exactly is f given, since a more elaborate description of f allows the algorithm to use more computational resources. Among the possible representations are black-box access to f (such as in computational learning theory), a representation of f in the desired computational model or a representation of f in some other model. One might conjecture that if f is given as its complete truth table (i.e., a list of f's values on each of its 2^n possible inputs), the most elaborate description conceivable, then any computational model can be efficiently computed, since the algorithm computing it can run poly(2^n) time. Several recent studies show that this is far from the truth - some models have efficient and simple algorithms that yield the desired result, others are believed to be hard, and for some models this problem remains open. In this thesis we will discuss the computational complexity of this question regarding several common types of computational models. We shall present several new hardness results and efficient algorithms, as well as new proofs and extensions for known theorems, for variants of decision trees, formulas and branching programs

Computational Complexity of Minimal Trap Spaces in Boolean Networks

Trap spaces of a Boolean network (BN) are the sub-hypercubes closed by the function of the BN. A trap space is minimal if it does not contain any smaller trap space. Minimal trap spaces have applications for the analysis of dynamic attractors of BNs with various update modes. This paper establishes computational complexity results of three decision problems related to minimal trap spaces of BNs: the decision of the trap space property of a sub-hypercube, the decision of its minimality, and the decision of the belonging of a given configuration to a minimal trap space. Under several cases on Boolean function specifications, we investigate the computational complexity of each problem. In the general case, we demonstrate that the trap space property is coNP-complete, and the minimality and the belonging properties are $\Pi_2^{\text P}$-complete. The complexities drop by one level in the polynomial hierarchy whenever the local functions of the BN are either unate, or are specified using truth-table, binary decision diagrams, or double-DNF (Petri net encoding): the trap space property can be decided in P, whereas the minimality and the belonging are coNP-complete. When the BN is given as its functional graph, all these problems can be decided by deterministic polynomial time algorithms