Relationships between the number of inputs and other complexity measures of Boolean functions

Average case complexity of DNFs and Shannon semi-effect for narrow subclasses of boolean functions

In this paper we establish some bounds on the complexity of disjunctive normal forms of boolean function from narrow subclasses (e.g. functions takes value 0 in a limited number of points). The bounds are obtained by reduction the initial problem to a simple set covering problem. The nature of the complexity bounds provided is tightly connected with Shannon effect and semi-effect for this classes.

任意次数的初等对称布尔函数的证书复杂度

Certificate Complexity of Elementary Symmetric Boolean Functions of Arbitrary Degree

本文解决了初等对称布尔函数证书复杂度的开放问题，给出了任意次数$d$（$1\le d\le n$）下的完整确定结果。

State complexity of catenation combined with a boolean operation: a unified approach

In this paper we study the state complexity of catenation combined with symmetric difference. First, an upper bound is computed using some combinatoric tools. Then, this bound is shown to be tight by giving a witness for it. Moreover, we relate this work with the study of state complexity for two other combinations: catenation with union and catenation with intersection. And we extract a unified approach which allows to obtain the state complexity of any combination involving catenation and a binary boolean operation.

The relation between tree size complexity and probability for Boolean functions generated by uniform random trees

We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives And and OR and the leaves are labelled with variables or negated variables. We study limiting distribution when the tree size tends to infinity and derive a relation between the tree size complexity and the probability of a function. This is done by first expressing trees representing a particular function as expansions of minimal trees representing this fu…

Query complexity of Boolean functions on slices

We study the deterministic query complexity of Boolean functions on slices of the hypercube. The $k^{th}$ slice $\binom{[n]}{k}$ of the hypercube $\{0,1\}^n$ is the set of all $n$-bit strings with Hamming weight $k$. We show that there exists a function on the balanced slice $\binom{[n]}{n/2}$ requiring $n - O(\log \log n)$ queries. We give an explicit function on the balanced slice requiring $n - O(\log n)$ queries based on independent sets in Johnson graphs. On the weight-2 slice, we show that hard functions are closely related to Ramsey graphs. Further we describe a simple way of transformi…

Topological aspects of Boolean functions

We discuss ways in which tools from topology can be used to derive lower bounds for the circuit complexity of Boolean functions.

立方体中间层上布尔函数的查询复杂度

Query complexity of Boolean functions on the middle slice of the cube

本文研究布尔函数在中间层切片上的查询复杂度，证明存在一个函数即使已知输入中0和1数量相同，仍需查询除7位外的所有输入位，回答了Byramji的问题，并给出候选函数及与差异类型问题的联系。

旋转对称布尔函数的块灵敏度与证书复杂度

Block Sensitivity and Certificate Complexity of Rotation Symmetric Boolean Functions

该研究分析旋转对称布尔函数的块灵敏度与证书复杂度，明确奇次、二次单项式旋转对称布尔函数的相关测度关系，还确定连续四次族Rₙ的测度，得到灵敏度小于块灵敏度且二者与证书复杂度相等的无穷多类函数。

On the Relationship between Energy Complexity and other Boolean Function Measures

In this work we investigate into energy complexity, a Boolean function measure related to circuit complexity. Given a circuit $\mathcal{C}$ over the standard basis $\{\vee_2,\wedge_2,\neg\}$, the energy complexity of $\mathcal{C}$, denoted by $\mathrm{EC}(\mathcal{C})$, is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function $f$, denoted by $\mathrm{EC}(f)$, is the minimum of $\mathrm{EC}(\mathcal{C})$ over all circuits $\mathcal{C}$ computing $f$. This concept has attracted lots of attention in literature. Recently, Dinesh, Otiv, and Sar…

集体渠化布尔函数

Collectively canalizing Boolean functions

本文研究源于系统生物学的集体渠化布尔函数，分析渠化变量集的数量与类型对函数行为的影响，提出渠化强度新度量，并将其与平均敏感度相联系。

加权布尔型CSP的复杂性

The Complexity of Weighted Boolean #CSP

本文给出了加权布尔约束满足问题实例的配分函数计算复杂性的一个二分法。该问题由非负函数的有限集合F参数化，该非负函数可以用于将权重分配给问题实例的配置（可行解）。经典的约束满足问题对应于0，1值函数的特殊情况。我们证明了配分函数，即所有配置的权重之和，可以在多项式时间内计算，如果（1）F中的每个函数都是"乘积型“，或（2）F中的每个函数都是"纯仿射”。对于每一个其他的固定集合F，计算配分函数是FP^{#P}-完全的。

Nearest neighbor representations of Boolean functions

A nearest neighbor representation of a Boolean function is a set of positive and negative prototypes in $R^n$ such that the function has value 1 on an input iff the closest prototype is positive. For $k$-nearest neighbor representation the majority classification of the $k$ closest prototypes is considered. The nearest neighbor complexity of a Boolean function is the minimal number of prototypes needed to represent the function. We give several bounds for this measure. Separations are given between the cases when prototypes can be real or are required to be Boolean. The complexity of parity is…

Gap theorems for robust satisfiability: Boolean CSPs and beyond

A computational problem exhibits a "gap property" when there is no tractable boundary between two disjoint sets of instances. We establish a Gap Trichotomy Theorem for a family of constraint problem variants, completely classifying the complexity of possible ${\bf NP}$-hard gaps in the case of Boolean domains. As a consequence, we obtain a number of dichotomies for the complexity of specific variants of the constraint satisfaction problem: all are either polynomial-time tractable or $\mathbf{NP}$-complete. Schaefer's original dichotomy for $\textsf{SAT}$ variants is a notable particular case. …

Complexity of the CNF-satisfiability problem

This paper is devoted to the complexity of the Boolean satisfiability problem. We consider a version of this problem, where the Boolean formula is specified in the conjunctive normal form. We prove an unexpected result that the CNF-satisfiability problem can be solved by a deterministic Turing machine in polynomial time.