Testing Basic Boolean Formulae

Theory Comput., 2019

A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. Therefore, k-monotone functions are natural generalizations of the classical monotone functions, which are the 1-monotone functions. Motivated by the recent interest in k-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of k-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are k-monotone (or are close to being k-monotone) from functions that are far from being k-monotone. Our results include the following: 1. We demonstrate a separation between testing k-monotonicity and testing monotonicity, on the hypercube domain {0, 1}d, for k ≥ 3; 2. We demonstrate a separation between testing and learning on {0, 1}d, for k = ω(log d): testi...

Lecture Notes in Computer Science, 2011

In a well-known result on graph property testing, [GT03] showed that every testable graph property has a "canonical" testing algorithm in which a set of vertices is selected uniformly at random and the edges queried are the complete graph over the selected vertices. In this paper we define a similar-in-spirit canonical form for Boolean function testing algorithms, and show that under some mild conditions on the function class and testing algorithm, property testers for Boolean functions can be transformed into this canonical form.

ArXiv, 2017

We continue the study of k-monotone Boolean functions in the property testing model, initiated by Canonne et al. (ITCS 2017). A function f : {0, 1} → {0, 1} is said to be kmonotone if it alternates between 0 and 1 at most k times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. In property testing, the fact that 1-monotonicity can be locally tested with polyn queries led to a previous conjecture that k-monotonicity can be tested with poly(n) queries. In this work we disprove the conjecture, and show that even 2-monotonicity requires an exponential in √ n number of queries. Furthermore, even the apparently easier task of distinguishing 2-monotone functions from functions that are far from being n.01-monotone also requires an exponential number of queries. Our results follow from constructions of families that are hard for a canonical tester that ...

Discrete Applied Mathematics, 1999

Let f : {0; 1} n → {0; 1} be a monotone Boolean function whose value at any point x ∈ {0; 1} n can be determined in time t. Denote by c = I ∈C i∈I xi the irredundant CNF of f, where C is the set of the prime implicates of f. Similarly, let d = J ∈D j∈J xj be the irredundant DNF of the same function, where D is the set of the prime implicants of f. We show that given subsets C ⊆ C and D ⊆ D such that (C ; D ) = (C; D), a new term in (C\C ) ∪ (D\D ) can be found in time O(n(t +n))+m o(log m) , where m=|C |+|D |. In particular, if f(x) can be evaluated for every x ∈ {0; 1} n in polynomial time, then the forms c and d can be jointly generated in incremental quasi-polynomial time. On the other hand, even for the class of ∧; ∨-formulae f of depth 2, i.e., for CNFs or DNFs, it is unlikely that uniform sampling from within the set of the prime implicates and implicants of f can be carried out in time bounded by a quasi-polynomial 2 polylog(·) in the input size of f. We also show that for some classes of polynomial-time computable monotone Boolean functions it is NP-hard to test either of the conditions D = D or C = C. This provides evidence that for each of these classes neither conjunctive nor disjunctive irredundant normal forms can be generated in total (or incremental) quasi-polynomial time. Such classes of monotone Boolean functions naturally arise in game theory, networks and relay contact circuits, convex programming, and include a subset of ∧; ∨-formulae of depth 3.

Theoretical Computer Science, 2003

Andreev et al. 3] gave constructions of Boolean functions (computable by polynomial-size circuits) with large lower bounds for read-once branching program (1-b.p.'s): a function in P with the lower bound 2 n;polylog(n) , a function in quasipolynomial time with the lower bound 2 n;O(log n) , and a function in LINSPACE with the lower bound 2 n;log n;O(1). W e p o i n t out alternative, much simpler constructions of such Boolean functions by applying the idea of almost k-wise independence more directly, without the use of discrepancy set generators for large a ne subspaces our constructions are obtained by derandomizing the probabilistic proofs of existence of the corresponding combinatorial objects. The simplicity of our new constructions also allows us to observe that there exists a Boolean function in AC 0 2] (computable by a depth 3, polynomial-size circuit over the basis f^ 1g) with the optimal lower bound 2 n;log n;O(1) for 1-b.p.'s.

Information and Computation, 2008

We investigate the complexity of finding prime implicants and minimum equivalent DNFs for Boolean formulas, and of testing equivalence and isomorphism of monotone formulas. For DNF related problems, the complexity of the monotone case differs strongly from the arbitrary case. We show that it is DP-complete to check whether a monomial is a prime implicant for an arbitrary formula, but the equivalent problem for monotone formulas is in L. We show PP-completeness of checking if the minimum size of a DNF for a monotone formula is at most k, and for k in unary, we show that the complexity of the problem drops to coNP. In [Uma01] a similar problem for arbitrary formulas was shown to be Σ p 2 -complete. We show that calculating the minimum equivalent DNF for a monotone formula is possible in output-polynomial time if and only if P = NP. Finally, we disprove a conjecture from [Rei03] by showing that checking whether two formulas are isomorphic has the same complexity for arbitrary formulas as for monotone formulas.

SIAM Journal on Discrete Mathematics, 2019

A function f : {0, 1} n → {0, 1} is said to be k-monotone if it flips between 0 and 1 at most k times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. Our work is part of a renewed focus in understanding testability of properties characterized by freeness of arbitrary order patterns as a generalization of monotonicity. Recently, Canonne et al. (ITCS 2017) initiate the study of k-monotone functions in the area of property testing, and Newman et al. (SODA 2017) study testability of families characterized by freeness from order patterns on real-valued functions over the line [n] domain. We study k-monotone functions in the more relaxed parametrized property testing model, introduced by Parnas et al. (JCSS, 72(6), 2006). In this process we resolve a problem left open in previous work. Specifically, our results include the following.

IEEE Transactions on Computers, 2000

This correspondence considers the efficiency of some algorithms for the evaluation of monotonic Boolean functions. It is shown that algorithms based on the criterion of maximizing the local information gain about the Boolean function with n variables may sometimes require a number of computational steps which is n/log n times the computational steps of the optimal algorithm.

Information Processing Letters, 2011

We study the problem of testing isomorphism (equivalence up to relabelling of the variables) of two Boolean functions f,g: {0, 1}n → {0, 1}. Our main focus is on the most studied case, where one of the functions is given (explicitly) and the other function may be queried. We prove that for every k ≤ n, the worst-case query complexity

Discrete Applied Mathematics, 2008

In this paper we examine the problem of determining the self-duality of a monotone boolean function in disjunctive normal form (DNF). We show that the self-duality of monotone boolean functions with n disjuncts such that each disjunct has at most k literals can be determined in O(2 k 2 k 2 n) time. This implies an O(n 2 log n) algorithm for determining the self-duality of log n-DNF functions. We also consider the version where any two disjuncts have at most c literals in common. For this case we give an O(n 4(c+1) ) algorithm for determining self-duality.