Monotone boolean functions as combinatorially piecewise linear maps

We study Boolean circuits as a representation of Boolean functions and consider different equivalence, audit, and enumeration problems. For a number of restricted sets of gate types (bases) we obtain efficient algorithms, while for all other gate types we show these problems are at least NP-hard.

RAIRO - Theoretical Informatics and Applications, 1993

On the positive and the inversion complexity of Boolean functions Informatique théorique et applications, tome 27, n o 4 (1993), p. 283-293. <http://www.numdam.org/item?id=ITA_1993__27_4_283_0> © AFCET, 1993, tous droits réservés. L'accès aux archives de la revue « Informatique théorique et applications » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Informatique théorique et Applications/Theoretical Informaties and Applications (vol. 27, n° 4, 1993, p. 283 à 293) ON THE POSITIVE AND THE INVERSION COMPLEXITY OF BOOLEAN FUNCTIONS (*) by V. DiciüNAS (*) Communicated by I. WEGENER Abstract.-We study unbounded fan-in circuits over the bases P = M\J {-i f\feM) and M' = M\j{-\ } with the inputs x x ,. . ., x n (positive and exmonotone circuits respectively). It was proved by Santha and Wilson [5] that any depth-d exmonotone circuit computing PARITY n requires at least Q(dn lid) négations. Using similar arguments we establish relations between the chain complexity ch (ƒ) (defined by Wegener in [6]) of a Boolean function f the size of the optimal positive circuit computing f and the minimal number of négations in any exmonotone circuit computing f It allows us to unify and slightly improve the lower bounds of Markov [3], Santha and Wilson [5] and Wegener [6], We also give almost matching upper bounds for symmetrie functions. Résumé.-Nous étudions les circuits à degré entrant borné sur les bases P-M[J {-i f:feM} et M' = Af U { ~i } sur les entrées x lt. . ., x n , où M est l'ensemble des fonctions booléennes monotones (que nous appelons circuits positifs e£ exmonotones respectivement). Santha et Wilson [5] ont prouvé que tout circuit exmonotone de profondeur d calculant PARITY n nécessitait au moins Q(dn lld) négations. En utilisant des arguments semblables, nous établissons des relations entre la complexité en chaine ch(f) d'une fonction booléenne f {définie par Wegener en [6]), la taille d'un circuit positif optimal calculant f et le nombre minimum de négations dans tout circuit exmonotone calculant f Ceci permet d'unifier et d'améliorer sensiblement les bornes inférieures de Markov [3], Santha et Wilson [5] et de Wegener [6]. Nous donnons aussi des bornes supérieures «presque exactes» pour les fonctions symétriques.

computational complexity, 2010

Every Boolean function on n variables can be expressed as a unique multivariate polynomial modulo p for every prime p. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: functions with low degree modulo p must have high complexity in every other characteristic q. More precisely, we show the following results about Boolean functions f : {0, 1} n → {0, 1} which depend on all n variables, and distinct primes p, q:

Information Processing Letters, 1999

This paper introduces the notions of balanced and strongly balanced Boolean functions and examines the complexity of these functions using harmonic analysis on the hypercube. The results are applied to derive a lower bound related to AC 0 functions. : S 0 0 2 0 -0 1 9 0 ( 9 9 ) 0 0 0 6 1 -7

Information Processing Letters, 2010

In this note, we present improved upper bounds on the circuit complexity of symmetric Boolean functions. In particular, we describe circuits of size 4.5n + o(n) for any symmetric function of n variables, as well as circuits of size 3n for MOD n 3 function.

Theory of Computing Systems / Mathematical Systems Theory, 2007

Any Boolean function can be defined by a Boolean circuit, provided we may use sufficiently strong functions in its gates. On the other hand, what Boolean functions can be defined depends on these gate functions: Each set B of gate functions defines the class of Boolean functions that can be defined by circuits over B. Although these classes have been known since the 1920s, their computational complexity was never investigated. In this paper we will study how difficult it is to decide for a Boolean function f and a class B, whether f is in B. Moreover, we will provide such a decision algorithm with additional information: How difficult is it to decide whether or not f is in B, provided we already know a circuit for f, but with gates from another class A? Given such a circuit, we know that f is in A. Is the problem harder if we do not have a concrete representation for f, but still know that it is from A? For nearly all possible combinations, we show that this is not the case, and that the problem is either in P or coNP-complete.

liafa.fr

We investigate the structure of "worst-case" quasi reduced ordered decision diagrams (or boolean graphs) and boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, then, introduce a notion of complexity which leads to the concept of "hard" boolean functions as functions whose boolean graphs are "worst-case" graphs. So we exhibit the surprising relation between hard functions and the Storage Access function (also known as Multiplexer). We also show some interesting properties of the hard functions and their graphs like the degree of the polynomial representation or the preservation of the hardness nature of the graph through variable permutations.

2011

Golumbic et al. (Discrete Appl. Math. 154:1465-1477, 2006) defined the readability of a monotone Boolean function f to be the minimum integer k such that there exists an ∧ − ∨-formula equivalent to f in which each variable appears at most k times. They asked whether there exists a polynomial-time algorithm, which given a monotone Boolean function f , in CNF or DNF form, checks whether f is a read-k function, for a fixed k. In this paper, we partially answer this question already for k = 2 by showing that it is NP-hard to decide if a given monotone formula represents a read-twice function. It follows also from our reduction that it is NP-hard to approximate the readability of a given monotone Boolean function f : {0, 1} n → {0, 1} within a factor of O(n). We also give tight sublinear upper bounds on the readability of a monotone Boolean function given in CNF (or DNF) form, parameterized by the number of terms in the CNF and the maximum size in each term, or more generally the maximum number of variables in the intersection of any constant number of terms. When the variables of the DNF can be ordered so that each term consists of a set of consecutive variables, we give much tighter logarithmic bounds on the readability.

Journal of Artificial Intelligence Research

In this paper, we focus on a less usual way to represent Boolean functions, namely on representations by switch-lists, which are closely related to interval representations. Given a truth table representation of a Boolean function f the switch-list representation of f is a list of Boolean vectors from the truth table which have a different function value than the preceding Boolean vector in the truth table. The main aim of this paper is to include this type of representation in the Knowledge Compilation Map by Darwiche and Marquis and to argue that switch-lists may in certain situations constitute a reasonable choice for a target language in knowledge compilation. First, we compare switch-list representations with a number of standard representations (such as CNF, DNF, and OBDD) with respect to their relative succinctness. As a by-product of this analysis, we also give a short proof of a longstanding open question proposed by Darwiche and Marquis, namely the incomparability of MODS ...

Theory of computing systems, 2018

The classification of Boolean functions plays an underpinning role in logic design and synthesis of VLSI circuits. This paper considers a underpinning question in Boolean function classification: how many distinct classes are there for k-input Boolean functions. We exploit various group algebraic properties to efficiently compute the number of unique equivalent classes. We have reduced the computation complexity from 2 m m! to (m + 1)!. We present our analysis for NPN classification of Boolean functions with up to ten variables and compute the number of NP and NPN equivalence classes for 3-10 variables. This is the first time to report the number of NP and NPN classifications for Boolean functions with 9-10 variables. We demonstrate the effectiveness of our method by both theoretical proofs and numeric experiments.

ABSTRACT A new invariant of the set of n-variable Boolean functions with respect to the action of AGL(n,2) is studied. Application of this invariant to prove ane nonequiv- alence of two Boolean functions is outlined. The value of this invariant is computed for PSap type bent functions.

Discrete Mathematics &amp; Theoretical Computer Science

Any attempt to find connections between mathematical properties and complexity has a strong relevance to the field of Complexity Theory. This is due to the lack of mathematical techniques to prove lower bounds for general models of computation.\par This work represents a step in this direction: we define a combinatorial property that makes Boolean functions &#39;&#39;\emphhard&#39;&#39; to compute in constant depth and show how the harmonic analysis on the hypercube can be applied to derive new lower bounds on the size complexity of previously unclassified Boolean functions.

Journal on Satisfiability, Boolean Modeling and Computation, 2019

The solution-graph of a Boolean formula on n variables is the subgraph of the hypercube H n induced by the satisfying assignments of the formula. The structure of solution-graphs has been the object of much research in recent years since it is important for the performance of SAT-solving procedures based on local search. Several authors have studied connectivity problems in such graphs focusing on how the structure of the original formula might affect the complexity of the connectivity problems in the solution-graph. In this paper we study the complexity of the isomorphism problem of solution-graphs of Boolean formulas. We consider the classes of formulas that arise in the CSP-setting and investigate how the complexity of the isomorphism problem depends on the formula type. We observe that for general formulas the solution-graph isomorphism problem can be solved in exponential time while in the cases of 2CNF formulas, as well as for CPSS formulas, the problem is in the counting complexity class C = P, a subclass of PSPACE. We also prove a strong property on the structure of solution-graphs of Horn formulas showing that they are just unions of partial cubes. In addition, we give a PSPACE lower bound for the problem on general Boolean functions. We prove that for 2CNF, as well as for CPSS formulas the solution-graph isomorphism problem is hard for C = P under polynomial time many-one reductions, thus matching the given upper bound.

Theoretical Computer Science, 2001

Recent results of Bucciarelli show that the semilattice of degrees of parallelism of firstorder boolean functions in PCF has both infinite chains and infinite antichains. By considering a simple subclass of Sieber's sequentiality relations, we identify levels in the semilattice and derive inexpressibility results concerning functions on different levels. This allows us to further explore the structure of the semilattice of degrees of parallelism: we identify semilattices characterized by simple level properties, and show the existence of new infinite hierarchies which are in a certain sense natural with respect to the levels.

Computational Complexity, 1995

Vie consider planar circuits, formulas and multilective planar circuits. It is shown that planar circuits and formulas are incomparable. An ~(n log n) lower bound is given for the multilective planar circuit complexity of a decision problem and an 12(n 3/2) lower bound is given for the multilective planar circuit complexity of a multiple output function.