Some Remarks on the size of Boolean Functions

1996

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Information Processing Letters, 1996

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.

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.

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.

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.

Theoretical Computer Science, 2011

A unate gate is a logical gate computing a unate Boolean function, which is monotone in each variable. Examples of unate gates are AND gates, OR gates, NOT gates, threshold gates, etc. A unate circuit C is a combinatorial logic circuit consisting of unate gates. Let f be a symmetric Boolean function of n variables, such as the Parity function, MOD function, and Majority function. Let m 0 and m 1 be the maximum numbers of consecutive 0's and consecutive 1's in the value vector of f , respectively, and let l = min{m 0 , m 1 } and m = max{m 0 , m 1 }. Let C be a unate circuit computing f . Let s be the size of the circuit C , that is, C consists of s unate gates. Let e be the energy of C , that is, e is the maximum number of gates outputting ''1'' over all inputs to C . In this paper, we show that there is a tradeoff between the size s and the energy e of C . More precisely, we show that (n + 1 − l)/m ≤ s e .

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:

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.