The Complexity of the Descriptiveness of Boolean Circuits over Different Sets of Gates

Abstract

Abstract. Any Boolean function can be defined by a Boolean circuit, provided we may use sufficiently strong functions in its gates. On the other hand, it depends on these gate functions, what Boolean functions can be defined: Each set B of gate functions defines the class of Boolean functions that can be defined by circuits over B. Although these classes are 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.

The Complexity of Deciding if a Boolean Function Can Be Computed by Circuits over a Restricted Basis

“…also [Hås88,Weg87] Next, we consider the case that the input functions are not given by their truth-table but in a succinct way, i. e., by a Boolean circuit over basis {∧, ∨, ¬} or any other complete basis. As mentioned in the introduction, Böhler and Schnoor [BS07], studying this kind of input representation, identified a few tractable cases of the non-uniform memberhip problems and showed that all other cases are coNP-complete. The lower bound for the non-uniform problem of course immediately translates to the uniform problem; however, since the algorithms given in [BS07] do not rely on the Galois connection described in Sect.…”

“…As mentioned in the introduction, Böhler and Schnoor [BS07], studying this kind of input representation, identified a few tractable cases of the non-uniform memberhip problems and showed that all other cases are coNP-complete. The lower bound for the non-uniform problem of course immediately translates to the uniform problem; however, since the algorithms given in [BS07] do not rely on the Galois connection described in Sect. 2 above but make use of particular properties of the individual clones, no upper bound for the uniform membership problem GEN can be obtained from that paper.…”

“…Thus we immediately obtain that GEN ∈ coNP. Hardness follows from [BS07], where it is shown that the membership problem for, say, N 2 is already coNP-hard.…”

“…One possibility for a more succinct description of Boolean functions is of course to use general circuits (i.e., over any complete basis) as representation of the functions they compute. This approach was already followed in a paper by Böhler and Schnoor [BS07]. They looked at the list of clones from [Pos41], and studied for each of them the membership problem, i.e., for each fixed clone B we have a membership problem MEM(B): Given f , decide if f ∈ B.…”

“…When using the input representation of [BS00] by truth-tables we will see that for all except 8 clones the non-uniform membership problem is in AC 0 , while for the remaining 8 it is in qAC 0 . Böhler and Schnoor [BS07] showed that, if the input function is represented by a Boolen circuit, the non-uniform clone membership problem is tractable only for a very small number of degenerated clones, and coNP-complete in all other cases. For our study of the uniform membership problem, the coNP lower bound of course remains valid.…”