On computing Boolean connectives of characteristic functions

Abstract

This paper is a study of the existence of polynomial time Boolean connective functions for languages. A language L has an AND function if there is a polynomial timefsuch that f(x,y) E L r x E L and y E L. L has an OR function if there is a polynomial time g such that g(x,y) E L r x G L or y E L. While all NP complete sets have these functions, Graph Isomorphism, which is probably not complete, is also shown to have both AND and OR functions. The results in this paper characterize the complete sets for the classes D P and pSAT[O(log n)] in terms of AND and OR, and relate these functions to the structure of the Boolean hierarchy and the query hierarchies. Also, this paper shows that the complete sets for the levels of the Boolean hierarchy above the second level cannot have AND or OR unless the polynomial hierarchy collapses. Finally, most of the structural properties of the Boolean hierarchy and query hierarchies are shown to depend only on the existence of AND and OR functions for the NP complete sets.

Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets

“…Although DP is closed under intersection but seems to lack closure under union (unless the polynomial hierarchy collapses to DP [Kad88,CK90b,Cha91]) and thus Theorem 3.10 in particular applies to DP, we note that the known results about Boolean hierarchies over NP [CGH + 88, KSW87] in fact even for the DP case imply stronger results than those given by our Theorem 3.10, due to the very special structure of DP. Indeed, since, e.g., E k (DP) = E 2k (NP) for any k ≥ 1 (and the same holds for the other hierarchies), it follows immediately that all the level-wise equivalences among the Boolean hierarchies (and also their ability to capture the Boolean closure) that are known to hold for NP also hold for DP even in the absence of the assumption of closure under union.…”

“…For example, graph minimal uncolorability is known to be complete for DP [CM87]. Note that DP clearly is closed under intersection, but is not closed under union unless the polynomial hierarchy collapses (due to [Kad88], see also [CK90b,Cha91]). 1.…”

“…The symmetric difference hierarchy also characterizes the Boolean closure of NP [KSW87]. In fact, these equalities are known to hold for all classes that contain Σ * and ∅ and are closed under union and intersection [Hau14, CGH + 88, KSW87, BBJ + 89, GNW90,CK90b,Cha91]. In Section 3, we prove that both the symmetric difference hierarchy (SDH) and the Boolean hierarchy (CH) remain equal to the Boolean closure (BC) even in the absence of the assumption of closure under union.…”

Toward the complexity of the existence of wonderfully stable partitions and strictly core stable coalition structures in enemy-oriented hedonic games

“…Chang and Kadin [10] define the following property: A problem A has AND ω functions 3 if there exists a polynomial-time computable function f such that for all n ∈ N and for all instances x 1 , x 2 , . .…”

Amplifying ZPP^SAT[1] and the Two Queries Problem

“…We define OR(L) and AND(L) as follows: [7] observed that SAT∧SAT has ANDs but does not have ORs unless PH collapses. Similarly, SAT∨SAT has ORs but not ANDs unless PH collapses.…”