4,548 research outputs found
AND and/or OR: Uniform Polynomial-Size Circuits
We investigate the complexity of uniform OR circuits and AND circuits of
polynomial-size and depth. As their name suggests, OR circuits have OR gates as
their computation gates, as well as the usual input, output and constant (0/1)
gates. As is the norm for Boolean circuits, our circuits have multiple sink
gates, which implies that an OR circuit computes an OR function on some subset
of its input variables. Determining that subset amounts to solving a number of
reachability questions on a polynomial-size directed graph (which input gates
are connected to the output gate?), taken from a very sparse set of graphs.
However, it is not obvious whether or not this (restricted) reachability
problem can be solved, by say, uniform AC^0 circuits (constant depth,
polynomial-size, AND, OR, NOT gates). This is one reason why characterizing the
power of these simple-looking circuits in terms of uniform classes turns out to
be intriguing. Another is that the model itself seems particularly natural and
worthy of study.
Our goal is the systematic characterization of uniform polynomial-size OR
circuits, and AND circuits, in terms of known uniform machine-based complexity
classes. In particular, we consider the languages reducible to such uniform
families of OR circuits, and AND circuits, under a variety of reduction types.
We give upper and lower bounds on the computational power of these language
classes. We find that these complexity classes are closely related to tallyNL,
the set of unary languages within NL, and to sets reducible to tallyNL.
Specifically, for a variety of types of reductions (many-one, conjunctive truth
table, disjunctive truth table, truth table, Turing) we give characterizations
of languages reducible to OR circuit classes in terms of languages reducible to
tallyNL classes. Then, some of these OR classes are shown to coincide, and some
are proven to be distinct. We give analogous results for AND circuits. Finally,
for many of our OR circuit classes, and analogous AND circuit classes, we prove
whether or not the two classes coincide, although we leave one such inclusion
open.Comment: In Proceedings MCU 2013, arXiv:1309.104
Polynomial-time reducibilities and āalmost allā oracle sets
AbstractIt is shown for every k>0 and for almost every tally setT, {A|A ā©½PkāttT} ā {A|A ā©½P(k+1)āttT}. In contrast, it is shown that for every set A, the following holds: (a) for almost every set B,A ā©½ Pm B if and only if A ā©½ P(logn)āT B; and (b) for almost every set B, A ā©½Ptt B if and only ifA ā©½PTB
On symmetric differences of NP-hard sets with weakly P-selective sets
AbstractThe symmetric differences of NP-hard sets with weakly-P-selective sets are investigated. We show that if there exist a weakly-P-selective set A and an NP-ā©½Pm-hard set H such that H - AĻµPbtt(sparse) and A ā HĻµPm(sparse) then P = NP. So no NP-ā©½Pm-hard set has sparse symmetric difference with any weakly-P-selective set unless P = NP. The proof of our main result is an interesting application of the tree prunning techniques (Fortune 1979; Mahaney 1982). In addition, we show that there exists a P-selective set which has exponentially dense symmetric difference with every set in Pbtt(sparse)
Classes of representable disjoint NP-pairs
For a propositional proof system P we introduce the complexity class of all disjoint -pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical -pairs associated with these proof systems hard or complete for . Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for and the reductions between the canonical pairs exist
- ā¦