136 research outputs found
Complexity of equivalence relations and preorders from computability theory
We study the relative complexity of equivalence relations and preorders from
computability theory and complexity theory. Given binary relations , a
componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \,
[xRy \lra f(x) Sf(y)]. Here is taken from a suitable class of effective
functions. For us the relations will be on natural numbers, and must be
computable. We show that there is a -complete equivalence relation, but
no -complete for .
We show that preorders arising naturally in the above-mentioned
areas are -complete. This includes polynomial time -reducibility
on exponential time sets, which is , almost inclusion on r.e.\ sets,
which is , and Turing reducibility on r.e.\ sets, which is .Comment: To appear in J. Symb. Logi
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
Complexity-class-encoding sets
Properties of sets which are complex because they encode complexity classes areexplored. It is shown that not all sets with inherent complexity are of this type, although this is the only type of set for which well-developed techniques exist for proving inherent complexity.Possibilities for the complexity of encoding sets are discussed, first with referenceto an āalmost everywhereā vs. āinfinitely many argumentsā classification, and later with reference to the density of the set of arguments on which the problem is complex.It is shown that relative complexity relationships among sets of this type are highlystructured, in contrast to the wide variation possible among arbitrary recursive sets
On the relative complexity of hard problems for complexity classes without complete problems
AbstractWe show that any recursive sequence of recursive sets which is ascending with respect to the standard polynomial time reducibility notions has no minimal upper bound. As a consequence, any complexity class with certain natural closure properties possesses either complete problems or no easiest hard problems. A further corollary is that, assuming P ā NP, the partial ordering of the polynomial time degrees of NP-sets is not complete, and that there are no degree invariant approximations to NP-complete problems
Structural Average Case Complexity
AbstractLevin introduced an average-case complexity measure, based on a notion of āpolynomial on average,ā and defined āaverage-case polynomial-time many-one reducibilityā among randomized decision problems. We generalize his notions of average-case complexity classes, Random-NP and Average-P. Ben-Davidet al. use the notation of ćC,Ā Fć to denote the set of randomized decision problems (L,Ā Ī¼) such thatLis a set in C andĪ¼is a probability density function in F. This paper introduces AverćC,Ā Fć as the class of randomized decision problems (L,Ā Ī¼) such thatLis computed by a type-C machine onĪ¼-average andĪ¼is a density function in F. These notations capture all known average-case complexity classes as, for example, Random-NP= ćNP,Ā P-compć and Average-P=AverćP,Ā āć, where P-comp denotes the set of density functions whose distributions are computable in polynomial time, and ā denotes the set of all density functions. Mainly studied are polynomial-time reductions between randomized decision problems: manyāone, deterministic Turing and nondeterministic Turing reductions and the average-case versions of them. Based on these reducibilities, structural properties of average-case complexity classes are discussed. We give average-case analogues of concepts in worst-case complexity theory; in particular, the polynomial time hierarchy and Turing self-reducibility, and we show that all known complete sets for Random-NP are Turing self-reducible. A new notion of āreal polynomial-time computationsā is introduced based on average polynomial-time computations for arbitrary distributions from a fixed set, and it is used to characterize the worst-case complexity classesĪpkandĪ£pkof the polynomial-time hierarchy
Notes on conformal invariance of gauge fields
In Lagrangian gauge systems, the vector space of global reducibility
parameters forms a module under the Lie algebra of symmetries of the action.
Since the classification of global reducibility parameters is generically
easier than the classification of symmetries of the action, this fact can be
used to constrain the latter when knowing the former. We apply this strategy
and its generalization for the non-Lagrangian setting to the problem of
conformal symmetry of various free higher spin gauge fields. This scheme allows
one to show that, in terms of potentials, massless higher spin gauge fields in
Minkowski space and partially-massless fields in (A)dS space are not conformal
for spin strictly greater than one, while in terms of curvatures, maximal-depth
partially-massless fields in four dimensions are also not conformal, unlike the
closely related, but less constrained, maximal-depth Fradkin--Tseytlin fields.Comment: 38 page
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