Results 1  10
of
28
Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
Abstract

Cited by 71 (23 self)
 Add to MetaCart
This paper is dedicated to the memory of Ronald V. Book, 19371997.
Degrees of random sets
, 1991
"... An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrando ..."
Abstract

Cited by 47 (4 self)
 Add to MetaCart
(Show Context)
An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrandom and weakly nrandom sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of nrandomness and weak nrandomness including a new definition in terms of a forcing relation analogous to the characterization of ngeneric sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than nrandomness. Chapter III is concerned with intrinsic properties of nrandom sequences. The main results are that an (n + 1)random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that nrandom sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is nrandom then A is nrandom relative to B. It follows that any countable distributive lattice can be embedded
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
Abstract

Cited by 46 (11 self)
 Add to MetaCart
(Show Context)
We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
On the Structure of Degrees of Inferability
 Journal of Computer and System Sciences
, 1993
"... Degrees of inferability have been introduced to measure the learning power of inductive inference machines which have access to an oracle. The classical concept of degrees of unsolvability measures the computing power of oracles. In this paper we determine the relationship between both notions. ..."
Abstract

Cited by 31 (18 self)
 Add to MetaCart
Degrees of inferability have been introduced to measure the learning power of inductive inference machines which have access to an oracle. The classical concept of degrees of unsolvability measures the computing power of oracles. In this paper we determine the relationship between both notions. 1 Introduction We consider learning of classes of recursive functions within the framework of inductive inference [21]. A recent theme is the study of inductive inference machines with oracles ([8, 10, 11, 17, 24] and tangentially [12]; cf. [10] for a comprehensive introduction and a collection of all previous results.) The basic question is how the information content of the oracle (technically: its Turing degree) relates with its learning power (technically: its inference degreedepending on the underlying inference criterion). In this paper a definitive answer is obtained for the case of recursively enumerable oracles and the case when only finitely many queries to the oracle are allo...
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
Abstract

Cited by 30 (3 self)
 Add to MetaCart
This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Learning via Queries and Oracles
 In Proc. 8th Annu. Conf. on Comput. Learning Theory
, 1996
"... Inductive inference considers two types of queries: Queries to a teacher about the function to be learned and queries to a nonrecursive oracle. This paper combines these two types  it considers three basic models of queries to a teacher, namely QEX[Succ], QEX[!] and QEX[+], together with members ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
Inductive inference considers two types of queries: Queries to a teacher about the function to be learned and queries to a nonrecursive oracle. This paper combines these two types  it considers three basic models of queries to a teacher, namely QEX[Succ], QEX[!] and QEX[+], together with membership queries to some oracle. The results for these three models of queryinference are very similar: If an oracle is already omniscient for queryinference, then it is already omniscient for EX. There is an oracle of trivial EXdegree, which allows nontrivial queryinference. Furthermore, queries to a teacher can not overcome differences between oracles and the queryinference degrees are a proper refinement of the EXdegrees. 1 Introduction One famous example of learning via queries to a teacher is the game Mastermind. The teacher first selects the code  a quadruple of colours  that should be learned. Then the learner tries to figure out the code. In each round, the learner makes one gue...
ResourceBounded Baire Category: A Stronger Approach
 Proceedings of the Tenth Annual IEEE Conference on Structure in Complexity Theory
, 1996
"... This paper introduces a new definition of resourcebounded Baire category in the style of Lutz. This definition gives an almostall/almostnone theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new definition than in previous definitions. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
This paper introduces a new definition of resourcebounded Baire category in the style of Lutz. This definition gives an almostall/almostnone theory of various complexity classes. The meagerness/comeagerness of many more classes can be resolved in the new definition than in previous definitions. For example, almost no sets in EXP are EXPcomplete, and NP is PFmeager unless NP = EXP. It is also seen under the new definition that no recrandom set can be (recursively) ttreducible to any PFgeneric set. We weaken our definition by putting arbitrary bounds on the length of extension strategies, obtaining a spectrum of different theories of Baire Category that includes Lutz's original definition. 1
Noisy Inference and Oracles
, 1996
"... A learner noisily infers a function or set, if every correct item is presented infinitely often while in addition some incorrect data ("noise") is presented a finite number of times. It is shown that learning from a noisy informant is equal to finite learning with Koracle from a usual i ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
A learner noisily infers a function or set, if every correct item is presented infinitely often while in addition some incorrect data ("noise") is presented a finite number of times. It is shown that learning from a noisy informant is equal to finite learning with Koracle from a usual informant. This result has several variants for learning from text and using different oracles. Furthermore, partial identification of all r.e. sets can cope also with noisy input.
The Arithmetical Complexity of Dimension and Randomness
"... Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension dim(A) ∈ [0, 1] and a strong dimension Dim(A) ∈ [0, 1]. Let DIM α and DIM α str be the classes of all sequenc ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension dim(A) ∈ [0, 1] and a strong dimension Dim(A) ∈ [0, 1]. Let DIM α and DIM α str be the classes of all sequences of dimension α and of strong dimension α, respectively. We show that DIM 0 is properly Π 0 2, and that for all ∆ 0 2computable α ∈ (0, 1], DIM α is properly Π 0 3. To classify the strong dimension classes, we use a more powerful effective Borel hierarchy where a coenumerable predicate is used rather than a enumerable predicate in the definition of the Σ 0 1 level. For all ∆ 0 2computable α ∈ [0, 1), we show that DIM α str is properly in the Π 0 3 level of this hierarchy. We show that DIM 1 str is properly in the Π 0 2 level of this hierarchy. We also prove that the class of Schnorr random sequences and the class of computably random sequences are properly Π 0 3.