The set A is low for Martin-Lof random if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of Ambos-Spies and Kucera [2], showing that each low for Martin-Lof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering

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 degree---depending 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...

Inductive inference considers two types of queries: Queries to a teacher about the function to be learned and queries to a non-recursive 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 query-inference, then it is already omniscient for EX. There is an oracle of trivial EX-degree, which allows nontrivial query-inference. Furthermore, queries to a teacher can not overcome differences between oracles and the query-inference degrees are a proper refinement of the EX-degrees. 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...

We investigate the model recBC of learning of r.e. sets, where changes in hypotheses only count when there is an extensional difference. We study the learnability of collections that are uniformly r.e. We prove that, in contrast with the case of uniformly recursive collections, identifiability does not imply recursive BC-identifiability. This answers a question of D. de Jongh. In contrast to the model of recursive identifiability, we prove that the BC-model separates the notions of finite thickness and finite elasticity. 1 Introduction In this paper we consider a model of learning where two hypotheses about the data under consideration are considered equal when they denote the same object, i.e. when they are extensionally the same. This model was first defined for identification of functions in Feldman [6], Barzdin [3]. The first reference for this model in the context of set learning (learning from text) seems to be Osherson and Weinstein [14]. The model, and similar ones, ha...

The present work investigates Gold style algorithmic learning from input-output examples whereby the learner has access to oracles as additional information. Furthermore this access has to be robust, that means that a single learning algorithm has to succeed with every oracle which meets a given specification. The first main result considers oracles of the same Turing degree: Robust learning with any oracle from a given degree does not achieve more than learning without any additional information. The further work considers learning from function oracles which describe the whole class of...

Abstract. Strong jump traceability has been studied by various authors. In this paper we study a variant of strong jump traceability by looking at a partial relativization of traceability. We discover a new subclass H of the c.e. K-trivials with some interesting properties. These sets are computationally very weak, but yet contains a cuppable member. Surprisingly they cannot be constructed using cost functions, and is the first known example of a subclass of the K-trivials which does not contain any promptly simple member. Furthermore there is a single c.e. set which caps every member of H, demonstrating that they are in fact very far away from being promptly simple. 1.

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Abstract. In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e. strongly jump traceables is Π 0 4-complete. §1. Introduction. One of the fundamental concerns of computability theory is in understanding the relative difficulty of computational problems as measured by Turing reducubility (≤T). The equivalence classes of the preordering ≤T are called Turing degrees, and it is long recognized that the fundamental operator on the structure of the Turing degrees is the jump operator. For a set A, the

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