Simulating teams with many conjectures

We study the learnability of enumerable families \(\mathcal{L}\) of uniformly recursive languages in dependence on the number of allowed mind changes, i.e., with respect to a well-studied measure of efficiency. We distinguish between exact learnability (\(\mathcal{L}\) has to be inferred w.r.t. \(\mathcal{L}\)) and class preserving learning (\(\mathcal{L}\) has to be inferred w.r.t. some suitable chosen enumeration of all the languages from \(\mathcal{L}\)) as well as between learning from positive and from both, positive and negative data.The measure of efficiency is applied to prove the superiority of class preserving learning algorithms over exact learning. We considerably improve results obtained previously and establish two infinite hierarchies. Furthermore, we separate exact and class preserving learning from positive data that avoids overgeneralization. Finally, language learning with a bounded number of mind changes is completely characterized in terms of recursively generable finite sets. These characterizations offer a new method to handle overgeneralizations and resolve an open question of Mukouchi (1992).KeywordsLanguage LearningRecursive FunctionInductive InferencePositive DataNegative DataThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider the inductive inference model of Gold [15]. Suppose we are given a set of functions that are learnable with certain number of mind changes and errors. What can we consistently predict about those functions if we are allowed fewer mind changes or errors? In [20] we relaxed the notion of exact learning by considering some higher level properties of the input-output behavior of a given function. in this context, a learner produces a program that describes the property of a given function. Can we predict generic properties such as threshold or modality if we allow fewer number of mind changes or errors? These questions were completely answered in [20] when the learner is restricted to a single IIM. In this paper we allow a team of IIMs to collaborate in the learning process. The learning is considered to be successful if any one of the team member succeeds. A motivation for this extension is to understand and characterize properties that are learnable for a given set of functions in a team environment.

The present work investigates to which extent semantical knowledge can support the learning of basic mathematical concepts. The considered learning criteria are learning characteristic or enumerable indices for languages from positive data where the learner has to converge either syntactically (Ex) or semantically (BC). The considered classes are the classes of all monoids of a given group, all ideals of a given ring or all subspaces of a given vector space. The following is shown: (a) Learnability depends much on the amount of semantic knowledge given at the synthesis of the learner where this knowledge is represented by programs for the algebraic operations, codes for prominent elements of the algebraic structure (like 0 and 1 in fields) and certain parameters (like the dimension of finite dimensional vector spaces). For several natural examples good knowledge of the semantics may enable to keep ordinal mind change bounds while restricted knowledge may either allow only BC-convergence or even not permit learnability at all. (b) A recursive commutative ring is Noetherian iff the class of its ideals is BC-learnable. Such a BC-learner can be synthesized from programs for addition and multiplication. In many Noetherian rings, one can Ex-learn characteristic indices for the ideals with an ordinal bound on the number of mind changes. But there are also some Noetherian rings where it is impossible to Ex-learn the ideals or to learn characteristic indices for them.

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base T, a hierarchy of sentences that are generalised logical consequences of T is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies.

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and “nearly” minimal size. i.e.. within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. Alim-computable functionis (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be “not-so-nearly” minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for whichnotations forconstructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.

From 25.07. to 30.07.2010, the Dagstuhl Seminar 10302 ``Learning paradigms in dynamic environments '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

Learning by the Process of Elimination

The most elementary study of man-machine communication deals with functions that are computable by an oracle machine. Such a machine is not supposed to work just by itself-it may interact with the external environment too. As far as learning programs are concerned, we may consider that the environment which provides data to the learner plays the role of an oracle. According to the various learning paradigms, the oracle can be used in quite different ways. It is fruitful to look at the paradigms from this point of view. Indeed, the various concepts of reducibility of one function or one set to another (e.g. Turing, truth-table and bounded truth-table reducibilities), that have been extensively studied in the theory of recursive functions, bare some relation to this aspect of learning. For most of the existing learning paradigms, the type of interaction between a learner and his environment is rather plain, but we describe a paradigm, dedicated to learning to control physical systems, for which this interaction is more complex. The reducibility notions are then extremely useful. We state a few results which illuminate the strength of reducibility concepts for studying learnability. >

This paper aims at solving several open questions in current neutrino physics: the neutrino mass hierarchy, the Dirac CP violating phase, the absolute mass of neutrinos, the nature of neutrinos (Dirac or Majorana), the Majorana matrix and the absolute value of the effective Majorana neutrino mass. In the research presented in this paper, we have shown that the precise definition of the mass splittings between neutrino mass eigenstates, done in the latest analysis of experimental data, can be of crucial importance for defining the nature of neutrino mass hierarchy. The Standard Model has three generations of fundamental matter particles. Three generations of the charged lepton mass show a hierarchical structure: mτ > mμ > me. Owing to that, there is a belief and it is considered that neutrinos may follow such hierarchical structure. In our calculations, we have also included the latest data obtained, based on the processing of measurement results, which showed that even with such data, obtained results favor the normal neutrino mass hierarchy. As for the individual neutrino mass calculated in this paper, in today’s neutrino physics it is only known that neutrino mass scale is bounded only from above, and both the Dirac and the Majorana character of neutrinos are compatible with all observations. Among some of the questions resolved in this paper, which are related to the properties of neutrinos, a positive answer was also given to the question of whether light neutrinos are self-conjugate particles or not.

This paper is motivated by the open question whether the union of two disjoint NP-complete sets always is NP-complete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities.Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions of m-equivalent disjoint sets. Under the hypothesis that NE ≠ coNE, we construct degrees in NP where our main question has a positive answer, i.e., these degrees are closed under unions of disjoint sets.KeywordsChromatic NumberSIAM JournalPolynomial HierarchyMathematical System TheoryPolynomial Time ReducibilityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents a metod of inductive inference of realvalued functions from given pairs of observed data of (x,h(x)), where h is a target function to be inferred. Eac of suc observed data inevitably involves some ranges of errors, and hence it is usually represented by a pair of rational numbers sow te approximate value and te error bound, respectively. On the other hand, a real number called a recursive real number can be represented by a pair of two of rational numbers, which converges to the real number and converges to zero, respectively. These sequeces show an approximate value of te real number and an error bound at each point. Such a real number can also be represented by a sequence of closed intervals wit rational end points converges to a singleton interval with the real number as both end points.In this paper, we propose a notion of recursive real-valued functions that can enjoy the merits of the both representations of the recursive real numbers. Then we present an algorithm which approximately infers real-valued functions from numerical data with some error bounds, and show that there exists a rich set of real-valued functions which is approximately inferable in the limit from such numerical data. We also discuss the precision of the guesses from the machine when sufficient data have not yet given.

Multi-sensor systems (MSS) have been increasingly applied in pattern classification while searching for the optimal classification framework is still an open problem. The development of the classifier ensemble seems to provide a promising solution. The classifier ensemble is a learning paradigm where many classifiers are jointly used to solve a problem, which has been proven an effective method for enhancing the classification ability. In this paper, by introducing the concept of Meta-feature (MF) and Trans-function (TF) for describing the relationship between the nature and the measurement of the observed phenomenon, classification in a multi-sensor system can be unified in the classifier ensemble framework. Then an approach called Genetic Algorithm based Classifier Ensemble in Multi-sensor system (GACEM) is presented, where a genetic algorithm is utilized for optimization of both the selection of features subset and the decision combination simultaneously. GACEM trains a number of classifiers based on different combinations of feature vectors at first and then selects the classifiers whose weight is higher than the pre-set threshold to make up the ensemble. An empirical study shows that, compared with the conventional feature-level voting and decision-level voting, not only can GACEM achieve better and more robust performance, but also simplify the system markedly.

Randomizing polynomials represent a function f(x) by a low-degree randomized mapping p(x, r) over a finite field F such that, for any input x, the output distribution of p(x, r) depends only on the value of f(x). We study the class of functions f which admit an efficient representation by constant-degree randomizing polynomials. It is known that this class contains NC1 as well as log-space classes contained in NC2. Whether it contains all polynomial-time computable functions is a wide open question. A positive answer would have major and unexpected consequences, including the existence of efficient constant-round multiparty protocols with unconditional security, and the equivalence of (polynomial-time) cryptography and cryptography in NC0.We obtain evidence for the limited power of randomizing polynomials by showing that a useful subclass of constant-degree randomizing polynomials cannot efficiently capture functions beyond NC. Concretely, we consider randomizing polynomials over fields F of a small characteristic in which each monomial has degree (at most) 2 in the random inputs r and constant degree in x. This subclass captures most constructions of randomizing polynomials from the literature. Our main result is that all functions f which can be efficiently represented by such randomizing polynomials over fields of a small characteristic are in non-uniform NC. (The same holds over arbitrary fields given a quadratic residuosity oracle.) This result is obtained in two steps: (1) we observe that computing f as above reduces to counting roots of degree-2 multivariate polynomials; (2) we design parallel algorithms for the latter problem. These parallel root counting algorithms may be of independent interest.On the flip side, our main result provides an avenue for obtaining new parallel algorithms via the construction of randomizing polynomials. This gives an unexpected application of cryptography to algorithm design. We provide several examples for the potential usefulness of this approach.

Charles Sims has asked whether or not the lower central subgroup γ n (F) of a free group F coincides with the normal closure in F of the basic commutators of weight n. This question has a positive answer in weights at most 5, but remains an open question in general. In earlier work with Gaglione and Spellman, it was shown that γ n (F) is the normal closure in F of the basic commutators of weights n through 2n − 4. Here, we specialize to the case where F has rank 2 and show that γ6(F) is the normal closure in F of the basic commutators of weights 6 and 7.

We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/ log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log2 k/ log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we obtain a lower bound of $$\Omega \left( {\sqrt {\log k/\log \log k} } \right)$$ for flow sparsifiers and a lower bound of $$\Omega \left( {\sqrt {\log k} /\log \log k} \right)$$ for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist $$\tilde O\left( {\sqrt {\log k} } \right)$$ cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than $$\tilde \Omega \left( {\sqrt {\log k} } \right)$$ would imply a negative answer to this question.