learning efficiency of very simple grammars from positive data the class of very simple grammars is known to be polynomial time identifiable in the limit from positive data this paper gives an even more general discussion on the efficiency of identification of very simple grammars from positive data which includes both positive and negative results in particular we present an alternative efficient inconsistent learning algorithm for very simple grammars

reflective inductive inference of recursive functions in this paper we investigate reflective inductive inference of recursive functions a reflective iim is a learning machine that is additionally able to assess its own competence first we formalize reflective learning from arbitrary and from canonical example sequences here we arrive at four different types of reflection reflection in the limit optimistic pessimistic and exact reflection then we compare the learning power of reflective iims with each other as well as with the one of standard iims for learning in the limit for consistent learning of three different types and for finite learning

hamilton equations for elasticae in the euclidean 3 space abstract the variational problem on spatial curves defined by the integral of the squared curvature whose solutions are the elasticae or nonlinear splines is analyzed from the hamiltonian point of view using a procedure developed by munoz masqueand pozo coronado j munoz masque l m pozo coronado j phys a 31 1998 6225 6242 the symmetry of the problem under rigid motions is then used to reduce the euler lagrange equations to a first order dynamical system

rich classes inferable from positive data inductive inference from positive data is shown to be remarkably powerful using the framework of elementary formal systems an elementary formal system efs for short is a kind of logic program on consisting of finitely many axioms any context sensitive language is definable by a restricted efs called a length bounded efs length bounded efss with at most n axioms are considered and it is shown that inductive inference from positive data works successfully for their models as well as for their languages from this it follows that any class of logic programs such as prolog programs corresponding to length bounded efss can be inferred from positive facts

iterative learning from positive data and negative counterexamples a model for learning in the limit is defined where a so called iterative learner gets all positive examples from the target language tests every new conjecture with a teacher oracle if it is a subset of the target language and if it is not then it receives a negative counterexample and uses only limited long term memory incorporated in conjectures three variants of this model are compared when a learner receives least negative counterexamples the ones whose size is bounded by the maximum size of input seen so far and arbitrary ones a surprising result is that sometimes absence of bounded counterexamples can help an iterative learner whereas arbitrary counterexamples are useless we also compare our learnability model with other relevant models of learnability in the limit study how our model works for indexed classes of recursive languages and show that learners in our model can work in non u shaped way never abandoning the first right conjecture

language learning from texts mindchanges limited memory and monotonicity abstract the paper explores language learning in the limit under various constraints on the number of mindchanges memory and monotonicity we define language learning with limited long term memory and prove that learning with limited memory is exactly the same as learning via set driven machines when the order of the input string is not taken into account further we show that every language learnable via a set driven machine is learnable via a conservative machine making only justifiable mindchanges we get a variety of separation results for learning with bounded number of mindchanges or limited memory under restrictions on monotonicity a surprising result is that there are families of languages that can be monotonically learned with at most one mindchange but can neither be weak monotonically nor conservatively learned many separation results have a variant if a criterion a can be separated from b then often it is possible to find a family l of languages such that l is a and b learnable but while it is possible to restrict the number of mindchanges or long term memory on criterion a this is impossible for b

iterative learning of simple external contextual languages it is investigated for which choice of a parameter q denoting the number of contexts the class of simple external contextual languages is iteratively learnable on the one hand the class admits for all values of q polynomial time learnability provided an adequate choice of the hypothesis space is given on the other hand additional constraints like consistency and conservativeness or the use of a one one hypothesis space changes the picture iterative learning limits the long term memory of the learner to the current hypothesis and these constraints further hinder storage of information via padding of this hypothesis it is shown that if q 3 then simple external contextual languages are not iteratively learnable using a class preserving one one hypothesis space while for q 1 it is iteratively learnable even in polynomial time it is also investigated for which choice of the parameters the simple external contextual languages can be learnt by a consistent and conservative iterative learner

polynomial time identification of very simple grammars from positive data this paper concerns a subclass of simple deterministic grammars called very simple grammars and studies the problem of identifying the subclass in the limit from positive data the class of very simple languages forms a proper subclass of simple deterministic languages and is incomparable to the class of regular languages this class of languages is also known as the class of left szilard languages of context free grammars after providing some properties of very simple languages we show that the class of very simple grammars is polynomial time identifiable in the limit from positive data in the following sense that is we show that there effectively exists an algorithm that given a target very simple grammar g over alphabet identifies a very simple grammar g equivalent to g in the limit from positive data satisfying the property that the time for updating a conjecture is bounded by o m and the total number of prediction errors made by the algorithm is bounded by o n where n is the size of g m max n 1 3 and n is the total length of all positive data provided

multi pattern languages we investigate languages consisting of words following one of the given finitely many patterns the issues concerning such multi pattern languages are relevant in inductive inference theory of learning and term rewriting we obtain results about decidability characterization hierarchies and special classes of multi pattern languages some open problems are also presented

on the role of procrastination in machine learning the traditional model of inductive inference is enhanced to allow learning machines to procrastinate about how many trials they will need to complete an inference or about how accurate their solution will be hierarchies of classes of learnable phenomena represented by sets of recursive functions isomorphic to the structure of the constructive ordinals are revealed the existence of such hierarchies indicates a potential advantage of procrastination as a learning technique tradeoffs between the two types of procrastination are shown not to exist in general one of the new classes of sets of inferrible functions introduced in this paper turns out to have the somewhat unusual property of being closed under finite unions a new technique based on a hardest to lear set relative to a class is used

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

first order jk clausal theories are pac learnable abstract we present positive pac learning results for the nonmonotonic inductive logic programming setting in particular we show that first order range restricted clausal theories that consist of clauses with up to k literals of size at most j each are polynomial sample polynomial time pac learnable with one sided error from positive examples only in our framework concepts are clausal theories and examples are finite interpretations we discuss the problems encountered when learning theories which only have infinite nontrivial models and propose a way to avoid these problems using a representation change called flattening finally we compare our results to pac learnability results for the normal inductive logic programming setting

the hamiltonian theory of elastica the total squared curvature of the curve 7 for a brief historical discussion see truesdell 7 it is worth noting here though that euler solved the problem for curves in the plane the problem of elastica has recently generated renewed interest notable recent works include that by langer and singer 5 who were able to integrate the frenet equations for elastica with fixed arclength in spaces of constant curvature with the use of killing fields griffiths 4 section ii b iii derived the same integrals for the euler lagrange equations for elastica in r3 bryant and griffiths 2 performed marsden weinstein reduction on elastica in the plane using the griffiths formalism however none of these works has fully revealed the role played by the euclidean symmetry for elastica in space in this paper we find what this role is for elastica in r3 langer and singer showed that the euler lagrange equa tions are

some natural conditions on incremental learning the present study aims at insights into the nature of incremental learning in the context of gold s model of identification in the limit with a focus on natural requirements such as consistency and conservativeness incremental learning is analysed both for learning from positive examples and for learning from positive and negative examples the results obtained illustrate in which way different consistency and conservativeness demands can affect the capabilities of incremental learners these results may serve as a first step towards characterising the structure of typical classes learnable incrementally and thus towards elaborating uniform incremental learning methods

on a question of davenport and lewis and new character sum bounds in finite fields let be a nontrivial multiplicative character of fpn we obtain the following results 1 let e 0 be given if b nj 1 xj j xj nj 1 nj hj z j 1 n is a box satisfying n j 1 hj p 5 e n then for p p e and some absolute constant c 0 we have denoting a nontrivial multiplicative character x b x p e b 2 assume a b fp such that a p 9 e b p 9 e b b 1 4 4 1 then for a non principal multiplicative character x i y d x y p 2 4 i d we also improve a result of karacuba typeset by ams tex

topological separations in inductive inference re learning in the limit from positive data a major concern is which classes of languages are learnable with respect to a given learning criterion we are particularly interested herein in the reasons for a class of languages to be unlearnable we consider two types of reasons one type is called topological where it does not help if the learners are allowed to be uncomputable an example of gold s is that no class containing an infinite language and all its finite sub languages is learnable even by an uncomputable learner another reason is called computational where the learners are required to be algorithmic in particular two learning criteria might allow for learning different classes of languages from one another but with dependence on whether the unlearnability is of type topological or computational in this paper we formalize the idea of two learning criteria separating topologically in learning power this allows us to study more closely why two learning criteria separate in learning power for a variety of learning criteria concerning vacillatory monotone several kinds of iterative and feedback learning we show that certain learning criteria separate topologically and certain others which are known to separate are shown not to separate topologically showing that learning criteria do not separate topologically implies that any known separation must necessarily exploit algorithmicity of the learner

learning local languages and their application to dna sequence analysis this paper presents an efficient algorithm for learning in the limit a special type of regular languages called strictly locally testable languages from positive data and its application to identifying the protein spl alpha chain region in amino acid sequences first we present a linear time algorithm that given a strictly locally testable language learns its deterministic finite state automaton in the limit from only positive data this provides one with a practical and efficient method for learning a specific concept domain of sequence analysis we then describe several experimental results using the learning algorithm developed above following a theoretical observation which strongly suggests that a certain type of amino acid sequences can be expressed by a locally testable language we apply the learning algorithm to identifying the protein spl alpha chain region in amino acid sequences for hemoglobin experimental scores show an overall success rate of 95 correct identification for positive data and 96 for negative data

quantum algorithms for some hidden shift problems almost all of the most successful quantum algorithms discovered to date exploit the ability of the fourier transform to recover subgroup structures of functions especially periodicity the fact that fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation in this paper we present three examples of unknown shift problems that can be solved efficiently on a quantum computer using the quantum fourier transform for one of these problems the shifted legendre symbol problem we give evidence that the problem is hard to solve classically by showing a reduction from breaking algebraically homomorphic cryptosystems we also define the hidden coset problem which generalizes the hidden shift problem and the hidden subgroup problem this framework provides a unified way of viewing the ability of the fourier transform to capture subgroup and shift structure

comparison of identification criteria for machine inductive inference abstract a natural pl 1 hierarchy of successively more general criteria of success for inductive inference machines is described based on the size of sets of anomalies in programs synthesized by such machines these criteria are compared to others in the literature some of our results are interpreted as tradeoff results or as showing the inherent relative computational complexity of certain processes and others are interpreted from a positivistic mechanistic philosophical stance as theorems in philosophy of science the techniques of recursive function theory are employed including ordinary and infinitary recursion theorems

mind change optimal learning of bayes net structure from dependency and independency data this paper analyzes the problem of learning the structure of a bayes net in the theoretical framework of gold s learning paradigm bayes nets are one of the most prominent formalisms for knowledge representation and probabilistic and causal reasoning we follow constraint based approaches to learning bayes net structure where learning is based on observed conditional dependencies and independencies between variables of interest e g the data are of the form x is dependent on y given any assignment to variables s or of the form x is independent of y given any assignment to variables s applying learning criteria in this model leads to the following results 1 the mind change complexity of identifying a bayes net graph over variables v from either dependency data or from independency data are v 2 the maximum number of edges 2 there is a unique fastest mind change optimal bayes net learner for either data type convergence speed is evaluated using gold s dominance notion of uniformly faster convergence for dependency data the optimal learner conjectures a graph if it is the unique bayes net pattern that satisfies the observed dependencies with a minimum number of edges and outputs no guess otherwise for independency data the optimal learner conjectures a graph if it is the unique bayes net pattern that satisfies the observed dependencies with a maximum number of edges and outputs no guess otherwise we investigate the complexity of computing the output of the fastest mind change optimal learner for either data type and show that each of these two problems is np hard assuming p rp to our knowledge these are the first np hardness results concerning the existence of a uniquely optimal bayes net structure

on the intrinsic complexity of learning a new view of learning is presented the basis of this view is a natural notion of reduction we prove completeness and relative difficulty results an infinite hierarchy of intrinsically more and more difficult to learn concepts is presented our results indicate that the complexity notion captured by our new notion of reduction differs dramatically from the traditional studies of the complexity of the algorithms performing learning tasks

refutable language learning with a neighbor system we consider inductive language learning and machine discovery from examples with some errors in the present paper the error or incorrectness we consider is the one described uniformly in terms of a distance over strings firstly we introduce a notion of a recursively generable distance over strings and for a language l we define a k neighbor language l as a language obtained from l by i adding some strings not in l each of which is at most k distant from some string in l and by ii deleting some strings in l each of which is at most k distant from some string not in l then we define a k neighbor system of a base language class as the collection of k neighbor languages of languages in the class and adopt it as a hypothesis space we give formal definitions of k neighbor refutable inferability and discuss necessary and sufficient conditions on such kinds of inference finally as a concrete class inferable in the sense we introduced we consider a language class definable by elementary formal systems efss for short as a main result we show that the language class definable by the so called length bounded efss with at most n axioms is refutable and inferable from complete examples

learning in the limit with lattice structured hypothesis spaces we define a collection of language classes which are txtex learnable learnable in the limit from positive data the learners map any data input to an element of a fixed lattice and keep the least upper bound of all lattice elements thus obtained as the current hypothesis each element of the lattice is a grammar for a language and the learner climbs the lattice searching for the right element we call these classes in our collection lattice classes we provide several characterizations of lattice classes and their learners which suggests they are very natural in particular we show that any class of languages is a lattice class iff it is txtex learnable consistently conservatively set drivenly and strongly monotonically we show several language classes previously discussed in the literature to be lattice classes including the locally k testable classes the piecewise k testable classes the k reversible languages and the pattern languages we also show that lattice classes contain three previously known collections of language classes string extension language classes function distinguishable language classes and closed set systems finally the lattice perspective helps analyze the learning of these classes illustrations include query learning results in dependence on the lattice structure characterizations of closure properties and the vc dimension of lattice classes in terms of lattice properties

the influence of elasticity on analysis the classic heritage etude des domaines suivants le catenaire solutions doubles et le calcul variationnel l elastique fonctions elliptiques analyse qualitative solutions multiples geometrie differentielle des courbes gauches systemes en petites vibrations nombres propres et modes simples equations differentielles lineaires fonctions de bessel polynomes de laguerre transformees integrales oscillations resonantes equations aux derives partielles concept de solution discontinuites la definition moderne d une fonction base fonctionnelle singularites solutions generalisees plaques la courbure gaussienne theorie tridimensionnelle deformation des regions deformation et rotation locale tenseurs nombres et vecteurs propres decomposition polaire resolution spectrale isotropie et hemitropie groupes torsion d un cylindre solution principale d un systeme d equations differentielles et son etat d approximation d autres solutions

inference of k testable languages in the strict sense and application to syntactic pattern recognition the inductive inference of the class of k testable languages in the strict sense k tssl is considered a k tssl is essentially defined by a finite set of substrings of length k that are permitted to appear in the strings of the language given a positive sample r of strings of an unknown language a deterministic finite state automation that recognizes the smallest k tssl containing r is obtained the inferred automation is shown to have a number of transitions bounded by o m where m is the number of substrings defining this k tssl and the inference algorithm works in o kn log m where n is the sum of the lengths of all the strings in r the proposed methods are illustrated through syntactic pattern recognition experiments in which a number of strings generated by ten given source non k tssl grammars are used to infer ten k tssl stochastic automata which are further used to classify new strings generated by the same source grammars the results of these experiments are consistent with the theory and show the ability of stochastic k tssls to approach other classes of regular languages

generalized notions of mind change complexity gold introduced the notion of learning in the limit where a class s is learnable iff there is a recursive machine m which reads the course of values of a function f and converges to a program for f whenever f is in s an important measure for the speed of convergence in this model is the quantity of mind changes before the onset of convergence the oldest model is to consider a constant bound on the number of mind changes m makes on any input function such a bound is referred here as type 1 later this was generalized to a bound of type 2 where a counter ranges over constructive ordinals and is counted down at every mind change although ordinal bounds permit the inference of richer concept classes than constant bounds they still are a severe restriction therefore the present work introduces two more general approaches to bounding mind changes these are based on counting by going down in a linearly ordered set type 3 and on counting by going down in a partially ordered set type 4 in both cases the set must not contain infinite descending recursive sequences these four types of mind changes yield a hierarchy and there are identifiable classes that cannot be learned with the most general mind change bound of type 4 it is shown that existence of type 2 bound is equivalent to the existence of a learning algorithm which converges on every also nonrecursive input function and the existence of type 4 is shown to be equivalent to the existence of a learning algorithm which converges on every recursive function a partial characterization of type 3 yields a result of independent interest in recursion theory the interplay between mind change complexity and choice of hypothesis space is investigated it is established that for certain concept classes a more expressive hypothesis space can sometimes reduce mind change complexity of learning these classes the notion of mind change bound for behaviourally correct learning is indirectly addressed by employing the above four types to restrict the number of predictive errors of commission in finite error next value learning nv a model equivalent to behaviourally correct learning again natural characterizations for type 2 and type 4 bounds are derived their naturalness is further illustrated by characterizing them in terms of branches of uniformly recursive families of binary trees

mind change complexity of learning logic programs the present paper motivates the study of mind change complexity for learning minimal models of length bounded logic programs it establishes ordinal mind change complexity bounds for learnability of these classes both from positive facts and from positive and negative facts building on angluin s notion of finite thickness and wright s work on finite elasticity shinohara defined the property of bounded finite thickness to give a sufficient condition for learnability of indexed families of computable languages from positive data this paper shows that an effective version of shinohara s notion of bounded finite thickness gives sufficient conditions for learnability with ordinal mind change bound both in the context of learnability from positive data and for learnability from complete both positive and negative data let be a notation for the first limit ordinal then it is shown that if a language defining framework yields a uniformly decidable family of languages and has effective bounded finite thickness then for each natural number m 0 the class of languages defined by formal systems of length m is identifiable in the limit from positive data with a mind change bound of m is identifiable in the limit from both positive and negative data with an ordinal mind change bound of m the above sufficient conditions are employed to give an ordinal mind change bound for learnability of minimal models of various classes of length bounded prolog programs including shapiro s linear programs arimura and shinohara s depth bounded linearly covering programs and krishna rao s depth bounded linearly moded programs it is also noted that the bound for learning from positive data is tight for the example classes considered

on a generalized notion of mistake bounds this paper proposes the use of constructive ordinals as mistake bounds in the on line learning model this approach elegantly generalizes the applicability of the on line mistake bound model to learnability analysis of very expressive concept classes like pattern languages unions of pattern languages elementary formal systems and minimal models of logic programs the main result in the paper shows that the topological property of effective finite bounded thickness is a sufficient condition for on line learnability with a certain ordinal mistake bound an interesting characterization of the on line learning model is shown in terms of the identification in the limit framework it is established that the classes of languages learnable in the on line model with a mistake bound of are exactly the same as the classes of languages learnable in the limit from both positive and negative data by a popperian consistent learner with a mind change bound of this result nicely builds a bridge between the two models

using contextual representations to efficiently learn context free languages we present a polynomial update time algorithm for the inductive inference of a large class of context free languages using the paradigm of positive data and a membership oracle we achieve this result by moving to a novel representation called contextual binary feature grammars cbfgs which are capable of representing richly structured context free languages as well as some context sensitive languages these representations explicitly model the lattice structure of the distribution of a set of substrings and can be inferred using a generalisation of distributional learning this formalism is an attempt to bridge the gap between simple learnable classes and the sorts of highly expressive representations necessary for linguistic representation it allows the learnability of a large class of context free languages that includes all regular languages and those context free languages that satisfy two simple constraints the formalism and the algorithm seem well suited to natural language and in particular to the modeling of first language acquisition preliminary experimental results confirm the effectiveness of this approach

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

learning regular sets from queries and counterexamples the problem of identifying an unknown regular set from examples of its members and nonmembers is addressed it is assumed that the regular set is presented by a minimamy adequate teacher which can answer membership queries about the set and can also test a conjecture and indicate whether it is equal to the unknown set and provide a counterexample if not a counterexample is a string in the symmetric difference of the correct set and the conjectured set a learning algorithm l is described that correctly learns any regular set from any minimally adequate teacher in time polynomial in the number of states of the minimum dfa for the set and the maximum length of any counterexample provided by the teacher it is shown that in a stochastic setting the ability of the teacher to test conjectures may be replaced by a random sampling oracle ex a polynomial time learning algorithm is shown for a particular problem of context free language identification

pac learning non recursive prolog clauses abstract recently there has been an increasing amount of research on learning concepts expressed in subsets of prolog the term inductive logic programming ilp has been used to describe this growing body of research this paper seeks to expand the theoretical foundations of ilp by investigating the pac learnability of logic programs we focus on programs consisting of a single function free non recursive clause and focus on generalizations of a language known to be pac learnable namely the language of determinate function free clauses of constant depth we demonstrate that a number of syntactic generalizations of this language are hard to learn but that the language can be generalized to clauses of constant locality while still allowing pac learnability more specifically we first show that determinate clauses of log depth are not pac learnable regardless of the language used to represent hypotheses we then investigate the effect of allowing indeterminacy in a clause and show that clauses with k indeterminate variables are as hard to learn as dnf we next show that a more restricted language of clauses with bounded indeterminacy is learnable using k cnf to represent hypotheses and that restricting the locality of a clause to a constant allows pac learnability even if an arbitrary amount of indeterminacy is allowed this last result is also shown to be a strict generalization of the previous result for determinate function free clauses of constant depth finally we present some extensions of these results to logic programs with multiple clauses

learning languages from positive data and a finite number of queries a computational model for learning languages in the limit from full positive data and a bounded number of queries to the teacher oracle is introduced and explored equivalence superset and subset queries are considered for the latter one we consider also a variant when the learner tests every conjecture but the number of negative answers is uniformly bounded if the answer is negative the teacher may provide a counterexample we consider several types of counterexamples arbitrary least counterexamples the ones whose size is bounded by the size of positive data seen so far and no counterexamples a number of hierarchies based on the number of queries answers and types of answers counterexamples is established capabilities of learning with different types of queries are compared in most cases one or two queries of one type can sometimes do more than any bounded number of queries of another type still surprisingly a finite number of subset queries is sufficient to simulate the same number of equivalence queries when behaviourally correct learners do not receive counterexamples and may have unbounded number of errors in almost all conjectures

on the hidden shifted power problem we consider the problem of recovering a hidden element s of a finite field mathbb f q of q elements from queries to an oracle that for a given x in mathbb f q returns x s e for a given divisor e mid q 1 we use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm for example they use substantially fewer queries to the oracle

deforming convex curves with fixed elastic energy abstract this paper presents a curve flow which preserves the elastic energy of the evolving curve if the initial curve is a planar simple and smooth curve with positive curvature then the local and global existence of the flow is proved under this flow the evolving curve will converge to a finite circle in the c metric as time goes to infinity

mind change efficient learning this paper studies efficient learning with respect to mind changes our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem but also locally in subproblems after receiving some evidence formalizing this idea leads to the notion of strong mind change optimality we characterize the structure of language classes that can be identified with at most mind changes by some learner not necessarily effective a language class l is identifiable with mind changes iff the accumulation order of l is at most accumulation order is a classic concept from point set topology we show that accumulation order is related to other established notions of structural complexity such as thickness and intrinsic complexity to aid the construction of learning algorithms we show that the characteristic property of strongly mind change optimal learners is that they output conjectures languages with maximal accumulation order we illustrate the theory by describing strongly mind change optimal learners for various problems such as identifying linear subspaces one variable patterns and fixed length patterns