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

the query complexity of finding local minima in the lattice in this paper we study the query complexity of finding local minimum points of a boolean function this task occurs frequently in exact learning algorithms for many natural classes such as monotone dnf o log n term dnf unate dnf and decision trees on the negative side we prove that any possibly randomized algorithm that produces a local minimum of a function f chosen from a sufficiently rich concept class using a membership oracle for f must ask n2 membership queries in the worst case in particular this lower bound applies to the class of decision trees a simple algorithm is known that achieves this lower bound on the positive side we show that for the class o log n term dnf finding local minimum points requires only n log n membership queries and more generally tn membership queries for t term dnf with tn this efficient procedure improves the time and query complexity of known learning algorithms for the class o log n term dnf 2001 elsevier science

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

approximate minimization of weighted tree automata abstract this paper studies the following approximate minimization problem given a minimal weighted tree automaton a with n states recognizing a weighted tree language f can we construct a smaller automaton a with n n states recognizing a language f that is a good approximation of f the corresponding problem for weighted automata on strings was recently studied by balle et al 16 15 where the authors introduced a new canonical form for weighted automata called singular value automata inspired by spectral methods and showed that truncating such canonical form yields a solution for the problem satisfying a certain approximation criteria in this paper we take a similar approach and show that in the tree case one can obtain an analogous canonical form that we call singular value tree automata and use it to study the approximate minimization problem for weighted tree automata we first establish the existence of this canonical form for weighted tree automata and then provide bounds on the quality of the resulting approximation method based on truncation we also study the problem of computing the canonical form given a minimal weighted tree automaton and show that in the tree case this task is considerably more complicated than in the string case in particular computing the canonical form reduces to solving a system of polynomial equations by further reducing this problem to the computation of generalized partition functions for weighted tree automata we propose and analyze two methods for computing the canonical form based on iterative methods fixed point iteration and newton s method our analysis of newton s method unveils a connection between iterative methods and sequences of sets of trees satisfying a certain technical condition that might be of independent interest

closed constraint algebras and path integrals for loop group actions in this paper we study systems with a closed algebra of second class constraints we describe a construction of the reduced theory that resembles the conventional treatment of first class constraints it suggests in particular to compute the symplectic form on the reduced space by a fiber integral of the symplectic form on the original space this approach is then applied to a class of systems with loop group symmetry the chiral anomaly of the loop group action spoils the first class character of the constraints but not their closure proceeding along the general lines described above we obtain a 2 form from a fiber path integral this form is not closed as a relict of the anomaly examples of such reduced spaces are provided by d branes on group manifolds with wzw action

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

comparison results for pdes with a singular potential several comparison results are obtained for solutions to linear elliptic and parabolic equations with a singular potential solutions to these equations are singular in many cases and our results roughly say that they all have comparable singularities provided that they belong to an appropriate space we formulate the hypothesis on the potential in terms of an inequality which in the case of the well known inverse square potential is a consequence of an improvement of hardy s inequality due to vazquez and zuazua

parameterized learnability of juntas we study the parameterized complexity of learning k juntas and some variations of juntas we show the hardness of learning k juntas and subclasses of k juntas in the pac model by reductions from a w 2 complete problem on the other hand as a consequence of a more general result we show that k juntas are exactly learnable with improper equivalence queries and access to a w p oracle

parameterized regular expressions and their languages we study regular expressions that use variables or parameters which are interpreted as alphabet letters we consider two classes of languages denoted by such expressions under the possibility semantics a word belongs to the language if it is denoted by some regular expression obtained by replacing variables with letters under the certainty semantics the word must be denoted by every such expression such languages are regular and we show that they naturally arise in several applications such as querying graph databases and program analysis as the main contribution of the paper we provide a complete characterization of the complexity of the main computational problems related to such languages nonemptiness universality containment membership as well as the problem of constructing nfas capturing such languages we also look at the extension when domains of variables could be arbitrary regular languages and show that under the certainty semantics languages remain regular and the complexity of the main computational problems does not change

four one shot learners for regular tree languages and their polynomial characterizability we survey two existing algorithms for the inference of finite state tree automata from membership queries and a finite positive sample or equivalence queries and we suggest a reformulation of one of them which we deem necessary to ensure its termination we present two algorithms for the same two settings when the underlying description is not a deterministic but a residual canonical finite state automaton to this end we adapt all necessary notions to the residual tree case from our completed perspective we discuss the terms on which those four algorithms can be considered to be of polynomial complexity and also where there may be hidden exponentiality

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

learning of event recording automata in regular inference a regular language is inferred from answers to a finite set of membership queries each of which asks whether the language contains a certain word one of the most well known regular inference algorithms is the l algorithm due to dana angluin however there are almost no extensions of these algorithms to the setting of timed systems we extend angluin s algorithm for on line learning of regular languages to the setting of timed systems since timed automata can freely use an arbitrary number of clocks we restrict our attention to systems that can be described by deterministic event recording automata deras we present three algorithms tl s g tl n s g and tl s for inference of deras in tl s g and tl n s g we further restrict event recording automata to be event deterministic in the sense that each state has at most one outgoing transition per action learning such an automaton becomes significantly more tractable the algorithm tl n s g builds on tl s g by attempts to construct a smaller in number of locations automaton finally tl s is a learning algorithm for a full class of deterministic event recording automata which infers a so called simple dera which is similar in spirit to the region graph

lower semicontinuity of functionals via the concentration compactness principle in this paper we show the weak lower semicontinuity of some classes of functionals using the concentration compactness principle of p l lions these functionals involve an integral term and we do not know whether it can be handled by the de giorgi theorem the semicontinuity result allows us to prove several existence results for quasilinear elliptic equations

map learning with uninterpreted sensors and effectors this dissertation presents a set of methods by which a learning agent called a critter can learn a sequence of increasingly abstract and powerful interfaces to control a robot whose sensorimotor apparatus and environment are initially unknown the result of the learning is a rich hierarchical model of the robot s world its sensorimotor apparatus and environment the learning methods rely on generic properties of the robot s world such as almost everywhere smooth effects of actions on sensory features at the lowest level of the hierarchy the critter analyzes the effects of its actions in order to define control signals one for each of the robot s degrees of freedom it uses a generate and test approach to define sensory features that capture important aspects of the environment it uses linear regression to learn action models that characterize context dependent effects of the control signals on the learned features it uses these models to define high level control laws for finding and following paths defined using constraints on the learned features the critter abstracts these control laws which interact with the continuous environment to a finite set of actions that implement discrete state transitions at this point the critter has abstracted the robot s world to a finite state machine and can use existing methods to learn its structure

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

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

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

learning dynamics system identification for perceptually challenged agents from the perspective of an agent the input output behavior of the environment in which it is embedded can be described as a dynamical system inputs correspond to the actions executable by the agent in making transitions between states of the environment outputs correspond to the perceptual information available to the agent in particular states of the environment we view dynamical system identification as inference of deterministic finite state automata from sequences of input output pairs the agent can influence the sequence of input output pairs it is presented by pursuing a strategy for exploring the environment we identify two sorts of perceptual errors errors in perceiving the output of a state and errors in perceiving the inputs actually carried out in making a transition from one state to another we present efficient high probability learning algorithms for a number of system identification problems involving such errors we also present the results of empirical investigations applying these algorithms to learning spatial representations

a sufficient condition to polynomially compute a minimum separating dfa the computation of a minimal separating automaton msa for regular languages has been studied from many different points of view from synthesis of automata or grammatical inference to the minimization of incompletely specified machines or compositional verification in the general case the problem is np complete but this drawback does not prevent the problem from having a real application in the above mentioned fields in this paper we propose a sufficient condition that guarantees that the computation of the msa can be carried out with polynomial time complexity

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

restricted ambiguity of erasing morphisms a morphism h is called ambiguous for a string s if there is another morphism that maps s to the same image as h otherwise it is called unambiguous in this paper we examine some fundamental problems on the ambiguity of erasing morphisms we provide a detailed analysis of so called ambiguity partitions and our main result uses this concept to characterise those strings that have a morphism of strongly restricted ambiguity furthermore we demonstrate that there are strings for which the set of unambiguous morphisms depending on the size of the target alphabet of these morphisms is empty finite or infinite finally we show that the problem of the existence of unambiguous erasing morphisms is equivalent to some basic decision problems for nonerasing multi pattern languages

improved lower bounds for learning from noisy examples this paper presents a general information theoretic approach for obtaining lower bounds on the number of examples required for probably approximately correct pac learning in the presence of noise this approach deals directly with the fundamental information quantities avoiding a bayesian analysis the technique is applied to several different models illustrating its generality and power the resulting bounds add logarithmic factors to or improve the constants in previously known lower bounds

the power of a pebble exploring and mapping an unknown environment is a fundamental problem that is studied in a variety of contexts many results have focused on finding efficient solutions to restricted versions of the problem in this paper we consider a model that makes very limited assumptions about the environment and solve the mapping problem in this general setting we model the environment by an unknown directed graph g and consider the problem of a robot exploring and mapping g the edges emanating from each vertex are numbered from 1 to d but we do not assume that the vertices of g are labeled since the robot has no way of distinguishing between vertices it has no hope of succeeding unless it is given some means of distinguishing between vertices for this reason we provide the robot with a pebble a device that it can place on a vertex and use to identify the vertex later in this paper we show 1 if the robot knows an upper bound on the number of vertices then it can learn the graph efficiently with only one pebble 2 if the robot does not know an upper bound on the number of vertices n then log log n pebbles are both necessary and sufficient in both cases our algorithms are deterministic

lie group valued moment maps we develop a theory of quasi hamiltonian g spaces for which the moment map takes values in the group g itself rather than in the dual of the lie algebra the theory includes counterparts of hamiltonian reductions the guillemin sternberg symplectic cro

a concentration compactness lemma with applications to singular eigenvalue problems let omega subset of or equal to r n be any open set we study the nonlinear eigenvalue problem delta p u lambda v x u p 2 u u epsilon d 0 1 p omega where 1 p n and v is an element of l loc 1 omega may have strong singularities and an indefinite sign the key ingredient is a precised concentration compactness lemma related to v dependent limiting problems this work follows extends and simplifies that of a tertikas 1998 j funct anal 154 42 66 dealing with the positive linear case for omega r n c 1999 academic press

fast identification of geometric objects with membership queries we investigate the number of membership queries that are needed to identify polygons i e intersections of halfplanes over a two dimensional grid 0 n 1 2 we exhibit a learning algorithm that learns 100 correctly while requiring no random examples and not more membership queries than previous algorithms needed in addition to their random examples for probably almost correctly learning even for moderate values of furthermore the learning algorithm in this paper uses only grid points for its membership queries this appears to be appropriate in situations where the probing device has only a limited resolution and for applications to the set of pixels on a two dimensional screen the learning algorithm that is described in this paper overcomes a fundamental obstacle related to the zigzag borderline that is generated by a halfplane over a discrete grid this obstacle had thwarted previous attempts to show that in angluin s model for on line learning with equivalence and membership queries one can learn in an efficient manner even those classes of geometric objects which cannot be learnt fast by equivalence queries alone such as rectangles in general position

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

on multiple solutions of a singular quasilinear equation on unbounded domain abstract variational methods are used to prove existence and multiplicity of solutions for the following quasilinear partial differential equation p u x s u p s 2 u k x u r 2 u in where 0 s p r n may be unbounded in particular r n is permitted

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

efficient learning of context free grammars from positive structural examples in this paper we introduce a new normal form for context free grammars called reversible context free grammars for the problem of learning context free grammars from positive only examples a context free grammar g n s p s is said to be reversible if 1 a a and b a in p implies a b and 2 a email protected and a email protected in p implies b c we show that the class of reversible context free grammars can be identified in the limit from positive samples of structural descriptions and there exists an efficient algorithm to identify them from positive samples of structural descriptions where a structural description of a context free grammar is an unlabelled derivation tree of the grammar this implies that if positive structural examples of a reversible context free grammar for the target language are available to the learning algorithm the full class of context free languages can be learned efficiently from positive samples

predicting neutron diffusion eigenvalues with a query based adaptive neural architecture a query based approach for adaptively retraining and restructuring a two hidden layer artificial neural network ann has been developed for the speedy prediction of the fundamental mode eigenvalue of the neutron diffusion equation a standard nuclear reactor core design calculation which normally requires the iterative solution of a large scale system of nonlinear partial differential equations pdes the approach developed focuses primarily upon the adaptive selection of training and cross validation data and on artificial neural network ann architecture adjustments with the objective of improving the accuracy and generalization properties of ann based neutron diffusion eigenvalue predictions for illustration the performance of a bare bones feedforward multilayer perceptron mlp is upgraded through a variety of techniques namely nonrandom initial training set selection adjoint function input weighting teacher student membership and equivalence queries for generation of appropriate training data and a dynamic node architecture dna implementation the global methodology is flexible in that it ran wrap around any specific training algorithm selected for the static calculations i e training iterations with a fixed training set and architecture finally the improvements obtained are carefully contrasted against past works reported in the literature

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

a path integral approach to the kontsevich quantization formula we give a quantum field theory interpretation of kontsevich s deformation quantization formula for poisson manifolds we show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory its batalin vilkovisky quantization yields a superconformal field theory the associativity of the star product and more generally the formality conjecture can then be understood by field theory methods as an application we compute the center of the deformed algebra in terms of the center of the poisson algebra

a note on the number of queries needed to identify regular languages we consider the number of queries required to identify a regular set given an oracle for the set and some auxiliary information about the set if the auxiliary information is n the number of states of the canonical finite state acceptor for the language then the upper and lower bounds on the number of queries are exponential in n if the auxiliary information consists of a set of strings guaranteed to reach every live state of the canonical acceptor for the language then the upper and lower bounds are polynomial in n and the size of the given set of strings as a corollary the problem considered by pao and carr 1978 is shown to be solvable in a polynomial number of queries

the crystallizing substochastic sequential machine extractor cryssmex this letter presents an algorithm cryssmex for extracting minimal finite state machine descriptions of dynamic systems such as recurrent neural networks unlike previous algorithms cryssmex is parameter free and deterministic and it efficiently generates a series of increasingly refined models a novel finite stochastic model of dynamic systems and a novel vector quantization function have been developed to take into account the state space dynamics of the system the experiments show that 1 extraction from systems that can be described as regular grammars is trivial 2 extraction from high dimensional systems is feasible and 3 extraction of approximative models from chaotic systems is possible the results are promising and an analysis of shortcomings suggests some possible further improvements some largely overlooked connections of the field of rule extraction from recurrent neural networks to other fields are also identified

learning regular omega languages we provide an algorithm for learning an unknown regular set of infinite words using membership and equivalence queries three variations of the algorithm learn three different canonical representations of omega regular languages using the notion of families of dfas one is of size similar to l a dfa representation recently learned using l 7 the second is based on the syntactic forc introduced in 14 the third is introduced herein we show that the second can be exponentially smaller than the first and the third is at most as large as the first two with up to a quadratic saving with respect to the second

models of active learning in group structured state spaces we investigate the problem of learning the transition dynamics of deterministic discrete state environments we assume that an agent exploring such an environment is able to perform actions from a finite set of actions in the environment and to sense the state changes the question investigated is whether the agent can learn the dynamics without visiting all states such a goal is unrealistic in general hence we assume that the environment has structural properties an agent might exploit in particular we assume that the set of all action sequences forms an algebraic group we introduce a learning model in different variants and study under which circumstances the corresponding group structured environments can be learned efficiently by experimenting with group generators actions it turns out that for some classes of such environments the choice of actions given to the agent determines if efficient learning is possible negative results are presented even without efficiency constraints for rather general classes of groups showing that even with group structure learning an environment from partial information is far from trivial however positive results for special subclasses of abelian groups turn out to be a good starting point for the design of efficient learning algorithms based on structured representations

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

existence and asymptotic behavior of ground states for quasilinear singular equations involving hardy sobolev exponents abstract we study the existence and decaying rate of solutions for the quasilinear problem p u x f u x g u in r n u 0 in r n u x 0 where p stands for the p laplacian operator 1 p n r n 0 is continuous and not identically zero 0 is a parameter x is the euclidean norm of x 0 p f g 0 0 are continuous and nondecreasing f has sublinear growth and the hardy sobolev exponent p p n n p bounds the growth of g we deal with variational methods and the lower and upper solutions technique