tradeoffs in the inductive inference of nearly minimal size programs inductive inference machines are algorithmic devices which attempt to synthesize in the limit programs for a function while they examine more and more of the graph of the function there are many possible criteria of success we study the inference of nearly minimal size programs our principal results imply that nearly minimal size programs can be inferred in the limit without loss of inferring power provided we are willing to tolerate a finite but not uniformly bounded number of anomalies in the synthesized programs on the other hand there is a severe reduction of inferring power in inferring nearly minimal size programs if the maximum number of anomalies allowed is any uniform constant we obtain a general characterization for the classes of recursive functions which can be synthesized by inferring nearly minimal size programs with anomalies we also obtain similar results for popperian inductive inference machines the exact tradeoffs between mind change bounds on inductive inference machines and anomalies in synthesized programs are obtained the techniques of recursive function theory including the recursion theorem are employed

confident and consistent partial learning of recursive functions abstract partial learning is a criterion where the learner infinitely often outputs one correct conjecture while every other hypothesis is issued only finitely often this paper addresses two variants of partial learning in the setting of inductive inference of functions first confident partial learning requires that the learner also on those functions which it does not learn singles out exactly one hypothesis which is output infinitely often second essentially class consistent partial learning is partial learning with the additional constraint that on the functions to be learnt almost all hypotheses issued are consistent with all the data seen so far the results of the present work are that confident partial learning is more general than explanatory learning incomparable with behaviourally correct learning and closed under union essentially class consistent partial learning is more general than behaviourally correct learning and incomparable with confident partial learning furthermore it is investigated which oracles permit to learn all recursive functions under these criteria for confident partial learning some non high oracles are omniscient for essentially class consistent partial learning all pa complete and all oracles of hyperimmune turing degree are omniscient

learning and classifying abstract we define and study a learning paradigm that sits between identification in the limit and classification more precisely we expect a learner to determine in the limit which members of a finite set d of possible data belong to a target language l where d is arbitrary so as d becomes larger and larger the task becomes closer and closer to identifying l but as d is always finite and l can be infinite it can still be expected that ex and bc learning are often more difficult than performing this classification task the paper supports this intuition and makes it precise taking into account desirable constraints on how the learner behaves such as bounding the number of mind changes and being conservative special attention is given to various forms of consistency in particular we might not only require consistency between the members of d to classify the current data and a language l but also consistency between larger sets of possible data to classify supersets of d and the same and l whereas in the classical paradigms of inductive inference or classification only the available data can grow here both the available data and the set of possible data to classify can grow we provide a fairly comprehensive set of results many of which are optimal that demonstrate the fruitfulness of the approach and the richness of the paradigm

growth functions on fuchsian groups and the euler characteristic let g be a finitely generated group and 2 a finite generating set for g then 2 determines a norm on g called the word norm defined by letting ig be the minimal length ofg as a word in 27 following milnor 15 we define the growth series g z 1 a2z a z where a is the number of elements of g of word norm exactly n much of the early work e g 1 16 17 23 on growth series was concerned with asymptotic properties of the coefficients of g z this work culminated in gromov s theorem 12 that g has polynomial growth if and only if it is virtually nilpotent and in grigorchuk s theorem 11 that there are finitely generated groups whose growth is neither polynomial nor exponential if g is a coxeter group and 2 is the standard generating set for g bourbaki 3 showed that the growth series g z is the series of a rational function f z and serre 20 showed that f 1 i z g where z g is the rational euler characteristic of g note that if g is finite then g 1 order g l z g but g z is not defined at one if g is infinite if g is a compact hyperbolic or irreducible euclidean coxeter group and 27 is the standard generating set for g serre showed that f 1 z f z and so the poles of f are algebraic units in contrast to the work on the asymptotic properties of the coefficients of g z these results raised the possibility of a beautiful theory of the exact structure of growth functions on groups however a decade went by before there was much further work on this subject in two papers 5 and 6 cannon studied growth functions for cocompact hyperbolic groups if g is a cocompact hyperbolic group and x is any finite generating set for g he showed that the coefficients a of the growth series g z satisfy a linear recursion and hence g z is the power series of a rational function f z he also computed some examples when g is a closed surface group or a compact hyperbolic coxeter group if g is a closed surface group and 2 is a geometric generating set see below for a definition for g coming from a one relator presentation then f is reciprocal f z f 1 z f 1 l x g the

finite identification of functions by teams with success ratio 1 2 and above abstract consider a scenario in which an algorithmic machine m is being fed the graph of a computable function m is said to finitely identify just in case after inspecting a finite portion of the graph of it emits its first conjecture which is a program for and it never abandons this conjecture thereafter a team of machines is a multiset of such machines a team is said to be successful just in case each member of some nonempty subset of predetermined size of the team is successful the ratio of the number of machines required to be successful to the size of the team is referred to as the success ratio of the team the present paper investigates the finite identification of computable functions by teams of learning machines the results presented complete the picture for teams with success ratio 1 2 and greater it is shown that at success ratio 1 2 introducing redundancy in the team can result in increased learning power in particular it is established that larger collections of functions can be learned by employing teams of 4 machines and requiring at least 2 to be successful than by employing teams of 2 machines and requiring at least 1 to be successful surprisingly it is also shown that introducing further redundancy at success ratio 1 2 does not yield any extra learning power in particular it is shown that the collections of functions that can be finitely identified by a team of 2m machines requiring at least m to be successful is the same as the collections of functions that can be finitely identified by a team of 4 machines requiring at least 2 to be successful if is even and the collections of functions that can be identified by a team of 2 machines requiring at least 1 to be successful if is odd these latter results require development of sophisticated simulation techniques

robust behaviorally correct learning intuitively a class of functions is robustly learnable if not only the class itself but also all of the transformations of the class under natural transformations such as via general recursive operators are learnable fulk showed the existence of a nontrivial class which is robustly learnable under the criterion ex however several of the hierarchies such as the anomaly hierarchies for ex and bc do not stand robustly fulk left open the question about whether bc and ex can be robustly separated in this paper we resolve this question positively

non commutative graded algebras and their hilbert series abstract this paper begins to develop a theory of non commutative graded algebras and their hilbert series which parallels the extensive already existing theory on commutative graded algebras we begin with a discussion of order ideals of monomials which give us vector space bases for these algebras we develop next the concept of weak summand which is a special relationship a subalgebra may have to an algebra and the concept of strongly free sets strongly free sets have many properties which are analogous to the properties of regular sequences in commutative algebras we discuss combinatorially free sets which are useful for constructing examples of many of the preceding ideas we conclude with an application of our results to algebraic topology by showing that there is a finite cw complex with only nine positive dimensional cells whose loop space has an irrational poincare series

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

partial learning of recursively enumerable languages this paper studies several typical learning criteria in the model of partial learning of r e sets in the recursion theoretic framework of inductive inference its main contribution is a complete picture of how the criteria of confidence consistency and conservativeness in partial learning of r e sets separate also in relation to basic criteria of learning in the limit thus this paper constitutes a substantial extension to prior work on partial learning further highlights of this work are very insightful characterisations of some of the inference criteria studied leading to interesting consequences about the structural properties of the collection of classes learnable under these criteria in particular a class is consistently partially learnable iff it is a subclass of a uniformly recursive family

classes with easily learnable subclasses in this paper we study the question of whether identifiable classes have subclasses which are identifiable under a more restrictive criterion the chosen framework is inductive inference in particular the criterion of explanatory learning ex of recursive functions as introduced by gold inform comput 10 1967 447 among the more restrictive criteria is finite learning where the learner outputs on every function to be learned exactly one hypothesis which has to be correct the topic of the present paper are the natural variants a and b below of the classical question whether a given learning criterion like finite learning is more restrictive than ex learning a does every infinite ex identifiable class have an infinite finitely identifiable subclass b if an infinite ex identifiable class s has an infinite finitely identifiable subclass does it necessarily follow that some appropriate learner ex identifies s as well as finitely identifies an infinite subclass of s these questions are also treated in the context of ordinal mind change bounds

inductive inference of approximations in this paper we investigate inductive inference identification criteria which permit infinitely many errors in explanations but which require that the density of these errors be no more than a certain prespectified amount we introduce three hierarchies of such criteria each of which has the same order type as the real unit interval these three hierarchies are progressively more strict in the way they measure density of errors of explanations the strictest of the three turns out to have all of its members save one incomparable to the identification criterion which permits finitely many errors in explanations

on generalized characters of nilpotent groups abstract we extend the results of g r robinson more on bounds on norms of generalized characters with applications to p local bounds and blocks bull london math soc 37 4 2005 555 565 to a general finite nilpotent group rather than a p group this enables us to obtain bounds on the number of complex irreducible characters of a finite group g which do not vanish identically on all non identity elements of a nilpotent subgroup n

optimal language learning from positive data email protected s original paper on inductive inference introduced a notion of an optimal learner intuitively a learner identifies a class of objects optimally iff there is no other learner that requires as little of each presentation of each object in the class in order to identify that object and for some presentation of some object in the class requires less of that presentation in order to identify that object beick considered this notion in the context of function learning and gave an intuitive characterization of an optimal function learner jantke and beick subsequently characterized the classes of functions that are algorithmically optimally identifiable herein email protected s notion is considered in the context of language learning it is shown that a characterization of optimal language learners analogous to email protected s does not hold it is also shown that the classes of languages that are algorithmically optimally identifiable cannot be characterized in a manner analogous to that of jantke and beick other interesting results concerning optimal language learning include the following it is shown that strong non u shapedness a property involved in email protected s characterization of optimal function learners does not restrict algorithmic language learning power it is also shown that for an arbitrary optimal learner f of a class of languages l f optimally identifies a subclass k of l iff f is class preserving with respect to k

learning recursive functions a survey studying the learnability of classes of recursive functions has attracted considerable interest for at least four decades starting with gold s 1967 model of learning in the limit many variations modifications and extensions have been proposed these models differ in some of the following the mode of convergence the requirements intermediate hypotheses have to fulfill the set of allowed learning strategies the source of information available to the learner during the learning process the set of admissible hypothesis spaces and the learning goals a considerable amount of work done in this field has been devoted to the characterization of function classes that can be learned in a given model the influence of natural intuitive postulates on the resulting learning power the incorporation of randomness into the learning process the complexity of learning among others on the occasion of rolf wiehagen s 60th birthday the last four decades of research in that area are surveyed with a special focus on rolf wiehagen s work which has made him one of the most influential scientists in the theory of learning recursive functions

on the relative sizes of learnable sets abstract measure and category or rather their recursion theoretical counterparts have been used in theoretical computer science to make precise the intuitive notion for most of the recursive sets we use the notions of effective measure and category to discuss the relative sizes of inferrible sets and their complements we find that inferable sets become large rather quickly in the standard hierarchies of learnability on the other hand the complements of the learnable sets are all large

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

the growth of free products abstract we improve estimates for the exponential growth rates of free products with or without amalgamation hnn extensions one relator groups and groups of positive deficiency some examples of groups for which the growth rates are realized are given

on the inference of approximate programs abstract a more powerful criterion of successful inductive inference is studied this criterion considers the inference successful when the result is a program which computes the function to be inferred everywhere except on a sparse but infinite subset of its domain in previous work sparse was taken to mean finite relationships with previously defined criteria are examined royer 19 considered the same issues and independently obtained some of our results

learning pattern languages over groups this article studies the learnability of classes of pattern languages over automatic groups it is shown that the class of bounded unions of pattern languages over finitely generated abelian automatic groups is explanatorily learnable for patterns in which variables occur at most n times it is shown that the classes of languages generated by such patterns as well as their bounded unions are for finitely generated automatic groups explanatorily learnable by an automatic learner in contrast automatic learners cannot learn the unions of up to two arbitrary pattern languages over the integers furthermore there is an algorithm which given an automaton describing a group g generates a learning algorithm m g such that either m g explanatorily learns all pattern languages over g or there is no learner for this set of languages at all not even a non recursive one for some automatic groups non learnability results of natural classes of pattern languages are provided

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

variants of some of the brauer fowler theorems abstract brauer and fowler noted restrictions on the structure of a finite group g in terms of c g t for an involution t g we consider variants of these themes we first note that for an arbitrary finite group g of even order we have g k f c g t 4 for each involution t g where f denotes the fitting subgroup of g and k f denotes the number of conjugacy classes of f in particular for such a group g we have g f g c g t 4 for each involution t g this result requires the classification of the finite simple groups the groups sl 2 2 n illustrate that the above exponent 4 cannot be replaced by any exponent less than 3 we do not know at present whether the exponent 4 can be improved in general though we note that the exponent 3 suffices when g is almost simple we are however able to prove that every finite group g of even order contains an involution u with g f g c g u 3 the proof of this fact reduces to proving two residual cases one in which g is almost simple where the classification of the finite simple groups is needed and one when g has a sylow 2 subgroup of order 2 for the latter result the classification of finite simple groups is not needed though the feit thompson odd order theorem is we also prove a very general result on fixed point spaces of involutions in finite irreducible linear groups which does not make use of the classification of the finite simple groups and some other results on the existence of non central elements not necessarily involutions with large centralizers in general finite groups lastly we prove without using the classification of finite simple groups that if g is a finite group and t g is an involution then all prime divisors of g f g are less than or equal to c g t 1

machine induction without revolutionary changes in hypothesis size this paper provides a beginning study of the effects on inductive inference of paradigm shifts whose absence is approximately modeled by various formal approaches to forbidding large changes in the size of programs conjectured one approach calledseverely parsimonious requires all the programs conjectured on the way to success to be nearly i e within a recursive function of minimal size it is shown that this very conservative constraint allows learning infinite classes of functions butnotinfinite r e classes of functions another approach callednon revolutionary requires all conjectures to be nearly the same size as one another this quite conservative constraint is nonetheless shown to permit learning some infinite r e classes of functions allowing up to one extrabounded sizemind change towards a final program learned certainly does not appear revolutionary however somewhat surprisingly for scientific inductive inference it is shown that there are classes learnablewiththe non revolutionary constraint respectively with severe parsimony up to i 1 mind changes and no anomalies which classes cannotbe learned with no size constraint an unbounded finite number of anomalies in the final program but with no more thanimind changes hence in some cases the possibility of one extra mind change is considerably more liberating than removal of very conservative size shift constraints the proofs of these results are also combinatorially interesting

on the power of inductive inference from good examples abstract the usual information in inductive inference available for the purposes of identifying an unknown recursive function f is the set of all input output examples x f x n en in contrast to this approach we show that it is considerably more powerful to work with finite sets of good examples even when these good examples are required to be effectively computable the influence of the underlying numberings with respect to which the identification has to be realized to the capabilities of inference from good examples is also investigated it turns out that nonstandard numberings can be much more powerful than godel numberings

enlarging learnable classes we study which classes of recursive functions satisfy that their union with any other explanatorily learnable class of recursive functions is again explanatorily learnable we provide sufficient criteria for classes of recursive functions to satisfy this property and also investigate its effective variants furthermore we study the question which learners can be effectively extended to learn a larger class of functions we solve an open problem by showing that there is no effective procedure which does this task on all learners which do not learn a dense class of recursive functions however we show that there are two effective extension procedures such that each learner is extended by one of them

on the complexity of inductive inference the notion of the complexity of performing an inductive inference is defined some examples of the tradeoffs between the complexity of performing an inference and the accuracy of the inferred result are presented an axiomatization of the notion of the complexity of inductive inference is developed and several results are presented which both resemble and contrast with results obtainable for the axiomatic computational complexity of recursive functions

a paradigm for class identification problems the following problem arises in many applications involving classification identification and inference there is a set of objects x and a particular x spl isin x is chosen unknown to us based on information obtained about x in a sequential manner one wishes to decide whether x belongs to one class of objects a sub 0 or a different class of objects a sub 1 the authors study a general paradigm applicable to a broad range of problems of this type which they refer to as problems of class identification or discernibility they consider various types of information sequences and various success criteria including discernibility in the limit discernibility with a stopping criterion uniform discernibility and discernibility in the cesaro sense they consider decision rules both with and without memory necessary and sufficient conditions for discernibility are provided for each case in terms of separability conditions on the sets a sub 0 and a sub 1 they then show that for any sets a sub 0 and a sub 1 various types of separability can be achieved by allowing failure on appropriate sets of small measure applications to problems in language identification system identification and discrete geometry are discussed

complexity issues for vacillatory function identification it was previously shown by barzdin and podnieks that one does not increase the power of learning programs for functions by allowing learning algorithms to converge to a finite set of correct programs instead of requiring them to converge to a single correct program in this paper we define some new subtle but natural concepts of mind change complexity for function learning and show that if one bounds this complexity for learning algorithms then by contrast with barzdin and podnieks result there are interesting and sometimes complicated trade offs between these complexity bounds bounds on the number of final correct programs and learning power

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

the synthesis of language learners abstract an index for an r e class of languages by definition is a procedure which generates a sequence of grammars defining the class an index for an indexed family of languages by definition is a procedure which generates a sequence of decision procedures defining the family studied is the metaproblem of synthesizing from indices for r e classes and for indexed families of languages various kinds of language learners for the corresponding classes or families indexed many positive results as well as some negative results are presented regarding the existence of such synthesizers the negative results essentially provide lower bounds for the positive results the proofs of some of the positive results yield as pleasant corollaries subset principle or tell tale style characterizations for the learnability of the corresponding classes or families indexed for example the indexed families of recursive languages that can be behaviorally correctly identified from positive data are surprisingly characterized by angluin s condition 2 the subset principle for circumventing overgeneralization

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

learning by switching type of information the present work is dedicated to the study of modes of data presentation in the range between text and informant within the framework of inductive inference in this study the learner alternatingly requests sequences of positive and negative data we define various formalizations of valid data presentations in such a scenario we resolve the relationships between these different formalizations and show that one of these is equivalent to learning from informant we also show a hierarchy formed for each of the formalizations studied by considering the number of switches between requests for positive and negative data

hypothesis spaces for learning in this paper we survey some results in inductive inference showing how learnability of a class of languages may depend on the hypothesis space chosen additionally optimal hypothesis spaces usable for every learnable class are considered we also discuss results which consider how learnability is effected if one requires learning using every suitable hypothesis space

effectivity questions for kleene s recursion theorem abstract the present paper investigates the quality of numberings measured in three different ways a the complexity of finding witnesses of kleene s recursion theorem in the numbering b for which learning notions from inductive inference the numbering is an optimal hypothesis space c the complexity needed to translate the indices of other numberings to those of the given one in all three cases one assumes that the corresponding witnesses or correct hypotheses are found in the limit and one measures the complexity with respect to the best criterion of convergence which can be achieved the convergence criteria considered are those of finite explanatory vacillatory and behaviourally correct convergence the main finding is that the complexity of finding witnesses for kleene s recursion theorem and the optimality for learning are independent of each other furthermore if the numbering is optimal for explanatory learning and also allows to solve kleene s recursion theorem with respect to explanatory convergence then it also allows to translate indices of other numberings with respect to explanatory convergence

anomalous learning helps succinctness abstract it is shown that allowing a bounded number of anomalies mistakes in the final programs learned by an algorithmic procedure can considerably succinctify those final programs naturally only those contexts are investigated in which the presence of anomalies is not actually required for successful inference learning the contexts considered are certain infinite subclasses of the class of characteristic functions of finite sets for each finite set d these subclasses have a finite set containing d this latter prevents the anomalies from wiping out all the information in the sets featured in these subclasses and shows the context to be fairly robust some of the results in the present paper are shown to be provably more constructive than others the results of this paper can also be interpreted as facts about succinctness of coding finite sets which facts have interesting consequences for learnability of decision procedures for finite sets

on duality in learning and the selection of learning teams previous work in inductive inference dealt mostly with finding one or several machines iims that successfully learn collections of functions herein we start with a class of functions and considerthe learner setof all iims that are successful at learning the given class applying this perspective to the case of team inference leads to the notion ofdiversificationfor a class of functions this enable us to distinguish between several flavours of iims all of which must be represented in a team learning the given class

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

learning read once formulas with queries a read once formula is a boolean formula in which each variable occurs at most once such formulas are also called formulas or boolean trees this paper treats the problem of exactly identifying an unknown read once formula using specific kinds of queries the main results are a polynomial time algorithm for exact identification of monotone read once formulas using only membership queries and a polynomial time algorithm for exact identification of general read once formulas using equivalence and membership queries a protocol based on the notion of a minimally adequate teacher 1 the results of the authors improve on valiant s previous results for read once formulas 26 it is also shown that no polynomial time algorithm using only membership queries or only equivalence queries can exactly identify all read once formulas

kolmogorov numberings and minimal identification abstract identification of programs for computable functions from their graphs by algorithmic devices is a well studied problem in learning theory freivalds and chen consider identification of minimal and nearly minimal programs for functions from their graphs to address certain problems in minimal identification for godel numberings freivalds later considered minimal identification in kolmogorov numberings kolmogorov numberings are in some sense optimal numberings and have some nice properties we prove certain separation results for minimal identification in every kolmogorov numbering in addition we also compare minimal identification in godel numberings versus minimal identification in kolmogorov numberings

the power of vacillation in language learning some extensions are considered of gold s influential model of language learning by machine from positive data studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars the main theorem of this paper is that there are classes of languages that can be learned if convergence in the limit to up to n 1 exactly correct grammars is allowed but which cannot be learned if convergence in the limit is to no more than n grammars where the no more than n grammars can each make finitely many mistakes this contrasts sharply with results of barzdin and podnieks and later case and smith for learnability from both positive and negative data a subset principle from a 1980 paper of angluin is extended to the vacillatory and other criteria of this paper this principle provides a necessary condition for avoiding overgeneralization in learning from positive data it is applied to prove another theorem to the effect that one can optimally eliminate half of the mistakes from final programs for vacillatory criteria if one is willing to converge in the limit to infinitely many different programs instead child language learning may be sensitive to the order or timing of data presentation it is shown though that for the vacillatory success criteria of this paper there is no loss of learning power for machines which are insensitive to order in several ways simultaneously for example partly set driven machines attend only to the set and length of sequence of positive data not the actual sequence itself a machine m is weakly n ary order independent stackrel rm def leftrightarrow for each language l on which for some ordering of the positive data about l m converges in the limit to a finite set of grammars there is a finite set of grammars d of cardinality leq n such that m converges to a subset of this same d for each ordering of the positive data for l the theorem most difficult to prove in the paper implies that machines which are simultaneously partly set driven and weakly n ary order independent do not lose learning power for converging in the limit to up to n grammars several variants of this theorem are obtained by modifying its proof and some of these variants have application in this and other papers along the way it is also shown for the vacillatory criteria that learning power is not increased if one restricts the sequence of positive data presentation to be computable some of these results are nontrivial lifts of prior work for the n 1 case done by the blums wiehagen osherson stob and weinstein schafer and fulk

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

learning languages from positive data and a limited number of short counterexamples we consider two variants of a model for learning languages in the limit from positive data and a limited number of short negative counterexamples counterexamples are considered to be short if they are smaller than the largest element of input seen so far negative counterexamples to a conjecture are examples which belong to the conjectured language but do not belong to the input language within this framework we explore how when learners using n short arbitrary negative counterexamples can be simulated or simulate using least short counterexamples or just no answers from a teacher we also study how a limited number of short counterexamples fairs against unconstrained counterexamples and also compare their capabilities with the data that can be obtained from subset superset and equivalence queries possibly with counterexamples a surprising result is that just one short counterexample can sometimes be more useful than any bounded number of counterexamples of arbitrary sizes most of the results exhibit salient examples of languages learnable or not learnable within corresponding variants of our models

on the power of recursive optimizers problems of the effective synthesis of fastest programs modulo a recursive factor for recursive functions given by input output examples or an arbitrary program are investigated in contrast to the non existence result proved by alton 1974 1976 we show various existence results thereby we deal in detail with the influence of the recursive factor in dependence of the concrete formalization of a fastest program in particular we shall show that even for function classes containing arbitrarily complex functions the effective synthesis of fastest programs modulo a simple recursive operator can be achieved sometimes

on the intrinsic complexity of learning recursive functions the intrinsic complexity of learning compares the difficulty of learning classes of objects by using some reducibility notion for several types of learning recursive functions both natural complete classes are exhibited and necessary and sufficient conditions for completeness are derived informally a class is complete iff both its topological structure is highly complex while its algorithmic structure is easy some self describing classes turn out to be complete furthermore the structure of the intrinsic complexity is shown to be much richer than the structure of the mind change complexity though in general intrinsic complexity and mind change complexity can behave orthogonally

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

probabilistic inductive inference inductive inference machines construct programs for total recursive functions given only example values of the functions probabilistic inductive inference machines are defined and for various criteria of successful inference it is asked whether a probabilistic inductive inference machine can infer larger classes of functions if the inference criterion is relaxed to allow inference with probability at least p 0 p ex and bc it is shown that any class of functions that can be inferred from examples with probability exceeding 1 2 can be inferred deterministically and that for probabilities p 1 2 there is a discrete hierarchy of inferability parameterized by p the power of probabilistic inference strategies is characterized by equating the classes of probabilistically inferable functions with those classes that can be inferred by teams of inductive inference machines a parallel model of inference or by a third model called frequency inference