a class of elliptic partial differential equations with exponential nonlinearities on demontre l existence d au moins une infinite denombrable de familles a 1 parametre de solutions non triviales continues de holder pour une classe de problemes aux valeurs limites de dirichlet d ordre 2 fortement non lineaires a 2 dimensions et plus

distortion theorems of plane quasiconformal mappings abstract the authors prove a conjecture on elliptic integrals and obtain sharp bounds for k r and k by using teichmuller s module theorem the authors obtain a distortion theorem of k quasiconformal mappings on the plane

strong jump traceability i the computably enumerable case abstract recent investigations in algorithmic randomness have lead to the discovery and analysis of the fundamental class k of reals called the k trivial reals defined as those whose initial segment complexity is identical with that of the sequence of all 1 s there remain many important open questions concerning this class such as whether there is a combinatorial characterization of the class and whether it coincides with possibly smaller subclasses such as the class of reals which are not sufficiently powerful as oracles to cup a turing incomplete martin lof random real to the halting problem hidden here is the question of whether there exist proper natural subclasses of k we show that the combinatorial class of computably enumerable strongly jump traceable reals defined via the jump operator by figueira nies and stephan santiago figueira andre nies frank stephan lowness properties and approximations of the jump electr notes theor comput sci 143 2006 45 57 is such a class and show that like k it is an ideal in the computably enumerable degrees this is the first example of a class of reals defined by a cost function construction which forms a proper subclass of k further we show that every c e strongly jump traceable set is not martin lof cuppable thus giving a combinatorial property which implies non ml cuppability

presentations of computably enumerable reals we study the relationship between a computably enumerable real and its presentations ways of approximating the real by enumerating a prefix free set of binary strings

on computational complexity and honest polynomial degrees abstract it is shown that no 20 set a with a and a both semilow has minimal honest polynomial hp degree the technique extends to show that the hp degrees and polynomial degrees of low r e sets are dense in particular r e sets of minimal hp degree must have high computational complexity in the sense of blum and marques j symbolic logic 38 1973 579 593 these results also hold for tally degrees and polynomial honest m degrees

the degrees of r e sets without the universal splitting property it is shown that every nonzero r e degree contains an r e set without the universal splitting property that is if 8 is any r e nonzero degree there exist r e sets 0 tb ta with deg a 8 such that if ao u al is an r e splitting of a then ao t b some generalizations are discussed

algebraic structures computable without delay abstract in this article we suggest a new systematic approach to studying algorithms on algebraic structures via primitive recursion the approach is designed to fill the gap between abstract computable structure theory and feasible in the sense of polynomial time computational or automatic algebra

limiting partial combinatory algebras from every partial combinatory algebra a we construct another partial combinatory algebra a lim a in a lim a every representable partial numerical function n is exactly of the form limt t n for some representable partial numerical function t n of a the partial combinatory algebra a lim a is a quotient of a by a partial equivalence relation and is equipped with a limit structure in the sense that each element of a lim a is the limit of a countable sequence of a elements in this paper we discuss the limit structures for a in terms of barendregt s range property if the range of a combinator is finite then it is a singleton moreover we repeat the construction a lim transfinite times to interpret infinitary calculi finally we attempt to interpret type free calculus by introducing another partial applicative structure which has an asynchronous application operator that allows a parallel limit operation

the friedman embedding theorem abstract in this paper we will present an explicit construction of harvey friedman which to every finitely generated group g associates a 2 generator subgroup k g sym n such that g embeds into k g and such that if g h then k g k h

beyond recursive real functions all recursive real functions are continuous in fact all theb recursive real functions are continuous for any oracleb simply because turing machines computing them are finite objects but simple discontinuous functions such as step functions have to be in some sense easy if they have recursive values and break points although they are not computable by the usual definition it seems unfair to label them noncomputable in the entire region just because of a few break points in this paper we investigate the properties of broader classes ofalmost everywhere recursive weakly almost everywhere recursive andrecursively approximablereal valued functions which capture these easy step functions and many other nonrecursive functions recursive versions of the classical lusin and egoroff theorems are proved we also characterize the property of the limit of a recursive sequence of functions and show that different notions of convergence uniform pointwise or in measure will result in different characterizations of the limiting function

oracle dependent properties of the lattice of np sets abstract we consider under the assumption p np questions concerning the structure of the lattice of np sets together with the sublattice p we show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize the two questions which we consider are whether every infinite np set contains an infinite p subset and whether there exists an np simple set we construct several oracles all of which make p np and which in addition make the above mentioned statements either true or false in particular we give a positive answer to the question raised by bennett and gill 1981 whether an oracle b exists making p b np b and such that every infinite set in np b has an infinite subset in p b the constructions of the oracles are finite injury priority arguments

dimension free quasiconformal distortion in n space most distortion theorems for k quasiconformal mappings in rn n 2 depend on both n and k in an essential way with bounds that become infinite as n tends to oo the present authors obtain dimension free versions of four well known distortion theorems for quasiconformal mappings namely bounds for the linear dilatation the schwarz lemma the e distortion theorem and the r1 quasisymmetry property of these mappings they show that the upper estimates they have obtained in each of these four main results remain bounded as n tends to oo with k fixed the proofs are based on a dimensioncancellation property of the function t h r 1 r t k t 0 k 0 where r t is the capacity of a teichmiiller extremal ring in rn the authors also prove a dimension free distortion theorem for the absolute cross ratio under k quasiconformal mappings of n from which several other distortion theorems follow as special cases

definable sets in ordered structures i this paper introduces and begins the study of a well behaved class of linearly ordered structures the minimal structures the definition of this class and the corresponding class of theories the strongly dlminlmal theories is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories theorems 2 1 and 2 3 respectively provide characterizations of 41 minimal ordered groups and rings several other simple results are collected in 3 the primary tool in the analysis of minimal structures is a strong analogue of ii forking symmetry given by theorem 4 2 this result states that any parametri cally definable unary function in an yminimal structure is piecewise either constant or an ore er preserving or reversing bijection of intervals the results that follow include the existence and uniqueness of prime models over sets theorem 5 1 and a characterization of all bto categorical aminimal structures theorem 6 1 1 introduction the class of linearly ordered structures has long been an im portant subject of concern to model theorists impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order among those theories that have been approached successfully are peano arithmetic the theory of ordered abelian groups that of real closed fields and that of linear order itself yet very little has been done in the way of developing a general model theory for ordered structures in this paper we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another let l be a finitary first order language and w an l structure a set of n tuples a c x is said to be parametrically definable if there is some l formula

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

smooth manifolds for semilinear wave equations on r2 on the existence of almost periodic breathers abstract in this paper we prove several nonexistence and existence results for certain real solutions to semilinear wave equations in one space dimension these solutions represent time almost periodic free vibrations which decay toward a constant equilibrium solution as the spatial variable goes to infinity our nonexistence results are of two kinds we first prove that the requirements of almost periodicity in time and of spatial decay at infinity force such solutions to be trivial whenever their spectrum does not interact with the nonlinearity in a certain sense we also prove non existence results without requiring any spectral conditions but this is at the expense of having to impose more stringent conditions on the nonlinearities such as certain convexity properties in the vicinity of the equilibrium solution finally we prove an existence result for time almost periodic free vibrations whose profiles decay exponentially rapidly toward a constant equilibrium solution to accomplish this we convert the wave equation into a dynamical system in which the original spatial variable plays the role of time we then embed that dynamical system into an appropriate banach space of time almost periodic functions so that the one parameter family of stable and unstable manifolds which we construct carry the solutions that we seek the major difficulty to overcome in our construction is a small divisor problem it is related to the fact that the spectrum of the infinitesimal generator for the linearized flow is a pure point spectrum without gaps we discuss several examples and we also stress some important qualitative differences which distinguish the almost periodic case from the purely periodic one

automorphisms of the lattice of recursively enumerable sets orbits an important program in the study of the structure of 8 the lattice of r e sets is the classilication of the orbits under aut j the automorphism group of 8 obviously the class of linite sets of a given cardinality and the class of inllnite zoinlmite recursive sets form orbits but other than these relatively few orbits are known harrington has shown that the creative sets form an orbit the most important result in the area is still soare s proof sol that maximal sets sets which are coatoms in 8 modulo the ideal of llnite sets form an orbit the proof of this result contains an intricate and powerful construction which has since been used to construct automorphisms by maass ml maass and stob mst and stob st it remains the most powerful method of constructing automorphisms the principal result of this paper is the existence of a new orbit the hemimaximal sets a r e set h is hemimaximal if there are a maximal set 44 and disjoint r e sets m m1 such that j40 u m1 44 each mi is non recursive and il4 h in section 1 we prove that the hemimaximal sets form an orbit by extending soare s proof for maximal sets and detail the properties of splittings of maximal sets which led us to this result in sec tion 2 we study the degrees of hemimaximal sets all degrees mentioned in this paper are r e we show that hemimaximal sets exist in every high degree and that for every degree a 0 there is a hemimaximal set h such

on the orbits of computably enumerable sets received by the editors april 6 2007 2000 mathematics subject classification primary 03d25 the first author s research was partially supported by nsf grants dms 96 34565 99 88716 02 45167 the second author s research was partially supported by the marsden fund of new zealand some of the work involved was done partially while the first and second authors were visiting the institute for mathematical sciences national university of singapore in 2005 these visits were supported by the institute the third author s research was partially supported by dms 96 22290 and dms 99 71137

reducibilities among equivalence relations induced by recursively enumerable structures in this paper we investigate the dependence of recursively enumerable structures on the equality relation which is fixed to a specific r e equivalence relation we compare r e equivalence relations on the natural numbers with respect to the amount of structures they permit to represent from a given class of structures such as algebras permutations and linear orders in particular we show that for various types of structures represented there are minimal and maximal elements

independence in computable algebra abstract we give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis we apply the condition to differentially closed real closed and difference closed fields with the relevant notions of independence to cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind we will then apply our techniques to derive new corollaries on the number of computable presentations of these structures the condition also implies classical and new results on vector spaces algebraically closed fields torsion free abelian groups and archimedean ordered abelian groups

hyper polynomial hierarchies and the polynomial jump abstract assuming that the polynomial hierarchy ph does not collapse we show the existence of ascending sequences of ptime turing degrees of length 1 ck in pspace such that successors are polynomial jumps of their predecessors moreover these ptime degrees are all uniformly hard for ph this is analogous to the hyperarithmetic hierarchy which is defined similarly but with the computable turing degrees the lack of uniform least upper bounds for ascending sequences of ptime degrees causes the limit levels of our hyper polynomial hierarchy to be inherently non canonical this problem is investigated in depth and various possible structures for hyper polynomial hierarchies are explicated as are properties of the polynomial jump operator on the languages which are in pspace but not in ph

quantifying the amount of verboseness we study the fine structure of the classification of sets of natural numbers a according to the number of queries which are needed to compute the n fold characteristic function of a a complete characterization is obtained relating the question to finite combinatorics in order to obtain an explicit description we consider several interesting combinatorial problems

the computable dimension of ordered abelian groups abstract let g be a computable ordered abelian group we show that the computable dimension of g is either 1 or that g is computably categorical if and only if it has finite rank and that if g has only finitely many archimedean classes then g has a computable presentation which admits a computable basis

index sets in computable analysis 01 classes in a space x where x equals 0 1 0 1 or the real line real are given an effective enumeration pe x and the computably continuous functions are given an effective enumeration fe x the notion of index sets associated with 01 classes and with computably continuous functions is developed the complexity of various problems of analysis is determined by the complexity of the associated index set

fa presentable groups and rings we consider structures which are fa presentable it is known that an fa presentable finitely generated group is virtually abelian we strengthen this result by showing that an arbitrary fa presentable group is locally virtually abelian as a consequence we prove that any fa presentable ring is locally finite this is a significant restriction and allows us to say a great deal about the structure of fa presentable rings in particular we show that any fa presentable ring with identity and no zero divisors is finite

relatively recursive reals and real functions abstract intuitively a real number is recursive if we can get as accurate an approximation as we like using a mechanical procedure such as a turing machine a real function is recursive if its value at a point x in its domain can be approximated effectively given an approximation to x however since there are only countably many turing machines there must be uncountably many non recursive reals and functions in this paper we study some of these non recursive reals and functions using a more recursion theoretic approach via the degree of unsolvability in particular we are interested in reals and real functions that are relatively recursive in where is the jump of the recursive degree inspired by the shoenfield limit lemma we show that a real is recursive if and only if it is the limit of a recursive sequence of rationals we then give three characterizations of a recursive function which are stated in terms of turing machine uniform convergence and sequential computability with uniform continuity a proof of their equivalence using a finite injury priority argument is given with the new definitions we can now give an upper bound to the difficulty of some uncomputable analysis operator such as differentiations and root findings

categoricity properties for computable algebraic fields we examine categoricity issues for computable algebraic fields such fields behave nicely for computable dimension we show that they cannot have finite computable dimension greater than 1 however they behave less nicely with regard to relative computable categoricity we give a structural criterion for relative computable categoricity of these fields and use it to construct a field that is computably categorical but not relatively computably categorical finally we show that computable categoricity for this class of fields is 0 complete

ideals in computable rings abstract we show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to wkl 0 over rca 0 and that the existence of a nontrivial proper finitely generated ideal in a commutative ring with identity which is not a field is equivalent to aca 0 over rca 0 we also prove that there are computable commutative rings with identity where the nilradical is 1 0 complete and the jacobson radical is 2 0 complete respectively

theory of interaction abstract theory of interaction aims to provide a foundational framework for computation and interaction it proposes four fundamental principles that characterize the common features of all models of computation and interaction these principles suffice to support a model independent treatment of the two most important relationships in computer science the equality between processes and the relative expressiveness between models based on the two relationships the theory of equality the theory of expressiveness and the theory of completeness are developed

online presentations of finitely generated structures abstract we systematically investigate into the online content of finitely generated structures the online content of a potentially infinite algebraic or combinatorial structure is perhaps best reflected by its pr degrees to be defined we confirm a natural conjecture by showing that the pr degrees of a finitely generated structure must be dense remarkably we show that pr degrees of an f g structure do not have to be upwards dense as an application of our techniques we refute a natural conjecture about honestly generated structures to be stated

effectively categorical abelian groups abstract we study effective categoricity of computable abelian groups of the form i h where h is a subgroup of q such groups are called homogeneous completely decomposable it is well known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite we study n 0 categoricity in this class of groups for n 1 we introduce a new algebraic concept of s independence which is a generalization of the well known notion of p independence we develop the theory of s independent sets we apply these techniques to show that every homogeneous completely decomposable group is 3 0 categorical we prove that a homogeneous completely decomposable group of infinite rank is 2 0 categorical if and only if it is isomorphic to the free module over the localization of z by a computably enumerable set of primes p with the semi low complement within the set of all primes we apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers including the free abelian group

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

random elements in effective topological spaces with measure following a suggestion of zvonkin and levin we generalize martin lof s definition of infinite random sequences over a finite alphabet via randomness tests to effective topological spaces with a measure we show that under weak computability conditions there is a universal randomness test we prove a theorem on randomness preserving functions which corrects and extends a result by schnorr and apply it to a number of examples in particular we show that a real number is random if and only if it has a random b ary representation for any b 2 we show that many computable continuously differentiable real functions preserve randomness especially all computable analytic functions which are not constant on any open subset of their domain preserve randomness finally we introduce a new randomness concept for subsets of natural numbers which we characterize in terms of random sequences surprisingly it turns out that there are infinite co r e random sets

the computability definability and proof theory of artinian rings abstract we show that in the context of reverse mathematics wk l 0 weak konig s lemma implies the statement ar t 0 which says that every artinian ring is noetherian over rc a 0 recursive comprehension axiom to achieve this goal we prove a general computable full structure theorem for computable artinian rings similar to the classical version found in most algebra texts

a characterization of c e random reals a real is computably enumerable if it is the limit of a computable increasing converging sequence of rationals a real is random if its binary expansion is a random sequence our aim is to offer a self contained proof based on the papers calude et al in m morvan c meinel d krob eds proc 15th symp on theoretical aspects of computer science paris springer berlin 1998 pp 596 606 chaitin j assoc comput mach 22 1975 329 slaman manuscript 14 december 1998 2 pp solovay unpublished manuscript ibm thomas j watson research center yorktown heights new york may 1975 215 pp of the following theorem a real is c e and random if and only if it is a chaitino real i e the halting probability of some universal self delimiting turing machine

computing power of turing machines in the framework of unsharp quantum logic we present recursion theoretical characterizations of the computational power of turing machines in the framework of unsharp quantum logic for unsharp quantum logic let a lattice ordered quantum multiple valued qmv algebra be its truth value lattice when lattice ordered qmv algebras satisfy a locally finite condition that is every non zero element has a finite order and its meet operation is computable we prove that 1 0 1 0 l d t e or l w t e 2 0 where l d t e respectively l w t e denotes the class of languages accepted by these turing machines in the depth first method respectively the width first method for sharp quantum logic similar results are obtained for a general orthomodular truth value lattice

computing constraint sets for differential fields kronecker s theorem and rabin s theorem are fundamental results about computable elds f and the decidability of the set of irreducible polynomials overf we adapt these theorems to the setting of dierential elds k with constrained pairs of dierential polynomials over k assuming the role of the irreducible polynomials we prove that two of the three basic aspects of kronecker s theorem remain true here and that the reducibility in one direction but not the other from rabin s theorem also continues to hold

lowness properties and randomness abstract the set a is low for martin lof randomness if each random set is already random relative to a a is k trivial if the prefix complexity k of each initial segment of a is minimal namely n k a n k n o 1 we show that these classes coincide this answers a question of ambos spies and kucera in p cholak s lempp m lerman r shore eds computability theory and its applications current trends and open problems american mathematical society providence ri 2000 each low for martin lof random set is 2 0 our class induces a natural intermediate 3 0 ideal in the r e turing degrees which generates the whole class under downward closure answering a further question in p cholak s lempp m lerman r shore eds computability theory and its applications current trends and open problems american mathematical society providence ri 2000 we prove that each low for computably random set is computable

if not empty np p is topologically large abstract in the classical cantor topology or in the superset topology np and consequently classes included in np are meagre however in a natural combination of the two topologies we prove that np p if not empty is a second category class while np complete sets form a first category class these results are extended to different levels in the polynomial hierarchy and to the low and high hierarchies p immune sets in np np simple sets p bi immune sets and np effectively simple sets are all second category if not empty it is shown that if c is any of the above second category classes then for all b np there exists an a c such that a is arbitrarily close to b infinitely often

the rational points of a definable set let x r n be a set that is definable in an o minimal structure over r this article shows that in a suitable sense there are very few rational points of x which do not lie on some connected semialgebraic subset of x of positive dimension

refuting learning revisited we consider within the framework of inductive inference the concept of refuting learning as introduced by mukouchi and arikawa where the learner is not only required to learn all concepts in a given class but also has to explicitly refute concepts outside the class in the first part of the paper we consider learning from text and introduce a concept of limit refuting learning that is intermediate between refuting learning and reliable learning we give characterizations for these concepts and show some results about their relative strength and their relation to confident learning in the second part of the paper we consider learning from texts that for some k contain all positive k formulae that are valid in the standard structure determined by the set to be learned in this model the following results are shown for the language with successor any countable axiomatizable class can be limit refuting learned from 1 texts for the language with successor and order any countable axiomatizable class can be reliably learned from 1 texts and can be limit refuting learned from 2 texts whereas the axiomatizable class of all finite sets cannot be limit refuting learned from 1 texts for the full language of arithmetic which contains in addition plus and times for any even k there is an axiomatizable class that can be limit refuting learned from k 1 texts but not from k texts a similar result with k 3 in place of k 1 holds with respect to the language of presburger s arithmetic

bounds for quasiconformal distortion functions abstract several new inequalities are proved for the distortion function k r appearing in the quasiconformal schwarz lemma other related special functions are studied and applications are given to quasiconformal maps in the plane some open problems are solved too

the π theory of the computably enumerable turing degrees is undecidable we show the undecidability of the 3 theory of the partial order of enumerable turing degrees 0 introduction recursively enumerable henceforth called enumerable sets arise naturally in many areas of mathematics for instance in the study of elementary theories as solution sets of polynomials or as the word problems of finitely generated subgroups of finitely presented groups putting the enumerable sets into context with each other in various ways yields structures whose study has for long been a mainstay of computability theory if the sets are related in the most elementary way namely by inclusion one obtains a distributive lattice e with very complex algebraic properties another way to compare sets is to look at the information content turing reducibility is a very general but the most widely accepted concept of relative computability a setx of natural numbers is turing reducible to y iff the answer to n x can be determined by a turing machine computation which can use answers to oracle questions y y during the computation for more restricted notions of relative computability one would for instance place a priori 1991 mathematics subject classification 03d25 03d35

degrees of dowd type generic oracles for each positive integer r m dowd 1992 inform and comput 96 65 76 introduced r generic oracles we call them r dowd oracles they are different from n genericity of arithmetical forcing an oracle d is r dowd if every r query tautology with respect to d is forced by a polynomial sized portion of d we propose the study of degrees and complexity of 1 dowd oracles dowd also stated that no r dowd oracle is recursively enumerable however this is false we show among others the following there exists a primitive recursive 1 dowd oracle for every oracle a there exists a 1 dowd oracle d that is turing equivalent to a for every 1 dowd oracle d there exists a 1 dowd oracle e such that e is polynomial time many one equivalent to d and e is not 2 dowd problems are formulated and analogy between the jump operator and the operation of taking the set of 1 query tautologies is discussed

the complexity of computable categoricity we show that the index set complexity of the computably categorical structures is 11 complete demonstrating that computable categoricity has no simple syntactic characterization as a consequence of our proof we exhibit for every computable ordinal a computable structure that is computably categorical but not relatively 0 categorical

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

an improved zero one law for algorithmically random sequences abstract results on random oracles typically involve showing that a class ifx p x has lebesgue measure one i e that some property p x holds for almost every x a potentially more informative approach is to show that p x is true for every x in some explicitly defined class of random sequences or languages in this note we consider the algorithmically random sequences originally defined by martin lof and their generalizations the n random sequences our result is an effective form of the classical zero one law for each n 1 if a class ifx p x is closed under finite variation and has arithmetical complexity 0 n 1 or 0 n 1 roughly the property p can be expressed with n 1 alternations of quantifiers then either p holds for every n random sequence or else holds for none of them this result has been used by book and mayordomo to give new characterizations of complexity classes of the form almost or the languages which can be or reduced to almost every oracle where or is a reducibility

the isomorphism problem for torsion free abelian groups is analytic complete we prove that the isomorphism problem for torsion free abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy namely 1 complete

lower bounds and the hardness of counting properties rice s theorem states that all nontrivial language properties of recursively enumerable sets are undecidable borchert and stephan math logic quart 46 4 2000 489 504 started the search for complexity theoretic analogs of rice s theorem and proved that every nontrivial counting property of boolean circuits is up hard hemaspaandra and rothe theoret comput sci 244 1 2 2000 205 217 improved the up hardness lower bound to upo 1 hardness the present paper raises the lower bound for nontrivial counting properties from upo 1 hardness to fewp hardness i e from constant ambiguity nondeterminism to polynomial ambiguity nondeterminism furthermore we prove that no relativizable technique can raise this lower bound to fewp 1 ttp hardness we also prove a rice style theorem for np namely that every nontrivial language property of np sets is np hard and for a broad class of leaf language classes we prove a sufficient condition for the natural analog of rice s theorem to hold

on the complexity of radicals in noncommutative rings this article expands upon the recent work by downey et al 2007 3 who classified the complexity of the nilradical and jacobson radical of commutative rings in terms of the arithmetical hierarchy let r be a computable not necessarily commutative ring with identity then it follows from the definitions that the prime radical of r is 11 and the levitzki radical of r is 20 we show that these upper bounds for the complexity of the prime and levitzki radicals are optimal by constructing two noncommutative computable rings with identity such that the prime radical of one is 11 complete while the levitzki radical of the other is 20 complete

recursive linear orders with incomplete successivities a recursive linear order is said to have intrinsically complete successivities if in every recursive copy the succ essivities form a complete set we show theorem 1 that there is a recursive linear order with intrinsically complete successivities but theorem 2 that this cannot be a discrete linear oder we investigate the related issues of intrinsically non low and non semilow successivities in discrete linear orders we show also theorem 3 that no recursive linear order has intrinsically wtt complete successivities