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

a new approach to the theory of functional integral equations of fractional order abstract the aim of the paper is to present a new approach to the theory of functional integral equations of fractional order that approach depends on converting of the mentioned equations to the form of functional integral equations of volterra stieltjes type it turns out that the study of functional integral equations of volterra stieltjes type is more convenient and effective than the study of functional integral equations of fractional order an example illustrating our approach is also discussed

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

extension theorems orbits and automorphisms of the computably enumerable sets we prove an algebraic extension theorem for the computably enumerable sets e using this extension theorem and other work we then show if a and are automorphic via then they are automorphic via a where l a and e a is 0 3 we give an algebraic description of when an arbitrary set is in the orbit of a computably enumerable set a we construct the first example of a definable orbit which is not a 0 3 orbit we conclude with some results which restrict the ways one can increase the complexity of orbits for example we show that if a is simple and is in the same orbit as a then they are in the same 0 6 orbit and furthermore we provide a classification of when two simple sets are in the same orbit

new results on a continuously differentiable exact penalty function the main motivation of this paper is to weaken the conditions that imply the correspondence between the solution of a constrained problem and the unconstrained minimization of a continuously differentiable function in particular a new continuously differentiable exact penalty function is proposed for the solution of nonlinear programming problems under mild assumptions a complete equivalence can be established between the solution of the original constrained problem and the unconstrained minimization of this penalty function on a perturbation of the feasible set this new penalty function and its exactness properties allow us to define globally and superlinearly convergent algorithms to solve nonlinear programming problems as an example a newton type algorithm is described which converges locally in one iteration in case of quadratic programming problems

a continuously differentiable exact penalty function for nonlinear programming problems with inequality constraints in this paper it is shown that given a nonlinear programming problem with inequality constraints it is possible to construct a continuously differentiable exact penalty function whose global or local unconstrained minimizers correspond to global or local solutions of the constrained problem

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

iteration complexity of first order augmented lagrangian methods for convex programming this paper considers a special class of convex programming cp problems whose feasible regions consist of a simple compact convex set intersected with an affine manifold we present first order methods for this class of problems based on an inexact version of the classical augmented lagrangian al approach where the subproblems are approximately solved by means of nesterov s optimal method we then establish a bound on the total number of nesterov s optimal iterations i e the inner iterations performed throughout the entire inexact al method to obtain a near primal dual optimal solution we also present variants with possibly better iteration complexity bounds than the original inexact al method which consist of applying the original approach directly to a perturbed problem obtained by adding a strongly convex component to the objective function of the cp problem

differentiable exact penalty functions for nonlinear second order cone programs we propose a method for solving nonlinear second order cone programs socps based on a continuously differentiable exact penalty function the construction of the penalty function is given by incorporating a multipliers estimate in the augmented lagrangian for socps under the nondegeneracy assumption and the strong second order sufficient condition we show that a generalized newton method has global and superlinear convergence we also present some preliminary numerical experiments

on an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems in this paper we study a new exact and smooth penalty function for the nonlinear mixed discrete programming problem by augumenting only one variable no matter how many constraints through the smooth and exact penalty function we can transform the nonlinear mixed discrete programming problem into an unconstrained optimization model we demonstrate that under mild conditions when the penalty parameter is sufficiently large optimizers of this penalty function are precisely the optimizers of the nonlinear mixed discrete programming problem alternatively under some mild assumptions the local exactness property is also presented the numerical results demonstrate that the new penalty function is an effective and promising approach as important applications we solve an increasingly popular search engine advertising problem via the new proposed penalty function

simulating alternating tree automata by nondeterministic automata new results and new proofs of the theorems of rabin mcnaughton and safra abstract we give a proof that alternating tree automata can be simulated by nondeterministic tree automata which yields new complexity results and a unified proof of the theorems of rabin mcnaughton and safra we also give a simple axiomatic framework for uniformizing strategies

computational complexity of recursively enumerable sets we survey a variety of recent notions and results for classifying the computational complexity of a recursively enumerable r e set these complexity theoretic notions are shown to be equivalent to various recursion theoretic notions and are used to relate the complexity properties of an r e set a to its algebraic structure in the appropriate lattice and to the information it encodes

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

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

a robust sequential quadratic programming method the sequential quadratic programming method developed by wilson han and powell may fail if the quadratic programming subproblems become infeasible or if the associated sequence of search directions is unbounded this paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems furthermore questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed

a sequential quadratic programming method for potentially infeasible mathematical programs abstract the paper describes a modification to the wilson han powell sequential quadratic programming technique that is applicable to potentially infeasible nonlinear programs the method is designed to locate stationary points of a nonlinear program in a generalized sense definition 2 8 these stationary points include the familiar kuhn tucker and fritz john points but also include what we call external and strong external stationary points the external and strong external stationary points are not feasible thus the problem may be infeasible and yet have stationary points loosely speaking an external stationary point is one that satisfies a certain first order necessary optimality condition for being as close to feasibility as possible the proposed technique is similar to those described by j v burke and s p han a modified sequential quadratic programming method manuscript and m sahba j optim theory appl 52 1987 291 309 however it is shown to possess substantially enhanced stability properties the global convergence properties of the method are described along with some continuity results for the quadratic programming subproblems

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

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

coding in the partial order of enumerable sets abstract we develop methods for coding with first order formulas into the partial order e of enumerable sets under inclusion first we use them to reprove and generalize the unpublished result of the first author that the elementary theory of e has the same computational complexity as the theory of the natural numbers relativized versions of the coding methods show that the p o of 0 p and 0 q sets are not elementarily equivalent for natural numbers p q as a further application definability of the class of quasimaximal sets in e is obtained on the other side we prove theorems limiting coding and definability in e thereby establishing a sharp contrast between e and other structures occurring in computability theory

isomorphism of lattices of recursively enumerable sets let w 0 1 2 and for a c w let ea be the lattice of subsets of w which are recursively enumerable relative to the oracle a let ea be ea where i is the ideal of finite subsets of w it is established that for any a b c w ea is effectively isomorphic to eb if and only if a t b where a is the turing jump of a a consequence is that if a t b then ea eb a second consequence is that ea can be effectively embedded into eb preserving least and greatest elements if and only if a t b

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

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

a linesearch based derivative free approach for nonsmooth constrained optimization in this paper we propose new linesearch based methods for nonsmooth constrained optimization problems when first order information on the problem functions is not available in the first part we describe a general framework for bound constrained problems and analyze its convergence toward stationary points using the clarke jahn directional derivative in the second part we consider inequality constrained optimization problems where both objective function and constraints can possibly be nonsmooth in this case we first split the constraints into two subsets difficult general nonlinear constraints and simple bound constraints on the variables then we use an exact penalty function to tackle the difficult constraints and we prove that the original problem can be reformulated as the bound constrained minimization of the proposed exact penalty function finally we use the framework developed for the bound constrained case to solve the penalized problem moreover we prove that every accumulation poin

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

invariance in boldsymbol mathcal e and boldsymbol mathcal e _ pi we define g a substructure of e the lattice of 0 1 classes and show that a quotient structure of g g is isomorphic to e the result builds on the 0 3 isomorphism machinery and allows us to transfer invariant classes from e to e though not in general orbits further properties of g and ramifications of the isomorphism are explored including degrees of equivalence classes and degree invariance

an augmented lagrangian function with improved exactness properties in this paper we introduce a new exact augmented lagrangian function for the solution of general nonlinear programming problems for this lagrangian function a complete equivalence between its unconstrained minimization on an open set and the solution of the original constrained problem can be established under mild assumptions and without requiring the boundedness of the feasible set of the constrained problem moreover we describe an unconstrained algorithmic model which is globally convergent toward kkt pairs of the original constrained problem the algorithmic model can be endowed with a superlinear rate of convergence by a proper choice of the search direction in the unconstrained minimization without requiring strict complementarity

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

some connections between bounded query classes and non uniform complexity let a x be the characteristic function of a consider the function cka x1 xk a x1 a xk we show that if cka can be computed in polynomial time with fewer than k queries to some set x then a p poly a generalization of this result has applications to bounded query classes circuits and enumerability in particular we obtain the following 1 assuming 3p 3p there are functions computable using f n 1 queries to sat that are not computable using f n queries to sat for f n o log n 2 if cka restricted to length n inputs can be computed by an unbounded fanin oracle circuit of size s n and depth d n with k 1 queries to some set x then a can be computed with an unbounded fanin non oracle circuit of size no k s n and depth d n o 1 3 assuming that ph 4p 4p and e 1 sat is not 2ne enumerable

exact penalty functions in constrained optimization in this paper formal definitions of exactness for penalty functions are introduced and sufficient conditions for a penalty function to be exact according to these definitions are stated thus providing a unified framework for the study of both nondifferentiable and continuously differentiable penalty functions in this framework the best known classes of exact penalty functions are analyzed and new results are established concerning the correspondence between the solutions of the constrained problem and the unconstrained minimizers of the penalty functions

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

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

post s program and incomplete recursively enumerable sets abstract a set a of nonnegative integers is recursively enumerable r e if a can be computably listed it is shown that there is a first order property q x definable in e the lattice of r e sets under inclusion such that i if a is any r e set satisfying q a then a is nonrecursive and turing incomplete and ii there exists an r e set a satisfying q a this resolves a long open question stemming from post s program of 1944 and it sheds light on the fundamental problem of the relationship between the algebraic structure of an r e set a and the turing degree of information that a encodes

solvability of volterra stieltjes operator integral equations and their applications abstract we investigate a class of operator integral equations of volterra stieltjes type and we study the solvability of those equations in the space of continuous functions equations in question create a generalization of numerous integral equations considered in nonlinear analysis the main tool used in our considerations is the technique associated with measures of noncompactness we show the applicability of our existence result in the study of a few integral equations of volterra type

the global convergence of augmented lagrangian methods based on ncp function in constrained nonconvex optimization abstract in this paper we present the global convergence properties of the primal dual method using a class of augmented lagrangian functions based on ncp function for inequality constrained nonconvex optimization problems we construct four modified augmented lagrangian methods based on different algorithmic strategies we show that the convergence to a kkt point or a degenerate point of the original problem can be ensured without requiring the boundedness condition of the multiplier sequence

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

serial and parallel computation of karush kuhn tucker points via nonsmooth equations in this paper a robust iterative algorithm is presented for computing karush kuhn tucker kkt points of nonlinear programs this algorithm is a variation of the ne sqp method for solving the nonlinear complementarily problem it makes use of a merit function that combines the original objective function of the nonlinear program with a residual function of the kkt conditions formulated as a system of nonsmooth equations global convergence and a q quadratic rate of convergence of the algorithm are established under some standard constraint qualifications in nonlinear programming theory but without the strict complementarily condition parallel implementations of the algorithm are discussed

the technique of volterra stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders our study in this paper concerns the infinite systems of nonlinear integral equations of volterra stieltjes type we show that some assumptions imposed on terms of the mentioned systems guarantee the solvability of those systems in the space c i c 0 with i 0 1 and c 0 being the classical sequence space as a particular and very important case of the investigated systems of nonlinear volterra stieltjes integral equations we obtain infinite systems of nonlinear integral equations of fractional orders thus the approach applied in the paper enables us to obtain rather general results concerning the existence of solutions of infinite systems of nonlinear fractional integral equations

characterization of recursively enumerable sets with supersets effectively isomorphic to all recursively enumerable sets we show that the lattice of supersets of a recursively enumerable r e set a is effectively isomorphic to the lattice of all r e sets if and only if the complement a of a is infinite and e wie n a finite i 0 i e a is semilow 5 it is obvious that the condition a semilow 5 is necessary for the other direction a certain uniform splitting property the outer splitting property is derived from semilow 5 and this property is used in an extension of soare s automorphism machinery for the construction of the effective isomorphism since this automorphism machinery is quite complicated we give a simplified proof of soare s extension theorem before we add new features to this argument

automorphisms of the lattice of recursively enumerable sets promptly simple sets we show that for every coinfinite r e set a there is a complete r e set b such that l a eff l b and that every promptiy simple set is automorphic in e to a complete set

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

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

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

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

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

some properties of nonlinear volterra stieltjes integral operators we study some properties of the nonlinear volterra stieltjes integral operator of general form which is defined with help of the stieltjes integral with the kernel depending on two variables the results obtained in the paper generalize those obtained earlier for the volterra stieltjes integral operators having less general form we prove also an existence result for nonlinear integral equation of volterra stieltjes type

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