infinitely often autoreducible sets a set a is autoreducible if one can compute for all x the value a x by querying a only at places y neq x furthermore a is infinitely often autoreducible if for infinitely many x the value a x can be computed by querying a only at places y neq x for all other x the computation outputs a special symbol to signal that the reduction is undefined it is shown that for polynomial time turing and truth table autoreducibility there are a b c in the class exp of all exponential time computable sets such that a is not infinitely often turing autoreducible b is turing autoreducible but not infinitely often truth table autoreducible and c is truth table autoreducible with g n 1 queries but not infinitely often turing autoreducible with g n queries here n is the length of the input g is nondecreasing and there exists a polynomial p such that p n bounds both the computation time and the value of g at input of length n furthermore connections between notions of infinitely often autoreducibility and notions of approximability are investigated the hausdorff dimension of the class of sets which are not infinitely often autoreducible is shown to be 1

stable subharmonic solutions in periodic reaction diffusion equations we construct an example of a t periodic semilinear reaction diffusion equation whose elliptic part is in divergence form for each t and which admits stable periodic solutions of higher minimal period thus there is no generic convergence to t periodic solutions this is in sharp contrast to the autonomous case where the existence of a lyapunov function implies quasiconvergence for all bounded solutions

a uniform approach to obtain diagonal sets in complexity classes abstract a uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given we obtain some known results as well as some new ones as corollaries of our main theorem the new results concern the complexity classes p np co np pspace apt almost polynomial time r random polynomial time and the polynomial hierarchy

the p t degrees of the recursive sets lattice embeddings extensions of embeddings and the two quantifier theory abstract ambos spies 1984a showed that the two basic nondistributive lattices can be embedded in rp t the polynomial time turing degrees of the recursive sets we introduce more general techniques to extend his results to show that every recursive lattice can be embedded in rp t in addition to lattice theoretic representation theorems we use the scheme of priority style arguments coupled with looking back techniques presented in shinoda and slaman 1988 1990 we also generalize the density type results of ladner 1975 and many others to settle the full extension of the embedding problem for rp t combined with the logical analysis of sentences with one alternation of quantifiers shore 1978 lerman 1983 these results suffice to decide the full theory of rp t they also give a strong nonhomogeneity result the p time degrees of the sets recursive in and if desired p time above two distinct sets a and b are almost never isomorphic the situation for the p time many one degrees is quite different we decide the extension of the embedding problem differently than for rp t but not the theory

kolmogorov complexity and degrees of tally sets we show that either epm tally epbtt tally or epm tally ep1 tt tally ep2 tt tally ep3 tt tally where erp tally denotes the class of sets which are equivalent to a tally set under rp reductions furthermore the question of whether or not emp tally ebttp tally is equivalent to the question of whether or not ne predicates can be solved in deterministic exponential time the proofs use the techniques of generalized kolmogorov complexity as corollaries to some of the main results we obtain new results about the kolmogorov complexity of sets in p

sublattices of the polynomial time degrees we show that any countable distributive lattice can be embedded in any interval of polynomial time degrees furthermore the embeddings can be chosen to preserve the least or the greatest element this holds for both polynomial time bounded many one and turing reducibilities as well as for all of the common intermediate reducibilities

a remark on stable subharmonic solutions of time periodic reaction diffusion equations abstract this paper is concerned with a time periodic reaction diffusion equation it is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation such solutions are called subharmonic solutions in this paper for any domain time period 0 and integer n 2 we construct an example of a time periodic reaction diffusion equation on with a minimal period which possesses a stable solution of minimal period n

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

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

resource bounded immunity and simplicity revisiting the 30 years old notions of resource bounded immunity and simplicity we investigate the structural characteristics of various immunity notions strong immunity almost immunity and hyperimmunity as well as their corresponding simplicity notions we also study limited immunity and simplicity called k immunity and feasible k immunity and their simplicity notions finally we propose the k immune hypothesis as a working hypothesis that guarantees the existence of simple sets in np

separation of np completeness notions we use hypotheses of structural complexity theory to separate various np completeness notions in particular we introduce an hypothesis from which we describe a set in np that is mbox leq rm p rm t complete but not mbox leq rm p tt complete we provide fairly thorough analyses of the hypotheses that we introduce

separating and collapsing results on the relativized probabilistic polynomial time hierarchy the probabilistic polynomial time hierarchy bph is the hierarchy generated by applying the bp operator to the meyer stockmeyer polynomial time hierarchy ph where the bp operator is the natural generalization of the probabilistic complexity class bpp the similarity and difference between the two hierarchies bph and ph is investigated oracles a and b are constructed such that both ph a and ph b are infinite while bph a is not equal to ph a at any level and bph b is identical to ph b at every level similar separating and collapsing results in the case that ph a is finite having exactly k levels are also considered

index sets and presentations of complexity classes this paper draws close connections between the ease of presenting a given complexity class b and the position of the index sets ib lcub i l mi b rcub and jb lcub i miis total l mi b rcub in the arithmetical hierarchy for virtually all classes b studied in the literature the lowest levels attainable are ib 03 and jb 02 the first holds iff b is 02 presentable and the second iff b is recursively presentable a general kind of priority argument is formulated and it is shown that every property enforcible by it is not recursively presentable it follows that the classes of p immune and p biimmune languages in exponential time are not recursively presentable it is shown that for all b with ib 03 many members of b do not provably in true 2 arithmetic belong to b a class h is exhibited such that ih 03 is open and ih 03 implies that the polynomial hierarchy is infinite

arithmetic theories for computational complexity problems this paper considers a number of arithmetic theories and shows how the strength of these theories relates to certain open problems in complexity theory concerning the polynomial time hierarchy these results are proved quite generally and hold for a wide class of subrecursive hierarchies they can be used to characterize certain properties of functions provable in these theories

the position of index sets of identifiable sets in the arithmetical hierarchy we prove that every set of partial recursive functions which can be identified by an inductive inference machine is included in some identifiable function set with index set in 3 3 an identifiable set is presented with index set in 2 3 but neither in 2 nor in 2 furthermore we show that there is no nonempty identifiable set with index set in 1 in 1 it is possible to locate this king of set in the last part of the paper we show that the problem to identify all partial recursive functions and the halting problem are of the same degree of unsolvability

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

almost every set in exponential time is p bi immune a set a is p bi immune if neither a nor its complement has an infinite subset in p we investigate here the abundance of p bi immune languages in linear exponential time e using resource bounded measure we prove that the class of p bi immune sets has measure 1 in e this implies that almost every language in e is p bi immune a bit further we show that every p random pseudorandom language is e bi immune regarding the existence of p bi immune sets in np we show that if np does not have measure 0 in e then np contains a p bi immune set another consequence is that the class of 1 tt p complete languages for e has measure 0 in e in contrast using the approach of resource bounded category it is shown that in e the class of p bi immune languages lacks the property of baire the baire category analogue of lebesgue measurability

a downward collapse within the polynomial hierarchy downward collapse also known as upward separation refers to cases where the equality of two larger classes implies the equality of two smaller classes we provide an unqualified downward collapse result completely within the polynomial hierarchy in particular we prove that for k 2 if rm p sigma p k 1 rm p sigma p k 2 then sigma p k pi p k rm ph we extend this to obtain a more general downward collapse result

genericity randomness and polynomial time approximations polynomial time safe and unsafe approximations for intractable sets were introduced by meyer and paterson technical report tm 126 laboratory for computer science mit cambridge ma 1979 and yesha siam j comput 12 1983 pp 411 425 respectively the question of which sets have optimal safe and unsafe approximations has been investigated extensively duris and rolim lecture notes in comput sci 841 springer verlag berlin new york 1994 pp 38 51 and ambos spies proc 22nd icalp springer verlag berlin new york 1995 pp 384 392 showed that the existence of optimal polynomial time approximations for the safe and unsafe cases is independent using the law of the iterated logarithm for p random sequences which has been recently proven in proc 11th conf computational complexity ieee computer society press piscataway nj 1996 pp 180 189 we extend this observation by showing that both the class of polynomial time delta levelable sets and the class of sets which have optimal polynomial time unsafe approximations have p measure 0 hence typical sets in e in the sense of p measure do not have optimal polynomial time unsafe approximations we will also establish the relationship between resource bounded genericity concepts and the polynomial time safe and unsafe approximation concepts

structural properties of bounded relations with an application to np optimization problems we introduce the notion bounded relation which comprises most resource bounded reducibilities which can be found in the literature including non uniform bounded reducibilities such as tp poly we state conditions on bounded relations which are again satisfied for most bounded reducibilities and which imply that every countable partial ordering can be embedded into every proper interval of the recursive degrees as corollaries we obtain that every countable partial ordering can be embedded into every proper interval of rec tp poly as well as into every proper interval between either two maximization or two minimization problems in the structures npo e and npo l for the two latter structures we show further that the result about embeddings of partial orderings extends to embeddings of arbitrary countable distributive lattices where in addition the least or the greatest element of the lattice can be preserved among other corollaries we obtain that for both structures every non trivial np optimization problem bounds a minimal pair in connection with embeddings into npo we introduce a representation of maximization or minimization problems by polynomial time computable functions we consider decidability issues w r t this representation and show for example that there is no effective procedure which decides for a subrecursive index of a polynomial time computable function whether the corresponding maximization problem is approximable within a constant factor

on the relative complexity of hard problems for complexity classes without complete problems abstract we show that any recursive sequence of recursive sets which is ascending with respect to the standard polynomial time reducibility notions has no minimal upper bound as a consequence any complexity class with certain natural closure properties possesses either complete problems or no easiest hard problems a further corollary is that assuming p np the partial ordering of the polynomial time degrees of np sets is not complete and that there are no degree invariant approximations to np complete problems

minimal pairs and complete problems two sets are said to form a minimal pair for polynomial many one reductions if neither set is in p and the only sets which are polynomially reducible via polynomial many one reductions to both sets are in p we consider the question of which complete sets can be the top that is the supremum of a minimal pair our main result is that any elementary recursive set which is hard for deterministic exponential time cannot be the top of a minimal pair hence in particular any complete set for an elementary recursive class containing exponential time cannot be such a supremum for np complete sets the problem remains open we show here that it is oracle dependent this is the first example of an oracle dependent property which is completely degree theoretic

non mitotic sets we study the question of the existence of non mitotic sets in np we show under various hypotheses that 1 tt mitoticity and m mitoticity differ on np t autoreducibility and t mitoticity differ on np this contrasts the situation in the recursion theoretic setting where ladner showed that autoreducibility and mitoticity coincide 2 tt autoreducibility does not imply weak 2 tt mitoticity from this it follows that autoreducibility and mitoticity are not equivalent for all reducibilities between 2 tt and t although the notions coincide for m and 1 tt reducibility

reductions on np and p selective sets abstract p selective sets are used to distinguish polynomial time bounded reducibilities on np in particular we consider different kinds of positive reductions these preserve membership in np and are not a priori closed under complements we show that the class of all sets which are both p selective and have positive reductions to their complements is p this is used to show that if dext next then there exists a set in np p that is not positive reducible to its complement various similar results are obtained we also show that p is the class of all sets which are both p selective and positive truth table self reducible from this result it follows that various naturally defined apparently intractible problems are not p selective unless p np

complex analytic quotients of algebraic g varieties it is shown that any compact semistable quotient of a normal variety by a complex reductive lie group g in the sense of heinzner and snow is a good quotient this reduces the investigation and classification of such complex analytic quotients to the corresponding questions in the algebraic category as a consequence of our main result we show that every compact space in nemirovski s class fancyscript q g has a realisation as a good quotient and that every complete algebraic variety in fancyscript q g is unirational with finitely generated cox ring and at worst rational singularities in particular every compact space in class fancyscript q t where t is an algebraic torus is a toric variety

diagonalization uniformity and fixed point theorems abstract we derive new fixed point theorems for subrecursive classes together with a theorem on the uniformity of certain reductions from a general formulation of the technique of delayed diagonalization this formulation extends the main theorem of u schoning theoret comput sci 18 1982 95 103 to cases which involve infinitely many diagonal classes c k and which allow each c k to contain uncountably many members the main technical work ties the familiar concept of a witness function directly to the often studied cantor set topology on languages and provides a delay construction which refines those due to schoning s breidtbart and d schmidt our a e fixed point theorems do not require that the programming system for the subrecursive class in question be well behaved we compare them to results which do the other theorem is similar to the uniform boundedness theorem of classical analysis and extends work of j grollmann and a selman

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

tally np sets and easy census functions abstract we study the question of whether every p set has an easy i e polynomial time computable census function we characterize this question in terms of unlikely collapses of language and function classes such as p1 fp where p1 is the class of functions that count the witnesses for tally np sets we prove that every p1ph function can be computed in fp p1 p1 consequently every p set has an easy census function if and only if every set in the polynomial hierarchy does we show that the assumption p1 fp implies p bpp and ph modk p for each k 2 we also relate a set s property of having an easy census function to other well studied properties of sets such as rankability and scalability the closure of the rankable sets under p isomorphisms finally we prove that it is no more likely that the census function of any set in p can be approximated more precisely can be n enumerated in time n for fixed and than that it can be precisely computed in polynomial time

genericity and randomness over feasible probability measures this paper investigates the notion of resource bounded genericity developed by ambos spies fleischhack and huwig ambos spies neis and terwijn have recently shown that every language that is t n random over the uniform probability measure is t n generic it is shown here that in fact every language that is t n random over any strongly positive t n computable probability measure is t n generic roughly speaking this implies that when genericity is used to prove a resource bounded measure result the result is not specific to the underlying probability measure

diagonalizations over polynomial time computable sets abstract a formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets the class of polynomial time computable functions respectively like e g p immunity it is shown that there are sets called p generic which have all properties enforceable by such diagonalizations we study the behaviour and the complexity of p generic sets in particular we show that the existence of p generic sets in np is oracle dependent even if we assume p np

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

strong separations of the polynomial hierarchy with oracles constructive separations by immune and simple sets abstract in this paper we show that the techniques introduced by furst 1984 which connected oracle separation results for the relativized polynomial time hierarchy to the problem of proving lower bounds for constant depth circuits and the subsequent probabilistic arguments introduced by yao 1985 hastad 1986 and ko 1989 in order to prove the existence of relativized polynomial time hierarchies with different structures can be adapted for resolving the main problems related to the existence of immune and simple sets in the relativized polynomial time hierarchy in particular we construct oracles which witness 1 for any k the existence of a p k immune set in p k 2 for any k the existence of a k simple set 3 for any k the existence of a p k immune set in a relativized polynomial time hierarchy for which p k p k p k 4 for any k 1 the existence of a p k 1 immune set in a relativized polynomial time hierarchy for which p k p k

on an optimal propositional proof system and the structure of easy subsets of taut in this paper we develop a connection between optimal propositional proof systems and structural complexity theory specifically there exists an optimal propositional proof system if and only if there is a suitable recursive presentation of the class of all easy polynomial time recognizable subsets of taut as a corollary we obtain the result that if there does not exist an optimal propositional proof system then for every theory t there exists an easy subset of taut which is not t provably easy

compact kahler quotients of algebraic varieties and geometric invariant theory abstract given an action of a complex reductive lie group g on a normal variety x we show that every analytically zariski open subset of x admitting an analytic hilbert quotient with projective quotient space is given as the set of semistable points with respect to some g linearised weil divisor on x applying this result to hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to git semistability in the sense of mumford and hausen it follows that the number of compact momentum map quotients of a given algebraic hamiltonian g variety is finite as further corollary we derive a projectivity criterion for varieties with compact kahler quotient

the density and complexity of polynomial cores for intractable sets let a be a recursive problem not in p lynch has shown that a then has an infinite recursive polynomial complexity core this is a collection c of instances of a such that every algorithm deciding a needs more than polynomial time almost everywhere on c we investigate the complexity of recognizing the instances in such a core and show that every recursive problem a not in p has an infinite core recognizable in subexponential time we further study how dense the core sets for a can be under various assumptions about the structure of a our main results in this direction are that if p np then np complete problems have polynomially nonsparse cores recognizable in subexponential time and that exptime complete problems have cores of exponential density recognizable in exponential time

relativized polynomial time hierarchies having exactly k levels it is proved that for every integer k geqq 0 there is an oracle a k relative to which the polynomial time hierarchy collapses so that it has exactly k levels furthermore sets b k and c k may be constructed so that relative to b k the polynomial time hierarchy has exactly k levels and the class pspace coincides with the polynomial time hierarchy and relative to c k the polynomial time hierarchy has exactly k levels and the class pspace is different from the polynomial time hierarchy

minimal pairs for p abstract we prove a general minimal pair theorem which yields as corollaries many results about minimal pairs for p obtained previously by various authors furthermore it yields the new counterintuitive result that there exist arbitrarily complex minimal pairs we also investigate the question of whether minimal pairs in np are low

relating polynomial time to constant depth going back to the seminal paper of furst saxe and sipser 1984 analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower bounds for circuit classes in this note we show how separating oracles can be obtained uniformly from circuit lower bounds without the need of carrying out a particular diagonalization our technical tool is the leaf language approach to the definition of complexity classes

a second step towards complexity theoretic analogs of rice s theorem abstract rice s theorem states that every nontrivial language property of the recursively enumerable sets is undecidable borchert and stephan 1997 initiated the search for complexity theoretic analogs of rice s theorem in particular they proved that every nontrivial counting property of circuits is up hard and that a number of closely related problems are spp hard the present paper studies whether their up hardness result itself can be improved to spp hardness we show that their up hardness result cannot be strengthened to spp hardness unless unlikely complexity class containments hold nonetheless we prove that every p constructibly bi infinite counting property of circuits is spp hard we also raise their general lower bound from unambiguous nondeterminism to constant ambiguity nondeterminism

a note on deterministic and nondeterministic time complexity it is proved that there exists for every set b e np co np a set a np co np such that a and b have the same deterministic time complexity note that b is easier nondeterministically than both a and a in that b e np while a np co np we also prove there exists for each b np co np a set a np co np such that the deterministic time complexity of a is less than that of b on some infinite set in this case then a is easier than b deterministically while b is easier than a nondeterministically

a general method to construct oracles realizing given relationships between complexity classes we present a method to prove oracle results of the following type let k 1 ldots k 2n and l 1 ldots l 2m be complexity classes our method provides a general framework for constructing an oracle a such that k a 2i 1 neq k a 2i for i 1 ldots n and l a 2j 1 l a 2j for j 1 ldots m using that method we prove several results of this kind the hardest of them is the existence of an oracle a such that p a neq np a p a bpp a and both conp a sets and np a sets are p a separable we exhibit also two theorems that cannot be proved by that method

constructing np intermediate problems by blowing holes with parameters of various properties the search for natural np intermediate problems is one of the holy grails within computational complexity ladner s original diagonalization technique for generating np intermediate problems blowing holes has a serious shortcoming it creates problems with a highly artificial structure by arbitrarily removing certain problem instances in this article we limit this problem by generalizing ladner s method to use parameters with various characteristics this allows one to define more fine grained parameters resulting in np intermediate problems where we only blow holes in a controlled subset of the problem we begin by fully characterizing the problems that admit np intermediate subproblems for a broad and natural class of parameterizations and extend the result further such that structural csp restrictions based on parameters that are hard to compute such as tree width are covered thereby generalizing a result by grohe for studying certain classes of problems including csps parameterized by constraint languages we consider more powerful parameterizations first we identify a new method for obtaining constraint languages such that csp are np intermediate the sets can have very different properties compared to previous constructions by for instance bodirsky grohe and provides insights into the algebraic approach for studying the complexity of infinite domain csps second we prove that the propositional abduction problem parameterized by constraint languages admits np intermediate problems this settles an open question posed by nordh zanuttini we provide a framework to obtain np intermediate problems based on diagonalization we categorize different types of parameters that are usable in our framework we construct np intermediate constraint satisfaction problems we resolve the open question whether boolean abduction admits intermediate problems

lattice embeddings for abstract bounded reducibilities we give an abstract account of resource bounded reducibilities as exemplified by the polynomially time or logarithmically space bounded reducibilities of turing truth table and many one type we introduce a small set of axioms that are satisfied for most of the specific resource bounded reducibilities appearing in the literature some of the axioms are of a more algebraic nature such as the requirement that the reducibility under consideration is a reflexive relation while others are formulated in terms of recursion theory and for example are related to delayed computations of arbitrary recursive sets the main technical result shown is that for any reducibility that satisfies these axioms every countable distributive lattice can be embedded into any proper interval of the structure induced on the recursive sets this extends a corresponding result for polynomially time bounded reducibilities due to ambos spies inform and control 65 1985 pp 63 84 as well as a result on embeddings of partial orderings for axiomatically described reducibilities due to mehlhorn j comput system sci 12 1976 pp 147 178

separating the low and high hierarchies by oracles abstract the relativized low and high hierarchies within np are considered an oracle a is constructed such that the low and high hierarchies relative to a are infinite and for each k an oracle ak is constructed such that the low and high hierarchies relative to ak have exactly k levels

the extended low hierarchy is an infinite hierarchy balcazar book and schoning introduced the extended low hierarchy based on the sigma levels of the polynomial time hierarchy as follows for k geq 1 level k of the extended low hierarchy is the set elow k a mid sgm k a subseteq sgm k 1 a oplus sat allender and hemachandra and long and sheu introduced refinements of the extended low hierarchy based on the delta and theta levels respectively of the polynomial time hierarchy for k geq 2 edlow k a mid dlt k a subseteq dlt k 1 a oplus sat and etlow k a mid tht k a subseteq tht k 1 a oplus sat this paper shows that the extended low hierarchy is properly infinite by showing for k geq 2 that elow k subsetneq etlow k 1 subsetneq edlow k 1 subsetneq elow k 1 the proofs use the circuit lower bound techniques of h aa stad and ko as corollaries to the constructions for k geq 2 oracle sets b k c k and d k such that ph b k sgm k b k supsetneq dlt k b k ph c k dlt k c k supsetneq tht k c k and ph d k tht k d k supsetneq sgm k 1 d k are obtained

uniformly hard languages ladner j assoc comput mach 22 1975 155 showed that there are no minimal recursive sets under polynomial time reductions given any recursive set a ladner constructs a set b such that b strictly reduces to a but b does not lie in p the set b does have very long sequences of input lengths of easily computable instances we examine whether ladner s results hold if we restrict ourselves to uniformly hard languages which have no long sequences of easily computable instances under a hard to disprove assumption we show that there exists a minimal recursive uniformly hard set under honest many one polynomial time reductions

resource bounded randomness and weakly complete problems we introduce and study resource bounded random sets based on lutz s concept of resource bounded measure 5 6 we concentrate on nc randomness c 2 which corresponds to the polynomial time bounded p measure of lutz and which is adequate for studying the internal and quantative structure of e dtime 2lin first we show that the class of nc random sets has p measure 1 this provides a new simplified approach to p measure 1 results next we compare randomness with genericity in the sense of 1 2 and we show that nc 1 random sets are nc generic whereas the converse fails from the former we conclude thatnc random sets are not p btt complete for e our technical main results describe the distribution of the nc random sets under p m reducibility we show that every nc random set in e has nk random predecessors in e for any k 1 whereas the amount of randomness of the successors is bounded we apply this result to answer a question raised by lutz 8 we show that the class of weakly complete sets has measure 1 in e and that there are weakly complete problems which are not p btt complete for e

the recursion theoretic structure of complexity classes abstract we prove easy recursion theoretic results which have as corollaries generalizations of existing diagonalization theorems on complexity classes roughly speaking almost no reasonable time space or even abstract complexity class can be expressed as the non trivial union of two recursively presentable classes which are closed under finite variations e g unless np p np p np complete languages and consequently the non trivial complement of one complexity class in another e g npp provided np p is almost never recursively presentable