the structure of generalized complexity cores structural properties of proper hard cores generalized complexity cores are studied the setting involves a countable class c of sets of strings that is closed under finite union and under finite variation for any set a let ca denote the collection of all c c such that c a any such a has a proper hard core h h a and for every c ca h c is finite the collection of all proper hard cores of a with respect to c modulo finite differences forms a lattice that is isomorphic to one of just three possible lattices a set b is c levelable if for any c ca there exists a c ca such that c c is infinite the structure of the lattice of hard cores of a with respect to c depends only on the relationship between a and c if a is finite or in c then the lattice is the trivial one point lattice if a is infinite not in c and not c levelable then the lattice is isomorphic to the lattice of all subsets of modulo finite subsets where lcub0 1rcub if a is c levelable then the lattice is isomorphic to a special lattice f f

oracles for deterministic versus alternating classes we construct oracles that force all possible relationships between textit np and textit exp k textit dtime 2 o n k we generalize these results to obtain a theorem about oracles that force relationships between deterministic and nondeterministic and alternating classes from which many corollaries follow the corollaries are interesting because we compare a powerful type of machine e g nondeterministic alternating to a less powerful type of machine that can use more time one of our corollaries is that the result textit dtime n log n subseteq sigma 2 text time n does not relativize

a classification of complexity core lattices abstract lynch 1975 has shown that every recursive set a not in p contains an infinite polynomial complexity core a set of elements c a such that any algorithm deciding a needs more than polynomial time almost everywhere on c actually any a not in p contains infinitely many different cores the collection of which forms a lattice under inclusion we study the structure of this lattice proving that surprisingly there are only three possibilities assuming the lattice is not trivial which happens if a is in p its shape depends only on whether a is almost p immune or not it is known that the natural intractable sets usually do not have this property

dwell time based observer design for unknown input switched linear systems without requiring strong detectability of subsystems this paper investigates the state observer design of a class of unknown input switched linear systems via mode dependent dwell time switchings the distinguishing feature of the proposed method is that strong detectability condition of subsystems of the switched systems is unnecessarily required first a time varying coordinate transformation is introduced to design a suitable reduced order observer for each subsystem then computable sufficient conditions on the synthesis of the observers are proposed in the framework of a mode dependent dwell time technique since the observer of an individual subsystem cannot be designed due to unavailability of strong detectability condition of the subsystem the state of the switched system is estimated under the condition of confining the dwell time by a pair of upper and lower bounds restricting the growth of the lyapunov function of the active subsystem and forcing energy of the overall switched system to decrease at switching instants finally an example is presented to demonstrate the effectiveness of the proposed method

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

state deadbeat response and observability in multi modal systems this paper deals with two aspects of multi modal systems first we show that any completely ontrollable multi modal system with state dimension n not exceeding 3 is capable through feedback of state deadbeat response we conjecture that the result holds for all n as is the case for the classical single mode system certain properties of multi modal systems indicate that they differ significantly from the single mode systems for example the controllable set is not in general a subspace and furthermore the number of steps necessary to reach all states in the controllable set is not bounded by the state dimension in this paper we obtain bounds for this number in the case of a completely controllable system with n leqq 3 and use them to establish state deadbeat response the second portion of this paper refines the controllability canonical form for a multi modal system this is accomplished through the introduction of a notion of observability dual to controllability for these systems

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

on the size of classes with weak membership properties abstract it is shown that the following classes have resource bounded measure 0 in e the class of p selective sets the class of p multiselective sets the class of cheatable sets the class of easily countable sets the class of easily approximable sets the class of near testable sets the class of nearly near testable sets the class of locally self reducible sets these are corollaries of a more general result stating that the class of sets that are p isomorphic to p quasi approximable sets has measure 0 in e by considering the recent approach of allender and strauss for measuring in subexponential classes we obtain similar results with respect to p for classes having weak logarithmic time membership properties

a uniform approach to define complexity classes abstract complexity classes are usually defined by referring to computation models and by putting suitable restrictions on them following this approach many proofs of results are tightly bound to the characteristics of the computation model and of its restrictions and therefore they sometimes hide the essential properties which insure the obtained results in order to obtain more general results a uniform family of computation models which encompasses most of the complexity classes of interest is introduced as a first initial set of results derivable from the proposed approach we will give a sufficient and necessary condition for proving separations of relativized complexity classes a characterization of complexity classes with complete languages and a sufficient condition for proving strong separations of relativized complexity classes examples of applications of these results to some specific complexity classes are then given additional results related to separations by sparse oracles can be found in bovet 1991

reachability realization and stabilizability of switched linear discrete time systems abstract in this paper the reachability realization of a switched linear discrete time system which is a collection of linear time invariant discrete time systems along with some maps for switching among them is addressed the main contribution of this paper is to prove that for a switched linear discrete time system there exists a basic switching sequence such that the reachable controllable state set of this basic switching sequence is equal to the reachable controllable state set of the system hence the reachability controllability can be realized by using only one switching sequence we also discuss the stabilizability of switched systems and obtain a sufficient condition for stabilizability two numeric examples are given to illustrate the results

on reachability and stabilization of switched linear systems reachability and stabilization issues for switched linear control systems are addressed a necessary and a sufficient condition are presented for reachability under mild assumptions we prove that the switched linear control systems are stabilizable in addition we show that an element driven switching stabilization strategy can be explicitly constructed

extending downward collapse from 1 versus 2 queries to m versus m 1 queries the top part of figure 1 1 shows some classes from the truth table bounded query and boolean hierarchies it is well known that if either of these hierarchies collapses at a given level then all higher levels of that hierarchy collapse to that same level this is a standard upward translation of equality that has been known for over a decade the issue of whether these hierarchies can translate equality downwards has proven vastly more challenging in particular with regard to figure 1 1 consider the following claim psigkmtt psigkmponett implies diffmsigk codiffmsigk bh sigmak this claim if true says that equality translates downwards between levels of the bounded query hierarchy and the boolean hierarchy levels that before the fact are immediately below them until recently it was not known whether ever held except for the degenerate cases m 0 and k 0 then hemaspaandra hemaspaandra and hempel siam j comput 28 1999 pp 383 393 proved that holds for all m for k 2 buhrman and fortnow j comput system sci 59 1999 pp 182 199 then showed that when k 2 holds for the case m 1 in this paper we prove that for the case k 2 holds for all values of m since there is an oracle relative to which for k 1 holds for all m fails see buhrman and fortnow our achievement of the k 2 case cannot be strengthened to k 1 by any relativizable proof technique the new downward translation we obtain also tightens the collapse in the polynomial hierarchy implied by a collapse in the bounded query hierarchy of the second level of the polynomial hierarchy

stabilization of switched linear systems in this note we study the stabilization problem of systems that switch among a finite set of controllable linear systems with arbitrary switching frequency for both cases of known and unknown switching functions feedback laws are designed to achieve exponential stability for the later case a method combining on line adaptive estimation and feedback stabilization is developed in the controller design

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

necessary and sufficient conditions for stabilization of discrete time planar switched systems abstract this paper discusses the stabilization problem for single input discrete time planar switched systems we establish necessary and sufficient conditions for the stabilization of planar switched systems a series of linear inequalities are presented for describing the set of all common quadratic lyapunov functions our results are not only easily testable but also constructive

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

complexity classes without machines on complete languages for up this paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines often counting classes probabilistic classes and intersection classes lack such enumerations concentrating on the counting class up we show that there are relativizations for which upa has no complete languages and other relativizations for which pb upb npb and upb has complete languages among other results we show that p up if and only if there exists a set s in p of boolean formulas with at most one satisfying assignment such that s sat is not in p p up coup if and only if there exists a set s in p of uniquely satisfiable boolean formulas such that no polynomial time machine can compute the solutions for the formulas in s if up has complete languages then there exists a set r in p of boolean formulas with at most one satisfying assignment so that sat r is complete for up finally we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class bpp there is a relativized world where bppa has no complete languages if bpp has complete languages then it has a complete language of the form b majority where b p and majority f f is true for at least half of all assignments is the canonical pp complete set

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

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

poincare series of fiber products and weak complete intersection ideals abstract we study the poincare series of modules over a fiber product of commutative local rings we introduce the notion of a weak complete intersection ideal these are the ideals with the property that every differential in their minimal free resolutions can be represented by a matrix whose entries are in the ideal itself we show that many properties of the poincare series that are known to hold for a fiber product over the maximal ideal also hold for those over weak intersection ideals

relating equivalence and reducibility to sparse sets for various polynomial time reducibilities r this paper asks whether being r reducible to a sparse set is a broader notion than being r equivalent to a sparse set although distinguishing equivalence and reducibility to sparse sets for many one or 1 truth table reductions would imply that p ne np this paper shows that for k truth table reductions k geq 2 equivalence and reducibility to sparse sets provably differ though gavalda and watanabe have shown that for any polynomial time computable unbounded function f cdot some sets f n truth table reducible to sparse sets are not even turing equivalent to sparse sets this paper shows that extending their result to the 2 truth table case would provide a proof that p ne np additionally this paper studies the relative power of different notions of reducibility and proves that disjunctive and conjunctive truth table reductions to sparse sets are surprisingly powerful refuting a conjecture of ko

passivity and stability of switched systems a multiple storage function method this paper presents a concept of passivity for switched systems using multiple storage functions this passivity property is invariant under compatible feedback interconnection branicky s stability theorem of multiple lyapunov functions is generalized by relaxing the non increasing condition on values of lyapunov like functions using this result we show that a passive switched system is stable in the sense of lyapunov moreover asymptotic stability is reached if all subsystems are asymptotically detectable

the boolean hierarchy i structural properties in this paper we study the complexity of sets formed by boolean operations union intersection and complement on np sets these are the sets accepted by trees of hardware with np predicates as leaves and together these form the boolean hierarchy we present many results about the structure of the boolean hierarchy separation and immunity results natural complete languages and structural asymmetries between complementary classes we show that in some relativized worlds the boolean hierarchy is infinite and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels we prove natural languages variations of vertex cover complete for the various levels of the boolean hierarchy we show the following structural asymmetry though no set in the boolean hierarchy is text d text p immune there is a relativized world in which the boolean hierarchy contains text cod text p immune sets thus this paper explores the structural properties of the

complexity measures for public key cryptosystems a general theory of public key cryptography is developed that is based on the mathematical framework of complexity theory two related approaches are taken to the development of this theory and th

quantum and classical complexity classes separations collapses and closure properties we study the complexity of quantum complexity classes such as eqp bqp and nqp quantum analogs of p bpp and np respectively using classical complexity classes such as zpp wpp and c p the contributions of this paper are threefold first via oracle constructions we show that no relativizable proof technique can improve the best known classical upper bound for bqp bqp awpp journal of computer and system sciences 59 2 1999 240 to bqp wpp and the best known classical lower bound for eqp p eqp to zpp eqp second we prove that there are oracles a and b such that relative to a corp is immune to nqp and relative to b bqp is immune to pc p extending a result of de graaf and valiant technical report quant ph 0211179 quantum physics 2002 we construct a relativized world where eqp is immune to modpkp third motivated by the fact that counting classes e g lwpp awpp etc are the best known classical upper bounds on quantum complexity classes we study properties of these counting classes we prove that wpp is closed under polynomial time truth table reductions while we construct an oracle relative to which wpp is not closed under polynomial time turing reductions the latter result implies that proving the equality of the similar appearing classes lwpp and wpp would require nonrelativizable proof techniques we also prove that both awpp and app are closed under tup reductions we use closure properties of wpp and awpp to prove interesting consequences in terms of the complexity of the polynomial hierarchy of the following hypotheses nqp bqp and eqp nqp

connected sums of gorenstein local rings given surjective homomorphisms r t s of local rings and ideals in r and s that are isomorphic to some t module v the connected sum r ts is defined to be ring obtained by factoring out the diagonal image of v in the fiber product r t s when t is cohen macaulay of dimension d and v is a canonical module of t it is proved that if r and s are gorenstein of dimension d then so is r ts this result is used to study how closely an artinian ring can be approximated by a gorenstein ring mapping onto it when t is regular it is shown that r ts almost never is a complete intersection ring the proof uses a presentation of the cohomology algebra extr ks k k as an amalgam of the algebras extr k k and ext s k k over isomorphic polynomial subalgebras generated by one element of degree 2

defying upward and downward separation upward and downward separation results link the collapse of small and large classes and are a standard tool in complexity theory we study the limitations of upward and downward separation pp we show that the exponential time limited nondeterminism hierarchy does not robustly possess downward separation we show that probabilistic classes do not robustly possess hartmanis immerman sewelson upward separation though np is known to robustly possess hartmanis immerman sewelson upward separation we show that np does not robustly possess hartmanis immerman sewelson upward separation with respect to strong immunity separation on the other hand we provide a structural sufficient condition for upward separation

dissipativity theory for switched systems a framework of dissipativity theory for switched systems using multiple storage functions and multiple supply rates is set up each subsystem of a switched system is associated with a storage function to describe the energy stored in the subsystem and is associated with a supply rate that represents energy coming from outside the subsystem when the subsystem is active the exchange of energy between the active subsystem and an inactive subsystem is characterized by cross supply rates stability is reached when all supply rates can be made negative as long as the total exchanged energy between the active subsystem and any inactive subsystems is finite in some sense two special forms of dissipativity passivity and l 2 gain are addressed for both cases asymptotic stability is guaranteed under certain negative output feedback plus asymptotic zero state detectability switched passivity conditions and switched l 2 gain inequalities are respectively derived which are generalizations of classical ones feedback invariance of passivity and a small gain theorem are also given

improved overshoot estimation in pole placements and its application in observer based stabilization for switched systems in this note we first establish an improved and generalized overshoot estimation in pole placements then this estimation is applied to the observer based output feedback stabilization problem for switched linear systems with arbitrary switching frequency our results complement those results of cheng about the state feedback stabilization of switched linear systems

hard core theorems for complexity classes nancy lynch proved that if a decision problem a is not solvable in polynomial time then there exists an infinite recursive subset x of its domain on which the decision is almost everywhere complex in this paper general theorems of this kind that can be applied to several well known automata based complexity classes including a common class of randomized algorithms are proved

a constructive approach to reachability realization of discrete time switched linear systems this paper studies the reachability of discrete time switched linear systems a new constructive approach to designing switching sequences is developed so that not only the reachable subspace is realized but also the required number of switchings is reduced significantly detailed comparison between the obtained result and that reported in the open literature is also made which verifies the advantage of the proposed approach

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

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

sparse sets in np p exptime versus nexptime this paper investigates the structural properties of sets in np p and shows that the computational difficulty of lower density sets in np depends explicitly on the relations between higher deterministic and nondeterministic time bounded complexity classes the paper exploits the recently discovered upward separation method which shows for example that there exist sparse sets in np p if and only if exptime nexptime in addition the paper uses relativization techniques to determine logical possibilities limitations of these proof techniques and exhibits one of the first natural structural differences between relativized np and conp

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

on simple and creative sets in np abstract two structurally defined types of np sets are studied k simple sets are defined and shown to exist in np other properties of these sets are investigated k creative sets as previously defined by joseph and young 1985 are next considered a new condition is given which implies that a set is k creative several previously considered np complete sets are proved to be k creative

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

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

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

on sets truth table reducible to sparse sets we study sets that are truth table reducible to sparse sets in polynomial time the principal results are as follows 1 for every integer k 0 there is a set l and a sparse set s such that l

exponential time and subexponential time sets abstract in this paper we prove that the symmetric difference of a p k parity hard set for e and a subexponential time computable set is still p k parity hard for e this remains true for p m hard set for e since a 1 parity reduction is a many one reduction in addition we show that this property fails to hold for some other types of reductions we introduce and study the notions of e complete kernel and e hard kernel

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

stability and l2 gain analysis for switched delay systems a delay dependent method in this paper we study stability and l 2 gain for a class of switched systems with time varying delays sufficient conditions for exponential stability and weighted l 2 gain are developed for a class of switching signals with average dwell time these conditions are delay dependent and are given in the form of linear matrix inequalities lmis as a special case of such switching signals we can obtain exponential stability and normal l 2 gain under arbitrary switching signals the state decay estimate is explicitly given two examples illustrate the effectiveness and applicability of the proposed method

reachability and controllability of switched linear discrete time systems this paper investigates the reachability and controllability issues for switched linear discrete time systems geometric characterization of controllability is presented for reversible systems the controllable sets and the reachable sets are identified in wonham s geometric approach and verifiable conditions for reachability and controllability are also presented

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

on infinite words determined by l systems abstract a deterministic l system generates an infinite word if each word in its derivation sequence is a prefix of the next yielding as a limit we generalize this notion to arbitrary l systems via the concept of prefix languages a prefix language is a language l such that for all x y l x is a prefix of y or y is a prefix of x every infinite prefix language determines a single infinite word where c is a class of l systems e g 0l dt0l we denote by c the class of infinite words determined by the prefix languages in c this allows us to speak of infinite 0l words infinite dt0l words etc we categorize the infinite words determined by a variety of l systems showing that the whole hierarchy collapses to just three distinct classes of infinite words pd0l d0l and cd0l our results are obtained with the help of a pumping lemma which we prove for t0l systems

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

ascent and descent of the golod property along algebra retracts abstract we study ascent and descent of the golod property along an algebra retract we characterise trivial extensions of modules fibre products of rings to be golod rings we present a criterion for a graded module over a graded affine algebra of characteristic zero to be a golod module

immunity relativizations and nondeterminism it is known 8 13 that there is a recursive set a such that text np a contains a set that is text p a immune that is there is an infinite set l in text np a such that no infinite subset of l is in text p a the first result generalizes this fact to situations where the running times of the machines specifying the larger class bound the size of the sets of strings queried in the computation trees of the machines specifying the smaller class the second result is of a different type for relativized complexity classes specified by bounds on the number of oracle queries and the number of nondeterministic steps allowed in computations it is known 14 that one can describe a set a such that there is an infinite hierarchy of classes relative to a where each new class in the hierarchy is obtained by increasing the amount of nondeterminism here it is shown that the conditions allowing infinite hierarchies to exist also allow for each i geqq 0 the existence of a set

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

completeness approximation and density polynomial time approximations to exponential time computable problems exp are considered from the point of view of structure infinitely often speedable and almost everywhere complex problems are studied using the notions of polynomial time productivity and immunity in particular the existence of a polynomial time immune set which is not polynomial time approximable at all but which is polynomial time tt complete in exp is proven the relationship between completeness and approximability is also studied it is shown that being polynomial time m complete in exp does not provide any control of the probability of erroneous approximations

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

controllability of switched bilinear systems the controllability of switched bilinear systems sblss is considered three kinds of controllabilities including weak controllability approximate controllability and global controllability are investigated one by one sets of easily verifiable sufficient conditions are obtained for each case which are applicable to a large class of switched bilinear systems

computing solutions uniquely collapses the polynomial hierarchy is there an np function that when given a satisfiable formula as input outputs one satisfying assignment uniquely that is can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments we show that if there is such a nondeterministic function then the polynomial hierarchy collapses to zppnp and thus in particular to npnp as the existence of such a function is known to be equivalent to the statement every np function has an np refinement with unique outputs our result provides the strongest evidence yet that np functions cannot be refined we prove our result via a result of independent interest we say that a set a is npsv selective npmv selective if there is a 2 ary partial np function with unique values a 2 ary partial np function that decides which of its inputs if any is more likely to belong to a this is a nondeterministic analog of the recursion theoretic notion of the semi recursive sets and the extant complexity theoretic notion of p selectivity our hierarchy collapse result follows by combining foo the easy observation that every set in np is npmv selective with the following result if a in np is npsv selective then a in np cap conp poly relatedly we prove that if a in np is npsv selective then a is low 2 we prove that the polynomial hierarchy collapses even further namely to np if all conp sets are npmv selective this follows from a more general result we prove every self reducible npmv selective set is in np

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

immunity and pseudorandomness of context free languages we discuss the computational complexity of context free languages concentrating on two well known structural properties immunity and pseudorandomness an infinite language is reg immune resp cfl immune if it contains no infinite subset that is a regular resp context free language we prove that i there is a context free reg immune language outside reg n and ii there is a reg bi immune language that can be computed deterministically using logarithmic space we also show that iii there is a cfl simple set where a cfl simple language is an infinite context free language whose complement is cfl immune similar to the reg immunity a reg primeimmune language has no polynomially dense subsets that are also regular we further prove that iv there is a context free language that is reg n bi primeimmune concerning pseudorandomness of context free languages we show that v cfl contains reg n pseudorandom languages finally we prove that vi against reg n there exists an almost 1 1 pseudorandom generator computable in nondeterministic pushdown automata equipped with a write only output tape and vii against reg there is no almost 1 1 weakly pseudorandom generator computable deterministically in linear time by a single tape turing machine

the existence and density of generalized complexity cores if c is a class of sets and a is not in c then an infinite set h is a proper hard core for a with respect to c if h a and for every c e c such that c a c c h is finite it is shown that if c is a countable class of sets of strings that is closed under finite union and finite variation then every infinite set not in c has a proper hard core with respect to c in addition the density of such generalized complexity cores is studied

downward translations of equality in this paper we construct oracles relative to which dtime t n equals ntime t n and dtime t n does not equal ntime t n for t n sufficiently smaller than t n a stronger result than this is also obtained though for fewer t n expressedin two parts for t n 2n there is an oracle relative to which dtime t n equals ntime t n and ntime 2n contains a set not in dtime t n for any t n growing more slowly than t n for t n 22n 0 1 there is an oracle relative to which dtime t n equals ntime t n and ntime log t n contains a set not in dtime t n for t n growing more slowly than t n these results expand on those obtained by dekhtyar 1976 book et al 1982 and allender 1989

completeness and weak completeness under polynomial size circuits this paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for espace under nonuniform many one reductions that are computed by polynomial size circuits p poly many one reductions every weakly p poly many one complete problem is shown to have a dense exponential nonuniform complexity core an exponential lower bound on the space bounded kolmogorov complexities of weakly p poly turing complete problems is established more importantly the p poly many one complete problems are shown to be unusually simple elements of espace in the sense that they obey nontrivial upper bounds on nonuniform complexity size of nonuniform complexity cores and space bounded kolmogorov complexity that are violated by almost every element of espace more generally a small span theorem for p poly many one reducibility in espace is proven and used to show that every p poly many one degree including the complete degree has measure 0 in espace in contrast almost every element of espace is weakly p many one complete all upper and lower bounds are shown to be tight

brief paper periodic stabilizability of switched linear control systems stabilizability via direct observer based state feedback control for discrete time switched linear control systems slcss is investigated in this paper for an slcs the control factors are not only the control input but also the switching signal and they need to be designed in order to stabilize the system as a result stabilization design for slcss is more complicated than that for non switched ones differently from the existing approaches a periodic switching signal and piecewise constant linear state feedback control are adopted to achieve periodic stabilizability for such systems it is highlighted that multiple feedback controllers need to be designed for one subsystem for discrete time slcss it is proved that reachability implies periodic stabilizability via state feedback a necessary and sufficient criterion for periodic stabilizability is also established two stabilization design algorithms are presented for real application moreover it is proved that reachability and observability imply periodic stabilizability via observer based feedback for discrete time slcss periodic detectability as the dual concept of periodic stabilizability is discussed and the stabilization design algorithms via observer based feedback are presented as well

some remarks on witness functions for nonpolynomial and noncomplete sets in np abstract we present two results about witness functions for sets in np and conp first any set that has a polynomially computable function which witnesses that it is not in conp must be at least np hard it follows from this result that any set in np conp that has a polynomially computable function which witnesses this fact must already be complete for np second if b is any set for which there is a polynomially computable function which witnesses that it is not complete for np by witnessing that some fixed set in np is not in p b then b must already be in np conp thus for two sets in np conp there are no polynomially computable functions which witness that one is not polynomially reducible to the other in proving the first result we introduce the notion of a k creative set and prove that all k creative sets are np complete since these sets seem not to be all polynomially isomorphic we counter the conjecture of berman and hartmanis that all np complete sets are isomorphic to sat with our own conjecture that not all k creative sets are isomorphic to sat the proofs we give are recursion theoretic in style but straightforward

unambiguous computation boolean hierarchies and sparse turing complete sets this paper studies for up two topics that have been intensely studied for np boolean hierarchies and the consequences of the existence of sparse turing complete sets unfortunately as is often the case the results for np draw on special properties of np that do not seem to carry over straightforwardly to up for example it is known for np and more generally for any class containing sigma and emptyset and closed under union and intersection that the symmetric difference hierarchy the boolean hierarchy and the boolean closure all are equal we prove that closure under union is not needed for this claim for any class cal k that contains sigma and emptyset and is closed under intersection e g up us and fewp the symmetric difference hierarchy over cal k the boolean hierarchy over cal k and the boolean closure of cal k all are equal on the other hand we show that two hierarchies the hausdorff hierarchy and the nested difference hierarchy which in the np case are equal to the boolean closure fail to be equal for the up case in some relativized worlds regarding sparse turing complete sets for up we prove that if up has sparse turing complete sets then the levels of the unambiguous polynomial hierarchy are simpler than one would otherwise expect they collapse one level in terms of their location in the promise unambiguous polynomial hierarchy we obtain related results under the weaker assumption that up has sparse turing hard sets

on inefficient special cases of np complete problems abstract every intractable set a has a polynomial complexity core a set h such that for any p subset s of a or of s h is finite a complexity core h of a is proper if h a it is shown here that if p np then every currently known i e either invertibly paddable or k creative np complete set a and its complement have proper polynomial complexity cores that are nonsparse and are accepted by deterministic machines in time 2 cn for some constant c turning to the intractable class d ext c 0 d time 2 cn it is shown that every set that is p m complete for d ext has an infinite proper polynomial complexity core that is nonsparse and recursive

optimal approximations and polynomially levelable sets a set a not in p is polynomially levelable if any algorithm for a has speedup to a polynomial infinitely often in a precisely if given any algorithm m for a and polynomial p it is possible to find another algorithm m for a and polynomial p such that m runs in time p x on infinitely many inputs x in a on which the running time of m exceeds p x intuitively this condition states that among polynomial time computable approximations to a there is no optimal one or one giving correct answers on a maximally large subset of a it appears that most naturally occurring intractable sets are polynomially levelable we prove this for sets not in p that are either paddable self reducible or complete for a deterministic time class we also discuss levelability preserving reductions and give a simple reducibility characterization of nonlevelable sets

technical communique observer driven switching stabilization of switched linear systems in this work we investigate the observer driven switching stabilization problem for a class of switched linear systems by incorporating the switched observer presented in sun wang xie and yu 2006 into the control loop we construct an observer driven path wise switching law for the switched linear system we prove that any stabilizable switched linear system is exponentially stable under the switching law and establish a weak version of the separation principle for switching and observer design

separating complexity classes with tally oracles abstract long and selman 1986 proved that for most familiar pairs of complexity classes separating the classes with a tally oracle is no easier than truly separating the classes refuting a claim in the literature we prove for the first time that for many familiar pairs of complexity classes separating the classes with a tally oracle is easier than truly separating the classes

brief paper design of switching sequences for controllability realization of switched linear systems in this paper we study the problem of designing switching sequences for controllability of switched linear systems each controllable state set of designed switching sequences coincides with the controllable subspace both aperiodic and periodic switching sequences are considered for the aperiodic case a new approach is proposed to construct switching sequences and the number of switchings involved in each designed switching sequence is shown to be upper bounded by d d d 1 1 here d is the dimension of the controllable subspace d email protected i 1 m where a i b i are subsystems for the periodic case we show that the controllable subspace can be realized within d switching periods

simplicity immunity relativizations and nondeterminism a diagonalization method is discussed by which a recursive oracle a can be constructed such that np a has a set l a which is both simple and p a immune

controllability and stabilizability of switched linear systems in this paper we study the conjecture made in ieee trans automat control 47 8 2002 1401 concerning the existence of basic switching sequence for switched systems and give it an affirmative answer and a construction method this paper proves that for switched linear system there exists a switching sequence such that the controllable state set of this switching sequence is equal to the controllable state set of the system hence the controllability can be realized by using only one switching sequence then the result is extended to the systems with time delay in controls furthermore we define the stabilizability of switched linear systems and based on certain transformation of state coordinate we give a necessary and sufficient condition for the stabilizability two examples are presented to illustrate the obtained results

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

accessibility of switched linear systems this note considers the controllability of switched linear systems the structure of accessibility lie algebra is revealed some accessibility properties are proved certain necessary and sufficient conditions for local or global weak or normal controllability of a large class of switched linear systems are obtained

controllability and observability of hybrid systems abstract a special class of hybrid systems is considered whose state space is a cross product space of a euclidean space and a finite state space such models may be used to represent systems subject to known abrupt parameter variations such as commu tated networks they may also be used to approximate some types of time varying systems controllability observability and stability of hybrid systems are inves tigated in particular a necessary and sufficient algebraic condition a simple algebraic criterion and a computationally simple algebraic sufficient test for controllability and observability are derived moreover a simple sufficient stability condition is provided

simplicity relativizations and nondeterminism relativizations of complexity classes in which simple sets exist are considered a recursive oracle is constructed relative to which a simple set exists for np some other general theorems are proven showing that the time bounds are not a crucial hypothesis bounds on the way in which the oracle is accessible namely the number of queries and or the number of nondeterministic steps are shown to be the fundamental hypothesis as a result simple sets are shown to exist in many different relativized complexity classes