a note on the circuit complexity of pp in this short note we show that for any integer k there are languages in the complexity class pp that do not have boolean circuits of size nk

likelihood ration test wald test and kuhn tucker test in linear models with inequality constraints on the regression parameters likelihood ratio test wald test and kuhn tucker test in linear models with inequality constraints on the regression parameters this paper considers the problem of testing statistical hypotheses in linear regression models with inequality constraints on the regression coefficients the kuhn tucker multiplier test statistic is defined and its relationships with the likelihood ratio test and the wald test are examined it is shown in particular that these relationships are the same as in the equality constrained case it is emphasized however that their common asymptotic distribution is a mixture of chi square distributions under the null hypothesis

gap definability as a closure property gap definability and the gap closure operator were defined by s fenner l fortnow and s kurth j comput system sci 48 116 148 1994 few complexity classes were known at that time to be gap definable in this paper we give simple characterizations of both gap definability and the gap closure operator and we show that many complexity classes are gap definable includingp p p p1 pspace exp nexp mp middle bitp andbp p if a class is closed under union and intersection and contains and then it is gap definable if and only if it containsspp its gap closure is the closure of this class together withsppunder union and intersection on the other hand we give some examples of classes which are reasonable and gap definable but not closed under union resp intersection complement finally we show that a complexity class such aspspaceorpp if it is not equal tospp contains a maximal gap definable many one reduction closed subclass which is properly betweensppand the class of allpspace incomplete pp incomplete sets with respect to containment the gap closure of the class of all incomplete sets inpspace resp pp ispspace resp pp

testing the nullity of garch coefficients correction of the standard tests and relative efficiency comparisons this article is concerned with testing the nullity of coefficients in generalized autoregressive conditionally heteroscedastic garch models the problem is nonstandard because the quasi maximum likelihood estimator is subject to positivity constraints this article establishes the asymptotic null and local alternative distributions of wald score and quasi likelihood ratio tests efficiency comparisons under fixed alternatives are considered two cases of special interest are tests of the null hypothesis of one coefficient equal to zero and tests of the null hypothesis of no conditional heteroscedasticity finally the proposed approach is used in the analysis of financial data and suggests reconsidering the preeminence of garch 1 1 among garch models

some results on the statistical properties of an inequality constrained least squares estimator in a linear model with two regressors abstract the inequality constrained least squares estimator in this paper is from the regression model y x 1 1 x 2 2 e where both of the regression coefficients are subject to non negative inequality constraints the first and second moments of this estimator are derived the regressors are permitted to be non orthogonal a number of conclusions are drawn from comparisons of the inequality constrained least squares estimator with the ordinary least squares and equality constrained least squares estimators

hypothesis testing with a restricted parameter space abstract this paper considers hypothesis tests when the parameter space is restricted under the alternative hypothesis multivariate one sided tests are a leading example optimal tests called directed tests are derived using a weighted average power criterion exact results are established first for gaussian linear regression models with known variance asymptotic analogues are then established for dynamic nonlinear models simulation is used to compare the tests discussed in the paper the d w directed test is found to perform best in an overall sense for multivariate one sided alternatives with the likelihood ratio test being a close second the d w and likelihood ratio tests are found to perform best for mixed one and two sided alternatives

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

new collapse consequences of np having small circuits we show that if a self reducible set has polynomial size circuits then it is low for the probabilistic class zpp np as a consequence we get a deeper collapse of the polynomial time hierarchy ph to zpp np under the assumption that np has polynomial size circuits this improves on the well known result in karp and lipton proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 stating a collapse of ph to its second level sigmap 2 under the same assumption furthermore we derive new collapse consequences under the assumption that complexity classes like up fewp and c p have polynomial size circuits finally we investigate the circuit size complexity of several language classes in particular we show that for every fixed polynomial s there is a set in zpp np which does not have o s n size circuits

a unifying approach to the decoupling of linear multivariate systems a unifying approach to the decoupling problem diagonal triangular block decoupling is introduced the key element is the interactor a polynomial matrix of unique structure associated with the transfer matrix of the system necessary and sufficient conditions for block decoupling are derived and the stability questions are resolved using the block coupling zeros the results obtained for decoupling via state feedback state feedback plus input dynamics and constant output feedback are conceptually simple unify and expand those existing and easily lead to constructive decoupling algorithms

solvability of the decoupling problem for linear constant a b c d quadruples with regular output feedback the aim of this paper is to deal with the general decoupling problem of linear constant a b c d quadruples under the assumption of regular output feedback in its most general setting this problem was hitherto unsolved here we give a complete characterization of solutions by regular output feedback by means of necessary and sufficient conditions of existence they generalize the result previously obtained by denham 4 for the linear constant a b c triples

the size of spp derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class spp 1 p spp 0 2 ph spp in other words spp is small by being a negligible subset of exponential time or large by containing the entire polynomial time hierarchy this addresses an open problem about the complexity of the graph isomorphism problem it is not weakly complete for exponential time unless ph is contained in spp it is also shown that the polynomial time hierarchy is contained in sppnp if np does not have p measure 0

a test of the null of integer integration against the alternative of fractional integration abstract this paper proposes a test of the null of integer integration against the alternative of fractional integration the null of integer integration is satisfied if the series is either i 0 or i 1 we reject the null if the kpss test rejects i 0 and a unit root test rejects i 1 we suggest a new unit root test a lower tail kpss test applied to the differenced data to use in this procedure we provide critical values under standard asymptotics and fixed b asymptotics we prove the consistency of this testing procedure against i d alternatives with 0 d 1 and simulation evidence on the size and power of the test in finite samples is provided

model following using measured output feedback this article considers the partial state feedback model following problem psfmfp introduced by howze et al 5 here we give a more general statement of this problem and show that its solution is closely related to the existence of certain d a invariant controllability subspaces where d is an output map connected with the measurable output we also give a procedure for the synthesis of a dynamic compensator using less integrators than in 5

decoupling by dynamic measurement feedback with stability necessary and sufficient conditions the noninteracting control problem of linear time invariant multivariable systems by dynamic measurement feedback is solved in the state space geometric approach the generalization of w m wonham s edp 1979 is considered the problem of the reduction of compensator s order is not treated but the solution derived is able to solve simultaneous problems like the decoupling and the disturbance decoupling problems

impossibility of classically simulating one clean qubit model with multiplicative error the one clean qubit model or the deterministic quantum computation with one quantum bit model is a restricted model of quantum computing where all but a single input qubits are maximally mixed it is known that the probability distribution of measurement results on three output qubits of the one clean qubit model cannot be classically efficiently sampled within a constant multiplicative error unless the polynomial time hierarchy collapses to the third level t morimae k fujii and j f fitzsimons phys rev lett 112 130502 2014 prltao0031 900710 1103 physrevlett 112 130502 it was open whether we can keep the no go result while reducing the number of output qubits from three to one here we solve the open problem affirmatively we also show that the third level collapse of the polynomial time hierarchy can be strengthened to the second level one the strengthening of the collapse level from the third to the second also holds for other subuniversal models such as the instantaneous quantum polynomial model m bremner r jozsa and d j shepherd proc r soc a 467 459 2011 prlaaz1364 502110 1098 rspa 2010 0301 and the boson sampling model s aaronson and a arkhipov stoc 2011 p 333 we additionally study the classical simulatability of the one clean qubit model with further restrictions on the circuit depth or the gate types

transducing markov sequences a markov sequence is a basic statistical model representing uncertain sequential data and it is used within a plethora of applications including speech recognition image processing computational biology radio frequency identification rfid and information extraction the problem of querying a markov sequence is studied under the conventional semantics of querying a probabilistic database where queries are formulated as finite state transducers specifically the complexity of two main problems is analyzed the first problem is that of computing the confidence probability of an answer the second is the enumeration of the answers in the order of decreasing confidence with the generation of the top k answers as a special case or in an approximate order thereof in particular it is shown that enumeration in any subexponential approximate order is generally intractable even for some fixed transducers and a matching upper bound is obtained through a proposed heuristic due to this hardness a special consideration is given to restricted yet common classes of transducers that extract matches of a regular expression subject to prefix and suffix constraints and it is shown that these classes are indeed significantly more tractable

model matching of discrete linear systems an analysis of the exact model matching problems of discrete time linear multivariable systems is presented the approach is based on input output polynomial models and all solutions to the problem are expressed in a parametric form compact necessary and sufficient conditions for the existence of causal and stable solutions are given a simple design procedure is proposed which provides a stable causal solution it consists essentially of solving a linear equation in polynomial matrices and yields the compensator in a form suitable for direct realisation

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

competing provers yield improved karp lipton collapse results via competing provers we show that if a language a is self reducible and has polynomial size circuits then s2a s2 building on this we strengthen the kamper afk theorem namely we prove that if np np conp poly then the polynomial hierarchy collapses to s2np conp we also strengthen yap s theorem namely we prove that if np conp poly then the polynomial hierarchy collapses to s2np under the same assumptions the best previously known collapses were to zppnp and zppnpnp respectively siam journal on computing 28 1 1998 311 journal of computer and system sciences 52 3 1996 421 building on proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 journal of computer and system sciences 39 1989 21 theoretical computer science 85 2 1991 305 theoretical computer science 26 3 1983 287 it is known that s2 zppnp proceedings of the 42nd ieee symposium on foundations of computer science ieee computer society press silver spring md 2001 pp 620 629 that result and its relativized version show that our new collapses indeed improve the previously known results the kamper afk theorem and yap s theorem are used in the literature as bridges in a variety of results ranging from the study of unique solutions to issues of approximation and so our results implicitly strengthen those results

on model following using measured output feedback when not all the state variables of a plant are available for measurement the problems involved in synthesizing a compensating control for the plant so that the impulse response matrix of the resulting closed loop system coincides with the impulse response matrix of a prespecified linear model and the resulting closed loop system the model following system is internally stable are considered for two cases the case where the dimension of the dynamical compensator is preassigned and the case whore the dimension is one of unknown parameters necessary and sufficient conditions for the existence of a solution to these problems arc given under certain assumptions the results give a procedure for synthesizing model following stable systems in particular it is easy to use these results to synthesize a model following stable system by static output feedback

probabilistic data exchange the work reported here lays the foundations of data exchange in the presence of probabilistic data this requires rethinking the very basic concepts of traditional data exchange such as solution universal solution and the certain answers of target queries we develop a framework for data exchange over probabilistic databases and make a case for its coherence and robustness this framework applies to arbitrary schema mappings and finite or countably infinite probability spaces on the source and target instances after establishing this framework and formulating the key concepts we study the application of the framework to a concrete and practical setting where probabilistic databases are compactly encoded by means of annotations formulated over random boolean variables in this setting we study the problems of testing for the existence of solutions and universal solutions materializing such solutions and evaluating target queries for unions of conjunctive queries in both the exact sense and the approximate sense for each of the problems we carry out a complexity analysis based on properties of the annotation for various classes of dependencies finally we show that the framework and results easily and completely generalize to allow not only the data but also the schema mapping itself to be probabilistic

simple tests of alternative specifications in stochastic frontier models abstract previous work on stochastic production frontiers has generated a family of models of varying degrees of complexity since this family is nested in the sense that the more general models contain the less general we can test the restrictions that distinguish the model in this paper we provide tests of these restrictions based on the results of estimating the simpler restricted models some of our tests are lm tests however in other cases the lm test fails so we provide alternative simple tests

determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy abstract it is shown that determining whether a quantum computation has a non zero probability of accepting is at least as hard as the polynomial time hierarchy this hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition or whether a given quantum bit has positive expectation value at the end of a quantum computation this result is achieved by showing that the complexity class nqp of adleman demarrais and huang a quantum analog of np is equal to the counting class co c equals p

minimax lq exact and inexact model matching problems abstract a mixed h 2 h optimal problem is presented for both disturbance attenuation and model mismatching minimization its solution is in the form of a two degrees of freedom controller whose feedback and feedforward paths can be independently designed if the conditions for exact model matching are not met an inexact model matching problem is addressed this solution provides an effective model matching design methodology for non minimum phase plants

counting classes are at least as hard as the polynomial time hierarchy in this paper it is shown that many natural counting classes such as pp c p and text mod k text p are at least as computationally hard as ph the polynomial time hierarchy in the following sense for each bf k of the counting classes above every set in bf k ph is polynomial time randomized many one reducible to a set in bf k with two sided exponentially small error probability as a consequence of the result it is seen that all the counting classes above are computationally harder than ph unless ph collapses to a finite level some other consequences are also shown

the complexity of two problems on arithmetic circuits by using arithmetic circuits encoding multivariate polynomials may be drastically more efficient than writing down the list of monomials via the study of two examples we show however that such an encoding can be hard to handle with a turing machine even if the degree of the polynomial is low namely we show that deciding whether the coefficient of a given monomial is zero is hard for p p under strong nondeterministic turing reductions as a result this problem does not belong to the polynomial hierarchy unless this hierarchy collapses for polynomials over fields of characteristic k 0 this problem is mod kp complete this gives a conp m o d k p algorithm for deciding an upper bound on the degree of a polynomial given by a circuit in fields of characteristic k 0

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

one complexity theorist s view of quantum computing the complexity of quantum computation remains poorly understood while physicists attempt to find ways to create quantum computers we still do not have much evidence one way or the other as to how useful these machines will be the tools of computational complexity theory should come to bear on these important questions quantum computing often scares away many potential researchers in computer science because of the apparent background need in quantum mechanics and the alien looking notation used in papers on the topic this paper will give an overview of quantum computation from the point of view of a complexity theorist we will see that one can think of bqp as yet another complexity class and study its power without focusing on the physical aspects behind it

kuhn tucker likelihood ratio and wald tests for nonlinear models with inequality constraints on the parameters abstract this paper considers the problem of testing statistical hypothesis in nonlinear regression models with inequality constraints on the parameters first the kuhn tucker test procedure is defined next it is shown that the distribution of the kuhn tucker the likelihood ratio and the wald test statistics converges to the same mixture of chi square distributions under the null hypothesis to illustrate these results two examples are considered 1 the problem of testing that individual effects are missing in an error component model and 2 the problem of testing equilibrium for a model of markets in disequilibrium

a score test against one sided alternatives abstract a score type statistic t s is introduced for testing h 0 against k 0 and more general one sided hypotheses when nuisance parameters may be present is a vector parameter the main advantages of t s are that it requires estimation of the model only under the null hypothesis and that it is asymptotically equivalent to the likelihood ratio statistic these are precisely the reasons for the popularity of the score tests for testing against two sided alternatives in this sense t s preserves the main attractive features of the classical two sided score test the theoretical results are presented in a general framework where the likelihood based score function is replaced by an estimating function so that the test is applicable even if the exact population distribution is unknown computation of t s is simplified by the fact that it can be computed easily once the corresponding two sided statistic has been computed the relevance and simplicity of t s are illustrated by discussing

the model following problem for linear constant a b c d quadruples the aim of this note is to deal with the model following problem mfp for linear constant a b c d quadruples in the first part we tackle the mfp with state feedback and we extend the results previously given by morse 7 for a b c triples the main result of this paper is given in the second part where we give necessary and sufficient conditions under which solutions using regular output feedback may exist such conditions were lacking up until now

modified lagrange multiplier tests for problems with one sided alternatives abstract a modified lagrange multiplier test statistic is proposed which takes explicit account of the one sided nature of the alternative in problems where the null hypothesis specifies that the true value of the parameter vector lies on the boundary of the parameter space computation of this statistic requires only the constrained maximum likelihood estimator conditions for the consistency of tests based on this statistic are examined and it is shown that the distribution of the statistic is not affected of nuisance parameters are allowed to lie on the boundary of the parameter space

locally best invariant tests of the error covariance matrix of the linear regression model summary this paper considers a class of hypothesis testing problems concerning the covariance matrix of the disturbances in the classical linear regression model a test that is locally best invariant against one sided alternative hypotheses is constructed and shown to be identical to a one sided version of the lagrange multiplier test

complexity classes of equivalence problems revisited to determine if two lists of numbers are the same set we sort both lists and see if we get the same result the sorted list is a canonical form for the equivalence relation of set equality other canonical forms arise in graph isomorphism algorithms to determine if two graphs are cospectral have the same eigenvalues we compute their characteristic polynomials and see if they are equal the characteristic polynomial is a complete invariant for cospectrality finally an equivalence relation may be decidable in p without either a complete invariant or canonical form blass and gurevich 1984 asked whether these conditions on equivalence relations having an fp canonical form having an fp complete invariant and being in p are distinct they showed that this question requires non relativizing techniques to resolve we extend their results and give new connections to probabilistic and quantum computation

complexity of counting the optimal solutions following the approach of hemaspaandra and vollmer we can define counting complexity classes c for any complexity class c of decision problems in particular the classes p kp with k 1 corresponding to all levels of the polynomial hierarchy have thus been studied however for a large variety of counting problems arising from optimization problems a precise complexity classification turns out to be impossible with these classes in order to remedy this unsatisfactory situation we introduce a hierarchy of new counting complexity classes opt kp and opt kp logn with k 1 we prove several important properties of these new classes like closure properties and the relationship with the p kp classes moreover we establish the completeness of several natural counting complexity problems for these new classes