stability and mcmillan degree for rational matrix interpolants abstract given a set of tangential interpolation conditions for a rational matrix function one can ask to find an interpolant which 1 meets a stability or norm side constraint as in nevanlinna pick interpolation or 2 has the minimal possible mcmillan degree among all interpolants with no stability side constraint as in the partial realization problem in this paper a connection between the problems is presented specifically it is shown that a minimal degree solution of a given set of interpolation conditions together with an associated mirror image set of interpolation conditions is automatically stable whenever it is unique this result extends recent results of antoulas and anderson for the scalar version of the problem

redundancy in logic i cnf propositional formulae a knowledge base is redundant if it contains parts that can be inferred from the rest of it we study some problems related to the redundancy of a cnf formula in particular any cnf formula can be made irredundant by deleting some of its clauses what results is an irredundant equivalent subset we study the complexity of problems related to irredundant equivalent subsets verification checking existence of an irredundant equivalent subset with a given size checking necessary and possible presence of clauses in irredundant equivalent subsets and uniqueness we also consider the problem of redundancy with different definitions of equivalence

laurent interpolation for rational matrix functions and a local factorization principle abstract a regular rational matrix function is constructed when a finite part of the laurent series of the function and of its inverse are given at a finite number of points provided an obvious necessary condition is satisfied the construction is such that the function will have at most one extra pole or zero possibly of high multiplicity this construction is further applied to develop local principles for factorizations of rational matrix functions

compiling propositional weighted bases in this paper we investigate the extent to which knowledge compilation can be used to improve model checking and inference from propositional weighted bases we first focus on the compilability issue for both problems deriving mainly non compilability results in the case preferences are subject to change then we present a general notion of c normal weighted base that is parametrized by a tractable class c for the clausal entailment problem we show how every weighted base can be turned compiled into a query equivalent c normal base whenever c is a complete class for propositional logic both negative and positive results are presented on the one hand complexity results are identified showing that the inference problem from a c normal weighted base is as difficult as in the general case when the prime implicates horn cover or renamable horn cover classes are targeted on the other hand we show that both the model checking and the clausal inference problem become tractable whenever dnnf normal bases are considered moreover we show that the set of all preferred models of a dnnf normal weighted base can be computed in time polynomial in the output size and as a consequence model checking is also tractable for such bases finally we sketch how our results can be used in model based diagnosis in order to compute the most likely diagnoses of a system

a behavioral approach to linear exact modeling the behavioral approach to system theory provides a parameter free framework for the study of the general problem of linear exact modeling and recursive modeling the authors present the solution of the continuous time polynomial exponential time series modeling problem both recursive and nonrecursive solutions are provided and classified according to properties like complexity and controllability it is shown in particular that recursive modeling corresponds to updating by means of a cascade inter connection of systems as a special case the solution of several other problems such as rational interpolation realization and modeling of arbitrary discrete time time series is obtained

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

structured matrices and unconstrained rational interpolation problems we describe a fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept we use the interpolation data to implicitly define a convenient non hermitian structured matrix and then apply a computationally efficient procedure for its triangular factorization this leads to a transmission line interpretation that makes evident the interpolation properties we further discuss connections with the lagrange interpolating polynomial as well as questions regarding the minimality and the admissible degrees of complexity of the solutions

matrix polynomial solutions of tangential lagrange sylvester interpolation conditions of low mcmillan degree abstract a matrix polynomial of mcmillan degree n 1 which satisfies n tangential lagrange interpolation conditions is constructed explicitly in realization form analogous results are given for more general bitangential lagrange sylvester interpolation conditions a corollary is a type of euclidean algorithm for matrix polynomials and mcmillan degree the main tool is the recent parametrization of all matrix polynomial solutions of a set of lagrange sylvester interpolation conditions due to ball gohberg and rodman

disjunctive merging quota and gmin merging operators when aggregating information from a group of agents accepting the pieces of information shared by all agents is a natural requirement in this paper we investigate such a unanimity condition in the setting of propositional merging we discuss two interpretations of the unanimity condition we show that the first interpretation is captured by existing postulates for merging but the second interpretation is not and this leads to the introduction of a new disjunction postulate disj it turns out that existing operators satisfying disj do not perform well with respect to the standard criteria used to evaluate merging operators logical properties computational complexity and strategy proofness to fill this gap we introduce two new families of propositional merging operators quota operators and gmin operators which satisfy disj and achieve interesting trade offs with respect to the logical computational and strategy proofness criteria

asymptotic formulas for the triple gamma function γ3 by means of its integral representation abstract there is an abundant literature on inequalities for the gamma function and its various related functions as well as their approximations only very recently several authors began to investigate various inequalities for the double gamma function 2 and its approximation here in this sequel to some of these recent works we aim at presenting an integral representation of the triple gamma function 3 which is then used to derive an asymptotic formula for 3 as a by product of the results presented here integral representations and asymptotic formulas for the gamma function and the double gamma function 2 are also given the methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple gamma functions n n 4

bitangential interpolation problems for symmetric rational matrix functions abstract we develop the zero pole structure of rational matrix functions s that satisfy an identity of the form s t v s v for some square matrix v for which v vt where 1 as an application we describe the set of symmetric or antisymmetric solutions of a set of bitangential interpolation conditions solutions of bitangential interpolation conditions satisfying other types of symmetries are also discussed

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

model reduction via truncation an interpolation point of view in this paper we focus our attention on linear time invariant continuous time linear systems with one input and one output siso lti systems we consider the problem of constructing a reduced order system via truncation of the original system given a siso strictly proper transfer function t s of mcmillan degree n and a strictly proper siso transfer function t over cap s of mcmillan degree n n we prove that t over cap s can always be constructed via truncation of the system t s the proof is mainly based on interpolation theory and more precisely on multipoint pade interpolation moreover new results about krylov subspaces are developed c 2003 elsevier inc all rights reserved

rational interpolation and state variable realizations abstract the problem is considered of passing from interpolation data for a real rational transfer function matrix to a minimal state variable realization of the transfer function matrix the tool is a loewner matrix which is a generalization of the standard hankel matrix of linear system realization theory and which possesses a decomposition into a product of generalized observability and controllability matrices

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

null pole subspaces of nonregular rational matrix functions abstract given a not necessarily regular rational matrix function w and a subset of the extended complex plane we associate with it a free module l w over the ring of scalar rational functions which are analytic on called the null pole subspace of w over in the scalar case the information encoded in this module is equivalent to the knowledge of all the poles and zeros including their multiplicities at all points of in the general matrix case the null pole subspace encodes the more complicated zero pole structure of a rational matrix function which is the key tool for the understanding of many factorization problems in this paper we show how various other modules which have been introduced in the literature in connection with pole zero structure e g pole module zero module can be read off from the null pole subspace and how conversely the null pole subspace can be recovered from its various pieces pole module null module a null pole coupling operator and left annihilator we give an analytic coordinate dependent description of these modules which has connection with realization theory and which is useful for computations we also solve the converse problem of finding a rational matrix function which has a given admissible module as its null pole subspace

on two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

column reduced rational matrix functions with given null pole data in the complex plane abstract explicit formulas are given for rational matrix functions which have a prescribed null pole structure in the complex plane and are column reduced at infinity a full parametrization of such functions is obtained the results are specified and developed further for matrix polynomials

on the succinctness of some modal logics one way of comparing knowledge representation formalisms that has attracted attention recently is in terms of representational succinctness i e we can ask whether one of the formalisms allows for a more economical encoding of information than the other proving that one logic is more succinct than another becomes harder when the underlying semantics is stronger we propose to use formula size games as put forward by adler and immerman 2003 1 but we present them as games for one player called spoiler games that are played on two sets of models and that directly link the length of a play in which spoiler wins the game with the size of a formula i e a formula that is true in the first set of models but false in all models of the second set using formula size games we prove the following succinctness results for m dimensional modal logic where one has a set i i 1 i m of indices for m modalities 1 on general kripke models and also on binary trees a definition c f i c i f with c i makes the resulting logic exponentially more succinct for m 1 2 several modal logics use such abbreviations c f e g in description logics the construct corresponds to adding role disjunctions and an epistemic interpretation of it is everybody in c knows indeed we show that on epistemic models i e s 5 models the logic with c f becomes more succinct for m 3 3 the results for the logic with everybody knows also hold for a logic with somebody knows and 4 on epistemic models public announcement logic is exponentially more succinct than epistemic logic if m 3 the latter settles an open problem raised by lutz 2006 18

minimal partial realization from generalized orthonormal basis function expansions a solution is presented for the problem of realizing a discrete time lti state space model of minimal mcmillan degree such that its first n expansion coefficients in terms of generalized orthonormal basis match a given sequence the basis considered also known as the hambo basis can be viewed as a generalization of the more familiar laguerre and two parameter kautz constructions allowing general dynamic information to be incorporated in the basis for the solution of the problem use is made of the properties of the hambo operator transform theory that underlies the basis function expansion as corollary results compact expressions are found by which the hambo transform and its inverse can be computed efficiently the resulting realization algorithms can be applied in an approximative sense for instance for computing a low order model from a large basis function expansion that is obtained in an identification experiment

inequalities involving γ x and γ 1 x we prove that the double inequality 1 x 1 x x x 1 x 1 2 0 with the best possible constants 0 and 3 2 the right hand side refines 1 x 1 1 x 2 x 0 which was proved in 1974 by w gautschi moreover we present sharp inequalities for the product the sum the difference and the ratio of x and 1 x

partial realization of descriptor systems abstract we address the partial realization problem for linear descriptor systems a general solution to this problem using the markov parameters of the system defined via its laurent series is provided for proper descriptor systems we also discuss numerically feasible algorithms for computing a partial minimal realization based on the unsymmetric lanczos and arnoldi processes applications to model reduction for examples from computational fluid dynamics and mechanical systems with holonomic constraints are also given

inequalities and monotonicity properties for the gamma function we present a survey of some recent results on the ratio of two gamma functions and prove the following inequalities for the product 1 1 x 1 x 2 1 x 1 1 x 2 0 and x 1 x 2 1 finally we study the possible extension to k variables of the lower bound above

first order logical filtering logical filtering is the process of updating a belief state set of possible world states after a sequence of executed actions and perceived observations in general it is intractable in dynamic domains that include many objects and relationships still potential applications for such domains e g semantic web autonomous agents and partial knowledge games encourage research beyond intractability results in this paper we present polynomial time algorithms for filtering belief states that are encoded as first order logic fol formulas our algorithms are exact in many cases of interest they accept belief states in fol without functions permitting arbitrary arity for predicates infinite universes of elements and equality they enable natural representation with explicit references to unidentified objects and partially known relationships still maintaining tractable computation previous results focus on more general cases that are intractable or permit only imprecise filtering our algorithms guarantee that belief state representation remains compact for strips actions among others with unbounded size domains this guarantees tractable exact filtering indefinitely for those domains the rest of our results apply to expressive modeling languages such as partial databases and belief revision in fol

on the complexity of propositional knowledge base revision updates and counterfactuals abstract we study the complexity of several recently proposed methods for updating or revising propositional knowledge bases under the principle of minimal change in particular we derive complexity results for the following problem given a knowledge base t an update p and a formula q decide whether q is derivable from t p the updated or revised knowledge base note that this problem includes the evaluation of the counterfactual p q over t that is a conditional statement if p then q where p is known or expected to be false we consider the general case where t is an arbitrary propositional formula or theory as well as restricted versions of this problem in particular where t is a conjunction of horn clauses or where the size of the update p is bounded by a constant

rational interpolation and recursive solution of lowner vandermonde systems of equations abstract rational interpolation problems for functions having numerator degree higher than denominator degree are connected with solutions of systems of equations the coefficient matrix of which has a mixed structure the first columns are of vandermonde type whereas the last columns form a lowner matrix here three term recursions for the rational interpolants are developed which can be translated into recurrence formulas for the solutions of homogeneous systems with such a coefficient matrix on this base an o n2 algorithm for the solution of n n nonhomogeneous lowner vandermonde systems is obtained

a recursive method for solving unconstrained tangential interpolation problems an efficient recursive solution is presented for the one sided unconstrained tangential interpolation problem the method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data the recursive procedure admits a physical interpretation in terms of discretized transmission lines in this framework the generating system is constructed as a cascade of first order sections singular steps occur only when the input data is contradictory i e only when the interpolation problem does not have a solution various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants the algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general hermite matrix interpolation problem a recursive method is proposed to compute a column reduced generating system that can be used to solve the minimal tangential interpolation problem

foundations of instance level updates in expressive description logics in description logic dl aboxes are used for describing the state of affairs in an application domain we consider the problem of updating aboxes when the state changes assuming that update information is described at an atomic level i e in terms of possibly negated abox assertions that involve only atomic concepts and roles we analyze such basic abox updates in several standard dls in particular addressing questions of expressibility and succinctness can updated aboxes always be expressed in the dl in which the original abox was formulated and if so what is the size of the updated abox it turns out that dls have to include nominals and the constructor of hybrid logic for updated aboxes to be expressible and that this still holds when updated aboxes are approximated moreover the size of updated aboxes is exponential in the role depth of the original abox and the size of the update we also show that this situation improves when updated aboxes are allowed to contain additional auxiliary symbols then dls only need to include nominals for updated aboxes to exist and the size of updated aboxes is polynomial in the size of both the original abox and the update

the size of a revised knowledge base abstract in this paper we address a specific computational aspect of belief revision the size of the propositional formula obtained by means of the revision of a formula with a new one in particular we focus on the size of the smallest formula which is logically equivalent to the revised knowledge base the main result of this paper is that not all formalizations of belief revision are equal from this point of view for some of them we show that the revised knowledge base can be represented by a polynomial size formula we call these results compactability results on the other hand for other ones the revised knowledge base does not always admit a polynomial space representation unless the polynomial hierarchy collapses at a sufficiently low level non compactability results we also show that the time complexity of query answering for the revised knowledge base has definitely an impact on being able to represent the result of the revision compactly nevertheless formalisms with the same complexity may have different compactability properties we also study compactability properties for a weaker form of equivalence called query equivalence which allows to introduce new propositional symbols moreover we extend our analysis to the special case in which the new formula has constant size and to the case of sequences of revisions i e iterated belief revision a complete analysis along these four coordinates is shown

preprocessing of intractable problems some computationally hard problems e g deduction in logical knowledge bases are such that part of an instance is known well before the rest of it and remains the same for several subsequent instances of the problem in these cases it is useful to preprocess off line this known part so as to simplify the remaining on line problem in this paper we investigate such a technique in the context of intractable i e np hard problems recent results in the literature show that not all np hard problems behave in the same way for some of them preprocessing yields polynomial time on line simplified problems we call them compilable while for other ones their compilability implies some consequences that are considered unlikely our primary goal is to provide a sound methodology that can be used to either prove or disprove that a problem is compilable to this end we define new models of computation complexity classes and reductions we find complete problems for such classes completeness meaning they are the less likely to be compilable we also investigate preprocessing that does not yield polynomial time on line algorithms but generically decreases complexity this leads us to define hierarchies of compilability that are the analog of the polynomial hierarchy a detailed comparison of our framework to the idea of parameterized tractability shows the differences between the two approaches

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

on compact representations of propositional circumscription we prove that unless the polynomial hierarchy collapses at the second level the size of a purely propositional representation of the circumscription circ t of a propositional formula t grows faster than any polynomial as the size of t increases we then analyze the significance of this result in the related field of closed world reasoning

some two sided inequalities for multiple gamma functions and related results there is an abundant literature on inequalities for the euler s gamma function c and its various related functions yet only very recently several authors began to study inequalities for the barnes double gamma function c 2 here in this paper we aim at presenting several two sided inequalities for the multiple gamma functions c n n 2 3 4 5 in our investigation of these two sided inequalities for the multiple gamma functions c n n 2 3 4 5 we employ and extend a method based upon taylor s formula and express email protected n 1 x as series involving the zeta functions we also give a more convenient explicit form of the multiple gamma functions c n email protected n n being the set of positive integers the main two sided inequalities for the multiple gamma functions c n n 2 3 4 5 which we have presented in this paper are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple gamma functions c n when email protected 6

sylvester and lyapunov equations and some interpolation problems for rational matrix functions abstract the main theme of this paper is the interrelations between interpolation problems for rational matrix functions and the linear matrix equations of sylvester and lyapunov type with suitable controllability and observability conditions using these interrelations we are able to solve certain interpolation problems for rational matrix functions which include for example the problem of constructing a rational matrix function of minimal size with prescribed zeros poles and their multiplicities as to matrix equations we prove for example a result which is in some sense the inverse to the well known chen wimmer inertia theorem an application to system theory is also included

inversion of mosaic hankel matrices via matrix polynomial systems abstract heinig and tewodors 18 give a set of components whose existence provides a necessary and sufficient condition for a mosaic hankel matrix to be nonsingular when this is the case they also give a formula for the inverse in terms of these components by converting these components into a matrix polynomial form we show that the invertibility conditions can be described in terms of matrix rational approximants for a matrix power series determined from the entries of the mosaic matrix in special cases these matrix rational approximations are closely related to pade and various well known matrix type pade approximants we also show that the inversion components can be described in terms of unimodular matrix polynomials these are shown to be closely related to the v and w matrices of antoulas used in his study of recursiveness in linear systems finally we present a recursion which allows for the efficient computation of the inversion components of all nonsingular principal mosaic hankel submatrices including the components for the matrix itself

regular rational matrix functions with prescribed null and pole data except at infinity regular rational matrix functions are constructed in realized form with prescribed null and pole data in the complex and with no restrictions at infinity the construction is based on extension theorems for null and pole pairs an application to an interpolation for regular matrix polynominals is included

on the zeros and poles of a transfer function abstract the poles and zeros of a linear transfer function can be studied by means of the pole module and the transmission zero module these algebraic constructions yield finite dimensional vector spaces whose dimensions are the number of poles and the number of multivariable zeros of the transfer function in addition these spaces carry the structure of a module over a ring of polynomials which gives them a dynamical or state space structure the analogous theory at infinity gives finite dimensional spaces which are modules over the valuation ring of proper rational functions following ideas of wedderburn and forney we introduce new finite dimensional vector spaces which measure generic zeros which arise when a transfer function fails to be injective or surjective a new exact sequence relates the global spaces of zeros the global spaces of poles and the new generic zero spaces this sequence gives a structural result which can be summarized as follows the number of zeros of any transfer function is equal to the number of poles when everything is counted appropriately the same result unifies and extends a number of results of geometric control theory by relating global poles and zeros of general possibly improper transfer functions to controlled invariant and controllability subspaces including such spaces at infinity

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and

minimal mcmillan degree rational matrix functions with prescribed local zero pole structure abstract given a rational m n matrix function w z and a subset of the complex plane c we give an explicit algorithm using column operations to produce a rational matrix function h z of minimal possible mcmillan degree which has the same left zero pole structure on as w z i e for which h wq where q and q 1 are rational n n matrix functions with no poles in

a new result on passivity preserving model reduction abstract the problem of model reduction with preservation of passivity is investigated the approach is based on positive real interpolation and is inspired by the similarity between lowner and pick matrices the former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions it follows that interpolation of the original set of data together with an appropriately defined mirror image set of data yields automatically positive real interpolants subsequently we show how this result can be implemented using a krylov projection procedure the ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large scale systems

on canonical forms for linear constant systems let f 0 h be a reachable system we introduce the notion of a nice selection and that of a minimal nice selection of column vectors of the reachability matrix g fg these notions lead to a unified treatment and a parametrization of all the and all the t canonical forms of f 0 h which are sometimes called first and second control canonical forms this global approach shows that the and the canonical forms are local coordinates of the quasi projeetive algebraic variety of all reachable systems of fixed dimension modulo basis change in the state space more over the canonical forms of f g h we mentioned above are in bijective correspondence with the so called and and matrix fraction representations of its transfer function finally we investigate a canonical form for proper rational matrix fractions which turns out to be a matrix fraction representation

a subadditive property of the gamma function abstract let minx 0 2x x and log 2 log 0 946850 we prove that the function x x is subadditive on 0 if and only if 0