the universe of propositional approximations the idea of approximate entailment has been proposed by schaerf and cadoli tractable reasoning via approximation artif intell 74 2 1995 249 310 as a way of modelling the reasoning of an agent with limited resources in that framework a family of logics parameterised by a set of propositional letters approximates classical logic as the size of the set increases the original proposal dealt only with formulas in clausal form but in finger and wassermann approximate and limited reasoning semantics proof theory expressivity and control j logic comput 14 2 2004 179 204 one of the approximate systems was extended to deal with full propositional logic giving the new system semantics an axiomatisation and a sound and complete proof method based on tableaux in this paper we extend another approximate system by schaerf and cadoli presented in a subsequent work m cadoli m schaerf the complexity of entailment in propositional multivalued logics ann math artif intell 18 1 1996 29 50 and then take the idea further presenting a more general approximation framework of which the previous ones are particular cases and show how it can be used to formalise heuristics used in theorem proving

parallel solutions of very large sparse lyapunov equations by balanced bbd decompositions in this paper a method for the parallel solution of large sparse matrix equations is presented the method is based on the balanced bordered block diagonal bbd decomposition which is applied in conjunction with the successive over relaxation sor iterative method to solve large lyapunov equations in the kronecker sum representation a variety of experimental results will be presented including solutions of lyapunov equations with matrices as large as 1993 spl times 1993

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

is intractability of nonmonotonic reasoning a real drawback abstract several studies about computational complexity of nonmonotonic reasoning nmr showed that nonmonotonic inference is significantly harder than classical monotonic inference this contrasts with the general idea that nmr can be used to make knowledge representation and reasoning simpler not harder in this paper we show that to some extent nmr fulfills the representation goal in particular we prove that nonmonotonic formalisms such as circumscription and default logic allow for a much more compact and natural representation of propositional knowledge than propositional calculus proofs are based on a suitable definition of a compilable inference problem and on non uniform complexity classes some results about intractability of circumscription and default logic can therefore be interpreted as the price one has to pay for having such an extra compact representation on the other hand intractability of inference and compactness of representation are not equivalent notions we exhibit intractable nonmonotonic formalisms whose nonmonotonic assumptions are representable by few propositional formulae finally sometimes nmr really makes reasoning simpler we present prototypical scenarios where closed world reasoning and well founded semantics account for a faster complete and unsound approximation of classical reasoning

solution of lyapunov equations by alternating direction implicit iteration abstract in this report a new procedure is presented for solving the lyapunov matrix equation first the system is reduced to tridiagonal form with gaussian similarity transformations then the resulting system is solved with alternating direction implicit adi iteration a matrix commutation property essential for model problem convergence of adi iteration applied to elliptic difference equations is not needed for this application all stable lyapunov matrix equations are model adi problems

a study of wavelets for the solution of electromagnetic integral equations the use of wavelet basis functions for the efficient solution of electromagnetic integral equations is studied it has previously been demonstrated that the use of wavelets for expansion and testing functions produces a sparse moment method matrix here this effect is examined and analyzed in terms of the radiation receiving characteristics of the wavelet basis functions the limitations of wavelets as an efficient solution technique are discussed and a comparison is made to other fast algorithms

on the use of coifman intervallic wavelets in the method of moments for fast construction of wavelet sparsified matrices orthonormal wavelets have been successfully used as basis and testing functions for the integral equations and extremely sparse impedance matrices have been obtained however in many practical problems the solution domain is confined in a bounded interval while the wavelets are originally defined on the entire real line to overcome this problem periodic wavelets have been described in the literature nonetheless the unknown functions must take on equal values at the endpoints of the bounded interval in order to apply periodic wavelets as the basis functions we present the intervallic coifman wavelets coiflets for the method of moments mom the intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets the intervallic wavelets form an orthonormal basis and preserve the same multiresolution analysis mra of other usual unbounded wavelets the coiflets possesses a special property that their scaling functions have many vanishing moments as a result the zero entries of the matrices are identified directly without using a truncation scheme with an artificially established threshold further the majority of matrix elements are evaluated directly without performing numerical integration procedures such as gaussian quadrature for an n spl times n matrix the number of actual numerical integrations is reduced from n sup 2 to the order of 3n 2l 1 when the coiflets of order l is employed the construction of intervallic wavelets is presented numerical examples of scattering problems are discussed and the relative error of this method is studied analytically

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

what makes propositional abduction tractable abduction is a fundamental form of nonmonotonic reasoning that aims at finding explanations for observed manifestations this process underlies many applications from car configuration to medical diagnosis we study here the computational complexity of deciding whether an explanation exists in the case when the application domain is described by a propositional knowledge base building on previous results we classify the complexity for local restrictions on the knowledge base and under various restrictions on hypotheses and manifestations in comparison to the many previous studies on the complexity of abduction we are able to give a much more detailed picture for the complexity of the basic problem of deciding the existence of an explanation it turns out that depending on the restrictions the problem in this framework is always polynomial time solvable np complete conp complete or s 2 p complete based on these results we give an a posteriori justification of what makes propositional abduction hard even for some classes of knowledge bases which allow for efficient satisfiability testing and deduction this justification is very simple and intuitive but it reveals that no nontrivial class of abduction problems is tractable indeed tractability essentially requires that the language for knowledge bases is unable to express both causal links and conflicts between hypotheses this generalizes a similar observation by bylander et al for set covering abduction

reasoning under minimal upper bounds in propositional logic reasoning from the minimal models of a theory as fostered by circumscription is in the area of artificial intelligence an important method to formalize common sense reasoning however as it appears minimal models may not always be suitable to capture the intuitive semantics of a knowledge base aiming intuitively at an exclusive interpretation of disjunctions of atoms i e if possible then assign at most one of the disjuncts the value true in a model in this paper we consider an approach which is more lenient and also admits non minimal models such that inclusive interpretation of disjunction also may be possible in cases where minimal model reasoning adopts an exclusive interpretation nonetheless in the spirit of minimization the approach aims at including only positive information that is necessary this is achieved by closing the set of admissible models of a theory under minimal upper bounds in the set of models of the theory which we refer to as curbing we demonstrate this method on some examples and investigate its semantical and computational properties we establish that curbing is an expressive reasoning method since the main reasoning tasks are shown to be pspace complete on the other hand we also present cases of lower complexity and in particular cases in which the complexity is located just as for ordinary minimal model reasoning at the second level of the polynomial hierarchy or even below

on the amount of nondeterminism and the power of verifying the relationship between nondeterminism and other computational resources is investigated based on the guess then check model gc systematic techniques are developed to construct natural complete languages for the classes defined by this model this improves a number of previous results in the study of limited nondeterminism connections of the model gc to computational optimization problems are exhibited

compactly representing utility functions using weighted goals and the max aggregator weighted propositional formulas can be used to model preferences over combinatorial domains each formula represents a goal we would like to see satisfied the weight of a formula represents the importance of the goal in question and to assess the desirability of a given alternative we aggregate the weights of the goals satisfied by that alternative one of several options is to aggregate by using the maximum of the weights of the satisfied goals this approach gives rise to a family of preference representation languages one for each of a range of possible restrictions we can impose on either formulas or weights we analyze the properties of these languages and establish results regarding their expressivity and absolute and relative succinctness we also study the computational complexity of the problem of finding the best and the worst alternative for a given set of weighted goals and of finding an alternative that is optimal for a group of agents for a range of different notions of collective optimality proposed in social choice theory and welfare economics

a theory of diagnosis from first principles suppose one is given a description of a system together with an observation of the system s behaviour which conflicts with the way the system is meant to behave the diagnostic problem is to determine those components of the system which when assumed to be functioning abnormally will explain the discrepancy between the observed and correct system behaviour we propose a general theory for this problem the theory requires only that the system be described in a suitable logic moreover there are many such suitable logics e g first order temporal dynamic etc as a result the theory accommodates diagnostic reasoning in a wide variety of practical settings including digital and analogue circuits medicine and database updates the theory leads to an algorithm for computing all diagnoses and to various results concerning principles of measurement for discriminating among competing diagnoses finally the theory reveals close connections between diagnostic reasoning and nonmonotonic reasoning

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

solving low density subset sum problems the subset sum problem is to decide whether or not the 0 l integer programming problem s n i l a i x i m i x i 0 or 1 has a solution where the a i and m are given positive integers this problem is np complete and the difficulty of solving it is the basis of public key cryptosystems of knapsack type an algorithm is proposed that searches for a solution when given an instance of the subset sum problem this algorithm always halts in polynomial time but does not always find a solution when one exists it converts the problem to one of finding a particular short vector v in a lattice and then uses a lattice basis reduction algorithm due to a k lenstra h w lenstra jr and l lovasz to attempt to find v the performance of the proposed algorithm is analyzed let the density d of a subset sum problem be defined by d n log 2 max i a i then for almost all problems of density d d n it is proved that the lattice basis reduction algorithm locates v extensive computational tests of the algorithm suggest that it works for densities d d c n where d c n is a cutoff value that is substantially larger than 1 n this method gives a polynomial time attack on knapsack public key cryptosystems that can be expected to break them if they transmit information at rates below d c n as n

implicates and prime implicates in random 3 sat abstract it has been observed previously that random 3 sat exhibits a phase transition at a critical ratio of constraints to variables where the average frequency of satisfiability falls abruptly from near 1 to near 0 in this paper we look beyond satisfiability to implicates and prime implicates of non zero length and show experimentally that for any given length these exhibit their own phase transitions all of these phase transitions appear to share the same critical point as the well known satisfiability phase transition we also find a rich regular pattern in which phase transitions for longer implicates or prime implicates are less steep at a given problem size and all the phase transitions sharpen with increasing problem size implicates correspond in a one to one way to nogoods and prime implicates correspond similarly to minimal nogoods knowledge about these phase transitions helps us to understand more about the behavior of search algorithms and knowledge compilation approaches in the context of random 3 sat

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

stable and high order accurate boundary treatments for the elastic wave equation on second order form stable and high order accurate boundary treatments for the elastic wave equation on second order form

optimal alternating direction implicit parameters for nonsymmetric systems of linear equations the determination of optimal parameters for the well known alternating direction implicit adi method leads to a minimization problem for rational functions of given degree l on compact sets e and f that contain the eigenvalues of the matrices involved if e and f are real intervals the solution of this problem was given by zolotarev more than 100 years ago and by wachspress in the 1960s however in the complex case e g if the linear system arises from the discretization of non self adjoint elliptic differential equations little is known about this problem in this paper two approaches to the zolotarev problem in the complex plane are considered first generalized leja points are considered as an example for rational functions which are asymptotically minimal in addition for rectangular eigenvalue domains e and f the exact minimal solutions for l 2 are constructed and it is shown by numerical examples that the results obtained using the optimal parameters of degree 2 are comparable to tho

reducing belief revision to circumscription and vice versa nonmonotonic formalisms and belief revision operators have been introduced as useful tools to describe and reason about evolving scenarios both approaches have been proven effective in a number of different situations however little is known about their relationship previous work by winslett has shown some correlations between a specific operator and circumscription in this paper we greatly extend winslett s work by establishing new relations between circumscription and a large number of belief revision operators this highlights similarities and differences between these formalisms furthermore these connections provide us with the possibility of importing results in one field into the other one

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

reasoning with models we develop a model based approach to reasoning in which the knowledge base is represented as a set of models satisfying assignments rather than a logical formula and the set of queries is restricted we show that for every propositional knowledge base kb there exists a set of characteristic models with the property that a query is true in kb if and only if it is satisfied by the models in this set we characterize a set of functions for which the model based representation is compact and provides efficient reasoning these include cases where the formula based representation does not support efficient reasoning in addition we consider the model based approach to abductive reasoning and show that for any propositional kb reasoning with its model based representation yields an abductive explanation in time that is polynomial in its size some of our technical results make use of the monotone theory a new characterization of boolean functions recently introduced the notion of restricted queries is inherent in our approach this is a wide class of queries for which reasoning is efficient and exact even when the model based representation kb provides only an approximate representation of the domain in question moreover we show that the theory developed here generalizes the model based approach to reasoning with horn expressions and captures even the notion of reasoning with horn approximations

k approximating circuits in this paper we define and study the k approximating circuits a circuit accepting a given set of inputs a is k approximated by accepting inputs that lifter from one of a by at most k bits we show that the existence of polynomial size k approximating circuits depends on the relation between k and the number of inputs

on the parameterized complexity of vertex cover and edge cover with connectivity constraints abstract we investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems namely vertex cover and edge cover specifically we impose the additional requirement that each connected component of a solution have at least t vertices resp edges from the solution for a fixed positive integer t and call the problem t total vertex cover resp t total edge cover in both cases the parameter k is the size of the solution we show that both problems remain fixed parameter tractable with these restrictions with running times of the form o c k for some constant c 0 in each case where the o notation hides polynomial factors for each fixed t 2 t total vertex cover has no polynomial kernel unless conp np poly for each fixed t 2 t total edge cover has a linear vertex kernel of size t 1 t k these results significantly improve earlier work on these problems we illustrate the utility of the technique used to solve t total vertex cover by applying it to derive an o c k time fpt algorithm for the t total edge dominating set problem our no poly kernel result for t total vertex cover and the known np hardness result for t total edge cover are in stark contrast to the fact that vertex cover has a 2 k vertex kernel and that edge cover is solvable in polynomial time this illustrates how even the slightest connectivity requirement results in a drastic change in the tractability of problems the curse of connectivity

interface waves in almost incompressible elastic materials we study the problem of two elastic half planes in contact and the stoneley interface wave that may exist at the interface between two different elastic materials emphasis being put on the case when the half planes are almost incompressible we show that numerical simulations involving interface waves require an unexpectedly high number of grid points per wavelength as the materials become more incompressible let and be the lame parameters and densities of the first and second half plane respectively a theoretical study shows that if k is a real constant k k k and 0 then for an accurate solution the required number of grid points per wavelength scales as 1 p where p is the order of accuracy of the numerical method this requirement becomes very restrictive close to the incompressible limit especially for lower order methods i e a small p the theoretical findings are supported by numerical experiments that illustrate the demanding resolution requirement as well as the superiority of higher order methods the scaling is also seen to hold for a more general choice of lame parameters numerical experiments when one of the half planes is a vacuum are also presented where the higher resolution requirement is illustrated in a numerical solution of lamb s problem

fixed parameter tractability and completeness i basic results for many fixed parameter problems that are trivially soluable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as k feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is soluable in time bounded by a polynomial of degree c where c is a constant independent of k we establish the main results of a completeness program which addresses the apparent fixed parameter intractability of many parameterized problems in particular we define a hierarchy of classes of parameterized problems fpt subseteq w 1 subseteq w 2 subseteq cdots subseteq w sat subseteq w p and identify natural complete problems for w t for t geq 2 in other papers we have shown many problems complete for w 1 dominating set is shown to be complete for w 2 and thus is not fixed parameter tractable unless independent set clique irredundant set and many other natural problems in w 2 are also fixed parameter tractable we also give a compendium of currently known hardness results as an appendix

improved kernel results for some fpt problems based on simple observations abstract in this paper we study the kernelization algorithms for several parameterized problems including parameterized co path set problem parameterized path contractibility problem and parameterized connected dominating set on g 7 graphs problem based on simple observations we give simple kernelization algorithms with kernel sizes 4 k 3 k 4 and o k 2 respectively which improves the previous best results 6 k 5 k 3 and o k 3 respectively

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

propositional circumscription and extended closed world reasoning are p p 2 complete abstract circumscription and the closed world assumption with its variants are well known nonmonotonic techniques for reasoning with incomplete knowledge their complexity in the propositional case has been studied in detail for fragments of propositional logic one open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is p 2 complete i e complete for a class of the second level of the polynomial hierarchy we answer this question by proving these problems p 2 complete and we show how this result applies to other variants of closed world reasoning

threshold dominating sets and an improved characterization of w 2 the threshold dominating set problem is that of determining for a graph g v e whether there is a subset v v of size k such that for each vertex v v there are at least r elements of the closed neighborhood n v that belong to v we consider the complexity of the problem parameterized by the pair k r it is trivial to observe that this is hard for w 2 it can also be easily shown to belong to a natural extension w 2 of w 2 defined in terms of circuit families of depth bounded by a function of the parameter we prove membership in w 2 and thus w 2 completeness using this as a starting point we prove that w 2 w 2

the intractability of computing the hamming distance given a string x and a language l the hamming distance of x to l is the minimum hamming distance of x to any string in l the edit distance of a string to a language is analogously defined first we prove that there is a language in ac0 such that both hamming and edit distance to this language are hard to approximate they cannot be approximated with factor o n 1 3 e for any e 0 unless p np n denotes the length of the input string second we show the parameterized intractability of computing the hamming distance we prove that for every t n there exists a language in ac0 for which computing the hamming distance is w t hard moreover there is a language in p for which computing the hamming distance is wp hard then we show that the problems of computing the hamming distance and of computing the edit distance are in some sense equivalent by presenting approximation ratio preserving reductions from the former to the latter and vice versa finally we define hamp to be the class of languages to which the hamming distance can efficiently i e in polynomial time be computed we show some properties of the class hamp on the other hand we give evidence that a characterization in terms of automata or formal languages might be difficult

maximum likelihood decoding of reed solomon codes is np hard maximum likelihood decoding is one of the central algorithmic problems in coding theory it has been known for over 25 years that maximum likelihood decoding of general linear codes is np hard nevertheless it was so far unknown whether maximum likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure in this paper we prove that maximum likelihood decoding is np hard for the family of reed solomon codes we moreover show that maximum likelihood decoding of reed solomon codes remains hard even with unlimited preprocessing thereby strengthening a result of bruck and naor

the complexity of selecting maximal solutions many important computational problems involve finding a maximal with respect to set inclusion solution in some combinatorial context we study such maximality problems from the complexity point of view and categorize their complexity precisely in terms of tight upper and lower bounds our results give characterizations of conp dp p2 fpnp fnp optp log n and fp p 2 in terms of subclasses of maximality problems an important consequence of our results is that finding an x minimal satisfying truth assignment for a given cnf boolean formula is complete for fnp optplog n solving an open question by papadimitriou proceedings of the 32nd ieee symposium on the foundations of computer science 1991 pp 163 169

boundary estimates for the elastic wave equation in almost incompressible materials we study the half plane problem for the elastic wave equation subject to a free surface boundary condition with particular emphasis on almost incompressible materials a normal mode analysis is developed to estimate the solution in terms of the boundary data showing that the problem is boundary stable the dependence on the material properties which is difficult to analyze by the energy method is made transparent by our estimates the normal mode technique is used to analyze the influence of truncation errors in a finite difference approximation our analysis explains why the number of grid points per wave length must be increased when the shear modulus mu becomes small compared to the first lame parameter lambda that is for almost incompressible materials when the surface waves are scaled to have unit wave length our analysis predicts that the grid size must be proportional to mu lambda 1 2 for a second order method for a fourth order method the grid size can be proportional t

the size of bdds and other data structures in temporal logics model checking temporal logic model checking is a verification method in which we describe a system the model and then we verify whether important properties expressed in a temporal logic formula hold in the system many model checking tools employ bdds or some other data structure to represent sets of states it has been empirically observed that the bdds used in these algorithms may grow exponentially as the model and formula increase in size we formally prove that no kind of data structure of polynomial size can represent the set of valid initial states for all models and all formulae this result holds for all data structures where a state can be checked in polynomial time therefore it holds not only for all types of bdds regardless of variable ordering but also for more powerful data structures such as rbcs mtbdds adds and sdds thus the size explosion of bdds is not a limit of these specific data representation structures but is unavoidable every formalism used in the same way would lead to an exponential size blow up

new wavelet based approach for time domain simulations a new numerical method for the time domain solution of maxwell s equations in linear media is proposed in this paper the field quantities derived from the spatial discretization of maxwell s equations are expanded in the time domain by wavelets on the interval this choice yields a new arrangement of the unknowns into a matrix instead of the usual vector and transforms the differential equations in time in a sylvester 1983 matrix equation the memory requirements are proportional to the number of spatial unknowns and the time evolution of the space quantities is obtained with better accuracy than in conventional marching on time techniques

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

computing circumscriptive databases i theory and algorithms abstract though circumscription was introduced by mccarthy over a decade ago there has been relatively little work on algorithms for computing circumscriptive databases in this paper we develop algorithms to compute the preferred models of circumscriptive databases at compile time using mixed integer linear programming techniques two advantages of this bottom up approach are that it makes efficient re use of previous computations and it provides much faster run time performance some other advantages of using linear programming to automate deduction at compile time are that its re optimization facilities elegantly accommodate database updates and also that it leads to a completely declarative formulation in which ordering of rules and literals in rule bodies plays no real role finally we plan to use a standard relational database system as our run time environment this should yield relatively fast run time processing and provide a more expressive query language in which aggregates and the like can be expressed easily

a new paradigm for public key identification the present paper investigates the possibility of designing zero knowledge identification schemes based on hard problems from coding theory zero knowledge proofs were introduced by goldwasser micali and rackoff 1985 their practical significance was soon demonstrated in the work of fiat and shamir 1986 who turned zero knowledge proofs of quadratic residuosity into efficient means of establishing user identities in the present paper we propose a new identification scheme based on error correcting codes which is zero knowledge and seems of practical value furthermore we describe several variants including one which has an identity based character the security of our schemes depends on the hardness of finding a word of given syndrome and prescribed small weight with respect to some randomly generated binary linear error correcting code this is of course not the first attempt to design a cryptographic scheme using tools from coding theory the difference is that identification protocols do not follow the public key paradigm based on trap door functions and described in the seminal diffie hellman paper 1976 rather they only require one way functions which opens the way to using in a rather direct manner simple combinatorial problems of the kind provided by coding theory the resulting schemes compare favorably to their number theoretic analogs

first order lub approximations characterization and algorithms one of the major approaches to approximation of logical theories is the upper and lower bounds approach introduced by selman and kautz 1991 1996 in this paper we address the problem of lowest upper bound lub approximation in a general setting we characterize lub approximations for arbitrary target languages both propositional and first order and describe algorithms of varying generality and efficiency for all target languages proving their correctness we also examine some aspects of the computational complexity of the algorithms both propositional and first order show that they can be used to characterize properties of whole families of resolution procedures discuss the quality of approximations and relate lub approximations to other approaches existing in the literature which are not typically seen in the approximation framework and which go beyond the knowledge compilation perspective that led to the introduction of lubs

sor for ax xb c we consider a new approach to the block sor method applied to linear systems of equations which can be written as a matrix equation ax xb c such systems arise for example from finite difference discretizations of seperable elliptic boundary value problems on rectangular domains on one hand this gives us an iterative method for the solution of such matrix equations e g lyapunov s matrix equation where b at and on the other hand the problem of choosing appropriate parameters for the block sor method can be written in a more compact form which may be helpful especially for non self adjoint problems i e if a and b are nonsymmetric using this technique we determine under more general assumptions than those of chin and manteuffel the optimal sor parameters for the model problem of a convection diffusion equation

default reasoning using classical logic abstract in this paper we show how propositional default theories can be characterized by classical propositional theories for each finite default theory we show a classical propositional theory such that there is a one to one correspondence between models for the latter and extensions of the former this means that computing extensions and answering queries about coherence set membership and set entailment are reducible to propositional satisfiability the general transformation is exponential but tractable for a subset which we call 2 dt a superset of network default theories and disjunction free default theories consequently coherence and set membership for the class 2 dt is np complete and set entailment is co np complete this work paves the way for the application of decades of research on efficient algorithms for the satisfiability problem to default reasoning for example since prepositional satisfiability can be regarded as a constraint satisfaction problem csp this work enables us to use csp techniques for default reasoning to illustrate this point we use the taxonomy of tractable csps to identify new tractable subsets for reiter s default logic our procedures allow also for computing stable models of extended logic programs

solution of large dense complex matrix equations utilizing wavelet like transforms this paper presents the wavelet like transforms which are quite different from the wavelet transform for the solution of large dense complex matrix equations from a purely numerical standpoint these wavelet like transforms are not true orthogonal transforms as the condition number of the resulting matrix changes after the thresholding these effects are illustrated through examples

fast solution methods in electromagnetics various methods for efficiently solving electromagnetic problems are presented electromagnetic scattering problems can be roughly classified into surface and volume problems while fast methods are either differential or integral equation based the resultant systems of linear equations are either solved directly or iteratively a review of various differential equation solvers their complexities and memory requirements is given the issues of grid dispersion and hybridization with integral equation solvers are discussed several fast integral equation solvers for surface and volume scatterers are presented these solvers have reduced computational complexities and memory requirements

semantical and computational aspects of horn approximations abstract selman and kautz proposed a method called horn approximation for speeding up inference in propositional knowledge bases their technique is based on the compilation of a propositional formula into a pair of horn formulae a horn greatest lower bound glb and a horn least upper bound lub in this paper we focus on glbs and address two questions that have been only marginally addressed so far 1 what is the semantics of the horn glbs 2 what is the exact complexity of finding them we obtain semantical as well as computational results the major semantical result is the set of minimal models of a propositional formula and the set of minimum models of its horn glbs are the same the major computational result is finding a horn glb of a propositional formula in cnf is np equivalent

krylov subspace methods for the sylvester equation abstract we describe galerkin and minimal residual algorithms for the solution of sylvester s equation ax xb c the algorithms use krylov subspaces forwhich orthogonal bases are generated by the arnoldi process for certain choices of krylov subspaces the computation of the solution splits into the solution of many independent subproblems this makes the algorithms suitable for implementation on parallel computers

stable and high order accurate difference methods for the elastic wave equation in discontinuous media in this paper we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media we consider both planar and curved interfaces where media parameters are allowed to be discontinuous the key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise we discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule conditions at layer interfaces are imposed weakly using penalties by deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability we present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials the numerical experiments verify high order accuracy and time stability

on the hardness of approximating stopping and trapping sets we prove that approximating the size of stopping and trapping sets in tanner graphs of linear block codes and more restrictively the class of low density parity check ldpc codes is np hard the ramifications of our findings are that methods used for estimating the height of the error floor of moderate and long length ldpc codes based on stopping and trapping set enumeration cannot provide accurate worst case performance predictions for most codes

minimum light number of lit only σ game on a tree let t be a tree with leaves each vertex of t is assigned a state either lit or off an assignment of states to all the vertices of t is called a configuration the lit only s game allows the player to pick a lit vertex and change the states of all its neighbours we prove that for any initial configuration one can make a sequence of allowable moves to arrive at a configuration in which the number of lit vertices is no greater than email protected email protected we also give examples to show that the bound email protected email protected cannot be relaxed to email protected email protected

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

horn approximations of empirical data abstract formal ai systems traditionally represent knowledge using logical formulas sometimes however a model based representation is more compact and enables faster reasoning than the corresponding formula based representation the central idea behind our work is to represent a large set of models by a subset of characteristic models more specifically we examine model based representations of horn theories and show that there are large horn theories that can be exactly represented by an exponentially smaller set of characteristic models we show that deduction based on a set of characteristic models requires only polynomial time as it does using horn theories more surprisingly abduction can be performed in polynomial time using a set of characteristic models whereas abduction using horn theories is np complete finally we discuss algorithms for generating efficient representations of the horn theory that best approximates a general set of models

the intractability of computing the minimum distance of a code it is shown that the problem of computing the minimum distance of a binary linear code is np hard and the corresponding decision problem is np complete this result constitutes a proof of the conjecture of berlekamp mceliece and van tilborg 1978 extensions and applications of this result to other problems in coding theory are discussed

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

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

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

semantic forgetting in answer set programming the notion of forgetting also known as variable elimination has been investigated extensively in the context of classical logic but less so in nonmonotonic logic programming and nonmonotonic reasoning the few approaches that exist are based on syntactic modifications of a program at hand in this paper we establish a declarative theory of forgetting for disjunctive logic programs under answer set semantics that is fully based on semantic grounds the suitability of this theory is justified by a number of desirable properties in particular one of our results shows that our notion of forgetting can be entirely captured by classical forgetting we present several algorithms for computing a representation of the result of forgetting and provide a characterization of the computational complexity of reasoning from a logic program under forgetting as applications of our approach we present a fairly general framework for resolving conflicts in inconsistent knowledge bases that are represented by disjunctive logic programs and we show how the semantics of inheritance logic programs and update logic programs from the literature can be characterized through forgetting the basic idea of the conflict resolution framework is to weaken the preferences of each agent by forgetting certain knowledge that causes inconsistency in particular we show how to use the notion of forgetting to provide an elegant solution for preference elicitation in disjunctive logic programming

fixed parameter tractability and completeness ii on completeness for w 1 abstract for many fixed parameter problems that are trivially solvable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is solvable in time bounded by a polynomial of degree c where c is a constant independent of k in a previous paper the w hierarchy of parameterized problems was defined and complete problems were identified for the classes w t for t 2 our main result shows that independent set is complete for w 1

on the relationship between circumscription and negation as failure abstract the aim of this paper is to investigate two powerful methods of handling negative information in logic based knowledge representation systems the logical minimization in the form of circumscription and the negation as failure rule formalized by various closures or completions of original theories we suggest a new more powerful form of the negation as failure rule and describe an important class of theories for which this form of negation as failure is equivalent to particular forms of circumscription these results establish a close relationship between the two important formalizations of nonmonotonic reasoning and provide a syntactic characterization of the corresponding circumscriptive theories this allows us to apply existing methods of deduction using various negation as failure rules to answering queries in a broad class of circumscriptive theories

an algorithm to compute circumscription abstract mccarthy s theory of circumscription appears to be the most powerful among various formalizations of nonmonotonic reasoning designed to handle incomplete and negative information in knowledge representation systems in this paper we describe a query answering algorithm for circumscriptive theories the closed world assumption cwa and its generalizations the generalized closed world assumption gcwa and the extended closed world assumption ecwa can be considered as special forms of circumscription consequently our algorithm also applies to answering queries in theories using the closed world assumption and its generalizations similarly since prioritized circumscription is equivalent to a conjunction of parallel circumscriptions our algorithm can be used to answer queries in theories circumscribed by prioritized circumscription

the parametrized complexity of some fundamental problems in coding theory the parametrized complexity of a number of fundamental problems in the theory of linear codes and integer lattices is explored concerning codes the main results are that maximum likelihood decoding and weight distrubution are hard for the parametrized complexity class w 1 the np completeness of these two problems was established by berlekamp mceliece and van tilborg in 1978 using by means of a reduction from three dimensional matching on the other hand our proof of hardness for w 1 is based on a parametric polynomial time transformation from perfect code in graphs an immediate consequence of our results is that bounded distance decoding is likely to be hard for binary linear codes concerning lattices we address the theta series problem of determining for an integer lattice l given by a set of generators and a positive integer k whether there is a vector x in l of euclidean norm k we prove here for the first time that theta series is np complete and show that it is also hard for w 1 furthermore we prove that the nearest vector problem for integer lattices is hard for w 1 these problems are the counterparts of weight distrubution and maximum likelihood decoding for lattices relations between all these problems and combinatorial problems in graphs are discussed