micro and macro models of granular computing induced by the indiscernibility relation in rough set theory rst and more generally in granular computing on information tables grc it a central tool is the pawlak s indiscernibility relation between objects of a universe set with respect to a fixed attribute subset let us observe that pawlak s relation induces in a natural way an equivalence relation on the attribute power set that identifies two attribute subsets yielding the same indiscernibility partition we call indistinguishability relation of a given information table i the equivalence relation that can be considered as a kind of global indiscernibility in this paper we investigate the mathematical foundations of indistinguishability relation through the introduction of two new structures that are respectively a complete lattice and an abstract simplicial complex we show that these structures can be studied at both a micro granular and a macro granular level and that are naturally related to the core and the reducts of i we first discuss the role of these structures in grc it by providing some interpretations then we prove several mathematical results concerning the fundamental properties of such structures

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

a global parallel algorithm for enumerating minimal transversals of geometric hypergraphs abstract we consider the problem of enumerating all minimal hitting sets of a given hypergraph v r where v is a finite set called the vertex set and r is a set of subsets of v called the hyperedges we show that when the hypergraph admits a balanced subdivision then a recursive decomposition can be used to obtain efficiently all minimal hitting sets of the hypergraph we apply this decomposition framework to get incremental polynomial time algorithms for finding minimal hitting sets and minimal set covers for a number of hypergraphs induced by a set of points and a set of geometric objects the set of geometric objects includes half spaces hyper rectangles and balls in fixed dimension a distinguishing feature of the algorithms we obtain is that they admit an efficient global parallel implementation in the sense that all minimal hitting sets can be generated in polylogarithmic time in v r and the total number of minimal transversals m using a polynomial number of processors for half spaces in r 2 we show that the above two enumeration problems can be solved using a different technique with amortized polynomial delay

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

bimonotone linear inequalities and sublattices of rn abstract a bimonotone linear inequality is a linear inequality with at most two nonzero coefficients that are of opposite signs if both different from zero a linear inequality defines a halfspace that is a sublattice of r n a subset closed with respect to componentwise maximum and minimum if and only if it is bimonotone veinott has shown that a polyhedron is a sublattice if and only if it can be defined by a finite system of bimonotone linear inequalities whereas topkis has shown that every sublattice of r n and of more general product lattices is the solution set of a system of nonlinear bimonotone inequalities in this paper we prove that a subset of r n is the solution set of a countable system of bimonotone linear inequalities if and only if it is a closed convex sublattice similarly we note that a subset of r n is closed and convex if and only if it is the solution set of a countable system of linear inequalities we also present necessary and or sufficient conditions for a sublattice to be the intersection of the cartesian product of its projections on the coordinate axes with the solution set of a possibly infinite system of bimonotone linear inequalities we provide explicit constructions of such systems of bimonotone linear inequalities under certain assumptions on the sublattice we obtain veinott s polyhedral representation theorem and a 0 1 version of birkhoff s representation theorem as corollaries we also point out a few potential pitfalls regarding properties of sublattices of r n

ground state sign changing solutions for the schrodinger kirchhoff equation in r3 abstract in this paper with the help of variational methods in association with the deformation lemma and miranda s theorem we investigate the existence of a least energy sign changing solution which has precisely two nodal domains for the following schrodinger kirchhoff equation in r 3 a b r 3 u 2 d x u v x u f u in r 3 u h 1 r 3 where a b 0 and the potential v r 3 r is locally holder continuous and not necessarily radially symmetric

invariant sets of descending flow in critical point theory with applications to nonlinear differential equations abstract properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a banach space are studied in this way several critical points can be found by constructing different invariant sets on which the functional is bounded below under suitable conditions the existence of at least four critical points of a functional is proved each critical point being in a certain invariant set the theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations in variant cases at least four solutions are obtained for these equations

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

tropical polar cones hypergraph transversals and mean payoff games we discuss the tropical analogues of several basic questions of convex duality in particular the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy we characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs we also give a tropical analogue of farkas lemma which allows one to check whether a linear inequality is implied by a finite family of linear inequalities here the certificate is a strategy of a mean payoff game we discuss examples showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone

a note on systems with max min and max product constraints we consider a system email protected x b where email protected r m x n is a non negative matrix and email protected r m is a non negative vector over the n dimensional variable l

infinitely many small solutions for the kirchhoff equation with local sublinear nonlinearities abstract we study the following nonlinear kirchhoff equation on the whole space 1 b r 3 u 2 v x u 2 d x u v x u f x u in r 3 where b 0 is a constant and the nonlinear term f x u is only locally defined for u small and satisfies some mild conditions it is proved that the above problem admits infinity many weak solutions provided that some additional assumptions are satisfied moreover we show that these solutions tend to zero in l r 3 our method relies upon a variant of the symmetric mountain pass lemma due to kajikiya and moser iteration method

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

an o mn algorithm for regular set covering problems a clutter l is a collection of m subsets of a ground set e l x1 xn with the property that for every pair ai aj l ai is neither contained nor contains aj a transversal of l is a subset of e l intersecting every member of l if we associate with each element xj e l a weight cj the problem of finding a transversal having minimum weight is equivalent to the following set covering problem min ctx mlx 1m xj 0 1 j 1 n where ml is the matrix whose rows are the incidence vectors of the subsets ai l and 1m denotes the vector with m ones a set covering problem is regular if there exists an ordering of the variables x1 xn such that for every feasible solution x with xi 1 xj 0 j i the vector x ej ei is also a feasible solution where ei is the ith unit vector the matrix m of a regular set covering problem is said to be regular a regular clutter is any clutter whose incidence matrix is regular in this paper we describe some properties of regular clutters and propose an algorithm which in o mn steps generates all the minimal transversals of a regular clutter l and produces the transversal having minimum weight

polynomial time recognition of 2 monotonic positive boolean functions given by an oracle we consider the problem of identifying an unknown boolean function f by asking an oracle the functional values f a for a selected set of test vectors a in 0 1 n furthermore we assume that f is a positive or monotone function of n variables it is not yet known whether or not the whole task of generating test vectors and checking if the identification is completed can be carried out in polynomial time in n and m where m min t f max f f and min t f respectively max f f denotes the set of minimal true respectively maximal false vectors of f to partially answer this question we propose here two polynomial time algorithms that given an unknown positive function f of n variables decide whether or not f is 2 monotonic and if f is 2 monotonic output both sets min t f and max f f the first algorithm uses o nm 2 n 2 m time and o nm queries while the second one uses o n 3 m time and o n 3 m queries

the nehari manifold for a kirchhoff type problem involving sign changing weight functions abstract this paper examines a class of kirchhoff type equations that involve sign changing weight functions using nehari manifold and fibering map the existence of multiple positive solutions is established

existence and multiplicity of solutions for an indefinite kirchhoff type equation in bounded domains we study the indefinite kirchhoff type problem where is a smooth bounded domain in and we require that f is sublinear at the origin and superlinear at infinity using the mountain pass theorem and ekeland variational principle we obtain the multiplicity of non trivial non negative solutions we improve and extend some recent results in the literature

on the dualization of hypergraphs with bounded edge intersections and other related classes of hypergraphs given a finite set v and integers k 1 and r 0 let us denote by the class of hypergraphs with k r bounded intersections i e in which the intersection of any k distinct hyperedges has size at most r we consider the problem given a hypergraph and a subfamily of its maximal independent sets mis either extend this subfamily by constructing a new mis or prove that there are no more mis that is it is known that for hypergraphs of bounded dimension as well as for hypergraphs of bounded degree where is a constant problem can be solved in incremental polynomial time in this paper we extend this result to any integers k r such that k r is a constant more precisely we show that for hypergraphs with k r const problem is nc reducible to the problem of generating a single mis for a partial subhypergraph of in particular this implies that is polynomial and we get an incremental polynomial algorithm for generating all mis furthermore combining this result with the currently known algorithms for finding a single maximally independent set of a hypergraph we obtain efficient parallel algorithms for incrementally generating all mis for hypergraphs in the classes and where is a constant we also show that for where k r const the problem of generating all mis of can be solved in incremental polynomial time and with space polynomial only in the size of

a fast parallel algorithm for the maximal independent set problem a parallel algorithm is presented that accepts as input a graph g and produces a maximal independent set of vertices in g on a p ram without the concurrent write or concurrent read features the algorithm executes in o log n 4 time and uses o n log n 3 processors where n is the number of vertices in g the algorithm has several novel features that may find other applications these include the use of balanced incomplete block designs to replace random sampling by deterministic sampling and the use of a dynamic pigeonhole principle that generalizes the conventional pigeonhole principle

ground state sign changing solutions for kirchhoff type problems in bounded domains abstract in the present paper we consider the existence of ground state sign changing solutions for a class of kirchhoff type problems 0 1 a b u 2 d x u f u x u 0 x where r n is a bounded domain with a smooth boundary n 1 2 3 a 0 b 0 and f c r r under some weak assumptions on f with the aid of some new analytical skills and non nehari manifold method we prove that 0 1 possesses one ground state sign changing solution u b and its energy is strictly larger than twice that of the ground state solutions of nehari type furthermore we establish the convergence property of u b as the parameter b 0 our results improve and generalize some results obtained by w shuai 2015 34

dual bounded generating problems partial and multiple transversals of a hypergraph we consider two generalizations of the notion of transversal to a finite hypergraph the so called multiple and partial transversals multiple transversals naturally arise in 0 1 programming while partial transversals are related to data mining and machine learning we show that for an arbitrary hypergraph the families of multiple and partial transversals are both dual bounded in the sense that the size of the corresponding dual hypergraph is bounded by a polynomial in the cardinality and the length of description of the input hypergraph our bounds are based on new inequalities of extremal set theory and threshold boolean logic which may be of independent interest we also show that the problems of generating all multiple and all partial transversals for a given hypergraph are polynomial time reducible to the generation of all ordinary transversals for another hypergraph i e to the well known dualization problem for hypergraphs as a corollary we obtain incremental quasi polynomial time algorithms for both of the above problems as well as for the generation of all the minimal binary solutions for an arbitrary monotone system of linear inequalities

existence and asymptotic behavior of nodal solutions for the kirchhoff type problems in r3 abstract in this paper we study the existence and asymptotic behavior of nodal solutions to the following kirchhoff problem a b r 3 u 2 d x u v x u f x u in r 3 u h 1 r 3 where v x is a smooth function a b are positive constants because the so called nonlocal term r 3 u 2 d x u is involved in the equation the variational functional of the equation has totally different properties from the case of b 0 under suitable construction conditions we prove that for any positive integer k the problem has a sign changing solution u k b which changes signs exactly k times moreover the energy of u k b is strictly increasing in k and for any sequence b n 0 n there is a subsequence b n s such that u k b n s converges in h 1 r 3 to w k as s where w k also changes signs exactly k times and solves the following equation a u v x u f x u in r 3 u h 1 r 3

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

an improvement on codebook search for vector quantization presents a simple but effective algorithm to speed up the codebook search in a vector quantization scheme when a mse criterion is used a considerable reduction in the number of operations is achieved this algorithm was originally designed for image vector quantization in which the samples of the image signal pixels are positive although it can be used with any positive negative signal with only minor modifications

dual bounded generating problems efficient and inefficient points for discrete probability distributions and sparse boxes for multidimensional data we show that x n y must hold for two finite sets x email protected r n whenever they can be separated by a nonnegative linear function such that x is above y and the componentwise minimum of any two distinct points in x is dominated by some point in y as a consequence we obtain an incremental quasi polynomial time algorithm for generating all maximal integer feasible solutions for a given monotone system of separable inequalities for generating all p inefficient points of a given discrete probability distribution and for generating all maximal hyper rectangles which contain a specified fraction of points of a given set in r n this provides a substantial improvement over previously known exponential time algorithms for these generation problems related to integer and stochastic programming and data mining furthermore we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities implying that for discrete probability distributions with independent coordinates both p efficient and p inefficient points can be separately generated in incremental polynomial time

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

degrees of acyclicity for hypergraphs and relational database schemes database schemes winch intuitively are collecuons of table skeletons can be wewed as hypergraphs a hypergraph is a generalization of an ordinary undirected graph such that an edge need not contain exactly two nodes but can instead contain an arbitrary nonzero number of nodes a class of acychc database schemes was recently introduced a number of basic desirable propemes of database schemes have been shown to be equivalent to acyclicity this shows the naturalness of the concept however unlike the situation for ordinary undirected graphs there are several natural noneqmvalent notions of acyclicity for hypergraphs and hence for database schemes various desirable properties of database schemes are constdered and it is shown that they fall into several equivalence classes each completely characterized by the degree of acycliclty of the scheme the results are also of interest from a purely graph theoretic viewpomt the original notion of aeyclicity has the countermtmtive property that a subhypergraph of an acychc hypergraph can be cyclic this strange behavior does not occur for the new degrees of acyelicity that are considered

a vector distribution model and an effective nearest neighbor search method for image vector quantization in this correspondence a modified version of hunt s 1980 image model is used to interpret the distribution of image data vectors the model suggests that the diagonal line of the coordinates system is a good approximation of the principal axis of the image data vector set the validity of the model is supported by experiments following this suggestion an effective nearest neighbor search method for vector quantization of image data is developed the method is based on partitioning the vector space using hyperplanes which are perpendicular to the diagonal direction of the coordinate system the validity of the method is assessed by analyzing its complexity and comparing its performance to those of existing algorithms on a number of images

complexity of dnf minimization and isomorphism testing for monotone formulas we investigate the complexity of finding prime implicants and minimum equivalent dnfs for boolean formulas and of testing equivalence and isomorphism of monotone formulas for dnf related problems the complexity of the monotone case differs strongly from the arbitrary case we show that it is dp complete to check whether a monomial is a prime implicant for an arbitrary formula but the equivalent problem for monotone formulas is in l we show pp completeness of checking if the minimum size of a dnf for a monotone formula is at most k and for k in unary we show that the complexity of the problem drops to conp in christopher umans christopher umans the minimum equivalent dnf problem and shortest implicants journal of computer and system sciences 63 4 2001 597 611 a similar problem for arbitrary formulas was shown to be 2 p complete we show that calculating the minimum equivalent dnf for a monotone formula is possible in output polynomial time if and only if p np finally we disprove a conjecture from steffen reith steffen reith on the complexity of some equivalence problems for propositional calculi in proceedings of the 28th international symposium on mathematical foundations of computer science mfcs vol 2747 lecture notes in computer science springer 2003 pp 632 641 by showing that checking whether two formulas are isomorphic has the same complexity for arbitrary formulas as for monotone formulas

on computing all abductive explanations from a propositional horn theory abduction is a fundamental mode of reasoning with applications in many areas of ai and computer science the computation of abductive explanations is an important computational problem which is at the core of early systems such as the atms and clause management systems and is intimately related to prime implicate generation in propositional logic many algorithms have been devised for computing some abductive explanation and the complexity of the problem has been well studied however little attention has been paid to the problem of computing multiple explanations and in particular all explanations for an abductive query we fill this gap and consider the computation of all explanations of an abductive query from a propositional horn theory or of a polynomial subset of them our study pays particular attention to the form of the query ranging from a literal to a compound formula to whether explanations are based on a set of abducible literals and to the representation of the horn theory either by a horn conjunctive normal form cnf or model based in terms of its characteristic models for these combinations we present either tractability results in terms of polynomial total time algorithms intractability results in terms of nonexistence of such algorithms unless p e np or semi tractability results in terms of solvability in quasi polynomial time established by polynomial time equivalence to the problem of dualizing a monotone cnf expression our results complement previous results in the literature and refute a longstanding conjecture by selman and levesque they elucidate the complexity of generating all abductive explanations and shed light on related problems such as generating sets of restricted prime implicates of a horn theory the algorithms for tractable cases can be readily applied for generating a polynomial subset of explanations in polynomial time

identifying the minimal transversals of a hypergraph and related problems the paper considers two decision problems on hypergraphs hypergraph saturation and recognition of the transversal hypergraph and discusses their significance for several search problems in applied computer science hypergraph saturation i e given a hypergraph cal h decide if every subset of vertices is contained in or contains some edge of cal h is shown to be co np complete a certain subproblem of hypergraph saturation the saturation of simple hypergraphs i e sperner families is shown to be under polynomial transformation equivalent to transversal hypergraph recognition i e given two hypergraphs cal h 1 cal h 2 decide if the sets in cal h 2 are all the minimal transversals of cal h 1 the complexity of the search problem related to the recognition of the transversal hypergraph the computation of the transversal hypergraph is an open problem this task needs time exponential in the input size it is unknown whether an output polynomial algorithm exists for several important subcases for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs output polynomial algorithms are presented computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice which is pointed out by identifying important applications in database theory boolean switching theory logic and artificial intelligence ai particularly in model based diagnosis

on a kirchhoff type problem in r n abstract in this paper we investigate the following kirchhoff type problem p r n u 2 u 2 d x u u u p 2 u in r n u h 1 r n where n 1 2 p 2 2 2 n n 2 if n 3 2 if n 1 2 and are two positive parameters by studying the decomposition of the nehari manifold to p and using the scaling technique we give a total description on the positive solutions to p we also make an observation on the sign changing solutions to p in the current paper

sign changing solutions for a class of kirchhoff type problem in bounded domains abstract we are interested in the existence of least energy sign changing solutions for a class of kirchhoff type problem in bounded domains because the so called nonlocal term b u 2 d x u is involving in the equation the variational functional of the equation has totally different properties from the case of b 0 combining constraint variational method and quantitative deformation lemma we prove that the problem possesses one least energy sign changing solution u b moreover we show that the energy of u b is strictly larger than the ground state energy finally we regard b as a parameter and give a convergence property of u b as b 0

a fast method of designing better codebooks for image vector quantization a maximum descent md method is proposed as an alternative to the generalized lloyd lbg algorithm for vector quantization codebook generation compared with the lbg algorithm the md algorithm requires far less computation and generates codebooks that are superior in quality this paper investigates and compares the performances of various partition techniques in the md algorithm in image vector quantization the md algorithm can produce codebooks with about 1 db improvement in peak signal to noise ratio in less than 1 100 of the time required by the lbg algorithm

multiplicity of sign changing solutions for kirchhoff type equations abstract we are concerned with sign changing solutions to nonlocal perturbation of the nonlinear schrodinger equation in the whole r 3 by using the method of invariant sets of descending flow we prove the existence of multiple sign changing solutions provided the perturbation is sufficiently small

the maximum latency and identification of positive boolean functions consider the problem of identifying min t f and max f f of a positive i e monotone boolean function f by using membership queries only where min t f max f f denotes the set of minimal true vectors maximal false vectors of f it is known that an incrementally polynomial algorithm exists if and only if there is a polynomial time algorithm to check the existence of an unknown vector u for given sets mt subseteq min t f and mf subseteq max f f that is u in 0 1 n setminus v v geq w rm for some w in mt cup v v leq w rm for some w in mf this paper introduces a measure for the difficulty to find an unknown vector which is called the maximum latency if the maximum latency is constant then an unknown vector can be found in polynomial time and there is an incrementally polynomial algorithm for identification several subclasses of positive functions are shown to have constant maximum latency e g 2 monotonic positive functions delta partial positive threshold functions and matroid functions while the class of general positive functions has lfloor n 4 rfloor 1 maximum latency and the class of positive k dnf functions has omega sqrt n maximum latency

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

horn extensions of a partially defined boolean function given a partially defined boolean function pdbf t f we investigate in this paper how to find a horn extension f 0 1 n mapsto 0 1 which is consistent with t f where t subseteq 0 1 n denotes a set of true boolean vectors or positive examples and f subseteq 0 1 n denotes a set of false boolean vectors or negative examples given a pdbf t f it is known that the existence of a horn extension can be checked in polynomial time as there are many horn extensions however we consider those extensions f which have maximal and minimal sets t f of the true vectors of f respectively for a pdbf t f there always exists the unique maximal i e maximum horn extension but there are in general many minimal horn extensions we first show that a polynomial time membership oracle can be constructed for the maximum extension even if its disjunctive normal form dnf can be very long our main contribution is to show that checking if a given horn dnf represents a minimal extension and generating a horn dnf of a minimal horn extension can both be done in polynomial time we also can check in polynomial time if a pdbf t f has the unique minimal horn extension however the problems of finding a horn extension f with the smallest t f and of obtaining a horn dnf whose number of literals is smallest are both np hard

version spaces and the consistency problem a version space is a collection of concepts consistent with a given set of positive and negative examples mitchell artificial intelligence 18 1982 203 226 proposed representing a version space by its boundary sets the maximally general g and maximally specific consistent concepts s for many simple concept classes the size of g and s is known to grow exponentially in the number of positive and negative examples this paper argues that previous work on alternative representations of version spaces has disguised the real question underlying version space reasoning we instead show that tractable reasoning with version spaces turns out to depend on the consistency problem i e determining if there is any concept consistent with a set of positive and negative examples indeed we show that tractable version space reasoning is possible if and only if there is an efficient algorithm for the consistency problem our observations give rise to new concept classes for which tractable version space reasoning is now possible e g 1 decision lists monotone depth two formulas and halfspaces

product code vector quantizers for waveform and voice coding memory and computation requirements imply fundamental limitations on the quality that can be achieved in vector quantization systems used for speech waveform coding and linear predictive voice coding lpc one approach to reducing storage and computation requirements is to organize the set of reproduction vectors as the cartesian product of a vector codebook describing the shape of each reproduction vector and a scalar codebook describing the gain or energy such shape gain vector quantizers can be applied both to waveform coding using a quadratic error distortion measure and to voice coding using an itakura saito distortion measure in each case the minimum distortion reproduction vector can be found by first selecting a shape code word and then based on that choice selecting a gain codeword several algorithms are presented for the design of shape gain vector quantizers based on a traning sequence of data or a probabilistic model the algorithms are used to design shape gain vector quantizers for both the waveform coding and voice coding application the quantizers are simulated and their performance is compared to that of previously reported vector quantization systems

standing waves for nonlinear schrodinger equations with a general nonlinearity for elliptic equations e2 u v x u f u 0 x rn n 3 we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of v as e 0 under conditions on f which we believe to be almost optimal

on the structure of armstrong relations for functional dependencies an armstrong relation for a set of functional dependencies fds is a relation that satisfies each fd implied by the set but no fd that is not implied by it the structure and size number of tuples of armstrong relatsons are investigated upper and lower bounds on the size of minimal sized armstrong relations are derived and upper and lower bounds on the number of distinct entries that must appear m an armstrong relation are given it is shown that the time complexity of finding an armstrong relation gwen a set of functional dependencies is precisely exponential in the number of attributes also shown s the falsity of a natural conjecture which says that almost all relations obeying a given set of fds are armstrong relations for that set of fds finally armstrong relations are used to generahze a result obtained by demetrovics using quite complicated methods about the possible sets of keys for a relauon

dual bounded generating problems all minimal integer solutions for a monotone system of linear inequalities we consider the problem of enumerating all minimal integer solutions of a monotone system of linear inequalities we first show that for any monotone system of r linear inequalities in n variables the number of maximal infeasible integer vectors is at most rn times the number of minimal integer solutions to the system this bound is accurate up to a polylog r factor and leads to a polynomial time reduction of the enumeration problem to a natural generalization of the well known dualization problem for hypergraphs in which dual pairs of hypergraphs are replaced by dual collections of integer vectors in a box we provide a quasi polynomial algorithm for the latter dualization problem these results imply in particular that the problem of incrementally generating all minimal integer solutions to a monotone system of linear inequalities can be done in quasi polynomial time

complexity of identification and dualization of positive boolean functions abstract we consider in this paper the problem of identifying min t and max f of a positive i e monotone boolean function by using membership queries only where min t max f denotes the set of minimal true vectors maximal false vectors of it is shown that the existence of an incrementally polynomial algorithm for this problem is equivalent to the existence of the following algorithms where and g are positive boolean functions an incrementally polynomial algorithm to dualize an incrementally polynomial algorithm to self dualize a polynomial algorithm to decide if and are mutually dual a polynomial algorithm to decide if is self dual a polynomial algorithm to decide if is saturated a polynomial algorithm in min max to identify min only some of these are already well known open problems in the respective fields other related topics including various equivalent problems encountered in hypergraph theory and theory of coteries used in distributed systems are also discussed

on the desirability of acyclic database schemes a class of database schemes called acychc was recently introduced it is shown that this class has a number of desirable properties in particular several desirable properties that have been studied by other researchers m very different terms are all shown to be eqmvalent to acydicity in addition several equivalent charactenzauons of the class m terms of graphs and hypergraphs are given and a smaple algorithm for determining acychclty is presented also given are several eqmvalent characterizations of those sets m of multivalued dependencies such that m is the set of murlvalued dependencies that are the consequences of a given join dependency several characterizations for a conflict free in the sense of lien set of muluvalued dependencies are provided