compact truncated toeplitz operators abstract a necessary and sufficient condition is found for a truncated toeplitz operator with bounded symbol to be compact on the model space

duality theorems for coinvariant subspaces of h1 abstract let be an inner function satisfying the connected level set condition of b cohn and let k 1 be the shift coinvariant subspace of the hardy space h 1 generated by we describe the dual space to k 1 in terms of a bounded mean oscillation with respect to the clark measure of namely we prove that k 1 z h 1 bmo the result yields a two sided estimate for the operator norm of a finite hankel matrix of size n n via bmo 2 n norm of its standard symbol where 2 n is the haar measure on the group c 2 n 1

xpath satisfiability in the presence of dtds we study the satisfiability problem associated with xpath in the presence of dtds this is the problem of determining given a query p in an xpath fragment and a dtd d whether or not there exists an xml document t such that t conforms to d and the answer of p on t is nonempty we consider a variety of xpath fragments widely used in practice and investigate the impact of different xpath operators on the satisfiability analysis we first study the problem for negation free xpath fragments with and without upward axes recursion and data value joins identifying which factors lead to tractability and which to np completeness we then turn to fragments with negation but without data values establishing lower and upper bounds in the absence and in the presence of upward modalities and recursion we show that with negation the complexity ranges from pspace to exptime moreover when both data values and negation are in place we find that the complexity ranges from nexptime to undecidable furthermore we give a finer analysis of the problem for particular classes of dtds exploring the impact of various dtd constructs identifying tractable cases as well as providing the complexity in the query size alone finally we investigate the problem for xpath fragments with sibling axes exploring the impact of horizontal modalities on the satisfiability analysis

the bergman space the bloch space and commutators of multiplication operators soit d le disque unite ouvert dans le plan complexe c soit h d l ensemble des fonctions analytiques bornees sur d on caracterise les fonctions feh d telles que t f t f tft f est un operateur compact ou t f est l operateur multiplication t f l a 2 l a 2 defini par t f g fg

on the tractability of minimal model computation for some cnf theories designing algorithms capable of efficiently constructing minimal models of conjunctive normal form theories cnfs is an important task in ai this paper provides new results along this research line and presents new algorithms for performing minimal model finding and checking over positive propositional cnfs and model minimization over propositional cnfs a cnf is positive if each of its clauses has at least a positive literal an algorithmic schema called the generalized elimination algorithm gea is presented that computes a minimal model of any positive cnf the schema generalizes the elimination algorithm ea 5 which computes a minimal model of positive head cycle free hcf cnf theories while the ea always runs in polynomial time in the size of the input hcf cnf the complexity of the gea depends on the complexity of the specific eliminating operator invoked therein which may in general turn out to be exponential therefore a specific eliminating operator is defined by which the gea computes in polynomial time a minimal model for a class of cnf that strictly includes head elementary set free hef cnf theories 14 which form in their turn a strict superset of hcf theories furthermore in order to deal with the high complexity associated with recognizing hef theories an incomplete variant of the gea called igea is proposed the resulting schema once instantiated with an appropriate elimination operator always constructs a model of the input cnf which is guaranteed to be minimal if the input theory is hef in the light of the above results the main contribution of this work is the enlargement of the tractability frontier for the minimal model finding and checking and the model minimization problems

a boundedness criterion for singular integral operators of convolution type on the fock space abstract we show that for an entire function belonging to the fock space f 2 c n on the complex euclidean space c n the integral operator s f z c n f w e z w z w d w z c n is bounded on f 2 c n if and only if there exists a function m l r n such that z r n m x e 2 x i 2 z 2 d x z c n here d w n e w 2 d w is the gaussian measure on c n with this characterization we are able to obtain some fundamental results of the operator s including the normality the c algebraic properties the spectrum and its compactness moreover we obtain the reducing subspaces of s in particular in the case n 1 we give a complete solution to an open problem proposed by k zhu for the fock space f 2 c on the complex plane c zhu 2015 30

reasoning with the finitely many valued łukasiewicz fuzzy description logic sroiq fuzzy description logics are a formalism for the representation of structured knowledge affected by imprecision or vagueness they have become popular as a language for fuzzy ontology representation to date most of the work in this direction has focused on the so called zadeh family of fuzzy operators or fuzzy logic which has several limitations in this paper we generalize existing proposals and show how to reason with a fuzzy extension of the logic sroiq the logic behind the language owl 2 under finitely many valued lukasiewicz fuzzy logic we show for the first time that it is decidable over a finite set of truth values by presenting a reasoning preserving procedure to obtain a non fuzzy representation for the logic this reduction makes it possible to reuse current representation languages as well as currently available reasoners for ontologies

bounded mean oscillation on the bidisk and operator bmo we identify several of the spaces in the inclusion chain of bmo spaces in two variables with certain bmo spaces of operator valued functions in one variable and discuss various interesting consequences of this identification our main result is a previously unknown strict inclusion in the chain of bmo spaces in two variables that translates into the fact that there exists a function b on the bidisk such that the associated little hankel operator b p 1p 2bp2p1 is bounded on products of normalized szego kernels kz1 kz2 z1 z2 d but does not extend to a bounded linear operator on h2 t2 in other words the well known result that boundedness of hankel operators in one variable can be tested on normalized szego kernels does not extend to little hankel operators in two variables in the framework of operator bmo functions this can be expressed as a new result about bmo spaces of hankel operator valued functions we also study an interesting link between the celebrated carleson counterexample l carleson mittag leffler report no 7 1974 for the bidisk and counterexamples to the operator carleson embedding theorem

a polynomial space construction of tree like models for logics with local chains of modal connectives the la logics logics with local agreement are polymodal logics defined semantically such that at any world of a model the sets of successors for the different accessibility relations can be linearly ordered and the accessibility relations are equivalence relations in a previous work we have shown that every la logic defined with a finite set of modal indices has an np complete satisfiability problem in this paper we introduce a class of la logics with a countably infinite set of modal indices and we show that the satisfiability problem is pspace complete for every logic of such a class the upper bound is shown by exhibiting a tree structure of the models this allows us to establish a surprising correspondence between the modal depth of formulae and the number of occurrences of distinct modal connectives more importantly as a consequence we can show the pspace completeness of gargov s logic dalla and nakamura s logic lgm restricted to modal indices that are rational numbers for which the computational complexity characterization has been open until now these logics are known to belong to the class of information logics and fuzzy modal logics respectively

on the failure of the finite model property in some fuzzy description logics fuzzy description logics dls are a family of logics which allow the representation of and the reasoning with structured knowledge affected by vagueness although most of the not very expressive crisp dls such as alc enjoy the finite model property fmp this is not the case once we move into the fuzzy case in this paper we show that if we allow arbitrary knowledge bases then the fuzzy dls alc under lukasiewicz and product fuzzy logics do not verify the fmp even if we restrict to witnessed models in other words finite satisfiability and witnessed satisfiability are different for arbitrary knowledge bases the aim of this paper is to point out the failure of fmp because it affects several algorithms published in the literature for reasoning under fuzzy dls

product of truncated hankel and truncated toeplitz operators abstract a truncated toeplitz operator is the compression of a classical toeplitz operator on the hardy space to a model space a truncated hankel operator is the compression of a hankel operator on the hardy space to the orthogonal complement of a model space we study the product of a truncated hankel operator and a truncated toeplitz operator and characterize when such a product is zero or compact

tractable reasoning via approximation abstract problems in logic are well known to be hard to solve in the worst case two different strategies for dealing with this aspect are known from the literature language restriction and theory approximation in this paper we are concerned with the second strategy our main goal is to define a semantically well founded logic for approximate reasoning which is justifiable from the intuitive point of view and to provide fast algorithms for dealing with it even when using expressive languages we also want our logic to be useful to perform approximate reasoning in different contexts we define a method for the approximation of decision reasoning problems based on multivalued logics our work expands and generalizes in several directions ideas presented by other researchers the major features of our technique are 1 approximate answers give semantically clear information about the problem at hand 2 approximate answers are easier to compute than answers to the original problem 3 approximate answers can be improved and eventually they converge to the right answer 4 both sound approximations and complete ones are described the method we propose is flexible enough to be applied to a wide range of reasoning problems in our research we considered approximation of several decidable problems with different worstcase complexity involving both propositional and first order languages in particular we defined approximation techniques for propositional logic fragments of first order logic concept description languages and modal logic in our research we also addressed the issue of representing the knowledge of a reasoner with limited resources and how to use such a knowledge for approximate reasoning purposes

polynomial time implication problems for unary inclusion dependencies unary inclusion dependencies are database constraints expressing subset relationships the decidability of implication for these dependencies together with embedded implicational dependencies such as functional dependencies are investigated as shown by casanova et al the unrestricted and finite implication problems are different for the class of functional and unary inclusion dependencies also for this class and for any fixed k finite implication has no k ary complete axiomatization for both of these problems complete axiomatizations and polynomial time decision procedures are provided linear time for unrestricted implication and cubic time for finite implication it follows that functional and unary inclusion dependencies form a semantically natural class of first order sentences with equality which although not finitely controllable is efficiently solvable and docile generalizing from these results it is shown that the interaction between functionaland inclusion dependencies characterizes 1 unrestricted implication of unary inclusion and all embedded implicational dependencies 2 finite implication of unary inclusion and all full implicational dependencies 3 finite implication of unary inclusion and all embedded tuple generating dependencies as a direct consequence of this analysis most of the applications of dependency implication are extended within polynomial time to database design problems involving unary inclusion dependencies such examples are tests for lossless joins and tests for complementarity of projective views finally if one additionally requires that

a pointwise finite dimensional reduction method for einstein lichnerowicz type systems the six dimensional case abstract we construct blowing up solutions for smooth perturbations of a focusing einstein lichnerowicz system on a closed 6 dimensional manifold our construction combines asymptotic analysis techniques with a finite dimensional reduction approach following the scheme of proof developed in premoselli 2016 the blow up examples constructed here highlight in particular the role of the coupling field in dimension 6

fuzzy description logics with general t norms and datatypes fuzzy description logics dls are a family of logics which allow the representation of and the reasoning within structured knowledge affected by vagueness although a relatively important amount of work has been carried out in the last years current fuzzy dls still present several limitations in this work we face two problems the common restriction to zadeh and lukasiewicz fuzzy logics and the inability to deal with datatypes different from fuzzy sets in particular we propose a semantics based on the use of a general left continuous t norm and an involutive negation specially focused on product logic and furthermore we show how to handle functional concrete roles relating individuals of the domain and strings real or integer numbers

dual truncated toeplitz operators abstract let u be a nonconstant inner function in this paper we study the dual truncated toeplitz operators on the orthogonal complement of the model space k u 2 this is a new class of toeplitz operator we show the product of two dual truncated toeplitz operators d f d g to be zero if and only if either f or g is zero we give a necessary and sufficient condition for the product of two dual truncated toeplitz operators to be a finite rank operator furthermore a necessary and sufficient condition is found for the product of two dual truncated toeplitz operators to be a dual truncated toeplitz operator the last two results are different from the classical toeplitz operator theory

an epistemic operator for description logics abstract description logics also called terminological logics or concept languages are fragments of first order logic that provide a formal account of the basic features of frame based systems however there are aspects of frame based systems such as nonmonotonic reasoning and procedural rules that cannot be characterized in a standard first order framework such features are needed for real applications and a clear understanding of the logic underlying them is necessary for principled implementations we show how description logics enriched with an epistemic operator can formalize such aspects the logic obtained is a fragment of a first order nonmonotonic modal logic we show that the epistemic operator formalizes procedural rules as provided in many knowledge representation systems and enables sophisticated query formulation including various forms of closed world reasoning we provide an effective procedure for answering epistemic queries posed to a knowledge base expressed in a description logic and extend this procedure in order to deal with rules we also address the computational complexity of reasoning with the epistemic operator identifying cases in which an appropriate use of the epistemic operator can help in decreasing the complexity of reasoning

matrix representations of truncated toeplitz operators abstract in this paper we describe matrix representations of truncated toeplitz operators on the model space k b where b is an infinite blaschke product satisfying some additional conditions our results are extensions of that obtained by cima ross and wogen in 2008

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

on reasoning about structural equality in xml a description logic approach we define a boolean complete dialect of description logic called dlfdreg that can be used to reason about structural equality in semistructured ordered data in the presence of document type definitions this application depends on the novel ability of dlfdreg to express functional dependencies over possibly infinite sets of feature paths defined by regular languages we also present a decision procedure for the associated logical implication problem the procedure underlies a mapping of such problems to satisfiability problems of datalog ns

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

the kl one family abstract the knowledge representation system kl one has been one of the most influential and imitated knowledge representation systems in the artificial intelligence community begun at bolt beranek and newman in 1978 kl one pioneered the development of taxonomic representations that can automatically classify and assimilate new concepts based on a criterion of terminological subsumption this theme generated considerab interest in both the formal community and a large community of potential users the kl one community has since expanded to include many systems at many institutions and in many different countries this paper introduces the kl one family and discusses some of the main themes explored by kl one and its successors we give an overview of current research describe some of the systems that have been developed and outline some future research directions

non compactness and multiplicity results for the yamabe problem on sn the aim of this paper is to show the existence of metrics ge on sn where ge is a perturbation of the standard metric g0 for which the yamabe problem possesses a sequence of solutions unbounded in l sn the metrics ge that we find are of class ck on sn with k n 34 we also prove some new multiplicity results

unitary equivalence to a complex symmetric matrix low dimensions abstract a matrix t m n c is uecsm if it is unitarily equivalent to a complex symmetric i e self transpose matrix we develop several techniques for studying this property in dimensions three and four among other things we completely characterize 4 4 nilpotent matrices which are uecsm and we settle an open problem which has lingered in the 3 3 case we conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above

a theorem of brown halmos type for bergman space toeplitz operators abstract we study the analogues of the brown halmos theorem for toeplitz operators on the bergman space we show that for f and g harmonic t f t g t h only in the trivial case provided that h is of class c 2 with the invariant laplacian bounded here the trivial cases are f or g holomorphic from this we conclude that the zero product problem for harmonic symbols has only the trivial solution finally we provide examples that show that the brown halmos theorem fails for general symbols even for symbols continuous up to the boundary

combining interval based temporal reasoning with general tboxes while classical description logics dls concentrate on the representation of static conceptual knowledge recently there is a growing interest in dls that additionally allow to capture the temporal aspects of conceptual knowledge such temporal dls are based either on time points or on time intervals as the temporal primitive whereas point based temporal dls are well investigated this is not the case for interval based temporal dls all known logics either suffer from rather limited expressive power or have undecidable reasoning problems in particular there exists no decidable interval based temporal dl that provides for general tboxes one of the most important expressive means in modern description logics in this paper for the first time we define an interval temporal dl that is equipped with general tboxes and for which reasoning is decidable and more precisely exptime complete

an algebraic semantics approach to the effective resolution of type equations abstract this article presents a syntactic calculus of partially ordered data type structures and its application to computation a syntax of record like terms and a type subsumption ordering are defined and shown to form a lattice structure a simple type as set interpretation of these term structures extends this lattice to a distributive one and in the case of finitary terms to a complete brouwerian lattice as a result a method for solving systems of type equations by iterated rewriting of type symbols is proposed which defines an operational semantics for kbl a knowledge base language it is shown that a kbl program can be seen as a system of equations thanks to the lattice properties of finite structures systems of simultaneous type equations are shown to admit least fixed point solutions an operational semantics for kbl is described as term rewriting fan out rewriting a particular rewriting computation order related to the conventional outermost term rewriting which rewrites symbols closer to the root first is defined and shown to be maximal correctness with respect to least fixed point semantics of kbl s operational semantics as defined by fan out rewriting is discussed finally extensions and further research directions are sketched

reasoning within expressive fuzzy rough description logics it is generally accepted that the management of imprecision and vagueness will yield more intelligent and realistic knowledge based applications description logics dls are suitable well known logics for managing structured knowledge that have gained considerable attention the last decade the current research progress and the existing problems of uncertain or imprecise knowledge representation and reasoning in dls are analyzed in this paper an integration between the theories of fuzzy dls and rough dls has been attempted by providing fuzzy rough dls based on fuzzy rough set theory the syntax semantics and properties of fuzzy rough dls are given it is proved that the satisfiability subsumption entailment and abox consistency reasoning in fuzzy rough dls may be reduced to the abox consistency reasoning in the corresponding fuzzy dls

symbols of truncated toeplitz operators abstract we consider three topics connected with coinvariant subspaces of the backward shift operator in hardy spaces h p properties of truncated toeplitz operators carleson type embedding theorems for the coinvariant subspaces factorizations of pseudocontinuable functions from h 1 these problems turn out to be closely connected and even in a sense equivalent the new approach based on the factorizations allows us to answer a number of challenging questions about truncated toeplitz operators posed by d sarason

a reproducing kernel thesis for operators on bergman type function spaces abstract in this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on bergman type spaces in particular the results in this paper apply to the weighted bergman space on the unit ball the unit polydisc and more generally to weighted fock spaces

modal logics description logics and arithmetic reasoning forthcoming in the journal of ariticial intelligence we introduce mathematical programming and atomic decomposition as the basic modal t box inference techniques for a large class of modal and description logics the class of description logics suitable for the proposed methods is strong on the arithmetical side in particular there may be complex arithmetical conditions on sets of accessible worlds role llers the atomic decomposition technique can deal with set constructors for modal parameters role terms and parameter role hierarchies specied in full propositional logic besides the standard modal operators a number of other constructors can be added in a relatively straightforward way examples are graded modalities qualied number restrictions and also generalized quantiers like most n more and many

spectral properties of truncated toeplitz operators by equivalence after extension abstract we study truncated toeplitz operators in model spaces k p for 1 p with essentially bounded symbols in a class including the algebra c r h as well as sums of analytic and anti analytic functions satisfying a separation condition using their equivalence after extension to toeplitz operators with 2 2 matrix symbols we establish fredholmness and invertibility criteria for truncated toeplitz operators with separated symbols and in particular we identify a class of operators for which semi fredholmness is equivalent to invertibility for symbols in c r h we extend to all p 1 the spectral mapping theorem for the essential spectrum stronger results are obtained in the case of operators with rational symbols or if the underlying model space is finite dimensional

zero products of toeplitz operators we prove that the product of finitely many toeplitz operators oil the hardy space is zero if and only if at least one of the operators is zero we use some new vector valued techniques that not only lead to a vector valued version of this result but also appear to be needed in the scalar case

on the complexity of verifying consistency of xml specifications xml specifications often consist of a type definition typically a document type definition dtd and a set of integrity constraints it has been shown previously that such specifications can be inconsistent and thus it is often desirable to check consistency at compile time it is known w fan and l libkin j acm 49 2002 pp 368 406 that for general keys foreign keys and dtds the consistency problem is undecidable however it becomes np complete when all keys are one attribute unary and tractable if no foreign keys are used in this paper we consider a variety of previously studied constraints for xml data and investigate the complexity of the consistency problem our main conclusion is that in the presence of foreign key constraints compile time verification of consistency is infeasible we look at absolute constraints that hold in the entire document and relative constraints that hold only in a part of the document for absolute constraints we prove decidability and establish complexity bounds for primary multiattribute keys and unary foreign keys and study unary constraints that involve regular expressions for relative constraints we prove that even for unary constraints the consistency problem is undecidable we also show that results continue to hold for extended dtds a more expressive typing mechanism for xml

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

reasoning with minimal models efficient algorithms and applications reasoning with minimal models is at the heart of many knowledge representation systems yet it turns out that this task is formidable even when very simple theories are considered in this paper we introduce the elimination algorithm which performs in linear time minimal model finding and minimal model checking for a significant subclass of positive cnf theories which we call positive head cycle free hcf theories we also prove that the task of minimal entailment is easier for positive hcf theories than it is for the class of all positive cnf theories finally we show how variations of the elimination algorithm can be applied to allow queries posed on disjunctive deductive databases and disjunctive default theories to be answered in an efficient way

on the complexity of integer programming a simple proof that integer programming ts in x s given the proof also estabhshes that there s a pseudopolynomial tune algorithm for integer programmmg with any fixed number of constraints

axiomatisations of functional dependencies in the presence of records lists sets and multisets we investigate functional dependencies in databases that support complex values such as records lists sets anu multisets therefore an abstract algebraic framework is proposed that classifies data models according to the underlying types they support this allows to emphasise the impact of the data types rather than the specifics of a particular data model the main results are finite minimal sound and complete sets of inference rules for the implication of functional dependencies in the presence of records and all combinations of lists sets and multisets the inference rules are similar to armstrong s original axioms for the relational data model thanks to the algebraic framework the completeness result however requires a deep analysis in the case of sets and in particular multisets

reasoning with rough description logics the current research progress and the existing problems of uncertain or imprecise knowledge representation and reasoning in description logics are analyzed in this paper approximate concepts are introduced to description logics based on rough set theory and a kind of new rough description logic rdl ac rough description logic based on approximate concepts is proposed based on approximate concepts the syntax semantics and properties of the rdl ac are given it is proved that the approximate concept satisfiability definitely satisfiability and possibly satisfiability reasoning problem and approximate concepts rough subsumption reasoning problem w r t rough tbox in rdl ac may be reduced to the concept satisfiability reasoning problem in almost standard alc the description logic that provides the boolean concept constructors plus the existential and universal restriction constructors the works of this paper provide logic foundations for approximate ontologies and theoretical foundations for reasoning algorithms of more expressive rough description logics including approximate concepts number restrictions nominals inverse roles and role hierarchies

bounded symbols and reproducing kernel thesis for truncated toeplitz operators compressions of toeplitz operators to coinvariant subspaces of h2 are called truncated toeplitz operators we study two questions related to these operators the first raised by sarason is whether boundedness of the operator implies the existence of a bounded symbol the second is the reproducing kernel thesis we show that in general the answer to the first question is negative and we exhibit some classes of spaces for which the answers to both questions are positive

on xml integrity constraints in the presence of dtds the article investigates xml document specifications with dtds and integrity constraints such as keys and foreign keys we study the consistency problem of checking whether a given specification is meaningful that is whether there exists an xml document that both conforms to the dtd and satisfies the constraints we show that dtds interact with constraints in a highly intricate way and as a result the consistency problem in general is undecidable when it comes to unary keys and foreign keys the consistency problem is shown to be np complete this is done by coding dtds and integrity constraints with linear constraints on the integers we consider the variations of the problem by both restricting and enlarging the class of constraints and identify a number of tractable cases as well as a number of additional np complete ones by incorporating negations of constraints we establish complexity bounds on the implication problem which is shown to be conp complete for unary keys and foreign keys

non compactness and infinite number of conformal initial data sets in high dimensions abstract on any closed riemannian manifold of dimension greater than 7 we construct examples of background physical coefficients for which the einstein lichnerowicz equation possesses a non compact set of positive solutions this yields in particular the existence of an infinite number of positive solutions in such cases

a rational reconstruction of nonmonotonic truth maintenance systems research note abstract the main contribution of this paper is a precise characterization of the inferences performed by nonmonotonic truth maintenance systems tmss using two standard nonmonotonic formalisms logic programming with the stable set semantics and autoepistemic logic the paper also contains an analysis of the role of dependency directed backtracking in dealing with contradictions and a proof that implementing a nonmonotonic tms is an np complete problem

fuzzy terms in this paper we will show how purely categorical constructions of terms are advantageous when investigating situations concerning uncertainty more specifically where uncertainty comes from and how uncertainty is integrated when dealing with terms over selected signatures there are basically two ways of invoking uncertainty for terms on one hand we may proceed by building composed monads where uncertainty is provided by some suitable monad composed with the traditional term monad on the other hand we can provide a strictly formal basis for term monads being created over categories themselves carrying uncertainty this is the distinction between computing with fuzzy and fuzzy computing and the fundamental question raised by these constructions is where uncertainty resides in language constructions for logic this paper also shows how the notion of signature often needs to be expanded to levels of signatures in particular when dealing with type constructors such levels allow us to strictly delineate e g primitive operations type terms and value level terms levels of signature will in this paper be exemplified by the construction of the signature of simply typed lambda calculus

combining horn rules and description logics in carin abstract we describe carin a novel family of representation languages that combine the expressive power of horn rules and of description logics we address the issue of providing sound and complete inference procedures for such languages we identify existential entailment as a core problem in reasoning in carin and describe an existential entailment algorithm for the alcnr description logic as a result we obtain a sound and complete algorithm for reasoning in non recursive carin alcnr knowledge bases and an algorithm for rule subsumption over alcnr we show that in general the reasoning problem for recursive carin alcnr knowledge bases is undecidable and identify the constructors of alcnr causing the undecidability we show two ways in which carin alcnr knowledge bases can be restricted while obtaining sound and complete reasoning

the complexity of existential quantification in concept languages abstract much of the research on concept languages which also are called terminological languages has focused on the computational complexity of subsumption the intractability results can be divided into two groups first it has been shown that extending the basic language jy with constructs containing some form of logical disjunction leads to co np hard subsumption problems secondly adding negation to jy makes subsumption pspace complete the main result of this paper is that extending jy with unrestricted existential quantification makes subsumption np complete this is the first proof of intractability for a concept language containing whether explicitly or implicitly no construct expressing disjunction unrestricted existential quantification is therefore alongside disjunction a source of computational complexity in concept languages

on rank one asymmetric truncated toeplitz operators on finite dimensional model spaces abstract in this paper we present some consequences of the description of matrix representations of asymmetric truncated toeplitz operators acting between finite dimensional model spaces we then describe all rank one asymmetric truncated toeplitz operators acting between finite dimensional model spaces

subsumption computed algebraically abstract this paper deals with terminological representation languages for kl one type knowledge representation systems such languages are based on the two primitive syntactic types called concepts and roles which are usually represented model theoretically as sets and binary relations respectively rather than following the model theoretic route we show that the semantics can be naturally accomodated in the context of an equational algebra of relations interacting with sets equational logic is then a natural vehicle for computing subsumptions both of concepts and of roles we thus propose the algebraic rather than model theoretic computation of subsumption

a variational analysis of einstein scalar field lichnerowicz equations on compact riemannian manifolds we establish new existence and non existence results for positive solutions of the einstein scalar field lichnerowicz equation on compact manifolds this equation arises from the hamiltonian constraint equation for the einstein scalar field system in general relativity our analysis introduces variational techniques in the form of the mountain pass lemma to the analysis of the hamiltonian constraint equation which has been previously studied by other methods

extending and implementing the stable model semantics a novel logic program like language weight constraint rules is developed for answer set programming purposes it generalizes normal logic programs by allowing weight constraints in place of literals to represent e g cardinality and resource constraints and by providing optimization capabilities a declarative semantics is developed which extends the stable model semantics of normal programs the computational complexity of the language is shown to be similar to that of normal programs under the stable model semantics a simple embedding of general weight constraint rules to a small subclass of the language called basic constraint rules is devised an implementation of the language the smodels system is developed based on this embedding it uses a two level architecture consisting of a front end and a kernel language implementation the front end allows restricted use of variables and functions and compiles general weight constraint rules to basic constraint rules a major part of the work is the development of an efficient search procedure for computing stable models for this kernel language the procedure is compared with and empirically tested against satisfiability checkers and an implementation of the stable model semantics it offers a competitive implementation of the stable model semantics for normal programs and attractive performance for problems where the new types of rules provide a compact representation

mixed integer programming methods for computing nonmonotonic deductive databases though the declarative semantics of both explicit and nonmonotonic negation in logic programs has been studied extensively relatively little work has been done on computation and implementation of these semantics in this paper we study three different approaches to computing stable models of logic programs based on mixed integer linear programming methods for automated deduction introduced by r jeroslow we subsequently discuss the relative efficiency of these algorithms the results of experiments with a prototype compiler implemented by us tend to confirm our theoretical discussion in contrast to resolution the mixed integer programming methodology is both fully declarative and handles reuse of old computations gracefully we also introduce compare implement and experiment with linear constraints corresponding to four semantics for explicit negation in logic programs the four valued annotated semantics blair and subrahmanian 1989 the gelfond lifschitz semantics 1990 the over determined models grant and subrahmanian 1989 the gelfond lifschitz semantics 1990 the over determined models grant and subrahmanian 1990 and the classical logic semantics gelfond and lifschitz 1990 argue for simultaneous use of two modes of negation in logic programs classical and nonmonotonic so we give algorithms for computing answer sets for such logic programs too

tarskian set constraints we investigate set constraints over set expressions with tarskian functional and relational operations unlike the herbrand constructor symbols used in recent set constraint formalisms the meaning of a tarskian function symbol is interpreted in an arbitrary first order structure we show that satisfiability of tarskian set constraints is decidable in nondeterministic doubly exponential time we also give complexity results and open problems for various extensions and restrictions of the language

autoepistemic logic autoepistemic logic is one of the principal modes of nonmonotonic reasoning it unifies several other modes of nonmonotonic reasoning and has important application in logic programming in the paper a theory of autoepistemic logic is developed this paper starts with a brief survey of some of the previously known results then the nature of nonmonotonicity is studied by investigating how membership of autoepistemic statements in autoepistemic theories depends on the underlying objective theory a notion similar to set theoretic forcing is introduced expansions of autoepistemic theories are also investigated expansions serve as sets of consequences of an autoepistemic theory and they can also be used to define semantics for logic programs with negation theories that have expansions are characterized and a normal form that allows the description of all expansions of a theory is introduced our results imply algorithms to determine whether a theory has a unique expansion sufficient conditions stratification that imply existence of a unique expansion are discussed the definition of stratified theories is extended and under some additional assumptions efficient algorithms for testing whether a theory is stratified are proposed the theorem characterizing expansions is applied to two classes of theories k1 theories and ae programs in each case simple hypergraph characterization of expansions of theories from each of these classes is given finally connections with stable model semantics for logic programs with negation is discussed in particular it is proven that the problem of existence of stable models is np complete authors abstract

any world assumptions in logic programming due to the usual incompleteness of information representation any approach to assign a semantics to logic programs has to rely on a default assumption on the missing information the stable model semantics that has become the dominating approach to give semantics to logic programs relies on the closed world assumption cwa which asserts that by default the truth of an atom is false there is a second well known assumption called open world assumption owa which asserts that the truth of the atoms is supposed to be unknown by default however the cwa the owa and the combination of them are extremal though important assumptions over a large variety of possible assumptions on the truth of the atoms whenever the truth is taken from an arbitrary truth space the topic of this paper is to allow any assignment i e interpretation over a truth space to be a default assumption our main result is that our extension is conservative in the sense that under the everywhere false default assumption cwa the usual stable model semantics is captured due to the generality and the purely algebraic nature of our approach it abstracts from the particular formalism of choice and the results may be applied in other contexts as well

generalized satisfiability for the description logic alc the standard reasoning problem concept satisfiability in the basic description logic alc is pspace complete and it is exptime complete in the presence of general concept inclusions several fragments of alc notably logics in the fl el and dl lite families have an easier satisfiability problem for some of these logics satisfiability can be decided in polynomial time we classify the complexity of the standard variants of the satisfiability problem for all possible boolean and quantifier fragments of alc with and without general concept inclusions

exp time tableaux for alc abstract the last years have seen two major advances in knowledge representation and reasoning first many interesting problems ranging from semi structured data to linguistics were shown to be expressible in logics whose main deductive problems are exptime complete second experiments in automated reasoning have substantially broadened the meaning of practical tractability instances of realistic size for pspace complete problems are now within reach for implemented systems still there is a gap between the reasoning services needed by the expressive logics mentioned above and those provided by the current systems indeed the algorithms based on tree automata which are used to prove exptime completeness require exponential time and space even in simple cases on the other hand current algorithms based on tableau methods can take advantage of such cases but require double exponential time in the worst case we propose a tableau calculus for the description logic alc for checking the satisfiability of a concept with respect to a tbox with general axioms and transform it into the first simple tableau based decision procedure working in single exponential time to guarantee the ease of implementation we also discuss the effects that optimizations propositional backjumping simplification semantic branching etc might have on our complexity result and introduce a few optimizations ourselves

magic sets for disjunctive datalog programs in this paper a new technique for the optimization of partially bound queries over disjunctive datalog programs with stratified negation is presented the technique exploits the propagation of query bindings and extends the magic set optimization technique originally defined for non disjunctive programs an important feature of disjunctive datalog programs is non monotonicity which calls for non deterministic implementations such as backtracking search a distinguishing characteristic of the new method is that the optimization can be exploited also during the non deterministic phase in particular after some assumptions have been made during the computation parts of the program may become irrelevant to a query under these assumptions this allows for dynamic pruning of the search space in contrast the effect of the previously defined magic set methods for disjunctive datalog is limited to the deterministic portion of the process in this way the potential performance gain by using the proposed method can be exponential as could be observed empirically the correctness of the method is established and proved in a formal way thanks to a strong relationship between magic sets and unfounded sets that has not been studied in the literature before this knowledge allows for extending the method and the correctness proof also to programs with stratified negation in a natural way the proposed method has been implemented in the dlv system and various experiments on synthetic as well as on real world data have been conducted the experimental results on synthetic data confirm the utility of magic sets for disjunctive datalog and they highlight the computational gain that may be obtained by the new method with respect to the previously proposed magic set method for disjunctive datalog programs further experiments on data taken from a real life application show the benefits of the magic set method within an application scenario that has received considerable attention in recent years the problem of answering user queries over possibly inconsistent databases originating from integration of autonomous sources of information

generalized fuzzy rough description logics description logics dls are a family of logics for representing structured knowledge which have proved to be very useful as ontology languages classical dls are not suitable to represent vague pieces of information the attempts to achieve a solution have led to the birth of fuzzy dls and rough dls in this work we provide a simple solution to join these two formalisms and define a fuzzy rough dl this logic is more general than other related approaches including tight and loose fuzzy rough approximations and being independent of the fuzzy logic operators considered we show the usefulness of our approach by presenting some uses case and we also describe how to extend two reasoning algorithms for fuzzy dls which are implemented in the fuzzy dl reasoners fuzzydl and delorean

the complexity of concept languages abstract a basic feature of terminological knowledge representation systems is to represent knowledge by means of taxonomies here called terminologies and to provide a specialized reasoning engine to do inferences on these structures the taxonomy is built through a representation language called a concept language or description logic which is given a well defined set theoretic semantics the efficiency of reasoning has often been advocated as a primary motivation for the use of such systems the main contributions of the paper are 1 a complexity analysis of concept satisfiability and subsumption for a wide class of concept languages 2 algorithms for these inferences that comply with the worst case complexity of the reasoning task they perform

decidability of shiq with complex role inclusion axioms motivated by medical terminology applications we investigate the decidability of an expressive and prominent description logic dl shiq extended with role inclusion axioms of the form r s t it is well known that a naive such extension leads to undecidability and thus we restrict our attention to axioms of the form r s r or s r r which is the most important form of axioms in the applications that motivated this extension surprisingly this extension is still undecidable however it turns out that by restricting our attention further to acyclic sets of such axioms we regain decidability we present a tableau based decision procedure for this dl and report on its implementation which promises to behave well in practice and provides important additional functionality in a medical terminology application

the effect of linear perturbations on the yamabe problem in conformal geometry the compactness conjecture asserts that the set of yamabe metrics on a smooth compact aspherical riemannian manifold left m g right is compact established in the locally conformally flat case by schoen lecture notes in mathematics vol 1365 pp 120 154 springer berlin 1989 surveys pure application and mathematics 52 longman science technology pp 311 320 harlow 1991 and for n le 24 by khuri marques schoen j differ geom 81 1 143 196 2009 it has revealed to be generally false for n ge 25 as shown by brendle j am math soc 21 4 951 979 2008 and brendle marques j differ geom 81 2 225 250 2009 a stronger version of it the compactness under perturbations of the yamabe equation is addressed here with respect to the linear geometric potential frac n 2 4 n 1 mathrm scal g mathrm scal g being the scalar curvature of left m g right we show that a priori l infty bounds fail for linear perturbations on all manifolds with n ge 4 as well as a priori gradient l 2 bounds fail for non locally conformally flat manifolds with n ge 6 and for locally conformally flat manifolds with n ge 7 in several situations the results are optimal our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g

two variable logic on data trees and xml reasoning motivated by reasoning tasks for xml languages the satisfiability problem of logics on data trees is investigated the nodes of a data tree have a label from a finite set and a data value from a possibly infinite set it is shown that satisfiability for two variable first order logic is decidable if the tree structure can be accessed only through the child and the next sibling predicates and the access to data values is restricted to equality tests from this main result decidability of satisfiability and containment for a data aware fragment of xpath and of the implication problem for unary key and inclusion constraints is concluded

making fuzzy description logic more general a version of fuzzy description logic based on the basic continuous t norm based fuzzy predicate logic bl is presented problems of satisfiability validity and subsumption of concepts are discussed and reduced to problems of fuzzy propositional logic known to be decidable for any continuous t norm for lukasiewicz t norm some stronger results are obtained

determining inference semantics for disjunctive logic programs abstract in a seminal paper gelfond and lifschitz 34 introduced simple disjunctive logic programs where in rule heads the disjunction operator is used to express incomplete information and defined the answer set semantics called gl semantics for short based on a program transformation called gl reduct and the minimal model requirement our observations reveal that the requirement of the gl semantics i e an answer set should be a minimal model of rules of the gl reduct may sometimes be too strong a condition and exclude some answer sets that would be reasonably acceptable to address this we present an alternative more permissive answer set semantics called the determining inference di semantics specifically we introduce a head selection function to formalize the operator and define answer sets as follows i given an interpretation i and a selection function sel we transform a disjunctive program into a normal program s e l i called a disjunctive program reduct ii given a base answer set semantics x for normal programs we define i to be a candidate answer set of w r t x if i is an answer set of s e l i under x and iii we define i to be an answer set of w r t x if i is a minimal candidate answer set the di semantics is general and applicable to extend any answer set semantics x for normal programs to disjunctive programs by replacing x with the gl n l p semantics defined by gelfond and lifschitz 33 we induce a di semantics for simple disjunctive programs and by replacing x with the well justified semantics defined by shen et al 65 we further induce a di semantics for general disjunctive programs we also establish a novel characterization of the gl semantics in terms of a disjunctive program reduct which reveals the essential difference of the di semantics from the gl semantics and leads us to giving a satisfactory solution to the open problem presented by hitzler and seda 36 about characterizing split normal derivatives of a simple disjunctive program such that answer sets of the normal derivatives are answer sets of under the gl semantics finally we give computational complexity results in particular we show that in the propositional case deciding whether a simple disjunctive program has some di answer set is np complete this is in contrast to the gl semantics and equivalent formulations such as the flp semantics 24 where deciding whether has some answer set is 2 p complete while brave and cautious reasoning are 2 p and 2 p complete respectively for both gl and di answer sets for general disjunctive programs with compound formulas as building blocks the complexity of brave and cautious reasoning increases under di semantics by one level of the polynomial hierarchy which thus offers higher problem solving capacity

attributive concept descriptions with complements abstract we investigate the consequences of adding unions and complements to attributive concept descriptions employed in terminological knowledge representation languages it is shown that deciding coherence and subsumption of such descriptions are pspace complete problems that can be decided with linear space

two theses of knowledge representation language restrictions taxonomic classification and the utility of representation services abstract levesque and brachman argue that in order to provide timely and correct responses in the most critical applications general purpose knowledge representation systems should restrict their languages by omitting constructs which require nonpolynomial worst case response times for sound and complete classification they also separate terminological and assertional knowledge and restrict classification to purely terminological information we demonstrate that restricting the terminological language and classifier in these ways limits these general purpose facilities so severely that they are no longer generally applicable we argue that logical soundness completeness and worst case complexity are inadequate measures for evaluating the utility of representation services and that this evaluation should employ the broader notions of utility and rationality found in decision theory we suggest that general purpose representation services should provide fully expressive languages classification over relevant contingent information approximate forms of classification involving defaults and rational management of inference tools

products of several toeplitz operators in this paper we give a necessary and sufficient condition for ta1ta2 tan ta1a2 an where the tai s are toeplitz operators on the hardy space of the unit disk we then show that ta1ta2ta3ta4ta5ta6 0 if and only if one of ai is identically zero a criteria for the compactness of ta1ta2 tan ta1a2 an is also given

enhancing disjunctive logic programming systems by sat checkers disjunctive logic programming dlp with stable model semantics is a powerful nonmonotonic formalism for knowledge representation and reasoning reasoning with dlp is harder than with normal v free logic programs because stable model checking deciding whether a given model is a stable model of a propositional dlp program is co np complete while it is polynomial for normal logic programs this paper proposes a new transformation m p which reduces stable model checking to unsat i e to deciding whether a given cnf formula is unsatisfiable the stability of a model m of a program p thus can be verified by calling a satisfiability checker on the cnf formula m p the transformation is parsimonious i e no new symbol is added and efficiently computable as it runs in logarithmic space and therefore in polynomial time moreover the size of the generated cnf formula never exceeds the size of the input and is usually much smaller we complement this transformation with modular evaluation results which allow for efficient handling of large real world reasoning problems the proposed approach to stable model checking has been implemented in dlv a state of the art implementation of dlp a number of experiments and benchmarks have been run using satz as satisfiability checker the results of the experiments are very positive and confirm the usefulness of our techniques

disjunctive stable models disjunctive logic programs have become a powerful tool in knowledge representation and commonsense reasoning this paper focuses on stable model semantics currently the most widely acknowledged semantics for disjunctive logic programs after presenting a new notion of unfounded sets for disjunctive logic programs we provide two declarative characterizations of stable models in terms of unfounded sets one shows that the set of stable models coincides with the family of unfounded free models i e a model is stable iff it contains no unfounded atoms the other proves that stable models can be defined equivalently by a property of their false literals as a model is stable iff the set of its false literals coincides with its greatest unfounded set we then generalize the well founded wpoperator to disjunctive logic programs give a fixpoint semantics for disjunctive stable models and present an algorithm for computing the stable models of function free programs the algorithm s soundness and completeness are proved and some complexity issues are discussed