a causal approach to nonmonotonic reasoning we introduce logical formalisms of production and causal inference relations based on input output logics of makinson and van der torre j philos logic 29 2000 383 408 these inference relations will be assigned however both standard semantics giving interpretation to their rules and natural nonmonotonic semantics based on the principle of explanation closure the resulting nonmonotonic formalisms will be shown to provide a logical representation of abductive reasoning and a complete characterization of causal nonmonotonic reasoning from mccain and turner proc aaai 97 providence ri 1997 pp 460 465 the results of the study suggest production and causal inference as general nonmonotonic formalisms providing an alternative representation for a significant part of nonmonotonic reasoning

domain permutation reduction for constraint satisfaction problems this paper is concerned with the constraint satisfaction problem csp it is well known that the general csp is np hard however there has been significant success in discovering subproblems which are tractable polynomial time solvable one of the most effective ways to obtain a tractable subproblem has been to force all of the constraint relations to lie in some tractable language in this paper we define a new way of identifying tractable subproblems of the csp let p be an arbitrary csp instance and c be any tractable language suppose there exists for each variable of p a permutation of the domain such the resultant permuted constraint relations of p all lie in c the domain permuted instance is then an instance of a tractable class and can be solved by the polynomial time algorithm for c solutions to p can be obtained by inverting the domain permutations the question for a given class of instances and language c whether such a set of domain permutations can be found efficiently is the key to this method s tractability one of the important contributions made in this paper is the notion of a lifted constraint instance which is a powerful tool to study this question we consider the open problem of discovering domain permutations which make instances max closed we show that for bounded arity instances over a boolean domain this problem is tractable while for domain size three it is intractable even for binary instances we give a simple proof verifying the tractability of discovering domain permutations which make instances row convex we refute a published result by giving a simple proof of the intractability of discovering domain permutations which make instances even with domain size four connected row convex we demonstrate that triangulated and stable marriage instances are reducible via domain permutations to max closed instances this provides a simple explanation for arc consistency deciding these instances we verify with a simple direct proof the tractability of identification of renamable horn instances and the intractability of finding the largest renamable horn subset of clauses of an instance of sat we describe natural tractable classes which properly extend the maximal relational classes arising from tractable constraint languages we believe that domain permutation reductions have a significant chance of being useful in practical applications

abduction versus closure in causal theories abstract there are two distinct formalizations for reasoning from observations to explanations as in diagnostic tasks the consistency based approach treats the task as a deductive one in which the explanation is deduced from a background theory and a minimal set of abnormalities the abductive method on the other hand treats explanations as sentences that when added to the background theory derive the observations we show that there is a close connection between these two formalizations in the context of simple causal theories domain theories in which a set of sentences are singled out as the explanatorily relevant causes of observations there are two main results which show that with certain caveats the consistency based approach can emulate abductive reasoning by adding closure axioms to a causal theory and that abductive techniques can be used in place of the consistency based method in the domain of logic based diagnosis it is especially interesting that in the latter case the abductive techniques generate only relevant explanations while diagnoses may have irrelevant elements

continuous dependence estimates for viscosity solutions of integro pdes we present a general framework for deriving continuous depen dence estimates for possibly polynomially growing viscosity solutions of fully nonlinear degenerate parabolic integro pdes we use this framework to pro vide explicit estimates for the continuous dependence on the coecients and the levy measure in the bellman isaacs integro pdes arising in stochas tic control dierenti al games moreover these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems finally we illustrate our results on some integro pdes arising when attempting to price european american options in an incomplete stock mar ket driven by a geometric levy process many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure pde methods

the hele shaw asymptotics for mechanical models of tumor growth models of tumor growth now commonly used present several levels of complexity both in terms of the biomedical ingredients and the mathematical description our first goal here is to formulate a free boundary model of hele shaw type a variant including growth terms starting from the description at the cell level and passing to the stiff limit in the pressure law of state in contrast with the classical hele shaw problem here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics a complete description requires the equation on the cell number density we then go on to consider a more complex model including the supply of nutrients through vasculature and we study the stiff limit for the involved coupled system

complexity results for explanations in the structural model approach we analyze the computational complexity of halpern and pearl s causal explanations in the structural model approach which are based on their notions of weak and actual cause in particular we give a precise picture of the complexity of deciding explanations partial explanations and partial explanations and of computing the explanatory power of partial explanations moreover we analyze the complexity of deciding whether all explanation or an partial explanation over certain variables exists we also analyze the complexity of deciding explanations and partial explanations in the case of succinctly represented context sets the complexity of deciding explanations in the general case of situations and the complexity of deciding subsumption and equivalence between causal models all complexity results are derived for the general case as well as for the restriction to the case of binary causal models in which all endogenous variables may take only two values to our knowledge no complexity results for explanations in the structural model approach have been derived so far our results give insight into the computational structure of halpern and pearl s explanations and pave the way for efficient algorithms and implementations

continuous dependence on the nonlinearity of viscosity solutions of parabolic equations abstract this paper establishes an upper bound for u v where u is a subsolution of u t f u d x u d 2 x u 0and v is a supersolution of v t g v d x v d 2 x v 0in 0 with neumann boundary conditions and where is an open convex set

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

the sharp bound for the hilbert transform on weighted lebesgue spaces in terms of the classical ap characteristic we show that the norm of the hilbert transform as an operator in the weighted space l p r for 2 p oo is bounded by a constant multiple of the first power of the classical ap characteristic of this result is sharp we also prove a bilinear imbedding theorem with simple conditions

short time uniqueness results for solutions of nonlocal and non monotone geometric equations we describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non monotone second order geometric equations arising in front propagation problems our method is based on some lower gradient bounds for the solution these estimates are crucial to obtain regularity properties of the front which allow to deal with nonlocal terms in the equations applications to short time uniqueness results for the initial value problems for dislocation type equations asymptotic equations of a fitzhugh nagumo type system and equations depending on the lebesgue measure of the fronts are presented

preferences and explanations in abductive reasoning preference criteria for selecting the best explanation are regarded as qualitative properties like being simpler or more plausible which are beyond the pure causal or deductive relationship between an explanandum and its explanations this paper is a contribution to the clarification of the relationship between preference criteria and structural properties of explanatory reasoning we present a detailed analysis of the connection between the logical properties satisfied by a logic based explanatory process and the structural properties satisfied by the criterion used for selecting the preferred explanations namely we characterize the postulates introduced in a previous work artificial intelligence 111 2 1999 131 169 as those satisfied by explanatory relations defined by preference relations over formulas several examples illustrating our results are analyzed including well known preference criteria like expectation orders preferential orders and other selection criteria that have appeared in the literature

on the spanning property of risk bonds priced by equilibrium we propose a method of pricing financial securities written on nontradable underlyings such as temperature or precipitation levels to this end we analyze a financial market where agents are exposed to financial and nonfinancial risk factors the agents hedge their financial risk in the stock market and trade a risk bond issued by an insurance company from the issuer s point of view the bond s primary purpose is to shift insurance risks related to noncatastrophic weather events to financial markets as such its terminal payoff and yield curve depend on an underlying climate or temperature process whose dynamics are independent of the randomness driving stock prices we prove that if the bond s payoff function is monotone in the external risk process it can be priced by an equilibrium approach the equilibrium market price of climate risk and the equilibrium price process are characterized as solutions of nonlinear backward stochastic differential equations bsdes transferring the bsdes into partial differential equations pdes we represent the bond prices as smooth functions of the underlying risk factors our analytical results make the model amenable to a numerical analysis

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

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

estimates for operator norms on weighted spaces and reverse jensen inequalities we examine the dependence on the a p norm of w of the operator norms of singular integrals maximal functions and other operators in l p w we also examine connections between some fairly general reverse jensen inequalities and the a p and rh p weight conditions

hard problems for simple default logics abstract we investigate the complexity of reasoning with a number of limited default logics surprising negative results the high complexity of simple three literal default rules as well as positive results a fast algorithm for skeptical reasoning with binary defaults are reported and sources of complexity are discussed these results impact on work on defeasible inheritance hierarchies as well as default reasoning in general

functional dependencies in horn theories abstract this paper studies functional dependencies in horn theories both when the theory is represented by its clausal form and when it is defined as the horn envelope of a set of models we provide polynomial algorithms for the recognition of whether a given functional dependency holds in a given horn theory as well as polynomial algorithms for the generation of some representative sets of functional dependencies we show that some problems of inferring functional dependencies e g constructing an irredundant fd cover are computationally difficult we also study the structure of functional dependencies that hold in a horn theory showing that every such functional dependency is in fact a single positive term boolean function and prove that for any horn theory the set of its minimal functional dependencies is quasi acyclic finally we consider the problem of condensing a horn theory prove that any horn theory has a unique condensation and develop an efficient polynomial algorithm for condensing horn theories

an assumption based tms abstract this paper presents a new view of problem solving motivated by a new kind of truth maintenance system unlike previous truth maintenance systems which were based on manipulating justifications this truth maintenance system is in addition based on manipulating assumption sets as a consequence it is possible to work effectively and efficiently with inconsistent information context switching is free and most backtracking and all retraction is avoided these capabilities motivate a different kind of problem solving architecture in which multiple potential solutions are explored simultaneously this architecture is particularly well suited for tasks where a reasonable fraction of the potential solutions must be explored

the complexity of logic based abduction abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations when the application domain is described by a logical theory we speak about logic based abduction candidates for abductive explanations are usually subjected to minimality criteria such as subset minimality minimal cardinality minimal weight or minimality under prioritization of individual hypotheses this paper presents a comprehensive complexity analysis of relevant decision and search problems related to abduction on propositional theories our results indicate that abduction is harder than deduction in particular we show that with the most basic forms of abduction the relevant decision problems are complete for complexity classes at the second level of the polynomial hierarchy while the use of prioritization raises the complexity to the third level in certain cases

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

abductive consequence relations abstract in this paper we present a systematic study of abductive consequence relations we show that a monotone abductive consequence relation satisfies the properties of a cumulative monotonic system as defined by kraus lehmann and magidor when the disjunction of all abductive explanations is the explanation used to justify the observations we also show that in general for this class of abductive consequence relations the or rule does not hold we present an example that shows that when there are preferences between different abductive explanations monotonicity does not hold we show that nonmonotonic abductive systems preserve a partial version of rational monotonicity and in fact are very similar to rational relations we also present semantic characterizations of both monotonic and nonmonotonic abductive systems in terms of cumulative models as defined by kraus lehmann and magidor

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

error estimates for approximate solutions to bellman equations associated with controlled jump diffusions we derive error estimates for approximate viscosity solutions of bellman equations associated to controlled jump diffusion processes which are fully nonlinear integro partial differential equations two main results are obtained i error bounds for a class of monotone approximation schemes which under some assumptions includes finite difference schemes and ii bounds on the error induced when the original levy measure is replaced by a finite measure with compact support an approximation process that is commonly used when designing numerical schemes for integro partial differential equations our proofs use and extend techniques introduced by krylov and barles jakobsen

higher order convergence rates in theory of homogenization ii oscillatory initial data abstract we establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic cauchy problems accompanied with rapidly oscillating initial data such result is new even for linear problems here we construct higher order initial layer and interior correctors which describe the oscillatory behavior near the initial and interior time zone of the domain to construct higher order correctors we develop a regularity theory in macroscopic scales and prove an exponential decay estimate for initial layer correctors the higher order expansion requires an iteration process successively correcting the initial layer then the interior this leads to a more complicated asymptotic expansion as compared to the non oscillating data case and this complexity is present even in the linear case a notable observation for fully nonlinear operators is that even if the given operator is space time periodic the interior correctors become aperiodic in the time variable as we proceed with the iteration process moreover each interior corrector of higher order is paired with a space time periodic version and the difference between the two decays exponentially fast with time

extended horn sets in propositional logic the class of horn clause sets in propositional logic is extended to a larger class for which the satisfiability problem can still be solved by unit resolution in linear time it is shown that to every arborescence there corresponds a family of extended horn sets where ordinary horn sets correspond to stars with a root at the center these results derive from a theorem of chandresekaran that characterizes when an integer solution of a system of inequalities can be found by rounding a real solution in a certain way a linear time procedure is provided for identifying hidden extended horn sets extended horn but for complementation of variables that correspond to a specified arborescence finally a way to interpret extended horn sets in applications is suggested

probabilistic horn abduction and bayesian networks this paper presents a simple framework for horn clause abduction with probabilities associated with hypotheses the framework incorporates assumptions about the rule base and independence assumptions amongst hypotheses it is shown how any probabilistic knowledge representable in a discrete bayesian belief network can be represented in this framework the main contribution is in finding a relationship between logical and probabilistic notions of evidential reasoning this provides a useful representation language in its own right providing a compromise between heuristic and epistemic adequacy it also shows how bayesian networks can be extended beyond a propositional language this paper also shows how a language with only unconditionally independent hypotheses can represent any probabilistic knowledge and argues that it is better to invent new hypotheses to explain dependence rather than having to worry about dependence in the language

on strongest neccessary and weakest sufficient conditions abstract given a propositional theory t and a proposition q a sufficient condition of q is one that will make q true under t and a necessary condition of q is one that has to be true for q to be true under t in this paper we propose a notion of strongest necessary and weakest sufficient conditions intuitively the strongest necessary condition of a proposition is the most general consequence that we can deduce from the proposition under the given theory and the weakest sufficient condition is the most general abduction that we can make from the proposition under the given theory we show that these two conditions are dual ones and can be naturally extended to arbitrary formulas we investigate some computational properties of these two conditions and discuss some of their potential applications

a formal model of diagnostic inference i problem formulation and decomposition abstract this paper which is part i of a two part series introduces a new model of diagnostic problem solving based on a generalization of the set covering problem the model formalizes the concepts of 1 1 whether or not a set of one or more disorders is sufficient to explain a set of occurring manifestations 2 2 what a solution is for a diagnostic problem and 3 3 how to generate all of the alternative explanations in a problem s solution in addition conditions for decomposing a diagnostic problem into independent subproblems are stated and proven this model is of interest because it captures several intuitively plausible features of human diagnostic inference it directly addresses the issue of multiple simultaneous causative disorders it can serve as a theoretical basis for expert systems for diagnostic problem solving and it provides a conceptual framework within which to view some recent ai work on diagnostic problem solving in general in part ii the concepts developed in this paper will be used to present algorithms for diagnostic problem solving

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

relevance from an epistemic perspective abstract in this paper we study relevance relations in the context of prepositional logical theories and subject matters or topics of interest which we take to be sets of atomic propositions in particular we are interested in answering questions like the following when is a sentence or theory relevant to a subject matter or when is a topic relevant to another topic or sentence given some background theory relevance is studied from a subjective or epistemic point of view that is we try to capture relevance relations from an agent s point of view relative to his or her deductive capabilities for this purpose we start out with the definition of regular belief which covers a wide range of belief models from the very weak to those closed under classical logical implication in the paper we consider one example from each of the extremes which are called explicit and implicit belief respectively we define a notion of prime implicates which applies to all models of regular belief and which has its usual meaning under implicit belief prime implicates turn out to be the right primitive from which all definitions of relevance are derived among the main technical contributions is a detailed comparison between relevance under implicit belief and three other approaches in the literature this investigation reveals that all four share a lot of common ground even though some have very different starting points we also study the complexity of determining relevance relations for implicit as well as explicit belief while intractability obtains often but not always for implicit belief the analogous problems for explicit belief are almost always tractable

abduction in logic programming a new definition and an abductive procedure based on rewriting a long outstanding problem for abduction in logic programming has been on how minimality might be defined without minimality an abductive procedure is often required to generate exponentially many subsumed explanations for a given observation in this paper we propose a new definition of abduction in logic programming where the set of minimal explanations can be viewed as a succinct representation of the set of all explanations we then propose an abductive procedure where the problem of generating explanations is formalized as rewriting with confluent and terminating rewrite systems we show that these rewrite systems are sound and complete under the partial stable model semantics and sound and complete under the answer set semantics when the underlying program is so called odd loop free we discuss an application of abduction in logic programming to a problem in reasoning about actions and provide some experimental results

singular solutions of fully nonlinear elliptic equations and applications we study the properties of solutions of fully nonlinear positively homogeneous elliptic equations near boundary points of lipschitz domains at which the solution may be singular we show that these equations have two positive solutions in each cone of mathbb r n and the solutions are unique in an appropriate sense we introduce a new method for analyzing the behavior of solutions near certain lipschitz boundary points which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below we also obtain a sharp phragmen lindelof result as well as a principle of positive singularities in certain lipschitz domains

a general maximal operator and the _ condition 1 let p v be two measures on r and let there be given for each cube q er afunction q supported in q we consider the maximal operator mf x sup ff q dv where the sup is extended over all cubes centered at x and obtain theorem 1 therearrangement inequality m y4 0 3 f dt here g denotes the non increasing rearrangement of g with respect to the measure a and is a nonincreas ing function given in terms of p v q from this one easily sees that m and tnus tne finiteness of this integral i e g l p i

extrapolation on lp μ we solve the extrapolation problem concerning bounded operators on lp that is we give end point estimates for sublinear operators t such that t lp lp is bounded with constant less than or equal to 1 p 1 m applications to the hardy littlewood maximal operator the hilbert transform and composition of operators are also given

a logic for default reasoning the need to make default assumptions is frequently encountered in reasoning about incompletely specified worlds inferences sanctioned by default are best viewed as beliefs which may well be modified or rejected by subsequent observations it is this property which leads to the non monotonicity of any logic of defaults in this paper we propose a logic for default reasoning we then specialize our treatment to a very large class of commonly occuring defaults for this class we develop a complete proof theory and show how to interface it with a top down resolution theorem prover finally we provide criteria under which the revision of derived beliefs must be effected

existence of viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games recently m g crandall and p l lions trans amer math soc 277 1983 1 42 introduced the class of viscosity solutions of these equations and proved uniqueness within this class this paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness

on some tractable classes in deduction and abduction abstract we address the identification of propositional theories for which entailment is tractable so that every query about logical consequences of the theory can be answered in polynomial time we map tractable satisfiability classes to tractable entailment classes including hierarchies of tractable problems and show that some initially promising conditions for tractability of entailment proposed by esghi 1993 and del val 1994 surprisingly only identify a subset of renamable horn we then consider a potential application of tractable entailment through a reduction due to esghi 1993 of certain abduction problems to a sequence of entailment problems besides clarifying the range of applicability of esghi s results from the semantic point of view we show that the reduction can almost trivially fail to be in any of the basic tractable classes discussed in the first part of the paper we leave open the question of how to more broadly identify tractable entailment classes as our examples suggest that there is room for progress in this area

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

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

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

some regularity and convergence results for parabolic hamilton jacobi bellman equations in bounded domains abstract we study parabolic hamilton jacobi bellman hjb equations in bounded domains with strong dirichlet boundary conditions we work under the assumption of the existence of a sufficiently regular barrier function for the problem to obtain well posedness and regularity of a related switching system and the convergence of its components to the hjb equation in particular we show existence of a viscosity solution to the switching system by a novel construction of sub and supersolutions and application of perron s method error bounds for monotone schemes for the hjb equation are then derived from estimates near the boundary where the standard regularisation procedure for viscosity solutions is not applicable and are found to be of the same order as known results for the whole space we deduce error bounds for some common finite difference and truncated semi lagrangian schemes

support set selection for abductive and default reasoning abstract of all the possible ways of computing abductive explanations the atms procedure is one of the most popular while this procedure is known to run in exponential time in the worst case the proof actually depends on the existence of queries with an exponential number of answers but how much of the difficulty stems from having to return these large sets of explanations here we explore abduction tasks similar to that of the atms but which return relatively small answers the main result is that although it is possible to generate some nontrivial explanations quickly deciding if there is an explanation containing a given hypothesis is np complete as is the task of generating even one explanation expressed in terms of a given set of assumption letters thus the method of simply listing all explanations as employed by the atms probably cannot be improved upon an interesting result of our analysis is the discovery of a subtask we call the support set selection task that is not only at the core of generating explanations but is also at the core of generating extensions in reiter s default logic moreover it is this subtask that accounts for the computational difficulty of both forms of reasoning this establishes for the first time a strong connection between computing abductive explanations and computing extensions in default logic

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

new extrapolation estimates given a sublinear operator t satisfying that t a lp cp 1 a lp for every measurable set a and every 1 0 1 r tf y dy1 log r m f x 1 log f x d x this estimate allows us to improve yano s extrapolation theorem and also to obtain that for every f l log l r 1 r tf y dylogr f 1 other types of extrapolation results are also given

a rearranged good inequality let tf be a maximal calderon zygmund singular integral mf the hardy littlewood maximal function and w an aoo weight we replace the good a inequality w x tf x 2r and mf x a by the rearrangement inequality tf t c mt t 2 tf 2t and show that it gives better estimates for tf in particular we obtain best possible weighted lp bounds previously unknown exponential integrability estimates and simplified derivations of known unweighted estimates for tf

continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations using the maximum principle for semicontinuous functions 3 4 we prove a general continuous dependence on the nonlinearities estimate for bounded holder continuous viscosity solutions of fully nonlinear degenerate elliptic equations furthermore we provide existence uniqueness and holder continuity results for bounded viscosity solutions of such equations our results are general enough to encompass hamilton jacobi bellman isaacs s equations of zero sum two player stochastic dierential games an immediate conse quence of the results obtained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations

error bounds for monotone approximation schemes for hamilton jacobi bellman equations we obtain error bounds for monotone approximation schemes of hamilton jacobi bellman equations these bounds improve previous results of krylov and the authors the key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate almost smooth supersolutions for the hamilton jacobi bellman equation

continuous dependence estimates and homogenization of quasi monotone systems of fully nonlinear second order parabolic equations abstract the aim of this paper is to extend the continuous dependence estimates proved by jakobsen and karlsen 2002 3 to quasi monotone systems of fully nonlinear second order parabolic equations as a by product of these estimates we get an holder estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation in the second part of the paper we employ similar techniques to study the periodic homogenization of quasi monotone systems of fully nonlinear second order uniformly parabolic equations finally some examples are discussed

the satisfiability problem for a class consisting of horn sentences and some non horn sentences in porportional logic in this paper the satisfiability problem for a class of proportional sentences is considered here a sentence is a set of clauses a clause is a set of literals first it is proposed that a class s 0 of propositional sentences which properly includes the class of propositional horn sentences a sentence c 1 c n is in s 0 if there are sets p 1 p n of positive literals such that 1 p 1 p 2 p n 2 p i c i for 1 i n and 3 c i p i is a horn clause for 1 i n then it is proposed that a new inference rule based on the resolution principle by which un satisfiability for s 0 in polynomial time can be decided

an almost linear time algorithm for graph realization given a 0 1 matrix m the graph realization problem for m is to find a tree t such that the columns of m are incidence vectors of paths in t or to show that no such t exists an algorithm is presented for this problem the time complexity of which is very nearly linear in the number of ones in m