decision lists and related boolean functions we consider boolean functions represented by decision lists and study their relationships to other classes of boolean functions it turns out that the elementary class of 1 decision lists has interesting relationships to independently defined classes such as disguised horn functions read once functions nested differences of concepts threshold functions and 2 monotonic functions in particular 1 decision lists coincide with fragments of the mentioned classes we further investigate the recognition problem for this class as well as the extension problem in the context of partially defined boolean functions pdbfs we show that finding an extension of a given pdbf in the class of 1 decision lists is possible in linear time this improves on previous results moreover we present an algorithm for enumerating all such extensions with polynomial delay

some new results concerning index 3 differential algebraic equations on introduit la notion de tractabilite d indice 3 on donne un critere matriciel pour tester cette tractabilite pour l equation differentielle algebrique f x t x t t 0

necessary conditions controllability and the value function for differential difference inclusions on deduit des conditions necessaires pour la commande optimale d un systeme contraint par des inclusions differentielles aux differences

lagrange type extremal trajectories in differential inclusions abstract necessary conditions for optimal or boundary solutions of differential inclusions usually state that such solutions are extremal in some sense there are several possible concepts of extremality which lead to different often difficult to compare necessary conditions in this paper we give a complete comparison of three classes of extremal trajectories two different lagrange type extremals and hamiltonian extremals in the second part we consider a nonconvex differential inclusion and prove that every boundary trajectory is a lagrangian extremal

necessary optimality conditions for differential difference inclusions with state constraints abstract we consider the mayer optimal control problem with dynamics given by a nonconvex differential difference inclusion whose trajectories are constrained to a closed set necessary optimality conditions in the form of the maximum principle are obtained

horn functions and submodular boolean functions abstract after providing a simple characterization of horn functions i e those boolean functions that have a horn dnf we study in detail the special class of submodular functions every prime implicant of such a function involves at most one complemented and at most one uncomplemented variable and based on this we give a one to one correspondence between submodular functions and partial preorders reflexive and transitive binary relations and in particular between the nondegenerate acyclic submodular functions and the partially ordered sets there is a one to one correspondence between the roots of a submodular function and the ideals of the associated partial preorder there is also a one to one correspondence between the prime implicants of the dual of the submodular function and the maximal antichains of the associated partial preorder based on these results we give graph theoretic characterizations for all minimum prime dnf representations of a submodular function the problem of recognizing submodular functions in dnf representation is conp complete

optimal compression of propositional horn knowledge bases complexity and approximation abstract horn formulae play a prominent role in artificial intelligence and logic programming in this paper we investigate the problem of optimal compression of propositional horn production rule knowledge bases the standard approach to this problem consisting in the removal of redundant rules from a knowledge base leads to an irredundant but not necessarily optimal knowledge base we prove here that the number of rules in any irredundant horn knowledge base involving n propositional variables is at most n 1 times the minimum possible number of rules therefore the quadratic time transformation of an arbitrary horn production rule base to an equivalent irredundant and prime one presented in 9 provides a reasonable approximation algorithm in order to formalize the optimal compression problem we define a boolean function of a knowledge base as being the function whose set of true points is the set of models of the knowledge base in this way the optimal compression of production rule knowledge bases becomes a problem of boolean function minimization in this paper we prove that the minimization of horn functions i e boolean functions associated to horn knowledge bases is np complete

the complexity of fault detection problems for combinational logic circuits in this correspondence we analyze the computational complexity of fault detection problems for combinational circuits and propose an approach to design for testability although major fault detection problems have been known to be in general np complete they were proven for rather complex circuits in this correspondence we show that these are still np complete even for monotone circuits and thus for unate circuits we show that for k level k 3 monotone unate circuits these problems are still np complete but that these are solvable in polynomial time for 2 level monotone unate circuits a class of circuits for which these fault detection problems are solvable in polynomial time is presented ripple carry adders decoder circuits linear circuits etc belong to this class a design approach is also presented in which an arbitrary given circuit is changed to such an easily testable circuit by inserting a few additional test points

second order necessary conditions for differential difference inclusion problems abstract we study an optimal control problem given by differential difference inclusions with end point constraints an approach concerning second order optimality conditions is proposed

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

pattern based modeling and solution of probabilistically constrained optimization problems we propose a new modeling and solution method for probabilistically constrained optimization problems the methodology is based on the integration of the stochastic programming and combinatorial pattern recognition fields it permits the fast solution of stochastic optimization problems in which the random variables are represented by an extremely large number of scenarios the method involves the binarization of the probability distribution and the generation of a consistent partially defined boolean function pdbf representing the combination f p of the binarized probability distribution f and the enforced probability level p we show that the pdbf representing f p can be compactly extended as a disjunctive normal form dnf the dnf is a collection of combinatorial p patterns each defining sufficient conditions for a probabilistic constraint to hold we propose two linear programming formulations for the generation of p patterns that can be subsequently used to derive a linear programming inner approximation of the original stochastic problem a formulation allowing for the concurrent generation of a p pattern and the solution of the deterministic equivalent of the stochastic problem is also proposed the number of binary variables included in the deterministic equivalent formulation is not an increasing function of the number of scenarios used to represent uncertainty results show that large scale stochastic problems in which up to 50 000 scenarios are used to describe the stochastic variables can be consistently solved to optimality within a few seconds

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

on online learning of decision lists a fundamental open problem in computational learning theory is whether there is an attribute efficient learning algorithm for the concept class of decision lists rivest 1987 blum 1996 we consider a weaker problem where the concept class is restricted to decision lists with d alternations for this class we present a novel online algorithm that achieves a mistake bound of o rdlog n where r is the number of relevant variables and n is the total number of variables the algorithm can be viewed as a strict generalization of the famous winnow algorithm by littlestone 1988 and improves the o r2dlog n mistake bound of balanced winnow our bound is stronger than a similar pac learning result of dhagat and hellerstein 1994 a combination of our algorithm with the algorithm suggested by rivest 1987 might achieve even better bounds

error free and best fit extensions of partially defined boolean functions abstract in this paper we address a fundamental problem related to the induction of boolean logic given a set of data represented as a set of binary truen vectors or positive examples and a set of falsen vectors or negative examples we establish a boolean function or an extension f so thatfis true resp false in every given true resp false vector we shall further require that such an extension belongs to a certain specified class of functions e g class of positive functions class of horn functions and so on the class of functions represents our a priori knowledge or hypothesis about the extensionf which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration the real world data may contain errors e g measurement and classification errors might come in when obtaining data or there may be some other influential factors not represented as variables in the vectors in such situations we have to give up the goal of establishing an extension that is perfectly consistent with the given data and we are satisfied with an extensionfhaving the minimum number of misclassifications both problems i e the problem of finding an extension within a specified class of boolean functions and the problem of finding a minimum error extension in that class will be extensively studied in this paper for certain classes we shall provide polynomial algorithms and for other cases we prove their np hardness

functions comuted by monotone boolean formulas with no repeated variables abstract if g is a monotone boolean function depending on all its variables the property that each prime implicant of g intersects each prime clause of g in a singleton is a well known necessary condition for g to be computable by a formula with no repeated variable and only using the connectives and or we prove that the condition is also sufficient our proof uses matroid theory

on the unique satisfiability problem unique sat is the problem of deciding whether a given boolean formula has exactly one satisfying truth assignment this problem is a typical moreover complete representative of a natural class of problems about unique solutions all these problems belong to the class dife l1 l2 l1 lz np studied by papadimitriou and yannakakis we consider the relationship between these two classes particularly whether unique sat is dife complete it is if np co np we construct an oracle relative to which unique sat is not complete for dif and another oracle relative to which unique sat is complete for dif e whereas np v co np

double horn functions abstract in this paper we define double horn functions which are the boolean functions f such that both f and its complement i e negation f are horn and investigate their semantical and computational properties double horn functions embody a balanced treatment of positive and negative information in the course of the extension problem of partially defined boolean functions pdbfs where a pdbf is a pair t f of disjoint sets t f 0 1 n of true and false vectors respectively and an extension of t f is a boolean function f that is compatible with t and f we derive syntactic and semantic characterizations of double horn functions and determine the number of such functions the characterizations are then exploited to give polynomial time algorithms i that recognize double horn functions from horn dnfs disjunctive normal forms and ii that compute the prime dnf from an arbitrary formula as well as its complement and its dual furthermore we consider the problem of determining a double horn extension of a given pdbf we describe a polynomial time algorithm for this problem and moreover an algorithm that enumerates all double horn extensions of a pdbf with polynomial delay however finding a shortest double horn extension in terms of the size of a formula representing it is shown to be intractable

measuring teachability using variants of the teaching dimension in a typical algorithmic learning model a learner has to identify a target object from partial information conversely in a teaching model a teacher has to give information that allows the learners to identify a target object we devise two variants of the classical teaching model for boolean concept classes based on the teaching dimension and describe them by teaching dimension like combinatorial parameters in the first model the learners choose consistent hypotheses with least complexity we show that 1 decision lists are the harder to teach the longer they are and that 2 term dnfs are the harder to teach the more terms they have this contrasts with the teachability results for these classes in the teaching dimension model in our second model the learners choose consistent hypotheses based on the assumption that the teacher is optimal we show that monomials can be taught with a linear number of examples whereas some 1 decision lists need exponentially many

the complexity of very simple boolean formulas with applications the concepts of textbf sat hardness and textbf sat completeness modulo npolylogn time and linear size reducibility denoted by textbf sat hard npolylogn n and textbf sat complete npolylogn n respectively are introduced regardless of whether textbf p textbf np or textbf p neq textbf np it is shown that intuitivelyeach textbf sat hard npolylogn n problem requires essentially at least as much deterministic time as andeach textbf sat complete npolylogn n problem requires essentially the same deterministic time as the satisfiability problem for 3cnf formulas it is proved that the leqq satisfiability tautology unique satisfiability equivalence and minimization problems are already textbf sat complete npolylogn n for very simple boolean formulas and for very simple systems of boolean equations these completeness results are used to characterize the deterministic time complexities of a number of problems for lattices propositional calculi

structure identification in relational data this paper presents several investigations into the prospects for identifying meaningful structures in empirical data namely structures permitting effective organization of the data to meet requirements of future queries we propose a general framework whereby the notion of identifiability is given a precise formal definition similar to that of learnability using this framework we then explore if a tractable procedure exists for deciding whether a given relation is decomposable into a constraint network or a cnf theory with desirable topology and if the answer is positive identifying the desired decomposition finally we address the problem of expressing a given relation as a horn theory and if this is impossible finding the best k horn approximation to the given relation we show that both problems can be solved in time polynomial in the length of the data

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

linear differential algebraic equations in spaces of integrable functions on developpe une l 2 theorie pour des equations differentielles algebriques lineaires t x t a t x t b t x t f t t a b x t r m

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

on a least squares collocation method for linear differential algebraic equations the aim of this note is to extend some results on least squares collocation methods and to prove the convergence of a least squares collocation method applied to linear differential algebraic equations some numerical examples are presented

least squares collocation for linear higher index differentialalgebraic equations differentialalgebraic equations with higher index give rise to essentially ill posed problems therefore their numerical approximation requires special care in the present paper we state the notion of ill posedness for linear differentialalgebraic equations more precisely based on this property we construct a regularization procedure using a least squares collocation approach by discretizing the pre image space numerical experiments show that the resulting method has excellent convergence properties and is not much more computationally expensive than standard collocation methods used in the numerical solution of ordinary differential equations or index 1 differentialalgebraic equations convergence is shown for a limited class of linear higher index differentialalgebraic equations

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

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems

the maximum principle for an optimal solution to a differential inclusion with end points constraints we derive pontryagin s maximum principle for a general optimal control problem using the set valued version of variational equation we achieve this aim by exploiting an adequate differential calculus of set valued maps furthermore the calmness condition is replaced by a surjectivity condition involving reachable sets of the set valued linearization of the initial control problem duality then provides both the adjoint differential inclusion and the maximum principle

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

algebraic structures with hard equivalence and minimization problems on considere la relation entre la forme dans laquelle un probleme algebrique est pose et la complexite de resoudre le probleme les problemes etudies sont l equivalence la minimisation et les problemes de minimisation approchee pour des formules comprenant des variables des parentheses des operateurs et optionnellement des constantes