weights of multipartitions and representations of ariki koike algebras ii canonical bases we consider the canonical basis for an integrable highest weight module of the quantum algebra uv sl e we show how the relationship between different canonical bases is controlled by the weight of a multipartition generalising a theorem of lascoux leclerc and thibon for the level 1 case

compiling problem specifications into sat we present a compiler that translates a problem specification into a propositional satisfiability test sat problems are specified in a logic based language called np spec which allows the definition of complex problems in a highly declarative way and whose expressive power is such as to capture all problems which belong to the complexity class np the target sat instance is solved using any of the various state of the art solvers available from the community the system obtained is an executable specification language for all np problems which shows interesting computational properties the performance of the system has been tested on a few classical problems namely graph coloring hamiltonian cycle job shop scheduling and on a real world scheduling application namely the tournament scheduling problem

convex sets with semidefinite representation we provide a sufficient condition on a class of compact basic semialgebraic sets bf k subset mathbb r n for their convex hull co k to have a semidefinite representation sdr this sdr is explicitly expressed in terms of the polynomials g j that define k examples are provided we also provide an approximate sdr that is for every fixed epsilon 0 there is a convex set bf k epsilon such that rm co bf k subseteq bf k epsilon subseteq rm co bf k epsilon bf b where b is the unit ball of mathbb r n and bf k epsilon has an explicit sdr in terms of the g j s for convex and compact basic semi algebraic sets k defined by concave polynomials we provide a simpler explicit sdr when the nonnegative lagrangian l f associated with k and any linear f in mathbb r x is a sum of squares we also provide an approximate sdr specific to the convex case

sufficient and necessary conditions for semidefinite representability of convex hulls and sets a set s subseteq mathbb r n is called semidefinite programming sdp representable or semidefinite representable if s equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality lmi clearly if s is sdp representable then s must be convex and semialgebraic it is describable by conjunctions and disjunctions of polynomial equalities or inequalities this paper proves sufficient conditions and necessary conditions for sdp representability of convex sets and convex hulls by proposing a new approach to constructing sdp representations the contributions of this paper are i for bounded sdp representable sets w 1 dots w m we give an explicit construction of an sdp representation for mathrm conv cup k 1 mw k this provides a technique for building global sdp representations from the local ones ii for the sdp representability of a compact convex semialgebraic set s we prove sufficient the boundary partial s is nonsingular and positively curved while necessary is partial s has nonnegative curvature at each nonsingular point in terms of defining polynomials for s nonsingular boundary amounts to them having nonvanishing gradient at each point on partial s and the curvature condition can be expressed as their strict versus nonstrict quasi concavity at those points on partial s where they vanish the gaps between them are partial s having or not having singular points either of the gradient or of the curvature s positivity a sufficient condition bypassing the gaps is when some defining polynomials of s satisfy an algebraic condition called sos concavity iii for the sdp representability of the convex hull of a compact nonconvex semialgebraic set t we find that the critical object is partial c t the maximum subset of partial t contained in partial mathrm conv t we prove sufficient for sdp representability partial c t is nonsingular and positively curved and necessary is partial c t has nonnegative curvature at nonsingular points the gaps between our sufficient and necessary conditions are similar to case ii the positive definite lagrange hessian pdlh condition which meshes well with constructions is also discussed

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

ultimate approximation and its application in nonmonotonic knowledge representation systems in this paper we study fixpoints of operators on lattices and bilattices in a systematic and principled way the key concept is that of an approximating operator a monotone operator on the product bilattice which gives approximate information on the original operator in an intuitive and well defined way with any given approximating operator our theory associates several different types of fixpoints including the kripke kleene fixpoint stable fixpoints and the well founded fixpoint and relates them to fixpoints of operator being approximated compared to our earlier work on approximation theory the contribution of this paper is that we provide an alternative more intuitive and better motivated construction of the well founded and stable fixpoints in addition we study the space of approximating operators by means of a precision ordering and show that each lattice operator o has a unique most precise we call it ultimale approximation we demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator o we then discuss applications of these results in logic programming

first order conditions for semidefinite representations of convex sets defined by rational or singular polynomials a set is called semidefinite representable or semidefinite programming sdp representable if it equals the projection of a higher dimensional set which is defined by some linear matrix inequality lmi this paper discusses the semidefinite representability conditions for convex sets of the form s mathcal d f x in mathcal d f x geq 0 here mathcal d x in mathbb r n g 1 x geq 0 ldots g m x geq 0 is a convex domain defined by some nice concave polynomials g i x they satisfy certain concavity certificates and f x is a polynomial or rational function when f x is concave over mathcal d we prove that s mathcal d f has some explicit semidefinite representations under certain conditions called preordering concavity or q module concavity which are based on the positivstellensatz certificates for the first order concavity criteria f u nabla f u t x u f x geq 0 quad forall x u in mathcal d when f x is a polynomial or rational function having singularities on the boundary of s mathcal d f a perspective transformation is introduced to find some explicit semidefinite representations for s mathcal d f under certain conditions in the special case n 2 if the laurent expansion of f x around one singular point has only two consecutive homogeneous parts we show that s mathcal d f always admits an explicitly constructible semidefinite representation

convergent hierarchy of sdp relaxations for a class of semi infinite convex polynomial programs and applications abstract in this paper we examine semidefinite linear programming approximations to a class of semi infinite convex polynomial optimization problems where the index sets are described in terms of convex quadratic inequalities we present a convergent hierarchy of semidefinite linear programming relaxations under a mild well posedness assumption we also provide additional conditions under which the hierarchy exhibits finite convergence these results are derived by first establishing characterizations of optimality which can equivalently be reformulated as linear matrix inequalities a separation theorem of convex analysis and a sum of squares polynomial representation of positivity of real algebraic geometry together with a special variable transformation play key roles in achieving the results finally as applications we present convergent relaxations for a broad class of robust convex polynomial optimization problems

propositional default logics made easier computational complexity of model checking liberatore and schaerf proceedings of the ecai 98 1998 give a proof that model checking for propositional normal default theories is in 2p and 2p o log n hard however the precise complexity is left as an open problem we solve this problem by proving that model checking for normal default theories is complete for 2p o log n this is the class of decision problems solvable in polynomial time with a logarithmic number of calls to an oracle in np additionally we analyse the computational cost of model checking w r t weak extensions stable expansions and n expansions and take a look at the complexity of model checking for disjunction free default theories furthermore we show that not only for disjunction free default theories but also for a larger class of default theories which we call default theories in extended horn normal form the complexity of model checking is in the case of normal default theories tractable additionally the complexity results are used to draw some interesting conclusions on translatability issues in particular there exists no function from default logic into logic programming which is polynomial faithful and modular unless conp 2p finally we give an overview of our results concerning model checking in case of disjunctive default logic and stationary default logic

a convex polynomial that is not sos convex a multivariate polynomial p x p x 1 x n is sos convex if its hessian h x can be factored as h x m t x m x with a possibly nonsquare polynomial matrix m x it is easy to see that sos convexity is a sufficient condition for convexity of p x moreover the problem of deciding sos convexity of a polynomial can be cast as the feasibility of a semidefinite program which can be solved efficiently motivated by this computational tractability it is natural to study whether sos convexity is also a necessary condition for convexity of polynomials in this paper we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sos convex

sums of squares on real algebraic curves given an affine algebraic variety v over with real points v compact and a non negative polynomial function f v with finitely many real zeros we establish a local global criterion for f to be a sum of squares in v we then specialize to the case where v is a curve the notion of virtual compactness is introduced and it is shown that in the local global principle compactness of v can be relaxed to virtual compactness the irreducible curves on which every non negative polynomial is a sum of squares are classified all results are extended to the more general framework of preorders moreover applications to the k moment problem from analysis are given in particular schmudgen s solution of the k moment problem for compact k is extended for dim k 1 to the case when k is virtually compact

translating default logic into standard autoepistemic logic since konolige s translation of default logic into strongly grounded autoepistemic logic several other variants of moore s original autoepistemic logic that embody default logic have been studied all these logics differ significantly from moore s autoepistemic logic standard ael in that expansions are subject to additional groundedness conditions hence the question naturally arises whether default logic can be translated into standard ael at all we show that a modular translation is not possible however we are able to construct a faithful polynomial time translation from default logic into standard ael which is nonmodular our translation exploits the self referentiality of ael it uses as an important intermediate step an embedding of marek s and truszczyn ski s nonmonotonic logic n into standard ael it follows from our results that the expressive power of standard ael is strictly greater than that of default logic

semidefinite representation of convex sets let s x in mathbb r n g 1 x geq 0 ldots g m x geq 0 be a semialgebraic set defined by multivariate polynomials g i x assume s is convex compact and has nonempty interior let s i x in mathbb r n g i x geq 0 and s resp s i be the boundary of s resp s i this paper as does the subject of semidefinite programming sdp concerns linear matrix inequalities lmis the set s is said to have an lmi representation if it equals the set of solutions to some lmi and it is known that some convex s may not be lmi representable helton and vinnikov in commun pure appl math 60 5 654 674 2007 a question arising from nesterov and nemirovski siam studies in applied mathematics society for industrial and applied mathematics siam philadelphia 1994 see helton and vinnikov in commun pure appl math 60 5 654 674 2007 and nemirovski in plenary lecture international congress of mathematicians icm madrid spain 2006 is given a subset s of mathbb r n does there exist an lmi representable set in some higher dimensional space mathbb r n n whose projection down onto mathbb r n equals s such s is called semidefinite representable or sdp representable this paper addresses the sdp representability problem the following are the main contributions of this paper i assume g i x are all concave on s if the positive definite lagrange hessian condition holds i e the hessian of the lagrange function for optimization problem of minimizing any nonzero linear function l t x on s is positive definite at the minimizer then s is sdp representable ii if each g i x is either sos concave 2 g i x w x t w x for some possibly nonsquare matrix polynomial w x or strictly quasi concave on s then s is sdp representable iii if each s i is either sos convex or poscurv convex s i is compact convex whose boundary has positive curvature and is nonsingular i e g i x 0 on s i s then s is sdp representable this also holds for s i for which s i s extends smoothly to the boundary of a poscurv convex set containing s iv we give the complexity of schmudgen and putinar s matrix positivstellensatz which are critical to the proofs of i iii

on semidefinite representations of non closed sets abstract spectrahedra are sets defined by linear matrix inequalities projections of spectrahedra are called semidefinitely representable sets both kinds of sets are of practical use in polynomial optimization since they occur as feasible sets in semidefinite programming there are several recent results on the question which sets are semidefinitely representable so far all results focus on the case of closed sets in this work we develop a new method to prove semidefinite representability of sets which are not closed for example the interior of a semidefinitely representable set is shown to be semidefinitely representable more general one can remove faces of a semidefinitely representable set and preserve semidefinite representability as long as the faces are parametrized in a suitable way

the number of simple modules for the hecke algebras of type g r p n with an appendix by xiaoyi cui abstract we derive a parameterization of simple modules for the cyclotomic hecke algebras of type g r p n with p 1 and n 3 over fields of any characteristic coprime to p we give explicit formulas for the number of simple modules over these cyclotomic hecke algebras

containment problems for polytopes and spectrahedra we study the computational question of whether a given polytope or spectrahedron s a as given by the positive semidefiniteness region of a linear matrix pencil a x is contained in another s b first we classify the computational complexity extending results on the polytope polytope case by gritzmann and klee to the polytope spectrahedron case for various restricted containment problems np hardness is shown we then study in detail semidefinite conditions to certify containment building upon work by ben tal and nemirovski and helton klep and mccullough in particular we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron it is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases including containment of a spectrahedron in a polyhedron moreover in the case of bounded s a the criteria will always succeed in certifying containment of some scale

proof of the modular branching rule for cyclotomic hecke algebras abstract we prove the modular branching rule of the cyclotomic hecke algebras which has remained open

solving logic program conflict through strong and weak forgettings we consider how to forget a set of atoms in a logic program intuitively when a set of atoms is forgotten from a logic program all atoms in the set should be eliminated from this program in some way and other atoms related to them in the program might also be affected we define notions of strong and weak forgettings in logic programs to capture such intuition reveal their close connections to the notion of forgetting in classical propositional theories and provide a precise semantic characterization for them based on these notions we then develop a general framework for conflict solving in logic programs we investigate various semantic properties and features in relation to strong and weak forgettings and conflict solving in the proposed framework we argue that many important conflict solving problems can be represented within this framework in particular we show that all major logic program update approaches can be transformed into our framework under which each approach becomes a specific conflict solving case with certain constraints we also study essential computational properties of strong and weak forgettings and conflict solving in the framework

modular branching rules and the mullineux map for hecke algebras of type a in a series of papers 21 22 23 kleshchev has recently proved a branching rule for modular representations of the symmetric group s r on r symbols fix a field f of arbitrary characteristic p then kleshchev s branching rule gives a precise description of the socle of the restriction of an irreducible fs r module to fs r 1 in 24 kleshchev applies this branching rule to give a purely combinatorial description of the mullineux map which describes the irreducible fs r module obtained by tensoring an arbitrary irreducible fs r module with the 1 dimensional sign representation in 13 see also 3 ford and kleshchev show with some technical combinatorics that kleshchev s description of the mullineux map is equivalent to a quite different algorithm conjectured by mullineux 27 in 1979 thus proving the so called mullineux conjecture the main purpose of this paper is to prove the quantum analogues of these results replacing the group algebra fs r with the corresponding hecke algebra h r over f at an arbitrary root of unity q f our main result is the hecke algebra analogue of kleshchev s modular branching rule it turns out that the branching rule for h r only depends on the integer where is the smallest natural number such that 1 q q 1 0 in f if q 1 is precisely the characteristic p of f and our branching rule specializes to the classical case considered by kleshchev as a consequence of this branching rule combining kleshchev s argument in 24 with the known block structure of h r from 9 we are able to deduce the quantum analogue of kleshchev s description of the corresponding mullineux map for h r the proof by ford and kleshchev in 13 that this is equivalent to mullineux s original algorithm does not depend on p being prime so our result is equivalent by 13 to a quantum version of the mullineux conjecture replacing p by we remark that there is now a quite different proof that kleshchev s algorithm gives the mullineux map using crystal bases this was first observed for fields of characteristic 0 in 25 modulo a conjecture which has now been proved 17 2 relating crystal bases for a certain affine quantum group to decomposition matrices of h r over fields of characteristic 0 the result in arbitrary characteristic can be deduced from the characteristic 0 case by an

np trees and carnap s modal logic in this paper we consider problems and complexity classes definable by interdependent queries to an oracle in np how the queries depend on each other is specified by a directed graph g we first study the class of problems where g is a general dag and show that this class coincides with d p 2 we then consider the class where g is a tree our main result states that this class is identical to p np o log n the class of problems solvable in polynomial time with a logarithmic number of queries to an oracle in np this result has interesting applications in the fields of modal logic and artificial intelligence in particular we show that the following problems are all p np o log n complete validity checking of formulas in carnap s modal logic checking whether a formula is almost surely valid over finite structures in modal logics k t and s4 a problem recently considered by halpern and kapron 1992 and checking whether a formula belongs to the stable set of beliefs generated by a propositional theory we generalize the case of dags to the case where g is a general possibly cyclic directed graph of np oracle queries and show that this class corresponds to p p 2 we show that such graphs are easily expressible in autoepistemic logic finally we generalize our complexity results to higher classes of the polynomial time hierarchy

ordering default theories and nonmonotonic logic programs first order theories are ordered under logical entailment based on the amount of information derived from theories in default logic on the other hand a theory contains default information as well as definite information to order default theories distinguishing different sorts of information is necessary to assess the information content of a default theory for this purpose we first introduce a multi valued interpretation of default theories using a ten valued bilattice it distinguishes between definite and credulous skeptical default information derived from a theory and is used for ordering default theories based on their information contents we then apply the technique to order nonmonotonic logic programs under the answer set semantics the results of this paper provide a method for comparing default theories or nonmonotonic logic programs in a manner different from the conventional extension model based viewpoint moreover they have important application to induction from nonmonotonic theories

exposed faces of semidefinitely representable sets a linear matrix inequality lmi is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite motivated by the fact that diagonal lmis define polyhedra the solution set of an lmi is called a spectrahedron linear images of spectrahedra are called semidefinitely representable sets part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming such as one can do on polyhedra by linear programming it is known that every face of a spectrahedron is exposed this is also true in the general context of rigidly convex sets we study the same question for semidefinitely representable sets lasserre proposed a moment matrix method to construct semidefinite representations for certain sets our main result is that this method can work only if all faces of the considered set are exposed this necessary condition complements sufficient conditions recently proved by lasserre helton and nie

graded hypothesis theories abstract reasoning about expert domains often involves imperfect knowledge in such cases any piece of information may prove to be useful this requires considering uncertain beliefs and sometimes when the expected information is missing making additional assumptions this paper presents a qualitative approach which allows for the simultaneous representation of uncertain and or incomplete information a running example illustrates the need for such a representation the proposed framework is a multimodal logic in which uncertainty is represented by means of a set of partially ordered symbolic grades expressed as modal operators assumptions are formulated as hypotheses in siegel and schwind s hypothesis theory we show that such hypotheses may be interpreted as constraints on the set of possible beliefs we thus obtain a very natural integration of multimodal graded logic and hypothesis theory we also study various properties of this formalism in presence of partial inconsistency

positive polynomials and projections of spectrahedra this article is concerned with different aspects of spectrahedra and their projections sets that are important in semidefinite optimization we prove results on the limitations of so called lasserre and theta body relaxation methods for semialgebraic sets and varieties as a special case we obtain the main result of netzer plaumann and schweighofer siam j optim 20 2010 pp 1944 1955 on nonexposed faces we also solve the open problems from that work we further give a unified account of several results on convex hulls of curves and images of polynomial maps we finally prove a positivstellensatz for projections of spectrahedra which exceeds the known results that only work for basic closed semialgebraic sets

convexity in semialgebraic geometry and polynomial optimization we review several and provide new results on the theory of moments sums of squares and basic semialgebraic sets when convexity is present in particular we show that under convexity the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence a highly desirable feature as convex problems are in principle easier to solve in addition if a basic semialgebraic set mathbf k is convex but its defining polynomials are not we provide two algebraic certificates of convexity which can be checked numerically the second is simpler and holds if a sufficient and almost necessary condition is satisfied it also provides a new condition for mathbf k to have semidefinite representation for this we use and extend some of the recent results from the author and helton and nie math program to appear finally we show that when restricting to a certain class of convex polynomials the celebrated jensen s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures

semantical considerations on nonmonotonic logic abstract commonsense reasoning is nonmonotonic in the sense that we often draw on the basis of partial information conclusions that we later retract when we are given more complete information some of the most interesting products of recent attempts to formalize nonmonotonic reasoning are the nonmonotonic logics of mcdermott and doyle 1 2 these logics however all have peculiarities that suggest they do not quite succeed in capturing the intuitions that prompted their development in this paper we reconstruct nonmonotonic logic as a model of an ideally rational agent s reasoning about his own beliefs for the resulting system called autoepistemic logic we define an intuitively based semantics for which we can show autoepistemic logic to be both sound and complete we then compare autoepistemic logic with the approach of mcdermott and doyle showing how it avoids the peculiarities of their nonmonotonic logic

on the relation between default and autoepistemic logic abstract default logic is a formal means of reasoning about defaults what normally is the case in the absence of contradicting information autoepistemic logic on the other hand is meant to describe the consequences of reasoning about ignorance what must be true if a certain fact is not known although the motivation and formal character of these two systems are different a closer analysis shows that they share a common trait which is the indexical nature of certain elements in the theory in this paper we compare the expressive power of the two systems first we give an effective translation of default logic into autoepistemic logic default theories can thus be embedded into autoepistemic logic we also present a more surprising result the reverse translation is also possible so that every set of sentences in autoepistemic logic can be effectively rewritten as a default theory the formal equivalence of these two differing systems is thus established this analysis gives an interpretive semantics to default logic and yields insight into the nature of defaults in autoepistemic reasoning

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

reducts of propositional theories satisfiability relations and generalizations of semantics of logic programs over the years the stable model semantics has gained a position of the correct two valued interpretation of default negation in programs however for programs with aggregates constraints the stable model semantics in its broadly accepted generalization stemming from the work by pearce ferraris and lifschitz has a competitor the semantics proposed by faber leone and pfeifer which seems to be essentially different our goal is to explain the relationship between the two semantics pearce ferraris and lifschitz s extension of the stable model semantics is best viewed in the setting of arbitrary propositional theories we propose here an extension of the faber leone pfeifer semantics or flp semantics for short to the full propositional language which reveals both common threads and differences between the flp and stable model semantics we use our characterizations of flp stable models to derive corresponding results on strong equivalence and on normal forms of theories under the flp semantics we apply a similar approach to define supported models for arbitrary propositional theories and to study their properties

the modular branching rule for affine hecke algebras of type a for the affine hecke algebra of type a at roots of unity we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule the latter generalizes work by vazirani

convex hulls of curves of genus one abstract let c be a real nonsingular affine curve of genus one embedded in affine n space whose set of real points is compact for any polynomial f which is nonnegative on c r we prove that there exist polynomials f i with f i f i 2 mod i c and such that the degrees deg f i are bounded in terms of deg f only using lasserre s relaxation method we deduce an explicit representation of the convex hull of c r in r n by a lifted linear matrix inequality this is the first instance in the literature where such a representation is given for the convex hull of a nonrational variety the same works for convex hulls of singular curves whose normalization is c we then make a detailed study of the associated degree bounds these bounds are directly related to size and dimension of the projected matrix pencils in particular we prove that these bounds tend to infinity when the curve c degenerates suitably into a singular curve and we provide explicit lower bounds as well

negation as failure careful closure procedure abstract the aim of this paper is a modification of minker s generalized closed world assumption that would allow application of the negation as failure rule with respect to a set p of not necessarily all predicates of a database db a careful closure procedure is introduced which when applied to a database db produces a new database db that is used to answer queries about predicates from db it is shown that db is consistent iff db is consistent if p is the set of all predicates from db and db does not contain functional symbols then db coincides with minker s gcwa the soundness and completeness of the careful closure procedure with respect to a minimal model style semantic is shown as an inference engine associated with db we propose a query evaluation procedure qep which is a combination of a method of splitting an indefinite database db into a disjunction of horn databases and clark s query evaluation procedure qep soundness of qep with respect to db is shown for a broad class of databases

generic spectrahedral shadows spectrahedral shadows are projections of linear sections of the cone of positive semidefinite matrices we characterize the polynomials that vanish on the boundaries of these convex sets when both the section and the projection are generic

a modular branching rule for the generalized symmetric groups abstract we give a modular branching rule for certain wreath products as a generalization of kleshchev s modular branching rule for the symmetric groups our result contains a modular branching rule for the complex reflection groups g m 1 n which are often called the generalized symmetric groups in splitting fields for z m z especially for m 2 which is the case of the weyl groups of type b we can give a modular branching rule in any field

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

complexity and expressive power of deterministic semantics for datalog abstract deterministic models are partial stable models which are not contradicted by any other stable model i e m is a deterministic model if there is no stable model n such that m n is not an interpretation for instanced the well founded model which coincides with the intersection of all partial stable models is a deterministic model as the well founded model is deterministic and unique for each program well founded model semantics has been proposed as the canonical deterministic semantics for partial stable models but the well founded model is not the unique deterministic model indeed the family of deterministic partial stable models is not in general a singleton and admits a minimum the well founded model and a maximum the max deterministic model the latter model is another candidate for a deterministic semantics the aim of this paper is to study the complexity and the expressive power of deterministic semantics in coherence with the deterministic nature of the model the expressive power of max deterministic semantics is shown to be able to express problems with unique solutions whereas the well founded model only captures a proper subset of the queries computable in polynomial time the so called fixpoint queries

classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems necessary and sufficient conditions for convexity and strong convexity respectively of connected sublevel sets that are defined by finitely many real valued c 1 1 maps are presented a novel characterization of strongly convex sets in terms of the so called local quadratic support is proved the results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous time nonlinear systems to be strongly convex an application of these conditions is a novel method to overapproximate attainable sets when strong convexity is present

np hardness of deciding convexity of quartic polynomials and related problems we show that unless p np there exists no polynomial time or even pseudo polynomial time algorithm that can decide whether a multivariate polynomial of degree four or higher even degree is globally convex this solves a problem that has been open since 1992 when n z shor asked for the complexity of deciding convexity for quartic polynomials we also prove that deciding strict convexity strong convexity quasiconvexity and pseudoconvexity of polynomials of even degree four or higher is strongly np hard by contrast we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time

free semidefinite representation of matrix power functions abstract consider the matrix power function x p defined over the cone of positive definite matrices s n it is known that x p is convex over s n if p 1 0 1 2 and x p is concave over s n if p 0 1 we show that the hypograph of x p admits a free semidefinite representation if p 0 1 is rational and the epigraph of x p admits a free semidefinite representation if p 1 0 1 2 is rational

circumscribing datalog expressive power and complexity abstract in this paper we study a generalization of datalog the language of function free definite clauses it is known that standard datalog semantics i e least herbrand model semantics can be obtained by regarding programs as theories to be circumscribed with all predicates to be minimized the extension proposed here called datalog circ consists in considering the general form of circumscription where some predicates are minimized some predicates are fixed and some vary we study the complexity and the expressive power of the language thus obtained we show that this language and actually its non recursive fragment is capable of expressing all the queries in db co np and as such is much more powerful than standard datalog whose expressive power is limited to a strict subset of ptime queries both data and combined complexities of answering datalog circ queries are studied data complexity is proved to be co np complete combined complexity is shown to be in general hard for co ne and complete for co ne in the case of herbrand bases containing k distinct constant symbols where k is bounded

possible world semantics and autoepistemic reasoning abstract the possible world semantics of modal logic has been employed to study the nonmonotonic reasoning formalism moore s autoepistemic logic only kripke structure which is also known as a standard model has been used to represent the autoepistemic expansions of modal theories here we argue that the mode of autoepistemic reasoning which is formalized by autoepistemic logic is not necessarily associated with a standard model another model of the possible world semantics the minimal model is helpful in knowledge representation and belief reasoning in particular the minimal model can be used as a structure to represent all the autoepistemic expansions of a modal theory we extend the minimal model with standard model to deal with both knowledge and belief and introduce a null value called unknown to provide a refined model for knowledge and belief we also present a notion of epistemic base of basic sentences to identify the possible worlds in a minimal model by reformulating moore s autoepistemic logic with respect to epistemic base some reasonable modal theories that do not have any autoepistemic expansion will have generalized autoepistemic expansions

on the number of constituents of induced modules of ariki koike algebras abstract we examine the crystal graph of the sl e module arising from an sl e categorification to study the defining endo functors of the categorification this yields lower bounds on the number of irreducible constituents of certain objects we use ariki s categorification result on ariki koike algebras to obtain a new lower bound on the number of constituents of their parabolically induced modules in particular this will imply reducibility of every induced module

a graph theoretic approach to default logic a network representation of propositional seminormal disjunction free default theories is presented leading to a graph theoretic approach to their analysis the problem of finding an extension is proved to be equivalent to that of determining a kernel for a corresponding graph allowing stronger complexity results as well as new conditions for the existence of extensions

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

morita equivalences of ariki koike algebras we prove that every ariki koike algebra is morita equivalent to a direct sum of tensor products of smaller ariki koike algebras which have q connected parameter sets a similar result is proved for the cyclotomic q schur algebras combining our results with work of ariki and uglov the decomposition numbers for the ariki koike algebras defined over fields of characteristic zero are now known in principle

on the lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems the lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions j b lasserre convexity in semialgebraic geometry and polynomial optimization siam j optim 19 1995 2014 2009 we give a new proof of the finite convergence property that does not require the assumption that the hessian of the objective be positive definite on the entire feasible set but only at the optimal solution in addition we show that the number of steps needed for convergence depends on more than the input size of the problem in particular the size of the semidefinite program that gives the exact reformulation of the convex polynomial optimization problem may be exponential in the input size

smooth hyperbolicity cones are spectrahedral shadows hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials a well understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral in this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone that is a spectrahedral shadow

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

polynomial matrix inequality and semidefinite representation consider a convex set s x in cal d g x succeq 0 where g x is a symmetric matrix whose every entry is a polynomial or rational function d rn is a domain on which g x is defined and g x succeq 0 means g x is positive semidefinite the set s is called semidefinite representable if it equals the projection of a higher dimensional set that is defined by a linear matrix inequality lmi this paper studies sufficient conditions guaranteeing semidefinite representability of s we prove that s is semidefinite representable in the following cases i d rn g x is a matrix polynomial and matrix sos concave ii d is compact convex g x is a matrix polynomial and strictly matrix concave on d iii g x is a matrix rational function and q module matrix concave on d explicit constructions of semidefinite representations are given some examples are illustrated

distinguished representations of non negative polynomials abstract let g 1 g r r x 1 x n such that the set k g 1 0 g r 0 in r n is compact we study the problem of representing polynomials f with f k 0 in the form f s 0 s 1 g 1 s r g r with sums of squares s i with particular emphasis on the case where f has zeros in k assuming that the quadratic module of all such sums is archimedean we establish a local global condition for f to have such a representation vis a vis the zero set of f in k this criterion is most useful when f has only finitely many zeros in k we present a number of concrete situations where this result can be applied as another application we solve an open problem from s kuhlmann et al positivity sums of squares and the multi dimensional moment problem ii adv geometry in press on one dimensional quadratic modules

the degenerate analogue of ariki s categorification theorem we explain how to deduce the degenerate analogue of ariki s categorification theorem over the ground field mathbb c as an application of schur weyl duality for higher levels and the kazhdan lusztig conjecture in finite type a we also discuss some supplementary topics including young modules tensoring with sign tilting modules and ringel duality

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

modal nonmonotonic logics ranges characterization computation many nonmonotonic formalism including default logic logic programming with stable models and autoepistemic logic can be represented faithfully by means of modal nonmonotonic logics in the family proposed by mcdermott and doyle in this paper properties of logics in this family are thoroughly investigated we present several results on characterization of expansions these results are applicable to a wide class of nonmonotonic modal logics using these characterization results algorithms for computing expansions for finite theories are developed perhaps the most important finding of this paper is that the structure of the family of modal nonmonotonic logics is much simpler than that of the family of underlying modal monotonic logics namely it is often the case that different monotonic modal logics collapse to the same nonmonotonic system we exhibit four families of logics whose nonmonotonic variants coincide 5 kd45 tw5 sw5 n wk and w5 d4wb these nonmonotonic logics naturally represent logics related to commonsense reasoning and knowledge representation such as autoepistemic logic reflexive autoepistemic logic default logic and truth maintenance with negation

on compact representations of propositional circumscription we prove that unless the polynomial hierarchy collapses at the second level the size of a purely propositional representation of the circumscription circ t of a propositional formula t grows faster than any polynomial as the size of t increases we then analyze the significance of this result in the related field of closed world reasoning

querying disjunctive databases through nonmonotonic logics abstract query languages for retrieving information from disjunctive databases are an interesting open area of research in this paper we study the expressive power of major nonmonotonic formalisms such as circumscription default logic autoepistemic logic and some logic programming languages used as query languages over disjunctive databases for this aim we define the semantics of query expressions formulated in different nonmonotonic logics the expressive power of the languages that we consider has been explored in the context of relational databases here we extend this study to disjunctive databases as a result we obtain a finer grained characterization of the expressiveness of those languages and interesting fragments thereof for instance we show that there exist simple queries that cannot be expressed by any preferential semantics including the minimal model semantics and the various forms of circumscription while they can be expressed in default and autoepistemic logic secondly we show that default logic autoepistemic logic and some of their fragments express the same class of boolean queries which turns out to be a strict subclass of the p 2 recognizable boolean queries the latter result is proved by means of a new technique based on a counting argument then we prove that under the assumption that the database consists of clauses whose length is bounded by some constant default logic and autoepistemic logic express all of the p 2 recognizable boolean queries while preference based logics cannot these results hold for brave reasoning we obtain dual results for cautious reasoning our results appear to be interesting both in the area of database theory and in the area of knowledge representation

linearization functors on real convex sets we prove that linearizing multilinear optimization problems leads to new functorial operations on real convex sets these operations are convex analogues of hom functors tensor products symmetric powers exterior powers and general schur functors on vector spaces and lead to novel constructions even for polyhedra we discuss their general theory and introduce mechanisms to compute them or approximate them in ways amenable to efficient computation

specht modules labelled by hook bipartitions i abstract brundan kleshchev and wang equip the specht modules s over the cyclotomic khovanov lauda rouquier algebra h n with a homogeneous z graded basis in this paper we begin the study of graded specht modules labelled by hook bipartitions n m 1 m in level 2 of h n which are precisely the hecke algebras of type b with quantum characteristic at least three we give an explicit description of the action of the khovanov lauda rouquier algebra generators 1 n 1 on the basis elements of s n m 1 m introducing certain specht module homomorphisms we construct irreducible submodules of these specht modules and thereby completely determine the composition series of specht modules labelled by hook bipartitions

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

graph theoretical structures in logic programs and default theories abstract in this paper we present a graph representation of logic programs and default theories we show that many of the semantics proposed for logic programs with negation can be expressed in terms of notions emerging from graph theory establishing in this way a link between the fields namely the stable models the partial stable models and the well founded semantics correspond respectively to the kernels semikernels and the initial acyclic part of an associated graph this link allows us to consider both theoretical existence uniqueness and computational problems tractability algorithms approximations from a more abstract and rather combinatorial point of view it also provides a clear and intuitive understanding about how conflicts between rules are resolved within the different semantics furthermore we extend the basic framework developed for logic programs to the case of default logic by introducing the notions of partial deterministic and well founded extensions for default theories these semantics capture different ways of reasoning with a default theory

relational queries computable in polynomial time we characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe we also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation this proves that the fixed point query hierarchy suggested by chandra and harel collapses at the first fixed point level it is also a general result showing that in finite model theory one application of fixed point suffices

a complete characterization of the gap between convexity and sos convexity our first contribution in this paper is to prove that three natural sum of squares sos based sufficient conditions for convexity of polynomials via the definition of convexity its first order characterization and its second order characterization are equivalent these three equivalent algebraic conditions henceforth referred to as sos convexity can be checked by semidefinite programming whereas deciding convexity is np hard if we denote the set of convex and sosconvex polynomials in n variables of degree d with cn d and cn d respectively then our main contribution is to prove that cn d cn d if and only if n 1 or d 2 or n d 2 4 we also present a complete characterization for forms homogeneous polynomials except for the case n d 3 4 which is joint work with g blekherman and is to be published elsewhere our result states that the set cn d of convex forms in n variables of degree d equals the set cn d of sos convex forms if and only if n 2 or d 2 or n d 3 4 to prove these results we present in particular explicit examples of polynomials in c2 6 c2 6 and c3 4 c3 4 and forms in c3 6 c3 6 and c4 4 c4 4 and a general procedure for constructing forms in cn d 2 cn d 2 from nonnegative but not sos forms in n variables and degree d although for disparate reasons the remarkable outcome is that convex polynomials resp forms are sos convex exactly in cases where nonnegative polynomials resp forms are sums of squares as characterized by hilbert

expressive power and complexity of partial models for disjunctive deductive databases abstract this paper investigates the expressive power and complexity of partial model semantics for disjunctive deductive databases in particular partial stable regular model maximal stable m stable and least undefined stable l stable semantics for function free disjunctive logic programs are considered for which the expressiveness of queries based on possibility and certainty inference is determined on the complexity side we determine the data and expression complexity of query evaluation the analysis pays particular attention to the impact of syntactical restrictions on programs in the form of limited use of disjunction and negation it appears that the considered semantics capture complexity classes at the lower end of the polynomial hierarchy in fact each class i p i p 1 i 3 is captured by some semantics using appropriate syntactical restrictions partial stable models have exactly the same expressive power and complexity as total stable models 2 p resp 2 p while a higher degree of expressiveness is obtained by the semantics which minimize undefinedness m stable regular and l stable semantics in particular l stable semantics has the highest expressive power 3 p resp 3 p an interesting result in this course is that in contrast with total stable models negation is for partial stable models more expressive than disjunction for the data complexity of queries we obtain completeness results for the classes i p i p i 3 and for the expression complexity completeness results for the analogous classes at the lower end of the weak exponential hierarchy the results of this paper complement and extend previous results and contribute to a more complete picture of the computational aspects of disjunctive logic programming and databases which supports in choosing an appropriate setting that fits the needs in practice

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

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