relative endotrivial modules and endo p permutation modules in the auslander reiten quiver abstract this note aims at describing the position of endo p permutation and relative endotrivial modules in the auslander reiten quiver of a group algebra kg the answer is that they lie at the end of their components or at relative ends unless the sylow subgroups are dihedral in which case they form entire components

minimal knowledge problem a new approach abstract in this paper we propose a new logic of minimal knowledge our approach falls into the general scheme of shoham s preference semantics it stems from an earlier work on logics of minimal knowledge by halpern and moses lin and shoham and lifschitz the novelty of our work is in a procedure for minimizing knowledge which we propose in this paper and which is different from earlier proposals we show that our logic preserves most desirable properties of earlier formalisms and at the same time avoids some of their drawbacks in addition to a semantic definition of our system we provide its equivalent syntactic characterization which relates our logic with the nonmonotonic modal logic s4f and allows us to use in our investigations standard modal logic techniques

cumulative default logic in defense of nonmonotonic inference rules abstract two problems of reiter s default logic have recently been discussed in the literature first inconsistencies between justifications of nonnormal defaults may lead to unintuitive results and second default logic is not cumulative i e the addition of theorems to the set of premises may change the derivable formulas to solve these two problems we strengthen the applicability condition for defaults and make the reasons for believing something an explicit part of the derived formulas the resulting new logic turns out to be semi monotonic if the additional expressiveness of nonnormal defaults is to be retained only some of the extensions of this logic are to be taken as acceptable sets of beliefs however those preserving priorities between defaults

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

nonmonotonicity and the scope of reasoning abstract circumscription default logic and autoepistemic logic capture aspects of the nonmonotonicity of human commonsense reasoning however perlis has shown that circumscription suffers from certain counterintuitive limitations concerning exceptions or counterexamples to defaults we observe that the unfortunate limitations of circumscription are even broader than perlis originally pointed out moreover these limitations are not peculiar to circumscription they appear to be endemic in nonmonotonic reasoning formalisms we develop a general solution involving restricting the scope of nonmonotonic reasoning and show that it remedies these problems in a variety of formalisms our solution has a number of attractive aspects in addition to its generality most importantly no modification of the underlying formalisms is required and the result is semantically compatible with existing approaches furthermore the necessary machinery is intuitively plausible and arguably useful for other purposes finally the solution is robust it is relatively tolerant of imprecise determinations of scope

negation by default and unstratifiable logic programs abstract the default approach to the theory of logic programs and deductive databases is based on the interpretation of negation by default rules default logic is a well suited formalism to express the closed world assumption and to define the declarative semantics of stratifiable logic programs the case of disjunctive consequences in rules is treated general logic programs may not have a meaning with respect to default semantics the contribution of the paper is to exhibit an interesting class of programs having a default semantics called effectively stratifiable programs this time disjunctive consequences are not considered effective stratification is a weaker constraint than stratification local stratification and weak stratification besides enlarging the class of stratifiable logic programs the paper contributes to provide a constructive definition of well founded models of logic programs the class of effectively stratifiable logic programs matches the class of programs having a total well founded model and in general the default semantics extends the well founded semantics

lexicographic priorities in default logic abstract resolving conflicts between default rules is a major subtask in performing default reasoning a declarative way of controlling the resolution of conflicts is to assign priorities to default rules and to prevent conflict resolution in ways that violate the priorities this work extends reiter s default logic with a priority mechanism that is based on lexicographic comparison given a default theory and a partial ordering on the defaults the preferred extensions are the lexicographically best extensions we discuss alternative ways of using lexicographic comparison and investigate their properties and relations between them the applicability of the priority mechanism to inheritance reasoning is investigated by presenting two translations from inheritance networks to prioritized default theories and relating them to inheritance theories presented earlier by gelfond and przymusinska and by brewka

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

a logic of knowledge and justified assumptions abstract in this paper we define the logic gk of knowledge and justified assumptions gk is best understood as a formalization of autoepistemic reasoning processes that are more general than those in moore s autoepistemic logic and is formally defined via a modification of shoham s preference semantics we show that gk includes not only moore s autoepistemic logic but also reiter s default logic to our knowledge gk is the first complete semantic unification of the two logics similarly to circumscription gk is based on the notion of logical minimization and thus provides a bridge between circumscription and fixed point nonmonotonic logics an outstanding problem in nonmonotonic logics as an application of this bridge we propose a formalization of logic programs with negation as failure in circumscription

nonmonotonic reasoning by inhibition nets abstract in this paper we will show that certain networks called inhibition nets may be regarded as cognitive agents drawing nonmonotonic inferences it will be proven that the system cl introduced by klm in artificial intelligence 44 1990 186 189 of nonmonotonic logic is both sound and complete with respect to the inferences drawn by finite hierarchical inhibition nets the latter class of inhibition nets is shown to correspond to the class of finite normal hierarchical logic programs concerning dynamics and also to the class of binary layered input driven artificial neural networks

evaluating the effect of semi normality on the expressiveness of defaults normal default logic ndl and semi normal default logic sndl are syntactically restricted variants of reiter s default logic dl although semi normal defaults have only a slightly more general form than normal defaults sndl does not share the nice properties of ndl e g semi monotonicity in this note we address the effect of semi normality on the expressive power of defaults using a classification method based on polynomial faithful and modular pfm translation functions the resulting classification indicates that sndl and dl are of equal expressive power which strictly exceeds that of ndl this strengthens an earlier result achieved by marek and truszczynski who establish the equivalence of weak semi normal defaults with general defaults furthermore it is established that prerequisite free fragments of dl and sndl pdl and psndl are of equal expressive power consequently psndl is less expressive than dl and sndl incomparable with ndl and more expressive than preferential entailment a generalized form of circumscription the latter result is in sharp contrast with imielinski s result which states that prerequisite free and semi normal default theories can be modularly translated into preferential entailment

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

results on translating defaults to circumscription abstract in this paper we define different concepts of translating a set of defaults into circumscription the most important of these modular translation requires that additive changes of specific facts not defaults of default theory translate to additive changes of the simple abnormality theory corresponding to circumscription we show that surprisingly enough an important class of defaults normal defaults with prerequisites does not have such a modular translation into circumscription abnormality theory we also establish the classes of defaults which are translatable into circumscription

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

vibration analysis of rotating twisted and open conical shells abstract based on a non linear strain displacement relationship of a non rotating twisted and open conical shell on thin shell theory a numerical method for free vibration of a rotating twisted and open conical shell is presented by the energy method where the effect of rotation is considered as initial deformation and initial stress resultants which are obtained by the principle of virtual work for steady deformation due to rotation then an energy equilibrium of equation for vibration of a twisted and open conical shell with the initial conditions is also given by the principle of virtual work in the two numerical processes the rayleigh ritz procedure is used and the two in plane and a transverse displacement functions are assumed to be algebraic polynomials in two elements the effects of characteristic parameters with respect to rotation and geometry such as an angular velocity and a radius of rotating disc a setting angle a twist angle curvature and a tapered ratio of cross section on vibration performance of rotating twisted and open conical shells are studied by the present method

reasoning by cases in default logic reiter s default logic is one of the most popular formalisms for describing default reasoning one important defect of default logic is however the inability to reason by cases over the years several solutions for this problem have been proposed all these proposals deal with deriving new propositions through reasoning by cases none however discuss the propositions that should no longer be derivable as a result of reasoning by cases this paper discusses the latter subject it shows that an intuitively plausible way of dealing with propositions that should no longer be derivable as a result of reasoning by cases can have far reaching consequences one of the consequences is that disjunctions must be viewed as describing possible extensions

relative projectivity and ideals in cohomology rings abstract in this paper we explore one aspect of the relationship between group cohomology and representation theory for a finite groupgand a fieldkin characteristicp 0 ideals in the cohomology ringh g k can sometimes be characterized by exact sequences ofkg modules in much the same way that elements of extkgare classically represented by exact sequences furthermore the maps on the stable category which represent the exact sequences have functorial properties which mimic the structure of the ideas in the ring the natural setting for this study is the relative homological algebra for the module category relative to the subcategory generated by a single module using the tensor product operation

transfers and regular elements in cohomology algebras of finite groups abstract a characterization of regular homogeneous elements in mod p cohomology algebras of finite groups will be given in terms of transfer maps defined by modules which are introduced by carlson peng and wheeler j algebra 204 1998 286 311

grounded fixpoints and their applications in knowledge representation in various domains of logic researchers have made use of a similar intuition that facts or models can be derived from the ground up they typically phrase this intuition by saying e g that the facts should be grounded or that they should not be unfounded or that they should be supported by cycle free arguments et cetera in this paper we formalise this intuition in the context of algebraical fixpoint theory we define when a lattice element x l is grounded for lattice operator o l l on the algebraical level we investigate the relationship between grounded fixpoints and the various classes of fixpoints of approximation fixpoint theory including supported minimal kripke kleene stable and well founded fixpoints on the logical level we investigate groundedness in the context of logic programming autoepistemic logic default logic and argumentation frameworks we explain what grounded points and fixpoints mean in these logics and show that this concept indeed formalises intuitions that existed in these fields we investigate which existing semantics are grounded we study the novel semantics for these logics that is induced by grounded fixpoints which has some very appealing properties not in the least its mathematical simplicity and generality our results unveil a remarkable uniformity in intuitions and mathematics in these fields

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

definability and commonsense reasoning abstract the definition of concepts is a central problem in commonsense reasoning many themes in nonmonotonic reasoning concern implicit and explicit definability implicit definability in nonmonotonic logic is always relative to the context the current theory of the world we show that fixed point equations provide a generalization of explicit definability which correctly captures the relativized context theories expressed within this logical framework provide implicit definitions of concepts moreover it is possible to derive these fixed points entirely within the logic

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

virtual projectivity and the stable module category abstract we introduce the concept of a nearly null map in the stable module category and relate it to the notion of virtual projectivity we show that the trivial module k is virtually m projective if and only if the map f n k in the triangle n f k m m is nearly null if these conditions hold then m generates the stable module category in fewer than l steps where l is the radical length of the group algebra we give examples to show that if m generates the stable module category then k is not necessarily virtually m projective we also give examples to show that for a given group the degree of virtual projectivity of k with respect to a module m is not bounded finally we develop a theory of virtual m variety of a module x this is a subset of the spectrum of the cohomology ring which controls virtual projectivity of x with respect to m

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

on embedding default logic into moore s autoepistemic logic abstract recently gottlob proved 2 that there does not exist a faithful modular translation of default logic into autoepistemic logic and presented a non modular translation gottlob s translation however is indirect it uses nonmonotonic logic n as an intermediate point quite complex and exploits sophisticated encoding of proof theory in autoepistemic formulas we provide a simpler and more intuitive non modular direct translation in addition our argument is purely model theoretic

default reasoning using classical logic abstract in this paper we show how propositional default theories can be characterized by classical propositional theories for each finite default theory we show a classical propositional theory such that there is a one to one correspondence between models for the latter and extensions of the former this means that computing extensions and answering queries about coherence set membership and set entailment are reducible to propositional satisfiability the general transformation is exponential but tractable for a subset which we call 2 dt a superset of network default theories and disjunction free default theories consequently coherence and set membership for the class 2 dt is np complete and set entailment is co np complete this work paves the way for the application of decades of research on efficient algorithms for the satisfiability problem to default reasoning for example since prepositional satisfiability can be regarded as a constraint satisfaction problem csp this work enables us to use csp techniques for default reasoning to illustrate this point we use the taxonomy of tractable csps to identify new tractable subsets for reiter s default logic our procedures allow also for computing stable models of extended logic programs

relative projectivity and the green correspondence for complexes abstract we investigate a version of the green correspondence for categories of complexes including homotopy categories and derived categories the correspondence is an equivalence between a category defined over a finite group g and the same for a subgroup h often the normalizer of a p subgroup of g we present a basic formula for deciding when categories of modules or complexes have a green correspondence and apply it to many examples in several cases the equivalence is an equivalence of triangulated categories and in special cases it is an equivalence of tensor triangulated categories

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

approximating operators and semantics for abstract dialectical frameworks we provide a systematic in depth study of the semantics of abstract dialectical frameworks adfs a recent generalisation of dung s abstract argumentation frameworks this is done by associating with an adf its characteristic one step consequence operator and defining various semantics for adfs as different fixpoints of this operator we first show that several existing semantical notions are faithfully captured by our definition then proceed to define new adf semantics and show that they are proper generalisations of existing argumentation semantics from the literature most remarkably this operator based approach allows us to compare adfs to related nonmonotonic formalisms like dung argumentation frameworks and propositional logic programs we use polynomial faithful and modular translations to relate the formalisms and our results show that both abstract argumentation frameworks and abstract dialectical frameworks are at most as expressive as propositional normal logic programs

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

relative projectivity and relative endotrivial modules this dissertation is concerned with modular representation theory of finite groups and more precisely with the study of classes of representations which we shall term relative endotrivial modules given a prime number p a finite group g of order divisible by p we shall say that a kg module m is endotrivial relatively to the kg module v if its endomorphism algebra endk m is isomorphic as a kg module to a direct sum of a trivial module and another module which is projective relatively to v i e in short endk m k v projective more accurately in the first part of the text projectivity relative to kg modules is used to define groups of relative endotrivial modules which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules however in order to achieve this goal we first need to develop the theory of projectivity relative to modules in particular with respect to standard group operations such as induction restriction and inflation then for finite groups having a cyclic sylow p subgroup using the structure of the group t g of endotrivial modules described in mt07 we give a complete classification of the groups of relative endotrivial modules we also study the case of groups that have a sylow p subgroup isomorphic to a klein group c2 c2 as well as the case of p nilpotent groups in a second part of the text it is shown how our new groups of relative endotrivial modules provide a natural context to generalise the dade group of a p group p to an arbitrary finite group the classification of endo permutation modules and the complete description of the structure of the dade group d p was completed in 2004 by s bouc with bou06 this adventure had started about 25 years earlier with the first papers and results by e dade in dad78a and dad78b in 1978 and the final classification was in fact achieved through the non effortless combined work of several co authors between 1998 and 2004 including s bouc j carlson n mazza and j thevenaz it is most interesting to note that crucial building pieces for this classification are indeed the endotrivial modules which are particular cases of endo permutation modules yet for an arbitrary finite group g no satisfying equivalent group structure to the dade group on a class of kg modules has been defined so far with the goal to fill this gap we turn the problem upside down in some sense and show how one can regard an endo permutation module as an endotrivial module of course not in the ordinary sense but in the relative sense this shall enable us to endow a set of isomorphism classes of endo p permutation modules with a group structure similar to that of the dade group we shall call this new group the generalised dade group of the group g explicitly compute its structure and show how it is closely related to that of the g stable points of the dade group of a sylow p subgroup of g

uniform semantic treatment of default and autoepistemic logics we revisit the issue of epistemological and semantic foundations for autoepistemic and default logics two leading formalisms in nonmonotonic reasoning we develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the lattice structure of the collection of all belief pairs for each logic we introduce a monotone operator on the lattice of belief pairs we then show that a whole family of semantics can be defined in a systematic and principled way in terms of fixpoints of this operator or as fixpoints of certain closely related operators our approach elucidates fundamental constructive principles in which agents form their belief sets and leads to approximation semantics for autoepistemic and default logics it also allows us to establish a precise one to one correspondence between the family of semantics for default logic and the family of semantics for autoepistemic logic the correspondence exploits the modal interpretation of a default proposed by konolige our results establish conclusively that default logic can be viewed as a fragment of autoepistemic logic a result that has been long anticipated at the same time they explain the source of the difficulty to formally relate the semantics of default extensions by reiter and autoepistemic expansions by moore these two semantics occupy different locations in the corresponding families of semantics for default and autoepistemic logics

transfer maps and virtual projectivity abstract when g is a finite group and h is a subgroup of g the concept of relative h projectivity is of fundamental importance in modular representation theory unfortunately in some ways the properties of relatively h projective modules are not as closely related to other ideas of representation theory as one might hope for example the variety of a module bears only a weak connection to relative projectivity the purpose of this paper is to introduce and study a related concept known as virtual h projectivity that seems to be much more closely related to the theory of varieties the study is actually carried out in the more general setting of virtual projectivity with respect to a finitely generated module w and the main tool is a transfer map tr w that generalizes the usual transfer map tr g h with respect to a subgroup h