Nonmonotonic Logic

We present an abstract framework for default reasoning, which includes Theorist, default logic, logic prog~mming, autoe~is~mic logic, non-monotonic modal logics, and certain instances of circumscription as special cases. The framework can be understood as a generalisation of Theorist. The generalisation allows any theory formulated in a monotonic logic to be extended by a defeasible set of assumptions. An assumption can be defeated (or "attacked") if its "contrary" can be proved, possibly with the aid of other conflicting assumptions. We show that, given such a framework, the standard semantics of most logics for default reasoning can be understood as sanctioning a set of assumptions, as an extension of a given theory, if and only if the set of assumptions is conflict-free (in the sense that it does not attack itself) and it attacks every assumption not in the set. We propose a more liberal, argumentation-theoretic semantics, based upon the notion of admissible extension in logic programming. We regard a set of assumptions, in general, as admissible if and only if it is conflict-free and defends itself (by attacking) every set of assumptions which attacks it. We identify conditions for the existence of extensions and for the equivalence of different semantics. (EJ 1997 Elsevier Science B.V.

Time and change are notions that seems unavoidable in some areas of work and investigation, languages that can deal with these notions are necessary. At the same time, methods for a proper time handling are quite complex, mainly because problem's complexity and variety of solutions. Between the languages developed to cover these expectations, under a specific view of time, are [Cobo and Augusto, 1999a] EMTLP and a metric temporal logic's fragment, bounded universal Horn formulae analyzed by Brzoska . Although both of them performed metric temporal programming, they face this fact from different perspectives. In this work we are going to try a comparison between them after a short overview over each. In this first stage we present a way of representing EMTPL's in first order logic using Brzoska's approximation as a bridge, and we also compare some aspects of both programming languages.

This paper revisits the CP=PC problem: the identification of the probability of a conditional with the corresponding conditional probability. Working with a fixed classical probability space and a minimal implicational backbone (rules (p1)-(p4) plus Modus Ponens), we prove a collapse theorem: if a global CP=PC identity is imposed for a single implicational connective and the logic validates a natural symmetry (A-B ⊢ B-A), then the probability assignment degenerates on the positive-probability fragment, trivializing the symmetric extension. We further show that moving to Łukasiewicz logic does not repair the identification: under truth-averaging semantics, the probability of A ⇒ B is a nonlinear expectation that generally diverges from Pr(B | A). The upshot is methodological: CP=PC should not be treated as a law of an implicational connective. Instead, we propose a division of roles: keep the implicational connective for inference, and introduce a dedicated conditional (B | A) whose semantics guarantees that its probability equals Pr(B | A). This reframes the Stalnaker program by separating inferential structure from probabilistic conditionalization and locating CP=PC in the semantics of an internal conditional rather than in the implicational fragment.

The proof of the Second Incompleteness Theorem consists essentially of proving the uniqueness and explicit definability of the sentence asserting its own unprovability. This turns out to be a rather general phenomenon: Every instance of self-reference describable in the modal logic of the standard proof predicate obeys a similar uniqueness and explicit definability law. The efficient determination of the explicit definitions of formulae satisfying a given instance of self-reference reduces to a simple algebraic problem--that of solving the corresponding fixed-point equation in the modal logic. We survey techniques for the efficient calculation of such fixed--points. GSdel elevated the s~atus of self-reference from that of a mere curiosity (e.g. the liar paradox) to that of a useful metamathematical tool (the Diagonalization Theorem). It is, however, a largely neglected tool whose occasional use in metamathematies elicits noises of surprise from the spectators. In the hope of reducing the level of this sm~prise we devote two papers to reviewing and extending our knowledge of self-referentiM sentences. It happens that one can go a long way in understanding such sentences by means of the most simple-minded of considerations. If we ~gree to refer to self-referentiM sentences as fixed--points, we can compress our simple-mindedness into the little motto: Calculate fixed-points! Calculations are possible because quite often a self-referentiM sentence is actually implicitly defined by the diagonalization through which it refers to itself. The transformation of the implicit definition into an explicit one constitutes a calculation. The dichotomy between the cMculations of this paper and those of the sequel is governed by the extent to which the calculations can be formalized in a given theory. In this paper, we consider a restricted class of self-referential sentences and present Mgorithms for explicit calculations of the members of this class. As this class of self-referentiM sentences has only three known ~pplications (the Incompleteness Theorems and ]55b's Theorem) and can be shown to have no others (Solovay's Completeness Theorem--see [21]~ we pursue the present calculation primarily for aesthetic reasons. The self-referential formulae with numerous applications do not seem to be so evplicitly cMeulable~ i.e. although we can give explicit definitions for them, ~ve c~nnot formalize the proof of the correctness of the definitions. These non-evlplicit calculations form the topic of the nex~ paper. 20 Craig Smorynsl~i ~odulo the results just cited, the first two sections of this paper are self-contMned. This is no longer true of Section 3~ where we assume some knowledge of modal model theory.

We are interested in systems which do not prescribe one single kind of preference, but in which varying kinds of preferences can be used simultaneously. In such systems it is essential to know the interaction among the kinds of preferences being used, and we therefore introduce and study a nonmonotonic logic to reason about sixteen strict and non-strict kinds of preferences, including ceteris paribus preferences. Moreover, we study "distinguished" preference orders based on specificity principles by showing when these distinguished preference orders are unique, and by presenting algorithms to calculate the distinguished preference orders.

The judicial syllogism represents one of the most significant and widely accepted applications of logic in the field of law. Alongside the legislative syllogism, it is a part of the broader conceptual framework commonly referred to as the legal syllogism. This logical structure is classified as a type of mediate deductive inference, which proceeds from general to particular statementsa reasoning process traditionally associated with the dictum de omni principle. Although intuitive and traditionally accepted for offering the proper structure for the application of law, the judicial syllogism is inherently static due to its foundation in classical mediate deductive reasoning. It does not accommodate the dynamic nature of judicial processes, where addressing quaestio juris and quaestio facti may modify the premises, thereby altering the conclusions. Therefore, this article intends to analyse the classical types of mediate deductive inferences, the static nature of the judicial syllogism and the shortcomings of this monotonic type of logic where the conclusion does not change once it is derived. After analysing the points in the construction of the judicial syllogism, the article also proposes a way of integrating nonmonotonic logic in the elaboration of the judicial syllogism in order to capture the actual dynamic of judiciary processes of applying law to particular cases, without altering the overall structure of the judicial syllogism. The operationalization of this theoretical framework could be of practical relevance in developing computational tools, especially in AI applications.

Default logic can be regarded as a mechanism to represent families of belief sets of a reasoning agent. As such, it is inherently second-order. In this paper, we study the problem of representability of a family of theories as the set of extensions of a default theory. We give a complete solution to the representability by means of normal default theories. We obtain partial results on representability by arbitrary default theories. We construct examples of denumerable families of non-including theories that are not representable. We also study the concept of equivalence between default theories.

Though the semantics of nonmonotonic logic programming has been studied extensively, relatively little work has been done on operational aspects of these semantics. In this paper, we develop techniques to compute the well-founded model of a logic program. We describe a prototype implementation and show, based on experimental results, that our technique is more efficient than the standard alternating fixpoint computation. Subsequently, we develop techniques to compute the set of all stable models of a deductive database. These techniques first compute the well-founded semantics and then use an intelligent branch and bound strategy to compute the stable models. We report on our implementation, as well as on experiments that we have conducted on the efficiency of our approach. monotonic reasoning, negation by failure. Index Term-Logic programming, deductive databases, non-I 'Throughout this paper, we will consider only deductive databases, i e., logic programs without function symbols.

This paper studies the semantics for the class of all defeasible (inheritance) networks, including cyclic and inconsistent networks using a transformation approach. First we show that defeasible networks can be translated, tractably, to default theories while preserving Horty's offpath credulous semantics for all consistent networks. Using the existing methods in dealing with the semantics of default logic, we are able to provide a tractable skeptical semantics, the well-founded semantics, and a new credulous semantics, the regular semantics, both of which are defined for any defeasible network. Furthermore, we show that these semantics are based on the same principle of specificity used by Horty in defining his credulous semantics of defeasible networks. 1 Introduction Two fundamental problems are to be addressed in this paper. First, the semantics of defeasible networks has previously been studied mainly under the assumption that such a network is acyclic and consis...

Strategies purporting to determine the meaning of inner states of belief-content in terms of their inferential role usually assume the inner structure of the human inferential competence to be that of first order logic plus identity. Considerations of computational complexity and cumbersomeness of representation tend to undermine the plausibility of combining such strategies with this assumption. In this paper I contend that appealing to rules of default reasoning won't make things turn out any better for the inferential role functionalist.

In order to reason in a non-trivializing way with contradictions, paraconsistent logics reject some classically valid inferences. As a way of recovering some of these inferences, Graham Priest (Priest (1991)) proposed to nonmonotonically strengthen the Logic of Paradox by allowing the selection of "less inconsistent" models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is applied to contradictory assumptions. In Priest's approach the inconsistent parts of models are compared in an extensional way by considering their inconsistent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known problem for Priest's extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In this paper, we demonstrate that Strong Reassurance holds for the extensional approach under a cardinality-based comparison of the inconsistent parts of models. Furthermore, we introduce and investigate the metatheory of the class of first-order nonmonotonic inconsistencytolerant construct over the extensional or quantitative comparisons of their respective models. Core model-theoretic properties for these logics, such as the Löwenheim-Skolem theorems, along with other nonmonotonic properties, are further studied.

Non-monotonic logics are examined and found to be inadequate as descriptions of reason maintenance systems (sometimes called truth maintenance systems). A logic is proposed that directly addresses the problem of characterizing the mental states of a reasoning agent attempting to reason with respect to some object theory. The proposed logic, propositional dynamic logic of derivation (PDLD), is given a semantics, and a sound and complete axiomatization. The descriptive power of PDLD is demonstrated by expressing various inferential control policies as PDLD formulae. In this note I will elaborate the propositional fragment of an axiomatic semantics of reason maintenance systems (RMS's) . The development of such a semantics stems from the desire to provide a declarative specification language for RMS's with particular emphasis on the description of the control of their reasoning processes, and to serve as a formal setting within which to compare and contrast the properties of different RMS's. * The research reported herein is funded in part by the Defense Advanced Research Projects Agency under contract number F30602-85-C-0033.

This paper presents a qualitative logic-based method for the steady-state analysis and revision of metabolic networks with inhibition. The approach is able to automatically revise an initial metabolic model -through the addition and removal of whole reactions or individual substrates, products and inhibitors -in order to ensure the existence of a steady-state behaviour consistent with a set of experimental observations. We show how this can be done in a nonmonotonic logic programming setting and discuss the challenges that arise when metabolic cycles or mutual inhibitions occur in the underlying network.

This paper presents a nonmonotonic ILP approach for the automatic revision of metabolic networks through the logical analysis of experimental data. The method extends previous work in two respects: by suggesting revisions that involve both the addition and removal of information; and by suggesting revisions that involve combinations of gene functions, enzyme inhibitions, and metabolic reactions. Our proposal is based on a new declarative model of metabolism expressed in a nonmonotonic logic programming formalism. With respect to this model, a mixture of abductive and inductive inference is used to compute a set of minimal revisions needed to make a given network consistent with some observed data. In this way, we describe how a reasoning system called XHAIL was able to correctly revise a state-of-the-art metabolic pathway in the light of real-world experimental data acquired by an autonomous laboratory platform called the Robot Scientist.

This paper makes two contributions towards the logical modelling of inhibition in metabolic networks. First it exposes the logical inconsistency of an existing state-of-the-art metabolic theory. Then it proposes a new model from which the required hypotheses are consistently inferred. Underlying this work is use of nonmonotonic logic programs to represent and reason about competing inhibitory effects on the net concentrations of metabolites in enzyme-catalysed biochemical networks.

This paper makes two contributions towards the logical modelling of inhibition in metabolic networks. First it exposes the logical inconsistency of an existing state-of-the-art metabolic theory. Then it proposes a new model from which the required hypotheses are consistently inferred. Underlying this work is use of nonmonotonic logic programs to represent and reason about competing inhibitory effects on the net concentrations of metabolites in enzyme-catalysed biochemical networks.

This paper presents a qualitative logic-based method for the steady-state analysis and revision of metabolic networks with inhibition. The approach is able to automatically revise an initial metabolic model -through the addition and removal of whole reactions or individual substrates, products and inhibitors -in order to ensure the existence of a steady-state behaviour consistent with a set of experimental observations. We show how this can be done in a nonmonotonic logic programming setting and discuss the challenges that arise when metabolic cycles or mutual inhibitions occur in the underlying network.

This paper presents a nonmonotonic ILP approach for the automatic revision of metabolic networks through the logical analysis of experimental data. The method extends previous work in two respects: by suggesting revisions that involve both the addition and removal of information; and by suggesting revisions that involve combinations of gene functions, enzyme inhibitions, and metabolic reactions. Our proposal is based on a new declarative model of metabolism expressed in a nonmonotonic logic programming formalism. With respect to this model, a mixture of abductive and inductive inference is used to compute a set of minimal revisions needed to make a given network consistent with some observed data. In this way, we describe how a reasoning system called XHAIL was able to correctly revise a state-of-the-art metabolic pathway in the light of real-world experimental data acquired by an autonomous laboratory platform called the Robot Scientist.

This paper describes a new logic-based approach for representing and reasoning about metabolic networks. First it shows how biological pathways can be elegantly represented in a logic programming formalism able to model full chemical reactions with substrates and products in different cell compartments, and which are catalysed by iso-enzymes or enzymecomplexes that are subject to inhibitory feedbacks. Then it shows how a nonmonotonic reasoning system called XHAIL can be used as a practical method for learning and revising such metabolic networks from observational data. Preliminary results are described in which the approach is validated on a state-ofthe-art model of Aromatic Amino Acid biosynthesis.

CF-induction is a sound and complete hypothesis finding procedure for full clausal logic which uses the principle of inverse entailment to compute a hypothesis that logically explains a set of examples with respect to a prior background theory. Currently, CF-induction computes hypotheses by applying combinations of several complex generalisation operators to an intermediate theory called a bridge formula. In this paper we propose an alternative approach whereby hypotheses are derived from a bridge formula using a single deductive operator and a single inductive operator. We show that our simplified procedure preserves the soundness and completeness of CF-induction.

Classical decision theop can be t e n restrictive when formulating decision models for practical situations. The principal group of restrictions, inherent in numerical decision theories, is that they make no provision for reasoning about the decision process itself. Classical procedures cannot reflect on what the decision is. what the options are, what methods ma! be used in making a decision, what knowledge ma) be relevant, and w forth. An approach that accommodates classical decision theory within a framework of first-order logic with nonmonotonic extensions is described. Among the benefits offered by the approach are: the potential to express qualitative urguments about the desirabilih of deckion options in the absence of probability or utilih parameters: to automate techniques for generating decision options; and to initiate, control. and terminate a decision process autonomously. The rationality of a logical decision framework is discussed. A particular benefit of implementing decision procedures in a nonmonotonic logic is that decision systems can adapt automatically as beliefs change. contradictions are encountered. or new knowledge is acquired.

It has recently been discovered that both quantum and classical propositional logics can be modelled by classes of non-orthomodular and thus non-distributive lattices that properly contain standard orthomodular and Boolean classes, respectively. In this paper we prove that these logics are complete even for those classes of the former lattices from which the standard orthomodular lattices and Boolean algebras are excluded. We also show that neither quantum nor classical computers can be founded on the latter models. It follows that logics are valuation-nonmonotonic in the sense that their possible models (corresponding to their possible hardware implementations) and the valuations for them drastically change when we add new conditions to their defining conditions. These valuations can even be completely separated by putting them into disjoint lattice classes by a technique presented in the paper.

It has recently been discovered that both quantum and classical propositional logics can be modelled by classes of non-orthomodular and thus non-distributive lattices that properly contain standard orthomodular and Boolean classes, respectively. In this paper we prove that these logics are complete even for those classes of the former lattices from which the standard orthomodular lattices and Boolean algebras are excluded. We also show that neither quantum nor classical computers can be founded on the latter models. It follows that logics are valuation-nonmonotonic in the sense that their possible models (corresponding to their possible hardware implementations) and the valuations for them drastically change when we add new conditions to their defining conditions. These valuations can even be completely separated by putting them into disjoint lattice classes by a technique presented in the paper.

Synopsis: Dynamic semantics violates numerous classical laws, including Non-Contradiction. Proponents of dynamic semantics have offered no explanation for this behavior, and some critics consider this result to be strong evidence against the tenability of the dynamic program. I defend and explain failures of Non-Contradiction by comparing dynamic semantics and classical, truthconditional semantics in terms of their idealizing assumptions. I demonstrate that dynamic semantics rejects context fixity, an idealizing assumption that truth-conditional semantics typically adopts. I then argue that any semantics which rejects context fixity should, by the classical semanticist's own lights, violate Non-Contradiction under certain circumstances. I then demonstrate that dynamic semantics violates Non-Contradiction in all and only those circumstances. I subsequently appeal to this insight to vindicate some of dynamic semantics' more controversial predictions. I close by suggesting that discussion of idealizing assumptions, common in the sciences, is similarly crucial to fruitful discussion in natural language semantics.

We argue for the value of examining the internal processes that robots might actually use to draw inferences in a timely way in a dynamic world. This requires a significantly different way of thinking about logic and reasoning, which in turn bears on some traditional logic-related problems such as omniscience and reasoning in the presence of a contradiction, as well as on a wide variety of other AI issues. A nonstandard internally-evolving notion of time seems to be the key that unlocks other tools.

When a reasoning system encounters a contradiction like p&q=>false, it may try to eliminate it by retracting some belief that supports either p or q. This paper addresses the issue of how to determine which of the possible supporting beliefs to retract. The problem is studied in the context of a heterogenous distributed environment in which we can not make assumptions about how agents derive their beliefs. The presented model considers two properties of beliefs: the authority of the agents that provided the belief in the first place and the costs incurred upon other agents that already used the belief (directly or indirectly) for their own decision making or action, if the belief is to be retracted. The model is implemented in an information infrastructure we have created for enterprise integration.

Generalising the ideas of [10] we define a simple extension of the notion of unfounded set, called assumption set, that applies to disjunctive logic programs with strong negation. We show that assumption-free interpretations of such extended logic programs coincide with equilibrium models in the sense of [13] and hence with the answer sets of [3, 4].

In this paper we show h o w a n umber of di erent f o r m ulations of nonmonotonic reasoning, probabilistic reasoning and design can be combined into a coherent logic-based abductive framework. This framework is based on allowing consistent assumptions to be used to prove a goal. Di erent frameworks are characterised by who chooses the assumptions, whether an adversary chooses the assumptions, nature chooses the assumptions, or one gets to choose whatever assumptions one likes.

Description logic programs (dl-programs) under the answer set semantics formulated by Eiter et al. have been considered as a prominent formalism for integrating rules and ontology knowledge bases. A question of interest has been whether dl-programs can be captured in a general formalism of nonmonotonic logic. In this paper, we study the possibility of embedding dl-programs into default logic. We show that dl-programs under the strong and weak answer set semantics can be embedded in default logic by combining two translations, one of which eliminates the constraint operator from nonmonotonic dl-atoms and the other translates a dl-program into a default theory. For dl-programs without nonmonotonic dl-atoms but with the negation-as-failure operator, our embedding is polynomial, faithful, and modular. In addition, our default logic encoding can be extended in a simple way to capture recently proposed weakly well-supported answer set semantics, for arbitrary dl-programs. These results reinforce the argument that default logic can serve as a fruitful foundation for query-based approaches to integrating ontology and rules. With its simple syntax and intuitive semantics, plus available computational results, default logic can be considered an attractive approach to integration of ontology and rules.

In this work we focus on extensions of Description Logics (DLs) of typicality by means of probabilities. We introduce a novel extension of the logic of typicality ALC + T R , able to represent and reason about typical properties and defeasible inheritance in DLs. The novel logic (ALCT P : Typical ALC with Probabilities as Proportions) allows inclusions of the form T(C) p D, with probability p representing a proportion, meaning that "all the typical Cs are Ds, and the probability that a C element is not a D element is 1 − p". We also compare and confront this novel logic with a similar one already presented in the literature (T CL , introduced in Lieto and Pozzato (2020, J. Exp. Theor. Artif. Intell., 32, 769-804)), inspired by the DISPONTE semantics and that allows inclusions of the form p : T(C) D with probability p, where p represents a degree of belief, whose meaning is that "we believe with a degree p that typical Cs' are also Ds.". We then show that the proposed ALCT P extension (like the previous T CL) can be applied in order to tackle a specific and challenging problem in the field of common-sense reasoning, namely the combination of prototypical concepts, that have been shown to be problematic to model for other symbolic approaches like fuzzy logic. We show that, for the proposed extension, the complexity of reasoning remains EXPTIME-complete as for the underlying standard monotonic DL ALC.

In multi agent systems, agents interact with each other through a dialogue protocol to inform each other the relevant facts in order to reach the common goals. In a dialogue the protocol defines the possible replies for a presented speech act and the strategy specifies which speech act is premier. The strategy defines how to achieve the agents' goals by choosing the best one among all choices. Compared with researches on dialogue protocol, there are few works on dialogue strategy. We proposed a formal model for dialogue strategy in order to select the best speech act and the proper content. This model in each step of dialogue can decide the appropriate act and content allowed by the dialogue protocol in order to satisfy the agents' goals. This model computes the priority of possible acts and contents using argumentation theory. The proposed model is illustrated through a common example of deliberation dialogue.

In this paper we introduce a prioritized default logic. We build this logic modularly from Standard Deontic Logic by the addition of default rules and priorities among them. Our main aim is to provide a logical framework to reason about scenarios where prescriptive and descriptive statements coexist and may be incomplete and contradictory. We motivate and illustrate the technical elements of our work with the use of examples (classical, and coming from software engineering). In addition, we present sound, complete, and terminating (with loop check) tableau-based proof calculi for credulous and sceptical reasoning in our logic.

Modus ponens (from A and "if A then C" infer C; short: MP) is one of the most basic inference rules. The probabilistic MP allows for managing uncertainty by transmitting assigned uncertainties from the premises to the conclusion (i.e., from P (A) and P (C|A) infer P (C)). In this paper, we generalize the probabilistic MP by replacing A by the conditional event A|H. The resulting inference rule involves iterated conditionals (formalized by conditional random quantities) and propagates previsions from the premises to the conclusion. Interestingly, the propagation rules for the lower and the upper bounds on the conclusion of the generalized probabilistic MP coincide with the respective bounds on the conclusion for the (non-nested) probabilistic MP.

Modus ponens (\emph{from $A$ and "if $A$ then $C$" infer $C$}, short: MP) is one of the most basic inference rules. The probabilistic MP allows for managing uncertainty by transmitting assigned uncertainties from the premises to the conclusion (i.e., from $P(A)$ and $P(C|A)$ infer $P(C)$). In this paper, we generalize the probabilistic MP by replacing $A$ by the conditional event $A|H$. The resulting inference rule involves iterated conditionals (formalized by conditional random quantities) and propagates previsions from the premises to the conclusion. Interestingly, the propagation rules for the lower and the upper bounds on the conclusion of the generalized probabilistic MP coincide with the respective bounds on the conclusion for the (non-nested) probabilistic MP.

We propose an architecture for a rule-based online management systems (RuleOMS). Typically, many domain areas face the problem that stakeholders maintain databases of their business core information and they have to take decisions or create reports according to guidelines, policies or regulations. To address this issue we propose the integration of databases, in particular relational databases, with a logic reasoner and rule engine. We argue that defeasible logic is an appropriate formalism to model rules, in particular when the rules are meant to model regulations. The resulting RuleOMS provides an efficient and flexible solution to the problem at hand using defeasible inference. A case study of an online child care management system is used to illustrate the proposed architecture.

Logic programs with abstract constraint atoms provide a unifying framework for studying logic programs with various kinds of constraints. Establishing strong equivalence between logic programs is a key property for program maintenance and optimization, and for guaranteeing the same behavior for a revised original program in any context. In this paper, we study strong equivalence of logic programs with abstract constraint atoms. We first give a general characterization of strong equivalence based on a new definition of program reduct for logic programs with abstract constraints. Then we consider a particular kind of program revision-constraint replacements addressing the question: under what conditions can a constraint in a program be replaced by other constraints, so that the resulting program is strongly equivalent to the original one.

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Abstract. A model is defined that predicts an agent's ascriptions of causality (and related notions of facilitation and justification) between two events in a chain, based on background knowledge about the normal course of the world. Background knowledge is represented by nonmonotonic consequence relations. This enables the model to handle situations of poor information, where background knowledge is not accurate enough to be represented in, eg, structural equations. Tentative properties of causality ascriptions are explored, ie, ...

Over recent years, various semantics have been proposed for dealing with updates in the setting of logic programs. The availability of different semantics naturally raises the question of which are most adequate to model updates. A systematic approach to face this question is to identify general principles against which such semantics could be evaluated. In this paper we motivate and introduce a new such principle-the refined extension principle-which is complied with by the stable model semantics for (single) logic programs. It turns out that none of the existing semantics for logic program updates, even though based on stable models, complies with this principle. For this reason, we define a refinement of the dynamic stable model semantics for Dynamic Logic Programs that complies with the principle. This work was partially supported by FEDER financed project FLUX (POSI/40958/SRI/2001) and by project SOCS (IST-2001-32530). Special thanks are due to Pascal Hitzler and Reinhard Kahle for helpful discussions.

This paper presents the notion of AND-OR reduction, which reduces a WF net to a smaller net by iteratively contracting certain well-formed subnets into single nodes until no more such contractions are possible. This reduction can reveal the hierarchical structure of a WF net, and since it preserves certain semantical properties such as soundness, it can help with analysing and understanding why a WF net is sound or not. The reduction can also be used to verify if a WF net is an AND-OR net. This class of WF nets was introduced in earlier work, and arguably describes nets that follow good hierarchical design principles. It is shown that the AND-OR reduction is confluent up to isomorphism, which means that despite the inherent non-determinism that comes from the choice of subnets that are contracted, the final result of the reduction is always the same up to the choice of the identity of the nodes. Based on this result, a polynomial-time algorithm is presented that computes this unique...