An extended framework for default reasoning

There are several ways to look at the fact that the account of default logic which can be provided for theories with non normal defaults is rather poor from a semantical point of view. An aspect of this problem seems to be that extensions of a default theory with non normal defaults need not be incompatible. Indeed, the extensions of a default theory gather the sentences which are consequences of that default theory. Accordingly, conclusions deserving the most confidence should be found in models common to all extensions of the default theory. But it happens that the first order models which gather those “most reliable conclusions” contradict the justification of each default of the theory! Let us illustrate this with the aid of an example inspired from Lukaszewicz [1984b]. Assume that on Sundays I go fishing unless I do not wake up early and represent it by the following semi-normal default.

This paper presents a brief overview on the development of a new method for remote sensing that combines polarimetrie and spectral features of signals received from distant objects, providing spectral analysis of polarimetric variables. The detailed theory and math models are developed for the case rain observation, however generalization for other objects is provided, and some examples of spectral polarimetric approach application in different fields are presented. The enhancement of the models and signal processing operational techniques development is made and checked for the case of remote sensing of the atmosphere.

This chapter contains sections titled: 5.1 Normal defaults, 5.2 Some theoretical properties, 5.3 A proof theory for normal default theories, 5.4 Limitations of normal default theories, Problems

Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics.Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap – for example in applications to software verification – they have fundamentally different perspectives on logic.In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations.

Theories of nonmonotonic reasoning are, on the face of it, of at least two sorts. In Circumscription , generic facts like Birds fly are taken to be essentially normative, and nonmonotonicity arises when individuals are assumed to be as normal as is consistent with available information about them. In theories like Default Logic such facts are taken to be rules of inference, and nonmonotonicity arises when available information is augmented by adding as many as possible of the inferences sanctioned by such rules. Depending on which of the two informal views is taken, different patterns of nonmonotonic reasoning are appropriate. Here it is shown that these different patterns of reasoning cannot be combined in a single theory of nonmonotonic reasoning. Introduction Nonmonotonic reasoning is that which lacks a monotonicity property which has been taken to be a characteristic of logical reasoning. In a theory of nonmonotonic reasoning, the consequences of a set of premises do not always accumulate as the set of premises is expanded. Nonmonotonicity has been researched in artificial intelligence because people reason nonmonotonically, and this in ways which seem directly related to intelligence. One important example of this is in commonsense reasoning about kinds, where jumping to conclusions enables intelligent agents to save time spent on gathering information. In just what way is such jumping to logically invalid conclusions reasonable?

In the first part we show why ordinals and ordinal notations are naturally connected with proof theoretical research. We introduce the program of ordinal analysis. The second part gives examples of applications of ordinal analysis.

This paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way.We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Löwenheim-Skolem theorem.The paper is completely self-contained and includes examples of application to particular many-valued formal systems.

With the “mathematical watershed”, Logic had been transformed into a foundational theory for mathematics, a theory of truth and proofs—far away from its philosophical status of theory of the intellectual process of reasoning. With the recent substitution of the traditional proofs-as-discourses paradigm by the proofs-as-programs one, Logic is now becomming a foundational theory for computing. One could interpret this new watershed as being “yet another technological drift”, bringing Logic always closer to practical ingeneering, always further from the human intellectual process of reasoning. This article promotes the dual point of view: enlightened by the contemporary analysis of the dynamic of proofs, which bring us to a new understanding of the semantic counterpart of processes operationality (including the links between semantic dereliction due to inconsistency and computational exuberance), Logic has never appeared so close to being, finally, the theory of reasoning.

A comparative study between the theories of default reasoning and open logic is given. Some concepts of open logic, such as new premises, rejections by facts, reconstructions, epistemic processes, and its limit are introduced to describe the evolution of hypotheses. An improved version of the limit theorem is given and proved. A model-theoretic interpretation of the closed normal defaults is given using the above concepts and the corresponding completeness is proved. Any extension of a closed normal defaalt theory is proved to be the limit of a δ-partial increasing epistemic process of that theory, and vice versa. It is proved that there exist two distinct extensions of a closed normal default theory iff there is an δ-non-monotonic epistemic process of that theory. The completeness of Reiter's proof is also given and proved, in terms of the epistemic processes. Finally, the work is compared with Gardenfors's theory of knowledge in flux.

With consideration of some practical difficulties of fault knowledge sharing and application in current aviation maintenance, the expression modeling and integration application of deep and shallow aircraft fault knowledge are studied. According to widely used knowledge management technology such as case-based reasoning and diagnostics expert system, ontology-based fault case knowledge model and model-based fault diagnosis knowledge model are provided. A concept of the deep and shallow knowledge of aircraft fault from the epistemological point of view is proposed, that is fault case knowledge on the basis of experience is the shallow fault knowledge, on the contrary fault diagnosis knowledge based on causality is the deep fault knowledge. An integration model and the integration application strategy of the deep and shallow aircraft fault knowledge are researched. A prototype of aircraft fault knowledge management system is developed and its application example in an aviation manufacturing enterprise shows that the integration application can benefit the aircraft maintenance.

Counterfactual conditions such as if A were not true, then C would not have been true have been formally studied by philosophers for causal claims for decades. Counterfactuals are often used informally in practice for diagnosing systems and identifying errors or misconfigurations. This paper develops a proof theory for counterfactual reasoning of Horn clauses, which have applications in domains including security and database and program analysis. The application to security that this paper focuses on is modeling and reasoning about probing attacks in Datalog-based trust management systems, where an attacker can apply counterfactual reasoning to obtain sensitive information embedded in the system. Our work is inspired by a Hilbert-style axiomatized system for counterfactual reasoning for Horn clauses, which are hard to use to construct proofs or study properties of the system. To alleviate this difficulty, we develop a sequent calculus from first principles. We show that the sequent calculus has cut elimination and is sound and complete with regard to the corresponding Hilbert style axiomatized system. We also show how to construct proofs that model practical counterfactual reasoning scenarios in trust management systems using our sequent calculus rules.

This paper deals with applying Life Cycle Cost (LCC) and Design To Cost (DTC) principles to aircraft engine programs. The dynamic driving elements of LCC are identified with an example of direct application to a deterministic computer model. This model was used as the principal tool to project operating and support costs for the XT701 turboshaft engine in conjunction with the U.S. Army Heavy Lift Helicopter Development that featured a specific DTC-related award fee in the contract. The overall methodology of LCC and DTC supported by the math model earned a Superior evaluation with an unprecedented 100 percent award fee for this kind of application. The customer audit, in support of the performance award, supports the conclusion that computer models can be used to enhance the LCC aspects of propulsion system development programs.

The first results in proof theory were obtained using axiomatic systems of logical deduction. The chapter develops propositional and predicate logic in such an axiomatic system. It shows how reasoning under assumption can be regained in an axiomatic system by proving the so-called “deduction theorem” propositional logic (and later, predicate logic as well). The deduction theorem provides the opportunity to introduce one of the main tools needed for proving properties of formal systems, namely, proof by induction on the natural numbers. The differences between minimal, intuitionistic, and classical systems of logic according to the properties of negation characterizing each of the systems are outlined. Independence of theorems such as the law of excluded middle are here given model-theoretically (in later chapters, these are also presented proof-theoretically). After presenting classical predicate logic, the systems of classical and intuitionistic arithmetic are presented. Gentzen’s first substantial result, the so-called Gödel-Gentzen translation, is discussed in the final section where it is proved that if intuitionistic arithmetic is consistent, so is classical arithmetic.

2. INTRODUCTION Since the inception of online social networks, the amount of social data that is being published on a daily basis has been increasing at an unprecedented rate. Smart, GPS-enabled, always-connected personal devices have taken the data generation to a new level by making it tremendously easy to generate and share social content like check-in information, likes, microblogs (e.g., Twitter), multimedia data, and so on. There is an enormous value in reasoning about such streaming data and deriving meaningful insights from it in real-time. Examples of potential applications include advertising, sentiment analysis, detecting natural disasters, social recommendations, personalized trends, spam detection, to name a few. There is thus an increasing need to build scalable systems to support such applications. Complex nature of social networks and their rapid evolution, coupled with the huge volume of streaming social data and the need for real-time processing, raise many computational challenges that have not been addressed in prior work. Social network data comprises of two major components. First, there is a network (linkage) component that captures the underlying interconnection structure among the entities in the social network. Second, there is content data that is typically associated with the nodes and the edges in the social network. The social network data stream contains updates to both these components. The structure of the network may itself change rapidly in many cases, especially when things like webpages and user tags (e.g., Twitter hashtags) are treated as nodes of the network. However, most of the social network data stream consists of updates to the data associated with the nodes and the edges, e.g., status updates and other content uploaded by the users, communication among the users, and so on. There is interest in performing a wide variety of queries and analytics over such data streams in real time. The queries can range from simple publish-subscribe queries, where a user is interested in being notified when something happens in his friend circle, to complex anomaly detection queries, where the goal is to identify anomalous behavior as early as possible. In this paper, we present an introduction to this new research area of stream querying and reasoning over social data. This area combines aspects from several well-studied research areas, chief among them, social network analysis, graph databases, and data streams. We provide a formal definition of the problem, survey the related prior work, and discuss some of the key research challenges that need to be addressed (and some of the solutions that have been proposed). We note that we use the term stream reasoning in this paper to encompass a broad range of tasks including various types of analytics, probabilistic reasoning, statistical inference, and logical reasoning. We contrast our use of this term with the recent work by Valle et al. [104, 105] who define this term more specifically to refer to integration of logical reasoning systems with data streams in the context of the Semantic Web. Given the vast amount of work on this and related topics, it is not our intention to be comprehensive in this brief overview. Rather we aim to cover some of the key ideas and representative work.

This paper focusses on the mathematical theory of state-based reasoning about program constructs solely through specifications of their parts, without any reliance on their implementation mechanism. That is, the semantic foundations of compositional state-based reasoning about concurrency. The main advantages of a purely semantic approach are that: it highlights the very concept of compositional state-based reasoning about concurrency without any syntactic overhead, and it serves as a basis for the encoding of the program semantics and corresponding proof rules inside tools such as PVS which support program verification.