Reasoning by symmetry in non-monotonic inference

Knowledge formalization and reasoning automation are central within Artificial Intelligence. Classical logic has been traditionally used for such purposes. However, it is better suited to deal with complete knowledge in ideal circumstances. In real situations, in which the knowledge is partial, classical logic is not sufficient since it is monotonic. No monotonic logics have been proposed to better cope with practical reasoning. A successful formalization of no monotonic reasoning is Reiter's default logic which extends classical logic with default rules. Unfortunately, default logic is undividable. One reason for that is the use of classical logic as its monotonic basis. In this work, we change the default logic monotonic basis and propose a new default logic based on the description logic ALC. This new default logic is decidable and useful to formalize practical reasoning on hierarchical ontology's with exceptions, specially the ones that deals with legal knowledge and reasoning. On the default counterpart, we add some restrictions to the application of defaults in order to obtain nice properties such as coherence and elimination of anomalous extensions. We present the main algorithms used to build the extensions for this logic with its complexity analysis.

Default theories that always have extensions

The past few decades have seen a resurgence of reasoning techniques in artificial intelligence involving both classical and non-classical logics. In his paper, ``Multi-valued Logics: A Uniform Approach to Reasoning in Artificial Intelligence'', Ginsberg has shown that through the use of bilattices, several reasoning techniques can be unified under a single framework. A bilattice is a structure that can be viewed as a class of truth values that can accommodate incomplete and inconsistent information and in certain cases default information. In bilattice theory, knowledge is ordered along two dimensions: truth/falsity and certainty/uncertainty. By defining the corresponding bilattices as truth spaces, Ginsberg has shown that the same theorem prover can be used to simulate reasoning in first order logic, default logic, prioritized default logic and assumption truth maintenance system. Although this is a significant contribution, Ginsberg's paper was lengthy and involved. This paper summarizes some of the essential concepts and foundations of bilattice theory. Furthermore, it discusses the connections of bilattice theory and several other existing multi-valued logics such as the various three-valued logics and Belnap's four-valued logic. It is noted that the set of four truth values in Belnap's logic form a lattice structure that is isomorphic to the simplest bilattice. Subsequently, Fitting proposed a conflation operation that can be used to select sub-sets of truth values from this and other bilattices. This method of selecting sub-sets of truth values provides a means for identifying sub-logic in a bilattice.

Reiter’s pioneering work on default reasoning aimed to formalize the notion of quickly “jumping to conclusions”. Unfortunately, his resulting formalism, default logic, proved more computationally complex than classical logics. The same has been true for the other major formalisms for nonmonotonic reasoning, circumscription [20], nonmonotonic logic [22], [21], and autoepistemic logic [25]. This has dramatically limited the applicability of formal methods vis a vis real problems involving defaults. The complexity of consistency checking is one of the two problems that must be addressed to reduce the complexity of default reasoning. We propose an approach to default reasoning based on approximate consistency checking, using a novel synthesis of limited contexts and fast incomplete checks, and argue that this combination overcomes the limitations of its component parts. Our approach trades correctness for speed, but we argue that the nature of default reasoning makes this trade relatively inexpensive and intuitively plausible. This approach not only accords well with Reiter’s original motivations, but converges to Default Logic in the computational limit. We describe a prototype implementation of a default reasoner based on these ideas, and a preliminary empirical evaluation.KeywordsConsistency CheckDefense Advance Research Project AgencyDefault LogicNonmonotonic ReasoningDefault ReasoningThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Chapter 1 Knowledge Representation and Classical Logic

The logic program formalism is commonly viewed as a modal or default logic. In this paper, we propose an alternative interpretation of the formalism as a terminological logic. A terminological logic is designed to represent two different forms of knowledge. A TBox represents definitions for a set of concepts. An ABox represents the assertional knowledge of the expert. In our interpretation, a logic program is a TBox providing definitions for all predicates; this interpretation is present already in Clark's completion semantics. We extend the logic program formalism such that some predicates can be left undefined and use classical logic as the language for the ABox. The resulting logic can be seen as an alternative interpretation of abductive logic program formalism. We study the expressivity of the formalism for representing uncertainty by proposing solutions for problems in temporal reasoning, with null values and open domain knowledge.

Connections between default reasoning and partial constraint satisfaction

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a system of beliefsS (and seen to be inS), basedboth on some being inS and on some restraints not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.

The axiomatic system P is an important standard for plausible, nonmonotonic inferences that is, however, known to be too weak to solve benchmark problems like irrelevance, or subclass inheritance so-called Drowning Problem. Spohn's ranking functions which provide a semantic base for system P have often been used to design stronger inference relations, like Pearl's system Z, or c-representations. While each c-representation shows excellent inference properties and handles particularly irrelevance and subclass inheritance properly, it is still an open problem which c-representation is the best. In this paper, we focus on the generic properties of c-representations and consider the skeptical inference relation c-inference that is obtained by taking all c-representations of a given knowledge base into account. In particular, we show that c-inference preserves the properties of solving irrelevance and subclass inheritance which are met by every single c-representation. Moreover, we characterize skeptical c-inference as a constraint satisfaction problem so that constraint solvers can be used for its implementation.

We investigate the connection between propositional proof systems and their canonical pairs. It is known that simulations between propositional proof systems translate to reductions between their canonical pairs. We focus on the opposite direction and study the following questions. Q1: For which propositional proof systems f and g does the implication [Formula: see text] hold, and for which does it fail? Q2: For which propositional proof systems of different strengths are the canonical pairs equivalent? Q3: What do (non-)equivalent canonical pairs tell about the corresponding propositional proof systems? Q4: Is every NP-pair (A, B), where A is NP-complete, strongly many-one equivalent to the canonical pair of some propositional proof system? In short, we show that Q1 and Q2 can be answered with 'for almost all', which generalizes previous results by Pudlák and Beyersdorff. Regarding Q3, inequivalent canonical pairs tell that the propositional proof systems are not "very similar," while equivalent, P -inseparable canonical pairs tell that they are not "very different." We can relate Q4 to the open problem in structural complexity that asks whether unions of disjoint NP-complete sets are NP-complete. This demonstrates a new connection between propositional proof systems, disjoint NP-pairs, and unions of disjoint NP-complete sets.

Handbook of Constraint Programming

This panel discussion consists of four position papers examining the potential of applying concepts and theories from the fields of artificial intelligence (AI) and expert system design in the research and development of distributed systems. The focus is on aspects of the operating systems and their stringent requirements as reflected in their complex decision-making processes. The panelists recognize that the AI field, with its constraints and limitations, may not provide the necessary technology for the implementation of real-time systems at the present time. They argue, however, that much insight can be gained in the development of new approaches and new algorithms using analytical and simulation models based on classical logic and modal logics. >

An important limitation of traditional logic programming as a knowledge representation tool, in comparison with classical logic, is that logic programming does not allow us to deal directly with incomplete information. In order to overcome this limitation, we extend the class of general logic programs by including classical negation, in addition to negation-as-failure. The semantics of such extended programs is based on the method of stable models. The concept of a disjunctive database can be extended in a similar way. We show that some facts of commonsense knowledge can be represented by logic programs and disjunctive databases more easily when classical negation is available. Computationally, classical negation can be eliminated from extended programs by a simple preprocessor. Extended programs are identical to a special case of default theories in the sense of Reiter.

We present a computational model of dialectical argumentation that could serve as a basis for studying elements of legal reasoning. Argumentation is well-suited to decisionmaking in the legal domain, where knowledge is incomplete, uncertain, and inconsistent, We model an argument both as information structure, i.e., argument units connecting claims with supporting data, and as dialectical process, i.e., an alternating series of moves made by opposing sides. Inspired by the legal domain, our model includes burden of proof as a key element, indicating the level of support that must be achieved by a particular side to an argument. Burden of proof acts as a move filter and termination criterion during argumentation and determines the eventual winner. We demonstrate our model by considering two examples that have been discussed previously in the artificial intelligence and legal reasoning literature. INTRODUCTION As the artificial intelligence (AI) and legal reasoning communities are well aware, most decisions are reached against a background of incomplete, uncertain, and inconsistent knowledge (i.e., weak theory domains; Porter, et. al., 1990). The most widely used AI methods for reasoning under uncertainty either rely on an absence of outright contradictions (e. g., probabilistic reasoning; Pearl, 1987) or are unable to support motivated decision-making in the face of inconsistent information (e.g., default reasoning; Ginsberg, 1987). Both solutions put the problem of deciding what to believe outside their respective domains of discourse. Choosing the proposition with highest Permission to copywithoutfee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appea, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republisb, requires a fee andlor specific permission.

It is claimed that the notion of preference is a fundamental modality in computing and is a generalization of the notion of minimality. A logic of feasible preference is presented. The non-monotonic behavior of negation in logic programming is modeled as a symbolic optimization problem. As a case study, for the class of logic programs with one or more stable models, we give a preferential transformation of logic programs that identifies their stable models as the optimal worlds in the intended model of the corresponding preferential theory. Minimization and minimization orderings are given explicit syntactic representations and their due status in the model theory. Preference logics gives a very elegant model theory for defaults, without any mention of fixpoints. Further, nonmonotonic reasoning is carried out in a monotonic logic, since members of the optimal worlds are not identified with theorems of a preferential theory. Preference logics have great potential to bring the areas of Symbolic Computation, Knowledge Representation and Classical Optimization closer.