A logic of normality and exception

The goal of this note is to provide a background and references for the invited lecture presented at Computer Science Logic 2006. We briefly discuss motivations that led to the emergence of nonmonotonic logics and introduce two major nonmonotonic formalisms, default and autoepistemic logics. We then point out to algebraic principles behind the two logics and present an abstract algebraic theory that unifies them and provides an effective framework to study properties of nonmonotonic reasoning. We conclude with comments on other major research directions in nonmonotonic logics.

Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one as in the standard development.

The default logic (Rei 80) and the autoepistemic logic (Mo 85) are non-monotonic logics. In default logic a default theory consists of a set of propositions sentences W and a set of defaults D. A sentence is defined to be derivable from a given default theory if it belongs to an extension of the default theory. A central question is: Given a default theory Δ = (D, W) and a sentence β, is there an extension of Δ which contains β?

Argumentative systems (Pollock, 1987; Vreeswijk, 1989; Prakken, 1993) are formalizations of the process of “defeasible reasoning”, i. e., reasoning to reach conclusions that could be discarded when new evidence appears. An argument for a conclusion p is a tentative piece of reasoning an agent would accept to explain p. If the agent gets new information, the conclusion p together with the argument that supported p may no longer be valid. In that way nonmonotonicity arises. The analysis of the relationships among arguments naturally captures many features of commonsense reasoning, which could be unclear or difficult to introduce in other frameworks, such as Default Logic (Reiter, 1980), Nonmonotonic Logic (McDermott & Doyle, 1980), Autoepistemic Logic (Moore, 1985) and Circumscription (McCarthy, 1980).

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.

Logic programs, deductive databases, and more generally non-monotonic theories, use various forms of default negation, not F , whose major distinctive feature is that not F is assumed “by default”, i.e., it is assumed in the absence of sufficient evidence to the contrary. The meaning of “sufficient evidence” depends on the specific semantics used. For example, in Reiter’s Closed World Assumption, CWA [32], not A is assumed for atomic A if A is not provable, or, equivalently, if there is a minimal model in which A is false. On the other hand, in Minker’s Generalized Closed World Assumption, GCWA [18, 16], or in McCarthy’s Circumscription, CIRC [17], not A is assumed only if A is false in all minimal models. In Clark’s Predicate Completion semantics for logic programs [7], this form of negation is called negation-as-failure because not A is derivable whenever attempts to prove A finitely fail. The more recent semantics proposed for logic programs and deductive databases, such as the stable semantics [14], well-founded semantics [34], partial stable or stationary semantics [27] and static semantics [30], propose even more sophisticated meanings for default negation, closely related to more general nonmonotonic formalisms such as Default Logic, DL [33], AutoEpistemic Logic, AEL [19], and AutoEpistemic logic of Beliefs, AEB [29]. Under all of these semantics, however, default negation “not ” significantly differs from classical negation “¬”. For example, the formula charged(x) ∧ ¬guilty(x) ⊃ acquitted(x) says that a person charged with a crime should be acquitted if he or she is actually proven to be not guilty. On the other hand, the formula charged(x) ∧ not guilty(x) ⊃ acquitted(x) says that one should be acquitted of any charges by default unless sufficient evidence of that person’s guilt is demonstrated.

Definability and commonsense reasoning

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?

The ability to “define” propositions using default assumptions about the same propositions is identified as a significant source of computational complexity in nonmonotonic reasoning. If such constructs are not allowed, i.e. the knowledge base is stratified, a notable computational advantage is obtained. This is demonstrated by developing an iterative algorithm for propositional stratified autoepistemic theories the complexity of which is dominated by required classical reasoning. Thus efficient subclasses of stratified nonmonotonic reasoning can be obtained by further restricting the form of sentences in the knowledge base. As an example quadratic and linear time algorithms are derived for specific subclasses of stratified autoepistemic theories. The results are shown to imply efficient reasoning methods for stratified cases of default logic, logic programs, truth maintenance systems, and nonmonotonic modal logics.

We consider the connections between belief revision, conditional logic and nonmonotonic reasoning, using as a foundation the approach to theory change developed by Alchourron, Gardenfors and Makinson (the AGM approach). This is first generalized to allow the iteration of theory change operations to capture the dynamics of epistemic states according to a principle of minimal change of entrenchment. The iterative operations of expansion, contraction and revision are characterized both by a set of postulates and by Grove’s construction based on total pre-orders on the set of complete theories of the belief logic. We present a sound and complete conditional logic whose semantics is based on our iterative revision operation, but which avoids Gardenfors’s triviality result because of a severely restricted language of beliefs and hence the weakened scope of our extended postulates. In the second part of the paper, we develop a computational approach to theory dynamics using Rott’s E-bases as a representation for epistemic states. Under this approach, a ranked E-base is interpreted as standing for the most conservative entrenchment compatible with the base, reflecting a kind of foundationalism in the acceptance of evidence for a belief. Algorithms for the computation of our iterative versions of expansion, contraction and revision are presented. Finally, we consider the relationship between nonmonotonic reasoning and both conditional logic and belief revision. Adapting the approach of Delgrande, we show that the unique extension of a default theory expressed in our conditional logic of belief revision corresponds to the most conservative belief state which respects the theory: however, this correspondence is limited to propositional default theories. Considering first order default theories, we present a belief revision algorithm which incorporates the assumption of independence of default instances and propose the use of a base logic for default reasoning which incorporates uniqueness of names. We conclude with an examination of the behavior of an implemented system on some of Lifschitz’s benchmark problems in nonmonotonic reasoning.

This chapter introduces the linguistic phenomena that are called “genericity” (both the so‐called reference to a kind and the characterizing statement types) and shows how they have figured into a wide range of fields, such as ethics and philosophy of science (both within philosophy), commonsense reasoning and nonmonotonic logic (within artificial intelligence), and conversational implicature (within pragmatics). It also discusses the main threads within the formal semantics literature about how generic statements ought to be described.

In this chapter I provide a brief introduction to the field of Logic-Based Artificial Intelligence (LBAI). I then discuss contributions to LBAI contained in the chapters and some of the highlights that took place at the workshop on LBAI from which the papers are drawn. The areas of LBAI represented in the book are: commonsense reasoning; knowledge representation; nonmonotonic reasoning; abductive and inductive reasoning; logic, probability and decision making; logic for causation and actions; planning and problem solving; logic, planning and high-level robotics; logic for agents and actions; theory of beliefs; logic and language; computational logic; system implementations; and logic applications to mechanical checking and data integration.

In this paper we consider the effects of Gelfond-Lifschitz reduct of theories in modal language on the properties of associated notions of expansion. It turns out that, as in the general case without Gelfond-Lifschitz reduct, various monotonic modal logics collapse in this situation to the same nonmonotonic logic.

In nonmonotonic reasoning, conclusions can be retracted when new pieces of information are incorporated into premises. This contrasts with classical reasoning which is monotonic, i.e., new premises can only increase the set of conclusions that can be drawn. Slightly weaker properties, such as cumulativity and rationality, seem reasonable counterparts of such a monotonicity property for nonmonotonic reasoning but intriguingly it turned out that some major nonmonotonic logics failed to be cumulative. These observations led to the study of variants in hope of restoring cumulativity but not losing other essential properties. In this paper, we take a fresh view on cumulativity by starting from a notion of rule entailment in the context of answer set programs. It turns out that cumulativity can be revived if the expressive precision of rules subject to answer set semantics is fully exploited when new premises are being incorporated. Even stronger properties can be established and we illustrate how the approach can be generalized for major nonmonotonic logics.

Compiling specificity into approaches to nonmonotonic reasoning