Unifying various approaches to default logic

Compiling specificity into approaches to nonmonotonic reasoning

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?

Semantics and complexity of abduction from default theories

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.

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.

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.

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).

A prerequisite to a theory of the way agents understand speech acts is a theory of how their beliefs and intentions are revised as a consequence of events. This process of attitude revision is an interesting domain for the application of nonmonotonic reasoning because speech acts have a conventional aspect that is readily represented by defaults, but that interacts with an agent's beliefs and intentions in many complex ways that may override the defaults. Perrault has developed a theory of speech acts, based on Rieter's default logic, that captures the conventional aspect; it does not, however, adequately account for certain easily observed facts about attitude revision resulting from speech acts. A natural theory of attitude revision seems to require a method of stating preferences among competing defaults. We present here a speech act theory, formalized in hierarchic autoepistemic logic (a refinement of Moore's autoepistemic logic), in which revision of both the speaker's and hearer's attitudes can be adequately described. As a collateral benefit, efficient automatic reasoning methods for the formalism exist. The theory has been implemented and is now being employed by an utterance-planning system.

A survey of complexity results for non-monotonic logics

This is the second in a series of workshops that are bringing together researchers from the theoretical end of both the logic programming and artificial intelligence communities to discuss their mutual interests. This workshop emphasizes the relationship between logic programming and non-monotonic reasoning.Luis' Moniz Pereira is Professor in the Department of Computer Science at the Universidade Nova Lisboa, Portugal. Anil Nerode is Professor and Director of the Mathematical Sciences Institute at Cornell University.Topics include: Stable Semantics. Autoepistemic Logic. Abduction. Implementation Issues. Well-founded Semantics. Truth Maintenance. Probabilistic Theories. Applications. Default Logic. Diagnosis. Complexity and Theory. Handling Inconsistency.

It is well known that, under certain conditions, it is possible to split logic programs under stable model semantics, i.e. to divide such a program into a number of different levels, such that the models of the entire program can be constructed by incrementally constructing models for each level. Similar results exist for other non-monotonic formalisms, such as auto-epistemic logic and default logic. In this work, we present a general, algebraic splitting theory for programs/theories under a fixpoint semantics. Together with the framework of approximation theory, a general fixpoint theory for arbitrary operators, this gives us a uniform and powerful way of deriving splitting results for each logic with a fixpoint semantics. We demonstrate the usefulness of these results, by generalizing Lifschitz and Turner's splitting theorem to other semantics for (non-disjunctive) logic programs.

Realism for agents with unconditional beliefs, desires and intentions (BDI agents) has been analyzed in modal logic. This paper provides a logical analysis of realism for agents with conditional beliefs and desires in a rule based approach analogous to Reiter's default logic. We distinguish two types of realism, which we call ‘a priori' and ‘a posteriori' realism. We analyze whether these two new properties are compatible with other properties discussed in the literature, such as existence of extensions. We show that Reiter's default logic is too strong, in the sense that a weaker notion of maximality of extensions is needed to satisfy realism. Finally we show that several existing approaches do not satisfy the new realism properties, and we introduce a new construction that does satisfy them.

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.

Preferential reasoning in the perspective of Poole default logic

In this paper we present a modal approach to “contrastive logic”, the logic of contrasts as these appear in natural language conjunctions such as ‘but’. We use a simple modal logic, which is an extension of the well-knownS5 logic, and base the contrastive operators proposed by Francez in [2] on the basic modalities that appear in this logic. We thus obtain a logic for contrastive operators that is more in accord with the tradition of intensional logic, and that, moreover — we argue — has some more natural properties. Particularly, attention is paid to nesting contrastive operators. We show that nestings of ‘but’ give quite natural results, and indicate how nestings of other contrastive operators can be done adequately. Finally, we discuss the example of the Hangman's Paradox and some similarities (and differences) with default reasoning.