Autoepistemic Theories Research Articles

Erratum to splitting an operator

In [Vennekens et al. 2006], we defined a class of stratified auto-epistemic theories and made the claim that the models of such theories under a number of different semantics (namely, (partial) expansions, (partial) extensions, Kripke-Kleene model and well-founded model) can be constructed in an incremental way, following the stratification of the theory. However, it turns out that this result only holds as long as the constructed models are consistent. This can be demonstrated by the following example. Let T be the theory {¬p; q ∧ ¬Kp}. This T is stratifiable with respect to the partition Σ0 = {p}, Σ1 = {q} of its alphabet. The least precise partial expansion, i.e., the Kripke-Kleene model, of this theory is the pair of possible world structures ({{q}}, {}). However, the stratified construction would first consider only the formula ¬p in alphabet Σ0 and construct the exact pair ({{}}, {{}}), i.e., p would be known to be false. Next, it would consider the formula q ∧ ¬Kp in alphabet Σ1 and substitute Kp by its truth value in the possible world structure {{}} for Σ0, thus arriving at q ∧ ¬f . The only partial expansion of this formula is the exact pair ({{q}}, {{q}}). Combining these possible world structures for Σ0 and Σ1, the stratified construction would finally yield ({{q}}, {{q}}), i.e., p is known to be false and q is known to be true, which is clearly not equal to the Kripke-Kleene model of T . The origin of this error lies in our definition of the operator D T (Definition 4.15 on page 790). This was defined as mapping every pair (P , S) ∈ BΣ to the possible world structure Q ∈ WΣ for which, for all i ∈ I:

Sequent calculi for skeptical reasoning in predicate default logic and other nonmonotonic logics

Sequent calculi for skeptical consequence in predicate default logic, predicate stable model logic programming, and infinite autoepistemic theories are presented and proved sound and complete. While skeptical consequence is decidable in the finite propositional case of all three formalisms, the move to predicate or infinite theories increases the complexity of skeptical reasoning to being ?11-complete. This implies the need for sequent rules with countably many premises, and such rules are employed.

Logic program semantics and circumscription of autoepistemic theories

In this paper we first demonstrate that all three prominent semantics for logic programs can be redefined in terms of alternating fixpoints of the Gelfond–Lifschitz transformation, and then show that the three semantics can be characterized by circumscription of autoepistemic theories.

On the impact of stratification on the complexity of nonmonotonic reasoning

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.

Autoepistemic circumscription and logic programming

We propose a framework of autoepistemic reasoning in which the underlying semantics is determined by the choice of a nonmonotonic inference mechanism and by specifying abelief constraint. While the latter makes the approach flexible in meeting possibly different applications, the former links the resulting semantics to a nonmonotonic reasoning formalism and thus allows adoption of existing techniques. In this paper we choosecircumscription as the underlying inference mechanism and use two different belief constraints to define two semantics,the stable circumscriptive semantics andthe well-founded circumscriptive semantics, for autoepistemic theories. The former coincides with Moore's autoepistemic logic for logic programs and is arguably more desirable in handling disjunctive autoepistemic theories. The latter is a reconstruction and extension of Przymusinski's iterative method for computing the leastAEL(circ) expansions for logic programs. We show that for logic programs the two construction methods coincide. However, while Przymusinski's construction method is restricted to logic programs only, the well-founded circumscriptive semantics is applicable to more general autoepistemic theories.

On Consistency and Completeness of Autoepistemic Theories

In this paper we expand the results on existence and uniqueness of stable autoepistemic expansions for finite stratified autoepistemic theories to the infinite case. We also introduce a notion of the closed world completion of autoepistemic theory T, which can be viewed as an autoepistemic version of Closed World Assumption. We describe a class of autoepistemic theories called strongly stratified and prove that closed world completions of such theories have consistent stable autoepistemic expansions.

Autoepistemic logic

Autoepistemic logic is one of the principal modes of nonmonotonic reasoning. It unifies several other modes of nonmonotonic reasoning and has important application in logic programming. In the paper, a theory of autoepistemic logic is developed. This paper starts with a brief survey of some of the previously known results. Then, the nature of nonmonotonicity is studied by investigating how membership of autoepistemic statements in autoepistemic theories depends on the underlying objective theory. A notion similar to set-theoretic forcing is introduced. Expansions of autoepistemic theories are also investigated. Expansions serve as sets of consequences of an autoepistemic theory and they can also be used to define semantics for logic programs with negation. Theories that have expansions are characterized, and a normal form that allows the description of all expansions of a theory is introduced. Our results imply algorithms to determine whether a theory has a unique expansion. Sufficient conditions (stratification) that imply existence of a unique expansion are discussed. The definition of stratified theories is extended and (under some additional assumptions) efficient algorithms for testing whether a theory is stratified are proposed. The theorem characterizing expansions is applied to two classes of theories, K 1 -theories and ae-programs. In each case, simple hypergraph characterization of expansions of theories from each of these classes is given. Finally, connections with stable model semantics for logic programs with negation is discussed. In particular, it is proven that the problem of existence of stable models is NP-complete.— Authors' Abstract

On the Classification and Existence of Structures in Default Logic1

We investigate possible belief sets of an agent reasoning with default rules. Besides of Reiter’s extensions which are based on a proof-theoretic paradigm (similar to Logic Programming), other structures for default theories, based on weaker or different methods of constructing belief sets are considered, in particular, weak extensions and minimal sets. The first of these concepts is known to be closely connected to autoepistemic expansions of Moore, the other to minimal stable autoepistemic theories containing the initial assumptions. We introduce the concept of stratifed collection of default rules and investigate the properties of the largest stratified subset of the family D, determined by W. We find a necessary and sufficient condition for a weak extension to be an extension in terms of stratification. We prove that for theories (D, W) without extension, the least fixed point of the associated operator (with weak extension or minimal set as a context) is an extension of suitably chosen (D’, W) with D’ ⊆ D. We investigate conditions for existence of extensions and introduce the notion of perfectly-stratified set of default rules and its variant of maximally perfectly-stratified set. Existence of such set of default rules turns out to be equivalent to the existence of extension. Finally, we investigate convergence of algorithm for computing extensions.