On Computing Belief Change Operations using Quantified Boolean Formulas

Abstract

In this paper, we show how an approach to belief revision and belief contraction can be axiomatised by means of quantified Boolean formulas. Specifically, we consider the approach of belief change scenarios, a general framework that has been introduced for expressing different forms of belief change. The essential idea is that for a belief change scenario (K, R, C), the set of formulas K, representing the knowledge base, is modified so that the sets of formulas R and C are respectively true in, and consistent with the result. By restricting the form of a belief change scenario, one obtains specific belief change operators including belief revision, contraction, update, and merging. For both the general approach and for specific operators, we give a quantified Boolean formula such that satisfying truth assignments to the free variables correspond to belief change extensions in the original approach. Hence, we reduce the problem of determining the results of a belief change operation to that of satisfiability. This approach has several benefits. First, it furnishes an axiomatic specification of belief change with respect to belief change scenarios. This then leads to further insight into the belief change framework.

Belief Merging in Dynamic Logic of Propositional Assignments

“…Our embeddings are somewhat simpler than the embeddings of belief change operations into QBF as done in [4] since DL-PA is a logic of programs. The same argument applies to embeddings of merging problems into MSO.…”

“…Table 2 collects some DL-PA expressions that are going to be convenient abbreviations. 4 The program vary(P) nondeterministically changes the truth value of some of the variables in P. Its length is linear in the cardinality of P. So the program vary(P A ); A? accesses all A-valuations that preserve the values of all those variables not occurring in A. Satisfiability of the Boolean formula A can be expressed in DL-PA by the formula vary(P A ); A?…”

Computational methods for database repair by signed formulae*

“…It has been shown that this language is useful for expressing a variety of computational paradigms, such as default reasoning [20], circumscribing inconsistent theories [21], paraconsistent preferential reasoning [6], and computations of belief revision operators (see [29], as well as Section 5 below). In this section we show how QBF solvers can be used for computing the ≤ i -preferred repairs of a given database.…”

A QBF-based formalization of abstract argumentation semantics

“…QBFs are formulas involving only propositional languages and quantifications over propositional variables. Their application is vast, covering many areas among which are planning [67], verification [12,59], and different computational paradigms for non-monotonic reasoning, such as default reasoning [15], circumscribing inconsistent theories [16] and computations of belief revision operators [36]. In our case, the use of signed theories and QBFs implies that decision problems like skeptical and credulous acceptance of arguments are a matter of logical entailment and satisfiability, which can be verified by existing QBF-solvers.…”