Propositional Update Operators Based on Formula/Literal Dependence

Abstract

We present and study a general family of belief update operators in a propositional setting. Its operators are based on formula/literal dependence, which is more fine-grained than the notion of formula/variable dependence that was proposed in the literature: formula/variable dependence is a particular case of formula/literal dependence. Our update operators are defined according to the "forget-then-conjoin" scheme: updating a belief base by an input formula consists in first forgetting in the base every literal on which the input formula has a negative influence, and then conjoining the resulting base with the input formula. The operators of our family differ by the underlying notion of formula/literal dependence, which may be defined syntactically or semantically, and which may or may not exploit further information like known persistent literals and pre-set dependencies. We argue that this allows to handle the frame problem and the ramification problem in a more appropriate way. We evaluate the update operators of our family w.r.t. two important dimensions: the logical dimension, by checking the status of the Katsuno-Mendelzon postulates for update, and the computational dimension, by identifying the complexity of a number of decision problems (including model checking, consistency and inference), both in the general case and in some restricted cases, as well as by studying compactability issues. It follows that several operators of our family are interesting alternatives to previous belief update operators. ACM Reference Format:Herzig, A., Lang, J., and Marquis, P. 2013. Propositional update operators based on formula/literal dependence.

On Quantifying Literals in Boolean Logic and its Applications to Explainable AI

“…Literal forgetting was employed in extended logic programs (Wang, Sattar, & Su, 2005) and disjunctive logic programs . It was also used to define more refined update operators using the "forget-thenconjoin" scheme that we discussed earlier (Herzig, Lang, & Marquis, 2013).…”

On The Reasons Behind Decisions

“…The necessary property for a decision is the set of all its necessary characteristics. 6 Pierre Marquis observed that the complete reason can be formulated using the notion of literal forgetting which is a more fine grained notion than variable forgetting (also known as existential quantification) [20,18,11]. 7 The consensus rule infers the term δ 1 ∧δ 2 from terms X ∧δ 1 and ¬X ∧δ 2 .…”

Dependence in Propositional Logic: Formula-Formula Dependence and Formula Forgetting -- Application to Belief Update and Conservative Extension

“…On the other hand, Herzig, Lang, and Marquis (2013) suggested that belief update should be based on the notion of dependence, and proposed the dependence-based update scheme, consisting of two steps: (1) forget every piece of φ which ¬ψ depends on; (2) expand the resulting belief base with ψ. This scheme has some resemblance to the so-called Levi Identity (Levi 1977) which defines an update operator from a given erasure operator.…”

“…Belief update, a type of belief change, studies how an agent modifies her belief base φ in the presence of new information ψ in a dynamically changing environment. Herzig, Lang, and Marquis (2013) suggested that belief update should be based on the notion of dependence, and proposed the dependence-based update scheme that consists of first removing any belief on which the negation of the new information depends, and then adding the new information.…”