AGM postulates in Answer Sets

Revising and updating beliefs and knowledge bases is an important topic in knowledge representation and reasoning that requires a solid theoretical basis. As a result, various researchers have proposed Answer Set Programming as one of their key components to set up their approaches. In the need to satisfy more general principles, this paper presents a new characterisation of a semantics. It consists in performing updates of epistemic states that meets well-accepted AGM revision postulates. Besides the formalism of properties that this framework shares with other equivalent update semantics, this proposal is also supported by a solver prototype as an important component of logic programming and automatic testbed of its declarative version. The solver may help compute agent’s knowledge bases for more complex potentially-industrial applications and frameworks. 1 Abductive Programs and MGAS Abduction is an alternative process to deductive reasoning in Classical Logic [4], whose formal definitions are omitted due to page-limit reasons. This simple and strong framework is the main core of a solid foundation for the update formulation, presented in the following sections. 2 Updating Epistemic States One of the latest proposals to meet most well-accepted principles for updates at the object level and in Minimal Generalised Answer Sets (MGAS) was first introduced in [3]. Such a proposal introduces a flexible foundation to set up the needed models for the desired properties. A deep analysis of the problem, the solution, justification, basic model-oriented properties and comparison with other approaches are available in [3]. By now, let us briefly introduce the semantics by a characterisation in Belief Revision. The semantics is formally expressed with the following set of definitions, borrowed and slightly modified from [3] to make it simpler and precise. Formally, an α-relaxed rule is a rule ρ that is weakened by a default-negated atom α in its body: Head(ρ) ← Body(ρ) ∪ {not α}. In addition, an α-relaxed program is a set of α-relaxed rules. ? This project is supported by a CONACYT Doctorate Grant. A generalised program is a set of rules of form ` ← > | ` ∈ A∗, where A∗ is a given set of literals. As a consequence, updating a program with another consists in transforming an ordered pair of programs into a single abductive program, as follows. Definition 1 (◦o-update Program). Given an updating pair of extended logic programs, denoted as Π1 ◦o Π2, over a set of atoms A; and a set of unique abducibles A∗, such that A ∩ A∗ = ∅; and the abductive program ΠA∗ = 〈Π ′ ∪ Π2,A 〉 with its corresponding α-relaxed program Π ′ such that α ∈ A∗, and its corresponding MGAS’s. Its ◦o-update program is Π ′ ∪Π2 ∪ΠG, where ΠG is a generalised program of M ∩ A∗ for some MGAS M of ΠA∗ and ◦o is the corresponding update operator. Last, the associated models S of an epistemic state (an updating pair) corresponds to the answer sets of a ◦o-update program as follows. Definition 2 (◦o-update Answer Set). Let Π◦o = (Π1 ◦o Π2) be an update pair over a set of atoms A. Then, S ⊆ A is an ◦o-update answer set of Π◦o if and only if S = S ′ ∩ A for some answer set S ′ of its ◦o-update program.