Proof of completeness of the SCIFF proof-procedure

This document provides the proof of completeness of the SCIFF proofprocedure. 1 Declarative semantics In the following we give semantics to a society instance by identifying sets of expectations which, together with the society’s knowledge base and the happened events, imply an instance of the goal if any and satisfy the integrity constraints. For notion of integrity constraint satisfaction we rely, in the following, upon a notion of entailment in a three-valued logic, since more general and capable of dealing with both open and closed society instances. Therefore, in the following, the symbol |= has to be interpreted as the notion of entailment in a three-valued setting. Furthermore, in this section, we consider negative literals of the kind ¬H() as new positive literals that have no definition in each open society instance. For closed society instances, we use Clark’s completion of the history, Comp(HAP), and negation is interpreted in the Closed World Assumption (CWA). Throughout this section, for the sake of simplicity, we always consider a ground version of society’s knowledge base and integrity constraints, and do not consider CLP-like constraints. We first recall the concept of ICS-consistent set of social expectations. Intuitively, given a society instance, a ICS-consistent set of social expectations consists of a set of expectations about social events that are compatible with P (i.e., the SOKB and the set HAP), and with ICS . Definition 1. (ICS-consistency) Given a (closed/open) society instance SHAP, an ICS-consistent set of social expectations ∆ is a set of expectations such that: SOKB ∪HAP ∪∆ |= ICS (1)