On existential declarations of independence in IF Logic

TL;DR

This paper investigates the impact of independence declarations in IF logic quantifiers, providing criteria to determine when such declarations affect truth values and showing all existential IF sentences are equivalent to first-order logic.

Contribution

It introduces a syntactical criterion for independence declarations in IF logic and proves that all existential IF sentences are equivalent to first-order sentences.

Findings

Existential IF sentences are equivalent to first-order sentences.

A syntactical criterion determines when independence declarations affect truth.

The fragment with knowledge memory has only first-order expressive power.

Abstract

We analyze the behaviour of declarations of independence between existential quantifiers in quantifier prefixes of IF sentences; we give a syntactical criterion for deciding whether a sentence beginning with such prefix exists such that its truth values may be affected by removal of the declaration of independence. We extend the result also to equilibrium semantics values for undetermined IF sentences. The main theorem allows us to describe the behaviour of various particular classes of quantifier prefixes, and to prove as a remarkable corollary that all existential IF sentences are equivalent to first-order sentences. As a further consequence, we prove that the fragment of IF sentences with knowledge memory has only first-order expressive power (up to truth equivalence).