The Expressive Power of k-ary Exclusion Logic

TL;DR

This paper explores the expressive power of k-ary exclusion logic, showing it captures certain second-order logic fragments and establishing a strict arity hierarchy, with novel translation techniques and implications for inclusion logic.

Contribution

It demonstrates that k-ary exclusion logic captures ESO[k] on the sentence level and establishes a strict arity hierarchy, introducing new translation methods and comparing it with inclusion logic.

Findings

EXC[k] captures ESO[k] on the sentence level.

A strict arity hierarchy exists for exclusion logic.

k-ary inclusion logic is weaker than EXC[k].

Abstract

In this paper we study the expressive power of k-ary exclusion logic, EXC[k], that is obtained by extending first order logic with k-ary exclusion atoms. It is known that without arity bounds exclusion logic is equivalent with dependence logic. By observing the translations, we see that the expressive power of EXC[k] lies in between k-ary and (k+1)-ary dependence logics. We will show that, at least in the case of k=1, the both of these inclusions are proper. In a recent work by the author it was shown that k-ary inclusion-exclusion logic is equivalent with k-ary existential second order logic, ESO[k]. We will show that, on the level of sentences, it is possible to simulate inclusion atoms with exclusion atoms, and this way express ESO[k]-sentences by using only k-ary exclusion atoms. For this translation we also need to introduce a novel method for "unifying" the values of certain variables in a team. As a consequence, EXC[k] captures ESO[k] on the level of sentences, and we get a strict arity hierarchy for exclusion logic. It also follows that k-ary inclusion logic is strictly weaker than EXC[k]. Finally we will use similar techniques to formulate a translation from ESO[k] to k-ary inclusion logic with strict semantics. Consequently, for any arity fragment of inclusion logic, strict semantics is more expressive than lax semantics.