The Expressive Power of k-ary Exclusion Logic

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. From the translations between them 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 unary exclusion logic, the both of these inclusions are proper. In a recent work by the author it was shown that k-ary inclusionexclusion 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 thus we get a strict arity hierarchy for exclusion logic. It also follows that k-ary inclusion logic is strictly weaker than EXC[k].