The Expressive Power of k-ary Exclusion Logic

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 when k = 1, 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 in 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 obtain a strict arity hierarchy for exclusion logic. It also follows that k-ary inclusion logic is strictly weaker than EXC [k]. Finally we use similar techniques to formulate a translation from ESO[k] to k-ary inclusion logic with an alternative strict semantics. Consequently, for any arity fragment of inclusion logic, strict semantics is strictly more expressive than lax semantics.