k元排它逻辑的表达能力
The Expressive Power of k-ary Exclusion Logic
摘要：
本文研究了用k元排它原子扩充一阶逻辑得到的k元排它逻辑EXC[k]的表达能力。排除逻辑与依赖逻辑是等价的，没有元界。通过观察翻译，我们发现EXC[k]的表达能力介于k元依赖逻辑和（k+1）元依赖逻辑之间。我们将证明，至少在k=1的情况下，这两个包含是适当的。在作者最近的工作中，证明了k元包含-排除逻辑与k元存在二阶逻辑ESO[k]是等价的。我们将证明，在句子的层次上，可以用排斥原子来模拟包含原子，并且这种方式只使用k元排斥原子来表达ESO[k]-句子。对于这种转换，我们还需要引入一种新的方法来“统一”团队中某些变量的值。因此，EXC[k]在句子级别上捕获ESO[k]，并且我们得到了排除逻辑的严格arity层次。它也遵循k元包含逻辑严格弱于EXC[k]。最后，我们将使用类似的技术来制定从ESO[k]到具有严格语义的k元包含逻辑的转换。因此，对于包含逻辑的任何arity片段，严格语义比宽松语义更具表达性。
英文摘要：
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.
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