Affiliation:
1. Humboldt-Universität Berlin, Berlin, Germany
Abstract
This article gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers. Precisely, this means that for every first-order formula φ(
y
,
z
) over the signature {<,+} there is a first-order formula ψ(
x
,
z
) which expresses over the structure 〈ℕ,<,+〉 (respectively, over initial segments of this structure) that the variable
x
is interpreted exactly by the number of possible interpretations of the variable
y
for which the formula φ(
y
,
z
) is satisfied. Applying this theorem, we obtain an easy proof of Ruhl's result that reachability (and similarly, connectivity) in finite graphs is not expressible in first-order logic with unary counting quantifiers and addition. Furthermore, the above result on Presburger arithmetic helps to show the failure of a particular version of the Crane Beach conjecture.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Mathematics,Logic,General Computer Science,Theoretical Computer Science
Reference36 articles.
1. Σ11-formulae on finite structures;Ajtai M.;Ann. Pure Appl. Logic,1983
2. Lecture Notes in Computer Science;Allender E.
3. Atserias A. 1999. Computational aspects of first-order logic on finite structures. M.S. thesis. Department of Computer Science University of California Santa Cruz Santa Cruz CA. Atserias A. 1999. Computational aspects of first-order logic on finite structures. M.S. thesis. Department of Computer Science University of California Santa Cruz Santa Cruz CA.
Cited by
42 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献