## Yijia Chen ; Michael Elberfeld ; Moritz Müller - The parameterized space complexity of model-checking bounded variable first-order logic

lmcs:3172 - Logical Methods in Computer Science, September 20, 2019, Volume 15, Issue 3 - https://doi.org/10.23638/LMCS-15(3:31)2019
The parameterized space complexity of model-checking bounded variable first-order logic

Authors: Yijia Chen ; Michael Elberfeld ; Moritz Müller

The parameterized model-checking problem for a class of first-order sentences (queries) asks to decide whether a given sentence from the class holds true in a given relational structure (database); the parameter is the length of the sentence. We study the parameterized space complexity of the model-checking problem for queries with a bounded number of variables. For each bound on the quantifier alternation rank the problem becomes complete for the corresponding level of what we call the tree hierarchy, a hierarchy of parameterized complexity classes defined via space bounded alternating machines between parameterized logarithmic space and fixed-parameter tractable time. We observe that a parameterized logarithmic space model-checker for existential bounded variable queries would allow to improve Savitch's classical simulation of nondeterministic logarithmic space in deterministic space $O(\log^2n)$. Further, we define a highly space efficient model-checker for queries with a bounded number of variables and bounded quantifier alternation rank. We study its optimality under the assumption that Savitch's Theorem is optimal.

Volume: Volume 15, Issue 3
Published on: September 20, 2019
Accepted on: September 20, 2019
Submitted on: March 7, 2017
Keywords: Computer Science - Logic in Computer Science