# The parameterized space complexity of model-checking bounded variable first-order logic

## TL;DR

This paper investigates the space complexity of model-checking for bounded-variable first-order logic, introducing a hierarchy and proposing space-efficient algorithms with implications for classical complexity bounds.

## Contribution

It defines the tree hierarchy for parameterized space complexity and develops a highly space-efficient model-checker for bounded-variable queries.

## Key findings

- Model-checking with bounded variables is complete for levels of the tree hierarchy.
- A space-efficient model-checker is proposed for bounded-variable, bounded alternation rank queries.
- Implications for classical space complexity, including Savitch's Theorem, are discussed.

## Abstract

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.

---
Source: https://tomesphere.com/paper/1703.01860