The parameterized space complexity of model-checking bounded variable first-order logic
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European Commission's projectAUTAR - A Unified Theory of Algorithmic Relaxations (EC-H2020-648276)
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(log2n). 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.
CitationChen, Y.; Elberfeld, M.; Muller, M. The parameterized space complexity of model-checking bounded variable first-order logic. "Logical methods in computer science", 2019, vol. 15, núm. 3, p. 31:1-31:29.