Nesting Depth of Operators in Graph Database Queries: Expressiveness Vs. Evaluation Complexity

TL;DR

This paper explores how controlling the nesting depth of operators in graph database queries impacts their expressiveness and evaluation complexity, establishing a hierarchy with strict inclusions and complexity bounds.

Contribution

It introduces a syntactic hierarchy for nesting depth in query languages, proves its strictness, and relates it to computational complexity and satisfiability problems.

Findings

Hierarchy of query languages based on nesting depth is strict.

Evaluation complexity increases with nesting depth, reaching the polynomial hierarchy.

Deciding equivalence at different levels is undecidable.

Abstract

Designing query languages for graph structured data is an active field of research, where expressiveness and efficient algorithms for query evaluation are conflicting goals. To better handle dynamically changing data, recent work has been done on designing query languages that can compare values stored in the graph database, without hard coding the values in the query. The main idea is to allow variables in the query and bind the variables to values when evaluating the query. For query languages that bind variables only once, query evaluation is usually NP-complete. There are query languages that allow binding inside the scope of Kleene star operators, which can themselves be in the scope of bindings and so on. Uncontrolled nesting of binding and iteration within one another results in query evaluation being PSPACE-complete. We define a way to syntactically control the nesting depth of iterated bindings, and study how this affects expressiveness and efficiency of query evaluation. The result is an infinite, syntactically defined hierarchy of expressions. We prove that the corresponding language hierarchy is strict. Given an expression in the hierarchy, we prove that it is undecidable to check if there is a language equivalent expression at lower levels. We prove that evaluating a query based on an expression at level i can be done in $\Sigma_i$ in the polynomial time hierarchy. Satisfiability of quantified Boolean formulas can be reduced to query evaluation; we study the relationship between alternations in Boolean quantifiers and the depth of nesting of iterated bindings.