  The challenge of answering graph queries over incomplete knowledge graphs is
gaining significant attention in the machine learning community. Neuro-symbolic
models have emerged as a promising approach, combining good performance with
high interpretability. These models utilize trained architectures to execute
atomic queries and integrate modules that mimic symbolic query operators.
However, most neuro-symbolic query processors are constrained to tree-like
graph pattern queries. These queries admit a bottom-up execution with constant
values or anchors at the leaves and the target variable at the root. While
expressive, tree-like queries fail to capture critical properties in knowledge
graphs, such as the existence of multiple edges between entities or the
presence of triangles. We introduce a framework for answering arbitrary graph
pattern queries over incomplete knowledge graphs, encompassing both cyclic
queries and tree-like queries with existentially quantified leaves. These
classes of queries are vital for practical applications but are beyond the
scope of most current neuro-symbolic models. Our approach employs an
approximation scheme that facilitates acyclic traversals for cyclic patterns,
thereby embedding additional symbolic bias into the query execution process.
Our experimental evaluation demonstrates that our framework performs
competitively on three datasets, effectively handling cyclic queries through
our approximation strategy. Additionally, it maintains the performance of
existing neuro-symbolic models on anchored tree-like queries and extends their
capabilities to queries with existentially quantified variables.
