# Solving Regular Tree Grammar Based Constraints

Yanhong A. Liu, Ning Li, and Scott D. Stoller

This paper describes the precise specification, design, analysis,
implementation, and measurements of an efficient algorithm for solving
regular tree grammar based constraints. The particular constraints are
for dead-code elimination on recursive data, but the method used for
the algorithm design and complexity analysis is general and applies to
other program analysis problems as well. The method is centered around
Paige's finite differencing, i.e., computing expensive set expressions
incrementally, and allows the algorithm to be derived and analyzed
formally and implemented easily. We propose higher-level
transformations that make the derived algorithm concise and allow its
complexity to be analyzed accurately. Although a rough analysis shows
that the worst-case time complexity is cubic in program size, an
accurate analysis shows that it is linear in the number of live program
points and in other parameters, including mainly the arity of data
constructors and the number of selector applications into whose
arguments the value constructed at a program point might flow. These
parameters explain the performance of the analysis in practice. Our
implementation also runs two to ten times as fast as a previous
implementation of an informally designed algorithm.
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