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Abstract:
In AI Planning, as well as Verification, a successful method is to compile the
application into boolean satisfiability (SAT), and solve it with
state-of-the-art DPLL-based procedures. There is a lack of formal understanding
why this works so well. Focussing on the Planning context, we identify a form
of problem structure concerned with the symmetrical or asymmetrical nature of
the cost of achieving the individual planning goals. We quantify this sort of
structure with a simple numeric parameter called AsymRatio, ranging between 0
and 1. We show empirically that AsymRatio correlates strongly with SAT solver
performance in a broad range of Planning benchmarks, including the domains used
in the 3rd International Planning Competition. We then examine carefully
crafted synthetic planning domains that allow to control the amount of
structure, and that are clean enough for a rigorous analysis of the
combinatorial search space. The domains are parameterized by size n, and by a
structure parameter k, so that AsymRatio is asymptotic to k/n. The CNFs we
examine are unsatisfiable, encoding one planning step less than the length of
the optimal plan. We prove upper and lower bounds on the size of the best
possible DPLL refutations, under different settings of k, as a function of n.
We also identify the best possible sets of branching variables (backdoors).
With minimum AsymRatio, we prove exponential lower bounds, and identify minimal
backdoors of size linear in the number of variables. With maximum AsymRatio, we
identify logarithmic DPLL refutations (and backdoors), showing a doubly
exponential gap between the two structural extreme cases. This provides a
concrete insight into the practical efficiency of modern SAT solvers.
