In this paper, we present a procedure for numeric planning based on Symbolic Pattern Planning (SPP). Given a numeric planning problem $ ext{Π}$, a pattern $ ext{⟨p⟩}$ is a sequence of actions used to define a formula encoding the subsequences of $ ext{⟨p⟩}$ executable from a starting state $ ext{S}$. Cardellini, Giunchiglia, and Maratea (2024a) follow the Planning as Satisfiability approach by defining, at each step $ ext{n} ext{≥} 0$, a formula $ ext{Π}^{ ext{⟨p⟩}}_n$ in which $(i)$ the pattern $ ext{⟨p⟩}$ is computed only for $ ext{n}=0$ in the initial state $ ext{I}$ of $ ext{Π}$, and then exploited at each step $ ext{n}$, $(ii)$ the starting state $ ext{S}$ is set to $ ext{I}$, and $(iii)$ the set $ ext{G}$ of goals is required to hold in the last state that can be reached by one of the subsequences of $ ext{⟨p⟩}$ concatenated $ ext{n}$ times. The procedure begins with $ ext{n}=0$, terminates as soon as $ ext{Π}^{ ext{⟨p⟩}}_n$ is satisfiable, and otherwise proceeds by incrementing $ ext{n}$. In this paper, possibly at each step, $(i)$ we symbolically search for an intermediate state $ ext{P}$ reachable from $ ext{I}$, closer to a goal state, $(ii)$ dynamically recompute the pattern $ ext{⟨p⟩}_h$ -- to be used in the next step -- in $ ext{P}$, $(iii)$ refine the pattern $ ext{⟨p⟩}_g$ used to reach $ ext{P}$, and $(iv)$ start the new search from the state $ ext{S}$ which can be either the initial state $ ext{I}$ or the last computed intermediate state $ ext{P}$, exploiting the computed patterns $ ext{⟨p⟩}_g$ and $ ext{⟨p⟩}_h$ to define the pattern $ ext{⟨p⟩}$ to be used in the search. In particular, at each step, we define a formula $ ext{Π}^{ ext{⟨p⟩}}_{S,P}$ encoding the existence of a state $ ext{P}'$ closer than $ ext{P}$ to a goal state, with $ ext{P}'$ reachable from the starting state $ ext{S}$ when using the pattern $ ext{⟨p⟩}$. We present different techniques for producing such formulas, each corresponding to a different strategy for exploring the search space. We prove their correctness and completeness, the latter under certain conditions.
Blogger's Review: This paper introduces the concept of symbolic pattern planning, offering a novel solution for numeric planning problems. By combining dynamic search and pattern recomputation, it significantly improves search efficiency and reachability, providing a solid theoretical foundation for future research. However, practical implementation may face challenges in balancing complexity and computational resources.