Many planning formalisms allow for mixing numeric with Boolean effects.
However, most of these formalisms are undecidable. In this paper, we will
analyze possible causes for this undecidability by studying the number of
different occurrences of actions, an approach that proved useful for metric
fluents before. We will start by reformulating a numeric planning problem known
as restricted tasks as a search problem. We will then show how an NP-complete
fragment of numeric planning can be found by using heuristics. To achieve this,
we will develop the idea of multi-valued partial order plans, a least
committing compact representation for (sequential and parallel) plans. Finally,
we will study optimization techniques for this representation to incorporate
soft preconditions.

通过研究动作的不同发生次数来分析导致规划正式化丧失可决定性的可能原因，使用多值偏序计划的概念实现了一个 NP 完全的数值规划片段，并研究了软性前提的优化技术。

数字规划中的多值部分有序计划

Multi-Valued Partial Order Plans in Numeric Planning

The recent approaches of extending the GRAPHPLAN algorithm to handle more
expressive planning formalisms raise the question of what the formal meaning of
"expressive power" is. We formalize the intuition that expressive power is a
measure of how concisely planning domains and plans can be expressed in a
particular formalism by introducing the notion of "compilation schemes" between
planning formalisms. Using this notion, we analyze the expressiveness of a
large family of propositional planning formalisms, ranging from basic STRIPS to
a formalism with conditional effects, partial state specifications, and
propositional formulae in the preconditions. One of the results is that
conditional effects cannot be compiled away if plan size should grow only
linearly but can be compiled away if we allow for polynomial growth of the
resulting plans. This result confirms that the recently proposed extensions to
the GRAPHPLAN algorithm concerning conditional effects are optimal with respect
to the "compilability" framework. Another result is that general propositional
formulae cannot be compiled into conditional effects if the plan size should be
preserved linearly. This implies that allowing general propositional formulae
in preconditions and effect conditions adds another level of difficulty in
generating a plan.

本论文介绍了一种名为 “编译方案” 的概念，并使用这种概念分析了一大类命题规划命题形式的表达能力，结果确认了 GRAPHPLAN 算法中有关条件效应的扩展是优化的，同时发现允许在先决条件和效果条件中使用一般命题公式会给生成计划带来额外的 Schwierigkeit。