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Intermediate Results on the Complexity of STRIPS$_{1}^{1}$

Stefan Edelkamp, Jiří Fink, Petr Gregor, Anders Jonsson, Bernhard Nebel

TL;DR

Light is shed on the question whether this small solution hypothesis for STRIPS$^1_1$ is true, calling a SAT solver for small instances, introducing the literal graph, and mapping it to Petri nets.

Abstract

This paper is based on Bylander's results on the computational complexity of propositional STRIPS planning. He showed that when only ground literals are permitted, determining plan existence is PSPACE-complete even if operators are limited to two preconditions and two postconditions. While NP-hardness is settled, it is unknown whether propositional STRIPS with operators that only have one precondition and one effect is NP-complete. We shed light on the question whether this small solution hypothesis for STRIPS$^1_1$ is true, calling a SAT solver for small instances, introducing the literal graph, and mapping it to Petri nets.

Intermediate Results on the Complexity of STRIPS$_{1}^{1}$

TL;DR

Light is shed on the question whether this small solution hypothesis for STRIPS is true, calling a SAT solver for small instances, introducing the literal graph, and mapping it to Petri nets.

Abstract

This paper is based on Bylander's results on the computational complexity of propositional STRIPS planning. He showed that when only ground literals are permitted, determining plan existence is PSPACE-complete even if operators are limited to two preconditions and two postconditions. While NP-hardness is settled, it is unknown whether propositional STRIPS with operators that only have one precondition and one effect is NP-complete. We shed light on the question whether this small solution hypothesis for STRIPS is true, calling a SAT solver for small instances, introducing the literal graph, and mapping it to Petri nets.
Paper Structure (9 sections, 5 theorems, 5 equations, 2 figures, 1 table)

This paper contains 9 sections, 5 theorems, 5 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. STRIPS$_{1}^{1}$
  3. Precursor Work on STRIPS$_{1}^{1}$
  4. Hypercubes
  5. SAT Encodings
  6. Shortest Plans in Good Tasks
  7. Literal Graph
  8. Compilation Schemes
  9. Conclusion

Key Result

Lemma 1

Let $\mathbf{p}$ be a good instance on $n$ variables. For every variable $v$ the graph $Q_n(\mathbf{p})$ contains at least $2^{n-3}$ bidirectional edges in the coordinate $v$.

Figures (2)

  • Figure 1: Sketch of four selected STRIPS$^1_1$ actions in the $5$-dimensional hypercube $Q_5$, green lines indicating the start and red arrows indicating the end of the actions, there are $2^{n-2} = 8$ edges associated edges with each action and there are overlaps. In other words, green edges correspond to fulfilled preconditions and red edges to action effects.
  • Figure 2: Result of compilation scheme for the above $(n=4)$ STRIPS$^1_1$ example into safe conservative Petri-Net and into a cooperative MAPF, the literals represent nodes that correspond to the literal graph, they are negated on the left-hand side nad positive on the right-hand side.

Theorems & Definitions (6)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Theorem 2