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

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.

💡 Research Summary

The paper investigates the computational complexity of the highly restricted planning fragment STRIPS 1 1, where each action has exactly one precondition and one effect. Building on Bylander’s seminal work, which established PSPACE‑completeness for STRIPS fragments with up to two preconditions and two effects and proved NP‑hardness for the 1‑precondition/1‑effect case, the authors aim to determine whether STRIPS 1 1 is NP‑complete, PSPACE‑hard, or lies somewhere in between.

To address this gap, the authors adopt three complementary approaches. First, they encode small‑scale instances (n = 5, 6, 7 variables) into propositional SAT and solve them with the modern SAT solver kissat. The encoding uses three groups of Boolean variables: (i) action‑selection variables indicating whether a particular pair of literals constitutes an action, (ii) path‑variables representing the state at each time step, and (iii) BFS‑variables that enforce reachability constraints and guarantee that a shorter solution does not exist. For n = 5 the solver proves that the longest optimal plan contains 30 states (29 actions), well below the theoretical maximum of 2⁵ = 32. For n = 6 the optimal length is 40 states (39 actions), again far under 2⁶ = 64, and the solver exhaustively verifies that no longer plan exists (the computation required about two weeks on a single core). For n = 7 a plan of 51 states (50 actions) was found, but an exact upper bound could not be established within reasonable time. These empirical results strongly suggest that the shortest‑plan length in STRIPS 1 1 grows sub‑exponentially, likely polynomially, with the number of variables.

Second, the paper introduces the “literal graph” Qₙ(p), a directed subgraph of the n‑dimensional hypercube whose vertices correspond to all possible truth assignments and whose edges correspond to the allowed STRIPS 1 1 actions. The authors define a “good” instance as one in which every variable appears in at least two actions with distinct preconditions. Under this assumption they prove two lemmas: (1) each variable coordinate contributes at least 2ⁿ⁻³ bidirectional edges, and (2) at least 2ⁿ⁻² vertices have degree at least n/4. These structural properties imply that the graph is densely bidirectional, making it difficult to embed a long simple (acyclic) path without revisiting vertices. Consequently, the existence of exponentially long shortest plans appears unlikely.

Third, the authors analyze the problem from a hypercube perspective. An action ⟨l₁, l₂⟩ flips exactly one bit (the effect) while requiring a specific bit (the precondition) to hold in the source state. Consequently each action covers 2ⁿ⁻² directed edges of the hypercube. Since the total number of distinct actions is bounded by (2ⁿ)·(2ⁿ) = 4ⁿ², and after eliminating duplicate variable usage the bound tightens to Θ(n²), the action space grows only quadratically in n. This stark contrast with the exponential number of vertices (2ⁿ) limits the length of any simple path that can be realized using distinct actions. The authors also relate this to the well‑studied “snake‑in‑the‑box” problem, where longest induced paths in a hypercube are known to be exponential, but those constructions rely on using many distinct edges per step—something not possible under the STRIPS 1 1 restriction.

To further solidify their findings, the authors present a translation of the literal graph into a conservative Petri net. Literals become places, actions become transitions, and the net’s safety guarantees that any firing sequence corresponds to a valid plan. Because the net is conservative, token counts are preserved, ensuring that any execution terminates after a bounded number of steps, again supporting the hypothesis of polynomial‑bounded plan lengths.

In summary, the paper combines SAT‑based computer‑assisted proofs, graph‑theoretic lemmas, hypercube combinatorics, and Petri‑net modeling to argue that STRIPS 1 1 does not admit exponentially long shortest plans. While a formal proof of NP‑completeness (or PSPACE‑hardness) remains open, the presented evidence strongly points toward a complexity class lower than PSPACE, possibly NP. The work narrows the gap left by Bylander’s original results and provides a solid foundation for future investigations into the exact complexity of this minimalistic planning fragment.