Proof System for Plan Verification under 0-Approximation Semantics

In this paper a proof system is developed for plan verification problems ${X}c{Y}$ and ${X}c{KW p}$ under 0-approximation semantics for ${\mathcal A}_K$. Here, for a plan $c$, two sets $X,Y$ of fluent literals, and a literal $p$, ${X}c{Y}$ (resp. ${X}c{KW p}$) means that all literals of $Y$ become true (resp. $p$ becomes known) after executing $c$ in any initial state in which all literals in $X$ are true.Then, soundness and completeness are proved. The proof system allows verifying plans and generating plans as well.

💡 Research Summary

The paper introduces a formal proof system for verifying plans under the 0‑approximation semantics of the action language A_K, a framework designed to handle incomplete knowledge, sensing actions, and conditional plans. The authors focus on two verification statements: {X} c {Y}, meaning that after executing plan c all literals in Y become true when the initial state satisfies X; and {X} c {KW p}, meaning that after executing c the truth value of literal p becomes known.

The work begins by recalling the syntax of A_K, which distinguishes fluent names and action names, and defines four kinds of propositions: initial‑knowledge, effect, executability, and knowledge (sensing) propositions. A state is represented as an a‑state (T,F), where T contains fluents known to be true and F contains fluents known to be false; any fluent not in T∪F is unknown. The 0‑approximation semantics is then defined. For a non‑sensing action a, the transition Res₀(a,σ) deterministically updates T and F based on the action’s certain effects (e⁺ₐ, e⁻ₐ). For a sensing action a, the transition Φ₀(a,σ) nondeterministically expands the knowledge by adding the set K(a) of fluents that become known, while preserving the original information. The authors prove a monotonicity lemma (Lemma 2.1) stating that if σ₁⊑σ₂ then the result of any action on σ₁ is a sub‑state of the result on σ₂. This property is crucial for later soundness arguments.

The core contribution is the construction of a proof system that mirrors the inductive definition of the extended transition function bΦ₀. The system consists of inference rules that correspond to the following operations: (1) constructing the initial a‑state from a set of literals X; (2) applying a non‑sensing action and checking that the resulting a‑state satisfies the goal literals; (3) applying a sensing action, generating all possible extensions, and recursively verifying the goal in each branch; (4) handling case (conditional) plans by selecting the branch whose guard is true; and (5) sequencing two plans c₁;c₂ by first verifying c₁ and then verifying c₂ from every reachable state of c₁.

Soundness (Theorem 3.1) is proved by showing that each inference rule preserves truth under the 0‑approximation semantics; the monotonicity lemma guarantees that if a rule holds for a more abstract state it also holds for any more concrete state. Completeness (Theorem 3.2) is established by constructing a derivation for any true verification statement, essentially by unfolding the semantics of the plan step by step and using the inference rules in reverse. Lemma 2.2 further simplifies verification: it suffices to consider the least initial a‑state (T_D,F_D) derived from the domain’s initial‑knowledge propositions, because any other initial state is an extension of this one.

Beyond verification, the authors argue that the proof system naturally supports plan generation. A proof tree can be read as a constructive description of a plan: each application of a rule adds an action or a conditional branch, and the tree’s leaves correspond to goal satisfaction. By storing proved triples {X} c {Y} in a graph (nodes = sets of literals, edges = actions), online queries about the existence of a plan from X′ to Y′ reduce to a simple path search, which is NL‑complete. This contrasts with earlier approaches that translate planning problems into SAT, QBF, or non‑monotonic logics, which do not directly yield new plans.

Complexity considerations are discussed: under 0‑approximation, the existence of polynomial‑size plans is NP‑complete, yet the proof system can verify short plans efficiently because it works on the compact representation of states rather than enumerating all possible worlds. The authors suggest that the system could be integrated into planners to perform offline pre‑computation of useful plan fragments, thereby accelerating online planning.

In conclusion, the paper provides a rigorously defined, sound, and complete proof system for plan verification under 0‑approximation semantics, and demonstrates its dual role as a plan synthesis mechanism. The work bridges the gap between formal verification techniques and practical planning, offering a foundation for future extensions to richer approximations, richer sensing models, and integration with automated reasoning tools.