An Undecidability Proof for the Plan Existence Problem

We extend, in Isabelle/HOL, the deep-and-shallow embedding methodology of our prior work from propositional to first-order modal logic (FML) with constant-domain Kripke semantics. Three embeddings of FML into classical higher-order logic (HOL) are provided side by side: a deep embedding, a heavyweight maximal-shallow embedding, and a lightweight minimal-shallow embedding. The minimal-shallow embedding is presented as an Isabelle/HOL locale, parametrised by an accessibility relation, a world-indexed interpretation, a universe of worlds, and a variable assignment; the locale form admits a global faithfulness theorem, stating that quantifying over all minimal-shallow interpretations recovers exactly deep validity. A central technical contribution is a mechanisation, for FML under constant-domain Kripke semantics, of the (countable) downward Löwenheim-Skolem theorem, which underpins the automation of our faithfulness proof between the deep and minimal-shallow embeddings. Deploying it inside an extension of the minimal-shallow locale resolves the surjectivity problem that arises against an uncountable domain of individuals -- where the locale's variable assignment, having countable domain V = nat, cannot be surjective onto the domain -- and thereby yields faithfulness over the full domain. Since prior work treats only the propositional fragment, we develop here the substitution machinery (free/bound-variable predicates, the fresh-variable function, capture-avoiding substitution, alphabetic renaming, the substitutability predicate, the substitution lemma, and size-based induction principles) needed for the first-order quantifiers.

In some real applications a plan may later become unfeasible due to newly imposed budget constraints, yet, at the same time, using only the original actions of the plan and their order is mandatory. In this paper, we study the problem of extracting, from a precomputed plan, a valid subplan that maximizes utility while respecting a cost bound. Each goal is given a utility value and the plan is reduced by removing actions that support low-utility goals, while preserving both executability and the original action order. We show the decision variant is NP-complete and propose two exact methods to solve it: one via oversubscription planning (OSP) and another via Integer Linear Programming (ILP). This paper extends our previous work published at ICAPS 2026 (Del Toro, Fuentetaja, and García-Olaya 2026b). While the core framework remains as introduced there, we further introduce a refined ILP formulation that significantly decreases the model size and improves computational efficiency.

Reasoning about necessity and possibility depends on assumptions about accessibility between worlds and about which objects exist at each one. The same inference may therefore hold under one modal system and fail under another. Evaluating language models on such problems requires testing whether their judgments follow the stated semantics rather than a familiar logic. We construct paired modal problems with identical premises and conjecture but different frame or domain conditions; automated reasoning verifies opposite labels. A balanced core prevents the semantic condition alone from revealing the answer. On this core, four of five recent models perform below the condition-only baseline under direct prompting. Yet enabling reasoning mode raises DeepSeek V4 Flash from 4.4% to 88.1% on unchanged prompts. Following stipulated modal semantics thus depends strongly on inference mode as well as model identity. When frame conditions are omitted, models often agree but fit different familiar logics best. We release the formulas, oracle artifacts, countermodels, and responses.