A Logic of Knowing How

In this paper, we propose a single-agent modal logic framework for reasoning about goal-direct “knowing how” based on ideas from linguistics, philosophy, modal logic and automated planning. We first define a modal language to express “I know how to guarantee phi given psi” with a semantics not based on standard epistemic models but labelled transition systems that represent the agent’s knowledge of his own abilities. A sound and complete proof system is given to capture the valid reasoning patterns about “knowing how” where the most important axiom suggests its compositional nature.

💡 Research Summary

The paper introduces a novel single‑agent modal logic for reasoning about “knowing how”, i.e., the ability to achieve a goal ϕ given a precondition ψ. The authors argue that traditional epistemic logics, which focus on “knowing that”, are insufficient for capturing this kind of procedural knowledge. To fill the gap they define a new binary modal operator Kh(ψ, ϕ) and give it a semantics based on labelled transition systems that model an agent’s knowledge of his own actions and their (possibly nondeterministic) effects.

The semantic framework consists of a set of states S, a set of action symbols Σ, a transition relation R⊆Σ×S×S, and a valuation V:S→2^P for propositional atoms. A state s satisfies Kh(ψ, ϕ) iff there exists a finite action sequence σ such that for every state s′ satisfying ψ, σ is strongly executable at s′ (i.e., each prefix of σ has at least one successor for the next action) and every possible execution of σ from s′ ends in a state that satisfies ϕ. Formally the truth condition has the quantifier pattern ∃σ ∀s ( s⊨ψ → (σ strongly executable at s ∧ ∀t (s σ→ t → t⊨ϕ) )). This ∃∀∀ pattern captures the intuition that the agent possesses a global guarantee: a single plan works from all ψ‑states, regardless of nondeterministic outcomes.

On top of this semantics the authors develop a proof system. The language includes the usual Boolean connectives and the Kh‑operator; they introduce an abbreviation Uϕ ≡ Kh(¬ϕ, ⊥) to express “cannot guarantee ϕ”. The most important axiom is a compositional principle:  Kh(ψ, χ) ∧ Kh(χ, ϕ) → Kh(ψ, ϕ), which formalises the idea that two sequential plans can be merged into one. Additional rules handle monotonicity, distribution over conjunction, and the interaction of U with Kh. The system is proved sound, complete (every semantically valid formula is derivable), and consistent.

The paper illustrates the logic with several planning‑style examples. In a grid world the agent knows it is somewhere in a set of “p‑states” but not the exact location; a plan “right then up” guarantees reaching a safe “q‑state” from any p‑state, so Kh(p, q) holds. A second example shows why a plan that works only from some p‑states but not all fails the Kh condition, highlighting the necessity of the universal quantifier over initial states. These examples also clarify the distinction between “de re” (knowledge of a plan) and “de dicto” (knowledge that a plan exists) interpretations of knowing‑how.

Philosophically, the authors position their approach between the “intellectualist” view (knowledge‑how reducible to knowledge‑that) and the “anti‑intellectualist” view (knowledge‑how as a distinct ability). By making the guarantee condition explicit, they avoid the pitfalls of both extremes: the ability to execute a plan is required, but the plan must succeed no matter which ψ‑state the agent actually occupies, thus excluding lucky or accidental successes.

The work connects directly to conformant planning in AI, where one seeks a single plan that succeeds under uncertainty about the initial state. The logical formalisation provides a bridge between planning algorithms and epistemic reasoning, suggesting future integration with automated theorem provers, model checkers, or planning systems.

Finally, the authors outline several avenues for further research: extending the logic to multi‑agent settings, incorporating probabilistic or temporal dimensions, handling dynamic updates of ψ (e.g., learning new preconditions), and exploring decidability and complexity issues. Overall, the paper offers a rigorous, compositional logic of “knowing how” that departs from the standard “knowledge as elimination of uncertainty” paradigm and opens new possibilities for both philosophical analysis and AI applications.