[Paper Review] Strategically knowing how

Raul Fervari, Andreas Herzig+2 more|arXiv (Cornell University)|May 15, 2017

Logic, Reasoning, and Knowledge8 references13 citations

TL;DR

This paper introduces a single-agent epistemic logic, SKH, that formalizes goal-directed knowing how by linking it to uniform strategies ensuring a goal. It provides a sound, complete, and decidable axiomatization, showing that knowing how to achieve φ entails knowing that one knows how to achieve φ, which captures strategic knowledge under uncertainty in planning scenarios.

ABSTRACT

In this paper, we propose a single-agent logic of goal-directed knowing how extending the standard epistemic logic of knowing that with a new knowing how operator. The semantics of the new operator is based on the idea that knowing how to achieve $ϕ$ means that there exists a (uniform) strategy such that the agent knows that it can make sure $ϕ$. We give an intuitive axiomatization of our logic and prove the soundness, completeness, and decidability of the logic. The crucial axioms relating knowing that and knowing how illustrate our understanding of knowing how in this setting. This logic can be used in representing both knowledge-that and knowledge-how.

Motivation & Objective

• To formalize goal-directed knowledge-how in a way that integrates with standard epistemic logic (knowing that), addressing the gap in formal treatments of knowing how in AI and epistemology.
• To resolve the de re vs. de dicto distinction in knowledge-how by ensuring that knowing how to achieve φ implies the agent knows a specific strategy to achieve φ.
• To provide a logical framework that supports reasoning about planning under uncertainty, where agents must choose actions based on incomplete information and potential outcomes.
• To establish a formal system that captures the interaction between knowing that and knowing how, particularly in scenarios involving non-deterministic actions and imperfect information.
• To ensure the logic is both mathematically robust and computationally tractable, with soundness, completeness, and decidability proven for the proposed system.

Proposed method

• Introduces a new epistemic modality ${\mathcal{K}h}$ to represent 'knowing how to achieve φ' as the existence of a uniform strategy such that the agent knows the strategy ensures φ.
• Defines the semantics of ${\mathcal{K}h}$ using a Kripke-style model with states, actions, and indistinguishability relations, where a strategy is a function mapping states to actions.
• Uses the concept of complete execution of a strategy from a state to define when a strategy guarantees a goal φ, ensuring that all possible outcomes of the strategy lead to φ.
• Constructs a canonical model based on maximal consistent sets of formulas, where truth of ${\mathcal{K}h}\phi$ is determined by whether all complete executions of a strategy from a state lead to φ.
• Employs the Axiom of Dependent Choice in the completeness proof to derive infinite execution sequences when ${\mathcal{K}h}\phi$ is not satisfied, leading to a contradiction that confirms the completeness of the system.
• Proves decidability via a small model property: any satisfiable formula φ has a model of size at most $2^{2|\phi|}$, ensuring finite search space for satisfiability checking.

Experimental results

Research questions

• RQ1 How can knowledge-how be formally captured in a way that distinguishes de re from de dicto readings, particularly in strategic contexts?
• RQ2 What logical axioms are needed to capture the interaction between knowing that and knowing how in goal-directed planning under uncertainty?
• RQ3 Can a logic of knowing how be both sound, complete, and decidable while preserving the strategic intuition that knowing how to achieve φ means having a uniform strategy that guarantees φ?
• RQ4 How does the presence of perfect recall (i.e., no forgetting) affect the validity of key axioms like ${\mathcal{K}h}\phi \to {\mathcal{K}h}{\mathcal{K}}\phi$?
• RQ5 In what ways does the proposed logic differ from prior approaches that combine ability and knowledge modalities, and why is a single ${\mathcal{K}h}$ modality more suitable for goal-directed knowledge?

Key findings

• The logic SKH is sound and complete with respect to its semantics, ensuring that all valid formulas are provable and all provable formulas are valid.
• The logic is decidable, with a small model property: any satisfiable formula φ has a model of size at most $2^{2|\phi|}$, enabling finite satisfiability checking.
• The key axiom $\mathtt{AxKhtoKhK}:{\mathcal{K}h}\phi \to {\mathcal{K}h}{\mathcal{K}}\phi$ holds, reflecting that if an agent knows how to achieve φ, then it knows that it knows how to achieve φ, which is justified under perfect recall.
• The logic distinguishes itself from prior systems by treating ${\mathcal{K}h}$ as a primitive modality rather than decomposing it into combinations of knowledge and ability, leading to a more coherent and intuitive semantics.
• The completeness proof relies on a canonical model construction using maximal consistent sets and the Axiom of Dependent Choice to derive a contradiction from the failure of ${\mathcal{K}h}\phi$ in a state where it should hold.
• The logic captures the essence of conformant planning: an agent knows how to achieve a goal if there exists a single strategy that guarantees the goal under all possible outcomes, even with uncertainty.