Extending Defeasible Reasoning Beyond Rational Closure

Abstract

The KLM framework is a well known extension of classical logic for incorporating defeasible reasoning. Central to the KLM framework is rational closure, recognized as the most conservative approach to defeasible reasoning. Rational closure operates through two complementary paradigms: model-theoretic and formula-theoretic. This paper concentrates on the model-theoretic dimension, known as minimal ranked entailment. Unfortunately, the practical implementation of minimal ranked entailment remains largely impractical due to high computational and storage demands. To address this, we present reduced minimal ranked entailment, an optimization that employs reduced ordered binary decision diagrams to eliminate redundant information, thereby enhancing computational efficiency and reducing memory requirements. We further demonstrate that this optimized approach significantly facilitates the practical application and development of model-theoretic extensions of rational closure. This is illustrated through our own Bayesian refinement of minimal ranked entailment, which conceptualizes defeasible entailment as a form of conditional probability.

Appendices

A Rational Closure Algorithms

Algorithm 6

BaseRank

Algorithm 7

RationalClosure

Algorithm 8

MinimalRank

Algorithm 9

MinimalEntailment

B Irredundant Base Rank Algorithms

Algorithm 10

IrredundantBaseRank

Algorithm 11

IrredundantRationalClosure

C Proof of Properties for Naive Bayes Entailment

Theorem 1

Naive bayes entailment is not LM-rational.

Reflexivity. Reflexivity stipulates that all formulas should be defeasible consequences of themselves, a condition generally met by any defeasible entailment relation.

$$\begin{aligned} (Ref) \ \ \mathcal {K}\mid \approx \alpha \mid \!\sim \alpha \end{aligned}$$

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Lemma 1

Naive bayes entailment satisfies reflexivity.

Proof

Naive bayes entailment identifies the most statistically probable minimal models consistent with any evidential fragment in $\llbracket \alpha \rrbracket$. It is therefore guaranteed to return a set of models that each satisfy $\alpha$.

Left Logical Equivalence. Left logical equivalence posits that if two formulas are logically equivalent, then their respective defeasible consequences must also be identical.

$$\begin{aligned} (LLE) \ \ \frac{\alpha \equiv \beta , \ \ \mathcal {K}\mid \approx \alpha \mid \!\sim \beta }{\mathcal {K}\mid \approx \beta \mid \!\sim \gamma } \end{aligned}$$

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Lemma 2

Naive bayes entailment satisfies left logical equivalence.

Proof

Should $\alpha \equiv \beta$, it follows by definition that $\llbracket \alpha \rrbracket = \llbracket \beta \rrbracket$, indicating identical evidential fragments. Naive bayes entailment will therefore return the same models for both $\alpha$ and $\beta$.

Right Weakening. Right weakening asserts that if $\beta$ is a defeasible consequence of $\alpha$, then any formula that is logically equivalent to $\beta$ must likewise be a defeasible consequence of $\alpha$.

$$\begin{aligned} (RW) \ \ \frac{\mathcal {K}\mid \approx \alpha \mid \!\sim \beta , \ \ \beta \models \gamma }{\mathcal {K}\mid \approx \alpha \mid \!\sim \gamma } \end{aligned}$$

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Lemma 3

Naive bayes entailment satisfies right weakening.

Proof

If $\mathcal {K}\mid \approx \alpha \mid \!\sim \beta$, then all models returned by naive bayes entailment satisfy $\beta$. Given that $\beta \models \gamma$, it follows by definition that $\llbracket \beta \rrbracket \subseteq \llbracket \gamma \rrbracket$, thereby ensuring that each model also satisfies $\gamma$.

And. And dictates that the conjunction of any defeasible consequences of a formula should likewise be a defeasible consequence of that formula.

$$\begin{aligned} (And) \ \ \frac{\mathcal {K}\mid \approx \alpha \mid \!\sim \beta , \ \ \mathcal {K}\mid \approx \alpha \mid \!\sim \gamma }{\mathcal {K}\mid \approx \alpha \mid \!\sim \beta \wedge \gamma } \end{aligned}$$

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Lemma 4

Naive bayes entailment satisfies and.

Proof

If every model returned by naive bayes entailment for $\alpha$ satisfies both $\beta$ and $\gamma$, then they are guaranteed to satisfy $\beta \wedge \gamma$.

Or. Or states that a defeasible consequence of two individual formulas should also be a defeasible consequence of their disjunction.

$$\begin{aligned} (Or) \ \ \frac{\mathcal {K}\mid \approx \alpha \mid \!\sim \gamma , \ \ \mathcal {K}\mid \approx \beta \mid \!\sim \gamma }{\mathcal {K}\mid \approx \alpha \vee \beta \mid \!\sim \gamma } \end{aligned}$$

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Lemma 5

Naive bayes entailment does not satisfy or.

Proof

Consider $\mathcal {K}= \{\texttt {a} \mid \!\sim \texttt {b}, \texttt {c} \rightarrow \texttt {a}, \texttt {c} \mid \!\sim \lnot \texttt {b}, \texttt {d} \rightarrow \texttt {c}, \texttt {d} \mid \!\sim \texttt {e}, \texttt {f} \rightarrow \texttt {a}, \texttt {f} \mid \!\sim \lnot \texttt {b} \}$, Fig. 7 shows a corresponding reduced $\mathcal {R}^{\mathcal {K}}_{RC}$. Naive bayes entailment identifies $\texttt {f}{} \texttt {e}\overline{\texttt {d}}{} \texttt {c}\overline{\texttt {b}}{} \texttt {a}$ and $\overline{\texttt {f}}{} \texttt {e}\overline{\texttt {d}}{} \texttt {c}\overline{\texttt {b}}{} \texttt {a}$ as most probable for c, making $\mathcal {K}\mid \approx _{NB} \texttt {c} \mid \!\sim \texttt {e}$ true. Naive bayes entailment also identifies $\texttt {f}{} \texttt {e}\overline{\texttt {d}}{} \texttt {c}\overline{\texttt {b}}{} \texttt {a}$ as most probable for f, making $\mathcal {K}\mid \approx _{NB} \texttt {f} \mid \!\sim \texttt {e}$ true. However, the most probable models for $\texttt {c} \vee \texttt {f}$ are $\texttt {f}{} \texttt {e}\overline{\texttt {d}}\overline{\texttt {c}}\overline{\texttt {b}}{} \texttt {a}$ and $\texttt {f}\overline{\texttt {e}}\overline{\texttt {d}}\overline{\texttt {c}}\overline{\texttt {b}}{} \texttt {a}$, making $\mathcal {K}\mid \approx _{NB} \texttt {c} \vee \texttt {f} \mid \!\sim \texttt {e}$ false.

Fig. 7.

Reduced minimal ranked interpretation $\mathcal {R}^{\mathcal {K}}_{RC}$.

Cautious Monotonicity. Cautious monotonicity posits that the integration of newly deduced information should not negate or undermine any previously inferable conclusions.

$$\begin{aligned} (CM) \ \ \frac{\mathcal {K}\mid \approx \alpha \mid \!\sim \beta , \ \ \mathcal {K}\mid \approx \alpha \mid \!\sim \gamma }{\mathcal {K}\mid \approx \alpha \wedge \beta \mid \!\sim \gamma } \end{aligned}$$

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Lemma 6

Naive bayes entailment does not satisfy cautious monotonicity.

Proof

Consider again the knowledge from the previous proof. Both $\mathcal {K}\mid \approx _{NB} \texttt {c} \mid \!\sim \texttt {e}$ and $\mathcal {K}\mid \approx _{NB} \texttt {c} \mid \!\sim \lnot \texttt {d}$ are true. However, naive bayes entailment identifies $\texttt {f}{} \texttt {e}\overline{\texttt {d}}{} \texttt {c}\overline{\texttt {b}}{} \texttt {a}$, $\overline{\texttt {f}}{} \texttt {e}\overline{\texttt {d}}{} \texttt {c}\overline{\texttt {b}}{} \texttt {a}$, $\texttt {f}{} \texttt {e}{} \texttt {d}{} \texttt {c}\overline{\texttt {b}}{} \texttt {a}$, and $\overline{\texttt {f}}{} \texttt {e}{} \texttt {d}{} \texttt {c}\overline{\texttt {b}}{} \texttt {a}$ as most probable for $\texttt {c} \wedge \texttt {e}$, making $\mathcal {K}\mid \approx _{NB} \texttt {c} \wedge \texttt {e} \mid \!\sim \lnot \texttt {d}$ false.

Rational Monotonicity. Rational monotonicity asserts that incorporating information which was not previously negated by existing knowledge should not cause the retraction of any prior conclusions.

$$\begin{aligned} (RM) \ \ \frac{\mathcal {K}\mid \approx \alpha \mid \!\sim \gamma , \ \ \mathcal {K}\not \mid \approx \alpha \mid \!\sim \lnot \beta }{\mathcal {K}\mid \approx \alpha \wedge \beta \mid \!\sim \gamma } \end{aligned}$$

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Lemma 7

Naive bayes entailment does not satisfy rational monotonicity.

Proof

Proving that naive bayes entailment doesn’t satisfy cautious monotonicity concurrently establishes its non-compliance with rational monotonicity.