Equilibria und weiteres Heiteres II

We investigate several technical and conceptual questions. Our main subject is the investigation of independence as a ternary relation in the context of non-monotonic logic. In the context of probability, this investigation was started by W.Spohn et al., and then followed by J.Pearl. We look at products of function sets, and thus continue our own investigation of independence in non-monotonic logic. We show that a finite characterization of this relation in our context is impossible, and indicate how to construct all valid rules.

💡 Research Summary

The paper investigates independence as a ternary relation within non‑monotonic logic, extending earlier work on probabilistic independence by Spohn, Pearl, and others. The authors introduce a formalism where three sets of variables X, Y, Z satisfy X | Y | Z if, for every x∈X, y∈Y, z∈Z, the probability distribution P obeys P(x,y,z)·P(y)=P(x,y)·P(y,z). This captures the intuitive notion that X and Z are independent once Y is fixed.

Beyond the probabilistic definition, the authors define a set‑theoretic analogue: for a subset A⊆X×Y×Z, X and Z are independent given Y if for any two tuples σ,τ∈A there exists a tuple ρ∈A that combines the X‑component of σ with the Z‑component of τ (the Y‑component is irrelevant). They prove that probabilistic independence implies set‑independence, but not conversely, by constructing explicit counter‑examples.

A side result shows that, regardless of the language’s cardinality, only countably many mutually inconsistent formulas can be defined. The proof proceeds by reducing formulas to conjunctions of literals and using a simple counting argument on the number of distinct literals that can appear in pairwise disjoint model sets.

The central contribution is the demonstration that no finite axiomatization can capture all valid ternary independence rules in this setting. The authors examine simple product spaces (X×Z, X×Z×W, X×Y×Z, etc.) and show that basic rules (symmetry, decomposition, weak union, contraction) are insufficient. They develop a systematic construction of new rules using “function trees,” “derivation trees,” and “universal trees.” These structures allow the generation of infinitely many valid inference patterns by iteratively applying basic rules and recombining variable fragments.

The paper also addresses “sub‑ideal” cases where full independence does not hold, proposing partial or weakened notions that still permit useful reasoning. In a later section, the authors extend the analysis to multisets, showing that even with multiplicities the lack of a finite inductive algorithm persists.

Finally, the authors revisit broader principles of non‑monotonic reasoning, clarifying the distinction between normality and consistency, discussing modularity of rule systems, and exploring inheritance mechanisms. They argue that many default reasoning frameworks can be understood through the lens of ternary independence, and that their construction method provides a complete catalogue of admissible rules for such systems.

Overall, the paper offers a deep theoretical exploration of ternary independence in non‑monotonic logic, proves the impossibility of a finite rule set, and supplies a constructive, algorithmic method for generating all valid independence rules, thereby extending the foundations of default and preferential reasoning.