Construction and learnability of canonical Horn formulas

Abstract

We describe an alternative construction of an existing canonical representation for definite Horn theories, the Guigues-Duquenne basis (or GD basis), which minimizes a natural notion of implicational size. We extend the canonical representation to general Horn, by providing a reduction from definite to general Horn CNF. Using these tools, we provide a new, simpler validation of the classic Horn query learning algorithm of Angluin, Frazier, and Pitt, and we prove that this algorithm always outputs the GD basis regardless of the counterexamples it receives.

On the Usability of Probably Approximately Correct Implication Bases

“…With this algorithm, it is possible to learn implicational theories from equivalence and membership oracles alone. Indeed, the resulting set H of implications is always the canonical basis equivalent to L [5]. Moreover, the algorithm always runs in polynomial time in |M | and the size of the sought implication basis [4,Theorem 2].…”

Learning definite Horn formulas from closure queries

“…An assignment x ∈ {0, 1} n satisfies the implication α → β, denoted x |= α → β, if it either falsifies the antecedent or satisfies the consequent, that is, x |= α or x |= β respectively, where now we are interpreting both α and β as positive terms (x |= α if and only if α ⊆ [x] if and only if [α] ≤ x, see Lemma 1 of [14]).…”

“…An assignment x is said to be closed iff x ⋆ = x, and similarly for variable sets. The following holds for every definite Horn function f (see Theorem 3 in [14]):…”

“…In this section we review briefly part of our results from our previous work [14]. We will skip many details as they can be found in the aforementioned article.…”

“…Then, a definite Horn CNF is saturated if all of its implications are. Moreover, any saturated definite Horn CNF must be irredundant (see Lemma 2 of [14] for a proof). A result from [19,21] states that Theorem 3 ( [19,21]).…”