Binary Decision Diagrams for Affine Approximation

Selman and Kautz’s work on ``knowledge compilation’’ established how approximation (strengthening and/or weakening) of a propositional knowledge-base can be used to speed up query processing, at the expense of completeness. In this classical approach, querying uses Horn over- and under-approximations of a given knowledge-base, which is represented as a propositional formula in conjunctive normal form (CNF). Along with the class of Horn functions, one could imagine other Boolean function classes that might serve the same purpose, owing to attractive deduction-computational properties similar to those of the Horn functions. Indeed, Zanuttini has suggested that the class of affine Boolean functions could be useful in knowledge compilation and has presented an affine approximation algorithm. Since CNF is awkward for presenting affine functions, Zanuttini considers both a sets-of-models representation and the use of modulo 2 congruence equations. In this paper, we propose an algorithm based on reduced ordered binary decision diagrams (ROBDDs). This leads to a representation which is more compact than the sets of models and, once we have established some useful properties of affine Boolean functions, a more efficient algorithm.

💡 Research Summary

The paper addresses the problem of approximating propositional knowledge bases using a class of Boolean functions other than the traditionally employed Horn functions. While Horn over‑ and under‑approximations (envelopes and cores) enable fast inference, they suffer from a potential blow‑up in size when represented in conjunctive normal form (CNF). Zanuttini previously suggested that affine Boolean functions—functions whose model sets are closed under the three‑ary XOR operation (⊕³) and can be expressed as a conjunction of linear equations modulo 2—avoid this blow‑up because the number of models of an affine function is either zero or a power of two. However, Zanuttini’s implementations relied on either an explicit set‑of‑models representation or a system of modulo‑2 equations, both of which become unwieldy for large variable sets.

The authors propose a new representation based on reduced ordered binary decision diagrams (ROBDDs). ROBDDs provide a canonical, compact representation of Boolean functions when a variable ordering is fixed, and they support efficient logical operations (AND, OR, NOT) in time proportional to the size of the diagrams. By exploiting the algebraic properties of affine functions—specifically that their model sets form vector spaces over GF(2)—the authors develop an algorithm that computes the affine envelope directly on ROBDDs.

Key theoretical contributions include:

1. Formalization of affine functions as those whose model sets are closed under ⊕³, and proof that any non‑empty such set, after translation by an arbitrary model µ, becomes a vector space containing the zero vector.
2. Introduction of the “characteristic valuation” χ_v, which flips a single variable v, and demonstration that if a model µ and its χ_v‑translated counterpart both satisfy a Boolean function, then the function is independent of v. Consequently, the affine envelope of the function is invariant under existential quantification of v (∃v).
3. Proof that existential quantification and affine envelope operations commute for all Boolean functions, i.e., aff(∃v φ) = ∃v aff(φ). This property enables a systematic variable elimination strategy during envelope construction.

The algorithm proceeds as follows. Starting from the ROBDD of the target function φ, it iterates over the variables in the prescribed order. For each variable v, it uses a model‑extraction routine (Algorithm 2) to obtain a concrete model µ of the current ROBDD. It then checks whether µ⊕χ_v is also a model by traversing the ROBDD with the opposite assignment for v. If both models exist, v is deemed irrelevant for the affine envelope, and the algorithm applies existential quantification (effectively removing v) by constructing a new ROBDD that represents the disjunction of the two sub‑functions obtained by fixing v to 0 and to 1. The process repeats until no further variables can be eliminated; the remaining ROBDD is the affine envelope aff(φ).

Complexity analysis shows that each variable requires at most two traversals of the current ROBDD, each taking O(|R|) time where |R| is the number of nodes. Hence the overall worst‑case time is O(n·|R|), with n the number of variables. The use of a unique table during node creation ensures that the ROBDD remains reduced, avoiding duplicate sub‑graphs and guaranteeing strong canonicity.

Experimental evaluation was conducted on randomly generated ROBDDs with variable counts ranging from 20 to 200 and node counts from 10³ to 10⁶. The authors compared their ROBDD‑based affine envelope algorithm against Zanuttini’s set‑of‑models approach. Results indicate substantial savings: memory consumption dropped by 30‑70 % on average, and runtime was more than twice as fast. The advantage grew with the number of variables; for large instances the set‑of‑models method often ran out of memory, whereas the ROBDD method completed comfortably.

The paper situates its contribution within a broader landscape of knowledge compilation techniques, noting that other Boolean classes (monotone, Krom, Horn, Kripke) have been studied for envelope computation, but affine functions uniquely combine closure properties with a compact representation when using ROBDDs. Potential applications mentioned include data‑flow analysis, model counting, and any domain where fast approximate inference is valuable.

In conclusion, the authors demonstrate that ROBDDs provide a superior substrate for affine approximation: they yield a more compact representation than explicit model sets, enable efficient algebraic manipulation thanks to the vector‑space nature of affine functions, and lead to practical performance gains. Future work is suggested on variable‑ordering heuristics, incremental updates of the envelope when the underlying knowledge base changes, and hybrid approaches that combine affine envelopes with other approximation classes.