Pairing Functions, Boolean Evaluation and Binary Decision Diagrams in Prolog

TL;DR

This paper introduces a novel encoding of Binary Decision Diagrams (BDDs) using pairing functions on natural numbers, enabling faithful boolean evaluation and efficient ranking/unranking of BDDs within Prolog.

Contribution

It presents a new method to encode BDDs as natural numbers via pairing functions, linking their structure to truth table representations and enabling efficient manipulation.

Findings

Encoding of BDDs as natural numbers preserves boolean evaluation

Derived ranking and unranking functions for BDDs

Implementation as a self-contained Prolog program

Abstract

A "pairing function" J associates a unique natural number z to any two natural numbers x,y such that for two "unpairing functions" K and L, the equalities K(J(x,y))=x, L(J(x,y))=y and J(K(z),L(z))=z hold. Using pairing functions on natural number representations of truth tables, we derive an encoding for Binary Decision Diagrams with the unique property that its boolean evaluation faithfully mimics its structural conversion to a a natural number through recursive application of a matching pairing function. We then use this result to derive {\em ranking} and {\em unranking} functions for BDDs and reduced BDDs. The paper is organized as a self-contained literate Prolog program, available at http://logic.csci.unt.edu/tarau/research/2008/pBDD.zip Keywords: logic programming and computational mathematics, pairing/unpairing functions, encodings of boolean functions, binary decision diagrams, natural number representations of truth tables