Hiding Propositional Constants in BDDs

Abstract

In [6] it is described how correctness of safety control systems of railways (interlockings) can be expressed using (large) propositional formulas. Interlockings are correct if these formulas are tautologies. Given the impressive capabilities to show propositional formulas tautologies ascribed to Binary Decision Diagrams (BDDs) [1], it seems natural to apply them to the formulas that we obtained. Somewhat to our dissatisfaction it appeared that the formulas had to be reduced with 98% for the plain BDD technique to yield results. The propositional formulas consist of 4000 lines, and BDDs could only be constructed of formulas covering up to 70 lines. Even if dynamic variable ordering is applied, formulas up till 70 lines can be handled [4]. In this note we describe a very straightforward and easy to implement extension to the BDD technique, using hiding functions. It turns out that these functions make it possible to check whether the sizable formulas describing the correctness of interlockings are tautologies which shows their strength. This strength is confirmed in [2,3] where it has been described how using partitioned transition relations the complexity of BDD based verifications can be significantly reduced. In essence these partitioned transition relations can be viewed as an application of the technique presented in this paper. At the end of the article we describe an experiment with randomised 3-SAT formulas. It is shown that if the number of occurrences of variables in 3-SAT formulas is small hiding functions apply very well. All experiments mentioned above indicate that hiding functions are useful if variables occur rather localised in formulas and/or have a relatively small number of occurrences.