Using Extended Logical Primitives for Efficient BDD Building

David Fernandez-Amoros,Ernesto Aranda-Escolástico,Ernesto Aranda-Escolástico,Sergio Bra,Sergio Bra,Ruben Heradio

doi:10.3390/math8081253

David Fernandez-Amoros, Ernesto Aranda-Escolástico + Show 4 more

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https://doi.org/10.3390/math8081253

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Abstract

Binary Decision Diagrams (BDDs) have been used to represent logic models in a variety of research contexts, such as software product lines, circuit testing, and plasma confinement, among others. Although BDDs have proven to be very useful, the main problem with this technique is that synthesizing BDDs can be a frustratingly slow or even unsuccessful process, due to its heuristic nature. We present an extension of propositional logic to tackle one recurring phenomenon in logic modeling, namely groups of variables related by an exclusive-or relationship, and also consider two other extensions: one in which at least n variables in a group are true and another one for in which at most n variables are true. We add XOR, atLeast-n and atMost-n primitives to logic formulas in order to reduce the size of the input and also present algorithms to efficiently incorporate these constructions into the building of BDDs. We prove, among other results, that the number of nodes created during the process for XOR groups is reduced from quadratic to linear for the affected clauses. the XOR primitive is tested against eight logical models, two from industry and six from Kconfig-based open-source projects. Results range from no negative effects in models without XOR relations to performance gains well into two orders of magnitude on models with an abundance of this kind of relationship.

Highlights

Formal methods use logical representations of models to enable complex analyses
The extension to logic that we propose in this paper could fit with small changes into what is known in the SAT community as Satisfiability Modulo Theories (SMTs) that is, using standard SAT
We have identified a major obstacle in Binary Decision Diagrams (BDDs) building and developed a mitigation strategy with a complexity reduction from a quadratic number of created nodes to a linear one for the occurrences of the XOR primitive, which can amount to a sizable amount of the translation of a model

Summary

Introduction

Formal methods use logical representations of models to enable complex analyses One such representation are Binary Decision Diagrams (BDDs), an evolution of Binary Decision Trees [1]. BDDs can represent the solutions of a propositional logic formula in a way which allows to compute properties of the solutions. In this regard, BDDs are a competing technology to SAT-solvers and #SAT-solvers. A SAT-solver is a program that takes a propositional formula and determines if it is satisfiable. It does so by running a heuristic tree search over the variables in the formula.

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