On the size of data structures used in symbolic model checking

TL;DR

This paper investigates the complexity and size growth of data structures like BDDs used in symbolic model checking, proving that exponential growth in their size is unavoidable in the worst case, regardless of data structure type or variable ordering.

Contribution

It formally proves that exponential size increases in data structures used for symbolic model checking are unavoidable, challenging assumptions about preprocessing benefits.

Findings

Preprocessing does not always exponentially improve performance.

Exponential growth of BDDs is unavoidable in the worst case.

The result applies to various data structures beyond BDDs.

Abstract

Temporal Logic Model Checking is a verification method in which we describe a system, the model, and then we verify whether some properties, expressed in a temporal logic formula, hold in the system. It has many industrial applications. In order to improve performance, some tools allow preprocessing of the model, verifying on-line a set of properties reusing the same compiled model; we prove that the complexity of the Model Checking problem, without any preprocessing or preprocessing the model or the formula in a polynomial data structure, is the same. As a result preprocessing does not always exponentially improve performance. Symbolic Model Checking algorithms work by manipulating sets of states, and these sets are often represented by BDDs. It has been observed that the size of BDDs may grow exponentially as the model and formula increase in size. As a side result, we formally prove that a superpolynomial increase of the size of these BDDs is unavoidable in the worst case. While this exponential growth has been empirically observed, to the best of our knowledge it has never been proved so far in general terms. This result not only holds for all types of BDDs regardless of the variable ordering, but also for more powerful data structures, such as BEDs, RBCs, MTBDDs, and ADDs.