On the size of binary decision diagrams representing Boolean functions

AbstractWe consider the size of the representation of Boolean functions by several classes of binary decision diagrams (BDDs) (also called branching programs), namely the classes of arbitrary BDDs of real time BDD (RBDD) (i.e. BDDs where each computation path is limited to the number of variables), of free BDDs (FBDDs) (also called read-once-only branching programs), of ordered BDDs (OBDDS) i.e. FBDDs where variables are tested in the same order along all paths), and binary decision trees (BDTs).Using well-known techniques, we first establish asymptotically sharp bounds as a function of n on the minimum size of arbitrary BDDs representing almost all Boolean functions of n variables and provide asymptotic lower and upper bounds, differing only by a factor of two, on the minimum size OBDDs representing almost all Boolean functions of n variables.We then, using a method to obtain exponential lower bounds on complexity of computation of Boolean functions by RBDD, FBDD and OBDD that originated in (Breitbart, 1968), present the highest such bounds to date and also present improved bounds on the relative economy of description of particular Boolean functions by the above classes of BDDs. For each nontrivial pair of BDD classes considered, we exhibit infinite families of Boolean functions representable much more concisely by BDDs in one class than by BDDs in the other

Restricted branching programs and hardware verification

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1995.Includes bibliographical references (p. 71-77).by Stephen John Ponzio.Ph.D

The Complexity of Equivalence and Containment for Free Single Variable Program Schemes

Non-containment for free single variable program schemes is shown to be NP-complete. A polynomial time algorithm for deciding equivalence of two free schemes, provided one of them has the predicates appearing in the same order in all executions, is given. However, the ordering of a free scheme is shown to lead to an exponential increase in size

Probabilistic representation and manipulation of Boolean functions using free Boolean diagrams

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (p. 145-149).by Amelia Huimin Shen.Ph.D

Explicit or Symbolic Translation of Linear Temporal Logic to Automata

Formal verification techniques are growing increasingly vital for the development of safety-critical software and hardware in practice. Techniques such as requirements-based design and model checking for system verification have been successfully used to verify systems for air traffic control, airplane separation assurance, autopilots, CPU logic designs, life-support, medical equipment, and other functions that ensure human safety. Formal behavioral specifications written early in the system-design process and communicated across all design phases increase the efficiency, consistency, and quality of the system under development. We argue that to prevent introducing design or verification errors, it is crucial to test specifications for satisfiability. We advocate for the adaptation of a new sanity check via satisfiability checking for property assurance. Our focus here is on specifications expressed in Linear Temporal Logic (LTL). We demonstrate that LTL satisfiability checking reduces to model checking and satisfiability checking for the specification, its complement, and a conjunction of all properties should be performed as a first step to LTL model checking. We report on an experimental investigation of LTL satisfiability checking. We introduce a large set of rigorous benchmarks to enable objective evaluation of LTL-to-automaton algorithms in terms of scalability, performance, correctness, and size of the automata produced. For explicit model checking, we use the Spin model checker; we tested all LTL-to-explicit automaton translation tools that were publicly available when we conducted our study. For symbolic model checking, we use CadenceSMV, NuSMV, and SAL-SMC for both LTL-to-symbolic automaton translation and to perform the satisfiability check. Our experiments result in two major findings. First, scalability, correctness, and other debilitating performance issues afflict most LTL translation tools. Second, for LTL satisfiability checking, the symbolic approach is clearly superior to the explicit approach. Ironically, the explicit approach to LTL-to-automata had been heavily studied while only one algorithm existed for LTL-to-symbolic automata. Since 1994, there had been essentially no new progress in encoding symbolic automata for BDD-based analysis. Therefore, we introduce a set of 30 symbolic automata encodings. The set consists of novel combinations of existing constructs, such as different LTL formula normal forms, with a novel transition-labeled symbolic automaton form, a new way to encode transitions, and new BDD variable orders based on algorithms for tree decomposition of graphs. An extensive set of experiments demonstrates that these encodings translate to significant, sometimes exponential, improvement over the current standard encoding for symbolic LTL satisfiability checking. Building upon these ideas, we return to the explicit automata domain and focus on the most common type of specifications used in industrial practice: safety properties. We show that we can exploit the inherent determinism of safety properties to create a set of 26 explicit automata encodings comprised of novel aspects including: state numbers versus state labels versus a state look-up table, finite versus infinite acceptance conditions, forward-looking versus backward-looking transition encodings, assignment-based versus BDD-based alphabet representation, state and transition minimization, edge abbreviation, trap-state elimination, and determinization either on-the-fly or up-front using the subset construction. We conduct an extensive experimental evaluation and identify an encoding that offers the best performance in explicit LTL model checking time and is constantly faster than the previous best explicit automaton encoding algorithm