Ingo Wegener

Branching programs and binary decision diagrams : theory and applications

Preface Introduction 1. Introduction 2. BPs and Decision Trees (DTs) 3. Ordered Binary Decision Diagrams (OBDDs) 4. The OBDD Size of Selected Functions 5. The Variable-Ordering Problem 6. Free BDDs (FBDDs) and Read-Once BPs 7. BDDs with Repeated Tests 8. Decision Diagrams (DDs) Based on Other Decomposition Rules 9. Integer-Valued DDs 10. Nondeterministic DDs 11. Randomized BDDs and Algorithms 12. Summary of the Theoretical Results 13. Applications in Verification and Model Checking 14. Further CAD Applications 15. Application in Optimization, Counting, and Genetic Programming Bibliography Index.

NC-ALGORITHMS FOR OPERATIONS ON BINARY DECISION DIAGRAMS

(Ordered) binary decision diagrams are a powerful representation for Boolean functions and are widely used in logical synthesis, verification, test pattern generation or as part of CAD tools. NC-al...

The size of reduced OBDD's and optimal read-once branching programs for almost all Boolean functions

Boolean functions are often represented by ordered binary-decision diagrams (OBDDs) introduced by Bryant (1986). Liaw and Lin (1992) have proved upper and lower bounds on the minimal OBDD size of almost all Boolean functions. Now tight bounds are proved for the minimal OBDD size for arbitrary or optimal variable orderings and for the minimal read-once branching program size of almost all functions. Almost all Boolean functions have a sensitivity of almost 1, i.e., the minimal OBDD size for an optimal variable ordering differs from the minimal OBDD size for a worst variable ordering by a factor of at most 1+/spl epsi/(n) where /spl epsi/(n) converges exponentially fast to 0. >

Optimal decision trees and one-time-only branching programs for symmetric Boolean functions*

Combinational complexity and depth are the most important complexity measures for Boolean functions. It has turned out to be very hard to prove good lower bounds on the combinational complexity or the depth of explicitly defined Boolean functions. Therefore one has restricted oneself to models where nontrivial lower bounds are easier to prove. Here decision trees, branching programs, and one-time-only branching programs are considered, where each variable may be tested on each path of computation only once. Efficient algorithms for the construction of optimal decision trees and optimal one-time-only branching programs for symmetric Boolean functions are presented. Furthermore, the following trade-off results are proved. An exponential lower bound on the decision tree complexity of some Boolean function is shown having linear formula size and linear one-time-only branching program complexity. Furthermore, a quadratic lower bound on the one-time-only branching program complexity of some Boolean function is shown having linear combinational complexity.

Optimal Attribute-Efficient Learning of Disjunction, Parity and Threshold Functions

Decision trees are a very general computation model. Here the problem is to identify a Boolean function f out of a given set of Boolean functions F by asking for the value of f at adaptively chosen inputs. For classes F consisting of functions which may be obtained from one function g on n inputs by replacing arbitrary n−k inputs by given constants this problem is known as attribute-efficient learning with k essential attributes. Results on general classes of functions are known. More precise and often optimal results are presented for the cases where g is one of the functions disjunction, parity or threshold.

Properties of complexity measures for prams and wrams

Abstract The computation of Boolean functions by parallel computers with shared memory (PRAMs and WRAMs) is considered. In particular, complexity measures for parallel computers like critical and sensitive complexity are compared with other complexity measures for Boolean functions like branching program depth and length of prime implicants and clauses. The relations between these complexity measures and their asymptotic behaviour are investigated for the classes of Boolean functions, monotone functions and symmetric functions.

On the complexity of slice functions

Abstract By a result of Berkowitz (1982), the monotone circuit complexity of slice functions cannot be much larger than the circuit (combinational) complexity of these functions for arbitrary complete bases. This result strengthens the importance of the theory of monotone circuits. We show in this paper that monotone circuits for slice functions can be understood as special circuits called set circuits. Here, disjunction and conjunction are replaced by set union and set intersection. All the main methods known for proving lower bounds on the monotone complexity of Boolean functions fail to work in their present form for slice functions. Furthermore, we show that the canonical slice functions of the Boolean convolution, the Nechiporuk Boolean sums, and the clique function can be computed with a linear number of gates.

Time-Space trade-offs for branching programs

Abstract Branching program depth and the logarithm of branching program complexity are lower bounds on time and space requirements for any reasonable model of sequential computation. In order to gain more insight to the complexity of branching programs and to the problems of time-space trade-offs one considers, on one hand, width-restricted and, on the other hand, depth-restricted branching programs. We present these computation models and the trade-off results already proved. We prove a new result of this type by presenting an effectively defined Boolean function whose complexity in depth-restricted one-time-only branching programs is exponential while its complexity even in width-2 branching programs is polynomial.

Optimal search with positive switch cost is NP-hard

Abstract We consider the problem of optimal search with positive switch cost. For vanishing switch cost we have efficient algorithms for the construction of an optimal strategy. We show here that already a very special subproblem with positive switch cost is NP-hard.

Distributed Hybrid Genetic Programming for Learning Boolean Functions

When genetic programming (GP) is used to find programs witli Boolean inputs and outputs, ordered binary decision diagrams (OBDDs) are often used successfully. In all known OBDD-based GP-systems the variable ordering, a crucial factor for the size of OBDDs, is preset to an optimal ordering of the known test function. Certainly this cannot be done in practical applications, where the function to learn and hence its optimal variable ordering are unknown. n nHere, the first GP-system is presented that evolves the variable ordering of the OBDDs and the OBDDs itself by using a distributed hybrid approach. For the experiments presented the unavoidable size increase compared to the optimal variable ordering is quite small. Hence, this approach is a big step towards learning well-generalizing Boolean functions.