We show that any language recognized by an NC ’ circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

, Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [24]). The sources considered output strings according to probability distributions in which no single string is too probable. The new model provides a fruitful viewpoint on problems studied previously as: ffl Extracting almost perfect bits from sources of weak randomness: the question of possibility as well as the question of efficiency of such extraction schemes are addressed. ffl Probabilistic Communication Complexity: it is shown that most functions have linear communication complexity in a very strong probabilistic sense. ffl Robustness of BPP with respect to sources of weak randomness (generalizing a result of Vazirani and Vazirani [27]). The paper has appeared in SIAM Journal o...

Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by ordering restrictions. In the context of EXOR-based logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important. We study the relation between (ordered, free) BDDs and FDDs. Both, BDDs and FDDs, result from DDs by defining the represented function in different ways. If the underlying DD is complete, the relation between both types of interpretation can be described by a Boolean transformation . This allows us to relate the FDD-size of f and the BDD-size of (f) also in the case that the corresponding DDs are free or ordered, but not (necessarily) complete. We use this property to derive...

We present a data structure for Boolean functions, which we call Parity--OBDDs or \Phi-- OBDDs, which combines the nice algorithmic properties of the well--known ordered binary decision diagrams (OBDDs) with a considerably larger descriptive power. Beginning from an algebraic characterization of the \Phi--OBDD complexity we prove in particular that the minimization of the number of nodes, the synthesis, and the equivalence test for \Phi--OBDDs, which are the fundamental operations for circuit verification, have efficient deterministic solutions. Several functions of pratical interest, i.e. the indirect storage access function, have exponential ODBB--size but are of polynomial size if \Phi--OBDDs are used. Keywords: data structures for Boolean functions, BDDs, circuit verification 1 Introduction Formal circuit verification is a fundamantal task. The following approach for verification is often used (for a survey see [8] and [21]). A data structure for representing Boolean functions is...

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Ordered Decision Diagrams (ODDs) as a means for the representation of Boolean functions are used in many applications in CAD. Depending on the decomposition type, various classes of ODDs have been defined, the most important being the Ordered Binary Decision Diagrams (OBDDs), the Ordered Functional Decision Diagrams (OFDDs) and the Ordered Kronecker Functional Decision Diagrams (OKFDDs). In this paper we clarify the computational power of OKFDDs versus OBDDs and OFDDs from a (more) theoretical point of view. We prove several exponential gaps between specific types of ODDs. Combining these results it follows that a restriction of the OKFDD concept to subclasses, such as OBDDs and OFDDs as well, results in families of functions which lose their efficient representation.

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Computational complexity is concerned with the complexity of solving problems and computing functions and not with the complexity of verifying circuit designs. The importance of formal circuit verification is evident. Therefore, a framework of a complexity theory for formal circuit verification with binary decision diagrams is developed. This theory is based on read-once projections. For many problems it is determined whether and how they are related with respect to read-once projections. It is proved that multiplication can be reduced to squaring but squaring is not a read-once projection of multiplication. This perhaps surprising result is discussed. For most of the common binary decision diagram models of polynomial size complete problems with respect to read-once projections are described. But for the class of functions with polynomial-size free binary decision diagrams (read-once branching programs) no complete problem with respect to read-once projection exists. Supported by DF...

We present a data structure --- the Mod-2-OBDD's that considerably extend OBDD's (ordered binary decision diagrams) as well as ESOP's (EXOR-sum-of-products). Many Boolean function of practical interest like hidden weighted bit function, indirect storage access function, important symmetric functions have exponential size optimal OBDD's and/or ESOP's (even multilevel EXOR-expressions) but (low degree) polynomial size Mod-2-OBDD's. The manipulation of Mod-2-OBDD's is at least as efficient as the manipulation of OBDD's. Apply operation, quantification, composition [Bry86] have the same complexity as in the case of OBDD's. Moreover, since the size of a minimal Mod-2-OBDD-representation of a Boolean function is, in general, smaller than the size of its optimal OBDD-representation manipulation is more efficient. Moreover, EXOR-operation and complementation can be performed much better -- namely in O(1) time. However, the price of O(1) time EXOR-apply operations is the can...