Ordered Binary-Decision Diagrams (OBDDS) represent Boolean functions as directed acyclic graphs. They form a canonical representation, making testing of functional properties such as satmfiability and equivalence straightforward. A number of operations on Boolean functions can be implemented as graph algorithms on OBDD

This paper presents lower bound results on Boolean function complexity under two different models. The first is an abstraction of tradeoffs between chip area and speed in very large scale integrated (VLSI) circuits. The second is the ordered binary decision diagram (OBDD) representation used as a data structure for symbolically representing and manipulating Boolean functions. These lower bounds demonstrate the fundamental limitations of VLSI as an implementation medium, and OBDDs as a data structure. They also lend insight into what properties of a Boolean function lead to high complexity under these models. Related techniques can be...

This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable properties of BDDs. Two algorithms are described for transforming a BED into a reduced ordered BDD. One is a generalized version of the BDD apply-operator while the other can exploit the structural information of the Boolean expression. This ability is demonstrated by verifying that two di erent circuit implementations of a 16-bit multiplier implement the same Boolean function. Using BEDs, this veri cation problem is solved in less than a second, while using standard BDD techniques this problem is infeasible. Generally, BEDs are useful in applications, for example tautology checking, where the end-result as a reduced ordered BDD is small.

We obtain the first non-trivial time-space tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant ε > 0. We also give the first separation result between the syntactic and semantic read-k models [BRS93] for k > 1 by showing that polynomial-size semantic read-twice branching programs can compute functions that require exponential size on any syntactic read-k branching program. We also show...

Decision Diagrams (DDs) are used in many applications in CAD. Various types of DDs, e.g. BDDs, FDDs, KFDDs, differ by their decomposition types. In this paper we investigate the different decomposition types and prove that there are only three that really help to reduce the size of DDs. 1 Introduction Decision Diagrams (DDs) are successfully applied in many fields of design automation, e.g. [17, 4, 1, 14, 7, 24, 11, 2, 9]. The most popular type of DD is the Ordered Binary Decision Diagram (OBDD) allowing efficient representation and manipulation of Boolean functions [5]. The more recent techniques have made it possible to handle (some) large functions without any basic variation of the OBDD concept itself. The dynamic variable ordering with sifting introduced by Rudell [21] allows to represent examples which could not be represented by any previous heuristic methods. Moreover, the variable ordering in [21] is handled by the package itself, alleviating the need for variable ordering ...

. The first result concerns branching programs having width (log n) O(1) . We give an \Omega\Gamma n log n= log log n) lower bound for the size of such branching programs computing almost any symmetric Boolean function and in particular the following explicit function: "the sum of the input variables is a quadratic residue mod p" where p is any given prime between n 1=4 and n 1=3 . This is a strengthening of previous nonlinear lower bounds obtained by Chandra, Furst, Lipton and by Pudl'ak. We mention that by iterating our method the result can be further strengthened to \Omega\Gamma n log n). The second result is a C n lower bound for read-once-only branching programs computing an explicit Boolean function. For n = \Gamma v 2 \Delta , the function computes the parity of the number of triangles in a graph on v vertices. This improves previous exp(c p n) lower bounds for other graph functions by Wegener and Z'ak. The result implies a linear lower bound for the space comp...

We first consider so-called (1; +s)-branching programs in which along every consistent path at most s variables are tested more than once. We prove that any such program computing a characteristic function of a linear code C has size at least 2\Omega\Gamma2/1 , where d 1 and d 2 are the minimal distances of C and its dual C : We apply this criterion to explicit linear codes and obtain a super-polynomial lower bound for s = o(n= log n): Then we introduce a natural generalization of read-k-times and (1; +s)- branching programs that we call semantic branching programs. These programs correspond to corrupting Turing machines which, unlike eraser machines, are allowed to read input bits even illegally, i.e. in excess of their quota on multiple readings, but in that case they receive in response an unpredictably corrupted value. We generalize the above-mentioned bound to the semantic case, and also prove exponential lower bounds for semantic read-once nondeterministic branching programs.

We show that many "critically constrained" Random 3SAT knowledge bases (KBs) can be compiled into disjunctive normal form easily by using a variant of the "Davis-Putnam" proof procedure. From these compiled KBs we can answer all queries about entailment of conjunctive normal formulas, also easily --- compared to a "bruteforce " approach to approximate knowledge compilation into unit clauses for the same KBs. We exploit this fact to develop an aggressive hybrid approach which attempts to compile a KB exactly until a given resource limit is reached, then falls back to approximate compilation into unit clauses. The resulting approach handles all of the critically constrained Random 3SAT KBs with average savings of an order of magnitude over the brute-force approach. Introduction Consider the task of reasoning from a propositional knowledge base (KB) F which is expressed as a conjunctive normal formula (CNF). We are given other, query CNFs Q 1 ; Q 2 ; : : : ; QN and asked, for each Q i ,...

Branching programs (b. p.'s) or decision diagrams are a general graph-based model of sequential computation. B.p.'s of polynomial size are a nonuniform counterpart of LOG.

In this paper, a simple technique which unifies the known approaches for proving lower bound results on the size of deterministic, nondeterministic, and randomized OBDDs and kOBDDs is described.