Ordered Binary Decision Diagrams (OBDDs) have found widespread use in CAD applications such as formal verification, logic synthesis, and test generation. OBDDs represent Boolean functions in a form that is both canonical and compact for many practical cases. They can be generated and manipulated by efficient graph algorithms. Researchers have found that many tasks can be expressed as series of operations on Boolean functions, making them candidates for OBDD-based methods. The success of OBDDs has inspired efforts to improve their efficiency and to expand their range of applicability. Techniques have been discovered to make the representation more compact and to represent other classes of functions. This has led to improved performance on existing OBDD applications, as well as enabled new classes of problems to be solved. This paper provides an overview of the state of the art in graph-based function representations. We focus on several recent advances of particular importance for forma...

We define the notion of a randomized branching program in the natural way similar to the definition of a randomized circuit. We exhibit an explicit function fn for which we prove that: 1) f n can be computed by polynomial size randomized read-once ordered branching program with a small one-sided error; 2) fn cannot be computed in polynomial size by deterministic readonce branching programs; 3) fn cannot be computed in polynomial size by deterministic read- k-times ordered branching program for k = o(n= log n) (the required deterministic size is exp \Gamma\Omega \Gamma n k \Delta\Delta ).

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.

To analyze the complexity of decision problems on graphs, one normally assumes that the input size is polynomial in the number of vertices. Galperin and Wigderson [GW83] and, later, Papadimitriou and Yannakakis [PY86] investigated the complexity of these problems when the input graph is represented by a polylogarithmically succinct circuit. They showed that, under such a representation, certain trivial problems become intractable and that, in general, there is an exponential blow up in problem complexity. Later, Balc'azar, Lozan, and Tor'an [Bal96, BL89, BLT92, Tor88] extended these results to problems whose inputs were structures other than graphs. In this paper, we show that, when the input graph is represented by a ordered binary decision diagram (OBDD), there is an exponential blow up in the complexity of most graph problems. In particular, we show that the GAP and AGAP problems become complete for PSPACE and EXP, respectively, when the graphs are succinctly represented by OBDDs. 1...

Bryant [5] has shown that any OBDD for the function MULn−1,n, i.e. the middle bit of the n-bit multiplication, requires at least 2 n/8 nodes. In this paper a stronger lower bound of essentially 2 n/2 /61 is proven by a new technique, using a universal family of hash functions. As a consequence, one cannot hope anymore to verify e.g. 128-bit multiplication circuits using OBDD-techniques because the representation of the middle bit of such a multiplier requires more than 3 · 10 17 OBDD-nodes. Further, a first non-trivial upper bound of 7/3 · 2 4n/3 for the OBDD-size of MULn−1,n is provided.

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This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps read-once branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...

Branching programs (b. p.'s) or decision diagrams are a general graph-based model of sequential computation. The b. p.'s of polynomial size are a nonuniform counterpart of LOG. Lower bounds for different kinds of restricted b. p.'s are intensively investigated. An important restriction are so called k-b. p.'s, where each computation reads each input bit at most k times. Although, for more restricted syntactic k-b.p.'s, exponential lower bounds are proven and there is a series of exponential lower bounds for 1-b. p.'s, this is not true for general (nonsyntactic) k-b.p.'s, even for k = 2. Therefore, so called (1; +k)-b. p.'s are investigated. For some explicit functions, exponential lower bounds for (1; +k)-b. p.'s are known. Investigating the syntactic (1; +k)-b. p.'s, Sieling has found functions f n;k which are polynomially easy for syntactic (1; +k)-b. p.'s, but exponentially hard for syntactic (1; +(k \Gamma 1))-b. p.'s. In the present paper, a similar hierarchy with res...

Abstract Branching programs are a well-established computation model for boolean functions, especially read-once branching programs have been studied intensively. Exponential lower bounds for deterministic and nondeterministic read-once branching programs are known for a long time. On the other hand, the problem of proving superpolynomial lower bounds for parity read-once branching programs is still open. In this paper restricted parity read-once branching programs are considered and an exponential lower bound on the size of well-structured parity graph-driven read-once branching programs for integer multiplication is proven. This is the first strongly exponential lower bound on the size of a nonoblivious parity read-once branching program model for an explicitly defined boolean function. In addition, more insight into the structure of integer multiplication is yielded.

We prove an exponential lower bound (2\Omega\Gamma n= log n) ) on the size of any randomized ordered read-once branching program computing integer multiplication. Our proof depends on proving a new lower bound on Yao's randomized one-way communication complexity of certain boolean functions. It generalizes to some other common models of randomized branching programs. In contrast, we prove that testing integer multiplication, contrary even to nondeterministic situation, can be computed by randomized ordered read-once branching program in polynomial size. It is also known that computing the latter problem with deterministic read-once branching programs is as hard as factoring integers.