Shuzo Yajima

Optimizing OBDDs Is Still Intractable for Monotone Functions

Optimizing the size of Ordered Binary Decision Diagrams is shown to be NP-complete for monotone Boolean functions. The same result for general Boolean functions was obtained by Bollig and Wegener recently.

Exponential Lower Bounds on the Size of OBDDs Representing Integer Divistion

An Ordered Binary Decision Diagram (OBDD) is a directed acyclic graph representing a Boolean function. The size of OBDDs largely depends on the variable ordering. In this paper, we show the size of the OBDD representing the i-th bit of the output of n-bit/n-bit integer division is Ω(2 (n-i)/8 ) for any variable ordering. We also show that V-OBDDs, Λ-OBDDs and ⊕-OBDDs representing integer division has the same lower bounds on the size. We develop new methods for proving lower bounds on the size of V-OBDDs, Λ-OBDDs and ⊕-OBDDs.

Size and Variable Ordering of OBDDs Representing Treshold Functions

An ordered binary decision diagram (OBDD) is a graph representation of a Boolean function, and it is considered as a restricted branching program. According to its good properties, an OBDD is widely used in computer aided logic design. In this paper, the size of ordered binary decision diagrams representing threshold functions is discussed. First, we prove an Ω(n2cn1-e) lower bound on the OBDD size necessary to represent any threshold function when the variable ordering can be chosen adaptively to minimize the OBDD size. Next, we show that it is not possible to find a good variable ordering only from the total order of weights, that is, for any variable ordering of this kind, there exists a threshold function that requires an exponential size OBDD, but is represented in polynomial size by the optimal variable ordering.

Tree-shellability of Boolean functions

In this paper, we define tree-shellable and ordered tree-shellable Boolean functions. A tree-shellable function is a positive Boolean function such that the number of prime implicants equals the number of paths from the root node to a 1-node in its binary decision tree representation. A tree-shellable function is easy to dualize and good for a kind of reliability computation. We show their basic properties and clarify the relations between several shellable functions, i.e. shellable, tree-shellable, ordered tree-shellable, aligned and lexico-exchange functions. We also discuss on tree-shellable quadratic functions.

Hardness of identifying the minimum ordered binary decision diagram

An ordered binary decision diagram (OBDD) is a graph representation of a Boolean function. We consider minimum OBDD identification problems: given positive and negative examples of a Boolean function, identify the OBDD with minimum number of nodes (or with minimum width) that is consistent with all the examples. We prove in this paper that the problems are NP-complete. The result implies that f(n)-width OBDD and f(n)-node OBDD are not learnable for some fixed f(n) under the PAC-learning model unless NP = RP. We also show that the problems are still NP-hard even if we restrict the functions to monotone functions.