Functional decomposition [1, 7, 14, 22] plays an important role in the analysis and design digital systems. It refers to the process of breaking a complex function into parts, which are less complex than the original function and are relatively indepen-dent to each other. In addition the behavior of the original function can be recon-structed if we compose these parts together. In logic synthesis, a complex function is decomposed into a set of sub-functions, such that each sub-function is easier to analyze, to understand, and to further synthesize. Functional decomposition has long been recognized as a pivotal role in LUT-based FPGA synthesis. It also has various applications to the minimization of circuit communication complexity.
1.1. Thesis Overview
Functional decomposition can be classified as follows:
• The function to be decomposed can be a completely or an incompletely speci-fied function.
• The function to be decomposed can be a single- or multiple-output function.
• A decomposition is disjoint if the sub-functions do not share common input variables; otherwise, it is non-disjoint.
• A decomposition is called Ashenhurst decomposition or simple decomposition if the topology of the decomposition is f (X) = h(g(XG), XH), where X = XG ∪ XH and g is a single-output function; a decomposition is called bi-decomposition if the topology is f (X) = h(g1(XA), g2(XB)), where X = XA∪ XB and h is a two-input gate.
In this thesis, our method focuses on completely specified functions. In addition the proposed method deals with multiple-output and non-disjoint decompositions.
Furthermore, we pay our attention to Ashenhurst decomposition.
The functional decomposition problem was first formulated by Ashenhurst in 1959 [1]. He visualized the decomposition feasibility with a decomposition chart, which is a two-dimensional Karnaugh map with rows corresponding to variables in the free set and columns corresponding to variables in the bound set. The decompo-sition chart is used to determine whether a given function f can be simply disjointly
1.1. Thesis Overview
decomposed with respect to a set of given bound set variables. Ashenhurst reduced the chart by merging all identical columns, and he showed that there exists a simple disjoint decomposition if and only if there are at most two distinct columns in the reduced decomposition chart. The disadvantage of Ashenhurst’s method is that we have to construct every chart with respect to every variable partition we consider. If we consider all the variable partitions, a function with n variables would have O(2n) charts since every variable can be in either the bound set or free set.
Roth and Karp proposed a cover-based method [22], trying to reduce the memory requirement of Ashenhurst’s method. They used covers to represent the functions.
By cover manipulations, the minterms of bound set variables will be mapped into equivalence classes. These equivalence classes are in one-to-one correspondence with the distinct columns in Ashenhurst’s method. Nevertheless, the process time of the algorithm is still a problem except we restrict the size of the bound set variables to be up-bounded by some constant k. A typical value of k is 5 which is the common input size of a LUT.
Recent progress on function manipulation using BDDs makes the BDD data structure becomes a popular tool to handle the functional decomposition problem.
Lai, Pedram, and Vrudhula [15] proposed a fast BDD-based method to implement the Roth-Karp algorithm. They used BDDs to represent Boolean functions, and showed that every variable ordering in the BDD implies a variable partition of the decomposition of the function. They ordered all the bound set variables above
1.1. Thesis Overview
the cut, and ordering all the free set variables in and below the cut. Based on Lai’s method, Stanion and Sechen proposed a method [25] to enumerate all the possible cuts in the BDD in order to find a good decomposition. The enumeration can be achieved by constructing a characteristic function for the set of cuts, then using a branch-and-bound procedure in order to find a cut which produces the best decomposition.
Identifying the common sub-functions in decomposing a function set is an im-portant issue in multiple-output functional decomposition. The first approach to multiple-output functional decomposition was proposed by Karp [14]. However Karp’s multiple-output approach works only for two-output functions. In order to overcome the limitation of Karp’s method, researchers proposed several ways to handle the multiple-output decomposition problem. Wurth, Eckl, and Antreich pre-sented an algorithm [29] based on the concept of a preferable decomposition function, which is a decomposition function valid for a single output and with the potential to be shared by other output functions to handle multiple-output decomposition.
They showed that the construction of all preferable decomposition functions can be achieved by partitioning the minterms of bound set variables into some global classes, and this information can be used to identify the common preferable decom-position functions of a multiple-output function.
Sawada, Suyama, and Nagoya proposed a BDD-based algorithm [24] to deal with the issue of identifying common sub-functions. They proposed a effective Boolean
1.1. Thesis Overview
resubstitution technique based on support minimization to effectively identify the common LUTs from a large amount of candidates. However when the size of the support variables is large, the examination becomes time-consuming and sometimes fails due to memory explosion.
Most prior work on functional decomposition used BDD as the underlying data structure. BDD can be used to handle the functional decomposition problem with proper variable ordering, which the size of the BDD is under an acceptable threshold.
However in some cases, the BDD-based algorithm has several limitations:
• Firstly, there are memory explosion problems for BDD-based algorithms. BDD can be of large size in representing a Boolean function. In the worst case, BDD size is exponential to the number of variables. When special variable ordering rules need to be imposed on BDDs for functional decomposition, the memory size needed in BDD computation may not be improved by changing a suitable variable ordering. Therefore it is typical that a function under decomposition using BDD as the underlying data structure can have just a few variables.
• Secondly, variable partitioning needs to be specified a priori, and cannot be au-tomated as an integrated part of the decomposition process. Decomposability under different variable partitions need to be analyzed case by case. In order to enumerate different variable partitions effectively and keep the size of BDD reasonably small, the set of bound set variables cannot be large. Otherwise, the computation time will easily over the pre-specified threshold.
1.1. Thesis Overview
• Thirdly, for BDD-based approaches, non-disjoint decomposition cannot be handled easily. In practical, decomposability needs to be analyzed by cases exponential to the number of joint (or common) variables.
• Finally, even though multiple-output decomposition [30] can be converted to single-output decomposition [12], BDD sizes may grow largely in this conver-sion.
Figure 1.1: Ashenhurst decomposition
The above limitations motivate the need for new data structures and computa-tion methods for funccomputa-tional decomposicomputa-tion. This thesis shows that, when Ashen-hurst decomposition [1] is considered, these limitations can be overcome through satisfiability (SAT) based formulation. Ashenhurst decomposition is a special case of functional decomposition, where, as illustrated in Figure 1.1, a function f (X) is decomposed into two sub-functions h(XH, XC, xg) and g(XG, XC) with f (X) = h(XH, XC, g(XG, XC)). For general functional decomposition, the function g can be