(1) Algebraic factorization F = A*B + C, where supp(A) and supp(B) are disjointFrom the above, it is clear that (1) is the special case of (2), and (3) is a special case of (4), if supports of A and B are disjoint.(2) Boolean factorization F = A*B + C, where supp(A) and supp(B) are not disjoint
(3) Simple algebraic decomposition F = A (op) B, where (op) is any operation and supp(A) and supp(B) are disjoint. Another name for this decomposition is disjoint-support bi-decomposition
(4) Non-algebraic (general) decomposition F = A (op) B, where (op) is still any operation, but supp(A) and supp(B) are not disjoint. Another name for this decomposition is bi-decomposition. If supp(A) = supp(F) or supp(B) = supp(F), this bi-decomposition is called weak; otherwise, it is strong. The concept of non-algebraic strong/weak bi-decomposition has been introduced in D.Bochmann, F.Dresig, B.Steinbach, “A new decomposition method for multilevel circuit design”. Proc. of Euro-DAC 1991, pp. 374 – 377.
(5) Curtis decomposition F = H( G(X), Y )
(5a) X and Y are disjoint sets of variablesNotice that, of all possible combinations of (5), (5b) subsumes (5a), and (5d) subsumes (5c); other combinations can happen, for example: (5a) can go with (5c), (5b) can go with (5d), etc.
(5b) X and Y are non-disjoint sets of variables
(5c) G has single binary output, also known as Ashenhurst decomposition
(5d) G has two or more binary outputs
The relationship of (4) and (5) is somewhat more complex in nature. On the one hand, (4) can be considered a special case of (5) if H is a two-input function, function G has two components, G1 = A and G2 = B and the variable set Y is empty. On the other hand, (5) can be considered a special case of (4), if we perform (4) to get the network on two-input gates, and then selectively collapse some of the gates to get function H, while the functions, which feed into H, will be considered as Gi's.
Therefore, structurally considered, (4) and (5) are equally powerful, but the actual efficiency depends on a particular implementation. If implementation is efficient it explores large search space in short time. Some trade-offs are possible. More search space with very long computation time; less search space but very quickly... The results also depend on the type of the benchmarks.
The remark about (4) and (5) being equally powerful is very important! Because Roth/Karp (I), Lai/Pedram (III), Chang/Marek-Sadowska (VII), etc. (see below) belong to (5), while Yang/Ciesielski (IX) and Mishchenko/Steinbach/Perkowski (X) belong to (4). I personally believe that (4) gives more freedom for nice trade-offs, and therefore, (4) will eventually outperform (5), including the case of decomposition for FPGA, which so far has been always approached using (5). But this is a very strong statement (which sounds more lilke my wishful thinking), and the future will show whether it is true.
Decomposition and Factorization Methods
Now we can review the decomposition methods and see how they fit into this framework. The order is chronological (roughly).
(I) Roth/Karp method [IBM J. Res. Dev. 1962] performs (5a)+(5c) or (5a)+(5d). As a particular case, Roth/Karp can end up with (3) if H has two inputs (but not with (4), because (4) is part of (5b), and in Roth/Karp we can only have (5a)!). In relation to the approaches described below, we can note that Roth/Karp method, we can have a column composed of only zeros [00...0] and only ones [11...1]. It means that one of the subfunctions Gi is a constant zero/one function.The decomposition methods (III) and (IV) can detect simple algebraic decompositions (3), if the bound set and the free set are separated in the variable order. For Stanion/Sechen, free set should be on top. For other methods, bound set should be on top. However, if the variable ordering does not separate bound set and free set, none of these methods can detect algebraic decomposition. BUT - and this is very important - Yang/Ciesielski method (IX) can detect algebraic decompositions even when the variable order is not favorable. For a simple example, consider F = ab + cd with variable order (a,c,b,d). You will see that by making a cut between (a,c,b) and d, you can still detect the algebraic OR-bi-decomposition.(II) Brayton/McMullen approach [ISCS'82] to perform algebraic factorization by deriving kernels and co-kernels clearly belongs to (1). Its generality is limited. It has been improved upon many times, but it is easy to implement, it is fast and reasonably efficient on many benchmarks. Therefore, as of today, it is perhaps the most widely-used factorization/decomposition method.
Several important improvements of this method have been proposed, in particular:
J. Rajski, J. Vasudevamurthy, “The Test-Preserving Concurrent Decomposition and Factorization of Boolean Expressions”, IEEE Trans. CAD, Vol.11 (6), June 1992, pp.778-793.
This method is implemented in SIS and gives very good results but it is still algebraic factorization (1).
H. Sawada, Sh. Yamashita, A. Nagoya. An Efficient Method for Generating Kernels on Implicit Cube Set Representations. Proc. of IWLS’ 99, pp. 260-263.
This method can generate all kernels using a ZDD representation. Even though the experimental results are not impressive, I believe, it is because this method is not properly integrated with other ideas related to (1), in particular with the Rajski/Vasudevamurthy method mentioned above.
(III) Lai/Pedram method [DAC'93] performs (5a)+(5c) or (5a)+(5d). Lai/Pedram with extended definition of cut set [DAC'93] also performs non-disjoint (in their paper, it is called "non-disjunctive") decomposition (5b)+(5c) or (5b)+(5d). As particular cases, the simple method can create (3) and the extended method can create (4).
(IV) Stanion/Sechen algorithm [DAC'95] performs (5b). In particular, when the cut is horizontal, it performs (5a). It can be both (5c) and (5d) depending on how many nodes are in the cut. If there are only two nodes, it is (5c), if more than two, it is (5d).
(IVa) Stanion/Sechen method [IEEE TCAD '94 "Boolean Division and Factorization Using Binary Decision Diagrams", Vol. 13, No.9, 1994, pp. 1179-1184.] This method belongs to (2). It is based on a specialized BDD operator "interval cofactor" to find good boolean divisors of a function. It uses don't-cares.
(V) Bertacco/Damiani algorithm originally published [ICCAD'97] to perform (5a)+(5c). Later an extension was proposed [IWLS'98] to handle (5a)+(5d). In my opinion, this extension is too complicated to be practical. In any case, this method cannot do (5b). It can only perform disjoint decomposition (calling it "disjunctive"). (You can find the above publications on Valeria Bertacco's webpage.)Notice that the concept of a cut used by Yang/Ciesielski (IX) is different from that of Lai/Pedram (III) and Stanion/Sechen (IV). In both cases, a cut denotes a subset of nodes. The previous authors specified a cut by enumerated a set of the BDDs nodes. In particular, they could have a cut composed of the BDD nodes that have both their predecessor and successor nodes in the BDD not belonging to the(VI) Sasao developed several decomposition approaches, in particular:
[IWLS'97] - (3)
[IWLS'98] - (5a)+(5c) (good approach but very slow)
[IWLS'99] - (5b)+(5d) (interesting approach but explores limited search space(?))
[IWLS'00] - (5a)+(5d) for symmetric functions only
(You can find these publications on Dr. Sasao's webpage.)(VII) Chang/Marek-Sadowska method [IEEE TCAD'96] is a curious mixture of many methods, which is more or less equivalent to (5b)+(5d). The distinctive feature of this approach that, unlike many of the above listed, it is capable of handling don't-cares.
(VIII) Files/Perkowski extended Curtis decomposition for multi-valued functions and created a number of new MDD-based algorithms to perform this decomposition with and without encoding intermediate signals. (Find the references on Craig's webpage: http://www.ece.pdx.edu/~cfiles/papers.html).
(VIIIa) Grygiel/Perkowski [DAC'99] used graph-coloring and multi-valued intermediate signals to perform decomposition of boolean functions. The represented functions using Labeled Rough Partitions (LRPs) and employed BDDs to implement the LRPs, but they did not use BDDs to direct the decomposition.
(It is possible to add Karplus here or in the chronological order. I do not have his paper, but generally it looks like (3) without EXOR gates.)
(IX) Yang/Ciesielski performed (4) using the generalized dominators. In the most general case, there is no connection between the approach with generalized 0-, 1-, or X-dominators and the Roth/Karp decomposition table, because Roth/Karp performs (5a), while Yang/Ciesielski perform (4), which, as discussed above, is equivalent to (5)=(5a)+(5b). In fact, Yang/Ciecielski is more general than Roth/Karp.
Disadvantages:
(a) does not always give preference to strong decomposition over the weak decomposition resulting in delay degradationAdvantages:
(b) the use of don't-cares is limited only to internally generated and only through the BDD minimization
(c) decompositions are limited to those detectable using one, albeit optimal, variable order and only horizontal cuts; meanwhile, other variables orders and non-horizontal cuts can possibly give better decompositions
(d) the common logic extraction performed on the resulting factored trees is not efficient because only completely specified functions of the tree nodes are used (representing nodes as incompletely-specified functions would produce freedom to find more compatible subfunctions)(a) as of today, it is the fastest way of detecting decompositions on the BDD structure
(b) it can find some of the decompositions that cannot be found by other BDD-based methods (see remark above about algebraic decomposition)
(c) is can be extended to make full use of don't-cares, non-horizontal cuts, and efficient common logic extraction. The latter is the goal of our present work directed towards bringing together (IX) and (X).
The cuts, as defined by Yang/Ciesielski, on the other hand, are specified
by an imaginary surface that intersects every BDD path as the most once,
and separates the BDD nodes into those above the surface (the cut nodes)
and those below the surface (the nodes that does not belong to the cut).
Notice that, in this case, it is not possible that a node belongs to the
cut while both its predecessors and
successors do not.
An additional restriction is that cuts defined by Yang/Ciesielski to generate 0- and 1- dominators always intesects one or more edges pointing to terminal nodes (this property may not be true for cuts used to generate x-dominators, as pointed out by C. Yang).
(X) Mishchenko/Steinbach/Perkowski [DAC'01] performed (4) using formulas with quantifiers.This classification and description may be incomplete and have inaccuracies, which I hope I will fix as time goes on.Disadvantages:
(a) excessive runtime for functions with more than 30 variablesAdvantages:
(b) the generated components A and B have the smallest possible support but they may not be well-decomposable, because cost function does not take into account the internal structure of the decomposed blocks(a) if the strong decomposition exists, it is always used before weak decomposition is used
(b) the netlist is well balanced resulting in short delay
(c) flexible in the use of don't cares: uses both the external and internally generated don't-cares, which are propagated as the decomposition proceeds
(d) some of the logic is efficiently shared across the logic cones
(e) the netlist is provably 100% testable for single stuck-at faults
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