Functional Decision Diagrams Research Articles

Novel Reduction Methods for Decision Diagrams

We propose a novel method of reduction for binary-based decision diagrams (DD) exploiting the similarities between Boolean functions. Conventional methods are able to remove redundant parts of DD that adhere to (or represent) identical structures. The proposed approach tries to take advantage of not only identical but also equivalent parts of a diagram, which produce identical results after a change in variable ordering. Basic types of DD, like Binary DD (BDD) or Functional DD (FDD) do not allow for the reduction of equivalent parts of a diagram as the notation of change in ordering would be lost. Therefore, we propose a new type of decision diagram able to incorporate different orders of avariables and thus allowing the reduction of equivalent DD parts – Multi-Variable Decision Diagram (MVDD). The only remaining obstacle is the determination of equivalency, for which we also propose a new heuristic. The paper also presents the reduction of Kronecker Functional DD (KFDD) and MVDD with regard to multiple parameters, where the optimization process relies heavily on evolutionary algorithms and Residual Variable (RV). The average added value of MVDD to the reduction of diagram size was 26.13% when compared to BDD and 12.27% when compared to KFDD while preserving similar values of power consumption and average path length in the diagram.

Reversible Circuit Synthesis Method Using Sub-graphs of Shared Functional Decision Diagrams

Abstract Reversible circuit synthesis methods based on decision diagrams achieve low quantum costs but do not account for quantum bit (qubit) limits for the application of reversible logic in quantum computing. Here, a synthesis method using sub-graphs of shared functional decision diagrams (SFDDs) is proposed for reducing the number of lines when synthesizing reversible circuits. An SFDD is partitioned into sub-graphs by exploiting the longest dominant-active paths, and the sub-graphs are mapped to reversible gate cascades. To further reduce the number of lines, template root matching is presented for reusing circuit lines. Experimental results indicate that the proposed method achieves the known minimum number of lines in many cases and has good scalability. Although the proposed method increases the quantum cost over a prior method based on functional decision diagrams, it significantly reduces the number of lines in most cases. Compared with the one-pass method using quantum multiple-valued decision diagrams, the proposed method reduces the quantum cost without increasing the number of lines in many cases. When compared with the lookup table-based method using a direct mapping flow, the method reduces the number of lines in a few cases. Thus, the method aids in the physical realization of a quantum circuit.

Synthesis of Reversible Circuits with Reduced Nearest-Neighbor Cost Using Kronecker Functional Decision Diagrams

Synthesis of Reversible Circuits with Reduced Nearest-Neighbor Cost Using Kronecker Functional Decision Diagrams

Reversible Circuits Synthesis from Functional Decision Diagrams by using Node Dependency Matrices

Decision diagrams are a data structure suitable for reversible circuit synthesis. Functional decision diagrams (FDDs) are particularly convenient in synthesis with Toffoli gates, since the functional expressions for decomposition rules used in them are similar to the functional expressions of Toffoli gates. The main drawback of reversible circuit synthesis based on decision diagrams is the usually large number of ancilla lines. This paper presents two methods for the reduction of the number of ancilla lines in reversible circuits derived from FDDs by selecting the order of implementation of nodes. In the first method, nodes are implemented by levels, starting from the bottom level to the top. The method uses appropriately defined level dependency matrices for choosing the optimal order of implementation of nodes at the same level. In this way, the optimization is performed level by level. The second method uses a diagram dependency matrix expressing mutual dependencies among all the nodes in the diagram. This method is computationally more demanding than the first method, but the reductions of both the number of lines and the Quantum cost of the circuits are larger.

Bi-Kronecker Functional Decision Diagrams: A Novel Canonical Representation of Boolean Functions

In this paper, we present a novel data structure for compact representation and effective manipulations of Boolean functions, called Bi-Kronecker Functional Decision Diagrams (BKFDDs). BKFDDs integrate the classical expansions (the Shannon and Davio expansions) and their bi-versions. Thus, BKFDDs are the generalizations of existing decision diagrams: BDDs, FDDs, KFDDs and BBDDs. Interestingly, under certain conditions, it is sufficient to consider the above expansions (the classical expansions and their bi-versions). By imposing reduction and ordering rules, BKFDDs are compact and canonical forms of Boolean functions. The experimental results demonstrate that BKFDDs outperform other existing decision diagrams in terms of sizes.

An improved KFDD based reversible circuit synthesis method

Since reversible logic has promising applications in domains like low-power design and quantum computing, it is necessary to efficiently design high-performance reversible logic circuits. In this paper, in order to reduce the number of qubits while achieving low quantum cost, the Kronecker functional decision diagram (KFDD) based reversible circuit synthesis method is improved by using two strategies. One is to generate a locally optimal reversible cascade for each of the KFDD nodes by performing transformations on the function represented by a node and using a gate library consisting of NOT, CNOT, Toffoli gates and mixed-polarity Peres gates. This strategy helps reduce the quantum cost and the number of qubits. The other is to map KFDD nodes to their locally optimal reversible cascades level by level via breadth-first traversal of the KFDD. This strategy reduces the number of qubits by using the circuit line labeled by an input variable of the KFDD as the target line of a reversible gate. Comparison results are presented in tabular and graphical forms. Compared to the existing decision diagram based reversible circuit synthesis methods, the proposed method can reduce both the quantum cost and the number of qubits in many cases. Compared to other state-of-the-art synthesis methods for reversible circuits, the proposed method can achieve much lower quantum cost but still incurs a high number of qubits for many functions. However, it can achieve the minimum number of qubits for a few irreversible functions while achieving much lower quantum cost. In addition, the proposed method is very time efficient.

Genetic algorithm for binary and functional decision diagrams optimization

Decision diagrams (DD) are a widely used data structure for discrete functions representation. The major problem in DD-based applications is the DD size minimization (reduction of the number of nodes), because their size is dependent on the variables order. Genetic algorithms are often used in different optimization problems including the DD size optimization. In this paper, we apply the genetic algorithm to minimize the size of both Binary Decision Diagrams (BDDs) and Functional Decision Diagrams (FDDs). In both cases, in the proposed algorithm, a Bottom-Up Partially Matched Crossover (BU-PMX) is used as the crossover operator. In the case of BDDs, mutation is done in the standard way by variables exchanging. In the case of FDDs, the mutation by changing the polarity of variables is additionally used. Experimental results of optimization of the BDDs and FDDs of the set of benchmark functions are also presented.

Complexity reduction of Toffoli networks based on FDD

Synthesis of switching functions by Toffoli gates has become a very important research topic in the last years, since Toffoli gates are used in the synthesis of reversible circuits. Early methods based on the truth-table representation of Boolean functions are applicable to functions with a relatively small number of variables. Later on, methods for synthesis by Toffoli gates based on decision diagrams (BDDs, FDDs or OKFDDs) were introduced and applied to the synthesis of both reversible and irreversible functions. This paper presents a method for the reduction of the number of lines and gates in the Toffoli gate realization of Boolean functions based on their Functional Decision Diagram (FDD) representation. Experiments show that, when the proposed reduction is used, the realization of the given function based on FDD will, on the average, be smaller in terms of the number of lines and the number of gates than the realizations based on an OKFDD, an optimal BDD or based on a FDD by using previously defined algorithms.

Hierarchical Verification of Galois Field Circuits

This paper proposes a hierarchical method for the formal hardware verification of Galois field architecture circuits. The reduced ordered functional decision diagram has been explored. The proposed method has been found to lead to significant gains in time and space, depending on the resources that are available. The theoretical claims that were made have been supported by experiments.

Ordered binary decision diagrams and the Shannon effect

We investigate the size and structure of ordered binary decision diagrams (OBDDs) for random Boolean functions. It was known that for most values of n, the expected OBDD size of a random Boolean function with n variables is equal to the worst-case size up to terms of lower order. Such a phenomenon is generally called strong Shannon effect. Here we show that the strong Shannon effect is not valid for all n. Instead it undergoes a certain periodic ‘phase transition’: If n lies within intervals of constant width around the values n=2 h + h, then the strong Shannon effect does not hold, whereas it does hold outside these intervals. Our analysis provides doubly exponential probability bounds and generalises to ordered Kronecker functional decision diagrams (OKFDDs).

OPKFDD 최소화를 위한 노드의 확장형 결정

OPKFDD(Ordered Pseudo-Kronecker Functional Decision Diagram)는 각 노드에서 다양한 확장방법(decomposition)을 취할 수 있는 Ordered-DD(Decision Diagram)의 한 종류로서 각 노드마다 Shannon, positive Davio, 그리고 negative Davio 확장중의 하나를 사용하도록 하며 다른 종류의 DD와 비교해서 작은 수의 노드로 함수를 표현할 수 있다. 그러나 각 노드마다 각기 다른 확장 방법을 선택할 수 있는 특징 때문에 입력 노드에 대한 확장 방법의 결정에 의해서 OPKFDD의 크기가 좌우되며 최소의 노드 수를 갖는 OPKFDD의 구성은 매우 어려운 문제로 알려져 있다. 본 논문에서는 DD 크기의 기준을 노드 수로 하여 기존의 OBDD(Ordered Binary Decision Diagram) 자료구조에서 각 노드의 확장방법을 결정하는 직관적(heuristic)인 방법을 제시하고, 주어진 입력변수 순서에 대해서 각 노드의 확장 방법을 결정하는 알고리즘을 제안하고 실험 결과를 제시한다. OPKFDD (Ordered Pseudo-Kronecker Functional Decision Diagram) is one of ordered-DDs (Decision Diagrams) in which each node can take one of three decomposition types : Shannon, positive Davio and negative Davio decompositions. Whereas OBDD (Ordered Binary Decision Diagram) uses only the Shannon decomposition in each node, OPKFDD uses the three decompositions and generates representations of functions with smaller number of nodes than other DDs. However, this leads to the extreme difficulty of getting an optimal solution for the minimization of OPKFDD. Since an appropriate decomposition type has to be chosen for each node, the size of the representation is decided by the selection of the decomposition type. We propose a heuristic method to generate OPKFDD efficiently from the OBDD of the given function and the algorithm of the decision of decomposition type for a given variable ordering. Experimental results demonstrate the performance of the algorithm.

A Unifying Approach to Edge-valued and Arithmetic Transform Decision Diagrams

This paper shows that binary decision diagrams (BDDs) and their generalizations are not only representations of switching and integer-valued functions, but also Fourier-like series expansions of them. Furthermore, it shows that edge-valued binary decision diagrams (EVBDDs) are related to arithmetic transform decision diagrams (ACDDs), which are the integer counterparts of the functional decision diagrams (FDDs). Finally, it shows that the complexity of multi-terminal binary decision diagrams (MTBDDs), EVBDDs and ACDDs of a function i>f depends on the structure of the truth-vector of i>f, partial arithmetic transform spectra of i>f and the arithmetic transform spectrum of i>f, respectively.

Efficient Algorithms for Creation of Linearly‐independent Decision Diagrams and their Mapping to Regular Layouts

A new kind of a decision diagrams are presented: its nodes correspond to all types of nonsingular expansions for groups of input variables, in particular pairs. The diagrams are called the Linearly Independent (LI) Decision Diagrams (LI DDs). There are 840 nonsigular expansions for a pair of variables, thus 840 different types of nodes in the tree. Therefore, the number of nodes in such (exact) diagrams is usually much smaller than the number of nodes in the well‐known Kronecker diagrams (which have only single‐variable Shannon, Positive Davio, and Negative Davio expansions in nodes). It is usually much smaller than 1/3 of the number of nodes in Kronecker diagrams. Similarly to Kronecker diagrams, the LI Diagrams are a starting point to a synthesis of multilevel AND/OR/EXOR circuits with regular structures. Other advantages of LI diagrams include: they generalize the well‐known Pseudo‐Kronecker Functional Decision Diagrams, and can be used to optimize the new type of PLAs called LI PLAs. Importantly, while the known decision diagrams used AND/EXOR or AND/OR bases, the new diagrams are AND/OR/EXOR‐based. Thus, because of a larger design space, multi‐level structures of higher regularity can be created with them. This paper presents both new concepts and new efficient synthesis algorithms.

Fast conversion algorithm for very large Booleanfunctions

Fixed polarity Reed-Muller (FPRM) expressions are alternatives to the traditional sum-of-products forms (SOPs) for the representation of Boolean functions. A fast algorithm is proposed to convert from SOPs to FPRM forms without generating disjoint cube covers or functional decision diagrams (FDDs). This procedure is based on the property of input redundancy and is tailored for very large multiple output Boolean functions. Test results for benchmark examples of up to 199 inputs and 99 outputs are given.

Testability and test compaction for decision diagram circuits

Novel node realisations are used to represent reduced ordered binary, functional and Kronecker decision diagrams as irredundant combinational circuits. These types of decision diagrams are compact and able to represent very large switching functions. The resulting circuits are completely testable in the single and multiple stuck-at fault models. Theorems on diagnostic properties of these circuits are formulated. The high complexity of functions represented by decision diagram circuits calls for efficient test pattern generation and test set compaction tools. Such tools were implemented using a genetic algorithm which maximises the number of potential faults covered by every test pattern. Experimental results for decision diagrams with up to 1763 inputs and 5329 nodes are presented. The efficiency of the genetic algorithm in this application is estimated.

OKFDD minimization by genetic algorithms with application to circuit design

A genetic algorithm (GA) is applied to minimize ordered Kronecker functional decision diagrams (OKFDDs), i.e. a graph-based representation of Boolean functions. OKFDDs are then used to derive circuits by substituting the nodes in the graph. The GA is first used to minimize the size of OKFDDs by choosing a good variable ordering and decomposition type list in parallel. Various GA parameters are studied, including population size and several genetic operators. Then the GA is used for the construction of OKFDD circuits that dependent on the chosen parameters trade off area efficiency versus testability. It is shown by experiments that the algorithm outperforms the best previously published heuristic approach. Finally, we apply our minimization algorithm to technology mapping for FPGAs. Also for this application experimental results are given to show the efficiency of our approach.

The complexity of the inclusion operation on OFDD's

Ordered functional decision diagrams (OFDD's) are a data structure for representation and manipulation of Boolean functions. A polynomial time algorithm for the test whether f/spl les/g for functions f and g given by OFDD's is presented. This is the last basic operation on OFDD's whose complexity was unknown. The result also solves an open question on the complexity of minimizing OFDD's for incompletely specified functions.

On variable ordering and decomposition type choice in OKFDDs

We present methods for the construction of small ordered Kronecker Functional Decision Diagrams (OKFDDs). OKFDDs are a generalization of ordered binary decision diagrams (OBDDs) and ordered functional decision diagrams (OFDDs) as well. Starting with an upper bound for the size of an OKFDD representing a tree-like circuit, we develop different heuristics to find good variable orderings and decomposition types for OKFDDs representing two-level and multilevel circuits, respectively. Experimental results are presented to show the efficiency of our approaches.

Ordered Kronecker functional decision diagrams-a data structure for representation and manipulation of Boolean functions

Ordered Kronecker functional decision diagrams (OKFDD's) are a data structure for efficient representation and manipulation of Boolean functions. OKFDD's are a generalization of ordered binary decision diagrams (OBDD)s) and ordered functional decision diagrams and thus combine the advantages of both. In this paper, basic properties of OKFDD's and their efficient representation and manipulation are given. Starting with elementary manipulation algorithms, we present methods for the construction of small OKFDD's. Our approach is based on dynamic variable ordering and decomposition-type choice. For changing the decomposition type, we use an efficient reordering-based method. We briefly discuss the implementation of PUMA, an OKFDD package, which was used in all our experiments. These experiments demonstrate the quality of our methods in comparison to sifting and interleaving for OBDD's.

On the Expressive Power of OKFDDs

Ordered Decision Diagrams (ODDs) as a means for the representation of Boolean functions are used in many applications in CAD. Depending on the decomposition type, various classes of ODDs have been defined, among them being the Ordered Binary Decision Diagrams (OBDDs), the Ordered Functional Decision Diagrams (OFDDs) and the Ordered Kronecker Functional Decision Diagrams (OKFDDs). Based on a formalization of the concept decomposition type we first investigate all possible decomposition types and prove that already OKFDDs, which result from the application of only three decomposition types, result in the most general class of ODDs. We then show from a (more) theoretical point of view that the generality of OKFDDs is really needed. We prove several exponential gaps between specific classes of ODDs, e.g. between OKFDDs on the one side and OBDDs, OFDDs on the other side. Combining these results it follows that a restriction of the OKFDD concept to subclasses, such as OBDDs and OFDDs as well, results in families of functions which lose their efficient representation.