Abstract-Given a set of logic primitives and a Boolean function, exact synthesis finds the optimum representation (e.g., depth or size) of the function in terms of the primitives. Due to its high computational complexity, the use of exact synthesis is limited to small networks. Some logic rewriting algorithms use exact synthesis to replace small subnetworks by their optimum representations. However, conventional approaches have two major drawbacks. First, their scalability is limited, as Boolean functions are enumerated to precompute their optimum representations. Second, the strategies used to replace subnetworks are not satisfactory. We show how the use of exact synthesis for logic rewriting can be improved. To this end, we propose a novel method that includes various improvements over conventional approaches: (i) we improve the subnetwork selection strategy, (ii) we show how enumeration can be avoided, allowing our method to scale to larger subnetworks, and (iii) we introduce XOR Majority Graphs (XMGs) as compact logic representations that make exact synthesis more efficient. We show a 45.8% geometric mean reduction (taken over size, depth, and switching activity), a 6.5% size reduction, and depth · size reductions of 8.6%, compared to the academic state-of-the-art. Finally, we outperform 3 over 9 of the best known size results for the EPFL benchmark suite, reducing size by up to 11.5% and depth up to 46.7%.
Exact synthesis is a versatile logic synthesis technique with applications to logic optimization, technology mapping, synthesis for emerging technologies, and cryptography. In recent years, advances in SAT solving have led to a heightened research effort into SAT-based exact synthesis. Advantages of exact synthesis include the use of various constraints (e.g., synthesis of emerging technology circuits). However, although progress has been made, its runtime remains unpredictable. This paper identifies two key points as hurdles to further progress. First, there are open questions regarding the design and implementation of exact synthesis systems, due to the many degrees of freedom. For example, there are different CNF encodings, different symmetry breaks to choose from, and different encodings may be suitable for different domains. Second, SAT-based exact synthesis is difficult to parallelize. Indeed, this is a common drawback of logic synthesis algorithms. This paper proposes four ways to close some open questions and to reduce runtime: 1) quantifying differences between CNF encoding schemes and their impacts on runtime; 2) demonstrating impact of symmetry breaking constraints; 3) showing how directed acyclic graph topology information can be used to decrease runtime; and 4) showing how topology information can be used to leverage parallelism.
We present a collection of modular open source C++ libraries for the development of logic synthesis applications. These libraries can be used to develop applications for the design of classical and emerging technologies, as well as for the implementation of quantum compilers. All libraries are well documented and well tested. Furthermore, being header-only, the libraries can be readily used as core components in complex logic synthesis systems.
We consider the problem of decomposing monotone Boolean functions into majority-of-three operations, with a particular focus on decomposing the majority-n function. When targeting monotone Boolean functions, Shannon's expansion can be expressed by a single majority-of-three operation. We exploit this property to transform binary decision diagrams (BDDs) for monotone functions into majority-inverter graphs (MIGs), using a simple one-to-one mapping. This process highlights desirable properties for further majority graph optimization, e.g., symmetries between the inputs of primitive operations, which are not apparent from BDDs. Although our construction yields a quadratic upper bound on the number of majority-3 operations required to realize majority-n, for small n the concrete values are much smaller compared to those obtained from previous constructions which have linear and quasi-linear asymptotic upper bounds. Further, we demonstrate that minimum size MIGs, for the monotone functions majority-5 and majority-7, can be obtained applying a small number of algebraic transformations to the BDD.
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