A Sound and Complete Axiomatization of Majority-n Logic

Manipulating logic functions via majority operators recently drew the attention of researchers in computer science. For example, circuit optimization based on majority operators enables superior results as compared to traditional logic systems. Also, the Boolean satisfiability problem finds new solving approaches when described in terms of majority decisions. To support computer logic applications based on majority a sound and complete set of axioms is required. Most of the recent advances in majority logic deal only with ternary majority (MAJ- 3) operators because the axiomatization with solely MAJ-3 and complementation operators is well understood. However, it is of interest extending such axiomatization to n-ary majority operators (MAJ-n) from both the theoretical and practical perspective. In this work, we address this issue by introducing a sound and complete axiomatization of MAJ-n logic. Our axiomatization naturally includes existing majority logic systems. Based on this general set of axioms, computer applications can now fully exploit the expressive power of majority logic.Comment: Accepted by the IEEE Transactions on Computer

Majority-Inverter Graph: A New Paradigm for Logic Optimization

In this paper, we propose a paradigm shift in representing and optimizing logic by using only majority (MAJ) and inversion (INV) functions as basic operations. We represent logic functions by Majority-Inverter Graph (MIG): a directed acyclic graph consisting of three-input majority nodes and regular/complemented edges. We optimize MIGs via a new Boolean algebra, based exclusively on majority and inversion operations, that we formally axiomatize in this work. As a complement to MIG algebraic optimization, we develop powerful Boolean methods exploiting global properties of MIGs, such as bit-error masking. MIG algebraic and Boolean methods together attain very high optimization quality. Considering the set of IWLS’05 benchmarks, our MIG optimizer (MIGhty) enables a 7% depth reduction in LUT-6 circuits mapped by ABC while also reducing size and power activity, with respect to similar AIG optimization. Focusing on arithmetic intensive benchmarks instead, MIGhty enables a 16% depth reduction in LUT-6 circuits mapped by ABC, again with respect to similar AIG optimization. Employed as front-end to a delay-critical 22-nm ASIC flow (logic synthesis + physical design) MIGhty reduces the average delay/area/power by 13%/4%/3%, respectively, over 31 academic and industrial benchmarks. We also demonstrate delay/area/power improve- ments by 10%/10%/5% for a commercial FPGA flow

Advanced System on a Chip Design Based on Controllable-Polarity FETs (invited paper)

Field-Effect Transistors (FETs) with on-line controllable-polarity are promising candidates to support next generation System-on-Chip (SoC). Thanks to their enhanced functionality, controllable-polarity FETs enable a superior design of critical components in a SoC, such as processing units and memories, while also providing native solutions to control power consumption. In this paper, we present the efficient design of a SoC core with controllable-polarity FET. Processing units are speeded-up at the datapath level, as arithmetic operations require fewer physical resources than in standard CMOS. Power consumption is decreased via embedded power-gating techniques and tunable high-performance/low-power devices operation. Memory cells are made smaller by merging the access interface with the storage circuitry. We foresee the advantages deriving from these techniques, by evaluating their impact on the design of SoC for a contemporary telecommunication application. Using a 22-nm vertically-stacked silicon nanowire technology, a coarse-grain evaluation at the block level estimates a delay and power reduction of 20% and 19% respectively, at a cost of a moderate area overhead of 15%, with respect to a state-of-art FinFET technology

Logic Synthesis for Established and Emerging Computing

Logic synthesis is an enabling technology to realize integrated computing systems, and it entails solving computationally intractable problems through a plurality of heuristic techniques. A recent push toward further formalization of synthesis problems has shown to be very useful toward both attempting to solve some logic problems exactly--which is computationally possible for instances of limited size today--as well as creating new and more powerful heuristics based on problem decomposition. Moreover, technological advances including nanodevices, optical computing, and quantum and quantum cellular computing require new and specific synthesis flows to assess feasibility and scalability. This review highlights recent progress in logic synthesis and optimization, describing models, data structures, and algorithms, with specific emphasis on both design quality and emerging technologies. Example applications and results of novel techniques to established and emerging technologies are reported

Exploiting Inherent Characteristic of Reversible Circuits for Faster Combinational Equivalence Checking

Reversible circuits implement invertible logic functions. They are of great interest to cryptography, coding theory, interconnect design, computer graphics, quantum computing, and many other fields. As for conventional circuits, checking the combinational equivalence of two reversible circuits is an important but difficult (coNP-complete) problem. In this work, we present a new approach for solving this problem significantly faster than the state-of-the-art. For this purpose, we exploit inherent characteristics of reversible computation, namely bi-directional (invertible) execution and the XOR-richness of reversible circuits. Bi-directional execution allows us to create an identity miter out of two reversible circuits to be verified, which naturally encodes the equivalence checking problem in the reversible domain. Then, the abundant presence of XOR operations in the identity miter enables an efficient problem mapping into XOR-CNF satisfiability. The resulting XOR-CNF formulas are eventually more compact than pure CNF formulas and potentially easier to solve. As previously anticipated, experimental results show that our equivalence checking methodology is more than one order of magnitude faster, on average, than the state-of-the-art solution based on established CNF-formulation and standard SAT solvers

Mapping Monotone Boolean Functions into Majority

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

Exploiting Circuit Duality to Speed Up SAT

In this paper, we establish a non-trivial duality between tautology and contradiction check to speed up circuit SAT. Tautology check determines if a logic circuit is true in every possible interpretation. Analogously, contradiction check determines if a logic circuit is false in every possible interpretation. A trivial transformation of a (tautology, contradiction) check problem into a (contradiction, tautology) check problem is the inversion of all outputs in a logic circuit. In this work, we show that exact logic inversion is not necessary. We give operator switching rules that selectively exchange tautologies with contradictions, and vice versa. Our approach collapses into logic inversion just for tautology and contradiction extreme points but generates non-complementary logic circuits in the other cases. This property enables computing benefits when an alternative, but equisolvable, instance of a problem is easier to solve than the original one. As a case study, we investigate the impact on SAT. There, our methodology generates a dual SAT instance solvable in parallel with the original one. This concept can be used on top of any other SAT approach and does not impose much overhead, except having to run two solvers instead of one, which is typically not a problem because multi-cores are widespread and computing resources are inexpensive. Experimental results show a 25% speed-up of SAT in a concurrent execution scenario. Also, statistical experiments confirmed that our runtime reduction is not of the random variation type

A Novel Basis for Logic Rewriting

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%

Inversion Minimization in Majority-Inverter Graphs

In this paper, we present new optimization techniques for the recently introduced Majority-Inverter Graph (MIG). Our optimizations exploit intrinsic algebraic properties of MIGs and aim at rewriting the complemented edges of the graph without changing its shape. Two exact algorithms are proposed to minimize the number of complemented edges in the graph. The former is a dynamic programming method for trees; the latter finds the exact solution with a minimum number of inversions using Boolean satisfiability (SAT). We also describe a heuristic rule based algorithm to minimize complemented edges using local transformations. Experimental results for the exact algorithm to fanout-free regions show an average reduction of 12.8% on the EPFL benchmark suite. Applying the heuristic method on the same instances leads to a total improvement of 60.2%

LUT Mapping and Optimization for Majority-Inverter Graphs

A Majority-Inverter Graph (MIG) is a directed acyclic graph in which every vertex represents a three-input majority operation and edges may be complemented to indicate operand inversion. MIGs have algebraic and Boolean properties that enable efficient logic optimization. They have been shown to obtain superior synthesis results as compared to state-of-the- art And-Inverter Graph (AIG) based algorithms. In this paper, we extend MIGs to Functionally Reduced MIGs (FRMIGs), analogous to the extension of AIGs to Functionally Reduced AIGs (FRAIGs). This enables the use of MIGs in a lossless synthesis design flow. We present an FRMIG based technology mapper for lookup tables (LUTs). Any MIG may be mapped to a k- LUT network. Using exact synthesis we may decompose the k- LUT network back into an equivalent MIG. We show how LUT mapping and exact k-LUT decomposition can be used to create an MIG optimization method. Finally, we present the results of applying our new optimization method and LUT mapper to both logic optimization and technology mapping