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

SATSoT: A Methodology to Map Controllable-Polarity Devices on a Regular Fabric Using SAT

Devices with controllable-polarity, such as Double-Gate Vertically-Stacked Nanowire FETs, have shown promising interests in recent years to implement XOR-based logic functions in an unprecedented compact way. Such a compactness is obtained at the cost of a denser interconnect, that can be mitigated by designing an efficient hyper-regular layout structure, called Sea-of-Tiles. In this paper, we propose a methodology, based on Boolean satisfiability, to map netlists of transistors on such a structure. The methodology endeavors to minimize the wiring complexity, by maximizing the sharing of the different terminals. We showed that its implementation, SATSoT, is able to automatically generate compact mappings with wiring complexities similar to manual layouts

Majority Logic Representation and Satisfiability

Majority logic is a powerful generalization of common AND/OR logic. Original two-level and multi-level logic networks can use majority operators as primitive connective, in place of AND/ORs. In such a way, Boolean functions have novel means for compact representation and efficient manipulation. In this paper, we focus on two-level logic representation. We define a Majority Normal Form (MNF), as an alternative to traditional Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF). After a brief investigation on the MNF expressive power, we study the problem of MNF-SATisfiability (MNF-SAT). We prove that MNF-SAT is NP-complete, as its CNF-SAT counterpart. However, we show practical restrictions on MNF formula whose satisfiability can be decided in polynomial time. We finally propose a simple algorithm to solve MNF- SAT, based on the intrinsic functionality of two-level majority logic. Although an automated MNF-SAT solver is still under construction, manual examples already demonstrate promising opportunities

From Defect Analysis to Gate-Level Fault Modeling of Controllable-Polarity Silicon Nanowires

Controllable-Polarity Silicon Nanowire Transistors (CP-SiNWFETs) are among the promising candidates to complement or even replace the current CMOS technology in the near future. Polarity control is a desirable property that allows the on-line configuration of the device polarity. CP-SiNWFETs result in smaller and faster logic gates unachievable with conventional CMOS implementations. From a circuit testing point of view, it is unclear if the current CMOS and FinFET fault models are comprehensive enough to model all the defects of CP-SiNWFETs. In this paper, we explore the possible manufacturing defects of this technology through analyzing the fabrication steps and the layout structure of logic gates. Using the obtained defects, we then evaluate their impacts on the performance and the functionality of CP-SiNWFET logic gates. Out of the results, we extend the current fault model to a new a hybrid model, including stuck-at ptype and stuck-at n-type, which can be efficiently used to test the logic circuits in this technology. The newly introduced fault model can be utilized to adequately capture the malfunction behavior of CP logic gates in the presence of nanowire break, bridge and float defects. Moreover, the simulations revealed that the current CMOS test methods are insufficient to cover all faults, i.e., stuck- Open. We proposed an appropriate test method to capture such faults as well

Majority-Inverter Graph: A Novel Data-Structure and Algorithms for Efficient Logic Optimization

In this paper, we present Majority-Inverter Graph (MIG), a novel logic representation structure for efficient optimization of Boolean functions. An MIG is a directed acyclic graph consisting of three-input majority nodes and regular/complemented edges. We show that MIGs include any AND/OR/Inverter Graphs (AOIGs), containing also the well- known AIGs. In order to support the natural manipulation of MIGs, we introduce a new Boolean algebra, based exclusively on majority and inverter operations, with a complete axiomatic system. Theoretical results show that it is possible to explore the entire MIG representation space by using only five primitive transformation rules. Such feature opens up a great opportunity for logic optimization and synthesis. We showcase the MIG potential by proposing a delay-oriented optimization technique. Experimental results over MCNC benchmarks show that MIG optimization reduces the number of logic levels by 18%, on average, with respect to AIG optimization performed by ABC academic tool. Employed in a traditional optimization-mapping circuit synthesis flow, MIG optimization enables an average reduction of {22%, 14%, 11%} in the estimated {delay, area, power} metrics, before physical design, as compared to academic/commercial synthesis flows

Unlocking Controllable-Polarity Transistors Opportunities by Exclusive-OR and Majority Logic Synthesis

For more than four decades, Complementary Metal-Oxide- Semiconductor (CMOS) Field Effect Transistors (FETs) have been the baseline technology for implementing digital computation systems. CMOS transistors natively implement Not-AND (NAND)- and Not- OR (NOR)-based logic operators. Nowadays, we observe a trend towards devices with an increased set of logic capabilities, i.e., with the ability to realize in a compact way specific logic operators as compared to the standard CMOS. In particular, controllable-polarity devices enable a native and compact realization of eXclusive-OR (XOR)- and MAJority (MAJ)- logic functions, and open a large panel of opportunities for future high-performance computing systems. However, main current logic synthesis tools exploit algorithms using NAND/NOR representations that are not able to fully exploit the capabilities of novel XOR- and MAJ-oriented technologies. In this paper, we review some recent work that aims at providing novel logic synthesis techniques that natively assess the logic capabilities of XOR- and MAJ-operators

Vertically-Stacked Silicon Nanowire Transistors with Controllable Polarity: a Robustness Study

Vertically-stacked Silicon NanoWire FETs (SiN- WFETs) with gate-all-around control are the natural and most advanced extension of FinFETs. At advanced technology nodes, due to Schottky contacts at channel interfaces, devices show an ambipolar behavior, i.e., the device exhibits n- and p-type charac- teristics simultaneously. This property, when controlled by an independent Double-Gate (DG) structure, can be exploited for logic computation, as it provides intrinsic XOR operation. Elec- trostatic doping of the transistor suppresses the need for dopant implantation at the source and drain regions, which potentially leads to a larger process variations immunity of the devices. In this paper, we propose a novel method based on Technology Computer-Aided Design (TCAD) simulations, enabling the predic- tion of emerging devices variability. This method is used within our DG-SiNWFET framework and shows that devices, whose polarity is controlled electrostatically, present better immunity to variations for some of their parameters, such as the off-current with 16× less standard deviation

A Surface Potential and Current Model for Polarity-Controllable Silicon Nanowire FETs

Silicon nanowire FET (SiNWFET) with dynamic polarity control has been experimentally demonstrated and has shown large potential in circuit applications. To fully explore its circuit-level opportunities, a physics-based compact model of the polarity-controllable SiNWFET is required. Therefore, in this paper, we extend the solution for conventional SiNWFETs to polarity-controllable SiNWFETs. By solving the current continuity equation, the potential distribution and drain current is obtained. The model shows good aoreement with TCAD simulation. It can be used as the core to develop the complete compact model for polarity-controllable SiNWFETs

An Efficient Manipulation Package for Biconditional Binary Decision Diagrams

Biconditional Binary Decision Diagrams (BBDDs) are a novel class of binary decision diagrams where the branching condition, and its associated logic expansion, is biconditional on two variables. Reduced and ordered BBDDs are remarkably compact and unique for a given Boolean function. In order to exploit BBDDs in Electronic Design Automation (EDA) applications, efficient manipulation algorithms must be developed and integrated in a software package. In this paper, we present the theory for efficient BBDD manipulation and its practical software implementation. The key features of the proposed approach are (i) strong canonical form pre-conditioning of stored BBDD nodes, (ii) recursive formulation of Boolean operations in terms of biconditional expansions, (iii) performance-oriented memory management and (iv) dedicated BBDD re-ordering techniques. Experimental results show that the developed BBDD package achieves an average node count reduction of 19.48% and a speed-up factor of 1.63x with respect to a state-of-art decision diagram manipulation package. Employed in the synthesis of datapath circuits, the BBDD manipulation package is capable to advantageously restructure arithmetic operations producing 11.02% smaller and 32.29% faster circuits as compared to a commercial synthesis flow

Majority-based Synthesis for Nanotechnologies

We study the logic synthesis of emerging nanotechnologies whose elementary devices abstraction is a majority voter. We argue that synthesis tools, natively supporting the majority logic abstraction, are the technology enablers. This is because they allow designers to validate majority-based nanotechnologies on large-scale benchmarks. We describe models and data-structures for logic design with majority-based nanotechnologies and we show results of applying new synthesis algorithms and tools. We conclude that new logic synthesis methods are required to achieve a fair assessment on emerging nanotechnologies