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

Towards Nonlinear Photonic Wires in Z-cut LiNbO3

International audienceUsing a modified Proton Exchange process we have realized Photonic Wires in X-cut LiNbO3. They exhibit highly confined mode, low propagation losses, low strain induced polarization coupling and no reduction of the nonlinear properties. We are now transferring this technique to Z-cut LiNbO3 in order to realize very efficient nonlinear devices in PPLN

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

A Circuit Synthesis Flow for Controllable-Polarity Transistors

Double-Gate (DG) controllable-polarity Field-Effect Transistors (FETs) are devices whose n- or p- polarity is on-line configurable by adjusting the second gate voltage. Such emerging transistors have been fabricated in silicon nanowires, carbon nanotuges and graphene technologies. Thanks to their enhanced functionality, DG controllable-polarity FETs implement arith- metic functions, such as XOR and MAJ, with limited physical resources enabling compact and high-performance datapaths. In order to design digital circuits with this technology, automated design techniques are of paramount importance. In this paper, we describe a design automation framework for DG controllable- polarity transistors. First, we present a novel dedicated logic representation form capable to exploit the polarity control during logic synthesis. Then, we tackle challenges at the physical level, presenting a regular layout technique that alleviates the interconnection issue deriving from the second gate routing. We use logic and physical synthesis tools to form a complete design automation flow. Experimental results show that the proposed flow is able to reduce the area and delay of digital circuits, based on 22-nm DG controllable-polarity SiNWFETs, by 22% and 42%, respectively, as compared to a commercial synthesis tool. With respect to a 22-nm FinFET technology, the proposed flow produces circuits, based on 22-nm DG controllable-polarity SiNWFETs, with 2.9x smaller area-delay product

Biconditional Binary Decision Diagrams: A Novel Canonical Logic Representation Form

In this paper, we present biconditional binary deci- sion diagrams (BBDDs), a novel canonical representation form for Boolean functions. BBDDs are binary decision diagrams where the branching condition, and its associated logic expansion, is biconditional on two variables. Empowered by reduction and ordering rules, BBDDs are remarkably compact and unique for a Boolean function. The interest of such representation form in modern electronic design automation (EDA) is twofold. On the one hand, BBDDs improve the efficiency of traditional EDA tasks based on decision diagrams, especially for arithmetic intensive designs. On the other hand, BBDDs represent the natural and native design abstraction for emerging technologies where the circuit primitive is a comparator, rather than a simple switch. We provide, in this paper, a solid ground for BBDDs by studying their underlying theory and manipulation properties. Thanks to an efficient BBDD software package implementation, we validate 1) speed-up in traditional decision diagrams applications with up to 4.4 gain with respect to other DDs, and 2) improved synthesis of circuits in emerging technologies, with about 32% shorter critical path than state-of-art synthesis techniques

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

Fault Modeling in Controllable Polarity Silicon Nanowire Circuits

Controllable polarity silicon nanowire transistors are among the promising candidates to replace current CMOS in the near future owing to their superior electrostatic characteristics and advanced functionalities. From a circuit testing point of view, it is unclear if the current CMOS and Fin-FET fault models are comprehensive enough to model all defects of controllable polarity nanowires. In this paper, we deal with the above problem using inductive fault analysis on three-independent-gate silicon nanowire FETs. Simulations revealed that the current fault models, i.e. stuck-open faults, are insufficient to cover all modes of operation. The newly introduced test algorithm for stuck open can adequately capture the malfunction behavior of controllable polarity logic gates in the presence of nanowire break and bridge on polarity terminals

The EPFL Combinational Benchmark Suite

In this paper, we present the EPFL combinational benchmark suite. We aim at completing existing benchmark suites by focusing only on natively combinational benchmarks. The EPFL combinational benchmark suite consists of 23 combinational circuits designed to challenge modern logic optimization tools. It is further divided into three parts. The first part includes 10 arithmetic benchmarks, e.g., square-root, hypotenuse, divisor, multiplier etc.. The second part consists of 10 random/control benchmarks, e.g., round-robin arbiter, lookahead XY router, alu control unit, memory controller etc.. The third part contains 3 very large circuits, featuring more than ten million gates each. All benchmarks have a moderate number of inputs/outputs ranging from few tens to about one thousand. The EPFL benchmark suite is available to the public and distributed in all Verilog, VHDL, BLIF and AIGER formats. In addition to providing the benchmarks, we keep track of the best optimization results, mapped into LUT-6, for size and depth metrics. Better logic implementations can be submitted online. After combinational equivalence checking tests, the best LUT-6 realizations will be included in the benchmark suite together with the author’s name and affiliation

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