128 IEEE TRANSACTIONS ON COMPUTERS, VOL. 70, NO. 1, JANUARY 2021 Karnaugh Map Method for Memristive and Spintronic Asymmetric Basis Logic Functions Vaibhav Vyas , Lucian Jiang-Wei, Peng Zhou, Student Member, IEEE, Xuan Hu , Student Member, IEEE, and Joseph S. Friedman , Senior Member, IEEE Abstract—The development of beyond-CMOS technologies with alternative basis logic functions necessitates the introduction of novel design automation techniques. In particular, recently proposed computing systems based on memristors and bilayer avalanche spin-diodes both provide asymmetric functions as basis logic gates - the implication and inverted-input AND, respectively. This article therefore proposes a method by which Karnaugh maps can be directly applied to systems with asymmetric basis logic functions. A set of identities is defined for these memristor and spintronic logic functions, enabling the formal demonstration of the Karnaugh map method and an explanation of the proposed technique. This method thus, enables the direct minimization of spintronic and memristive logic circuits without translation to conventional Boolean algebra, facilitating the further development of these novel computing paradigms. Preliminary analyses demonstrate that this Karnaugh map minimization approach can provide a 28 percent reduction in step count as compared to previous manual optimization. Index Terms—Boolean algebra, beyond-CMOS computing, asymmetric logic, emerging technologies, memristors, spintronics Ç 1INTRODUCTION MERGING computing technologies provide unconven- implementation. For example, [13] minimizes a large func- Etional basis logic sets that introduce new challenges for tion into NAND, OR, and parity gates, which are then real- computing system design and integration. In particular, ized with memristors; [14] minimizes a function into OR and stateful logic based on memristors efficiently provides the inverter gates; [15] uses NOT, NAND, and OR gates; [16] implication and NAND functions, while a single bilayer ava- uses majority gates; [17] minimizes the number of memris- lanche spin-diode device can perform either an inverted- tors (but not the step count); and [18] minimizes functions input AND (IAND) function or an OR function (see Figs. 1, 2, with the assistance of CMOS buffer circuits. Binary decision and 3). However, both the implication and IAND functions diagrams have also been used [19], as has an interpretation are non-commutative and asymmetric, inhibiting the use of of memristors as threshold logic elements [20]. As all of these conventional techniques for logic design. Many of these con- techniques minimize circuits with conventional symmetric ventional techniques are based on Maurice Karnaugh’s 1953 logic functions and then implement these circuits with mem- proposal for a map method for the minimization of Boolean ristors, the circuit that is realized does not fully exploit the logic [1]. Karnaugh’s map method enables logic minimiza- fundamental basis logic operations provided by memristors. tion based on the AND and OR functions, while various logic Though helpful in significantly reducing circuit complexity, concepts based on memristors and spintronic devices pro- none of the previous works provide a technique for realizing vide basis logic gates for which there is no efficient technique a fully minimized circuit with asymmetric logic functions for translation to AND and OR gates [2], [3], [4], [5], [6], [7], that takes into account the true physical capabilities of these [8], [9], [10], [11], [12], preventing direct application of the devices. Karnaugh map. A previous work in the related field of quantum comput- Logic minimization for stateful memristor logic has previ- ing shows the advantage of using Karnaugh maps as the ously been investigated, but has not resulted in a minimiza- starting point of logic manipulation for unusual logic func- tion technique that fully exploits asymmetric logic functions. tions which lead to a gate and operation count reduction In much of the previous work, logic minimization is compared to previous methods at the time. Similar to the performed with a basis set of conventional logic functions, asymmetric logic gates addressed in this paper, quantum cir- with each basis logic gate mapped to an efficient memristor cuits do not support conventional logic functions, and thus modifications of the conventional techniques are needed in order to process them natively, in an efficient manner. The authors are with the Department of Electrical and Computer Engineer- The Karnaugh map method is an effective visual approach ing, University of Texas at Dallas, Richardson, TX 75080 USA. E-mail: [email protected], {Lucian.Jiang-Wei, peng.zhou, xuan.hu, for minimizing simple logical computations, and adapting joseph.friedman}@utdallas.edu. the Karnaugh map method is therefore a natural first step Manuscript received 26 Sept. 2019; revised 9 Feb. 2020; accepted 5 Apr. 2020. towards enabling efficient logic minimization for unconven- Date of publication 13 Apr. 2020; date of current version 11 Dec. 2020. tional computing systems. For example, previous results (Corresponding author: Joseph S. Friedman.) with Karnaugh maps adapted for quantum circuits [21] illus- Recommended for acceptance by A. L. E. Digital Object Identifier no. 10.1109/TC.2020.2986970 trate the potential of Karnaugh maps for minimizing unusual 0018-9340 ß 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html_ for more information. Authorized licensed use limited to: Univ of Texas at Dallas. Downloaded on December 13,2020 at 22:26:17 UTC from IEEE Xplore. Restrictions apply. VYAS ET AL.: KARNAUGH MAP METHOD FOR MEMRISTIVE AND SPINTRONIC ASYMMETRIC BASIS LOGIC FUNCTIONS 129 Fig. 1. Schematic of memristive implication logic, where voltages applied Fig. 3. Bilayer avalanche spin-diode, where the magnetic fields due to to the memristors modulate the resistance state. input currents A and B modulate the output current. enables the development of an algebra and Karnaugh map logic functions. This paper therefore proposes modifications method that enables logic minimization for both stateful of Karnaugh’s map method that enable the reduction of com- memristor logic and bilayer avalanche spin-diode logic. plex asymmetric logic functions directly into minimized stateful memristor logic circuits and bilayer avalanche spin- 2.1 Stateful Memristor Logic diode logic circuits. The component devices are described in A memristor (or, more generally, a memristive device) is a Section 2, and their asymmetric logic functionality is two-terminal non-volatile device with a resistance that can explained. Section 3 formally details the algebraic underpin- be modified through application of a voltage across the two nings of the proposed Karnaugh map method through neces- terminals [22], [23]. In general, the resistance is on a spectrum sary proofs of logical identities. The modified map method is determined by the history of applied voltages; in the ideal proposed and explained in Section 4, and examples are pro- case, applied voltages above a threshold magnitude switch a vided to enable its practical use. The approach is used to memristor between purely resistive and conductive states. algorithmically design a one-bit full adder in Section 5, dem- The memristors shown in Fig. 1 switch to a conductive ‘1’ onstrating the utility of this Karnaugh map approach in the state when VP À VN >VTH , and a resistive ‘0’ state when manipulation of asymmetric logic. Finally, conclusions are VN À VP >VTH , where VN is the voltage at node N, VP is the presented in Section 6. voltage at node P, and VTH is the threshold voltage. In some physical implementations, the resistance state is a result of the growth and retraction of a metallic filament [24]. 2BACKGROUND As shown in Table 1, the implication function OUT ¼ A ! B V V The asymmetric basis logic functions provided by stateful is performed by applying COND to memristor A and SET V <V <V V memristor logic and bilayer avalanche spin-diode logic can be to memristor B, where COND TH SET and SET À V <V 000 performed compactly, but an applicable device count reduc- COND TH [25]. When memristor A is in the resistive V tion method requires the development of techniques tailored state, the SET voltage across memristor B is greater than V 1 to these functions. In particular, the implication function can TH , causing memristor B to switch to the conductive ‘ ’ state 1 be performed by two non-volatile memristors, while a NAND or remain in that state. If memristor A is in the conductive ‘ ’ V V function can be performed by three such memristors. In state, the voltage across memristor B is SET À COND; as this V bilayer avalanche spin-diode logic, a single device can per- is less than TH , no switching occurs. This implication func- form the IAND and OR functions. The implication and IAND tion (IMPLY) can thus be performed by two memristors in a functions are both non-commutative and asymmetric, and single step, while a NAND function can be performed simi- their integration with NAND and OR functions, respectively, larly with three memristors in two steps [7], [13]. Cascaded logic operations are performed in a unique manner within the stateful memristor logic paradigm: rather than each cascaded operation being performed by a distinct set of devices, the memristors are continually reused through a step-wise application of VSET and VCOND voltages. Table