Abstract
Among emerging nonvolatile storage technologies, redoxbased resistive switching Random Access Memory (ReRAM) is a prominent one. The realization of Boolean logic functionalities using ReRAM adds an extra edge to this technology. Recently, 7state ReRAM devices were used to realize ternary arithmetic circuits, which opens up the computing space beyond traditional binary values. In this manuscript, we report realization of multivalued and fuzzy logic operators with a representative application using ReRAM devices. Multivalued logic (MVL), such as Łukasiewicz logic generalizes Boolean logic by allowing more than two truth values. MVL also permits operations on fuzzy sets, where, in contrast to standard crisp logic, an element is permitted to have a degree of membership to a given set. Fuzzy operations generally model human reasoning better than Boolean logic operations, which is predominant in current computing technologies. When the available information for the modelling of a system is imprecise and incomplete, fuzzy logic provides an excellent framework for the system design. Practical applications of fuzzy logic include, industrial control systems, robotics, and in general, design of expert systems through knowledgebased reasoning. Our experimental results show, for the first time, that it is possible to model fuzzy logic natively using multistate memristive devices.
Introduction
Claude Shannon, in his landmark work^{1}, demonstrated that the twovalued logic system developed by George Boole^{2}, can be mimicked through operations of an electrical circuit. This resulted in widespread adoption of twovalued switching algebra or Boolean algebra. For every digital device in the modern world, Boolean algebra is used to perform the underlying computation. Boolean logic uses two truth values, true and false, even though in real life, we often require more than two truth values for describing an event. For example, we describe a day as ‘sunny’, ‘partly clouded’ or ‘clouded’, which means that the element weather cannot be discretely classified into the set ‘sunny’ or ‘clouded’ To capture such phenomena and their underlying logical process, multivalued logic system was introduced.
In 300 BC, Aristotle proposed the principle of noncontradiction which ruled out simultaneous existence of two contradictory propositions^{3}. This principle, also known as the law of excluded middle, is one of the classical laws of thought. In modern times, Jan Łukasiewicz introduced a third truth value, thereby first formally studying the field of multivalued logic (MVL) in 1920^{4}. In the following year, Emil Post published a system of functionally complete MVL algebra with additional truth degrees (n ≥ 2), ‘chained Post algebra’^{5}. Later on, finitevalued Łukasiewicz logic was rigorously axiomatized and extended to arbitrarily many truth values^{6,7}. Applications of MVL can be found in linguistics^{8}, circuit simulation and manufacturing testing of digital circuits^{9}.
An important offshoot of MVL that permits inference under vagueness and allows realvalued member elements is Fuzzy logic. The term fuzzy logic was introduced by Lotfi A. Zadeh in context of fuzzy set theory^{10}. In contrast to the classical logic systems that adheres to a set of elements with crisp truth values, fuzzy logic operates on fuzzy sets. In a fuzzy set, elements of the set can have a degree of membership. Operators from a MVL, e.g., Łukasiewicz logic, can be applied to the fuzzy set, akin to how Boolean logic operators are applied to the crisp set. In fuzzy logic, a linguistic model is built from a set of IF–THEN rules which describe the control model. Mamdani Warren demonstrated that fuzzy logic could be used for developing operational automated control systems^{11} and clinical practice decision support systems^{12}. The most well known application of fuzzy logic based control system was deployed in Sendai Subway 1000 series subway trains in Japan for speed control^{13}. The fuzzy controller based train had a higher relative smoothness of the starts and stops when compared to other trains, and was 10% more energy efficient than humancontrolled accelerated trains. Further applications of fuzzy logic include expert systems^{14}, robotics^{15} and diverse subdomains of machine intelligence^{16}.
Despite these wide ranging applications of MVL, the present computing technology is heavily based on Boolean logic. There have been prior studies on porting MVL to digital and analog circuits with promising results. MVL has been demonstrated to improve energyefficiency by reducing the switching activity of VLSI interconnects^{17,18}. MVL based arithmetic circuits are simpler and more efficient over corresponding Boolean logic based implementations^{19,20,21}. However, a limiting factor of MVL realization has been the inherent representation of information in binary format in semiconductor devices, thereby forcing a designer to switch between logic formats, which was clearly an inefficient solution. In this manuscript, we leverage the multistate memristive devices which can inherently operate in the multivalued domain. The multistate memristive devices used in this experiment are Redoxbased resisitve switches (ReRAMs), which are considered as one of the most promising emerging nonvolatile memory technologies^{22,23,24}. Implementation using passive crossbar configuration enables ultradense 4F ^{2} integration. Recently, TaO _{ x } based ReRAMs draw significant attention due to excellent performance in term of high endurance (>10^{12})^{25}, long retention (10 years)^{26}, multilevel switching capability (3bit)^{27} and fast read/write speed of below 200 ps ^{28}. Besides the memory applications, ReRAM based passive crossbar arrays offer the implementation of memoryintensive computing paradigms, i.e. the logic operations are directly processed in the memory and arithmetic tasks. This merges the boundaries between memory and arithmetic logic units and eases the vonNeumannbottleneck for computation^{29}. Furthermore, memristive crossbar arrays can enable the multiparallel search algorithms for pattern recognition tasks, widely required for neuromorphic applications^{30}.
The current paper reports the first implementation of Łukasiewicz logic using the ReRAMbased memristive devices. We do not impose any theoretical limit on the number of states for the memristive devices^{31} and hence this work can be used for realizing any application that uses finitevalued Łukasiewicz logic family L _{ n }. Before going to the implementation details, we briefly review the basics of Łukasiewicz logic. Formally, a finitelyvalued Łukasiewicz logic family L _{ n } can be defined over the following truth values.
The designated truth value is 1. Implication (IMP→) and negation (NEG¬) are the two operators used in Łukasiewicz logic, defined through the following functions.
In this paper, we also present a detailed case study to realize fuzzy logic control using Łukasiewicz logic and the implementation of a fuzzy logic controller using multistate ReRAM crossbar arrays.
Results
Device Properties
In this work, 2 μm × 2 μm Pt/W/TaO _{ x }/Pt crosspoint bipolar memristive devices arranged in word structure have been fabricated. Figure 1 shows the scanning electron microscope and transmission electron microscopy image of the devices used in this experiment. The ReRAM device stack of 25 nm Pt/13 nm W/7 nm TaO _{ x }/30 nm Pt is depicted in Fig. 1(b). Figure 1(c) shows the schematic of a single device along with the details of the stacked layers. Figure 2(a) shows the typical I–V characteristics of the ReRAM device with set current compliance of 1.0 mA, along with the electroforming curve. After the electroforming process, the device was toggled to high resistance state by applying the reset voltage. The maximum applied voltage V _{ stop } during RESET process defines the final resistive state of the device. This feature is also used in pulse mode operation, and can thus be used in memory and logic operations for controlling the multilevel states. To enable highly reproducible RESET operation, we have always applied a DC SET operation before each pulsed RESET operation (200 ns). Note that a nanosecond pulsed SET operations are also feasible, but have not been applied in this work. Figure 2(b) shows a very tight resistance distribution of low resistance state (LRS) and six multilevel resistive states. This confirms the excellent switching properties of these devices. For the multilevel programming, we have split the total applied amplitude across the bottom electrode and the top electrode. A fixed positive amplitude (+0.7V) is assigned at the bottom electrode while a varying negative amplitude (−0.7V to −1.5V) is applied to the top electrode. Under this configuration, the total applied amplitude across the cell varies from −1.4V to −2.2V for the given pulse width of 200 ns. For each V _{ stop } voltage, the cell is toggled to a different high resistance state (HRS). The final resistance state has been read by the means of a 120 μs pulse with amplitude V _{ READ } = 0.1V. Figure 2(c) shows the mean value of each resistive state from R0 to R4.
For Łukasiewicz logic family L _{3} (or threevalued Łukasiewicz logic family), three logic values are used. The three resistance states (R0, R1, R2) of the multilevel device correspond to logic values (0, 0.5, 1) respectively in the Łukasiewicz logic family L _{3}. Additionally, the intermediate results in L _{3} can be (1.5 or 2) which require two additional logic values. The resistance states (R3, R4) represent the intermediate logic values (1.5, 2) respectively. Corresponding to the logic values u = {0, 0.5, 1, 1.5, 2}, the operand voltage V _{ u } is {0, 0.2, 0.4, 0.6, 0.8}V respectively. The multivalued operands say u and v can be applied as operand voltages to the top electrode (TE) and the bottom electrode (BE). Note that, u and v are always from the multivaled logic set L _{3} {0, 0.5, 1}. A predefined OFFSET voltage V _{ OFFSET } is used for each pulse to allow equidistant voltage stepping. The voltage applied to the TE is V _{ TE } = −(V _{ OFFSET } + V _{ u }). Depending on the operation being realized, the actual voltage applied to the BE is V _{ BE } = V _{ OFFSET } ± V _{ v }. The effective potential difference across the device is V _{ eff } = V _{ TE } − V _{ BE }. If V _{ BE } = V _{ OFFSET } + V _{ v }, the resulting resistance state of the device is R _{ u+v }. Otherwise if V _{ BE } = V _{ OFFSET } − V _{ v }, the resulting resistance state of the device is R _{ u−v }. Fig. 3 demonstrates the multilevel operation of the device.
Proposed Łukasiewicz Logic Implementation
The developed implementation strategy for realization of the Łukasiewicz logic operates and stores the multivalued results directly in the ReRAM devices. The computed results are available as the resistive states of the device and can be read out. Before any operation, the devices are initialized to LRS. The realization of the NEG¬ and IMP→ operator for L _{3} is explained below.
NEG¬ operator
The negation operator works on a single operand. For computing ¬u, a constant voltage −(V _{ OFFSET } + V _{1}) is applied to the TE while the V _{ OFFSET } − V _{ u } is applied to the BE. The negated operand is stored in the ReRAM device as a corresponding resistive state.
IMP→ operator
The implication operator works on two operands. The flowchart for performing the implication operation u → v is shown in Fig. 4(a) and the steps are described below.
Step 1: In the beginning, all the devices are initialized to the LRS.
Step 2: To compute ¬u in device D _{0}, −(V _{ OFFSET } + V _{1}) is applied to the TE and V _{ OFFSET } −V _{ u } to the BE.
Step 3: The resistive state of the device D _{0} is read out by means of V _{ READ } = 0.1V.
Step 4: In the second device D _{1}, the negated sum of offset voltage V _{ OFFSET } and the voltage corresponding to read out value V _{¬u } is applied to the TE i.e V _{ TE } = −(V _{ OFFSET } + V _{¬u }) whereas at the BE, V _{ BE } = V _{ OFFSET } + V _{ v } is applied.
Step 5: In the following step, resistive state R of the D _{1} is read out. If the resistive state R < R2, then the operation is complete. Otherwise, device D _{1} is toggled to the LRS and the set operation is used to change the device state to R2 i.e., V _{ TE } = −1.1V and V _{ BE } = 0.7V is applied.
Figure 4(b) demonstrates the computation of 0 → 1 using the proposed method in terms of applied operation voltages and corresponding states. The overall operation requires seven steps. Initially, the device D _{0} and D _{1} are in LRS state, shown as R _{ D0} and R _{ D1} respectively. In cycle t1, 1 − u is computed by applying 0.7V to the BE and 1.1 − V to the TE of device D _{0}, shown as V _{ BE } and V _{ TE0} respectively. In cycle t2, the resistive state of D _{0} is read out, which is R2. In the next cycle, 1 − u + v is computed by applying 1.1V and −1.1 to the BE and TE of device D _{1}. In cycle t4, the current resistive state(R _{ D1}) of device D _{1} is read out. Since R _{ D1} = R4 > R2, device D _{1} is reset to the LRS and then set to R2 in cycles t5 and t6. The computation of 0 → 1 is complete. To verify the correctness of computation, we read out the state of the device D _{2} in the last cycle t7. The state R _{ D2} is R2 (corresponding to logic 1) which is the correct result for 0 → 1.
Proofofconcept
We demonstrate the realization of a fuzzy logic controller as a representative application of Łukasiewicz logic. A conventional fuzzy controller has three major steps, as shown in Fig. 5(a).

1.
Fuzzify input variables using fuzzy membership functions. If the inputs to the system are analog, analogtodigital converters would be used to convert the inputs to the corresponding values in Łukasiewicz logic. Then, these multivalued inputs are fuzzified using the membership functions.

2.
Execute all the fuzzy inference rules from the rule database to determine the fuzzy output functions.

3.
Defuzzify the fuzzy output functions to get crisp output value i.e a single multivalued output value.
To illustrate the working of a fuzzy logic control, we consider a fuzzy logic controller for regulating screen brightness. The screen brightness has to be regulated based on the ambient light AMBIENT and screen brightness BRIGHTNESS. Each of the variables can be represented using three gradations.

AMBIENT ∈ {Low (L), Moderate (M), High (H)}

BRIGHTNESS ∈ {Subdued (S), Regular (R), Bright (B)}
We use the fuzzy membership functions shown in Fig. 5(b) and (c) to fuzzify the input variables AMBIENT and BRIGHTNESS respectively. The fuzzy membership functions can be expressed in terms of Łukasiewicz operators^{32,33}. Therefore, we can realize the “Fuzzifier” block of a fuzzy control system (shown in Fig. 5(a)) using Łukasiewicz logic operations only. Let us consider the inverted notch membership function f _{(LOW,AMBIENT)} for the grade LOW of variable AMBIENT. f _{(LOW,AMBIENT)}. It can be expressed as (¬v → v) → 0, where v is the input. Similarly, the flipped notch membership function f _{(SUBDUED,BRIGHTNESS)} for the grade SUBDUED of variable BRIGHTNESS can be written as (v → ¬v) → 0. These membership functions can be simplified and expressed in terms of min(u, v) and ¬u (1 − u) operation as shown below.
The detailed procedure to simplify the functions is provided in Supplementary Discussion S1. The series of steps required to realize the inverted notch and flipped notch membership function using multistate ReRAM is presented in Fig. 6(a) and (b) respectively. We experimentally verified the correctness of computation. Figure 7 shows the experimental results when the input variable AMBIENT is 0 and also for value 1 for input BRIGHTNESS. In Supplementary Discussion S2, we have shown the experimental validation of membership function for other values of input AMBIENT.
The fuzzy inference engine determines the ACTION to be taken based on fuzzified inputs and be stated as a set of rules. The ACTION can be increase brightness (↑), decrease brightness (↓) or no action (↔). A rule for example can the following–if AMBIENT is Low and BRIGHTNESS is Subdued, then ACTION is no action (↔). Figure 5(d) presents multiple such control rules to determine the ACTION represented compactly as a rule table. To evaluate the output of a fuzzy rule, a suitable Tnorm function is used. We use Łukasiewicz Tnorm–max(0, u + v − 1) for evaluation of the fuzzy controller rules. The Łukasiewicz Tnorm can be expressed in terms of neg and min functions as ¬min(1, (1 − u) + (1 − v)). Figure 8(a) show the sequence of steps to realize Łukasiewicz Tnorm using the ReRAM devices. As a representative example, Fig. 9 shows the state transitions of the ReRAM devices during computation of Tnorm for inputs u = 1 and v = 1.
Once all the rules have been evaluated, the outputs of these is combined using a suitable Tconorm. The Tconorm should be the dual of the Tnorm used for rule evaluation. Therefore, we use Łukasiewicz Tconorm which is min(1, a + b), for combining the output of the rules. Figure 8(b) shows the steps for computation of the Łukasiewicz Tconorm using the multistate memristive devices. Tconorm evaluation for inputs u = 1 and v = 0.5 is given in Supplementary Fig. S3. To obtain crisp output, the combined fuzzy output in the end has to be defuzzied. The defuzzifier block shown in Fig. 5(a) computes crisp output using an appropriate defuzzification method^{34}. Note that the defuzzifier block has not been implemented in the presented prototype, which can be realized using conventional methods.
Discussion
Knowledgebased system is capable to reason with judgmental, imprecise, and qualitative knowledge as well as with formal knowledge of established theories. The design of such systems is an important challenge in the realm of Artificial Intelligence (AI). The incompleteness and uncertainty associated with the knowledgebase in is handled through fuzzy logic. Fuzzy logic allows linguistic variable^{35} to be assigned inexact or partial truth values for modeling logical reasoning. In this work, we have shown the realization of fuzzy logic control in terms of threevalued Łukasiewicz logic operands. We have demonstrated the feasibility of implementation of threevalued Łukasiewicz logic L _{3} using the multistates memristive devices.
Our demonstrated method can be scaled up for arbitrary nvalued Łukasiewicz logic L _{3}, n ≥ 3, depending on the number of resistive states available. For the realization of L _{ n }, the memristive device should support at least 2n states. From the perspective of area, the implementation of a highervalued logic system does not increase the area per device since it is dependent on the number of resistive states. However with increase in number of resistive states, the peripheral circuitry has to be more robust.todo.
Regarding the representation of numbers, it is well understood that for higher radix, the number of literals reduce in logarithmic order in comparison to lowerradix. For example, the efficiency of a nvalued representation of a truthvalue N compared to its corresponding Boolean representation is equal to \(\frac{{lo}{{g}}_{n}(N)+1}{{lo}{{g}}_{2}(N)+1}\). Implementation of a given fuzzy system in Boolean logic requires the treatment of every member with varied degree in a separate set and performing Boolean logic operations on those sets. Therefore, the computation steps do also increase in logarithmic proportion when using the Boolean logic in comparison to the fuzzy logic. Traditionally, Mamdanitype fuzzy systems use min and max functions for evaluation of fuzzy rules and combining the output of the rules^{11}. min and max functions can be expressed as in terms of Łukasiewicz logic operators:
Therefore, it is possible to use the multivalued Łukasiewicz logic for realization of Mamdani type fuzzy systems as well. In general, Łukasiewicz logic is capable of dealing with a wide range of approximate reasoning paradigms, since it can express evaluation function of multivalued logic classes described in terms of +, −, min and max ^{35,36,37}.
In past, Łukasiewicz logic arrays has been demonstrated for realization of fuzzy inference engines and expert systems^{38,39,40} with CMOSbased circuitry. However, such realizations did not have any multivalued storage devices for storing the intermediate results thereby requiring costly conversions toandfrom binary representation. In this work, we have experimentally realized a working fuzzy system by using Łukasiewicz logic, that does not use any intermediate binary/Boolean representation. Although implementation of fuzzy logic gates have been reported in the DNA computing paradigm^{41}, this is the first experimentally reported work on multivalued logic operators as well as a demonstrative application of that in fuzzy inference engine using memristive devices. Recently, an implementation of Boolean minimum and maximum gate has been demonstrated using memristive devices^{42} with their application restricted to the implementation of sorting networks.
Here, we have successfully demonstrated Łukasiewicz logic operation on 2 μm × 2 μm ReRAM devices. However, these devices are fully compatible to 4F ^{2} configuration in crossbar array in conjunction with a selector device and can be scaled down to 5 nm ^{43,44}. The integration of the selector device would prevent the problem of sneak paths in the crossbar array. Ultradense largescale multistate ReRAM crossbars can be controlled by peripheral control circuitry, as shown in Supplementary Fig. S4. The approach of implementing Łukasiewicz logic operation within the resistive memory device using the available multilevel states is a highly attractive option for future hybrid CMOS/ReRAM chips for enhancing its present functionality. Each multivalued operation requires a constant number of steps, 1 step for negation and 7 steps for implication (depicted in Fig. 4), to be realized, irrespective of the value of n in an nvalued logic system. For Boolean realization of the implication and negation operators, the number of steps would increase with the value of n ^{29,45,46}. Furthermore, parallel operations across multiple devices that share the same wordline, can be enabled by carefully packing operations that have the same input, similar to the strategy proposed by Bhattacharjee et al.^{47}. In contrast, to leverage such parallelism, the Boolean circuits corresponding to the implication and negation operations need to be replicated. Multilevel ReRAM devices reduces the complexity of state representation and thus, brings fundamental benefits across arithmetic and logical primitives. This capability has farreaching implications in modern InternetofThings (IoT) systems, which promotes local computing due to the bandwidth scarcity. By having multivalued and fuzzy logic primitives at the device level, efficient processinginmemory can be undertaken for application domains like public key cryptography, error correcting codes, industrial control and security. Note that, the energyefficiency can be further boosted by having shortpulse (subns) operations.
Conclusion
Fuzzy set allows its elements with certain degrees of membership, in contrast to a crisp set. Operators on the fuzzy set can realistically model reallife applications in, for example, industrial control, linguistics, decision variables and bioinformatics, and therefore, have grown in usage over last half century. The logic operations on the fuzzy set is performed through fuzzy inference system. In this manuscript, we demonstrated a practical fuzzy inference system, by realizing the fuzzy logic operations using the multistate TaO _{ x } devices. Further, each fuzzy operation is mapped to a series of the multivalued logic primitives, namely, Łukasiewicz logic. We showed a practical fuzzy inference system through a limited number of logical steps and 1 × 3 memristive crossbar array. The multistate TaO _{ x } devices enable computation entirely using multivalued elements for the operations, without need for any intermediate representations. Therefore, these devices provide a natural platform to undertake multivalued logic and thus, fuzzy inference operations. We believe that these results can greatly benefit scientific community and provide a direction to move forward in the field of fuzzy logic.
Methods
Device Fabrication
Crosspoint based Ta _{2} O _{5} ReRAM was fabricated on thermally grown SiO _{2} samples. In our design, each device shares a common bottom electrode (BE). The BE was patterned in 30 nmthick platinum (Pt) layers, grown by the sputtering process. After patterning the BE, switching layer of 7 nmthick TaO _{ x }, 13 nmthick tungsten (W), and 25 nmthick platinum (Pt) were sequentially deposited by the sputtering process. The TaO _{ x } layer was grown with reactive sputtering process with 76.6% Argon and 23.3% Oxygen at partial pressure of 2.3 × 10^{−2} mbar. The W ohmic electrode, and the Pt were grown with DC sputtering method. For the top electrode (TE) patterning, photolithography and reactive ion etching steps were performed. These steps lead to the Pt/W/TaO _{ x }/Pt memristive device stack. Figure 1 shows the scanning electron microscopy (SEM) of 1 × 3 crossbar array with 2 μm × 2 μm size cell with crosssectional tunneling electron microscopy (TEM) image and its corresponding schematic diagram. More experimental details can be found in reference^{48}.
Measurement Setup
The pristine state of the memristive devices was highly resistive (GΩ) and therefore an electroforming process was required. This process was carried out by applying a positive DC voltage on the TE for a given current compliance, while keeping the BE grounded. This turned the devices into low resistance state (LRS). Now, the memristive devices were sequentially switched to high resistance state (HRS) by the ‘reset process’ and low resistance state (LRS) by the ‘set process’. In this experiment, the ‘set’ process is performed with DC voltage while the reset operations were performed with 200 ns pulse width and 120 μs long pulse width at 0.1V was used to read the respective resistance states. More measurement details can be traced in reference^{48}.
Data availability
No datasets were generated or analysed during the current study.
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Acknowledgements
This work was supported by the German Research Foundation (DFG) within the framework of SFB 917, Nanoswitches and the authors thank Dr Daesung Park and Mr Sebastian Zischke from Central Facility for Electron Microscopy (GFE), RWTH Aachen for the crosssectional TEM images, and Anne Siemon from RWTH Aachen for critical review of the manuscript.
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Debjyoti Bhattacharjee designed the experiments, interpreted the data and wrote the manuscript. Wonjoo Kim prepared the devices, performed the measurements. Anupam Chattopadhyay conceived the idea, initiated and supervised the research and cowrote the manuscript; Rainer Waser initiated and supervised the research; Vikas Rana conceived the idea, supervised the research and cowrote the manuscript. All authors discussed the results and implications at all stages and contributed to the improvement of the manuscript text.
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Bhattacharjee, D., Kim, W., Chattopadhyay, A. et al. Multivalued and Fuzzy Logic Realization using TaOx Memristive Devices. Sci Rep 8, 8 (2018). https://doi.org/10.1038/s41598017183293
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