The integration of Quality of Service (QoS)-specific communications and Multi-access Edge Computing (MEC) in advanced 5G networks is driving the emergence of innovative applications and business models. An important outcome this development is the ability to execute real-time automated monitoring and control tasks as edge services on MEC servers. Such real-time control applications, which are often modeled as Directed Acyclic Graphs (DAGs) due to their intricate interdependencies, can be periodic and persistent or aperiodic and dynamic (e.g., event-triggered alarms). Tasks within these DAGs may operate at various QoS levels, where higher levels enhance accuracy and reliability, improving the overall application QoS. This paper proposes a QoS-aware anytime scheduling approach for dynamically scheduling an aperiodic DAG-structured application on an MEC system already supporting periodic real-time tasks. We propose the Meta-heuristic QoS-Aware DAG Scheduler (M-QADS), a metaheuristic approach that begins by creating initial base schedules and iteratively improves them through task QoS level enhancements. Extensive simulations using both randomly generated and standard benchmark DAGs, alongside comparisons with the baseline heuristic Enhanced QoS HEFT (EQ-HEFT), reveal that M-QADS outperforms EQ-HEFT across diverse scenarios, demonstrating its effectiveness.

The future of automotive communication hinges on integrating legacy real-time protocols with advanced Ethernet technologies to empower Software-Defined Vehicles (SDVs). SDVs propose the adoption of the zonal computing architectures, which centralize vehicle functions into distinct zones connected by a high-speed communication backbone. While on the one hand, different zones can employ distinct network protocols like CAN, FlexRay, etc., Time Sensitive Ethernet (IEEE802.1Q) is being projected as the most promising protocol for the central backbone network. With such a heterogeneous distributed platform, SDVs demand reliable and deterministic end-to-end communication strategies, especially for zone-to-backbone or interzonal traffic via the central backbone. Although, there exist a few strategies for message transmission across heterogeneous network domains, they are ad-hoc in nature and oblivious to the precise demands of the control applications they cater to. These drawbacks may lead to poor bandwidth utilization and/or network congestion. Opposed to these ad-hoc techniques, this work proposes an optimal SMT (Satisfiability Modulo Theories) formulation for i) multiplexing periodic zonal message frames onto a minimum number of Ethernet frames, taking into account (m, k)-firmness-based relaxations on specific message flows, and ii) routing Ethernet frames between specified source and destination switches. Through extensive experimental evaluations of the proposed formulation using Z3 solver demonstrate substantial performance improvements, showing at least 30% gain in frame utilization under diverse traffic conditions and timing constraints. Our results validate the proposed approach as a robust solution for ensuring efficient and predictable real-time communication in futuristic SDVs.

Adversarial attacks rely on transferability, where an adversarial example (AE) crafted on a surrogate classifier tends to mislead a target classifier. Recent ensemble methods demonstrate that AEs are less likely to mislead multiple classifiers in an ensemble. This paper proposes a new ensemble training using a Pairwise Adversarially Robust Loss (PARL) that by construction produces an ensemble of classifiers with diverse decision boundaries. PARL utilizes outputs and gradients of each layer with respect to network parameters in every classifier within the ensemble simultaneously. PARL is demonstrated to achieve higher robustness against black-box transfer attacks than previous ensemble methods as well as adversarial training without adversely affecting clean example accuracy. Extensive experiments using standard Resnet20, WideResnet28-10 classifiers demonstrate the robustness of PARL against state-of-the-art adversarial attacks. While maintaining similar clean accuracy and lesser training time, the proposed architecture has a 24.8% increase in robust accuracy (∊= 0.07) from the state-of-the art method. Code is available at: https://github.com/shubhishukla10/PARL

ABSTRACT Internet‐of‐Things‐enabled frameworks have eased the development of complex systems, but they throw a significant challenge for efficient resource utilization, thereby improving the system performance. An intelligent scheduler is essential for managing the resources and allocating the same resources to different requests or tasks. This work proposes a generic methodology to optimize system performance metrics such as throughput, utilization, and reward achieved. We present an integer linear programming formulation of the problem to find an optimal solution. We present offline heuristic methods to quickly find reasonable solutions, given the intractable nature of the problem. These heuristics yield promising outcomes, with deviations from optimal solutions below 20% in scenarios with task overlap and high utilization. In scenarios with minimal overlap and utilization, deviations remain under 10%. However, as variables and constraints increase in ILP, the demand for time and memory resources rises substantially. We conduct a comparative analysis of heuristic performance across various scenarios and large test cases. Additionally, we extend our methods to handle resources in online mode, presenting an extensive comparative study with encouraging results.

The rapid adoption of Electric Vehicles (EVs), driven by stringent environmental regulations and rising fuel costs, is reshaping the landscape of Vehicle Routing Problems (VRP). This shift has led to the Electric Vehicle Routing Problem (EVRP), which incorporates EV-specific operational constraints such as limited driving range, energy consumption, recharging strategies, and detour-related charging costs. The challenge becomes even more critical in modern mixed fleets , where Electric and Internal Combustion Engine Vehicles (ICEVs) coexist and must be co-routed efficiently. A widely adopted two-step strategy first uses Capacitated VRP (CVRP) algorithms to generate energy-oblivious routes, then makes EV routes energy-feasible via charging station insertion. While VRP and CVRP are extensively studied, methods for efficiently ensuring energy feasibility for EVs on fixed routes remain limited. This article introduces the Fixed Route Vehicle Charging Problem with Discrete Partial Charging (FRVCP-DPC) , extending FRVCP by allowing partial recharging up to predefined discrete levels. We develop a scalable optimal Dynamic Programming algorithm, Best Energy Feasible Route Generator (BEFRG) , to select detour points, charging stations, and charge levels that minimize total route time while maintaining energy feasibility. To evaluate BEFRG in dynamic traffic conditions, we introduce EFRGen , a traffic-aware EVRP simulator built on Simulation of Urban Mobility (SUMO) and OpenStreetMap (OSM). Experiments on the Montoya benchmark—spanning 120 instances with up to 320 demand points and 38 charging stations—show that BEFRG computes optimal solutions for all cases within one minute.

Cyber-Physical Systems controlling assembly line operations are central to manufacturing processes. Assembly line systems have diversified over time, depending on multiple factors, including the products being manufactured, the workstations and resources used, factory layouts, and so on. This diversity in assembly line configurations has added layers of complexity to the Assembly Line Balancing Problem (ALBP). While many powerful meta-heuristic techniques exist, their performance can vary significantly depending on the specific characteristics of the ALBP instance, such as the structure of the precedence graph, the distribution of task times, and the number of workstations. Recognizing the need for a more versatile solution, this article introduces a generic local search strategy called Flexible Meta-Heuristic (FMH), which includes a set of adjustable tuning parameters for adapting to specific scenarios. FMH combines and extends the strengths of Hill Climbing (HC), Simulated Annealing (SA), and Genetic Algorithm (GA) to provide effective solutions across a wide range of problems. Through extensive experiments using standard benchmarks and randomly generated datasets, FMH demonstrates high accuracy, deviating by at most 0.9% from best-known benchmark values. Additionally, FMH is significantly less resource-intensive, solving problems with up to 150 tasks in minutes where exact solvers can take hours, making it more scalable and applicable to large industrial scenarios. Our findings suggest that the algorithm’s flexibility and strategic hyper-parameter tuning contribute significantly to its effectiveness in solving diverse ALBPs.

As mobile robots navigate through a warehouse collecting items from storage locations and transporting them to designated drop-off points, they consume energy. In this paper, we introduce an intelligent Robot Load Sharing (RLS) algorithm designed to minimize energy usage during item transportation within a warehouse by a fleet of mobile robots, ensuring sustained energy levels along their routes. The energy consumption at each time slot is parameterized as a quadratic function of the load and the traveling distance for a team of robots. A load-sharing framework is presented where pairs of off-loading and on-loading robots are selected that decide on their meeting points and the load amount to be shared. Our algorithm is compared with different no-load-sharing scenarios and we observe a significant reduction in the total energy consumption of the warehouse.

Quantum computing is an emerging technology that has the potential to achieve exponential speedups over their classical counterparts. To achieve quantum advantage, quantum principles are being applied to fields such as communications, information processing, and artificial intelligence. However, quantum computers face a fundamental issue since quantum bits are extremely noisy and prone to decoherence. Keeping qubits error free is one of the most important steps towards reliable quantum computing. Different stabilizer codes for quantum error correction have been proposed in past decades and several methods have been proposed to import classical error correcting codes to the quantum domain. Design of encoding and decoding circuits for the stabilizer codes have also been proposed. Optimization of these circuits in terms of the number of gates is critical for reliability of these circuits. In this paper, we propose a procedure for optimization of encoder circuits for stabilizer codes. Using the proposed method, we optimize the encoder circuit in terms of the number of 2-qubit gates used. The proposed optimized eight-qubit encoder uses 18 CNOT gates and 4 Hadamard gates, as compared to 14 single qubit gates, 33 2-qubit gates, and 6 CCNOT gates in a prior work. The encoder and decoder circuits are verified using IBM Qiskit. We also present encoder circuits for the Steane code and a 13-qubit code, that are optimized with respect to the number of gates used, leading to a reduction in number of CNOT gates by 1 and 8, respectively.

In the past decade, the demand for Electric Vehicle (EV) charging has increased, leading to an irregular fluctuation in EV inflow at the Charging Stations (CS). The policy-makers, therefore, need to have an infrastructure planning mechanism that addresses this fluctuation by estimating the ideal location and capacity of these CSs. This problem is known as the Charging Station Siting and Sizing Problem (CSSSP). Due to the uncertainty in EV inflow, the possible non-availability of charging ports, and the resulting unpredictability in the queueing time, a limited number of EVs can get charged. To minimize this dissatisfaction with charging, we model the uncertainty in the EV demand in terms of a statistical distribution varying over time. We propose a queueing mechanism that accounts for the demand distribution over time but restricts the waiting time by a given threshold. With this mechanism, we propose an iterative heuristic algorithm where an initial allocation of ports is obtained using a policy. Then a statistical approximation approach is proposed to estimate the total unsatisfied EVs within a CS for the given port allocation as a derived random variable. Finally, an approach is proposed to intelligently rearrange the charging ports across the CSs, resulting in a fresh allocation. We repeat the steps until there is no further reduction in the unsatisfied demands. We validate the performance of the proposed method against the Monte Carlo simulation and provide a comparative analysis.

A significant transition in the characteristics of computational workloads coupled with the developments in computing as well as networking concepts and practices has presented various challenges and opportunities. In this work, we study dynamic offloading of soft real-time tasks in SDN based fog computing system. Preliminary results have been presented to highlight the impact of intra/inter-cluster offloading on the penalty due to deadline miss. Finally, the future research directions have been outlined.

Semantic hashing is a crucial component of content based search and retrieval systems. To achieve an effective semantic hashing for images, it is essential to map them to hash space in a way that preserves the semantic information. Most state-of-the-art deep semantic hashing approaches do not fully take into account the structural information and the inherent hierarchy in the dataset. Also, the distribution of hash codes is primarily driven by semantic information that comes from the supervision labels. We propose a semantic hashing framework which utilizes the hyperbolic metric learning to learn the structural and hierarchical information. This information is leveraged in the form of proxy labels for training the hashing network with the proposed novel Structure-Semantic Disagreement (SSD) loss. SSD enforces the model to learn to hash with semantic as well as structural information, leading to more robust and uniformly distributed hash codes. Tests on multiple public domain datasets establish the effectiveness of the proposed approach. Moreover, the developed SSD loss can also be applied to other classification models to improve the representation by enforcing the model to use the structure information more effectively.

The purpose of this study is to offer requisite models for unravelling the complex issues that arise in a supplier selection-order allocation problem by soothing the risk and disturbances. To do this, a new uncertain environment, interval type-2 Pythagorean fuzzy set (IT2PFS) is introduced to assist the experts for assuring secure and reliable outcomes in hesitant situations. The operational laws on IT2PFS under Dombi t-norm and t-conorm are defined, which are further utilized to develop geometric Bonferroni mean and Bonferroni mean operators based on Choquet integral (CI) under IT2PFS. The aggregation operators can ennoble the pliability of the information blending process through the adjustment of several parameters and can make good interactions among the criteria with their weights. Thereafter, to determine the weights of the criteria, an empirical method, namely the decision making trial and evaluation laboratory (DEMATEL) is amplified by conjoining CI and IT2PFS into it. A multicriteria decision making method, namely, the multiattribute border approximation area comparison (MABAC) and subsequently the CI based grey relational analysis method are then employed to classify the suppliers and derive their corresponding weights with reference to the sustainable criteria. Over and above, a new multiobjective optimization model is exhibited to uphold the purchasing managers for assembling suppliers keeping in mind the sustainability aspects and overall risk of suppliers by following Markowitz portfolio theory. A real-life supply chain management problem is demonstrated to elucidate the aptness and efficiency of the proposed work. Furthermore, the effectiveness and importance of the study are validated via the comparative analysis with existing approaches. The main contribution of the study is in handling the difficulty and confusion that arise during information gathering, information aggregation, suppliers evaluation and order allocation phases.

Due to the steep rise in global population, urbanization, and industrialization, most of the cities in the world today are witnessing increased carbon footprints and reduced per capita space. In such a scenario, vehicle sharing and carpooling systems, specifically with electric vehicles (EV), can significantly help due to the reduced cost of ownership, maintenance, and parking space. In this article, we study the challenging problem of optimal sizing and efficient routing for an electric vehicle-on-demand system. Users demand EVs at the pooling stations at different time instances with individual deadlines to reach the destinations. The objective is to fulfill all the demands respecting the deadlines with minimum investment, which essentially translates to minimizing the total number of EVs. We define the problem formally using mixed-integer linear programming formulation and propose a set of intelligent and efficient heuristic algorithms to solve it efficiently. The proposed algorithms’ performances are tested and validated in a simulated environment on a reasonable size city network with many EV demands. The results obtained show that the proposed heuristic algorithms are competent by reducing 200–360 EVs per day on a network of 282 charging ports, indicating their scalability to be implemented in real-world scenarios.

We propose WLAMr-DDH which is an efficient convolutional neural network based approach for deep semantic hashing for face image retrieval. The proposed model is a substantial improvement upon the end-to-end deep hash learning in the classification framework, where the model learns to generate hash codes and hash functions in a single stage. It combines the strong representation capabilities obtained from the proposed weighted lateral connections for multi-scale feature representation with our novel augmentation mask layer. The augmentation mask is a simple yet effective way to obtain weighted saliency maps at the final convolution layer which makes the prominent features to have more contribution in the final output. Also, with weight normalization via re-parametrization, which decouples the direction from magnitude, faster convergence is achieved. Results obtained from the proposed framework on publicly available datasets shows substantial improvements and outperforms several state-of-the-art methods by good margin. The approach is generic and WLAM block can be easily plugged in a multitude of image retrieval and hashing applications.

Power shortage is a serious issue especially in developing nations. Such power deficits are traditionally handled through rolling blackouts - a service area is divided into subareas, each of which is denied power during a designated time in the day. Today, smart grids provide the opportunity of avoiding complete blackouts, converting them to brownouts which allow selective provisioning of power supply to support essential loads while curtailing supply to less critical loads. We formulate the brownout based power distribution problem as aninteger linear programming (ILP)and show that solution strategies such asconventional dynamic programming (DP)impose substantial overheads. So, we propose thestreamlined DP-based priority level allocator (SDPA)which utilizes the discrete nature of power demands of each subarea and generates the overall optimal solution far quicker by focusing on a lower number of non-dominating partial DP-solutions.SDPAis found to be about 9 to 33 times faster thanDPand applicable to real-time brown-out based power distribution in moderate sized grids. However, evenSDPAmay fail to meet the real-time requirements of dynamic power imbalance mitigation in very large grids. So, a fast yet effective power adjustment approach namely,Proportionally Balanced Priority level Allocator (PBPA), has been designed and implemented. Experimental results show that although solutions provided byPBPAcould be less effective by upto 12 percent compared to optimal dynamic programming based schemes, being about 4 orders of magnitude faster, it can be deployed for real time allocations of power.

FlexRay offers high-speed reliable transmission in the vehicle domain. Thus, it is one of the most popular in-vehicle communication protocols for the applications that are safety-critical or applications with very low transmission time requirement. Due to the real-time constraint, the worst case end-to-end delay (EED) for such applications transmitting over a shared bus must be known. This paper addresses the problem of computation of EED for a given set of tasks with precedence constraint and corresponding messages, which are scheduled with slot multiplexing for event-triggered communication over the dynamic segment of FlexRay. First, computing precise delay has been shown to be an intractable problem. Subsequently, motivated by the intractability, an efficient estimation technique has been proposed to compute approximate EED. Unlike the common practice of considering the network in isolation, we present a comprehensive timing analysis, which considers the effects of task execution as well. Moreover, EED is allowed to be greater than interactivation time of task graph. The extensive simulation has been performed on the test cases generated with uniform as well as normal distribution.Note to Practitioners—In the past few decades, due to the large-scale use and sophistication of automotive electronic systems, various in-vehicle communication networks and protocols are also evolving. FlexRay is one such in-vehicle communication protocol, which provides the mechanism for the fault-tolerant high-speed communication over FlexRay bus. FlexRay is widely being employed for safety-critical applications in the automotive domain. For such applications, often the timing constraint is imposed with a bound on the worst case EED. Thus, timing properties of the designed network must be verified before applying it on the real domain such that the worst case EED for the propagation of an event at sensor end to actuator end is known. The proposed estimation technique can be used to compute approximate EED for an application scheduled with other applications over the dynamic segment of FlexRay. In addition, various guidelines for designing the network in order to reduce overestimation have also been presented.

Higher reliability and lower energy consumption are conflicting, yet among the most important design objectives for the real-time systems. Moreover, in the domain of real-time systems, non-preemptive scheduling is relatively unexplored with objectives such as reliability and energy. Thus we propose an active replication based framework to schedule a set of periodic real-time tasks in the non-preemptive heterogeneous environment such that the given reliability and timing constraints are satisfied whereas the energy consumption is minimized. First, we formulate the problem as a constraint optimization problem that provides an optimal solution; however, it does not scale well. Thus, we also propose heuristics which apply reservation of processors and reallocation of jobs, to compute suboptimal solution efficiently in terms of energy consumption as well as schedulability. Heuristics make use of the interplay of task-level reliability target, reliability of replicas, number of replicas, reliability of tasks, and energy consumption. We perform an experimental study on the test cases generated by extending UUnisort algorithm[1]and observe the effect of various simulation parameters on energy consumption and schedulability.

Fog computing is a non-trivial extension of cloud computing to overcome many inherent limitations, such as huge network bandwidth and high latency. Fog computing involves a significantly large number of fog instances, unlike few centralized cloud servers. This work is an attempt to address two crucial challenges in fog computing systems in an integrated manner, namely the pricing of the resources and offloading of real-time tasks to appropriate fog instances. We propose an integrated framework to devise the pricing of the resources to maximize the social welfare and an offloading mechanism to minimize the cost of execution of the workloads with the timing constraint.

With the evolution of the technology, digital microfluidic (DMF) biochips have become a vital part of biochemical research. Hence, it is required to consider the reliability of the different fluidic operations performed on a biochip. Sample preparation is an important process of any real-life bioprotocol implementation on a DMF biochip. In this process, sequence of mixing and dilution steps are determined to get the desired target concentrations. The mixers used for performing mix-split steps may incorporate some noise during mixing and can result in the erroneous concentrations of the reagents. Thus, the reliability analysis of the resultant target concentration is necessary and methods are required to be developed to reduce these concentration errors. In this paper, the error and reliability models are discussed to compare the reliabilities of the existing mixing algorithms. Simulation results show that for a given target ratio, reliability of common dilution operation sharing (Liuet al., ICCAD-2013) is higher. We also discuss the mixer assignment techniques and the heuristic approach is found to quickly provide the better order of mixer assignment in order to achieve highly reliable mixture after sample preparation.

In this work, we propose a multi-objective optimal scheduling strategy for air-conditioning devices to optimize both energy consumption and thermal comfort for the users. We propose a graph-based modeling for the problem and utilize Johnson's all elementary circuit finding algorithm to obtain the desired solutions. The proposed methodology has been experimented on test cases that mimic real-world scenario, and further, the applicability of Karp's minimum mean cycle algorithm is also studied in this problem set-up.

Existing electrical grid systems have a limited amount of real-time monitoring and controlling capabilities of energy generation and consumption facilities, which trigger various technical issues including voltage overloading, demand–supply mismatch, peak load consumption, etc. Some of the primary reasons for these key issues have been identified to be the inefficient utilization of energy infrastructure and uncoordinated power consumption pattern among the consumers. In this paper, we propose a coordinated load scheduling and controlling algorithm to schedule controllable appliances with the objective to minimize peak load consumption. For this purpose, we model the problem into the strip packing problem, a well-known NP-hard problem, and discuss the applicability of existing heuristics in our problem setup. We then discuss a new offline heuristic solution, named MinPeak, specifically designed for load scheduling problem. We have conducted comprehensive simulation studies using available benchmark data sets and have performed extensive comparative analyses of the proposed algorithm with some of the well-known heuristics for strip packing problem. Furthermore, experiments have been carried out using practical electricity consumption data to evaluate the performance of the algorithm in real life. The results obtained are very encouraging in terms of reducing peak load consumption and overall efficiency of the system.

Residential, commercial, and industrial buildings have been reported to consume a large portion of the generated energy. With the introduction of smart grid and its energy optimization techniques, it is now possible to efficiently manage and control consumers’ energy usage to fulfil their demands with the existing energy generation infrastructure, which otherwise seems to be a backbreaking challenge. This paper presents an efficient energy management solution for buildings with a large number of thermostatic devices (air conditioners) that maintain the temperature of different thermal zones in a predefined range. The primary objective of this paper is to schedule the thermostatic devices in order to reduce total energy consumption by these devices when they are in operation for a very long duration of time, while maintaining the other constraints. We formulate it as a graph problem where minimum mean cycle will provide the desired solution. The proposed methodology ensures that at no point in time the power consumption goes beyond a certain peak power consumption limit. We also enhance the methodology to reduce peak load consumption. Furthermore, a fast greedy approach has been developed to efficiently scale up the aforementioned scheduling scheme for a large number of devices. Experimental results show that significant improvements can be obtained by the proposed approaches over existing algorithms in reducing average energy consumption.

Motivated by recent high bandwidth communication systems, Inter-Symbol Interference (ISI) channels with 1-bit quantized output are considered under an average-power-constrained continuous input. While the exact capacity is difficult to characterize, an approximation that matches with the exact channel output up to a probability of error is provided. The approximation does not have additive noise, but constrains the channel output (without noise) to be above a threshold in absolute value. The capacity under the approximation is computed using methods involving standard Gibbs distributions. Markovian achievable schemes approaching the approximate capacity are provided. The methods used over the approximate ISI channel result in ideas for practical coding schemes for ISI channels with 1-bit output quantization.

We present a symbolic-event-propagation-based scheme to generate hazard-free tests for robust path delay faults. This approach identifies all robustly testable paths in a circuit and the corresponding complete set of test vectors. We address the problem of finding a minimal set of test vectors that covers all robustly testable paths. We propose greedy and simulated-annealing-based algorithms to find the same. Results on ISCAS89 benchmark circuits show a considerable reduction in test vectors for covering all robustly testable paths.

Present-day designers require deep reasoning methods to analyze circuit timing. This includes analysis of effects of dynamic behavior (like glitches) on critical paths, simultaneous switching, and identification of specific patterns and their timings. This paper proposes a novel approach that uses a combination of symbolic event propagation and temporal reasoning to extract timing properties of gate-level circuits. The formulation captures the complex situations like triggering of traditional false paths and simultaneous switching in a unified symbolic representation in addition to identifying false paths, critical paths, as well as conditions for such situations. This information is then represented as an event-time graph. A temporal logic on events is proposed that can be used to formulate a wide class of useful queries for various input scenarios. These include maximum/minimum delays, transition times, duration of patterns, etc. An algorithm is developed that retrieves answers to such queries from the event-time graph. A binary decision diagram-based implementation of this system has been made. Results on the International Symposium on Circuits and Systems (ISCAS)85 benchmarks are presented.

Present day designers require deep reasoning methods to analyze circuit timing. This includes analysis of effects of dynamic behavior (like glitches) on critical paths, simultaneous switching and identification of specific patterns and their timings. This paper proposes a novel approach that uses a combination of symbolic event propagation and temporal reasoning to extract timing properties of gate-level circuits. The formulation captures complex situations like triggering of traditional false paths and simultaneous switching in a unified symbolic representation in addition to identifying false paths, critical paths as well as conditions for such situations. This information is then represented as an event-time graph. A simple temporal logic on events is proposed that can be used to formulate a wide class of useful queries for various input scenarios. These include maximum/minimum delays, transition times, duration of patterns, etc. An algorithm is developed that retrieves answers to such queries from the event-time graph. A complete BDD based implementation of this system has been made. Results on the ISCAS85 benchmarks indicate very interesting properties of these circuits.