Polynomial-time algorithms for generation of prime implicants

AbstractA notion of a neighborhood cube of a term of a Boolean function represented in the canonical disjunctive normal form is introduced. A relation between neighborhood cubes and prime implicants of a Boolean function is established. Various aspects of the problem of prime implicants generation are identified and neighborhood cube-based algorithms for their solution are developed. The correctness of algorithms is proven and their time complexity is analyzed. It is shown that all presented algorithms are polynomial in the number of minterms occurring in the canonical disjunctive normal form representation of a Boolean function. A summary of the known approaches to the solution of the problem of the generation of prime implicants is also included

Determination of prime implicants by differential evolution for the dynamic reliability analysis of non-coherent nuclear systems

open4We present an original computational method for the identification of prime implicants (PIs) in non-coherent structure functions of dynamic systems. This is a relevant problem for dynamic reliability analysis, when dynamic effects render inadequate the traditional methods of minimal cut-set identification. PIs identification is here transformed into an optimization problem, where we look for the minimum combination of implicants that guarantees the best coverage of all the minterms. For testing the method, an artificial case study has been implemented, regarding a system composed by five components that fail at random times with random magnitudes. The system undergoes a failure if during an accidental scenario a safety-relevant monitored signal raises above an upper threshold or decreases below a lower threshold. Truth tables of the two system end-states are used to identify all the minterms. Then, the PIs that best cover all minterms are found by Modified Binary Differential Evolution. Results and performances of the proposed method have been compared with those of a traditional analytical approach known as Quine-McCluskey algorithm and other evolutionary algorithms, such as Genetic Algorithm and Binary Differential Evolution. The capability of the method is confirmed with respect to a dynamic Steam Generator of a Nuclear Power Plant.Di Maio, Francesco; Baronchelli, Samuele; Vagnoli, Matteo; Zio, EnricoDI MAIO, Francesco; Baronchelli, Samuele; Vagnoli, Matteo; Zio, Enric

SoK: Modeling for Large S-boxes Oriented to Differential Probabilities and Linear Correlations (Long Paper)

Automatic methods for differential and linear characteristic search are well-established at the moment. Typically, the designers of novel ciphers also give preliminary analytical findings for analysing the differential and linear properties using automatic techniques. However, neither MILP-based nor SAT/SMT-based approaches have fully resolved the problem of searching for actual differential and linear characteristics of ciphers with large S-boxes. To tackle the issue, we present three strategies for developing SAT models for 8-bit S-boxes that are geared toward differential probabilities and linear correlations. While these approaches cannot guarantee a minimum model size, the time needed to obtain models is drastically reduced. The newly proposed SAT model for large S-boxes enables us to establish that the upper bound on the differential probability for 14 rounds of SKINNY-128 is 2^{-131}, thereby completing the unsuccessful work of Abdelkhalek et al. We also analyse the seven AES-based constructions C1 - C7 designed by Jean and Nikolic and compute the minimum number of active S-boxes necessary to cause an internal collision using the SAT method. For two constructions C3 and C5, the current lower bound on the number of active S-boxes is increased, resulting in a more precise security analysis for these two structures

Automated synthesis and optimization of multilevel logic circuits.

With the increased complexity of Very Large Scaled Integrated (VLSI) circuits, multilevellogic synthesis plays an even more important role due to its flexibility and compactness.The history of symbolic logic and some typical techniques for multilevel logic synthesisare reviewed. These methods include algorithmic approach; Rule-Based approach; BinaryDecision Diagram (BDD) approach; Field Programmable Gate Array(FPGA) approachand several perturbation applications.One new kind of don't cares (DCs), called functional DCs has been proposed for multilevellogic synthesis. The conventional two-level cubes are generalized to multilevel cubes.Then functional DCs are generated based on the properties of containment. The conceptof containment is more general than unateness which leads to the generation of newDCs. A separate C program has been developed to utilize the functional DCs generatedas a Boolean function is decomposed for both single output and multiple output functions.The program can produce better results than script.rugged of SIS, developed by UC Berkeley,both in area and speed in less CPU time for a number of testcases from MCNC andIWLS'93 benchmarks.In certain applications ANDjXOR (Reed-Muller) logic has shown some attractive advantagesover the standard Boolean logic based on AND JOR operations. A bidirectionalconversion algorithm between these two paradigms is presented based on the concept of polarityfor sum-of-products (SOP) Boolean functions, multiple segment and multiple pointerfacilities. Experimental results show that the algorithm is much faster than the previouslypublished programs for any fixed polarity. Based on this algorithm, a new technique calledredundancy-removal is applied to generalize the idea to very large multiple output Booleanfunctions. Results for benchmarks with up to 199 inputs and 99 outputs are presented.Applying the preceding conversion program, any Boolean functions can be expressedby fixed polarity Reed-Muller forms. There are 2n polarities for an n-variable function andthe number of product terms depends on these polarities. The problem of exact polarityminimization is computationally extensive and current programs are only suitable whenn :::; 15. Based on the comparison of the concepts of polarity in the standard Boolean logicand Reed-Muller logic, a fast algorithm is developed and implemented in C language whichcan find the best polarity for multiple output functions. Benchmark examples of up to 25inputs and 29 outputs run on a personal computer are given.After the best polarity for a Boolean function is calculated, this function can be furthersimplified using mixed polarity methods by combining the adjacent product terms. Hence,an efficient program is developed based on decomposition strategy to implement mixedpolarity minimization for both single output and very large multiple output Boolean functions.Experimental results show that the numbers of product terms are much less thanthe results produced by ESPRESSO for some categories of functions

Anytime algorithms for ROBDD symmetry detection and approximation

Reduced Ordered Binary Decision Diagrams (ROBDDs) provide a dense and memory efficient representation of Boolean functions. When ROBDDs are applied in logic synthesis, the problem arises of detecting both classical and generalised symmetries. State-of-the-art in symmetry detection is represented by Mishchenko's algorithm. Mishchenko showed how to detect symmetries in ROBDDs without the need for checking equivalence of all co-factor pairs. This work resulted in a practical algorithm for detecting all classical symmetries in an ROBDD in O(|G|³) set operations where |G| is the number of nodes in the ROBDD. Mishchenko and his colleagues subsequently extended the algorithm to find generalised symmetries. The extended algorithm retains the same asymptotic complexity for each type of generalised symmetry. Both the classical and generalised symmetry detection algorithms are monolithic in the sense that they only return a meaningful answer when they are left to run to completion. In this thesis we present efficient anytime algorithms for detecting both classical and generalised symmetries, that output pairs of symmetric variables until a prescribed time bound is exceeded. These anytime algorithms are complete in that given sufficient time they are guaranteed to find all symmetric pairs. Theoretically these algorithms reside in O(n³+n|G|+|G|³) and O(n³+n²|G|+|G|³) respectively, where n is the number of variables, so that in practice the advantage of anytime generality is not gained at the expense of efficiency. In fact, the anytime approach requires only very modest data structure support and offers unique opportunities for optimisation so the resulting algorithms are very efficient. The thesis continues by considering another class of anytime algorithms for ROBDDs that is motivated by the dearth of work on approximating ROBDDs. The need for approximation arises because many ROBDD operations result in an ROBDD whose size is quadratic in the size of the inputs. Furthermore, if ROBDDs are used in abstract interpretation, the running time of the analysis is related not only to the complexity of the individual ROBDD operations but also the number of operations applied. The number of operations is, in turn, constrained by the number of times a Boolean function can be weakened before stability is achieved. This thesis proposes a widening that can be used to both constrain the size of an ROBDD and also ensure that the number of times that it is weakened is bounded by some given constant. The widening can be used to either systematically approximate an ROBDD from above (i.e. derive a weaker function) or below (i.e. infer a stronger function). The thesis also considers how randomised techniques may be deployed to improve the speed of computing an approximation by avoiding potentially expensive ROBDD manipulation.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

A survey of DA techniques for PLD and FPGA based systems

Programmable logic devices (PLDs) are gaining in acceptance, of late, for designing systems of all complexities ranging from glue logic to special purpose parallel machines. Higher densities and integration levels are made possible by the new breed of complex PLDs and FPGAs. The added complexities of these devices make automatic computer aided tools indispensable for achieving good performance and a high usable gate-count. In this article, we attempt to present in an unified manner, the different tools and their underlying algorithms using an example of a vending machine controller as an illustrative example. Topics covered include logic synthesis for PLDs and FPGAs along with an in-depth survey of important technology mapping, partitioning and place and route algorithms for different FPGA architectures.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31206/1/0000108.pd

Logic synthesis and optimisation using Reed-Muller expansions

This thesis presents techniques and algorithms which may be employed to represent, generate and optimise particular categories of Exclusive-OR SumOf-Products (ESOP) forms. The work documented herein concentrates on two types of Reed-Muller (RM) expressions, namely, Fixed Polarity Reed-Muller (FPRM) expansions and KROnecker (KRO) expansions (a category of mixed polarity RM expansions). Initially, the theory of switching functions is comprehensively reviewed. This includes descriptions of various types of RM expansion and ESOP forms. The structure of Binary Decision Diagrams (BDDs) and Reed-Muller Universal Logic Module (RM-ULM) networks are also examined. Heuristic algorithms for deriving optimal (sub-optimal) FPRM expansions of Boolean functions are described. These algorithms are improved forms of an existing tabular technique [1]. Results are presented which illustrate the performance of these new minimisation methods when evaluated against selected existing techniques. An algorithm which may be employed to generate FPRM expansions from incompletely specified Boolean functions is also described. This technique introduces a means of determining the optimum allocation of the Boolean 'don't care' terms so as to derive equivalent minimal FPRM expansions. The tabular technique [1] is extended to allow the representation of KRO expansions. This new method may be employed to generate KRO expansions from either an initial incompletely specified Boolean function or a KRO expansion of different polarity. Additionally, it may be necessary to derive KRO expressions from Boolean Sum-Of-Products (SOP) forms where the product terms are not minterms. A technique is described which forms KRO expansions from disjoint SOP forms without first expanding the SOP expressions to minterm forms. Reed-Muller Binary Decision Diagrams (RMBDDs) are introduced as a graphical means of representing FPRM expansions. RMBDDs are analogous to the BDDs used to represent Boolean functions. Rules are detailed which allow the efficient representation of the initial FPRM expansions and an algorithm is presented which may be employed to determine an optimum (sub-optimum) variable ordering for the RMBDDs. The implementation of RMBDDs as RM-ULM networks is also examined. This thesis is concluded with a review of the algorithms and techniques developed during this research project. The value of these methods are discussed and suggestions are made as to how improved results could have been obtained. Additionally, areas for future work are proposed

A computer-aided design for digital filter implementation

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