Algorithms in computer-aided design of VLSI circuits.

With the increased complexity of Very Large Scale Integrated (VLSI) circuits,Computer Aided Design (CAD) plays an even more important role. Top-downdesign methodology and layout of VLSI are reviewed. Moreover, previouslypublished algorithms in CAD of VLSI design are outlined.In certain applications, Reed-Muller (RM) forms when implemented withAND/XOR or OR/XNOR logic have shown some attractive advantages overthe standard Boolean logic based on AND/OR logic. The RM forms implementedwith OR/XNOR logic, known as Dual Forms of Reed-Muller (DFRM),is the Dual form of traditional RM implemented with AND /XOR.Map folding and transformation techniques are presented for the conversionbetween standard Boolean and DFRM expansions of any polarity. Bidirectionalmulti-segment computer based conversion algorithms are also proposedfor large functions based on the concept of Boolean polarity for canonicalproduct-of-sums Boolean functions. Furthermore, another two tabular basedconversion algorithms, serial and parallel tabular techniques, are presented forthe conversion of large functions between standard Boolean and DFRM expansionsof any polarity. The algorithms were tested for examples of up to 25variables using the MCNC and IWLS'93 benchmarks.Any n-variable Boolean function can be expressed by a Fixed PolarityReed-Muller (FPRM) form. In order to have a compact Multi-level MPRM(MMPRM) expansion, a method called on-set table method is developed.The method derives MMPRM expansions directly from FPRM expansions.If searching all polarities of FPRM expansions, the MMPRM expansions withthe least number of literals can be obtained. As a result, it is possible to findthe best polarity expansion among 2n FPRM expansions instead of searching2n2n-1 MPRM expansions within reasonable time for large functions. Furthermore,it uses on-set coefficients only and hence reduces the usage of memorydramatically.Currently, XOR and XNOR gates can be implemented into Look-Up Tables(LUT) of Field Programmable Gate Arrays (FPGAs). However, FPGAplacement is categorised to be NP-complete. Efficient placement algorithmsare very important to CAD design tools. Two algorithms based on GeneticAlgorithm (GA) and GA with Simulated Annealing (SA) are presented for theplacement of symmetrical FPGA. Both of algorithms could achieve comparableresults to those obtained by Versatile Placement and Routing (VPR) toolsin terms of the number of routing channel tracks

Computer aided synthesis and optimisation of electronic logic circuits

In this thesis, a variety of algorithms for synthesis and optimisation of combinational and sequential logic circuits are developed. These algorithms could be part of new commercial EGAD package for future VLSI digital designs. The results show that considerable saving in components can be achieved resulting in simpler designs that are smaller, cheaper, consume less power and easier to test. The purpose of generating different sets of coefficients related to Reed Muller (RM) is that they contain different number of terms; therefore the minimum one can be selected to design the circuits with reduced gate count. To widen the search space and achieve better synthesis tools, representations of Mixed Polarity Reed Muller (MPRM), Mixed Polarity Dual Reed Muller (MPDRM), and Pseduo Kronecker Reed Muller (PKRO RM) expansions are investigated. Efficient and fast combinatorial techniques and algorithms are developed for the following: â Bidirectional conversion between MPRM/ MPDRM form and Fixed Polarity Reed Muller forms (FPRM)/Fixed Polarity Dual Reed Muller forms (FPDRM) form respectively. The main advantages for these techniques are their simplicity and suitability for single and multi output Boolean functions. â Computing the coefficients of any polarity related to PKRO_RM class starting from FPRM coefficients or Canonical Sum of Products (CSOP). â Computing the coefficients of any polarity related to MPRM/or MPDRM directly from standard form of CSOP/Canonical Product of sums (CPOS) Boolean functions, respectively. The proposed algorithms are efficient in terms of CPU time and can be used for large functions. For optimisation of combinational circuits, new techniques and algorithms based on algebraic techniques are developed which can be used to generate reduced RM expressions to design circuits in RM/DRM domain starting from FPRM/FPDRM, respectively. The outcome for these techniques is expansion in Reed Muller domain with minimal terms. The search space is 3`" Exclusive OR Sum of Product (ESOP)/or Exclusive NOR Product of Sums (ENPOS) expansions. Genetic Algorithms (GAs) are also developed to optimise combinational circuits to find optimal MPRM/MPDRM among 3° different polarities without the need to do exhaustive search. These algorithms are developed for completely and incompletely specified Boolean functions. The experimental results show that GA can find optimum solutions in a short time compared with long time required running exhaustive search in all the benchmarks tested. Multi Objective Genetic Algorithm (MOGA) is developed and implemented to determine the optimal state assignment which results in less area and power dissipation for completely and incompletely specified sequential circuits. The goal is to find the best assignments which reduce the component count and switching activity simultaneously. The experimental results show that saving in components and switching activity are achieved in most of the benchmarks tested compared with recently published research. All algorithms are implemented in C++.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Combinational logic synthesis based on the dual form of Reed-Muller representation

In certain applications, AND/XOR (Reed-Muller), and ORlXNOR (Dual form of Reed-Muller) logic have shown some attractive advantages over the standard Sum of Products (SOP) and Product of Sums (POS). Bidirectional conversion algorithms between SOP and AND/XOR also between POS and ORlXNOR based on Sparse and partitioning techniques are presented for multiple output Boolean functions. The developed programs are tested for some benchmarks with up to 20 inputs and 40 outputs. A new direct method is presented to calculate the coefficients of the Fixed Polarity Dual Reed-Muller (FPDRM) from the truth vector of the POS. Any Boolean function can be expressed by FPDRM forms. There are 211 polarities for an n-variable function and the number of sum terms depends on these polarities. Finding the best polarity is costly interims of CPU time, in order to search for the best polarity which will lead to the minimum number of sums for a particular function. Therefore, an algorithm is developed to compute all the coefficients of the Fixed Polarity Dual Reed-Muller (FPDRM) with polarity p from any polarity q. This technique is used to find the best polarity of FPDRM among the 211 fixed polarities. The algorithm is based on the Dual- polarity property and the Gray code strategy. Therefore, there is no need to start from POS form to find FPDRM coefficients for all the polarities. The proposed methods are efficient in terms of memory size and CPU time. A fast algorithm is developed and implemented in C language which can convert between POSs and FPDRMs. The program was tested for up to 23 variables. A modified version of the same program was used to find the best polarity. For up to 13 variables the CPU time was less than 42 seconds. To search for the optimal polarity for large number of variables and to reduce the se arch time 0 ffinding the 0 ptimal polarity 0 fthe function, two new algorithms are developed and presented in this thesis. The first one is used to convert between P OS and Positive Polarity Dual Reed-Muller (PPDRM) forms. The second algorithm will find the optimal fixed polarity for the FPDRM among the 211 different polarities for large n-variable functions. The most popular minimization criterion of the FPDRM form is obtained by the exhaustive search of the entire polarity vector. A non-exhaustive method for FPDRM expansions is presented. The new algorithms are based on separation of the truth vector (T) of POSs around each variable Xi into two groups. Instead of generating all of the polarity sets and searching for the best polarity, this algorithm will find the optimal polarity using the separation and sparse techniques, which will lead to optimal polarity. Time efficiency and computing speed are thus achieved in this technique. The algorithms don't require a large size of memory and don't require a long CPU time. The two algorithms are implemented in C language and tested for some benchmark. The proposed methods are fast and efficient as shown in the experimental results and can be used for large number of variables.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Computer aided synthesis and optimisation of electronic logic circuits

In this thesis, a variety of algorithms for synthesis and optimisation of combinational and sequential logic circuits are developed. These algorithms could be part of new commercial EGAD package for future VLSI digital designs. The results show that considerable saving in components can be achieved resulting in simpler designs that are smaller, cheaper, consume less power and easier to test.The purpose of generating different sets of coefficients related to Reed Muller (RM) is that they contain different number of terms; therefore the minimum one can be selected to design the circuits with reduced gate count. To widen the search space and achieve better synthesis tools, representations of Mixed Polarity Reed Muller (MPRM), Mixed Polarity Dual Reed Muller (MPDRM), and Pseduo Kronecker Reed Muller (PKRO RM) expansions are investigated. Efficient and fast combinatorial techniques and algorithms are developed for the following:- Bidirectional conversion between MPRM/ MPDRM form and Fixed Polarity Reed Muller forms (FPRM)/Fixed Polarity Dual Reed Muller forms (FPDRM) form respectively. The main advantages for these techniques are their simplicity and suitability for single and multi output Boolean functions.- Computing the coefficients of any polarity related to PKRO_RM class starting from FPRM coefficients or Canonical Sum of Products (CSOP).- Computing the coefficients of any polarity related to MPRM/or MPDRM directly from standard form of CSOP/Canonical Product of sums (CPOS) Boolean functions, respectively. The proposed algorithms are efficient in terms of CPU time and can be used for large functions.For optimisation of combinational circuits, new techniques and algorithms based on algebraic techniques are developed which can be used to generate reduced RM expressions to design circuits in RM/DRM domain starting from FPRM/FPDRM, respectively. The outcome for these techniques is expansion in Reed Muller domain with minimal terms. The search space is 3`" Exclusive OR Sum of Product (ESOP)/or Exclusive NOR Product of Sums (ENPOS) expansions.Genetic Algorithms (GAs) are also developed to optimise combinational circuits to find optimal MPRM/MPDRM among 3° different polarities without the need to do exhaustive search. These algorithms are developed for completely and incompletely specified Boolean functions. The experimental results show that GA can find optimum solutions in a short time compared with long time required running exhaustive search in all the benchmarks tested.Multi Objective Genetic Algorithm (MOGA) is developed and implemented to determine the optimal state assignment which results in less area and power dissipation for completely and incompletely specified sequential circuits. The goal is to find the best assignments which reduce the component count and switching activity simultaneously. The experimental results show that saving in components and switchingactivity are achieved in most of the benchmarks tested compared with recentlypublished research. All algorithms are implemented in C++

Combinational logic synthesis based on the dual form of Reed-Muller representation.

In certain applications, AND/XOR (Reed-Muller), and ORlXNOR (Dualform of Reed-Muller) logic have shown some attractive advantages over thestandard Sum of Products (SOP) and Product of Sums (POS). Bidirectionalconversion algorithms between SOP and AND/XOR also between POS andORlXNOR based on Sparse and partitioning techniques are presented for multipleoutput Boolean functions. The developed programs are tested for somebenchmarks with up to 20 inputs and 40 outputs.A new direct method is presented to calculate the coefficients of the FixedPolarity Dual Reed-Muller (FPDRM) from the truth vector of the POS. AnyBoolean function can be expressed by FPDRM forms. There are 211 polarities foran n-variable function and the number of sum terms depends on these polarities.Finding the best polarity is costly interims of CPU time, in order to search for thebest polarity which will lead to the minimum number of sums for a particularfunction. Therefore, an algorithm is developed to compute all the coefficients ofthe Fixed Polarity Dual Reed-Muller (FPDRM) with polarity p from any polarity q.This technique is used to find the best polarity of FPDRM among the 211 fixedpolarities. The algorithm is based on the Dual- polarity property and the Gray codestrategy. Therefore, there is no need to start from POS form to find FPDRMcoefficients for all the polarities. The proposed methods are efficient in terms ofmemory size and CPU time. A fast algorithm is developed and implemented in Clanguage which can convert between POSs and FPDRMs. The program was testedfor up to 23 variables. A modified version of the same program was used to findthe best polarity. For up to 13 variables the CPU time was less than 42 seconds.To search for the optimal polarity for large number of variables and toreduce the se arch time 0 ffinding the 0 ptimal polarity 0 fthe function, two newalgorithms are developed and presented in this thesis. The first one is used toconvert between P OS and Positive Polarity Dual Reed-Muller (PPDRM) forms.The second algorithm will find the optimal fixed polarity for the FPDRM amongthe 211 different polarities for large n-variable functions. The most popularminimization criterion of the FPDRM form is obtained by the exhaustive search ofthe entire polarity vector. A non-exhaustive method for FPDRM expansions ispresented. The new algorithms are based on separation of the truth vector (T) ofPOSs around each variable Xi into two groups. Instead of generating all of thepolarity sets and searching for the best polarity, this algorithm will find the optimalpolarity using the separation and sparse techniques, which will lead to optimalpolarity. Time efficiency and computing speed are thus achieved in this technique.The algorithms don't require a large size of memory and don't require a long CPUtime. The two algorithms are implemented in C language and tested for somebenchmark. The proposed methods are fast and efficient as shown in theexperimental results and can be used for large number of variables

Mathematical modelling and simulations of the ion transport through confined geometries

In this dissertation, we focus on different aspects of modelling ion transport in confined geometries. The transport of the ions through pores was first investigated in the 19th century for cell membranes. In the last years, there has been a significant increase in research of ion transport in nanoscale devices, such as nanopores, nanowires and many more. Especially synthetic pores have the potential to be used as nanoscale diodes, switches or in DNA sequencing. In this thesis, we investigate different modelling approaches and discuss their use and validity in various situations. The transport properties of nanoscale pores are strongly determined by the confined geometry as well as surface charges. Depending on the experimental setup considered finite size, electrostatic as well as electrochemical properties have to be resolved on various scales. This leads to a variety of models ranging from microscopic approaches, such as Molecular Dynamics, to macroscopic models like mean field theory. Since finite size effects and fluid dynamics effects should not be neglected in confined geometries various extensions of the Poisson-Nernst-Planck (PNP) system were introduced in the literature such as density functional theory or the coupling to fluid dynamics. Another challenge in ion transport modelling is the multiscale nature of the synthetic nanopores as their length scale is sometimes 104 times larger than their radial dimension. In the first part of the thesis, we develop a multiscale method that investigates the asymptotic behaviour of the PNP equations for long and narrow nanopores. The significant difference in the radial and lateral length scale allows us to decouple the system and to solve the behaviour in the boundary layers close to the charged pore walls correctly. Two new asymptotic methods were developed to describe the transport problem inside the pore. This asymptotic approximation serves as the basis for the numerical solver. We investigate the quality of the approximations for a variety of pores with different computational experiments. We present comparison of the microscopic quantities such as concentrations and electric potential as well as macroscopic quantities such as current voltage characteristic of exemplary pores. In the second part of the thesis, we compare the simulations of the PNP system with Monte-Carlo methods in the case of ion-channels. We discuss the different modelling assumptions as well as the advantages of both methods. Yet again we present results of the numerical simulations and discuss regimes in which both methods are valid. In the last part, we investigate the optimal control problem for nanopores. Here we want to modify the surface charge of a nanopore to obtain a desired behaviour, such as current-voltage characteristics or rectification behaviour. Two method are derived and implemented as a solution of stated problem

First principles conceptual models of chemical reactivity: Quantitative curly arrows and frontier orbitals

The computational expense associated with evaluating the electrostatic potential at a series of points stems from the presence of the position vector of each point in the denominator of a complicated 3D integral. A multipole expansion of the potential is significantly less computationally demanding, and yields a good approximation to the exact potential far from the charge distribution, but penetration effects lead to erroneous potentials at short range. In this work we present a new, computationally efficient method for approximating molecular electrostatics, the Reduced Orbital Potential Approximation, which combines multipole information with the full electrostatic potential arising from a simple model density, which can be evaluated at a fraction of the cost of the full density and incorporates some of the penetration correction without the need for damping functions. A new tool for the chemical interpretation of ab initio wavefunctions is also introduced which aims to establish a rigorous link between accurate computations of the potential energy surface and widely employed chemical descriptions of change during a reaction, such as frontier orbitals and \curly arrows". To achieve this, the total energy is partitioned tensorially into a global potential energy containing no quantities associated with chemical bonding and a covalency energy, for which the necessary assumptions and approximations for the use of chemically intuitive notions of reactivity can be considered valid. The scheme is applied first to canonical orbitals and shown to provide quantitative bonding information in line with classical molecular orbital diagrams, and then to localised orbitals in an attempt to recover frontier orbitals and curly-arrows. An extension to the method is also explored which enforces the assumption that core orbitals are mere spectators to reactions. The method is shown to give good results for a simple test system, and then applied to an SN2 reaction