PALMINI---fast Boolean minimizer for personal computers

Nguyen, Lam B. Q.; Perkowdki, M. A.; Goldstein, N.B.

doi:10.1145/37888.37985

Cited by 36 publications

(18 citation statements)

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“…Use of the disjoint cubes representation of Boolean functions has been found advantageous in many algorithms used in digital logic design [3], [18], [37]- [39], [54] and [108].…”

Section: Example Iv5mentioning

confidence: 99%

“…Hence, the new algorithm for the generation of the disjoint cubes representation of Boolean functions is important for both classical [3], [43] and [54] and spectral [15], [21], [22], and [25] techniques of designing digital logic. Similar preprocessing algorithms were described and used previously in the PAL-MINI [37], UMINI [38], EXORCISM [18], and EXORCISM-MY [39] programs, but the new algorithm is more efficient since it does not use sorting and minimizes the number of cube calculus operations used.…”

Section: V2 Algorl1hm To Generate Disjoint Cubesmentioning

confidence: 99%

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Spectral Methods for Boolean and Multiple-Valued Input Logic Functions

Falkowski¹

2000

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Spectral techniques in digital logic design have been known for more than thirty years. They have been used for Boolean function classification, disjoint decomposition, parallel and serial linear decomposition, spectral translation synthesis (extraction of 2 linear pre-and post-filters), multiplexer synthesis, prime implicant extraction by spectral summation, threshold logic synthesis, estimation of logic complexity, testing, and state assignment.This dissertation resolves many important issues concerning the efficient application of spectral methods used in the computer-aided design of digital circuits. The main obstacles in these applications were, up to now, memory requirements for computer systems and lack of the possibility of calculating spectra direcdy from Boolean equations.By using the algorithms presented here these obstacles have been overcome. Moreover, the methods presented in this dissertation can be regarded as representatives of a whole family of methods and the approach presented can 1Y: easily adapted to other orthogonal transforms used in digital logic design. Algorithms are shown for Adding, Arithmetic, and Reed-Muller transforms. However, the main focus of this dissertation is on the efficient computer calculation of Rademacher-Walsh spectra of Boolean functions, since this particular ordering of Walsh transforms is most frequendy used in digital logic design.A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON-and DC-cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from non-disjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. By such an approach each spectral coefficient can be calculated separately or all the coefficients can be calculated in parallel. These advantages are absent in the existing methods. The possibility of calculating only some coefficients is very important since there are many spectral methods in digital logic design for which the values of only a few selected coefficients are needed.Most of the current methods used in the spectral domain deal only with completely specified Boolean functions. On the other hand, all of the algorithms introduced here are valid, not only for completely specified Boolean functions, but for functions with don't cares. Don't care minterms are simply represented in the form of disjoint cubes.The links between spectral and classical methods used for designing digital cir- A new algorithm is shown that converts the disjoint cube representation of Boolean functions into fixed-polarity Generalized Reed-Muller Expansions (GRME).Since the known fast algorithm that generates the GR..\1E, based on the factorization of 4 the Reed-Muller transform matrix, always starts from the truth table (mintenns) of a Boole...

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“…Use of the disjoint cubes representation of Boolean functions has been found advantageous in many algorithms used in digital logic design [3], [18], [37]- [39], [54] and [108].…”

Section: Example Iv5mentioning

confidence: 99%

Section: V2 Algorl1hm To Generate Disjoint Cubesmentioning

confidence: 99%

Spectral Methods for Boolean and Multiple-Valued Input Logic Functions

Falkowski¹

2000

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“…Because the size of the cubes resulting from a disjoint sharp operation is always smaller than the initial cubes, they are inserted after the initial cubes. Therefore, the next two cubes that are compared are cube [l] and cube [3] of Figure 3.b, and so on.…”

Section: Circuit Realization Based On Spectramentioning

confidence: 99%

“…In the beginning of the eighties the research in logic design was concentrated on the development of fast two-level AND-OR logic minimizers like the well known ESPRESSO [1,2], and PALMINI [3]. However, most logic functions have a smaller and often faster circuit realization as a multi-level circuit.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal and Nonorthogonal Expansions for Multi-Level Logic Synthesis for Nearly Linear Functions and their Application to Field Programmable Gate Array Mapping

Schäfer¹

2000

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The growing complexity of integrated circuits and the large variety of architectures of Field Programmable Gate Arrays (FPGAs) require sophi~ticated logic design tools. In 2 the beginning of the eighties the research in logic design was concentrated on the development of fast two-level AND-OR logic minimizers like the well known ESPRESSO. However, most logic functions have a smaller and often faster circuit realization as a multi-level circuit. Thus, synthesis tools emerged for the minimization of the circuit area in a multi-level realization. Most of these synthesis tools are based on the "unate paradigm". Therefore, the synthesis methods are only advantageous for functions having a minimal circuit realization based on AND-OR gates. However, many common functions have a minmal circuit realization having a mix of AND, OR and EXOR gates like counters, adders, multipliers, and parity generators. Therefore, the design of such functions with synthesis tools based on the "unated paradigm" is very inefficient Circuits incorporating the EXOR gate have received less attention than AND-OR circuits because the EXOR gate was perceived as slower and larger in terms of its circuit realization than the AND and the OR gate. However, the upcoming of Field Programmable Gate Arrays (FPGAs) like the Xilinx Table-Look-Up (TLU) architecture the Actel ACT™ series and the CLi 6000 series from Concurrent Logic, which allow the realization of the EXOR gate with the same speed and circuit cost as the AND and OR gate, eliminates the disadvantages of the EXOR gate over the AND and OR gate. Thus, there is a strong need for logic synthesis tools that take advantage of EXOR gates.The mapping to th~new FPGAs recently obtained an increased interest. The developed synthesis algorithms for FPGAs are based on the mapping and restructuring of the Directed Acyclic Graph (DAG) representation of the logic function. Even though the new FPGAs allow the realization of the EXOR gate without any speed and circuit size penalty in comparison to the AND and OR gate, the synthesis methods have been based on the "unate paradigm".To overcome the disadvantages of the current logic synthesis tools with respect to (nearly) linear functions and FPGA synthesis, this dissertation introduces an extended 3 theory of spectral methods for multiple-valued input, incompletely specified binary output logic. The spectral methods have not been popular in logic synthesis because of their four major drawbacks:(1) the computational complexity, especially if no Fast Transform exists, (2) the memory requirement to store the function in the necessary minterm representation, (3) they can not take efficiently advantage of incompletely specified functions, (4) suitable only for few applications in logic synthesis.To overcome the two last stated drawbacks, this dissertation introduces the T spectrum. The T spectrum separates the information obtained for the specified and not specified parts of the underlying function. Thus, it is possible to determine directly the contribution of the sp...

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“…These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PRESTO [4], [33], MINI [15], ESPRESSO [3], and others [10], [24] produce nonminimum ISOPs.…”

Section: Introductionmentioning

confidence: 99%

Worst and best irredundant sum-of-products expressions

Sasao

Butler

2001

IEEE Trans. Comput.

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AbstractÐIn an irredundant sum-of-products expression (ISOP), each product is a prime implicant (PI) and no product can be deleted without changing the function. Among the ISOPs for some function f, a worst ISOP (WSOP) is an ISOP with the largest number of PIs and a minimum ISOP (MSOP) is one with the smallest number. We show a class of functions for which the Minato-Morreale ISOP algorithm produces WSOPs. Since the ratio of the size of the WSOP to the size of the MSOP is arbitrarily large when n, the number of variables, is unbounded, the Minato-Morreale algorithm can produce results that are very far from minimum. We present a class of multiple-output functions whose WSOP size is also much larger than its MSOP size. For a set of benchmark functions, we show the distribution of ISOPs to the number of PIs. Among this set are functions where the MSOPs have almost as many PIs as do the WSOPs. These functions are known to be easy to minimize. Also, there are benchmark functions where the fraction of ISOPs that are MSOPs is small and MSOPs have many fewer PIs than the WSOPs. Such functions are known to be hard to minimize. For one class of functions, we show that the fraction of ISOPs that are MSOPs approaches 0 as n approaches infinity, suggesting that such functions are hard to minimize.

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PALMINI---fast Boolean minimizer for personal computers

Cited by 36 publications

References 18 publications

Spectral Methods for Boolean and Multiple-Valued Input Logic Functions

Spectral Methods for Boolean and Multiple-Valued Input Logic Functions

Orthogonal and Nonorthogonal Expansions for Multi-Level Logic Synthesis for Nearly Linear Functions and their Application to Field Programmable Gate Array Mapping

Worst and best irredundant sum-of-products expressions

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