5 Logika Boolean.ppt

7/28/2019 5 Logika Boolean.ppt

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Pendahuluan

Aljabar boolean, adalah sistem aljabar himpunanatau proposisi yang memenuhi aturan-aturan

ekivalen logis.

Misalkan B dengan operasi + dan *, atau suatukomplemen, dan dua elemen yang beda 0 dan 1

yang didefinisikan pada himpunan atau

proposisi, sehingga a,b dan c merupakan

elemen B yang mempunyai sifat-sifat identitas,

komutatif, distributif dan komplemen.


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aa*ba+b


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Basis Bilangan

Biner = 0,1 Oktal = 0,1,2,3,4,5,6,7 Desimal = 0,1,2,3,4,5,6,7,8,9 Hexadesimal =

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F


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Prinsip Dualitas

Apabila dalam sebuah aljabar boole B,operasi + dipertukarkan dengan * dan

sebaliknya, dan 0 dipertukarkan dengan 1

dan sebaliknya. (a*1)+(b*0)=b adalah (a+0)*(b+1)=b


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Fungsi Boolean

Dinyatakan dalam bentuk f(x), g(x) danlainnya.


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Pengujian Ekivalen

Bentuk Standar Bentuk Kanonik

Bentuk Sum-of-ProductSum = gabungan dan product = irisan.

Masukannya E sebuah ekspresi boole, danekspresi SoP ekivalen dengan E Gunakan kaidah2 ekivalen sehingga tersisa bentuk +

dan * saja.

Gunakan kaidah De Morgan. Gunakan kaidah Distribusi. Gunakan kaidah komutatif. Gunakan kaidah penyerapan dan identitas.


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Operasi Logika

Fungsi Boole Simbol Operator Nama

F0=0 Null

F1=xy x*y and

F2=xy x/y Inhibition

F3=x Transfer

F4=xy y/x Inhibition


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F5=y Transfer

F6=xy+xy xy XOR

F7=x+y x+y OR

F8=(x+y) xy NOR

F9=xy+xy xy Equivalence


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F10=y y Complement

F11=x + y xy Implication

F12=x X Complement

F13=x+y xy ImplicationF14=(xy) xy NAND

F15=1 Identity


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Penyederhanaan

Aljabar menggunakan logika ekivalen. Peta Karnaugh Metode Quine-Mc.Cluskey (untuk kasus

empat variabel atau lebih)

Espresso heuristic Metode Petrick's


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Peta Karnaugh

Peta Karnaugh digambarkan dengankotak bujur sangkar. Setiap kotak

merepresentasikan minterm(SoP).

Jumlah kotak=2 pangkat jumlah variabel.


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The Karnaugh map, also known as a Veitch diagram (KV-map orK-map for short), is a tool to facilitate the simplification ofBooleanalgebraIC expressions. The Karnaugh map reduces the need forextensive calculations by taking advantage of human pattern-recognition and permitting the rapid identification and elimination ofpotential race hazards.

The Karnaugh map was invented in 1952 by Edward W. Veitch. Itwas further developed in 1953 by Maurice Karnaugh, atelecommunications engineer at Bell Labs, to help simplify digitalelectronic circuits.

In a Karnaugh map the boolean variables are transferred (generallyfrom a truth table) and ordered according to the principles ofGraycode in which only one variable changes in between squares. Oncethe table is generated and the output possibilities are transcribed,the data is arranged into the largest even group possible and theminterm is generated through the axiom laws of boolean algebra.


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Dua variabely

X

0 1

0 xy

1 xy xy


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Contoh

y

X 0 1

0 xy

1 xy xy

= x + y


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Latihan

E= xy + xy E= xy + xy + xy E= xy + xy


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Tiga variabelyz

x

00 01 10 11

0 xyz xyz xyz

1 xyz xyz xyz xyz

= x + y + z


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Latihan

E= xyz + xyz + xyz + xyz E= xyz + xyz + xyz + xyz + xyz E= xyz + xyz + xyz + xyz + xyz


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Metode Quine-Mc.Cluskey

Dengan peta Karnaugh, untuk empatvariabel menjadi lebih kompleks.

Metode ini akan membantupenyelesaiannya.

Metode ini terdiri atas dua bagianMenentukan term-term sebagai kandidat.Memilih kandidat untuk mendapatkan

ekspresi dengan jumlah literal sedikit.


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ESPRESSO Heuristic

A radically different approach to this issue isfollowed in the ESPRESSO algorithm,developed by Brayton e.a. at the University ofCalifornia, Berkeley.[7] Rather than expanding a

logic function into minterms, the programmanipulates "cubes", representing the productterms in the ON-, DC- and OFF-coversiteratively. Although the minimization result is notguaranteed to be the global minimum, in practice

this is very closely approximated, while thesolution is always free from redundancy.


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Compared to the other methods, this one isessentially more efficient, reducing memory

usage and computation time by several orders of

magnitude. Its name reflects the way of instantly

making a cup of fresh coffee. There is hardly anyrestriction to the number of variables, output

functions and product terms of a combinational

function block. In general, e.g. tens of variables

with tens of output functions are readily dealtwith.


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The input for ESPRESSO is a function table ofthe desired functionality; the result is aminimized table, describing either the ON-coveror the OFF-cover of the function, depending onthe selected options. By default the productterms will be shared as much as possible by theseveral output functions, but the program can beinstructed to handle each of the output functionsseparately. This allows for efficientimplementation in two-level logic arrays such as

a PLA (Programmable Logic Array) or a PAL(Programmable Array Logic).


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The ESPRESSO algorithm proved so successfulthat it has been incorporated as a standard logic

function minimization step into virtually any

contemporary logic synthesis tool. For

implementing a function in multi-level logic, theminimization result is optimized by factorization

and mapped onto the available basic logic cells

in the target technology, whether this concerns

an FPGA (Field Programmable Gate Array) or anASIC (Application Specific Integrated Circuit).

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5 Logika Boolean.ppt