Post on 20-Dec-2020
Proiectare Logica Digital Logic Design
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Recapitulare
β’ Mr. Smith Problem (utilitatea algebrei booleene),
β’ Operatii logice: simboluri si notatii
β’ Algebra booleana: Teoreme Simple; Asociativitate, Comutativitate si Distributivitate; Reguli de simplificare; Principiul dualitatii, Teoremele De Morgan; Termeni de consens
β’ Demonstrarea identitatilor folosind teoremele aglebrei booleene sau tabela de adevar
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NOT Y=A' AND Y=AΒ·B OR Y=A+B
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George Boole (1815-1864)
If I do not take the car then I will take the umbrella if
it is raining or the weather forecast is bad.
I will take an umbrella with me if
it is raining or the weather forecast is bad.
Bad Forecast
NOT
OR
AND
Rain
Car
Take umbrella
Operatii logice: simboluri si notatii
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Operatie Tip Op Notatie Simbol clasic Simbol IEEE
wronex
NOT unar
π΄ , A' sau Β¬A
AND binar π΄ β π΅
OR binar π΄ + π΅
XOR binar π΄π΅
Teoreme Simple:
(π¨β²)β² = π¨ : Dubla negatie == afirmatie
π¨ β π¨β² = π: O propozitie SI negatia sa FALSA
π¨ + π¨β² = π : O propozitie SAU negatia sa ADEV.
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Asociativitate, Comutativitate si Distributivitate
Asociativitate: π¨ β π© β πͺ = π¨ β π© β πͺ = π¨ β π© β πͺ π¨ + π© + πͺ = π¨ + π© + πͺ = π¨ + π© + πͺ
Comutativitate: π¨ β π© = π© β π¨ π¨ + π© = π©+ π¨
Distributivitate: π¨ β π© + πͺ = π¨ β π© + π¨ β πͺ
π¨ + π© β πͺ = (π¨ + π©) β (π¨ + πͺ) Ciudata
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Reguli de simplificare Idempotenta: π¨ β π¨ = π¨ π¨ + π¨ = π¨ In care apar constantele booleene: π¨ β π = π π¨ β π = π¨ : 1 = elem. neutru fata de " β " π¨ + π = π¨ : 0 = elem. neutru fata de " + " π¨ + π = π Grupul cel mai larg de reguli de simplificare: π¨ + π¨ β π© = π¨ si duala sa π¨ β π¨ + π© = π¨ π¨ + π¨ β π©β² = π¨ si duala sa π¨ β π¨ + π©β² = π¨
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Principiul dualitatii β Dualitatea De Morgan
Daca intr-o identitate booleana interschimbam constantele
π β π
si operatorii " β " β "+"
atunci obtinem o noua identitate, numita identitatea duala.
Exemple:
π¨ β π = π si duala sa π¨ + π = π
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Teoremele De Morgan Un exemplu remarcabil al principiului dualitatii il reprezinta teoremele De Morgan
π¨ β π© β² = π¨β² +π©β² (π¨ + π©)β² = π¨β²π©β²
Nota: In locul operatorului NOT notat cu (') β single quote (de ex. π¨β²) se mai foloseste simbolul Β¬, adica Β¬π¨. La fel de uzitata este si notatia cu bara deasupra: π¨ .
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Termen de consens Teorema de consens: Fie functia π = π¨ β πͺ + π© β πͺβ². Un termen de consens este format din cele doua variabile diferite de C, adica π¨ β π©. Functia formata pin adaugarea acesti termen nu difera de cea initiala pentru orice valoare parametrilor: Adica π¨ β πͺ + π© β πͺβ² β‘ π¨ β πͺ + π© β πͺβ² + π¨ β π© Prin definitie un termen de consens este un termen
a carei prezenta nu schimba valoarea functiei. πͺ + π¨π©πͺβ² + π¨π© = πͺ + π¨π©πͺβ² Scopul introducerii acestor termeni de consens este simplificarea. Expresia de mai sus se simplifica: πͺ + π¨π©πͺβ² = πͺ + π¨π©
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Demonstrarea identitatilor A+AB=A
Se poate face folosind teoremele
sau tabela de adevar
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A B AB A+AB
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1
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Numere in virgula mobila
Numerele reale sunt reprezentate in sistemele de calcul sub forma numerelor in virgula mobila.
Cel mai cunoscut si raspandit standard este IEEE -754 lansat in 1985. El a impus reprezentarea numerelor pe 32 si 64 biti (respectiv numere in simpla si dubla precizie)
Acest standard a fost revizuit in 2008 si a devenit standard international cu numele ISO/IEC/IEEE 60559:2011 (se poate cumpara cu ~ 700 lei).
1. IEEE - Institute of Electrical and Electronics Engineers
2. ISO - International Organization for Standardization
3. IEC - International Electrotechnical Commission
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IEEE 754 Formatul IEEE contine un set de reprezentari ale
valorilor numerice si impune simboluri precum NaN, +/-Inf si altele.
Un numar real in acest format este reprezentat prin 3 numere intregi
β’ s β semnul (0 sau 1 nr + sau -)
β’ e β exponent
β’ m β mantisa (significand in EN). Ea memoreaza cifrele semnificative.
mai este precizat un numar
β’ b β baza de numeratie (2 sau 10) 14
π = βπ π ΓπΓ ππ
Formatele IEEE - 754 Cele mai raspandite sunt binari32 si binari64,
respectiv in simpla si dubla precizie (vezi aici)
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Nume Nume usual
Baza nr biti
mantisa
Nr cifre semnif in baza 10
Nr. biti exp.
Exp. bias
E min E max
binary16 Half precision
2 11 3.31 5 24β1 = 15 β14 +15
binary32 Single precision
2 24 7.22 8 27β1 = 127 β126 +127
binary64 Double precision
2 53 15.95 11 210β1 = 1023
β1022 +1023
binary128 Quadruple precision
2 113 34.02 15 214β1 = 16383
β16382 +16383
binary256 Octuple precision
2 237 71.34 19 218β1 = 262143
β262142 +262143
Bitul cel mai semnificativ al mantisei este omis in reprezentare. Ex. in binary64 : 1s+11e+(53-1)m=64biti
Binary64 online
Pentru reprezentarea in dubla precizie (binary64) exista o pagina online unde se poate vizualiza acest format (si aici pe cnic.ro).
binary64 foloseste 1 bit de semn, 11 biti pentru exponent si 52 biti pentru mantisa (NCM-1; adica se omite bitul cel mai semnificativ din mantisa deoarece acesta este intotdeauna 1)
Exemplu: β0.567810 se reprezinta astfel
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Numar cifre semnificative in baza 10
ππΆπ10 =Numar cifre semnificative in baza 10 NCE = Numar cifre binare exponent ππΆπ = Numar cifre binare mantisa
π΅πͺπΊππ = π₯π¨π ππ ππ΅πͺπ΄ β π΅πͺπ΄Γ πππππ π
Ex: β’ binary32 ππΆπ10 = 24 log10 2 β 24 Γ 0.301 β 7.22 β’ binary64 ππΆπ10 = 53log10 2 β 15.95
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Nume NCM NCE NCS10
binary16 11 5 3.31
binary32 24 8 7.22
binary64 53 11 15.95
binary128 113 15 34.02
binary256 237 19 71.34
Numar cifre semnificative in baza 10
ππΆπ10 = Numar cifre semnificative in baza 10
ππΆπ = Numar cifre in baza b al mantisei
π΅πͺπΊππ = π₯π¨π ππ π
π΅πͺπ΄ β π΅πͺπ΄Γ πππππ π
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Algoritm conversie in virgula mobila Se extrage semnul: if(x>=0)semn=0;else semn=1 Se calculeaza deplasarea: offset=ππ΅πͺπ¬βπ β π .
De ex. in binary64: NCE=11 offset=210 β1=1023
Se calculeaza exponentul real: er=[lg(|x|)/lg(2)] adica er=partea intreaga din
log10 π₯
log10 2
Se determina exponentul: exp=er+offset Se determina partea fractionara ce intra in mantisa:
man=|π|/πππ β π Se trece in baza 2 aceasta parte fractionara pe π΅πͺπ΄β π biti
EXCEPTIE cand x=0 semn=0;exp=0;man=0
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Algoritm conversie in virgula mobila
Exemplu x=-12.0625 semn=1
in binary64: offset=210 β 1=1023
Se calculeaza exponentul real: er=partea intreaga
din log10 π₯
log10 2=3
exponentul: exp=er+offset=102610=100000000102
partea fractionara a mantisei: man=|π|/πππ βπ=0.507812510=0.10000010β¦02 Se trece in baza 2 aceasta parte fractionara pe π΅πͺπ΄β π =52 biti
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Algoritm conversie in virgula mobila
Exemplu x=-12.0625 semn=1
= β1 Γ (23 + 22 + 2-4)
= β11 Γ 23 Γ (20 + 2-1 + 2-7)
= β11 Γ 21026β1023 Γ (1 + 2-1 + 2-7)
1100000000101000001000β¦0
Alte exemple β21.375; 47.9375; 7.6875 etc (local)
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mantisa 52 biti exponent 11 biti
semn 1 bit
Functii booleene: forme de reprezentare
Forma disjunctiva FD:
β’ π(π΄, π΅, πΆ, π·) = π΄π΅πΆ + πΆπ· + π΅
Suma de produse
Forma conjunctiva FC:
β’ g(A, B, C, D) = (A + B + C)(πΆ +π· )(A + B)
Produs de sume
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Forme canonice
Forma canonica disjunctiva FCD:
β’ π π΄, π΅, πΆ = π0π΄β²π΅β²πΆβ² + π1π΄
β²π΅β²πΆ + π2π΄β²π΅πΆβ² +β―+
π7π΄π΅πΆ = ππππ23β1π=0 , unde ππ se numeste
mintermenul i.
Forma canonica conjunctiva FCC:
β’ g(A, B, C) =(π0+π΄ + π΅ + πΆ) β (π1 + π΄ + π΅ + πΆβ²) +
β―+ (π7 + π΄β² + π΅β² + πΆβ²) = ππ23β1π=0 , unde ππ se
numeste maxtermenul i.
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π β π¨ = π; π + π¨ = π¨ dispar termenii pentru care ππ = π
π + π¨ = π; π β π¨ = π¨ dispar factorii pentru care ππ = π
De Morgan ππ π¨ + π© + πͺ = π¨β²π©β²πͺβ² β² ππ π¨ + π© + πͺβ² = π¨β²π©β²πͺ β²
Sum
a d
e p
rod
use
P
rod
us
de
su
me
Tabele de adevar - mintermeni
Forma analitica
β’ π π΄, π΅, πΆ, π· = π΄ π΅ πΆ + π΄ π΅πΆ + π΄ π΅πΆ + π΄π΅ πΆ = π1 +π2 +π3 +π5
Tabelul de adevar
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Mintermeni
Mintermen Notatie A B C f
A B C π0 0 0 0 0
A B C π1 0 0 1 1
A BC π2 0 1 0 1
A BC π3 0 1 1 1
AB C π4 1 0 0 0
AB C π5 1 0 1 1
ABC π6 1 1 0 0
ABC π7 1 1 1 0
Forme canonice: exemplu
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Fie functia f definita de tabelul: i A B C f mintermeni ππ Maxtermeni π΄π
Minfactori
0 0 0 0 1 A'B'C' A+B+C
1 0 0 1 0 A'B'C A+B+C'
2 0 1 0 0 A'BC' A+B'+C
3 0 1 1 1 A'BC A+B'+C'
4 1 0 0 1 AB'C' A'+B+C
5 1 0 1 0 AB'C A'+B+C'
6 1 1 0 1 ABC' A'+B'+C
7 1 1 1 1 ABC A'+B'+C'
FCD: π = π¨β²π©β²πͺβ² + π¨β²π©πͺ + π¨π©β²πͺβ² + π¨π©πͺβ² + π¨π©πͺ = π π, π, π, π, π FCC: π = (π¨ + π© + πͺβ²)(π¨ + π©β² + πͺ)(π¨β² + π© + πͺβ²) = π΄ π, π, π
Diagrame Venn
Diagrama in teoria multimilor
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Diagrama Venn
Diagrame Veitch: reprezentari canonice
2 variabile
3 variabile
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π π¨,π© π¨ π¨
π© π¨ π© π¨π©
π© π¨ π© π¨π©
π¨β¨π© π¨ = π π¨ = π
π© = π π π
π© = π π π
π π¨,π© = π¨β¨π© = π¨π© + π¨ π©
π π¨,π©, πͺ π¨ π© π¨ π© π¨π© π¨π©
πͺ π¨ π© πͺ π¨ π©πͺ π¨π©πͺ π¨π© πͺ
πͺ π¨ π© πͺ π¨ π©πͺ π¨π©πͺ π¨π© πͺ
π¨
π©
πͺ
π¨
π© π©
πͺ π¨ π©
πͺ
Diagrame Veitch: reprezentari canonice
3 variabile
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π π¨,π©, πͺ π¨ π© π¨ π© π¨π© π¨π©
πͺ π¨ π© πͺ π¨ π©πͺ π¨π©πͺ π¨π© πͺ
πͺ π¨ π© πͺ π¨ π©πͺ π¨π©πͺ π¨π© πͺ
π¨
π©
πͺ
π¨
π© π©
πͺ
π π¨,π©, πͺ π¨ π© π¨ π© π¨π© π¨π©
πͺ π π π π
πͺ π π π π
π π¨,π©, πͺ = π© + πͺ
Diagrame Veitch: reprezentari canonice
4 variabile
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π π¨,π©, πͺ, π« π¨ π© π¨ π© π¨π© π¨π©
πͺ π« π¨ π© πͺ π« π¨ π©πͺ π« π¨π©πͺ π« π¨π© πͺ π«
πͺ π« π¨ π© πͺ π« π¨ π©πͺ π« π¨π©πͺ π« π¨π© πͺ π«
πͺπ« π¨ π© πͺπ« π¨ π©πͺπ« π¨π©πͺπ« π¨π© πͺπ«
πͺπ« π¨ π© πͺπ« π¨ π©πͺπ« π¨π©πͺπ« π¨π© πͺπ«
π¨
π©
πͺ
π¨
π© π©
πͺ
π«
π«
π«
1
2 3
4
5
6 7 8 9 10 11 12
13 14 15
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De ce este incorecta diagrama Venn?
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1
2 3
4
5
6 7 8 9 10 11 12
13 14 15
16
π¨ π© πͺ π« 13 π¨ π©πͺ π« π¨π©πͺ π« π¨π© πͺ π« 11
π¨ π© πͺ π« π¨ π©πͺ π« π¨π©πͺ π« π¨π© πͺ π«
π¨ π© πͺπ« π¨ π©πͺπ« π¨π©πͺπ« 10 π¨π© πͺπ«
π¨ π© πͺπ« π¨ π©πͺπ« π¨π©πͺπ« π¨π© πͺπ«
A
B C
D
De ce este incorecta ACEASTA diagrama Venn?
Diagrame Venn: reprezentari canonice
4 variabile
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π π¨,π©, πͺ, π« π¨ π© π¨ π© π¨π© π¨π©
πͺ π« 0 1 1 0
πͺ π« 0 1 1 0
πͺπ« 1 0 0 1
πͺπ« 1 1 1 1
π¨
π©
πͺ
π¨
π© π©
πͺ
π π¨,π©, πͺ, π« = π©πͺ + π© πͺ + πͺπ«
π«
π«
π«
Tema de antrenament
Problemele 2.1-2.14 de la pagina 41 din cursul de referinta : B. Holdsworth and R.C. Woods, Digital Logic Design, 4th Edition, 1999
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