Computer System Structures cz:struktury počítačových systémů

Transcription

1 4..25 omputer System Structures cz:struktury počítčových systémů Version:. Lecturer: Richrd Šust ČVUT-FEL in Prgue, R suject 35SPS Mini-question: Minimize KM of prime detector F(w,x,y,z) = S(,2,3,5,7,,3) Krok - ssigning numers of input sttes Krok 2 cover KM '+.'.+..'+.'.' =.'+.( xor )+.'.' SPS 2

2 4..25 Mini-question: re functions equivlent? x<=( nd not ) or ( nd not ); x2<=( nd not ) xor ( nd ); x3<=( or ) nd (not or not ); x4<=( xor ) or ( nd not ); SPS 3 Mini-question: re functions equivlent? x<=( nd not ) or ( nd not ); x2<=( nd not ) xor ( nd ); x3<=( or ) nd (not or not ); x4<=( xor ) or ( nd not ); x, x3, x4 We solve y KM x2 xor of mps SPS 4 2

3 4..25 K-mps - Implicnts - w w y wx yz x z + xz + wx + yz z More solutions (coverges) cn exist F(w,x,y,z) = S(,2,3,5,7,8,9,,,3,5) x z + xz + wz + yz y wx yz z x x w y wx yz z wx yz m m m 3 m 2 m 4 m 5 m 7 m 6 m 2 m 3 m 5 m 4 m 8 m 9 m m SPS 6 x z + xz + wz + x y x w w z y wx yz x z + xz + wx + yz y z x x 3

4 4..25 Implicnt it covers ""s Prime implicnt lrger s possile Essentil prime implicnt it lone covers "" Two-level K-mp simplifiction Method: Grow implicnts into prime implicnts over ll "" (minimize numer of product terms) SPS 7 Exmple of Implicnts X 6 prime implicnts: '', ',, '',, ' essentil minimum cover: + ' + '' 5 prime implicnts:, ',, ', '' essentil minimum cover: 4 essentil implicnts SPS 8 4

5 4..25 lgorithm for two-level simplifiction K-mp lgorithm Step : choose "" Step 2: mximize covering implicnts to prime implicnts (numer of elements must e power of 2) Repet Steps nd 2 until finding ll prime implicnts Step 3: select essentil implicnts Step 4: cover remining "" y the smllest numer of prime implicnts SPS 9 X f(,,,) = m(4,5,6,8,9,,3) + d(,7,5) X X X X X X 2 primes round ''' X X 2 primes round ' X X X X X 3 primes round ''' X X 2 essentil primes X X minimum cover (3 primes) [Seungryoul M.: omintion Logic, KIST 22] SPS 5

6 4..25 Quine-Mcluskey Method Tulr method to systemticlly find ll prime implicnts f(,,,) = Sm(4,5,6,8,9,,3) + Sd(,7,5) Stge : Find ll prime implicnts Step : Fill olumn with ON-set nd -set minterm indices. Group y numer of 's. Step 2: pply Uniting Theorem E.g., vs. yields - vs. yields - When used in comintion, mrk with check. If cnnot e comined, mrk with str. These re the prime implicnts. Impliction Tle olumn I olumn II olumn III _* _* * _ _ * _* _* _ _ _ _* _ _ Repet until no further comintions cn e mde. Stge2: over ON-set [Seungryoul M.: omintion Logic, KIST 22] SPS Morse econ EE y Mcluskey : 5: 7: 8: 9: 2: 3: 4: 5: *.5:- *.9:- *5.7:- *5.3:- *7.5:- *8.9:- *8.2:- *9.3:- *2.3:- *2.4:- *3.5:- *4.5:- (.5).(9.3):-- (.9).(5.3):-- (5.7).(3.5):-- (5.3).(7.5):-- (8.9).(2.3):-- (8.2).(9.3):-- (2.3).(4.5):-- (2.4).(3.5):-- '+.+.' ='.(+)+. (.5).(9.3):-- (5.7).(3.5):-- (8.9).(2.3):-- SPS 2 6

7 4..25 Group Minimiztion - Groups shre dt - Gte Logic: Two-Level Simplifiction lock igrm Truth Tle N N 2 F = F 2 < F 3 > F F 2 F 3 SPS 4 7

8 c c c5 c6 c4 c2 c3 c c c2 c3 c4 c5 c6 to 7 segment control signl decoder Gte Logic:Two-Level Simplifiction = : F < : F 2 > : F 3 F = '.'.'.' + '..' '..' (isolted '' ) F2 = F3 = '.'. + '. + '...'.' + ' + ' F = F2'. F3' Notice: Equlity comprisons of inry numers require more complicted logic functions thn less or greter comprisons ecuse isolted cells complicte coverge. SPS 5 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Exmple: / 7-segment X X X X X X = ' ' = ' ' + + ' 2 = + ' + 3 = ' ' + ' + ' + ' 4 = ' ' + ' 5 = + ' ' + ' + ' 6 = + ' + ' + ' x x x x x x x x x x x x x x X X X X X X SPS 6 8

9 4..25 Group minimiztion = ' ' = ' + ' ' + 2 = + ' + 3 = ' ' + ' + ' + ' 4 = ' ' + ' 5 = + ' ' + ' + ' 6 = + ' + ' + ' totl: 5 minterms 9 minterms+ 6 shred = + ' + + ' ' + ' = ' + ' ' + + ' ' 2 = ' + ' + ' ' + + ' 3 = ' ' + '+ ' + ' 4 = ' ' + ' 5 = + ' ' + ' + ' 6 = + ' + ' + ' totl: 8 minterms 2 minterms (6)+6 shred 2 X 2 X X X X X X X X X X X SPS 7 oolen lger mthemtic foundtion of logic 9

10 4..25 Reminding of sic Lws. losure (cz: uzávěr množiny): set S is sid to e closed with respect to inry opertor if whenever y x = z S; for ll x, y S Ex: opertion + or * in S= Z ={ -2, -,,, 2...} 2. ssocitive lw: inry opertor on set S is sid to e ssocitive whenever (x y) z = x (y z) for ll x, y, z, S Ex: in Z ssoc. +, ut non-ssoc. -, (7-5)- 7-(5-) 3. ommuttive lw: x y = y x for ll x, y S 4. Identity element: set S is sid to hve n identity element if there exists n element e S; e x = x e = x for every x S Ex) The element with +, or with * on the set of integers. 5. Inverse element x y = e Ex) In Z, + (-) = 6. istriutive lw opertor is sid to e distriutive over opertor whenever x (y z) = (x y) (x z) SPS 9 George oole (85-864) orn to working clss prents in the English industril town of Lincoln, tught himself mthemtics. pproched logic in new wy reducing it to simple lger incorported logic into mthemtics; xiomtic foundtions of oolen lgeric systems ws performed y Huntington (94) In 93, "oolen lger" ws first suggested y Sheffer s nme for the system specified y Huntington postultes. ( Note: Sheffer invented Sheffer s stroke k NN opertion) The crter oole on the Moon SPS [Source: 2

11 4..25 oolen lger oolen lger n lgeric structure on set of elements with two inry opertions + nd. provided the following (Huntington) postultes re stisfied :. losure with respective to () the opertor + () the opertor. 2. n identity element with respect to () +, designted y : x + = + x = x. ()., designted y : x = x = x. 3. ommuttive with respect to () + : x + y = y + x () with respect to : x y = y x. 4. istriutive over () + : x (y+z) = (x y)+(x z) (). : x+(y.z) = (x+y).(x+z) 5. For every element x, there exists n element x (clled the complement of x) such tht () x + x = nd () x x =. 6. There exists t lest two elements x, y such tht x y. Note: Huntington postultes do not include the ssocitive lw ut it cn e derived SPS 2 Switching lger vs. Multiple Vlued oolen lger oolen lger is termed Switching lger when = {, } When > 2, the system is multiple vlued. Exmple: M = {(,, 2, 3), #, &} # & SPS 22

12 4..25 oolen Logic k Switching lger oolen logic is six-tuple (, +,., ',, ), where is set of elements {, }, is minimum element in nd mximum element in ' is unry opertion (complement) two inry opertions. + re defined on, stisfying xioms: xiom () () ommuttivity cz:komuttivit + = + = Identity cz:neutrlit + = = istriutivity cz:istriutivit + ( c) = ( + ) ( + c) ( + c) = + c omplementtion cz: Komplementrit + = = nnihiltor cz:gresivit + = = (Note: oolen logic is one exmple of oolen lger.. ) SPS 23 Eng: ulity of xioms Nme of lw OR version N version ommuttivity + = + = Identity + = = istriutivity + ( c) = ( + ) ( + c) ( + c) = + c omplementtion + = = nnihiltion + = = Idempotency + = = oule negtion ( ) = socitivity + ( + c) = ( + ) + c ( c) = ( ) c emorgn s (x + y) = x. y (x.y) = x + y sorption + ( ) = ( + ) = Merging (x y) + (x y ) = x (x + y) (x + y ) = x SPS 24 2

13 4..25 z: ulit xiomů teorémů Vlstnost N OR OR N Komuttivit + = + = Identit + = = istriutivit + ( c) = ( + ) ( + c) ( + c) = + c Komplementrit + = = gresivit + = = Idempotence + = = vojí negce ( ) = socitivit + ( + c) = ( + ) + c ( c) = ( ) c emorgn ( + ) = ( ) = + sorpce + ( ) = ( + ) = Sloučení (x y) + (x y ) = x (x + y) (x + y ) = x SPS 25 socitivity Lw: Multi-input Gtes c c Y=(.).c =..c c c Y=(+)+c = ++c c c c c Y=((.).c) = (..c) Y=((+)+c) = (++c) SPS 26 3

14 = - SoP +. =. +. =.(+) =. = (omment: lso covers.).(+) = - PoS.(+) =. +. = +. = (omment: lso covers +) + '. = + SPS + '. =(+')(+) (distriutive+) =.(+) = + (omment: ' is redundnt) + '.' = +' + '.' =(+')(+') (distriutive+) =.(+') = +' Vrints of sorption Lw 27 SPS Generl sorption Lw omment: we cn sustitute ny oolen functions for or in sorption lws Let f(x), g(x): n e ny logic function then f(x)+f(x).g(x) = f(x) f(x) + f(x).g(x) = f(x). + f(x).g(x) = f(x).(+g(x)) = f(x). = f(x) f(x).(f(x)+g(x)) = f(x) f(x).(f(x)+g(x)) = f(x). f(x) + f(x).g(x)= f(x) + f(x).g(x) = f(x) f(x) + f(x)'.g(x) = f(x)+g(x) f(x) + f(x)'.g(x) =(f(x)+f(x)')(f(x)+g(x)) (distriutive+) =.(f(x)+g(x)) = f(x)+g(x) Generlly, we cn sustitute logic functions insted of vriles in ny oolen lw 28 4

15 4..25 Non-Equlity (XOR) = xor =.'+'. Equlity = ( xor )' =.+'.' < (less thn, Inhiition of ) < =.' <= (less thn or equl, Impliction), <= = +' > (greter thn, Inhiition of ) > =.' >= (greter thn or equl, Impliction), >= = +' SPS erived Opertion 29 SPS Exmple: 2 it comprtor Y = >= = ( ).( >=)+> (sustitute logic expressions of derived functions) = ('.'+.)(+')+.' (st prenthesis nd the lst element contin the sme vriles thus we pply distriutive +) = (('.'+.)+.').((+') +.' ) (we remove prentheses y commuttive lw) = ('.'+.+.').((+') +.' ) (sorption lw) = ('.'+.+.').((+') +.' ) = ('.'+ ).((+') +.' ) (sorption lw gin) = ('+ ).((+') +.' ) (distriutive lw) = ('+ ).(+') +('+ )..' = ('+ ).(+') +'..' +..' (idempotency) = ('+ ).(+') +.' The sme result s 8th slide of the st lecture 3 5

16 4..25 SPS () ( + )' =.' emorgn s Theorem () (.)' = ' + ' Generlized emorgn's Theorem () ( + + z)' =.' z' () (. z)' = ' + ' + z' The crter de Morgn on the Moon [Source of the imge: (lso with the crter) Exmple : (. +.c +.c) y gtes = ((. +.c +.c)')' = ((.)'.(.c)'.(.c)')' [nnd gtes only] = (('+').('+c').('+c'))' = ('+')'+('+c')'+('+c')' = ((('+')'+('+c')'+('+c')')')' [nor-gtes only] 3 emorgn's theorem follows from KM e Morgn's theorem is SoP nd PoS coverge of sme elements SoP: etector of primes F(w,x,y,z) = S(,2,3,5,7,,3) PoS: etector of non-primes F'(w,x,y,z) = S(,4,6,8,9,,2,4,5) ('+).('++').('+'+).('++).'+.'.+..'+.'.' SPS 32 6

17 4..25 emorgn Equivlents of Gtes Opertion NN N OR NOR N OR XOR XNOR Y Y=( ) SPS 33 Negtion ules Externl Inverter Externl Inverter Integrted Inverter Integrted Inverter input logic circuit = output logic circuit + = oule negtion + = + = N + NOT = NN + = NN + NOT = N OR + NOT = NOR + = NOR + NOT = OR Note: Functions re equl from logic point of view, for further informtion, see hzrds in the next lecture. SPS 34 7

18 4..25 Exmple From NN gtes to NOR y emorgn : c c c c SPS 35 Identity & nnihiltor Lws cz:neutrlit & gresivit Gte s ontrol Element of Switch 8

19 4..25 Enle ctive-high Gte s ontrol Element of Switch. =.= Enle = Enle = t in Enle ctive-low t out t in disled mode t out= t in. =.= Enle=, not Enle = t out Enle=, not Enle = t in t out disled mode t in t out t in t out= Enle ctive-low t in Enle = + = += t in Enle = t in SPS t out disled mode t out t out= 37 Tri stte uffer Enle ctive-high Enle = Enle = t in t out t in high impednce t in t out Enle ctive-low Enle= not Enle = Enle= not Enle = t in t out t in t out t in high impednce SPS 38 9

20 ounter HL Hrdwre escription Lnguges for forml description nd design of electronic circuits Motivtion exmple- econ MHL2 - equtions (/3) STOP STRT STOP M "MHL2" = "" Equtions: SPS 4 2

21 4..25 Motivtion exmple- econ MHL2 - circuit (2/3) ircuit SPS 4 Motivtion exmple- econ MHL2 - VHL (3/3) "MHL2" = "" lirry ieee; use ieee.std_logic_64.ll; entity KOM is port ( X, X, X2, X3, X4, X5 : in std_logic; STOP, Y : out std_logic); end entity; rchitecture ehviorl of KOM is signl,, c, d, e, f : std_logic; signl Y, Y, Y, Y : std_logic; egin -- prezncime signly, n rozdil od progrmovni se v hrdwru nevytvori nove promenne <= X5; <= X4; c <= X3; d <= X2; e <= X; f <= X; Y <= (e nd f) or (not c nd e) or (not c nd f) or (d nd f); Y <= (not e nd f) or (c nd not d nd f) or (not c nd d nd f) or (c nd not d nd not e); Y <= (c nd e) or (d nd e) or f; Y <= ''; -- Spojíme, ztím tkto, příště si ukážeme lepší konstrukce -- Y <= (not nd not nd Y) -- =, = or (not nd nd Y) -- =, = or ( nd not nd Y) -- =, = or ( nd nd Y); -- =, = STOP <= nd nd f; end rchitecture; SPS 42 2

22 4..25 VHL VHSI (Very High Speed Integrted ircuit) HL Wht is VHL? design entry lnguge design simultion nd modeling lnguge netlist lnguge for expressing circuit connectivity etween hierrchy of locks stndrd lnguge SPS 44 22

23 4..25 VHL history 98: Initited in 98 y US o ( U.S eprtment of efense ) to ddress the hrdwre life-cycle crisis (note: d ws lso initited y o in 97) : evelopment of seline lnguge y Intermetrics, IM nd Texs Instruments 986: ll rights trnsferred to IEEE (red I-Triple-E) - Institute of Electricl nd Electronics Engineers * 987: IEEE Stndrd VHL 987 o requires comprehensive VHL descriptions supplied with every SI delivered to the o. 994: Revised stndrd (nmed VHL ) 29: Revised Stndrd (nmed VHL 76-28) SPS 45 Verilog versus VHL World mrket divided etween VHL & Verilog Verilog - introduced in 983 s simultion lnguge, fterwrds extended with support for synthesis dded. VHL mostly in Europe Verilog dominnt in US VHL More generl lnguge More high-level constructs etter extendiility y pckges Verilog: Not s generl s VHL etter for low-level circuits Simpler thn VHL, especilly for nd Jv progrmmers, ut it cn lso confuse them y illusion on similrity to SPS 46 23

24 4..25 VHL / Verilog Google trend VHL Verilog SPS 47 escriing omintionl Logic Using tflow esign Style EE 545 Introduction to VHL 24

25 4..25 Register Trnsfer Level (RTL) esign escription Tody s Topic omintionl Logic omintionl Logic Registers SPS 49 VHL esign Styles VHL esign Styles dtflow oncurrent sttements structurl omponents nd interconnects ehviorl Sequentil sttements Registers Stte mchines Test enches Register Trnsfer Level (RTL) stype SPS 5 25

26 prolemtic or impossile implementtion in hrdwre VHL dt types VHL for Specifiction VHL for Simultion file I/O flot 3.45 it, integer 234 string "234" VHL for Synthesis types derived from std_logic type chrcter '' oolen SPS 5 Multi-Vlued Logic Representtions MVL - 9 more sttes of signls MVL - 9 Unitilized U Wek H on t re - Wek L Forcing Wek Unknown W Forcing High Impednce Z Forcing Unknown X Wired Or/nd Impossile implement in yclone II SPS 52 26

27 4..25 it versus std_logic it vlues ''nd '' oolen - vlues TRUE FLSE only conditions std_logic MVL-9: enumrted type with vlues defined y literls '', '', 'X', 'Z','U', '-', 'L', 'H', 'W' Wrning: std_logic nd it re mutully unconvertile SPS 53 sic rules for source code. VHL is not cse sensitive, 2. VHL does not understnd dicritics, not even in comments. 3. Identifier must strt with letter, lst chrcter cnnot e n underscore, two connected underscores re not llowed 4. ll sttements end with semi-colon 5. omments precede with (--) The rest of line is treted s comment. rrige return termintes comment. 6. No method for commenting lock SPS 54 27

28 4..25 The most importnt rules Keep things simple, simple progrms re the est Prtition the design (ivide et Imper) Prefer known sfe-constructions; VHL is strongly type lnguge, mny times stricter thn Jv or Pscl Keep in mind tht synthesizle VHL code is not clssic progrm ut it descries circuits. To keep circuits simple, check results of your work (in Qurtus II: Locte->Locte in RTL View, Locte in Technology Mp View...) SPS 55 28