Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I MN=11, th nxt stt o th lip lop is 1. () omplt th ollowin tl (us on t rs whn possil). Prsnt stt Nxt stt Q Q + M N 1 0 1 1 () Usin th tl otin in () n Krnuh mps, riv n minimiz th input qutions or ountr ompos o M-N lip-lops, whih ounts in th ollowin squn: AB = 000, 001, 011, 111, 101, 100, () Usin th D lip-lop, implmnt th sm ountr ivn in (). Qustion 2. Th stt irm o squntil iruit is ivn s low. () Tult th rlt stt tl. () Ru th stt tl to minimum numr o stts usin row mthin. () Rpt () usin n implition hrt. () Drw th ru stt irm.
Answr 1. () onitions I MN = 00 thn Q + = 0 I MN = 01 thn Q + = Q I MN = 10 thn Q + = Q' I MN = 11 thn Q + = 1 Prsnt stt Q Nxt stt Q + M N MN inputs 0 X 1 X 1 0 X 0 1 1 X 1 oul 00 or 01 11 or 10 00 or 10 11 or 01 () Prsnt stt - nxt stt tl n rlvnt MN input onitions tls or th squn o AB = 000, 001, 011, 111, 101, 100,. For this ountr, thr MN lip-lops r rquir A B A + B + + M A N A M B N B M N 0 X 0 X 1 X 1 1 0 X 1 X X 1 0 X X X X X X X X X 1 1 1 1 1 X X 1 X 1 1 0 X X 0 X 1 1 X 1 0 X X 0 1 1 0 X X X X X X X X X 1 1 1 1 X 1 X 0 X 1 M A M B M AB 01 11 11 11 11 11 10 X X X 0 X X X X 0 1 X X 1 X X 0 X X X X N A N B N AB 01 11 11 11 11 11 10 0 X X X 0 X X X X X X X X 1 X X 1 1 X 1 0 X 1 1 1 0 From th Krnuh mps, n lso y insption: M A = B, N A = ; M B = A', N B = A' ; M = A', N = A'+B
M A M B M N A' N B' N ' lok () Input qutions or th D lip-lops or th ountr n riv usin th prsnt stt- nxt stt tl. A B A + B + + 1 1 0 X X X 1 1 1 1 1 0 1 1 1 1 0 X X X 1 1 1 1 D A D B D AB 01 11 11 11 11 11 10 X X X X X X 0 1 1 1 1 1 1 1 1 0 From th Krnuh mps, n lso y insption: D A = B+A ; D B = A' ; D = A' +B D A D B D A' B' ' lok
Answr 2. () It is mor onvnint to pply prours or stt rution usin stt tl rthr thn stt irm. Th stt tl or th ivn stt irm is tult in th ollowin tl. Prsnt stt Nxt stt () Whn two stts r quivlnt, on o thm n rmov without ltrin th input-output rltionships. Stp 1. By insptin th rows in th tl o (), it is possil to s tht stts n r quivlnt. Thy oth o to stts n n hv outputs o 0 n 1 or x = 0 n x = 1, rsptivly. Thror, stts n r quivlnt n on o ths stts n rmov. Stp 2. Th row with prsnt stt is rmov n stt in th nxt stt olumn is rpl y its quivlnt, stt. Th prour is shown in th nxt tl. Prsnt stt Nxt stt Stp 3. Prsnt stt now hs nxt stts n n outputs 0 n 1 or x = 0 n x = 1, rsptivly. Insption o th rows rvls tht stts n r quivlnt. So, stt n rmov n rpl y. Th inl ru stt tl is shown in th nxt tl. Prsnt stt Nxt stt () hkin h pir o stts or possil quivlny n on systmtilly y mns o n implition tl. In this tl, on th lt si lon th vrtil r list ll th stts xpt th irst, n ross th ottom
horizontlly r list ll th stts xpt th lst. Th rsult is isply o ll possil omintions o two stts with squr pl in th intrstion o row n olumn whr th two stts n tst or quivln. Stp 1. Two stts tht r not quivlnt r mrk with ross in th orrsponin squr. Stp 2. W ntr in th rminin squrs th pirs o stts tht r impli y th pir o stts rprsntin squrs. Som o th squrs hv ntris o impli stts tht must urthr invstit to trmin whthr thy r quivlnt or not. Thror, th nxt stp is to mk sussiv psss throuh th tl to trmin whthr ny itionl squrs shoul mrk with ross. A squr in th tl is ross out i it ontins t lst on impli pir tht is not quivlnt. - - - - - - - - - - - - Stp 3. Thr is no n to inlu sl impli pirs (- n ) in th tl. Th inl implition tl is shown low.
- - - - - - - - - - From th tl, w n onlu tht th quivlnt stts r (, ) n (, ) (ross stions o sh squrs). So n. Th ru stt tl is otin y rplin y n y. Th rsult is th sm s otin in () or this xmpl. Howvr, th implition tl provis muh mor rution thn th row mthin in mny ss. - Prsnt stt Nxt stt () Th ru stt irm is illustrt low.