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# Nonlinear & Neural Networks LAB. PART 1. 5. State Reduction ( )

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Nonlinear & Neural Networks LAB. PART 1. 5. State Reduction ( )

Nonlinear & Neural Networks LAB. State Reduction ( ) -. ;. -. 2 ; Row matching methode ; Implication chart

Nonlinear & Neural Networks LAB. State Diagram : X, Z 4bit 1010 or 0110 bit 1 X = 0010 0110 1100 1010 0011... Z = 0000 0001 0000 0001 0000... Row Matching Method

Nonlinear & Neural Networks LAB. Transition Table: PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S3S400 S2S5S600 S3S7S800 S4S9S1000 S5S11S1200 S6S13S1400 S7S0 00 S8S0 00 S9S0 00 S10S0 10 S11S0 00 S12S0 10 S13S0 00 S14S0 00

Nonlinear & Neural Networks LAB. Transition Table: PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S3S400 S2S5S600 S3S7S800 S4S9S1000 S5S11S1200 S6S13S1400 S7S0 00 S8S0 00 S9S0 00 S10S0 10 S11S0 00 S12S0 10 S13S0 00 S14S0 00 S 10

Nonlinear & Neural Networks LAB. Transition Table: PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S3S400 S2S5S600 S3S7S800 S4S9S 1000 S5S11S 1000 S6S13S1400 S7S0 00 S8S0 00 S9S0 00 S 10S0 10 S11S0 00 S13S0 00 S14S0 00

Nonlinear & Neural Networks LAB. Transition Table: PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S3S400 S2S5S600 S3S7S800 S4S9S 1000 S5S11S 1000 S6S13S1400 S7S0 00 S8S0 00 S9S0 00 S 10S0 10 S11S0 00 S13S0 00 S14S0 00 S 7

Nonlinear & Neural Networks LAB. Transition Table: PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S3S400 S2S5S600 S3S 7 00 S4S 7S 1000 S5S 7S 1000 S6S 7 00 S0 00 S 10S0 10 S 3

Nonlinear & Neural Networks LAB. Transition Table: PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S 3S400 S2S5S 300 S 7 00 S4S 7S 1000 S5S 7S 1000 S 7S0 00 S 10S0 10

Nonlinear & Neural Networks LAB. Transition Table: PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S 3S400 S2S5S 300 S 7 00 S4S 7S 1000 S5S 7S 1000 S 7S0 00 S 10S0 10 S 4

Nonlinear & Neural Networks LAB. PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S 3S 400 S2S 4S 300 S 7 00 S 4S 7S 1000 S 7S0 00 S 10S0 10 Final Reduced State Transition Table

Nonlinear & Neural Networks LAB. Corresponding State Diagram

Nonlinear & Neural Networks LAB. Implication Chart Method X, Z serial sequence 010 or 110 1 State transition table: PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S1S200 S1S3S400 S2S5S600 S3S0 00 S4S0 10 S5S0 00 S6S0 10

Nonlinear & Neural Networks LAB. Implication Chart Method Enumerate all possible combinations of states taken two at a time Naive Data Structure: Xij will be the same as Xji Also, can eliminate the diagonal Implication Chart Next States Under all Input Combinations

Nonlinear & Neural Networks LAB. Implication Chart Filling in the Implication Chart Entry Xij ?Row is Si, Column is Sj Si is equivalent to Sj if outputs are the same and next states are equivalent Xij contains the next states of Si, Sj which must be equivalent if Si and Sj are equivalent If Si, Sj have different output behavior, then Xij is crossed out Example: S0 transitions to S1 on 0, S2 on 1; S1 transitions to S3 on 0, S4 on 1; So square X contains entries S1-S3 (transition on zero) S2-S4 (transition on one) S1-S3 S2-S4 S0 S1

Nonlinear & Neural Networks LAB. Implication Chart Method Starting Implication Chart S2 and S4 have different I/O behavior This implies that S1 and S0 cannot be combined

Nonlinear & Neural Networks LAB. Implication Chart Method Results of First Marking Pass Second Pass Adds No New Information S3 and S5 are equivalent S4 and S6 are equivalent This implies that S1 and S2 are too! Reduced State Transition Table Reduced State Transition Table PRESENT STATE NEXT STATEOUTPUT X=0X=1X=0X=1 S0S 1 00 S 3S 400 S 3S0 00 S 4S0 10

Nonlinear & Neural Networks LAB. Multiple Input State Diagram Example State Diagram Symbolic State Diagram PRESENT STATE NEXT STATE OUTPUT 00 01 10 11 S0S0 S1 S2 S31 S1S0 S3 S1 S50 S2S1 S3 S2 S41 S3S1 S0 S4 S50 S4S0 S1 S2 S51 S5S1 S4 S0 S50

Nonlinear & Neural Networks LAB. Multiple Input Example Implication Chart Minimized State Table PRESENT STATE NEXT STATE OUTPUT 00 01 10 11 S 0S 0 S1 S2 S 31 S1S 0 S 3 S1 S 30 S2S1 S 3 S2 S 01 S 3S1 S 0 S 0 S 30

Nonlinear & Neural Networks LAB. Implication Chart Method Does the method solve the problem with the odd parity checker? Implication Chart S0 is equivalent to S2 since nothing contradicts this assertion!

Nonlinear & Neural Networks LAB. Implication Chart Method The detailed algorithm: 1.Construct implication chart, one square for each combination of states taken two at a time 2.Square labeled Si, Sj, if outputs differ than square gets "X". Otherwise write down implied state pairs for all input combinations 3.Advance through chart top-to-bottom and left-to-right. If square Si, Sj contains next state pair Sm, Sn and that pair labels a square already labeled "X", then Si, Sj is labeled "X". 4.Continue executing Step 3 until no new squares are marked with "X". 5.For each remaining unmarked square Si, Sj, then Si and Sj are equivalent.

Nonlinear & Neural Networks LAB. (e,g) e Row matching methode (State reduction) PS NSOutput X=0X=1X=0X=1 aab00 bcd00 cad00 def01 eaf01 fgf01 gaf01 PS NSOutput X=0X=1X=0X=1 aab00 bcd00 cad00 de f01 e af01 fe f01

Nonlinear & Neural Networks LAB. Row matching methode (State reduction) PS NSOutput X=0X=1X=0X=1 aab00 bcd00 cad00 de f01 e af01 fe f01 (d,f) d PS NSOutput X=0X=1X=0X=1 aab00 bcd 00 cad 00 d e d 01 e ad 01

Nonlinear & Neural Networks LAB.

Nonlinear & Neural Networks LAB. Present State Next StatePresent Output X=01 adc0 bfh0 ced1 dae0 eca1 ffb1 gbh0 hcg1 Implication chart (State reduction)

Nonlinear & Neural Networks LAB.

Present State Next State Output X=01 a a c 0 bfh0 c c a 1 ffb1 gbh0 hc g1 (a,d) a (c,e) c

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