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Encoding

Encoding. V ariant I A = 00 B = 01 C = 10 D = 11. V ariant II A = 00 B = 11 C = 01 D = 10. V ariant II. V ariant I. 3 sta tes - 3 encodings. 4 sta tes - 3 encodings. Encodings. How to encode ?. Can we check all possible encodings?. 5 sta tes -. 140 encodings.

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Encoding

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  1. Encoding Variant I A = 00 B = 01 C = 10 D = 11 Variant II A = 00 B = 11 C = 01 D = 10 Variant II Variant I

  2. 3 states - 3 encodings 4 states - 3 encodings Encodings How to encode? Can we check all possible encodings? 5 states - 140 encodings 7 states - 840 encodings More than 10 million encodings 9 states -

  3. Partition reminder… Product of partitionsa•b is the largest(with respect to relation ) partition, that is not larger than aandb. a = b = a• b=

  4. Sum of partitions… Sum of partitionsa+b is the smallest (with respect to relation) partition, which is not smaller than aandb.

  5. Substitution Property of a partition Partition on set of states of machine M=<S, V, δ> has the substitution property (closed partition), when: Partition has the substitution property when elements of a block under any input symbol transit to themselves or to other block of partition 

  6. Theorem Given is automatonMwith set of statesS, |S| = n. To encode states we need Q1, ..., Qkmemory elements (flip-flops). If partition  exist with substitution property and if ramongkencoding variablesQ1, ..., Qk, wherer = log2(,), is subordinated to blocks of partition  such that all states included in one block are denoted with the same variables Q1, ..., Qr , thenfunctionsQ’1, ..., Q’r, are independent on remaining (k – r) variables. Conversely, if firstrvariables of the next state Q’1, ..., Q’r (1  r < k) can be determined from the values of inputs and first rvariablesQ1, ..., Qr independently on values of the remaining variables, then there exists partition  with substitution property such that two states si, sj are in the same block of partition if and only if they are denoted by the same value of the first r variables.

  7. x q1 Q1 q2 Q2 z f1(x,Q1) D1 f2(x,Q1,Q2) D2 f0(x,Q2) Serial Decomposition Given is automatonMwith set of statesS. Sufficient and necessary condition of serial decomposition of M into two serially connected automata M1, M2 is existence of partition with substitution property and partition  such    = 0.

  8. q1 f1(x,Q1) D1 Q1 z f0(x,Q1,Q2) x q2 Q2 f2(x,Q2) D2 Parallel Decomposition AutomatonM jest decomposable into two sub-automataM1, M2 working in parallel iff in the set of states S of this automaton there exist two non-trivial partitions 1, 2with substitution property such that 1 2 = (0)

  9. in s S11,0 S11,1 S12,0 S12,1 S11,0 S11,1 S12,0 S12,1 x s 0 1 s21 s21 s23 s23 s21 0 0 1 0 s22 s23 s22 s22 s23 0 1 0 1 s11 s11 s12 s12 s12 s11 s23 s22 s23 s21 s23 1 1 0 0 Serial Decomposition - Example s12 s11 s21 s22 s23    = (0) State of the predecessor machine State of primary input x

  10. x q1 Q1 q2 Q2 z f1(x,Q1) D1 f2(x,Q1,Q2) D2 f0(x,Q2) x s S11,0 S11,1 S12,0 S12,1 S11,0 S11,1 S12,0 S12,1 x s 0 1 s21 s21 s23 s23 s21 0 0 1 0 s22 s23 s22 s22 s23 0 1 0 1 s11 s11 s12 s12 s12 s11 s23 s22 s23 s21 s23 1 1 0 0 s12 s11 Serial Decomposition – Example continued s21 s22 s23 S11=ABE S12 =CDF S21=AD M1 = BC EF

  11. x s S11,0 S12,0 S11,1 S12,1 x s 0 1 M1 s21 s21 s21 s23 s23 s21 s21 s23 s22 s23 s23 s21 s21 out y s22 s23 s21 x s23 s22 s22 s23 s23 s23 s22 s23 M2(2) Parallel Decomposition-Example s11 s12 s21 s22 s23 Knowing both partitions we can create table 2, next combining columns with the same input X we obtain the table of one of machines 1 2 = (0) ABE CDF ABE CDF AC BD M2 Combining columns EF

  12. x 2 y M1 M2() out M1 out y x M2(2) Decomposition Schemata Serial Decomposition Parallel Decomposition

  13. A,B B,D A,C A,E C,F E,F C,D A,D A,F Calculating a closed partition We create a graph of pairs of successors for various initial nodes. E F A,B A,C A,D

  14. Dekompozycja z autonomicznym zegarem Some automata have a decomposition in which we use the autonomous clock - and sub-automaton that is not dependent on inputs. Partition iof set of statesSof automatonMis compatible with input, if for each stateSj S and for allvl  V (Sj,v1), (Sj,v2), ..., (Sj,vl), ..., (Sj,vp), are in one block of partition i. A sufficient and necessary condition of existence of decomposition of automaton M, with an autonomous clock with log2() states is that there exists a closed partition  and a non-trivial, compatible with input partition iof the set of states Sof this machine such that   i

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