1. Encoding xsxs 0101 AAB00 BAC00 CDC00 DAB01 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. Encodings 3 states - 3 encodings 4 states - 3 encodings 5 states - 140 encodings 7 states -840 encodings 9 states - More than 10 million encodings How to encode? Can we check all possible encodings?

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

5. Substitution Property of a partition Partition  on set of states of machine M= 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 automaton M with set of states S, |S| = n. To encode states we need Q 1,..., Q k memory elements (flip-flops). If partition  exist with substitution property and if r among k encoding variables Q 1,..., Q k, where r =  log 2  (,  ) , is subordinated to blocks of partition  such that all states included in one block are denoted with the same variables Q 1,..., Q r, then functions Q’ 1,..., Q’ r, are independent on remaining (k – r) variables. Conversely, if first r variables of the next state Q’ 1,..., Q’ r (1  r < k) can be determined from the values of inputs and first r variables Q 1,..., Q r independently on values of the remaining variables, then there exists partition  with substitution property such that two states s i, s j are in the same block of partition if and only if they are denoted by the same value of the first r variables.

7. Serial Decomposition Given is automaton M with set of states S. Sufficient and necessary condition of serial decomposition of M into two serially connected automata M 1, M 2 is existence of partition  with substitution property and partition  such    = 0. f 1 (x,Q 1 )D1D1 f 2 (x,Q 1,Q 2 )D2D2 f 0 (x,Q 2 ) x q1q1 Q1Q1 Q2Q2 q2q2 z

8. Parallel Decomposition Automaton M jest decomposable into two sub-automata M 1, M 2 working in parallel iff in the set of states S of this automaton there exist two non-trivial partitions  1,  2 with substitution property such that  1   2 =  (0) f 0 (x,Q 1,Q 2 ) x f 2 (x,Q 2 )D2D2 q2q2 z f 1 (x,Q 1 )D1D1 q1q1 Q1Q1 Q2Q2

9. Serial Decomposition - Example xsxs 0101 AAF00 BEC01 CCE01 DFA10 EBF11 FDE00 s 11 s 12 s 11 10 xsxs 1 0 0 S 11,0 1 1 0 S 11,1 0 0 1 S 12,0 0 1 0 S 12,1 s 21 s 22 s 23 S 12,0 s 23 s 21 S 12,1 s 23 s 22 s 23 s 22 s 23 s 22 s 23 s 21 S 11,1S 11,0 in s    =  (0) s 11 s 12 s 21 s 22 s 23 State of the predecessor machine State of primary input x

10. f 1 (x,Q 1 )D1D1 f 2 (x,Q 1,Q 2 ) D2D2 f 0 (x,Q 2 ) x q1q1 Q1Q1 Q2Q2 q2q2 z s 11 s 12 s 11 10 xsxs 1 0 0 S 11,0 1 1 0 S 11,1 0 0 1 S 12,0 0 1 0 S 12,1 s 21 s 22 s 23 S 12,0 s 23 s 21 S 12,1 s 23 s 22 s 23 s 22 s 23 s 22 s 23 s 21 S 11,1S 11,0 xsxs Serial Decomposition – Example continued xsxs 0101 AAF00 BEC01 CCE01 DFA10 EBF11 FDE00 s 21 s 22 s 23 s 11 s 12 S 11 =ABE S 21 =AD S 12 =CDF M 1 = BC EF

11. Parallel Decomposition- Example  1   2 =  (0) s 23 s 22 s 23 s 21 s 23 s 22 s 23 s 21 10 xsxs s 23 s 21 s 23 S 11,1 s 23 s 21 s 23 S 12,1 s 22 s 23 s 22 s 21 S 12,0S 11,0 xsxs s 11 s 12 s 21 s 22 s 23 xsxs 0101 AAF00 BEC01 CCE01 DFA10 EBF11 FDE00 AC BD EF ABE CDF out x M2(2)M2(2) y M1M1 Combining columns Knowing both partitions we can create table 2, next combining columns with the same input X we obtain the table of one of machines M2M2

12. Decomposition Schemata M1M1 M2()M2() out x 2 y x M2(2)M2(2) y M1M1 Serial Decomposition Parallel Decomposition

13. Calculating a closed partition xsxs 01 AAF BEC CCE DFA EBF FDE A,BA,EC,FC,D F E B,DA,CE,FA,DA,F A,B A,C A,D We create a graph of pairs of successors for various initial nodes.

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  i of set of states S of automaton M is compatible with input, if for each state S j  S and for all v l  V  (S j,v 1 ),  (S j,v 2 ),...,  (S j,v l ),...,  (S j,v p ), are in one block of partition  i. A sufficient and necessary condition of existence of decomposition of automaton M, with an autonomous clock with  log 2  (  )  states is that there exists a closed partition  and a non- trivial, compatible with input partition  i of the set of states S of this machine such that    i