EncodingEncodingx

s0 1 0 1

A A B 0 0

B A C 0 0

C D C 0 0

D A B 0 1

Variant I

A = 00B = 01C = 10D = 11

Variant I

A = 00B = 01C = 10D = 11

Variant II

A = 00B = 11C = 01D = 10

Variant II

A = 00B = 11C = 01D = 10

21211 QQxQQD

2121212 QQxQxQQQxD

21QxQy

2121 QQxQxD

xD2

21QxQy

Variant IIVariant II

Variant IVariant I

Encodings

3 states - 3 encodings3 states - 3 encodings

4 states - 3 encodings4 states - 3 encodings

5 states - 5 states - 140 encodings140 encodings

7 states - 7 states - 840 encodings840 encodings

9 states - 9 states - More than 10 million encodingsMore than 10 million encodings

How to encode? How to encode?

Can we check all possible encodings?Can we check all possible encodings?

Partition reminder…

b =

Product of partitions a • b is the largest (with respect to

relation ) partition, that is not larger than a and b.

)3,5;2,6 ;1,4()3,5,6;1,2,4(a =

a • b = )3,5;6;2 ;1,4(

Sum of partitions…

Sum of partitions a + b is the smallest (with respect to

relation ) partition, which is not smaller than a and b.

9,8,7;6,5;4,3;2,1a

9;8,7;5,4;3,2;6,1b

9,8,7;6,5,4,3,2,1 ba

Substitution Property of a partition

Partition on set of states of machine M=<S, V, δ> has the substitution property (closed partition), when:

Partition on set of states of machine M=<S, V, δ> has the substitution property (closed partition), when:

mpipib

kkjip bvsvsbbssVvm

)),((),,(( ),,( ,

Partition has the substitution property when elements of a block under any input symbol transit to themselves or to other block of partition

Partition has the substitution property when elements of a block under any input symbol transit to themselves or to other block of partition

Theorem

Given is automaton M with set of states S, |S| = n.To encode states we need Q1, ..., Qk memory elements (flip-flops).

If partition exist with substitution property and if r among k encoding variables Q1, ..., Qk, where

r = log2(,), is subordinated to blocks of partition such that all states

included in one block are denoted with the same variables Q1, ..., Qr , 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 Q1, ...,

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.

Given is automaton M with set of states S, |S| = n.To encode states we need Q1, ..., Qk memory elements (flip-flops).

If partition exist with substitution property and if r among k encoding variables Q1, ..., Qk, where

r = log2(,), is subordinated to blocks of partition such that all states

included in one block are denoted with the same variables Q1, ..., Qr , 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 Q1, ...,

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.

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 M1, M2 is existence of partition with

substitution property and partition such = 0.

Given is automaton M with set of states S. 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.

f1(x,Q1) D1 f2(x,Q1,Q2) D2 f0(x,Q2)xx

q1q1 Q1

Q1 Q2Q2q2

q2 zz

Parallel Decomposition

Automaton M jest decomposable into two sub-automata M1, M2

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)

Automaton M jest decomposable into two sub-automata M1, M2

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)

f0(x,Q1,Q2)xx

f2(x,Q2) D2

q2q2

zzf1(x,Q1) D1

q1q1

Q1Q1

Q2Q2

Serial Decomposition - ExampleSerial Decomposition - Example

xs

0 1 0 1

A A F 0 0

B E C 0 1

C C E 0 1

D F A 1 0

E B F 1 1

F D E 0 0

s11s12s12

s12s11s11

10x

s

1

0

0

S11,0

1

1

0

S11,1

0

0

1

S12,0

0

1

0

S12,1

s21

s22

s23

S12,0

s23

s23

s21

S12,1

s23s22s23

s22s23s22

s23s21s21

S11,1S11,0in

s

FDCEBA ,,;,,1

= (0) = (0)

FECBDAτ ,;,;,

s11s11 s12

s12

s21s21 s22

s22 s23s23

State of the predecessor

machine

State of the predecessor

machine State of primary input x

State of primary input x

f1(x,Q1) D1 f2(x,Q1,Q2) D2f0(x,Q2)

xxq1

q1 Q1Q1 Q2

Q2q2q2 zz

s11s12s12

s12s11s11

10x

s

1

0

0

S11,0

1

1

0

S11,1

0

0

1

S12,0

0

1

0

S12,1

s21

s22

s23

S12,0

s23

s23

s21

S12,1

s23s22s23

s22s23s22

s23s21s21

S11,1S11,0x

s

Serial Serial Decomposition – Decomposition –

Example continuedExample continued

xs

0 1 0 1

A A F 0 0

B E C 0 1

C C E 0 1

D F A 1 0

E B F 1 1

F D E 0 0

FDCEBA ,,;,,1

FECBDAτ ,;,;,s21

s21 s22s22 s23

s23

s11s11 s12

s12

S11=ABES11=ABE

S21=ADS21=AD S12 =CDFS12 =CDF

M1 =M1 =

BCBC

EFEF

Parallel Parallel Decomposition-Decomposition-

ExampleExample

1 2 = (0)1 2 = (0)

s23s22s23

s21s23s22

s23s21s21

10x

s

s23

s21

s23

S11,1

s23

s21

s23

S12,1

s22s22s23

s23s23s22

s21s21s21

S12,0S11,0x

s

FDCEBA ,,;,,1 s11

s11 s12s12

FEDBCA ,;,;,2 s21

s21 s22s22 s23

s23

xs

0 1 0 1

A A F 0 0

B E C 0 1

C C E 0 1

D F A 1 0

E B F 1 1

F D E 0 0

ACAC

BDBD

EFEF

ABE CDF ABE CDFABE CDF ABE CDF

outxx

M2(2)yy

M1

Combining columns

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

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

Decomposition Schemata

M1 M2() outxx

22 yy

outxx

M2(2)

yy

M1

Serial DecompositionSerial Decomposition

Parallel DecompositionParallel Decomposition

Calculating a closed partition

xs

0 1

A A F

B E C

C C E

D F A

E B F

F D E

A,BA,B

A,EA,E C,FC,F

C,DC,DFF EE

B,DB,D

A,CA,C E,FE,F

A,DA,D A,FA,F

A,BA,B

A,CA,C

A,DA,D

FD,C,;EB,A,1

FE,;DB,;CA,2

1

We create a graph of pairs of successors for various initial nodes.

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 Sj S and for all vl 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 i of the set of states S of this

machine such that i

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 Sj S and for all vl 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 i of the set of states S of this

machine such that i