Composition of State
Machines
Mealy machines
Products of Mealy machines
Products of semiautomata
Products of transformation semigroups
Products of incomplete machines
Mealy machine
Output function
states
Input alphabet
Transition function
Output alphabet
The action of Mealy Machine
Processing word
Mealy machine as black box
Restricted Parallel Connection
Restricted direct product
Full parallel connection
Full direct product
General parallel connection
General direct product
=
Cascade connection
Cascade product
Alternative interpretation of
cascade connection
Wreath connection
Wreath product
Restricted direct product of state
machines
Example
Full direct product of state
machines
Example
Cascade product of state machines
Example
(0,0) σ
(0,1) σ
(1,1) σ
0
0
1
0 σ
1 σ
1 σ
Wreath product of state machines
Example
F°((0,0),( α, σ))=(F(0, α(0)),F’(0, σ))=(1,1)
F°((1,0),( α, σ))=(F(1, α(0)),F’(0, σ))=(1,1)
F°((0,1),( α, σ))=(F(0, α(1)),F’(1, σ))=(1,0)
F°((1,1),( α, σ))=(F(1, α(1)),F’(1, σ))=(1,0)
Set of all mappings
0
1
α
σ
σ
β
σ
τ
γ
τ
σ
δ
τ
τ
Example 2
F°((q’,q),(f, σ))=(F’(q’, f(q)),F(q, σ))
F°((0,0),( α, σ))=(F’(0, α(0)),F(0, σ))=(1,1)
F°((1,0),( α, σ))=(F’(1, α(0)),F(0, σ))=(0,1)
F°((0,1),( α, σ))=(F’(0, α(1)),F(1, σ))=(1,1)
F°((1,1),( α, σ))=(F’(1, α(1)),F(1, σ))=(0,1)
‘
Set of all mappings
α
0
1
σ
σ
Product of transformation
semigroups
TS(M) product TS(M’)
TS(M)
TS(M’)
TS(M product M’)
M
M’
M product M’
All state machines and transformation semigroups will be assumed to be
complete in this section.
Transformation semigroup of
restricted direct product
iff
Restricted direct product of
transformation semigroups
Theorem
TS(M) product TS(M’)
TS(M)
TS(M’)
TS(M product M’)
M
M’
M product M’
product – restricted
direct product
Full direct product of transformation
semigroups
Direct product of
two semigroups
Theorem
TS(M) product TS(M’)
TS(M)
TS(M’)
TS(M product M’)
M
M’
M product M’
product – full direct
product
Theorem
Cascade and wreath products
There is no simple straightforward construction that yields the
transformation semigroup B from a suitable combination of A and A'.
What we will do here is to show that B can be covered by the wreath
product of the transformation semigroups A and A'.
Wreath product of transformation
semigroups
then
Theorem
TS(M) product TS(M’)
TS(M)
TS(M’)
TS(M product M’)
M
M’
M product M’
product –
cascade/wreath
product
Associativity. Theorems
Products of incomplete machines
We now extend our definitions of products of state
machines and transformation semigroups to include the
incomplete cases.
Restriction of state machine
Example
Restriction of transformation
semigroup
:
where
Is defined by
Products of incomplete machines
suppose
Products of incomplete
transformation semigroups
All things are complete
Theorems
1
2
3