Composition of State Machines

Mealy machinesProducts of Mealy machinesProducts of semiautomata

Products of transformation semigroupsProducts of incomplete machines

Mealy machine

statesInput alphabet

Output alphabetTransition function

Output function

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

σ

σ

σ

σ11(1,1)

σ10(0,1)

σ00(0,0)

Wreath product of state machines

Example

Set of all mappings

ττδ

στγ

τσβ

σσα

10

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)

Example 2

Set of all mappings

σσα

10

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)

F°((q’,q),(f, σ))=(F’(q’, f(q)),F(q, σ))

‘

Product of transformation semigroups

All state machines and transformation semigroups will be assumed to be complete in this section.

M M’

M product M’

TS(M) TS(M’)

TS(M product M’)

TS(M) product TS(M’)

Transformation semigroup of restricted direct product

iff

Restricted direct product of transformation semigroups

Theorem

M M’

M product M’

TS(M) TS(M’)

TS(M product M’)

TS(M) product TS(M’) product – restricted direct product

Full direct product of transformation semigroups

Direct product of two semigroups

Theorem

M M’

M product M’

TS(M) TS(M’)

TS(M product M’)

TS(M) product TS(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

M M’

M product M’

TS(M) TS(M’)

TS(M product M’)

TS(M) product TS(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

:

whereIs defined by

Products of incomplete machines

suppose

Products of incomplete transformation semigroups

TheoremsAll things are complete

3

2

1