03 Composition of state machinescs.ioc.ee/~margo/aat/03_Composition_of_state_machines.pdfcascade...
Transcript of 03 Composition of state machinescs.ioc.ee/~margo/aat/03_Composition_of_state_machines.pdfcascade...
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Composition of State Machines
Mealy machinesProducts of Mealy machinesProducts of semiautomata
Products of transformation semigroupsProducts of incomplete machines
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Mealy machine
statesInput alphabet
Output alphabetTransition function
Output function
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The action of Mealy Machine
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Processing word
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Mealy machine as black box
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Restricted Parallel Connection
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Restricted direct product
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Full parallel connection
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Full direct product
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General parallel connection
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General direct product
=
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Cascade connection
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Cascade product
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Alternative interpretation of cascade connection
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Wreath connection
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Wreath product
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Restricted direct product of state machines
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Example
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Full direct product of state machines
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Example
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Cascade product of state machines
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Example
σ
σ
σ
σ11(1,1)
σ10(0,1)
σ00(0,0)
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Wreath product of state machines
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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)
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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))
‘
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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’)
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Transformation semigroup of restricted direct product
iff
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Restricted direct product of transformation semigroups
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Theorem
M M’
M product M’
TS(M) TS(M’)
TS(M product M’)
TS(M) product TS(M’) product – restricted direct product
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Full direct product of transformation semigroups
Direct product of two semigroups
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Theorem
M M’
M product M’
TS(M) TS(M’)
TS(M product M’)
TS(M) product TS(M’) product – full direct product
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Theorem
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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'.
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Wreath product of transformation semigroups
then
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Theorem
M M’
M product M’
TS(M) TS(M’)
TS(M product M’)
TS(M) product TS(M’) product –cascade/wreath product
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Associativity. Theorems
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Products of incomplete machines
We now extend our definitions of products of state machines and transformation semigroups to include the incomplete cases.
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Restriction of state machine
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Example
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Restriction of transformation semigroup
:
whereIs defined by
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Products of incomplete machines
suppose
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Products of incomplete transformation semigroups
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TheoremsAll things are complete
3
2
1