Algebraic Decomposition of Finite State Automata and Formal Models - - PowerPoint PPT Presentation

SLIDE 1

Attila Egri-Nagy, Chrystopher L. Nehaniv

Algebraic Decomposition of Finite State Automata and Formal Models of Understanding

University of Hertfordshire, United Kingdom

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Outline

Krohn-Rhodes Theory Formal models of understanding, applications Implementations Examples

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Krohn-Rhodes Theory

Automata are studied as algebraic objects (semigroup) and the main concern is the ’factorization‘

• f automata.

Partially and informally: Theorem 1 (Krohn-Rhodes Decomposition Theorem) A finite automaton A can be represented homomorphically by a cascade product of components from {AF, AG1, . . . , AGn}. where F is the flip-flop monoid (the smallest semigroup with an identity and two right-zero element), and G1, . . . , Gn denote simple groups dividing the characteristic semigroup of A.

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hierarchical composition, wreath product

f1 ∈S1 f2 :A1 →S2 f3 :A2×A1 →S3 (A1, S1) (A2, S2) (A3, S3) b1 ∈A1 b2 ∈A2 b3 ∈A3

a1 ∈A1 a1 a2 ∈A2

(A3, S3) ≀ (A2, S2) ≀ (A1, S1) (a3, a2, a1) · (f3, f2, f1) = (b3, b2, b1) = (a3 · f3(a2, a1), a2 · f2(a1), a1 · f1)

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The significance of the theorem

Rephrasing the theorem for a practical computer scientist: For all systems for which we can give a finite state automata description a hierarchical model can be generated automatically. Hierarchical implies:

information flow between levels are restricted

enabling modularity (also within one level with parallel components)

generalization and specialization are natural

• perations realized by taking subsets of levels in

either direction up or down the hierarchy

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Practical Applications

artificial intelligence: creating representations of

the environment on the fly

automated object-oriented programming biology: well-defined complexity measure,

understanding metabolic networks

physics: top level coordinates correspond to

conserved quantities The possible users can equally be humans, robots or software.

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mathematicians vs. programmers

A mathematical proof sometimes provides a clear algorithm but usually efficiency and computational feasibility are not considered. problematic points: ∃, ∀ when the sets are huge. the only problem in semigroup theory: there are so many elements... A finite state machine with n states may end up with a semigroup with nn elements, e.g. 1010 = 10 billion and sometimes we deal with subsets, and their number is

21010 so it’s not just the question of memory-upgrade...

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Implementations I.

V ∪ T-technique (Krohn, Rhodes 1965.)

iteratively decomposes a semigroup into two

possibly overlapping parts (using the Green class picture)

the number of hierarchical components is big =

⇒

practically inapplicable

it is implemented in a computer algebra system,

called GAP.

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Implementations II.

Holonomy decomposition (Zeiger 1968, Eilenberg 1974, Nehaniv 2004)

works with a detailed study of how the automaton’s

characteristic semigroup acts on the subsets of the state set

implemented using breadth-first search, O(nn) applying techniques from the theory of formal

languages, O(2n)

implemented as a standalone software (due to its

experimental nature) in Java.

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Example I

state set: residue classes mod 6 input symbols: adding one +1, doubling x2

1 2

+1,x2

3

+1

4

x2 +1 x2 x2 +1 x2

5

+1 +1 x2

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Example I - images

I = images ∪ singletons ∪ state set.

The images as they are generated:

{0,1,2,3,4,5} {0,2,4}

x2

{1,3,5}

+1

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Example I - subduction, tiling, skeleton

P ≤ Q ⇐ ⇒ ∃s ∈ S1, P ⊆ Q · s (P, Q ∈ I). P ≡ Q ⇐ ⇒ P ≤ Q and Q ≤ P. P ≺ Q, ifP ⊂ Q, P < QandP is maximal

{5} {4} {1} {3} {2} {0} {0,2,4} {1,3,5} {0,1,2,3,4,5} 2 1

C2 ≀ S3

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Example II – randomly generated automaton

2

y

3

z

1

x

4

x,y

5

z x

6

y,z y x z x,y z z x,y

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Example II

{1,2,3,4,5,6} {1,2,3,4,5}

x

{1,2,3,4,6}

y

{1,3,4,5,6}

z

{1,3,4,5}

x

{1,3,5,6}

z

{1,2,3,4}

x

{1,2,4,6}

y

{3,4,5,6}

z

{1,3,4}

x

{3,5,6}

z

{1,2,3}

x

{1,2,6}

y

{3,4,6}

z

{2,3,4,5}

x

{2,3,4,6}

y

{1,4,5,6}

z

{1,3,4,6}

y

{1,2,4,5}

x

{1,4}

x

{1,2,4}

y

{3,5}

z

{1,2}

x y

{3,4}

z

{2,3,4}

x

{2,4,6}

y

{4,5,6}

z

{1,2,3,6}

y

{1,3,5}

x

{1,3,6}

z

{1,4,6}

y

{2,4,5}

x y

{1,4,5}

z y

{1,2,3,5}

x y

{3,4,5}

z

{1}

x

{3}

z

{4,6}

y

{5,6}

z

{2,3,6}

y

{1,3}

x

{1,6}

y

{3,6}

z x z

{4,5}

x

{1,5}

z z

{2,6}

y

{2,3,5}

x y

{4}

x y

{5}

z

{1,5,6}

z

{2,4}

x y y

{2,3}

x y

{2,5}

x

{2}

y

{1,2,5}

x

{6}

y z

The bottleneck of the current implementation is that I is fully calculated.

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Example II – long decomposition

15 hierarchical levels for 6 states

{6} {5} {2} {3} {4} {1} {1,2} {1,4} {1,3} {4,5} {4,6} {3,4} {5,6} {2,3} {2,5} {3,6} {3,5} {2,6} {1,5} {2,4} {1,6} {1,2,6} {1,2,4} {1,2,3} {1,2,5} {1,4,6} {1,4,5} {1,3,5} {4,5,6} {2,3,4} {1,3,6} {3,4,6} {3,4,5} {1,3,4} {2,3,5} {2,3,6} {3,5,6} {2,4,5} {1,5,6} {2,4,6} {1,2,3,6} {1,2,4,6} {1,2,4,5} {1,2,3,4} {1,2,3,5} {1,4,5,6} {1,3,4,6} {2,3,4,5} {1,3,5,6} {2,3,4,6} {3,4,5,6} {1,3,4,5} {1,2,3,4,6} {1,2,3,4,5} {1,3,4,5,6} {1,2,3,4,5,6} 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

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Thank You!

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