2. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
3. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
4. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
5. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
6. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
7. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
8. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
9. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
10. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
11. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
12. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
13. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
14. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
15. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
16. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
17. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
18. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
19. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
20. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
21. Disambiguation Of Stochasic Processes
Competence Response Of Transgenic B. subtilis
Bacillus subtilis
Strains
Comptence Circuit
(Wild Type)
Comptence Response
Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)
23. Basic Idea: Negate & Subtract!
Just Like Real Numbers, · · · Well Almost
x + (−y) = 0 ⇔ x = y
Example: 0.375 + (−0.375) = 0
24. Basic Idea: Negate & Subtract!
Just Like Real Numbers, · · · Well Almost
x + (−y) = 0 ⇔ x = y
Example: 0.375 + (−0.375) = 0
25. Symbols Can Be Anything
Individual Symbols Need Not Have Algebraic Properties
Manipulate the information content in symbolic streams. . .
+
−
No matter what the symbols are. . .
26. Quantization Of Observed Sequences
How Do We Get Symbols?
σ6
σ5
σ8
σ7 σ1 σ1 σ3 · · ·
· · · σ5 σ3 σ3 · · ·
σ2
σ1 (a) Phase Space Quantization
σ3
σ4
σ5 Binary
Quantization
σ4
σ3
σ2 (b) Data Range Discretization
σ1
Continuous 8 symbol
Signal Quantization
27. Probabilistic Finite State Automata As Stochastic
Models
Syntactic Definition: The Anatomy
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
Alphabet)
σ1 |0.8 σ1 |0.3
σ2 |0.1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
at state q3
PFSA
Can act as symbolic generators
28. Probabilistic Finite State Automata As Stochastic
Models
Syntactic Definition: The Anatomy
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
Alphabet)
σ1 |0.8 σ1 |0.3
σ2 |0.1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
at state q3
PFSA
Can act as symbolic generators
29. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
30. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
31. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
32. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
33. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
34. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
35. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1σ1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
36. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1σ1σ1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
37. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1σ1σ1σ2
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
38. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1σ1σ1σ2σ1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
39. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1σ1σ1σ2σ1σ2
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
40. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1σ1σ1σ2σ1σ2σ2
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
41. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1σ1σ1σ2σ1σ2σ2σ1
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
42. Probabilsitic Automata As Stcoachstic Models
Can Act As Generators
σ1 |0.9 σ2 |0.2
States
q2
q1
Transition
(Events From
σ1 |0.3 Alphabet)
σ1 |0.8
σ2 |0.1 σ1σ1σ2σ2σ1σ1σ1σ1σ2σ1σ2σ2σ1σ2
q3 Alphabet
Symbol
σ2 |0.7
Probability of
generating σ2
PFSA at state q3
43. Causal States Of Dynamical Processes
Equivalence Classes Of The Nerode Relation
Future-equivalent Strings
Go To The Same ”State“
Equality Of Distributions over...
future symbols
future substrings
of length 2
future substrings What Does
of length 3
”Equivalence Of Futures“
future substrings
of length 4 Mean ?
all lengths of future substrings
44. Causal States Of Dynamical Processes
Equivalence Classes Of The Nerode Relation
Future-equivalent Strings
Go To The Same ”State“
Equality Of Distributions over...
future symbols
future substrings
of length 2
future substrings What Does
of length 3
”Equivalence Of Futures“
future substrings
of length 4 Mean ?
all lengths of future substrings
45. Stochastic Processes As Probabilistic Dynamical
Systems
Measures On Infinite Strings
Present
Dynamics completely
specified by the
probabilites:
p(τ1 Σω )
p(τ2 Σω )
Past p(τ3 Σ
ω )
Observation
.
.
τ .
All Possible
Infinite
Futures (Σω )
Quantized
Stochastic Process −− − − − −
− − − − −→ Probability space (Σω , B, P)
Finite Encoding
Probability space (Σω , B, P) −− − − − − − − −
− − − − − − − −→ PFSA
46. Stochastic Processes As Probabilistic Dynamical
Systems
Measures On Infinite Strings
Present
Dynamics completely
specified by the
probabilites:
p(τ1 Σω )
p(τ2 Σω )
Past p(τ3 Σ
ω )
Observation
.
.
τ .
All Possible
Infinite
Futures (Σω )
Quantized
Stochastic Process −− − − − −
− − − − −→ Probability space (Σω , B, P)
Finite Encoding
Probability space (Σω , B, P) −− − − − − − − −
− − − − − − − −→ PFSA
47. Stochastic Processes As Probabilistic Dynamical
Systems
Measures On Infinite Strings
Present
Dynamics completely
specified by the
probabilites:
p(τ1 Σω )
p(τ2 Σω )
Past p(τ3 Σ
ω )
Observation
.
.
τ .
All Possible
Infinite
Futures (Σω )
Quantized
Stochastic Process −− − − − −
− − − − −→ Probability space (Σω , B, P)
Finite Encoding
Probability space (Σω , B, P) −− − − − − − − −
− − − − − − − −→ PFSA
48. Stochastic Processes As Probabilistic Dynamical
Systems
Measures On Infinite Strings
Present
Dynamics completely
specified by the
probabilites:
p(τ1 Σω )
p(τ2 Σω )
Past p(τ3 Σ
ω )
Observation
.
.
τ .
All Possible
Infinite
Futures (Σω )
Quantized
Stochastic Process −− − − − −
− − − − −→ Probability space (Σω , B, P)
Finite Encoding
Probability space (Σω , B, P) −− − − − − − − −
− − − − − − − −→ PFSA
49. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
50. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
51. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
52. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
53. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
54. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
55. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
56. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
57. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
58. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
59. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
60. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
61. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
62. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
63. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
64. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
65. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
66. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
67. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
68. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
69. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
70. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
71. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
72. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
73. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
74. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
75. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
76. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
77. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
78. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
79. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
80. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
81. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
82. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
83. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
84. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
85. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
86. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
87. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
88. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
89. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
90. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
91. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
92. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
93. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
94. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
95. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
96. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
97. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
98. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
99. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
100. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
101. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
102. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
103. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
104. Relationship Between Space of Automata & Space Of
Measures
Multiple PFSA May Map To Same Measure
Any two PFSA can be
Dynamics-equivalent made structure equivalent
PFSA
Space Of PFSA (A ) Non-minimal realizations exist
Structue is not that important
measure 2
measure 1
Space Of measures on Each measure defines
infinite strings (P) a probability space: (Σω , B, p1 )
105. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
Any string
1(0.1)
terminating in 10
0(0.9)
q2
q1
1(0.2)
0(0.8)
Any string
terminating in 1
1(0.7)
q3
0(0.3)
Any string
terminating in 00
• Causality Structure
How many causal states?
Interdependence between causal states
• Probability Morph
Σ⋆ Immediate future from causal states
(Set of all
finite strings on Analogous to tangent vectors
alphabet Σ)
106. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
Any string
1(0.1)
terminating in 10
0(0.9)
q2
q1
1(0.2)
0(0.8)
Any string
terminating in 1
1(0.7)
q3
0(0.3)
Any string
terminating in 00
• Causality Structure
How many causal states?
Interdependence between causal states
string:
ω′ 01
1
• Probability Morph
string:
ω00 Σ⋆ Immediate future from causal states
(Set of all
0 string: finite strings on Analogous to tangent vectors
ω′ 00 alphabet Σ)
(A)
Transition
Structure
induced by
state definitions
107. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
Any string
1(0.1)
terminating in 10
0(0.9)
q2
q1
1(0.2)
0(0.8)
Any string
terminating in 1
1(0.7)
q3
0(0.3)
Any string
terminating in 00
0(0.9)
1(0.1)
1(0.1)
• Causality Structure
0(0.9)
1(0.2) How many causal states?
(B)
Symbol Interdependence between causal states
0(0.8) probabilities
identical from
1(0.2) same state
0(0.8)
• Probability Morph
Σ⋆ Immediate future from causal states
(Set of all
finite strings on Analogous to tangent vectors
alphabet Σ)
108. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
Any string
1(0.1)
terminating in 10
0(0.9)
q2
q1
1(0.2)
0(0.8)
Any string
terminating in 1
1(0.7)
q3
0(0.3)
Any string
terminating in 00
0(0.9)
1(0.1)
1(0.1)
• Causality Structure
0(0.9)
1(0.2) How many causal states?
(B)
Symbol Interdependence between causal states
string: 0(0.8) probabilities
identical from
ω′ 01
1(0.2) same state
0(0.8)
1
• Probability Morph
string: Immediate future from causal states
ω00 Σ⋆
(Set of all
0 string: finite strings on Analogous to tangent vectors
ω′ 00 alphabet Σ)
(A)
Transition
Structure
induced by
state definitions
109. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
And the exact graphs are not even that important.....
1(0.1)
0(0.9)
q1 q2
1(0.2)
Any string 0(0.8)
terminating in 1
1(0.7) q3
0(0.3)
0(0.9)
1(0.1)
1(0.1)
0(0.9)
Σ⋆
(Set of all finite
strings on alphabet Σ)
State splitting produces
110. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
And the exact graphs are not even that important.....
1(0.1)
0(0.9)
q1 q2
1(0.2)
Any string 0(0.8)
terminating in 1
1(0.7) q3
0(0.3)
1(0.1)
0(0.9)
1(0.1) 0(0.9)
1(0.1) 0(0.9)
0(0.9) 1(0.1)
Σ⋆
(Set of all finite
strings on alphabet Σ)
State splitting produces
111. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
And the exact graphs are not even that important.....
1(0.1)
0(0.9)
q1 q2
1(0.2)
Any string 0(0.8)
terminating in 1
1(0.7) q3
0(0.3)
1(0.1)
0(0.9)
1(0.1) 0(0.9)
1(0.1) 0(0.9)
0(0.9) 1(0.1)
Σ⋆
(Set of all finite
strings on alphabet Σ)
State splitting produces
112. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
And the exact graphs are not even that important.....
1(0.1)
0(0.9)
q1 q2
1(0.2)
Any string 0(0.8)
terminating in 1
1(0.7) q3
0(0.3)
1(0.1)
0(0.9)
1(0.1) 0(0.9)
1(0.1) 0(0.9)
0(0.9) 1(0.1)
Σ⋆
(Set of all finite
strings on alphabet Σ)
State splitting produces
113. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
And the exact graphs are not even that important.....
1(0.1)
0(0.9)
q1 q2
1(0.2)
Any string 0(0.8)
terminating in 1
1(0.7) q3
0(0.3)
1(0.1)
0(0.9)
1(0.1) 0(0.9)
1(0.1) 0(0.9)
0(0.9) 1(0.1)
Σ⋆
(Set of all finite
strings on alphabet Σ)
State splitting produces
114. PFSAs Are Encodings Of Measure Spaces
· · · And Not Just Graphs!
And the exact graphs are not even that important.....
1(0.1)
0(0.9)
q1 q2
1(0.2)
Any string 0(0.8)
terminating in 1
1(0.7) q3
0(0.3)
1(0.1)
0(0.9)
1(0.1) 0(0.9)
1(0.1) 0(0.9)
0(0.9) 1(0.1)
Σ⋆
(Set of all finite
strings on alphabet Σ)
State splitting produces