Classification among Hidden Markov...
Transcript of Classification among Hidden Markov...
Classification among Hidden Markov Models
S. Akshay - Hugo Bazille - Eric Fabre - Blaise Genest
18/06/2019
1/18
Summary
1 Introduction of the problem
2 Different classifications
3 Limit sure classifiability
4 Variants
5 Conclusion
2/18
Introduction of the problem (1)
Framework
Stochastic systems,
Partial information.
⇒ Hidden Markov Models:
xstart y
z
a, 12a, 1
2
a, 12
b, 12
b, 14
b, 14
a, 12
3/18
Introduction of the problem (2)
Classification
Given two systems A1,A2 and an observation w , decide which oneproduced it.
Encompasses diagnosis, opacity...
Can we classify...
For sure?
Almost sure?
Limit sure?
We cannot?
x
y
b, 110
a, 910
b
x ′
y ′
b, 910
a, 110
b
4/18
Introduction of the problem (2)
Classification
Given two systems A1,A2 and an observation w , decide which oneproduced it.
Encompasses diagnosis, opacity...
Can we classify...
For sure?
Almost sure?
Limit sure?
We cannot?
x
y
b, 110
a, 910
b
x ′
y ′
b, 910
a, 110
b
4/18
What is a classifier?
Function f : Σ∗ → {⊥, 1, 2}
What is a good classifier?
Accurate?
Reactive?
No error?
5/18
Sure classification
Informally: ability to distinguish after some time.Formally: ∀w ∈ Σ∞,∃v ,w = vv ′, v ∈ L1, v 6∈ L2.
x
y
b, 110
a, 910
c
x ′
y ′
b, 910
d , 110
b
Theorem
Sure classification is decidable in PTIME, by deciding if L∞1 ∩ L∞2 = ∅.
6/18
Almost sure classification
Informally: ability to distinguish after some time with probability 1.Formally: P(w ∈ Σ∞, ∃v ,w = vv ′, v ∈ L1, v 6∈ L2) = 1.
x
y
b, 110
a, 910
c
x ′
y ′
b, 910
a, 110
b
Theorem
Almost sure classification is PSPACE-complete, by deciding ifP(L∞1 ∩ L∞2 ) = 0.
7/18
Limit sure classifiability
Informally: classify with arbitrarily high precision.Formally: there is a classifier f that eventually answers correctly withprobability > 1− ε for all ε > 0.
x
y
b, 110
a, 910
b, 14
c, 34
x ′
y ′
b, 910
a, 110
b, 14
c, 34
x ′′
y ′′
b, 910
a, 110
b, 12
c, 12
8/18
Main result on limit sure classifiability
Theorem
Limit sure classifiability is decidable in PTIME.
We want arbitrarily high precision:
Transient components ”do notmatter”,
We mainly study BSCCs.
x
y
b, 110
a, 910
b, 14
c, 34
x ′
y ′
b, 910
a, 110
b, 12
c, 12
9/18
Main result on limit sure classifiability
Theorem
Limit sure classifiability is decidable in PTIME.
We want arbitrarily high precision:
Transient components ”do notmatter”,
We mainly study BSCCs.
x
y
b, 110
a, 910
b, 14
c, 34
x ′
y ′
b, 910
a, 110
b, 12
c, 12
9/18
Study of BSCC
Two BSCCs are problematic if they:
1 Are co-reachable,
2 Have the same stochastic language.
How to check this? State by state?
x
y
b, 12 b, 1
2
a, 12
a, 12
x ′
y ′
a, 12 b, 1
2
a, 12
b, 12
Lx 6≡ Lx ′
Lx 6≡ Ly ′
...
But L 12x+ 1
2y ≡ L 1
2x ′+ 1
2y ′ .
10/18
Study of BSCC
Two BSCCs are problematic if they:
1 Are co-reachable,
2 Have the same stochastic language.
How to check this? State by state?
x
y
b, 12 b, 1
2
a, 12
a, 12
x ′
y ′
a, 12 b, 1
2
a, 12
b, 12
Lx 6≡ Lx ′
Lx 6≡ Ly ′
...
But L 12x+ 1
2y ≡ L 1
2x ′+ 1
2y ′ .
10/18
How to be smart? (1)
Number of possible distributions: infinite!
Consider stationnary distributions σX on beliefs X .
xstart y
z
a, 12a, 1
2
a, 12
b, 12
b, 14
b, 14
a, 12
x y
12
12
34
14
For a belief X , σX is computable in PTIME.
11/18
Interest of stationary distributions
Theorem
The following are equivalent:
1 One cannot classify between A1,A2,
2 There exists an X in a BSCC of twin beliefs such that(A1, σ
1X ) ≡ (A2, σ
2X ).
One problem solved, but... still an exponential number of beliefs!
12/18
Interest of stationary distributions
Theorem
The following are equivalent:
1 One cannot classify between A1,A2,
2 There exists an X in a BSCC of twin beliefs such that(A1, σ
1X ) ≡ (A2, σ
2X ).
One problem solved, but... still an exponential number of beliefs!
12/18
How to be smart? (2)
Have only a limited number of beliefs?A = A1 × A2, for a BSCC Di of A and (y1, y2) ∈ Di ,
X1 = {x1 | (x1, y2) ∈ Di},X2 = {x2 | (y1, x2) ∈ Di}.
Theorem
NSC: check equivalence for such X1,X2.
Polynomial number of such beliefs,
Each check with Linear Programming: PTIME!
13/18
How to be smart? (2)
Have only a limited number of beliefs?A = A1 × A2, for a BSCC Di of A and (y1, y2) ∈ Di ,
X1 = {x1 | (x1, y2) ∈ Di},X2 = {x2 | (y1, x2) ∈ Di}.
Theorem
NSC: check equivalence for such X1,X2.
Polynomial number of such beliefs,
Each check with Linear Programming: PTIME!
13/18
With tries?
User has a reset button:
Can try again and again,
Chooses the system randomly every time.
Attack-classifiability
Decide if there exists a reset strategy such that:
1 It will finish with probability 1,
2 It is limit sure classifiable after the last reset.
1− ε attacker-classifiability
With ε fixed, decide if there exists a reset strategy such that:
1 It will finish with probability 1,
2 Classification will be correct with probability 1− ε after last reset.
14/18
With tries?
User has a reset button:
Can try again and again,
Chooses the system randomly every time.
Attack-classifiability
Decide if there exists a reset strategy such that:
1 It will finish with probability 1,
2 It is limit sure classifiable after the last reset.
1− ε attacker-classifiability
With ε fixed, decide if there exists a reset strategy such that:
1 It will finish with probability 1,
2 Classification will be correct with probability 1− ε after last reset.
14/18
With tries?
User has a reset button:
Can try again and again,
Chooses the system randomly every time.
Attack-classifiability
Decide if there exists a reset strategy such that:
1 It will finish with probability 1,
2 It is limit sure classifiable after the last reset.
1− ε attacker-classifiability
With ε fixed, decide if there exists a reset strategy such that:
1 It will finish with probability 1,
2 Classification will be correct with probability 1− ε after last reset.
14/18
An example
x
y
b, 110
a, 910
b
x ′
y ′
b, 910
a, 110
b
Not attack classifiable,
∀ε, 1− ε attack classifiable.
15/18
An example
x
y
b, 110
a, 910
b
x ′
y ′
b, 910
a, 110
b
Not attack classifiable,
∀ε, 1− ε attack classifiable.
15/18
Results on these variants
Theorem
Attack-classifiability is PSPACE-complete.
1− ε attacker-classifiability is undecidable.
Idea of proofs:Attack-classifiability:
Find subpart of the systems that are classifiable,
Hardness: reduction from language inclusion for finite automata.
1− ε attacker-classifiability:
Reduction from 0 and 1 isolation problem for PFA.
16/18
Results on these variants
Theorem
Attack-classifiability is PSPACE-complete.
1− ε attacker-classifiability is undecidable.
Idea of proofs:Attack-classifiability:
Find subpart of the systems that are classifiable,
Hardness: reduction from language inclusion for finite automata.
1− ε attacker-classifiability:
Reduction from 0 and 1 isolation problem for PFA.
16/18
Summary
Classifiabilities
1 Sure: a word in only one language.
2 Almost Sure: a word in only one language with probability 1.
3 Limit Sure: probability of error decreases to 0.
4 Attack: Limit Sure with tries.
5 1− ε Attack: decide with a fixed threshold of error with tries.
Class Sure Almost Sure Limit Sure Attack 1− ε attack
Cplxt PTIME PSPACE PTIME PSPACE undecidable
17/18
Strong links with:
Distance 1 problem: determine if
supW∈Σ∞ |P1(W )− P2(W )| = 1
AFF-diagnosability and ε-diagnosability.
Questions time!
18/18
Strong links with:
Distance 1 problem: determine if
supW∈Σ∞ |P1(W )− P2(W )| = 1
AFF-diagnosability and ε-diagnosability.
Questions time!
18/18