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

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What is a classifier?

Function f : Σ∗ → {⊥, 1, 2}

What is a good classifier?

Accurate?

Reactive?

No error?

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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 = ∅.

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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.

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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

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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 ′ .

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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.

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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!

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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!

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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!

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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!

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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.

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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.

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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.

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An example

x

y

b, 110

a, 910

b

x ′

y ′

b, 910

a, 110

b

Not attack classifiable,

∀ε, 1− ε attack classifiable.

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An example

x

y

b, 110

a, 910

b

x ′

y ′

b, 910

a, 110

b

Not attack classifiable,

∀ε, 1− ε attack classifiable.

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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.

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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.

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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

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Strong links with:

Distance 1 problem: determine if

supW∈Σ∞ |P1(W )− P2(W )| = 1

AFF-diagnosability and ε-diagnosability.

Questions time!

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Strong links with:

Distance 1 problem: determine if

supW∈Σ∞ |P1(W )− P2(W )| = 1

AFF-diagnosability and ε-diagnosability.

Questions time!

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