Switched Probabilistic I/O Automata Ling Cheung 1 Nancy Lynch 2 - - PowerPoint PPT Presentation

Introduction Composition Switched PIOA Conclusion

Switched Probabilistic I/O Automata

Ling Cheung1 Nancy Lynch2 Roberto Segala3 Frits Vaandrager1

1Nijmegen Institute for Computing and Information Sciences

University of Nijmegen, the Netherlands

2MIT Computer Science and Artificial Intelligence Laboratory, U.S.A. 3Dipartimento di Informatica, Universit`

a di Verona, Italy

ICTAC 2004, Guiyang, China

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion

Outline

1

Introduction Basics Randomization

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion

Outline

1

Introduction Basics Randomization

2

The trouble with composition What is parallel composition? How much does the daemon know? Global choice vs local choice

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion

Outline

1

Introduction Basics Randomization

2

The trouble with composition What is parallel composition? How much does the daemon know? Global choice vs local choice

3

Switched PIOA The Switched PIOA model Implementing parallel compositions

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion

Outline

1

Introduction Basics Randomization

2

The trouble with composition What is parallel composition? How much does the daemon know? Global choice vs local choice

3

Switched PIOA The Switched PIOA model Implementing parallel compositions

4

Summary and future work Summary Future work

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Basics Randomization

To NIII Colloquium Attendees:

Thank you all for coming to my talk!

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Basics Randomization

For this talk . . .

We need very little probability theory: discrete distributions. Examples:

fair coin: {Head, 1

2, Tail, 1 2};

fair dice: {i, 1

6 | 1 ≤ i ≤ 6}.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Basics Randomization

For this talk . . .

We need very little probability theory: discrete distributions. Examples:

fair coin: {Head, 1

2, Tail, 1 2};

fair dice: {i, 1

6 | 1 ≤ i ≤ 6}.

Underlying model: nondeterministic automata with asynchronous composition. (In our paper: input/output distinction, combination of synchronous and asynchronous compositions, etc.)

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Basics Randomization

For this talk . . .

We need very little probability theory: discrete distributions. Examples:

fair coin: {Head, 1

2, Tail, 1 2};

fair dice: {i, 1

6 | 1 ≤ i ≤ 6}.

Underlying model: nondeterministic automata with asynchronous composition. (In our paper: input/output distinction, combination of synchronous and asynchronous compositions, etc.) Total order semantics: if both actions a and b occur, one must precede the other.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Basics Randomization

Schedulers and trace distributions

History-dependent, randomized schedulers transform nondeterministic choices into probabilistic choices. ·

a

• b
• ·

a

• b
• ·

· ·

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Basics Randomization

Schedulers and trace distributions

History-dependent, randomized schedulers transform nondeterministic choices into probabilistic choices. ·

a p

• b

1−p

• ·

a q

• b

1−q

• ·

· ·

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Basics Randomization

Schedulers and trace distributions

History-dependent, randomized schedulers transform nondeterministic choices into probabilistic choices. ·

a p

• b

1−p

• ·

a q

• b

1−q

• ·

· · Each scheduler induces a trace distribution: a discrete distributions on finite traces. {aa, pq, ab, p(1 − q), b, 1 − p}

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Nondeterministic parallel composition

· P

a

·

• ·

Q

b

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Nondeterministic parallel composition

· P

a

·

• ·

Q

b

·

• The interleaving axiom:

· PQ

a

• b
• ·

b

• ·

a

• ·

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Probabilistic parallel composition

· P

a

·

• ·

Q

b

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Probabilistic parallel composition

· P

a

·

• ·

Q

b

·

• What is a probabilistic behavior of PQ?

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Probabilistic parallel composition

· P

a

·

• ·

Q

b

·

• What is a probabilistic behavior of PQ?

Quick answer: bias factor θ. · PQ

a θ

• b

1−θ

• ·

b

• ·

a

• ·

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Probabilistic parallel composition

· P

a

·

• ·

Q

b

·

• What is a probabilistic behavior of PQ?

Quick answer: bias factor θ. Imagine a coin-flipping daemon. · PQ

a θ

• b

1−θ

• ·

b

• ·

a

• ·

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

What is the value of θ?

· PQ

a θ

• b

1−θ

• ·

b

• ·

a

• ·

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

What is the value of θ?

· PQ

a θ

• b

1−θ

• ·

b

• ·

a

• ·

·

• Fixed θ: parameterized composition operator θ.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

What is the value of θ?

· PQ

a θ

• b

1−θ

• ·

b

• ·

a

• ·

·

• Fixed θ: parameterized composition operator θ.

Limitations: static parameter, not commutative, not associative.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

What is the value of θ?

· PQ

a θ

• b

1−θ

• ·

b

• ·

a

• ·

·

• Fixed θ: parameterized composition operator θ.

Limitations: static parameter, not commutative, not associative. Variable θ: a supply of coins with different biases; imaginary daemon chooses a coin based on his knowledge.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

How much does the daemon know?

There are two scenarios:

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

How much does the daemon know?

There are two scenarios: Scenario 1: context-independent P

• Q
• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

How much does the daemon know?

There are two scenarios: Scenario 1: context-independent P

• Q
• Scenario 2: context-dependent

P

• Q
• Env
• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Scenario 1: context-independent composition

How much does the daemon know?

Daemon, P and Q all inside a big black box. P

• Q
• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Scenario 1: context-independent composition

How much does the daemon know?

Daemon, P and Q all inside a big black box. P

• Q
• Daemon knows the histories of P and Q, but nothing about the
• utside world.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Scenario 1: context-independent composition

How much does the daemon know?

Daemon, P and Q all inside a big black box. P

• Q
• Daemon knows the histories of P and Q, but nothing about the
• utside world.

Problem: non-associativity.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-associativity: P(QR)

Context-independent composition

· P

a ·

• ·

Q

b ·

• ·

R c

p

• d

1−p

• ·

·

• Inner daemon: R, 1.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-associativity: P(QR)

Context-independent composition

· P

a ·

• ·

Q

b ·

• ·

R c

p

• d

1−p

• ·

·

• Inner daemon: R, 1.

Outer daemon: QR, 1; if c, then P, 1, else QR, 1.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-associativity: P(QR)

Context-independent composition

· P

a ·

• ·

Q

b ·

• ·

R c

p

• d

1−p

• ·

·

• Inner daemon: R, 1.

Outer daemon: QR, 1; if c, then P, 1, else QR, 1. Result: {cab, p, dba, 1 − p}. ·

b

·

a

·

• d

1−p

• c

p

• ·

a

·

b

·

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-associativity: (PQ)R

Context-independent composition

Claim: {cab, p, dba, 1 − p} not possible! · P

a ·

• ·

Q

b ·

• ·

R c

p

• d

1−p

• ·

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-associativity: (PQ)R

Context-independent composition

Claim: {cab, p, dba, 1 − p} not possible! · P

a ·

• ·

Q

b ·

• ·

R c

p

• d

1−p

• ·

·

• Outer daemon: R, 1.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-associativity: (PQ)R

Context-independent composition

Claim: {cab, p, dba, 1 − p} not possible! · P

a ·

• ·

Q

b ·

• ·

R c

p

• d

1−p

• ·

·

• Outer daemon: R, 1.

Inner daemon: {P, q, Q, 1 − q}.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-associativity: (PQ)R

Context-independent composition

Claim: {cab, p, dba, 1 − p} not possible! · P

a ·

• ·

Q

b ·

• ·

R c

p

• d

1−p

• ·

·

• Outer daemon: R, 1.

Inner daemon: {P, q, Q, 1 − q}. ·

• c

p

• d

1−p

• ·
• a

q

• b

1−q

• ·
• a

q

• b

1−q

• ·

b

• ·

a

• ·

b

• ·

a

• ·

· · ·

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-associativity: (PQ)R

Context-independent composition

Claim: {cab, p, dba, 1 − p} not possible! · P

a ·

• ·

Q

b ·

• ·

R c

p

• d

1−p

• ·

·

• Outer daemon: R, 1.

Inner daemon: {P, q, Q, 1 − q}. Conclusion: inner daemon doesn’t know enough. ·

• c

p

• d

1−p

• ·
• a

q

• b

1−q

• ·
• a

q

• b

1−q

• ·

b

• ·

a

• ·

b

• ·

a

• ·

· · ·

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Scenario 2: context-dependent composition

How much does the daemon know?

Daemon sees the outside world. P

• Q
• Env
• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Scenario 2: context-dependent composition

How much does the daemon know?

Daemon sees the outside world. P

• Q
• Env
• Daemon knows the histories of P, Q and Env.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Scenario 2: context-dependent composition

How much does the daemon know?

Daemon sees the outside world. P

• Q
• Env
• Daemon knows the histories of P, Q and Env.

Problem: violation of the interleaving axiom! I.e., there exists Env such that (ab) Env ∼ = (a.b + b.a) Env .

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-interleaving semantics

Context-dependent composition

(ab) Env: · P

a ·

• ·

Q

b ·

• ·

Env

c p

• d

1−p

• ·

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Non-interleaving semantics

Context-dependent composition

(ab) Env: · P

a ·

• ·

Q

b ·

• ·

Env

c p

• d

1−p

• ·

·

• (a.b + b.a) Env:

·

a q

• b

1−q

• ·

b

• ·

a

• ·

·

• ·

Env

c p

• d

1−p

• ·

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

The (same) counterexample

Context-dependent composition

·

b

·

a

·

• d

1−p

• c

p

• ·

a

·

b

·

The trace distribution {cab, p, dba, 1 − p} is possible in (ab) Env, but not in (a.b + b.a) Env.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

The (same) counterexample

Context-dependent composition

·

b

·

a

·

• d

1−p

• c

p

• ·

a

·

b

·

The trace distribution {cab, p, dba, 1 − p} is possible in (ab) Env, but not in (a.b + b.a) Env. Conclusion: we have a non-interleaving, but total order semantics.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

What’s wrong?

Something is wrong with our understanding of parallel composition.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

What’s wrong?

Something is wrong with our understanding of parallel composition. In context-independent composition, the problem shows up as non-associativity.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

What’s wrong?

Something is wrong with our understanding of parallel composition. In context-independent composition, the problem shows up as non-associativity. In context-dependent composition, the same problem leads to difference between ab and a.b + b.a.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Two types of nondeterministic choices: global vs. local

Global choice: ab, resolved by a daemon. · P

a ·

• ·

Q

b ·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Parallel Composition Knowledge Global vs. Local

Two types of nondeterministic choices: global vs. local

Global choice: ab, resolved by a daemon. · P

a ·

• ·

Q

b ·

• Local choice: a.b + b.a, resolved by a local scheduler.

·

a q

• b

1−q

• ·

b

• ·

a

• ·

·

• Behavior varies depending on the perspective!

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Dissecting the problem, Part I: eliminate global choices.

To better understand the problem, we developed the model of Switched PIOA. · P

a

·

• ·

Q

b

·

• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Dissecting the problem, Part I: eliminate global choices.

To better understand the problem, we developed the model of Switched PIOA. · P

a

·

· ·

• ·

Q

b

·

· ·

• active states (foreground) vs. inactive states (background);

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Dissecting the problem, Part I: eliminate global choices.

To better understand the problem, we developed the model of Switched PIOA.

• P

a

·

·

• goQ

goP

• ·
• goQ

goP

• ·

Q

b

·

• goP

goQ

• ·
• goP

goQ

• active states (foreground) vs. inactive states (background);

control exchange via special actions (e.g. goP, goQ);

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Dissecting the problem, Part I: eliminate global choices.

To better understand the problem, we developed the model of Switched PIOA.

• P

a

·

·

• goQ

goP

• ·
• goQ

goP

• ·

Q

b

·

• goP

goQ

• ·
• goP

goQ

• active states (foreground) vs. inactive states (background);

control exchange via special actions (e.g. goP, goQ); Every decision is made locally, so no more daemons.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Due to the absence of global choices . . .

Parallel composition in Switched PIOA: easy to define;

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Due to the absence of global choices . . .

Parallel composition in Switched PIOA: easy to define; commutative and associative;

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Due to the absence of global choices . . .

Parallel composition in Switched PIOA: easy to define; commutative and associative; deep/semantic compositionality of trace distribution semantics;

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Due to the absence of global choices . . .

Parallel composition in Switched PIOA: easy to define; commutative and associative; deep/semantic compositionality of trace distribution semantics; That’s all very nice, but parallel processes don’t really exchange control . . .

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Dissecting the problem, Part II: reintroduce global choices.

Control-exchange should not be taken semantically.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Dissecting the problem, Part II: reintroduce global choices.

Control-exchange should not be taken semantically. Switch PIOA is an implementation tool for various composition operators.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Dissecting the problem, Part II: reintroduce global choices.

Control-exchange should not be taken semantically. Switch PIOA is an implementation tool for various composition operators. Examples:

fixed bias factor θ;

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Dissecting the problem, Part II: reintroduce global choices.

Control-exchange should not be taken semantically. Switch PIOA is an implementation tool for various composition operators. Examples:

fixed bias factor θ; context-independent.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing biased composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing biased composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Local schedulers: always return control after one local move.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing biased composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Local schedulers: always return control after one local move.

Arbiter:

usually schedule {goP, θ, goQ, 1 − θ};

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing biased composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Local schedulers: always return control after one local move.

Arbiter:

usually schedule {goP, θ, goQ, 1 − θ}; if finalP then goQ, 1 and vice versa.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing biased composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Local schedulers: always return control after one local move.

Arbiter:

usually schedule {goP, θ, goQ, 1 − θ}; if finalP then goQ, 1 and vice versa.

Examples: goP . a. finalP . goQ . b. finalQ, θ;

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing biased composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Local schedulers: always return control after one local move.

Arbiter:

usually schedule {goP, θ, goQ, 1 − θ}; if finalP then goQ, 1 and vice versa.

Examples: goP . a. finalP . goQ . b. finalQ, θ; goQ . b. finalQ . goP . a. finalP, 1 − θ.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing context-independent composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Cheung, Lynch, Segala, Vaandrager

Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing context-independent composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Local schedulers: no scheduling restrictions (“run to

completion”).

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion The Model Explicit Implementation

Implementing context-independent composition

• P

doneP

• finalP
• Arb

goQ

• goP
• Q

doneQ

• finalQ
• Local schedulers: no scheduling restrictions (“run to

completion”). Arbiter: if finalP then goQ, 1 and vice versa.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Summary Future Work

To summarize . . .

Parallel composition is trickier than we thought.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Summary Future Work

To summarize . . .

Parallel composition is trickier than we thought. Switched PIOA is a probabilistic model without global choices.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Summary Future Work

To summarize . . .

Parallel composition is trickier than we thought. Switched PIOA is a probabilistic model without global choices. Parallel composition in Switched PIOA is well-behaved.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Summary Future Work

To summarize . . .

Parallel composition is trickier than we thought. Switched PIOA is a probabilistic model without global choices. Parallel composition in Switched PIOA is well-behaved. Switched PIOA can be used to study various “real” parallel composition operators.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Summary Future Work

Future work

Philosophical: is there a “most intuitive” parallel composition

• perator?

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Summary Future Work

Future work

Philosophical: is there a “most intuitive” parallel composition

• perator?

Technical: “decomposing” trace distribution semantics. Trace Distribution

• Trace Set
• Trace Likelihood

Trace

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Summary Future Work

Future work

Philosophical: is there a “most intuitive” parallel composition

• perator?

Technical: “decomposing” trace distribution semantics. Trace Distribution

• Trace Set
• Trace Likelihood

Trace

Practical: modeling communication and/or security protocols in Switched PIOA.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Summary Future Work

Future work

Philosophical: is there a “most intuitive” parallel composition

• perator?

Technical: “decomposing” trace distribution semantics. Trace Distribution

• Trace Set
• Trace Likelihood

Trace

Practical: modeling communication and/or security protocols in Switched PIOA. – End –

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

5

Appendix Trace Set Semantics Trace Likelihood Semantics Getting Stuck

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics

Loosely speaking, a trace distribution is a discrete probability distribution over the set of finite traces.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics

Loosely speaking, a trace distribution is a discrete probability distribution over the set of finite traces. To go from trace distribution to trace set, we forget probabilities by: DiscDistr(Traces)

support Powerset(Traces)

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics

Loosely speaking, a trace distribution is a discrete probability distribution over the set of finite traces. To go from trace distribution to trace set, we forget probabilities by: DiscDistr(Traces)

support Powerset(Traces)

That is, schedulers return sets of possible transitions, rather than discrete distributions over possible transitions.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics: Example

· Early

a

• a
• ·

c

• ·

d

• ·

·

• ·

Late

a

·

c

• d
• ·

·

• Equivalent in semantics: trace, trace set, trace distribution.

Not equivalent in semantics: bisimulation.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics: Example

· Early′

a

• a
• ·

e

• f
• c
• ·

e

• f
• d
• ·

·

• ·

Late′

a

·

e

• f
• c
• d
• ·

·

• Add input e,f -loops.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics: Example

· Early′

a

• a
• ·

e

• f
• c
• ·

e

• f
• d
• ·

·

• ·

Late′

a

·

e

• f
• c
• d
• ·

·

• Add input e,f -loops.

Equivalent in semantics: trace. Not equivalent in semantics: trace set, trace distribution, bisimulation.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics: Example

· Early′

a

• a
• ·

e

• f
• c
• ·

e

• f
• d
• ·

·

• ·

Late′

a

·

e

• f
• c
• d
• ·

·

• Trace set {aec, afd} not possible in Early′.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics: Example

· Early′

a

• a
• ·

e

• f
• c
• ·

e

• f
• d
• ·

·

• ·

Late′

a

·

e

• f
• c
• d
• ·

·

• Trace set {aec, afd} not possible in Early′.

Consider the case in which: Early′ chooses the left-hand branch; and

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics: Example

· Early′

a

• a
• ·

e

• f
• c
• ·

e

• f
• d
• ·

·

• ·

Late′

a

·

e

• f
• c
• d
• ·

·

• Trace set {aec, afd} not possible in Early′.

Consider the case in which: Early′ chooses the left-hand branch; and environment performs f .

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics: Example

· Early′

a

• a
• ·

e

• f
• c
• ·

e

• f
• d
• ·

·

• ·

Late′

a

·

e

• f
• c
• d
• ·

·

• Trace set {aec, afd} not possible in Early′.

Consider the case in which: Early′ chooses the left-hand branch; and environment performs f . At this point, Early′ does not have the option to perform d.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Trace Set Semantics: Example

· Early′

a

• a
• ·

e

• f
• c
• ·

e

• f
• d
• ·

·

• ·

Late′

a

·

e

• f
• c
• d
• ·

·

• Trace set {aec, afd} not possible in Early′.

Consider the case in which: Early′ chooses the left-hand branch; and environment performs f . At this point, Early′ does not have the option to perform d. Important: Early′ cannot choose between inputs e and f .

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

What’s the lesson here?

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

What’s the lesson here?

It’s not about the numbers . . .

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

What’s the lesson here?

It’s not about the numbers . . . Examples of ”undesirable” properties of trace distribution semantics can be reproduced in trace set semantics.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

What’s the lesson here?

It’s not about the numbers . . . Examples of ”undesirable” properties of trace distribution semantics can be reproduced in trace set semantics. Key: each trace distribution contains a collection of traces, rather than a single trace. In some cases, this allows us to observe branching structure.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

An Alternative: Trace Likelihood Semantics

Each behavior is represented by a pair α, p.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

An Alternative: Trace Likelihood Semantics

Each behavior is represented by a pair α, p. Intended meaning: under some scenario, trace α occurs with probability p.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

An Alternative: Trace Likelihood Semantics

Each behavior is represented by a pair α, p. Intended meaning: under some scenario, trace α occurs with probability p. Difficulty: what is a possible scenario?

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

An Alternative: Trace Likelihood Semantics

Each behavior is represented by a pair α, p. Intended meaning: under some scenario, trace α occurs with probability p. Difficulty: what is a possible scenario? Frequentist probabilities: prediction about a large number of experiments, not about a single experiment.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Getting Stuck . . .

my current strategy: stop thinking, start reading;

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Appendix Trace Set Trace Likelihood Getting Stuck

Getting Stuck . . .

my current strategy: stop thinking, start reading; do something concrete: modeling oblivious transfer.

Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata