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 Cheung 1 Nancy Lynch 2 Roberto Segala 3 Frits Vaandrager 1 1 Nijmegen Institute for Computing and Information Sciences University of Nijmegen, the Netherlands 2 MIT Computer Science and Artificial Intelligence Laboratory, U.S.A. 3 Dipartimento 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 Introduction 1 Basics Randomization Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Outline Introduction 1 Basics Randomization The trouble with composition 2 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 Introduction 1 Basics Randomization The trouble with composition 2 What is parallel composition? How much does the daemon know? Global choice vs local choice Switched PIOA 3 The Switched PIOA model Implementing parallel compositions Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Switched PIOA Conclusion Outline Introduction 1 Basics Randomization The trouble with composition 2 What is parallel composition? How much does the daemon know? Global choice vs local choice Switched PIOA 3 The Switched PIOA model Implementing parallel compositions Summary and future work 4 Summary Future work Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Basics Switched PIOA Randomization Conclusion To NIII Colloquium Attendees: Thank you all for coming to my talk! Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Composition Basics Switched PIOA Randomization Conclusion 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 Basics Switched PIOA Randomization Conclusion 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 Basics Switched PIOA Randomization Conclusion 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 Basics Switched PIOA Randomization Conclusion 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 Basics Switched PIOA Randomization Conclusion Schedulers and trace distributions History-dependent, randomized schedulers transform nondeterministic choices into probabilistic choices. · � � � � � � �������� � a b � � � p � 1 − p � � · · � � � � � � �������� � a b � � � q � 1 − q � � · · Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

� � Introduction Composition Basics Switched PIOA Randomization Conclusion Schedulers and trace distributions History-dependent, randomized schedulers transform nondeterministic choices into probabilistic choices. · � � � � � � �������� � a b � � � p � 1 − p � � · · � � � � � � �������� � a b � � � q � 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 Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion Nondeterministic parallel composition �� �� �� �� · · P Q �� �� �� �� a b � · � · Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

� � Introduction Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion Nondeterministic parallel composition �� �� �� �� · · P Q �� �� �� �� a b � · � · The interleaving axiom: �� �� P � Q · � � ��������� � a � b � �� � �� � � � � · · � � ��������� � b � a � � � � � � · · Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion Probabilistic parallel composition �� �� �� �� · · P Q �� �� �� �� a b � · � · Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

Introduction Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion Probabilistic parallel composition �� �� �� �� · · P Q �� �� �� �� a b � · � · What is a probabilistic behavior of P � Q ? Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

� � Introduction Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion Probabilistic parallel composition �� �� �� �� · · P Q �� �� �� �� a b � · � · What is a probabilistic behavior of P � Q ? Quick answer: bias factor θ . �� �� · P � Q � � � � � a b � �������� � � � � � � �� �� � · θ 1 − θ · � � ��������� � b � a � � � � � � · · Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

� � Introduction Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion Probabilistic parallel composition �� �� �� �� · · P Q �� �� �� �� a b � · � · What is a probabilistic behavior of P � Q ? Quick answer: bias factor θ . Imagine a coin-flipping daemon. �� �� · P � Q � � � � � a b � �������� � � � � � � �� �� � · θ 1 − θ · � � ��������� � b � a � � � � � � · · Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

� � Introduction Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion What is the value of θ ? �� �� P � Q · � � � � � � �������� a � b � � � � � �� �� � · θ 1 − θ · � � ��������� � � b a � � � � � � · · Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

� � Introduction Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion What is the value of θ ? �� �� P � Q · � � � � � � �������� a � b � � � � � �� �� � · θ 1 − θ · � � ��������� � � b a � � � � � � · · Fixed θ : parameterized composition operator � θ . Cheung, Lynch, Segala, Vaandrager Switched Probabilistic I/O Automata

� � Introduction Parallel Composition Composition Knowledge Switched PIOA Global vs. Local Conclusion What is the value of θ ? �� �� P � Q · � � � � � � �������� a � b � � � � � �� �� � · θ 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