Analysis of Probabilistic Basic Parallel Processes Rmi Bonnet 1 - PowerPoint PPT Presentation

Analysis of Probabilistic Basic Parallel Processes Rémi Bonnet 1 Stefan Kiefer 1 Anthony W. Lin 1 , 2 1 University of Oxford, UK 2 Academia Sinica, Taiwan FoSSaCS, Grenoble 9 April 2014 Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) The rules of a probabilistic Basic Parallel Process (pBPP): 0 . 7 X ֒ − → XY 0 . 3 X ֒ − → ε 0 . 6 Y ֒ − → YY 0 . 4 Y ֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic Basic Parallel Process (pBPP): 0 . 7 X ֒ − → XY 0 . 3 X ֒ − → ε 0 . 6 Y ֒ − → YY 0 . 4 Y ֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X ֒ − → XY 0 . 3 X ֒ − → ε 0 . 6 Y ֒ − → YY 0 . 4 Y ֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X Y X ֒ − → XY 0 . 3 X ֒ − → ε 0 . 6 Y ֒ − → YY 0 . 4 Y ֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X Y X ֒ − → XY 0 . 3 X ֒ − → ε X Y 0 . 6 Y ֒ − → YY 0 . 4 Y ֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X Y X ֒ − → XY 0 . 3 X ֒ − → ε ε X Y 0 . 6 Y ֒ − → YY 0 . 4 Y ֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X Y X ֒ − → XY 0 . 3 X ֒ − → ε ε X Y 0 . 6 Y ֒ − → YY ε 0 . 4 Y ֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X Y X ֒ − → XY 0 . 3 X ֒ − → ε ε X Y 0 . 6 Y ֒ − → YY ε Y Y 0 . 4 Y ֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X Y X ֒ − → XY 0 . 3 X ֒ − → ε ε X Y 0 . 6 Y ֒ − → YY ε Y Y 0 . 4 Y ֒ − → ε ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X Y X ֒ − → XY 0 . 3 X ֒ − → ε ε X Y 0 . 6 Y ֒ − → YY ε Y Y 0 . 4 Y ֒ − → ε ε ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Probabilistic Basic Parallel Processes (pBPP) A run as a growing tree: The rules of a probabilistic X Basic Parallel Process (pBPP): 0 . 7 X Y X ֒ − → XY 0 . 3 X ֒ − → ε ε X Y 0 . 6 Y ֒ − → YY ε Y Y 0 . 4 Y ֒ − → ε ε ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Runs are random. The order of the “nonterminals” is not important ⇒ SPNs. Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Which Nonterminals are Picked? Either randomly Markov chain either uniformly with multiplicities either uniformly without multiplicities Consider state XYX . With multiplicities: probability of scheduling X is 2 / 3. Without multiplicities: probability of scheduling X is 1 / 2. Both versions make sense. The same results hold. Or nondeterministically (by a scheduler) MDP The set of states is N Γ , an infinite set, where Γ := set of nonterminals. Notation: with states α i ∈ N Γ run: α 0 ⇒ α 1 ⇒ α 2 ⇒ . . . Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Coverability , a start state α 0 ∈ N Γ and a state φ ∈ N Γ . Given a pBPP A run α 0 ⇒ α 1 ⇒ . . . covers φ if there is i ≥ 0 with α i ≥ φ (componentwise). Given also a finite “target” set F = { φ 1 , . . . , φ k } ⊂ N Γ . A run covers F if it covers some φ j ∈ F . Coverability problem: Given a pBPP and α 0 and a target set F : Starting from α 0 , is F covered with probability 1 ? = Reachability of an upward-closed set F ↑ with probability 1 Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Applications of Coverability F := { Producer Consumer } Covering F = Transaction between a Producer and a Consumer can take place F := { Grantrequest } Covering F = (At least) one request is granted Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Examples for Coverability 1 1 X ֒ − → XY and Y ֒ − → ε start state α 0 = X target set F = { YYY } F is covered with probability 1, i.e., runs like X ⇒ XY ⇒ XYY ⇒ XY ⇒ XYY ⇒ XYYY ⇒ . . . have (together) probability 1. This is true even though there are runs that don’t cover F , like X ⇒ XY ⇒ X ⇒ XY ⇒ X ⇒ . . . (they have together probability 0) Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Examples for Coverability 0 . 7 1 X ֒ − → XX Y ֒ − → Y 0 . 3 X ֒ − → Y α 0 = X F = { XXX } Runs of the form X ⇒ Y ⇒ Y ⇒ . . . have probability 0 . 3, so the probability of covering F is < 1 (but positive). Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Reaching a Trap , an initial state α 0 , and a target set F ⊂ N Γ . Fix a pBPP Write Trap := those states from which one cannot reach F ↑ Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α 0 with probability 1 ⇐ ⇒ From α 0 one cannot reach a trap while avoiding F ↑ . One direction is easy: if one can reach a trap while avoiding F ↑ , then there is a finite path to do so. That path has a positive probability. The other direction is less immediate. Purely qualitative. Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Karp-Miller-Style Algorithm for Coverability Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α 0 with probability 1 ⇐ ⇒ From α 0 one cannot reach a trap while avoiding F ↑ . Idea: decide coverability using the “trap” criterion. Karp-Miller-Like Tree 0 . 7 1 Example: α 0 = X X ֒ − → XX Y ֒ − → Y 0 . 3 F = { XXX } X ֒ − → Y X Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Karp-Miller-Style Algorithm for Coverability Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α 0 with probability 1 ⇐ ⇒ From α 0 one cannot reach a trap while avoiding F ↑ . Idea: decide coverability using the “trap” criterion. Karp-Miller-Like Tree 0 . 7 1 Example: α 0 = X X ֒ − → XX Y ֒ − → Y 0 . 3 F = { XXX } X ֒ − → Y X XX Y Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

Karp-Miller-Style Algorithm for Coverability Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α 0 with probability 1 ⇐ ⇒ From α 0 one cannot reach a trap while avoiding F ↑ . Idea: decide coverability using the “trap” criterion. Karp-Miller-Like Tree 0 . 7 1 Example: α 0 = X X ֒ − → XX Y ֒ − → Y 0 . 3 F = { XXX } X ֒ − → Y X XX Y Rémi Bonnet, Stefan Kiefer , Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes