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


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Analysis of Probabilistic Basic Parallel Processes

Rémi Bonnet1 Stefan Kiefer1 Anthony W. Lin1,2

1University of Oxford, UK 2Academia Sinica, Taiwan

FoSSaCS, Grenoble 9 April 2014

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X X Y A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X X Y X Y A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X X Y X 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

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X X Y X 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

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X X Y X Y ε ε Y Y A run: X ⇒ XY ⇒ XYY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X X Y X Y ε ε Y 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

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SLIDE 11

Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X X Y X Y ε ε Y 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

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Probabilistic Basic Parallel Processes (pBPP)

The rules of a probabilistic Basic Parallel Process (pBPP): X

0.7

֒ − → XY X

0.3

֒ − → ε Y

0.6

֒ − → YY Y

0.4

֒ − → ε A run as a growing tree: X X Y X Y ε ε Y Y 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

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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 NΓ, an infinite set, where Γ := set of nonterminals. Notation: run: α0 ⇒ α1 ⇒ α2 ⇒ . . . with states αi ∈ NΓ

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Coverability

Given a pBPP , a start state α0 ∈ NΓ and a state φ ∈ NΓ. A run α0 ⇒ α1 ⇒ . . . covers φ if there is i ≥ 0 with αi ≥ φ (componentwise). Given also a finite “target” set F = {φ1, . . . , φk} ⊂ NΓ. A run covers F if it covers some φj ∈ F. Coverability problem: Given a pBPP and α0 and a target set F: Starting from α0, 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

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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

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Examples for Coverability

X

1

֒ − → XY and Y

1

֒ − → ε start state α0 = 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

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Examples for Coverability

X

0.7

֒ − → XX Y

1

֒ − →Y X

0.3

֒ − → Y α0 = 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

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Reaching a Trap

Fix a pBPP , an initial state α0, and a target set F ⊂ NΓ. Write Trap := those states from which one cannot reach F↑ Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α0 with probability 1 ⇐ ⇒ From α0 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

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Karp-Miller-Style Algorithm for Coverability

Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α0 with probability 1 ⇐ ⇒ From α0 one cannot reach a trap while avoiding F↑. Idea: decide coverability using the “trap” criterion. Karp-Miller-Like Tree Example: α0 = X X

0.7

֒ − → XX Y

1

֒ − →Y F = {XXX} X

0.3

֒ − → Y X

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Karp-Miller-Style Algorithm for Coverability

Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α0 with probability 1 ⇐ ⇒ From α0 one cannot reach a trap while avoiding F↑. Idea: decide coverability using the “trap” criterion. Karp-Miller-Like Tree Example: α0 = X X

0.7

֒ − → XX Y

1

֒ − →Y F = {XXX} X

0.3

֒ − → Y X XX Y

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Karp-Miller-Style Algorithm for Coverability

Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α0 with probability 1 ⇐ ⇒ From α0 one cannot reach a trap while avoiding F↑. Idea: decide coverability using the “trap” criterion. Karp-Miller-Like Tree Example: α0 = X X

0.7

֒ − → XX Y

1

֒ − →Y F = {XXX} X

0.3

֒ − → Y X XX Y

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Karp-Miller-Style Algorithm for Coverability

Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α0 with probability 1 ⇐ ⇒ From α0 one cannot reach a trap while avoiding F↑. Idea: decide coverability using the “trap” criterion. Karp-Miller-Like Tree Example: α0 = X X

0.7

֒ − → XX Y

1

֒ − →Y F = {XXX} X

0.3

֒ − → Y X XX Y ∈ Trap, so we’ve found a path to a trap

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Karp-Miller-Style Algorithm for Coverability

Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α0 with probability 1 ⇐ ⇒ From α0 one cannot reach a trap while avoiding F↑. The other example from before: α0 = X F = {YYY} X

1

֒ − → XY and Y

1

֒ − → ε X XY

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Karp-Miller-Style Algorithm for Coverability

Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α0 with probability 1 ⇐ ⇒ From α0 one cannot reach a trap while avoiding F↑. The other example from before: α0 = X F = {YYY} X

1

֒ − → XY and Y

1

֒ − → ε X XY

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Karp-Miller-Style Algorithm for Coverability

Proposition (builds on [Abdulla, Ben Henda, Mayr, ’07]) F is covered from α0 with probability 1 ⇐ ⇒ From α0 one cannot reach a trap while avoiding F↑. The other example from before: α0 = X F = {YYY} X

1

֒ − → XY and Y

1

֒ − → ε X XY So one cannot reach a trap while avoiding F↑. (In fact, one cannot reach a trap at all.)

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Karp-Miller Style

Try to find paths to a trap: Build a tree breadth-first starting from α0: nodes = states branches = runs Never include states that cover F. If a configuration is larger than a predecessor, prune. If a leaf is a trap, return “coverability with prob < 1”. If all leaves are pruned, return “coverability with prob 1”. Termination: Dickson’s lemma, König’s lemma Correctness: “Smaller is better”

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Results

That was a proof (sketch) of: Theorem The coverability problem is decidable: Given a pBPP , an initial state α0, a target set F. One can decide whether F covered with probability 1 (when starting in α0). This is in contrast to general stochastic Petri nets (as established in [Abdulla, Ben Henda, Mayr, ’07]).

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Results

That was a proof (sketch) of: Theorem The coverability problem is decidable: Given a pBPP , an initial state α0, a target set F. One can decide whether F covered with probability 1 (when starting in α0). This is in contrast to general stochastic Petri nets (as established in [Abdulla, Ben Henda, Mayr, ’07]). However: Theorem The coverability problem has nonelementary complexity (is Tower-hard).

  • Proof. Long. By a reduction from a 2-counter machine with a

Tower-budget.

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Results for MDPs

Theorem Given a pBPP , and α0 and F as before. One can decide whether there exists a scheduler such that F is covered with probability 1 (when starting in α0). If yes, then there is a memoryless deterministic scheduler, and one can compute such a scheduler.

  • Proof. By abstracting the state space (finite subset).

Relies on Petri-Net reachability no upper complexity bound. The MDP and Markov-Chain problems are rather different.

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Results for MDPs

Theorem Given a pBPP , and α0 and F as before. And k ∈ N. One can decide whether for all k-fair schedulers F is covered with probability 1 (when starting in α0). k-fairness means: if an X-nonterminal is present, then an X-nonterminal must be scheduled within k steps.

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Q-States Targets

Theorem Given a pBPP , and α0. Given also F = {X1, . . . , Xj}. F is covered with probability 1 in the Markov-Chain model ⇐ ⇒ F is covered with prob 1 for some scheduler ⇐ ⇒ F is covered with prob 1 for all k-fair schedulers. (k ≥ |Γ|) One can decide in polynomial time whether F is covered with probability 1.

  • Proof. Simple.

An analogous restriction leads to decidability for general VASSs [Abdulla, Ben Henda, Mayr, ’07] (no complexity bound).

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Semilinear Targets

Theorem Given a pBPP , and α0. Given also a semilinear target set S ⊆ NΓ. The following problems are undecidable: (a) Is S reached with probability 1? (b) Is S reached with probability 1 for some scheduler? (c) Is S reached with probability 1 for all 7-fair schedulers?

  • Proof. Reduction from 2-counter machines.

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes

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Concluding Remarks

pBPP are simple and natural stochastic Petri nets. The considered problems are qualitative in two senses. Laws of probability impose a special but natural kind of fairness. Variations quickly lead to multi-dimensional random walks.

Rémi Bonnet, Stefan Kiefer, Anthony W. Lin Analysis of Probabilistic Basic Parallel Processes