[PPT] - Fair Termination for Parameterized Probabilistic Concurrent Systems PowerPoint Presentation

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Fair Termination for Parameterized Probabilistic Concurrent Systems

(TACAS’17)

Ondˇ rej Leng´ al1 Anthony W. Lin2 Rupak Majumdar3 Philipp R¨ ummer4

1Brno University of Technology, Czech Republic 2Department of Computer Science, University of Oxford, UK 3MPI-SWS Kaiserslautern, Germany 4Uppsala University, Sweden

9 May 2019 (MOSCA’19)

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Outline

Parameterized probabilistic concurrent systems

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 2 / 21

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Outline

Parameterized probabilistic concurrent systems Liveness

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 2 / 21

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Outline

Parameterized probabilistic concurrent systems Liveness Fairness

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 2 / 21

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Outline

Parameterized probabilistic concurrent systems Liveness Fairness Regular model checking

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 2 / 21

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Motivating Example

Herman’s protocol (merging version)

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

◮ if no token:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

◮ if no token: return

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

◮ if no token: return ◮ if has token: flip a coin

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

◮ if no token: return ◮ if has token: flip a coin

heads: pass the token clockwise

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

◮ if no token: return ◮ if has token: flip a coin

heads: pass the token clockwise
tails: keep the token

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

◮ if no token: return ◮ if has token: flip a coin

heads: pass the token clockwise
tails: keep the token

if a process with a token gets another one:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

◮ if no token: return ◮ if has token: flip a coin

heads: pass the token clockwise
tails: keep the token

if a process with a token gets another one: merge them

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version) ring topology, leader election scheduler selects processes unstable configuration:

◮ > 1 tokens

stable configuration:

◮ 1 token (leader)

goal: | = ♦leader is elected Herman’s algorithm: when selected:

◮ if no token: return ◮ if has token: flip a coin

heads: pass the token clockwise
tails: keep the token

if a process with a token gets another one: merge them Pr(| = ♦leader is elected) = 1

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 3 / 21

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Motivating Example

Herman’s protocol (merging version)

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really?

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really?

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really?

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really?

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really?

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really?

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really?

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really? Fairness needed!

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really? Fairness needed! But which fairness?

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Motivating Example

Herman’s protocol (merging version) Pr(| = ♦leader is elected) = 1 really? Fairness needed! But which fairness? We use finitary fairness

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 4 / 21

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Setting

Liveness of Fair Parameterized Probabilistic Concurrent Systems

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21

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Setting

Liveness of Fair Parameterized Probabilistic Concurrent Systems

◮ Parameterized Concurrent Systems: N finite-state processes

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21

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Setting

Liveness of Fair Parameterized Probabilistic Concurrent Systems

◮ Parameterized Concurrent Systems: N finite-state processes ◮ Probabilistic: each process can flip a coin

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21

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Setting

Liveness of Fair Parameterized Probabilistic Concurrent Systems

◮ Parameterized Concurrent Systems: N finite-state processes ◮ Probabilistic: each process can flip a coin ◮ Fair: each process will have the opportunity to move

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21

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Setting

Liveness of Fair Parameterized Probabilistic Concurrent Systems

◮ Parameterized Concurrent Systems: N finite-state processes ◮ Probabilistic: each process can flip a coin ◮ Fair: each process will have the opportunity to move ◮ Liveness: a good configuration is always reachable with Pr = 1

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21

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Setting

Liveness of Fair Parameterized Probabilistic Concurrent Systems

◮ Parameterized Concurrent Systems: N finite-state processes ◮ Probabilistic: each process can flip a coin ◮ Fair: each process will have the opportunity to move ◮ Liveness: a good configuration is always reachable with Pr = 1

Examples: Herman’s protocol, Israeli-Jalfon protocol, population protocols, . . .

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21

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Setting

Liveness of Fair Parameterized Probabilistic Concurrent Systems

◮ Parameterized Concurrent Systems: N finite-state processes ◮ Probabilistic: each process can flip a coin ◮ Fair: each process will have the opportunity to move ◮ Liveness: a good configuration is always reachable with Pr = 1

Examples: Herman’s protocol, Israeli-Jalfon protocol, population protocols, . . . An infinite-state Markov Decision Process (MDP) F

1 2 1 2

1

1 2 1 2

Evil scheduler Random process

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21

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Setting

Liveness of Fair Parameterized Probabilistic Concurrent Systems

◮ Parameterized Concurrent Systems: N finite-state processes ◮ Probabilistic: each process can flip a coin ◮ Fair: each process will have the opportunity to move ◮ Liveness: a good configuration is always reachable with Pr = 1

Examples: Herman’s protocol, Israeli-Jalfon protocol, population protocols, . . . An infinite-state Markov Decision Process (MDP) F

1 2 1 2

1

1 2 1 2

Evil scheduler Random process Pr(s0 | = ♦F) ? = 1

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21

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Almost-Sure Liveness

Weakly-finite MDPs: for a fixed initial configuration, the set of reachable states is finite

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 6 / 21

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Almost-Sure Liveness

Weakly-finite MDPs: for a fixed initial configuration, the set of reachable states is finite Almost-sure liveness in weakly-finite MDPs:

nly distinguish = 0 and > 0 transitions

F

1 2 1 2

1

1 2 1 2

Evil scheduler Random process

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 6 / 21

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Almost-Sure Liveness

Weakly-finite MDPs: for a fixed initial configuration, the set of reachable states is finite Almost-sure liveness in weakly-finite MDPs:

nly distinguish = 0 and > 0 transitions

F Evil scheduler Angelic process

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 6 / 21

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Almost-Sure Liveness

Weakly-finite MDPs: for a fixed initial configuration, the set of reachable states is finite Almost-sure liveness in weakly-finite MDPs:

nly distinguish = 0 and > 0 transitions

F Evil scheduler Angelic process

Lemma

Pr(s0 | = ♦F) = 1 iff Proc. has winning strategy from all s ∈ Reach(s0).

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 6 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for liveness in param. prob. conc. systems under all schedulers

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 7 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for liveness in param. prob. conc. systems under all schedulers Regular Model Checking: Uppsala & Paris

◮ Bouajjani, Jonsson, Nilsson, and Touili [CAV’00]

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 7 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for liveness in param. prob. conc. systems under all schedulers Regular Model Checking: Uppsala & Paris

◮ Bouajjani, Jonsson, Nilsson, and Touili [CAV’00] ◮ usually safety of deterministic systems

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 7 / 21

SLIDE 49

Symbolic Framework: Regular Model Checking

Regular Model Checking for liveness in param. prob. conc. systems under all schedulers Regular Model Checking: Uppsala & Paris

◮ Bouajjani, Jonsson, Nilsson, and Touili [CAV’00] ◮ usually safety of deterministic systems

liveness in parameterized probabilistic concurrent systems:

◮ extension of Lin & R¨ ummer [CAV’16]

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 7 / 21

SLIDE 50

Symbolic Framework: Regular Model Checking

Regular Model Checking for liveness in param. prob. conc. systems under all schedulers Regular Model Checking: Uppsala & Paris

◮ Bouajjani, Jonsson, Nilsson, and Touili [CAV’00] ◮ usually safety of deterministic systems

liveness in parameterized probabilistic concurrent systems:

◮ extension of Lin & R¨ ummer [CAV’16]

this talk: embedding of fairness into the system

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 7 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 8 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking A configuration: a word over Σ: T N T N N

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 8 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking A configuration: a word over Σ: T N T N N A set of configurations: a finite automaton A over Σ T , N T T , N

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 8 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking A configuration: a word over Σ: T N T N N A set of configurations: a finite automaton A over Σ T , N T T , N Transition relation: a (length-preserving) transducer τ T / T N / N T / N N / T T / T T / T N / N

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 8 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness:

◮ Start, Good, τ1, and τ2 given

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness:

◮ Start, Good, τ1, and τ2 given ◮ Task: find

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness:

◮ Start, Good, τ1, and τ2 given ◮ Task: find

FA Inv over-approximating reachable states

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness:

◮ Start, Good, τ1, and τ2 given ◮ Task: find

FA Inv over-approximating reachable states, and
transducer P< encoding progress for Process

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21

SLIDE 60

Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness:

◮ Start, Good, τ1, and τ2 given ◮ Task: find

FA Inv over-approximating reachable states, and
transducer P< encoding progress for Process
Advice bits

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness: Start, Good, τ1, and τ2 given Advice bits: local conditions on FA Inv and transducer P< over Σ

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 10 / 21

SLIDE 62

Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness: Start, Good, τ1, and τ2 given Advice bits: local conditions on FA Inv and transducer P< over Σ

1 Start ⊆ Inv 2 τ∪(Inv) ⊆ Inv

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 10 / 21

SLIDE 63

Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness: Start, Good, τ1, and τ2 given Advice bits: local conditions on FA Inv and transducer P< over Σ

1 Start ⊆ Inv 2 τ∪(Inv) ⊆ Inv 3 P< is a strict preorder (i.e., irreflexive, transitive)

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 10 / 21

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Symbolic Framework: Regular Model Checking

Regular Model Checking for 2-player reachability games: Liveness: Start, Good, τ1, and τ2 given Advice bits: local conditions on FA Inv and transducer P< over Σ

1 Start ⊆ Inv 2 τ∪(Inv) ⊆ Inv 3 P< is a strict preorder (i.e., irreflexive, transitive) 4 For any evil transition from Inv \ Good to se, there is an angelic

transition from se that

goes to Inv and
progresses w.r.t. P<

∀x ∈ Inv \ Good, ∀y ∈ Σ∗ \ Good : (x →τ1 y) ⇒ (∃z ∈ Inv : (y →τ2 z ∧ z <P x))

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 10 / 21

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Finitary Fairness — [Alur & Henzinger’98]

k-Fairness

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

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Finitary Fairness — [Alur & Henzinger’98]

k-Fairness intuition: binds the scope of and ♦ operators to k steps.

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

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Finitary Fairness — [Alur & Henzinger’98]

k-Fairness intuition: binds the scope of and ♦ operators to k steps. weak (justice): ♦A ⇒ ♦B

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

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Finitary Fairness — [Alur & Henzinger’98]

k-Fairness intuition: binds the scope of and ♦ operators to k steps. weak (justice): ♦A ⇒ ♦B No (sub-)path of length k satisfies (A ∧ ¬B).

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

SLIDE 69

Finitary Fairness — [Alur & Henzinger’98]

k-Fairness intuition: binds the scope of and ♦ operators to k steps. weak (justice): ♦A ⇒ ♦B No (sub-)path of length k satisfies (A ∧ ¬B).

◮ A cannot hold for k consecutive steps without B holding.

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

SLIDE 70

Finitary Fairness — [Alur & Henzinger’98]

k-Fairness intuition: binds the scope of and ♦ operators to k steps. weak (justice): ♦A ⇒ ♦B No (sub-)path of length k satisfies (A ∧ ¬B).

◮ A cannot hold for k consecutive steps without B holding.

strong (compassion): ♦A ⇒ ♦B

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

SLIDE 71

Finitary Fairness — [Alur & Henzinger’98]

k-Fairness intuition: binds the scope of and ♦ operators to k steps. weak (justice): ♦A ⇒ ♦B No (sub-)path of length k satisfies (A ∧ ¬B).

◮ A cannot hold for k consecutive steps without B holding.

strong (compassion): ♦A ⇒ ♦B No path satisfies ψk ∧ ¬B. ψ0 = true ψi = ♦(A ∧ ψi−1)

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

SLIDE 72

Finitary Fairness — [Alur & Henzinger’98]

k-Fairness intuition: binds the scope of and ♦ operators to k steps. weak (justice): ♦A ⇒ ♦B No (sub-)path of length k satisfies (A ∧ ¬B).

◮ A cannot hold for k consecutive steps without B holding.

strong (compassion): ♦A ⇒ ♦B No path satisfies ψk ∧ ¬B. ψ0 = true ψi = ♦(A ∧ ψi−1)

◮ A cannot hold k times without B holding at some point.

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

SLIDE 73

Finitary Fairness — [Alur & Henzinger’98]

k-Fairness intuition: binds the scope of and ♦ operators to k steps. weak (justice): ♦A ⇒ ♦B No (sub-)path of length k satisfies (A ∧ ¬B).

◮ A cannot hold for k consecutive steps without B holding.

strong (compassion): ♦A ⇒ ♦B No path satisfies ψk ∧ ¬B. ψ0 = true ψi = ♦(A ∧ ψi−1)

◮ A cannot hold k times without B holding at some point.

Finitary fairness: if k-fair for some k

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21

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Encoding Finitary Fairness into RMC

Encoding Finitary Fairness into RMC:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 12 / 21

SLIDE 75

Encoding Finitary Fairness into RMC

Encoding Finitary Fairness into RMC: Fix some k

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 12 / 21

SLIDE 76

Encoding Finitary Fairness into RMC

Encoding Finitary Fairness into RMC: Fix some k Example for process selection (weak fairness)

◮ every process is selected at least once in k steps

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 12 / 21

SLIDE 77

Encoding Finitary Fairness into RMC

Encoding Finitary Fairness into RMC: Fix some k Example for process selection (weak fairness)

◮ every process is selected at least once in k steps

Append a counter to encoding of every process, initialized to maximum

◮ the maximum value is bounded

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 12 / 21

SLIDE 78

Encoding Finitary Fairness into RMC

Encoding Finitary Fairness into RMC: Fix some k Example for process selection (weak fairness)

◮ every process is selected at least once in k steps

Append a counter to encoding of every process, initialized to maximum

◮ the maximum value is bounded

When a process is selected, reset its counter to max. value

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 12 / 21

SLIDE 79

Encoding Finitary Fairness into RMC

Encoding Finitary Fairness into RMC: Fix some k Example for process selection (weak fairness)

◮ every process is selected at least once in k steps

Append a counter to encoding of every process, initialized to maximum

◮ the maximum value is bounded

When a process is selected, reset its counter to max. value When a process is not selected, decrement its counter

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 12 / 21

SLIDE 80

Encoding Finitary Fairness into RMC

Encoding Finitary Fairness into RMC: Fix some k Example for process selection (weak fairness)

◮ every process is selected at least once in k steps

Append a counter to encoding of every process, initialized to maximum

◮ the maximum value is bounded

When a process is selected, reset its counter to max. value When a process is not selected, decrement its counter Good configurations are also those where some counter = 0

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 12 / 21

SLIDE 81

Encoding Finitary Fairness into RMC

Encoding Finitary Fairness into RMC: Fix some k Example for process selection (weak fairness)

◮ every process is selected at least once in k steps

Append a counter to encoding of every process, initialized to maximum

◮ the maximum value is bounded

When a process is selected, reset its counter to max. value When a process is not selected, decrement its counter Good configurations are also those where some counter = 0 Generalized to arbitrary weak and strong fairness

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 12 / 21

SLIDE 82

Encoding Finitary Fairness into RMC

Example: Herman’s protocol: w/o fairness: N | T | T | N

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 13 / 21

SLIDE 83

Encoding Finitary Fairness into RMC

Example: Herman’s protocol: w/o fairness: N | T | T | N w/ fairness: N 1 1 0 | T 1 1 1 | T 1 1 0 | N 1 0 0

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SLIDE 84

Encoding Finitary Fairness into RMC

Example: Herman’s protocol: w/o fairness: N | T | T | N w/ fairness: N 1 1 0 | T 1 1 1 | T 1 1 0 | N 1 0 0 scheduler picks a process N 1 1 0 | T 1 1 1 | T 1 1 0 | N 1 0 0

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 13 / 21

SLIDE 85

Encoding Finitary Fairness into RMC

Example: Herman’s protocol: w/o fairness: N | T | T | N w/ fairness: N 1 1 0 | T 1 1 1 | T 1 1 0 | N 1 0 0 scheduler picks a process N 1 1 0 | T 1 1 1 | T 1 1 0 | N 1 0 0 process player decrements/resets counters N 1 0 0 | T 1 1 0 | T 1 1 1 | N 0 0 0

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 13 / 21

SLIDE 86

Encoding Finitary Fairness into RMC

Theorem

Let S be a regular representation of an MDP with finitary fairness constraints C. The presented transformation yields a regular representation of an MDP SF (without fairness constraints) such that (if C are realizable) Pr(Start | = ♦Good) = 1 iff Pr(StartF | = ♦GoodF) = 1

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 14 / 21

SLIDE 87

Case Studies: Population Protocols

Moran process

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21

SLIDE 88

Case Studies: Population Protocols

Moran process a model of genetic drift

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21

SLIDE 89

Case Studies: Population Protocols

Moran process a model of genetic drift linear array

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21

SLIDE 90

Case Studies: Population Protocols

Moran process a model of genetic drift linear array alleles A or B

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SLIDE 91

Case Studies: Population Protocols

Moran process a model of genetic drift linear array alleles A or B rules:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21

SLIDE 92

Case Studies: Population Protocols

Moran process a model of genetic drift linear array alleles A or B rules:

◮ . . . A . . . . . . A . . .

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21

SLIDE 93

Case Studies: Population Protocols

Moran process a model of genetic drift linear array alleles A or B rules:

◮ . . . A . . . . . . A . . . ◮ . . . A B . . . . . . A A . . . and . . . B A . . . . . . A A . . .

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21

SLIDE 94

Case Studies: Population Protocols

Moran process a model of genetic drift linear array alleles A or B rules:

◮ . . . A . . . . . . A . . . ◮ . . . A B . . . . . . A A . . . and . . . B A . . . . . . A A . . . ◮ . . . A B . . . . . . B B . . . and . . . B A . . . . . . B B . . .

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21

SLIDE 95

Case Studies: Population Protocols

Moran process a model of genetic drift linear array alleles A or B rules:

◮ . . . A . . . . . . A . . . ◮ . . . A B . . . . . . A A . . . and . . . B A . . . . . . A A . . . ◮ . . . A B . . . . . . B B . . . and . . . B A . . . . . . B B . . .

goal: A

∗

r B

∗

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21

SLIDE 96

Case Studies: Population Protocols

Moran process a model of genetic drift linear array alleles A or B rules:

◮ . . . A . . . . . . A . . . ◮ . . . A B . . . . . . A A . . . and . . . B A . . . . . . A A . . . ◮ . . . A B . . . . . . B B . . . and . . . B A . . . . . . B B . . .

goal: A

∗

r B

∗

Cell cycle switch — similar, but has an intermediate state

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Case Studies: Population Protocols

Clustering

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Case Studies: Population Protocols

Clustering linear array

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Case Studies: Population Protocols

Clustering linear array alleles A or B

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SLIDE 100

Case Studies: Population Protocols

Clustering linear array alleles A or B rules:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 16 / 21

SLIDE 101

Case Studies: Population Protocols

Clustering linear array alleles A or B rules:

◮ . . . A B . . . . . . B A . . . and . . . B A . . . . . . A B . . .

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 16 / 21

SLIDE 102

Case Studies: Population Protocols

Clustering linear array alleles A or B rules:

◮ . . . A B . . . . . . B A . . . and . . . B A . . . . . . A B . . . ◮ . . . A B . . . . . . B A . . . and . . . B A . . . . . . A B . . .

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 16 / 21

SLIDE 103

Case Studies: Population Protocols

Clustering linear array alleles A or B rules:

◮ . . . A B . . . . . . B A . . . and . . . B A . . . . . . A B . . . ◮ . . . A B . . . . . . B A . . . and . . . B A . . . . . . A B . . .

goal: A

∗ B ∗

r B

∗ A ∗

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 16 / 21

SLIDE 104

Case Studies: Population Protocols

Coin game a population of agents every agent has one currency: Dollars or Euros in each step, an agent either:

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 17 / 21

SLIDE 105

Case Studies: Population Protocols

Coin game a population of agents every agent has one currency: Dollars or Euros in each step, an agent either:

◮ keeps it currency or ◮ randomly selects k neighbours and changes currency to the majority

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 17 / 21

SLIDE 106

Case Studies: Population Protocols

Coin game a population of agents every agent has one currency: Dollars or Euros in each step, an agent either:

◮ keeps it currency or ◮ randomly selects k neighbours and changes currency to the majority

goal: D∗ or E∗

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 17 / 21

SLIDE 107

Evaluation

Encoding implemented in FAIRYTAIL

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 18 / 21

SLIDE 108

Evaluation

Encoding implemented in FAIRYTAIL Input:

◮ FAs for Start, Good ◮ transducers for τ1, and τ2

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 18 / 21

SLIDE 109

Evaluation

Encoding implemented in FAIRYTAIL Input:

◮ FAs for Start, Good ◮ transducers for τ1, and τ2

Output:

◮ FAs for StartF, GoodF ◮ transducers for τ F

1 , and τ F 2

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 18 / 21

SLIDE 110

Evaluation

Encoding implemented in FAIRYTAIL Input:

◮ FAs for Start, Good ◮ transducers for τ1, and τ2

Output:

◮ FAs for StartF, GoodF ◮ transducers for τ F

1 , and τ F 2

SLRP [Lin & R¨ ummer, CAV’16] used to find advice bits

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 18 / 21

SLIDE 111

Evaluation

Encoding implemented in FAIRYTAIL Input:

◮ FAs for Start, Good ◮ transducers for τ1, and τ2

Output:

◮ FAs for StartF, GoodF ◮ transducers for τ F

1 , and τ F 2

SLRP [Lin & R¨ ummer, CAV’16] used to find advice bits

◮ SYNTHESISE: use a SAT solver (Sat4j) to obtain a candidate

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 18 / 21

SLIDE 112

Evaluation

Encoding implemented in FAIRYTAIL Input:

◮ FAs for Start, Good ◮ transducers for τ1, and τ2

Output:

◮ FAs for StartF, GoodF ◮ transducers for τ F

1 , and τ F 2

SLRP [Lin & R¨ ummer, CAV’16] used to find advice bits

◮ SYNTHESISE: use a SAT solver (Sat4j) to obtain a candidate ◮ VERIFY: check the candidate is OK/refine SAT formula

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 18 / 21

SLIDE 113

Evaluation

Table: Results of experiments (timeout = 10 hours).

Case study Time Herman’s protocol (merge, line) 3.64 s Herman’s protocol (annih., line) 4.33 s Herman’s protocol (merge, ring) 4.31 s Herman’s protocol (annih., ring) 4.61 s Moran process (2 types, line) 2 m 48 s Moran process (3 types, line) 56 m 14 s Cell cycle switch (1 types, line) 43.94 s Cell cycle switch (2 types, line) 9 h 46 m Clustering (2 types, line) 10 m 30 s Clustering (3 types, line) T/O Coin game (k = 3, clique) 1 m 0 s

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 19 / 21

SLIDE 114

Solution to Herman’s protocol (merge, ring)

q0 N , 1 q1 N q2 N , 1 T T T q3 N T

Inv

r0 ( N , N ), ( 0 , 0 ), ( 0 , T ), ( 1 , 0 ), ( 1 , 1 ) r1 ( 1 , 1 ) ( T , T ) r2 ( N , N ), ( N , T ), ( T , N ), ( T , T ) ( 0 , 0 ), ( 0 , 1 ), ( 1 , 0 ), ( 1 , 1 ) ( N , T ) ( 0 , 1 )

P<

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SLIDE 115

Conclusion

A nice symbolic framework for reasoning about parameterized probabilistic concurrent systems.

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SLIDE 116

Conclusion

A nice symbolic framework for reasoning about parameterized probabilistic concurrent systems. In this talk extended with finitary fairness.

◮ a natural notion of fairness in such systems

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SLIDE 117

Conclusion

A nice symbolic framework for reasoning about parameterized probabilistic concurrent systems. In this talk extended with finitary fairness.

◮ a natural notion of fairness in such systems

Future work: many optimizations possible

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 21 / 21

SLIDE 118

Conclusion

A nice symbolic framework for reasoning about parameterized probabilistic concurrent systems. In this talk extended with finitary fairness.

◮ a natural notion of fairness in such systems

Future work: many optimizations possible more general systems (e.g., grid topology)

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 21 / 21

SLIDE 119

Conclusion

A nice symbolic framework for reasoning about parameterized probabilistic concurrent systems. In this talk extended with finitary fairness.

◮ a natural notion of fairness in such systems

Future work: many optimizations possible more general systems (e.g., grid topology) more general fairness

Leng´ al, Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 21 / 21