Motivating Example Herman’s protocol (merging version) = ♦ leader is elected ) = 1 Pr ( | 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
Setting Liveness of Fair Parameterized Probabilistic Concurrent Systems Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21
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
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
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
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
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
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) 1 1 2 2 Evil scheduler 1 F Random process 2 1 1 2 Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21
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) 1 1 2 2 Evil scheduler 1 F Random process 2 1 1 2 = ♦ F ) ? Pr ( s 0 | = 1 Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 5 / 21
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
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: only distinguish = 0 and > 0 transitions 1 1 2 2 Evil scheduler 1 F 2 Random process 1 1 2 Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 6 / 21
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: only distinguish = 0 and > 0 transitions Evil scheduler F Angelic process Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 6 / 21
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: only distinguish = 0 and > 0 transitions Evil scheduler F Angelic process Lemma Pr ( s 0 | = ♦ F ) = 1 iff Proc. has winning strategy from all s ∈ Reach ( s 0 ) . Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 6 / 21
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
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
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
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
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
Symbolic Framework: Regular Model Checking Regular Model Checking Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 8 / 21
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
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 , N T Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 8 / 21
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 , N T Transition relation: a (length-preserving) transducer τ T / T T / T N / N N / N N / T T / T T / N Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 8 / 21
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
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: ◮ Start , Good , τ 1 , and τ 2 given Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: ◮ Start , Good , τ 1 , and τ 2 given ◮ Task: find Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: ◮ Start , Good , τ 1 , and τ 2 given ◮ Task: find • FA Inv over-approximating reachable states Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: ◮ Start , Good , τ 1 , and τ 2 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
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: ◮ Start , Good , τ 1 , and τ 2 given ◮ Task: find � • FA Inv over-approximating reachable states, and Advice bits • transducer P < encoding progress for Process Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 9 / 21
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: Start , Good , τ 1 , and τ 2 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
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: Start , Good , τ 1 , and τ 2 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
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: Start , Good , τ 1 , and τ 2 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
Symbolic Framework: Regular Model Checking Regular Model Checking for 2-player reachability games: Liveness: Start , Good , τ 1 , and τ 2 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 s e , there is an angelic transition from s e that • goes to Inv and • progresses w.r.t. P < ∀ y ∈ Σ ∗ \ Good : ∀ x ∈ Inv \ Good , ( x → τ 1 y ) ⇒ ( ∃ z ∈ Inv : ( y → τ 2 z ∧ z < P x )) Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 10 / 21
Finitary Fairness — [Alur & Henzinger’98] k -Fairness Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21
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
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
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
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
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
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 ψ k ∧ � ¬ B . ψ 0 = true ψ i = ♦ ( A ∧ � ψ i − 1 ) Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 11 / 21
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 ψ k ∧ � ¬ B . ψ 0 = true ψ i = ♦ ( A ∧ � ψ i − 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
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 ψ k ∧ � ¬ B . ψ 0 = true ψ i = ♦ ( A ∧ � ψ i − 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
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
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
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
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
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
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
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
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
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
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 Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 13 / 21
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
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
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 S F (without fairness constraints) such that (if C are realizable) Pr ( Start | = ♦ Good ) = 1 iff Pr ( Start F | = ♦ Good F ) = 1 Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 14 / 21
Case Studies: Population Protocols Moran process Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21
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
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
Case Studies: Population Protocols Moran process a model of genetic drift linear array alleles A or B Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21
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
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
Case Studies: Population Protocols Moran process a model of genetic drift linear array alleles A or B rules: ◮ . . . A . . . � . . . A . . . ◮ . . . A B . . . � . . . A A . . . � . . . A A . . . A . . . and . . . B Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21
Case Studies: Population Protocols Moran process a model of genetic drift linear array alleles A or B rules: ◮ . . . A . . . � . . . A . . . ◮ . . . A B . . . � . . . A A . . . � . . . A A . . . A . . . and . . . B B . . . � . . . B B . . . and . . . B A . . . � . . . B ◮ . . . A B . . . Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21
Case Studies: Population Protocols Moran process a model of genetic drift linear array alleles A or B rules: ◮ . . . A . . . � . . . A . . . ◮ . . . A B . . . � . . . A A . . . � . . . A A . . . A . . . and . . . B B . . . � . . . B B . . . and . . . B A . . . � . . . B ◮ . . . A B . . . ∗ ∗ goal: A or B Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21
Case Studies: Population Protocols Moran process a model of genetic drift linear array alleles A or B rules: ◮ . . . A . . . � . . . A . . . ◮ . . . A B . . . � . . . A A . . . � . . . A A . . . A . . . and . . . B B . . . � . . . B B . . . and . . . B A . . . � . . . B ◮ . . . A B . . . ∗ ∗ goal: A or B Cell cycle switch — similar, but has an intermediate state Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 15 / 21
Case Studies: Population Protocols Clustering Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 16 / 21
Case Studies: Population Protocols Clustering linear array Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 16 / 21
Case Studies: Population Protocols Clustering linear array alleles A or B Leng´ al , Lin, Majumdar, R¨ ummer Fair Termination for Probabilistic Systems MOSCA’19 16 / 21
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
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