Controlling a population of identical NFA Nathalie Bertrand Inria - PowerPoint PPT Presentation

Controlling a population of identical NFA Nathalie Bertrand Inria Rennes joint work with Miheer Dewaskar (ex CMI student), Blaise Genest (IRISA) and Hugo Gimbert (LaBRI) SynCoP & PV workshops @ ETAPS 2018 Workshops SynCoP & PV, April 2018

Motivation Control of gene expression for a population of cells credits: G. Batt Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 2/ 16

Motivation Control of gene expression for a population of cells credits: G. Batt ◮ cell population ◮ gene expression monitored through fluorescence level ◮ drug injections affect all cells ◮ response varies from cell to cell ◮ obtain a large proportion of cells with desired gene expression level Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 2/ 16

Motivation Control of gene expression for a population of cells credits: G. Batt ◮ arbitrary nb of components ◮ cell population ◮ full observation ◮ gene expression monitored through fluorescence level ◮ uniform control ◮ drug injections affect all cells ◮ non-det. model for single ◮ response varies from cell to cell cell ◮ obtain a large proportion of cells ◮ global reachability objective with desired gene expression level Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 2/ 16

Problem formalisation ◮ population of N identical NFA ◮ uniform control policy under full observation ◮ resolution of non-determinism by an adversary Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 3/ 16

Problem formalisation ◮ population of N identical NFA ◮ uniform control policy under full observation ◮ resolution of non-determinism by an adversary b a , b a a b F a b a config: # copies in each state Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 3/ 16

Problem formalisation ◮ population of N identical NFA ◮ uniform control policy under full observation ◮ resolution of non-determinism by an adversary b a , b a a b F a a b a config: # copies in each state ◮ controller chooses the action ( e.g. a ) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 3/ 16

Problem formalisation ◮ population of N identical NFA ◮ uniform control policy under full observation b a , b ◮ resolution of non-determinism by an adversary a a b b a , b a b a a a b F a a b a config: # copies in each state ◮ controller chooses the action ( e.g. a ) ◮ adversary chooses how to move each individual copy ( a -transition) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 3/ 16

Problem formalisation ◮ population of N identical NFA ◮ uniform control policy under full observation b a , b ◮ resolution of non-determinism by an adversary a a b b a , b a b a a a b F a b a , b a b a a a b config: # copies in each state a b a ◮ controller chooses the action ( e.g. a ) ◮ adversary chooses how to move each individual copy ( a -transition) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 3/ 16

Problem formalisation ◮ population of N identical NFA ◮ uniform control policy under full observation b a , b ◮ resolution of non-determinism by an adversary a a b b a , b a b a a a b F a b a , b a b a a a b config: # copies in each state a b a ◮ controller chooses the action ( e.g. a ) ◮ adversary chooses how to move each individual copy ( a -transition) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 3/ 16

Problem formalisation ◮ population of N identical NFA ◮ uniform control policy under full observation b a , b ◮ resolution of non-determinism by an adversary a a b b a , b a b a a a b F a b a , b a b a a a b config: # copies in each state a b a ◮ controller chooses the action ( e.g. a ) ◮ adversary chooses how to move each individual copy ( a -transition) Question can one control the population to ensure that for all non-deterministic choices all NFAs simultaneously reach a target set? Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 3/ 16

Population control Fixed N : build finite 2-player game, identify global target states, decide if controller has a winning strategy for a reachability objective Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 4/ 16

Population control Fixed N : build finite 2-player game, identify global target states, decide if controller has a winning strategy for a reachability objective Challenge: Parameterized control ∀ N ∃ σ ∀ τ ( A N , σ, τ ) | = � F N ? b a , b a a b F a b a Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 4/ 16

Population control Fixed N : build finite 2-player game, identify global target states, decide if controller has a winning strategy for a reachability objective Challenge: Parameterized control ∀ N ∃ σ ∀ τ ( A N , σ, τ ) | = � F N ? b a , b a a b F a b a This talk ◮ decidability and complexity ◮ memory requirements for controller σ ◮ admissible values for N Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 4/ 16

Monotonicity property and cutoff Monotonicity property: the larger N , the harder for controller ∃ σ ∀ τ ( A N , σ, τ ) | = � F N ∀ M ≤ N ∃ σ ∀ τ ( A M , σ, τ ) | = � F M = ⇒ Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 5/ 16

Monotonicity property and cutoff Monotonicity property: the larger N , the harder for controller ∃ σ ∀ τ ( A N , σ, τ ) | = � F N ∀ M ≤ N ∃ σ ∀ τ ( A M , σ, τ ) | = � F M = ⇒ Cutoff: smallest N for which controller has no winning strategy Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 5/ 16

Monotonicity property and cutoff Monotonicity property: the larger N , the harder for controller ∃ σ ∀ τ ( A N , σ, τ ) | = � F N ∀ M ≤ N ∃ σ ∀ τ ( A M , σ, τ ) | = � F M = ⇒ Cutoff: smallest N for which controller has no winning strategy b q 1 winning σ if N < M A \ a 1 b play b then a i s.t. q i is empty . . A ∪{ b } F . winning τ for N = M b A \ a M always fill all q i ’s q M b cutoff is M A = { a 1 , · · · , a M } unspecified edges lead to a sink state Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 5/ 16

Lower bound on the cutoff b d a u c u a , b , c a F d c b u , d u , d u , d ··· 2 M bottom states a , b , c (here 6) ◮ ∀ N ≤ 2 M , ∃ σ, A N | = ∀ σ � F N accumulate copies in bottom states, then u / d to converge ◮ for N > 2 M controller cannot avoid reaching the sink state Cutoff O (2 |A| ) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 6/ 16

Lower bound on the cutoff b d a u c u a , b , c a F d c b u , d u , d u , d ··· 2 M bottom states a , b , c (here 6) ◮ ∀ N ≤ 2 M , ∃ σ, A N | = ∀ σ � F N accumulate copies in bottom states, then u / d to converge ◮ for N > 2 M controller cannot avoid reaching the sink state Cutoff O (2 |A| ) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 6/ 16

Lower bound on the cutoff b d a u c u a , b , c a F d c b u , d u , d u , d ··· 2 M bottom states a , b , c (here 6) ◮ ∀ N ≤ 2 M , ∃ σ, A N | = ∀ σ � F N accumulate copies in bottom states, then u / d to converge ◮ for N > 2 M controller cannot avoid reaching the sink state Cutoff O (2 |A| ) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 6/ 16

Lower bound on the cutoff b d a u c u a , b , c a F d c b u , d u , d u , d ··· 2 M bottom states a , b , c (here 6) ◮ ∀ N ≤ 2 M , ∃ σ, A N | = ∀ σ � F N accumulate copies in bottom states, then u / d to converge ◮ for N > 2 M controller cannot avoid reaching the sink state Cutoff O (2 |A| ) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 6/ 16

Lower bound on the cutoff b d a u c u a , b , c a F d c b u , d u , d u , d ··· 2 M bottom states a , b , c (here 6) ◮ ∀ N ≤ 2 M , ∃ σ, A N | = ∀ σ � F N accumulate copies in bottom states, then u / d to converge ◮ for N > 2 M controller cannot avoid reaching the sink state Cutoff O (2 |A| ) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 6/ 16

Lower bound on the cutoff b d a u c u a , b , c a F d c b u , d u , d u , d ··· 2 M bottom states a , b , c (here 6) ◮ ∀ N ≤ 2 M , ∃ σ, A N | = ∀ σ � F N accumulate copies in bottom states, then u / d to converge ◮ for N > 2 M controller cannot avoid reaching the sink state Cutoff O (2 |A| ) Controlling a population of NFA – Nathalie Bertrand Workshops SynCoP & PV, April 2018– 6/ 16