Controlling a population of identical NFA
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- Britton Goodman
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1 Controlling popultion of identicl NFA Nthlie Bertrnd Inri Rennes joint work with Miheer Dewskr (ex CMI student), Blise Genest (IRISA) nd Hugo Gimert (LBRI) LSV 20th nniversry LSV 20th nniversry, My 2017
2 Bck in Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
3 Bck in Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
4 Bck in Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
5 Bck in Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
6 Bck in Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
7 Bck in LSV = Loisirs Sport Vcnces Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
8 Motivtion Control of gene expression for popultion of cells credits: G. Btt Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
9 Motivtion Control of gene expression for popultion of cells cell popultion gene expression monitored through fluorescence level drug injections ffect ll cells response vries from cell to cell credits: G. Btt otin lrge proportion of cells with desired gene expression level Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
10 Motivtion Control of gene expression for popultion of cells cell popultion gene expression monitored through fluorescence level drug injections ffect ll cells response vries from cell to cell credits: G. Btt otin lrge proportion of cells with desired gene expression level ritrry n of components full oservtion uniform control NFA model for single cell glol rechility ojective Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
11 Prolem formlistion popultion of N identicl NFA uniform control policy under full oservtion resolution of non-determinism y n dversry Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
12 Prolem formlistion popultion of N identicl NFA uniform control policy under full oservtion resolution of non-determinism y n dversry, F config: # copies in ech stte Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
13 Prolem formlistion popultion of N identicl NFA uniform control policy under full oservtion resolution of non-determinism y n dversry, F config: # copies in ech stte controller chooses the ction (e.g. ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
14 Prolem formlistion popultion of N identicl NFA uniform control policy under full oservtion resolution of non-determinism y n dversry,, F config: # copies in ech stte controller chooses the ction (e.g. ) dversry chooses how to move ech individul copy (-trnsition) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
15 Prolem formlistion popultion of N identicl NFA uniform control policy under full oservtion resolution of non-determinism y n dversry,, F config: # copies in ech stte controller chooses the ction (e.g. ) dversry chooses how to move ech individul copy (-trnsition), Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
16 Prolem formlistion popultion of N identicl NFA uniform control policy under full oservtion resolution of non-determinism y n dversry,, F config: # copies in ech stte controller chooses the ction (e.g. ) dversry chooses how to move ech individul copy (-trnsition), Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
17 Prolem formlistion popultion of N identicl NFA uniform control policy under full oservtion resolution of non-determinism y n dversry,, F config: # copies in ech stte controller chooses the ction (e.g. ) dversry chooses how to move ech individul copy (-trnsition), Question cn one control the popultion to ensure tht for ll non-deterministic choices ll NFAs simultneously rech trget set? Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
18 Popultion control Fixed N: uild finite 2-plyer gme, identify glol trget sttes, decide if controller hs winning strtegy for rechility ojective N σ τ (A N, σ, τ) = F N? Chllenge: Prmeterized control Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
19 Popultion control Fixed N: uild finite 2-plyer gme, identify glol trget sttes, decide if controller hs winning strtegy for rechility ojective N σ τ (A N, σ, τ) = F N? Chllenge: Prmeterized control This tlk decidility nd complexity memory requirements for controller σ dmissile vlues for N Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
20 Monotonicity property nd cutoff Monotonicity property: the lrger N, the hrder for controller σ τ(a N, σ, τ) = F N = M N σ τ(a M, σ, τ) = F M Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
21 Monotonicity property nd cutoff Monotonicity property: the lrger N, the hrder for controller σ τ(a N, σ, τ) = F N = M N σ τ(a M, σ, τ) = F M Cutoff: smllest N for which controller hs no winning strtegy Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
22 Monotonicity property nd cutoff Monotonicity property: the lrger N, the hrder for controller σ τ(a N, σ, τ) = F N = M N σ τ(a M, σ, τ) = F M Cutoff: smllest N for which controller hs no winning strtegy q 1 A\ 1. F A {} A\ M q M A = { 1,, M } unspecified edges led to sink stte winning σ if N < M ply then i s.t. q i is empty winning τ for N = M lwys fill ll q i s cutoff is M Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
23 A nturl ttempt: the support gme Assumption: if stte q 2 or q 4 is empty, controller wins Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
24 A nturl ttempt: the support gme Assumption: if stte q 2 or q 4 is empty, controller wins Support gme: Eve chooses ction Adm chooses trnsfer grph (footprint of copies moves) 1,2,3 1,2,3,4 1,3,4 Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
25 A nturl ttempt: the support gme Assumption: if stte q 2 or q 4 is empty, controller wins Support gme: Eve chooses ction Adm chooses trnsfer grph (footprint of copies moves) 1,2,3 1,2,3,4 1,3,4 If Eve wins support gme then controller hs winning strtegy for ll N Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
26 Support gme is not equivlent to popultion gme controller lterntes nd ; dversry lwys fills q 2 nd q 4 in the -step Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
27 Support gme is not equivlent to popultion gme controller lterntes nd ; dversry lwys fills q 2 nd q 4 in the -step Ply in support gme is not relisle: Controller wins with () ω! Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
28 Support gme is not equivlent to popultion gme controller lterntes nd ; dversry lwys fills q 2 nd q 4 in the -step Ply in support gme is not relisle: Controller wins with () ω! Memoryless support-sed controllers re not enough! Exponentil memory on top of support my even e needed. Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
29 Cpcity gme: refining winning condition of support gme G H G G H ccumultor Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
30 Cpcity gme: refining winning condition of support gme G H G G H ccumultor Finite cpcity ply: ll ccumultors hve finitely mny entries Bounded cpcity ply: finite ound on # entries for ccumultors Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
31 Cpcity gme: refining winning condition of support gme G H G G H ccumultor Finite cpcity ply: ll ccumultors hve finitely mny entries Bounded cpcity ply: finite ound on # entries for ccumultors Bounded cpcity corresponds to relizle plys does not seem to e regulr Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
32 Cpcity gme: refining winning condition of support gme G H G G H ccumultor Finite cpcity ply: ll ccumultors hve finitely mny entries Bounded cpcity ply: finite ound on # entries for ccumultors Bounded cpcity corresponds to relizle plys does not seem to e regulr Cpcity gme: Eve wins ply if either it reches suset of F, or it does not hve finite cpcity. Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
33 Cpcity gme: refining winning condition of support gme G H G G H ccumultor Finite cpcity ply: ll ccumultors hve finitely mny entries Bounded cpcity ply: finite ound on # entries for ccumultors Bounded cpcity corresponds to relizle plys does not seem to e regulr Cpcity gme: Eve wins ply if either it reches suset of F, or it does not hve finite cpcity. Eve wins cpcity gme iff Controller hs winning strtegy for ll N Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
34 Solving the cpcity gme Nive solution set of plys with infinite cpcity is ω-regulr non-deterministic Büchi utomton guesses n ccumultor, nd checks it hs infinitely mny entries winning condition cn e determinized into prity condition exponentil lowup Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
35 Solving the cpcity gme Nive solution 2EXPTIME procedure in the size of NFA A Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
36 Solving the cpcity gme Nive solution 2EXPTIME procedure in the size of NFA A Better solution q G x y H x y enters ccumultor from q q t x G G seprtes pir (t, x) entries rise from seprted pirs trcking trnsfer grphs seprting new pirs is sufficient Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
37 Solving the cpcity gme Nive solution 2EXPTIME procedure in the size of NFA A Better solution q G x y H x y enters ccumultor from q q t x G G seprtes pir (t, x) entries rise from seprted pirs trcking trnsfer grphs seprting new pirs is sufficient Prity gme: cpcity gme enriched with trcking lists in sttes priorities reflect how the trcking list evolves (removls, shifts, etc.) # sttes = (simply!) exponentil in A # priorities = polynomil in A Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
38 Solving the cpcity gme Nive solution 2EXPTIME procedure in the size of NFA A Better solution q G x y H x y enters ccumultor from q q t x G G seprtes pir (t, x) entries rise from seprted pirs trcking trnsfer grphs seprting new pirs is sufficient Prity gme: cpcity gme enriched with trcking lists in sttes priorities reflect how the trcking list evolves (removls, shifts, etc.) # sttes = (simply!) exponentil in A # priorities = polynomil in A Prity gme is equivlent to cpcity gme. Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
39 Complexity of the popultion control prolem Theorem: The popultion control prolem is EXPTIME-complete. Upper ound : popultion control prolem cpcity gme cpcity gme to prity gme solving prity gme of size exp. nd poly. priorities Lower ound : encoding of poly spce lternting Turing mchine Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
40 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
41 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
42 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
43 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
44 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
45 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
46 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
47 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
48 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
49 Lower ound on the cutoff d u c c u d F,,c u,d u,d u,d,,c 2M ottom sttes (here 6) N 2 M, σ, A N = σ F N ccumulte copies in ottom sttes, then u/d to converge for N > 2 M controller cnnot void reching the sink stte Comined with counter, cutoff is even douly exponentil! Cutoff O(2 A ) Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
50 Summry of results Uniform control of popultion of identicl NFA prmeterized control prolem: gther ll copies in F (surprisingly) quite involved! tight results for complexity, cutoff, nd memory complexity: EXPTIME-complete decision prolem ound on cutoff: douly exponentil memory requirement: exponentil memory (orthogonl to supports) is needed nd sufficient for controller Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
51 Bck to motivtions Control of gene expression for popultion of cells credits: G. Btt Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
52 Bck to motivtions Control of gene expression for popultion of cells credits: G. Btt need for truely proilistic model MDP insted of NFA need for truely quntittive questions proportions nd proilities insted of convergence nd (lmost)-sure N mx σ P σ(a N = t lest 80% of MDPs in F ).7? Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
53 Thnks, nd hppy nniversry! Controlling popultion of NFA Nthlie Bertrnd LSV 20th nniversry, My / 15
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