Nonlinear real arithmetic and -satis ability Paolo Zuliani Nonlinear real arithmetic and -satis...

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Transcript of Nonlinear real arithmetic and -satis ability Paolo Zuliani Nonlinear real arithmetic and -satis...

  • Nonlinear real arithmetic and δ-satisfiability

    Paolo Zuliani

    School of Computing Science Newcastle University, UK

    (Slides courtesy of Sicun Gao, MIT)

    1 / 26

  • Introduction

    I We use hybrid systems for modelling and verifying biological system models

    I prostate cancer therapy I psoriasis UVB treatment

    I Hybrid systems combine continuous dynamics with discrete state changes

    2 / 26

  • Why Nonlinear Real Arithmetic and Hybrid Systems? (I)

    A prostate cancer model1

    dx

    dt =

    ( αx

    1 + e(k1−z)k2 − βx

    1 + e(z−k3)k4 −m1

    ( 1− z

    z0

    ) − c1

    ) x + c2

    dy

    dt =m1

    ( 1− z

    z0

    ) x +

    ( αy

    ( 1− d0

    z

    z0

    ) − βy

    ) y

    dz

    dt =− zγ − c3

    v =x + y

    I v - prostate specific antigen (PSA)

    I x - hormone sensitive cells (HSCs)

    I y - castration resistant cells (CRCs)

    I z - androgen

    1 A.M. Ideta, G. Tanaka, T. Takeuchi, K. Aihara: A mathematical model of intermittent androgen suppression

    for prostate cancer. Journal of Nonlinear Science, 18(6), 593–614 (2008)

    3 / 26

  • Why Nonlinear Real Arithmetic and Hybrid Systems? (I) Intermittent androgen deprivation therapy

    mode 1;

    𝑑𝑆1 𝑑𝑡

    = 𝑣01 + 𝑎𝑎1 ∗ 𝑡

    𝑑𝑆2 𝑑𝑡

    = 𝑣02

    𝑑𝑣1 𝑑𝑡

    = 𝑎𝑎1

    mode 2;

    𝑑𝑆1 𝑑𝑡

    = 𝑣01

    𝑑𝑆2 𝑑𝑡

    = 𝑣02

    mode 3;

    𝑑𝑆1 𝑑𝑡

    = 𝑣01 + 𝑎𝑑1 ∗ 𝑡

    𝑑𝑆2 𝑑𝑡

    = 𝑣02

    𝑑𝑣1 𝑑𝑡

    = 𝑎𝑑1

    mode 4;

    𝑑𝑆1 𝑑𝑡

    = 𝑣01 + 𝑎𝑑1 ∗ 𝑡

    𝑑𝑆2 𝑑𝑡

    = 𝑣02 + 𝑎𝑑2 ∗ 𝑡

    𝑑𝑣1 𝑑𝑡

    = 𝑎𝑑1

    𝑑𝑣2 𝑑𝑡

    = 𝑎𝑑2

    mode 5;

    𝑑𝑆2 𝑑𝑡

    = 𝑣02 + 𝑎𝑑2 ∗ 𝑡

    𝑑𝑣2 𝑑𝑡

    = 𝑎𝑑2

    𝑣1 = 𝑣𝑚𝑎𝑥

    𝑆1 ≥ 𝑆2 + 𝑣01 ∗ 𝑡𝑠𝑎𝑓𝑒

    𝑡 = 𝑡𝑟𝑒𝑎𝑐𝑡

    𝑣1 = 0

    on-therapy

    𝑑𝑥

    𝑑𝑡 =

    𝛼𝑥

    1 + 𝑒 𝑘1−𝑧 𝑘2 −

    𝛽𝑥

    1 + 𝑒 𝑧−𝑘3 𝑘4 − 𝑚1 1 −

    𝑧

    𝑧0 − 𝑐1 𝑥 + 𝑐2

    𝑑𝑦

    𝑑𝑡 = 𝑚1 1 −

    𝑧

    𝑧0 𝑥 + 𝛼𝑦 1 −

    𝑑0𝑧

    𝑧0 − 𝛽 𝑦

    𝑑𝑧

    𝑑𝑡 = −𝑧𝛾 + 𝑐3

    off-therapy

    𝑑𝑥

    𝑑𝑡 =

    𝛼𝑥

    1 + 𝑒 𝑘1−𝑧 𝑘2 −

    𝛽𝑥

    1 + 𝑒 𝑧−𝑘3 𝑘4 − 𝑚1 1 −

    𝑧

    𝑧0 − 𝑐1 𝑥 + 𝑐2

    𝑑𝑦

    𝑑𝑡 = 𝑚1 1 −

    𝑧

    𝑧0 𝑥 + 𝛼𝑦 1 −

    𝑑0𝑧

    𝑧0 − 𝛽 𝑦

    𝑑𝑧

    𝑑𝑡 = 𝑧0 − 𝑧 𝛾 + 𝑐3

    𝑥 + 𝑦 ≤ 𝑟0 𝑥 + 𝑦 ≥ 𝑟1

    4 / 26

  • Why Nonlinear Real Arithmetic and Hybrid Systems? (II)

    A model of psoriasis development and UVB treatment2

    dSC

    dt = γ1

    ω(1− SC+λSCd SCmax

    )SC

    1 + (ω − 1)( TA+TAd Pta,h

    )n − β1InASC −

    k1sω

    1 + (ω − 1)( TA+TAd Pta,h

    )nSC + k1TA

    dTA

    dt =

    k1a,sωSC

    1 + (ω − 1)( TA+TAd Pta,h

    )n +

    2k1sω

    1 + (ω − 1)( TA+TAd Pta,h

    )n + γ2GA− β2InATA− k2sTA− k1TA

    dGA

    dt = (k2a,s + 2k2s )TA− k2GA− k3GA− β3GA

    dSCd

    dt = γ1d (1−

    SC + SCd

    SCmax,t SCd − β1d InASCd − k1sdSCd −

    kpSC 2 d

    k2a + SC 2 d

    + k1dTAd )

    dTAd

    dt = k1a,sdSCd + 2k1sdSCd + γ2dTAd + k2dGAd − β2d InATAd − k2sdTAd − k1dTAd

    dGAd

    dt = (k2a,sd + 2k2sd )TAd − k2dGAd − k3dGAd − β3dGAd

    I Therapy episode: 48 hours of irradiation + 8 hours of rest

    I Therapy episode = multiply β1 and β2 by a constant InA

    2 H. Zhang, W. Hou, L. Henrot, S. Schnebert, M. Dumas, C. Heusèle, and J. Yang. Modelling epidermis

    homoeostasis and psoriasis pathogenesis. Journal of The Royal Society Interface, 12(103), 2015.

    5 / 26

  • Why Nonlinear Real Arithmetic and Hybrid Systems? (II)

    A model of psoriasis development and UVB treatment2

    dSC

    dt = γ1

    ω(1− SC+λSCd SCmax

    )SC

    1 + (ω − 1)( TA+TAd Pta,h

    )n − β1 InASC −

    k1sω

    1 + (ω − 1)( TA+TAd Pta,h

    )nSC + k1TA

    dTA

    dt =

    k1a,sωSC

    1 + (ω − 1)( TA+TAd Pta,h

    )n +

    2k1sω

    1 + (ω − 1)( TA+TAd Pta,h

    )n + γ2GA− β2 InATA− k2sTA− k1TA

    dGA

    dt = (k2a,s + 2k2s )TA− k2GA− k3GA− β3GA

    dSCd

    dt = γ1d (1−

    SC + SCd

    SCmax,t SCd − β1d InASCd − k1sdSCd −

    kpSC 2 d

    k2a + SC 2 d

    + k1dTAd )

    dTAd

    dt = k1a,sdSCd + 2k1sdSCd + γ2dTAd + k2dGAd − β2d InATAd − k2sdTAd − k1dTAd

    dGAd

    dt = (k2a,sd + 2k2sd )TAd − k2dGAd − k3dGAd − β3dGAd

    I Therapy episode: 48 hours of irradiation + 8 hours of rest

    I Therapy episode = multiply β1 and β2 by a constant InA

    2 H. Zhang, W. Hou, L. Henrot, S. Schnebert, M. Dumas, C. Heusèle, and J. Yang. Modelling epidermis

    homoeostasis and psoriasis pathogenesis. Journal of The Royal Society Interface, 12(103), 2015.

    5 / 26

  • Real-World Applications

    I Psoriasis Stratification to Optimise Relevant Therapy (PSORT)

    I Large (∼£5m) I Primarily biomarkers discovery I We use computational modelling for understanding psoriasis’

    mechanisms

    I Personalised ultraviolet B treatment of psoriasis through biomarker integration with computational modelling of psoriatic plaque resolution

    I Starts February 2017 I PIs: P.Z. and Nick Reynolds (Institute of Cellular Medicine) I Computational modelling to inform UVB therapies used

    in the clinic — real impact on people’s health!

    6 / 26

  • Real-World Applications

    I Psoriasis Stratification to Optimise Relevant Therapy (PSORT)

    I Large (∼£5m) I Primarily biomarkers discovery I We use computational modelling for understanding psoriasis’

    mechanisms

    I Personalised ultraviolet B treatment of psoriasis through biomarker integration with computational modelling of psoriatic plaque resolution

    I Starts February 2017 I PIs: P.Z. and Nick Reynolds (Institute of Cellular Medicine) I Computational modelling to inform UVB therapies used

    in the clinic — real impact on people’s health!

    6 / 26

  • Bounded Reachability

    I Reachability is a key property in verification, also for hybrid systems

    I Reachability is undecidable even for linear hybrid systems (Alur, Courcoubetis, Henzinger, Ho. 1993)

    I [Bounded Reachability] Does the hybrid system reach a goal state within a finite time and number of (discrete) steps?

    I “Can a 5-episode UVB therapy remit psoriasis for a year?”

    I Reasoning about nonlinear real arithmetic is hard . . .

    7 / 26

  • Bounded Reachability

    I Reachability is a key property in verification, also for hybrid systems

    I Reachability is undecidable even for linear hybrid systems (Alur, Courcoubetis, Henzinger, Ho. 1993)

    I [Bounded Reachability] Does the hybrid system reach a goal state within a finite time and number of (discrete) steps?

    I “Can a 5-episode UVB therapy remit psoriasis for a year?”

    I Reasoning about nonlinear real arithmetic is hard . . .

    7 / 26

  • Bounded Reachability

    I Reachability is a key property in verification, also for hybrid systems

    I Reachability is undecidable even for linear hybrid systems (Alur, Courcoubetis, Henzinger, Ho. 1993)

    I [Bounded Reachability] Does the hybrid system reach a goal state within a finite time and number of (discrete) steps?

    I “Can a 5-episode UVB therapy remit psoriasis for a year?”

    I Reasoning about nonlinear real arithmetic is hard . . .

    7 / 26

  • Type 2 Computability

    Turning machines operate on finite strings, i.e., integers, which cannot capture real-valued functions.

    I Real numbers can be encoded on infinite tapes.

    I Real numbers are functions over integers.

    I Real functions can be computed by machines that take infinite tapes as inputs, and output infinite tapes encoding the values.

    Definition (Name of a real number)

    A real number a can be encoded by an infinite sequence of rationals γa : N→ Q such that

    ∀i ∈ N |a− γa(i)| < 2−i .

    8 / 26

  • Type 2 Computability

    A function f (x) = y is computable if any name of x can be algorithmically mapped to a name of y

    ...

    ...

    ...

    M

    ... k input tapes

    work tapes

    output tape

    fM (y1, . . . , yk) = y

    y

    y1

    yk

    ... ...

    ...

    ... ...

    } }

    Writing on any finite segment of the output tape takes finite time.

    9 /