物理情報システムの検証手法
制御と形式手法の相互理解，そして協働へ

蓮尾 一郎
国立情報学研究所 (NII)，総合研究大学院大学
JST ERATO 蓮尾メタ数理システムデザインプロジェクト

物理情報システムの検証手法
制御と形式手法の相互理解，そして協働へ• 専門： 理論計算機科学，論理学，計算機科学における代数的・圏論的構造，
最近は制御理論も蓮尾 一郎
国立情報学研究所 (NII)，総合研究大学院大学
JST ERATO 蓮尾メタ数理システムデザインプロジェクト

Outline
制御理論 vs 形式手法・ソフトウェア科学
数学的相似・相互乗り入れ
例1（相似）： 確率的プログラムのマルチンゲールによる解析[Chakarov & Sankaranarayanan, CAVʼ13][Takisaka, Oyabu, Urabe & Hasuo, ATVAʼ18]

例2（乗り入れ）： 近似双模倣による離散化[Girard & Pappas, IEEE TAC ʼ07]

[Kido, Sedwards & Hasuo, IFAC ADHS ʼ18]

実システム応用に向けて
機械学習・AIとの協働
ソフトウェア工学，現実的解決
ERATO MMSD プロジェクトの取り組み
2

物理情報システム：
制御理論と形式手法・ソフトウェア科学の協働
物理情報システム Cyber-Physical System (CPS)“A mechanism that is controlled or monitored by computerbased algorithms, tightly integrated with the Internet and its
users” (Wikipedia)
関連： embedded system, networked system, sensor network,
robotics, systems of systems, Industrie 4.0, smart city,
hybrid systems
連続的物理プラント + 離散的デジタル制御
米国： NSF Key Area of Research (2006-)
3

制御理論 vs
形式手法・ソフトウェア科学
形式手法

制御理論
連続空間・連続時間（離散の場合も）微分方程式
連続量最適化
LMI, SDP, mixed integer, …

機械，電気

ソフトウェア科学
対象となる
システム

離散空間・離散時間（高々可算無限）モデリング オートマトン（グラフ）
プログラム
形式
最終的な
解き方

グラフアルゴリズム
SATソルバ，SMTソルバ

主な
応用対象

ソフトウェア4

… 共通性？
対比の項目は同じ

5

… 共通性？
対比の項目は同じ

制約充足問題翻訳
ソフトウェア
検証問題

spec ↦ automaton
model ↦ prop. formula
…

SAT
graph
reachability
…

最適化問題

制御問題

翻訳
5

LMI
SDP
mixed integerstochastic optimization…

Outline
制御理論 vs 形式手法・ソフトウェア科学
数学的相似・相互乗り入れ
例1（相似）： 確率的プログラムのマルチンゲールによる解析[Chakarov & Sankaranarayanan, CAVʼ13][Takisaka, Oyabu, Urabe & Hasuo, ATVAʼ18]

例2（乗り入れ）： 近似双模倣による離散化[Girard & Pappas, IEEE TAC ʼ07]

[Kido, Sedwards & Hasuo, IFAC ADHS ʼ18]

実システム応用に向けて
機械学習・AIとの協働
ソフトウェア工学，現実的解決
ERATO MMSD プロジェクトの取り組み
6

Probabilistic Programs
1 x := m
2 while x > 0 do
3i f prob(p) do
4
x := x 1
5else
6x := x + 1
7
fi
8 od

Programs withrandom assignment
x := Gaussian(0, 0.2)

probabilistic branching
if prob(0.2)(say, m = 16 and p = 0.2)
<latexit sha1_base64="JaB/5Fey3F2sEw+FzLG75bYBN20=">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</latexit>

Example: right (random walk)
Disclaimer: we disregard the Bayesian aspects
observations, conditioning, priors/posteriors, etc.
See e.g. [Gordon+, FOSE’14]. Languages: Church, Anglican, Venture, …
7

Hasuo (NII, Tokyo)

Reachability Probabilities in PP
Question What isPr(the program terminates)?
How do we define it?
Configuration graph

1 x := m
2 while x > 0 do
3i f prob(p) do
4
x := x 1
5else
6x := x + 1
7
fi
8 od
(say, m = 16 and p = 0.2)
<latexit 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zuPRMIjfOphBE898xayR6bcIyKCNpkjZSQQc6c5cGvN1otLhoI9uHpvOLbfMnYfXu5uj3e2x1/tDp8c1G8gv7Px7xv/sbG1Md74r40nG59vnGx8vRHee3jv5N5v7/1u9IfRn0b/PfqfCvqTe3Wff9vofEb/9//2g5Rz</latexit>

A state is a pair (program location, memory state)Transition by small-step operational semantics➜ MD (Markov chain) M

Pr(Termination) = Pr(M reaches })<latexit 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WXt4doXa0drX69Fa39d+/vaP9b+uXl/k25ON19V0J+8Vff5w1rrs6n+BVBNd+s=</latexit>

8

Hasuo (NII, Tokyo)

Reachability Probabilities in PP
Question What isPr(the program terminates)?
How do we define it?
Configuration graph

1 x := m
2 while x > 0 do
3i f prob(p) do
4
x := x 1
5else
6x := x + 1
7
fi
8 od
(say, m = 16 and p = 0.2)
<latexit 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zuPRMIjfOphBE898xayR6bcIyKCNpkjZSQQc6c5cGvN1otLhoI9uHpvOLbfMnYfXu5uj3e2x1/tDp8c1G8gv7Px7xv/sbG1Md74r40nG59vnGx8vRHee3jv5N5v7/1u9IfRn0b/PfqfCvqTe3Wff9vofEb/9//2g5Rz</latexit>

real- andint- valued
A state is a pair (program location,
memorystate) variables➜ infinite
statesTransition by small-step operationalsemantics➜ no “direct” automated analysis➜ MD (Markov chain) M

Pr(Termination) = Pr(M reaches })<latexit 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WXt4doXa0drX69Fa39d+/vaP9b+uXl/k25ON19V0J+8Vff5w1rrs6n+BVBNd+s=</latexit>

8

Hasuo (NII, Tokyo)

Probabilistic Control Flow Graph(pCFG) See e.g. [Chatterjee+, POPL’17]
1 x := m
2 while x > 0 do
3i f prob(p) do
4
x := x 1
5else
6x := x + 1
7
fi
8 od
(say, m = 16 and p = 0.2)
<latexit sha1_base64="JaB/5Fey3F2sEw+FzLG75bYBN20=">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</latexit>

Hasuo (NII, Tokyo)

Probabilistic Control Flow Graph(pCFG) See e.g. [Chatterjee+, POPL’17]
1 x := m
2 while x > 0 do
3i f prob(p) do
4
x := x 1
5else
6x := x + 1
7
fi
8 od
(say, m = 16 and p = 0.2)
<latexit sha1_base64="JaB/5Fey3F2sEw+FzLG75bYBN20=">AAAuIHiczVrdc9zGkadyl4u9l7vYucd7Gd9SFUrG0lyqfJeyQpbMlWirTFM0SctKCIoeALO7YwIYaGZAcgPhn8nfkoe8Xd3j3V9z3QPsApjB2s7HQ7aKXPT0b3o+uqc/ZhFkMVd6Z+d/7/3kH/7xp//0s3feHfzzz//lX3/x3vu/fKlELkP2dShiIV8FVLGYp+xrzXXMXmWS0SSI2TfB9QT539wwqbhIz/UiY5cJnaV8ykOqoenqvT/6AZvxtEh4yjM6Y2UxfhQm5aBujpXGSfB0Vl6keRKApL2YTfXlYPPuk71kc3A75zEjm3f7O5skEgNC+JRs+pkUwVb2oG4iBMF3o/EmECxWrGn70LRN+UBEA5+lUXvAwZaiC49sJnvj/9wkNI3IZra3s727+aCCrqY8uHpvuLO9Yz7EfRjXD8ON+nNy9b731I9EmCcs1WFMlboY72T6sqBS8zAGiX6uWEbDaxB/AY8pTZi6LMx2l+Q+tERkKiT8pZqY1naPgiZKLZIAkAnVc2XzsLGPd5Hr6a8vC55muWZpWA00zWOiBUHdkYhLFup4AQ80lBzmSsI5lTTUoJfOKEyzO+hPzudckSRXmgSM6DmDrZZA1DAUDO2xoBGLBvBpi5jSNFwE4s6rV4PfIFV79Qq8gdIJlQsZebdULQDigaEFYFeJB/oXU+9OgSxWEjK4315ldMMz2MxMacnDa9XdbJbOQP/zyyKgAYu7PBrHl8XdogRxE5FkYHhfUpTBFBn4dwuRoUkXN7tliwpzecPaDbshi+N2gxSaaoScsRh29xwmV8AJKGDesH9AkS0xJVRKcasewDcjPCU0DIWMYIcYueV6TnA+oDRJvhQRk+nA/1qxT+P4/FbgcKCbM5htqpfzhZ2O2NSPcY1KL2JW+Li15rGseLru6nJbOppJms15eOdJkMN/z1Dlkr3JwU4qA/3W52kY5xGrkepXZGsBy70jNAKdw0lCVdYOoghEHJVdK9D8+vce/huFUfX9RktWTwPpmAcSrKDAVuU2V/vmETWnGYPvkMahR2iuBQxMPZIJxVEN5sD72BP6F4b/0bZZMdkjxWjf298D66Oxnv+qXALN1sDcja3vkYuQSzi8HkHHkOQJwR0Bh/FxmHigsxS0A+vaw4M+MJ/7f81ncN9ssMYDpjIWok8lS58y8FN2G4okAa9VgGFpVRZmp4NpYaiyi8A9S8uLR5eFXwzHoLPhrv/4rf94+Mi3oTyqRcmkgGeLS6dghEsAiwsf98QGheKHEMfHL8qinnFQHJeOBKrLi/FlAxmOHcykXHEn7gzSmeSzuTbm0Z6N0mB4+GRiRP3xJeWKgS8qtj8GXRebvgL3l1Unwgjzvc2yXNNnB7pcwL9L8w86NyNDJ2d/Fe6ONcEuBI9mDKDTFuB+B5HF6PhgWTj+5giGgYNpGskczhiTg154ngILIzk0zWkKZ6BIy9L3fqD3agfhcBUn9oKmFR/+X60MZ8rTbWflEcTdlqCnNj/m0zb/yOkfZqIx86cToHoQgQWBSQVC10gIcAXZB9MrH37w0ejXmf7oSbFfGk+Mhy0RqSBPADHatyQjBwQ3KtOUxxaGZbw0ypszGq1VrsjqCYpstV3QVvqpiHnCtbLwt2xmjoK9VDbl0ymIetyY9FIc8GC9lZkW/hGYGqZUrRn5j11p7M2PELbndk0WXz17CmYIvQ/gfw8TOHMTqR+CsXAIjpurTpu2n1oagJoWSjvqbbMjw6/0ljHJRQTeHzNVCAGazYTkGA6YDrcBxLZn22Q4Bic+esTAWw934XFsnh7Bk/Gg1kzmYI0sqnymojfs4vKD3wzH3nB3/2ExfFT6kiktJHO8Z0Izoy8TUisfemXcVxcIW4z2LrrgU0hJtobjB/3g8mL3stUTpW4Nd23wFzHjKuaV3OVh+gI27hoSh9H2LngrH0T0DKNl7VWqHdbSnrTOM8gQjOCYolURnAN4Qny2sAGbN6KQsPifxrPmpCJhewOG1q3eqDyA6MXeXK3A09vI8S0xC9agg370OuEHhz3o6TrhhweI/ksDPfSD3HgGtVIMST4kEKHmN5BsavDN1wqT50RuX1MwxYQitsoHGMFqgUDOyNMpkwxzRJnHYO2IgfTIYEQKmeNcKMwmIQGCESCPCGnGNY0RRxUJeAoZFBFweiiYsnUAeIKOSgtrM56b5udfnuDZs2Pud6UxjAgcKnhUqFtiHsF66gw2/Q4WmEN+y9KbbZhWO5r2fO5jKQcF03aI+S2mrDhCPYCVClRDV9HQADZ7go+ZHdRXXcbTivHi1Go/FBLqAeB8enRksZ7dQSTDjOvZKzuvYWjUxy/OMWCDatpWhNN25uighGyBhHTWAfO0+MRSEKhmhQAlugicXiMjFdqF4JpXkKnZCRcEq19hmNkSF3PQHivoGwoMFbO91ZFyPOULtM+oleAuG2znBC40DkyS9n3FT81dV/7YAcnkgevEtrLENYLbiHJpxUmm+CxdJaX7lWfezbT/neBpf4uT4JpzyFIFBwrl1LvD02I1AOQUdYVs5fBS9nQFL42MNX1MWd/fq2Kt6acYHNqov2PNW9MTnVMTQAxlZ1K5bkMq0sJ8LpKmnEHC8Rw0njWhEqyraukL2YoldeyLofYMr6GONNFP1hRWtlbU1p1KSIBoo/g5pEGtmuNRpk3NMRdSJxz8QgUIZSjtYsKKTFhF/E1GiTv1iBmpstZAUIkm8+RhcfF2a/fBpZPIIKIsDPDt6KGPKfbb0t4LpbLl3mUUXB6EKAUBCWpRKVsN4BwGVs+oa3bELFTrIlomNB/DupxU8Ru8Nizrgx0WFdmHMeLN02szlp2lKJU2YgzVgzBC8KFXxuewM625VGQfxogxT71y5gtwfFVZXL4uHpbtDejNMGnclMinDrtxukHxlc1N29xjh5snDfdpVT9RkjBNb6jkNIDEEHOVU498ReD72PYlcbvOO3TPWVhV/2/rU2hZxMvVqYfj+tI9858+u8sagKEsxEEHcdCDmCRRA0DC1pcDuCouqjrbOSGHSdxAkbD98aokNO67tm9stZFBLzLoQcoVclXLAVL2ILXMlzk/IAxl+34aqxakIt0LF6UbPwqwcOlD768B9nnYa541GwtCsMG5QJlOYgppblWa1UA+hcoUvMnj1ULmLMWm3aYJfxnApkd2hY2nfylzt5Fp2i2xkTBCbQ0qxaReimhtQsUwMuyN+P5RsTrzC5i9Y3hnrVwJCYvfZrtc3dI2BERdu5BxvxOdtjWPtzod+NTxT5B0JBTgNywEXRrCufw6pq0aviIdzPO0g0HSSQ70CZQG7fygarBXAJG5C1y1OBc80bM3xqTQLvdg68vuQYO+gClLLP6d2mLG2zcThrSNLAuNZdxiGtF3MJS5gFmlRm+cs582inWCUABlR9Bcv81cz8E7gNAFqA5AmQzASWhonOI08AGwM0Z2UE63hDltWcwpc/nPWxnZcychOzuRImyCZUX2YUyw7KB6gyZwZpImrfJh1WIhs1mz/5lzHZHNznjKOoirFWFYzh23gNq3GbYi7UAaNRJTJ8ik0ZeU436nUTOWabKRNa4DsWYTHdOEtaZT0T1z/jG4MwiZpy1HgxH0qk45eqATF+pc2QdCtNKVA+fEJ3ljNrxvkAN3ECPTdgnBoieGYqujcim+s+5qAWlaHai4Xf0sgmIxQfNvWTRzzYKr0yYxw8hV0faCIK95lt60llQ3OGrtoCrSuTbNMRkI+Ey9CeGxvuyufuxlEQE25NZ54lzomLpieSF8sfIdurwsqquhpSgI6pqnucgVSWim7JCPxcX3y+qWH3hHYDlHey4mFy5aYpw5PRzZs+pZnBZXHSHrFrRmPd1p26Lsm4RmRHf2P3baR6+WxzyUxRHY+Ctb20fKhtTDvXJM8WhiI51TeaQcTD35ifuD3OeLzKQfFWLLn7zGSD3Vi/KBk4Hb2IP12FMbi25mPfzsHGBtj4N07+IM56AH6Xif48iwWufM0I687pE97z2w59aJ7UMdvmycHTzb3LM2+8zlHx+3+MdO+faStgqSl67zkc09v/mVAMs701Td5G5JkafRA2LqdmsH+CyFnLG1B3WDvVPJsjzB6Jk4Of5ZoudtgHZ+PqCRbtYYQe5qLSGqfh3DdK8Fe11AtVh1tkecdTVc0w7qRNq4usVCnk86ZV9F2kUT74Q0s81zRnhCZ9XVfJ5ybXnCCY1i2vrlZOJDMnPDClpG8erRyV5WvTBh4hp6fVv1+HY9+rk5Xktqdd7svFVGTPKbsnWHam96R09XxYeO41DzBqIcTeObSVqq2NjjcPy6uc47K93LAcycsZwajq+Kcen5kYAE0UMKMvqeINuNsI6GGI/Nu2nm6gop/BXRl/hke7Ru8mJNK6DpUmmQS3yw9tfeIDhp5Jw412hTHlRlb3Ott7MDyV77p5T6ut2PqZyx+/4Rfg3Iu/hGXFF0bqQJXl9SLbJu9+WrAV0o7F7nrYhI3KYm7CGr82IESrQ6Q8nk3l7ehNXl69/HevLsz16Noxt0avBVqxGpiZMHTFDxVloJxlw1O6frsLEG557sRLLWHZShehBoMvWsqg5l/cKQHWVZy10h4fJbogx6jaSzPGjOMxJ2eOKtUta9/csDY+V44sxQIMFEoaLvkvOQtqpJJJy7q6QrDTBLaavJP2UhGBe+7Dd/Uf8Y6r9aRZ9X6yCfrSCfrYMcriDOOuV1M3F4di5vUbXmzc+iR7FBbK5xKr4hHABkyi0EUrbBJgvTs3rPqiXK3eVkgXNoA5Gu7mYRiHf+GSj9k62xNxo/ePK6eDsqV7+IZcFwvHyH4sPf7O6QTHv4b/8Taq65qHwCvYuyvLhs3qroviVYv7KL3jJWGt/gg2KFEHwJIQf+3sQzZzagiofmkO75MgHd83hRca7Z4lbIqOYFU8XM+yGGh0pgKqQZ85A0komai1t8gRQGxR+n1J65C6s7wAyXAveqCzuPRMIjfOphBE898xayR6bcIyKCNpkjZSQQc6c5cGvN1otLhoI9uHpvOLbfMnYfXu5uj3e2x1/tDp8c1G8gv7Px7xv/sbG1Md74r40nG59vnGx8vRHee3jv5N5v7/1u9IfRn0b/PfqfCvqTe3Wff9vofEb/9//2g5Rz</latexit>• Locations +memory states (e.g. “x = 13”)

Hasuo (NII, Tokyo)

Our Problem
Given
a pCFG. n variables, state set L<latexit sha1_base64="TrDp5ZMxYMgmbAOoFevsrtRlvmA=">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</latexit>(Lebesgue measurable) target region<latexit sha1_base64="47CoD7UoRMA25SBMb9pnTyAUxXQ=">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</latexit>

initial configuration
AnswerPr(ReachC)
10

Hasuo (NII, Tokyo)

The Martingale-Based Approach
Use parametric certificates, i.e. functions<latexit 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va3tb+v/WPzT5tPNg82D2voT95o+vxhrfPZ/Ppf+iJugQ==</latexit>

to give approximate answers. Template-based synthesis
Qualitative questions
Pr(ReachC) =? 1 (almost sure reachability)
Pr(ReachC) >=? α (threshold reachability)
Quantitative questions
Pr(ReachC) >= ?? (lowerbound, “verification”)
Pr(ReachC) <= ?? (upperbound, “refutation”)
Exp(StepsToReachC), upperbound, lowerbound, …
“Concentration” [Chaterjee+, POPL’16]
Find B s.t. (x > B ➜ Pr(StepsToReachC > x) < ae-bx)
11

Hasuo (NII, Tokyo)

Disclaimer
For the coming definitions, the setting is:tr

A Markov chain (S, S ! DS), and C ✓ S<latexit 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2vHa1+uRWt/Wfvb2t/X/rG1vXW2db51UUPffafp87u1zm8r+SchYHLy</latexit>• a pCFG:
finite locations + memory states• a MC:
all the information as explicit states➜ infinite states
12

Hasuo (NII, Tokyo)

Additive Ranking Supermartingale[McIver+ PSSE’04] [Chakarov+ CAV’13]

Def.
tr
Let (S, S ! DS) be a MC,⌘ : S ! R be a function.

Overapprox.
Exp(Steps to C)

Ranking supermartingale

X⌘ : S ! R is defined by
X(X⌘)(s) =tr(s)(s0 ) · ⌘(s0 )[0, +∞)valued

2

s0

=

X
s0

<latexit 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ytB/DAb/DB6y0hp/IIPihVC8COEDPj705GxWZ8qHhgj3fdkDGfPo2XJuWXLeyHDiufPFDPfhxgeHgJTAU3ZCEkjmaiFuMcPSGFQfDml9s1dWNUBZrgSuF9e2I1IKEaEz0YYwZMRvrqGthkfERFCm8yQMhKIudMcuLVm68MlQ8Ee3Hw8nNhfGbsPb/fGk93x5Ou94YuD6gvkn278YuOXG1sbk41fb7zY+GrjdOObjeDRPzx68ejVo38a//P4X8f/Nv73Evroo6rP3290fsb/8V9mKO+u</latexit>

Pr(s ! s0 ) · ⌘(s0 )= Exp(⌘(s0 ) | s ! s0 ).

0

1
2

1
2

𝔼

the value of 𝑓
decreases at least 1
at the next state1

The system eventually visits

almost surelyDef.
tr
Let (S, S ! DS) be a MC, C ✓ S.
⌘: S ! R

0 is a ranking supermartingale for C if

• 8s 2 S \ C.
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ODTnW2SqhH+efoJ1ddcVDyD3nlRXFw2X1V0vxLEr7LBD6O3DKXCL/igWCEEP0LIgL87HWmbdanknjbSXUdEcPY8XJSca7a4S4Rf8dxAMv19iObhITDp0ZSNkNSSiZwnd/gBKQyKL6fkrr4LqzrADJcCd8sLuxHxkxHhwQgjeDzCV9fQFvARSXxoExlSWgLRd5oDu9ZsfbikKdiDq3eHE/MrY/vh9c54sj2efLkzfLZXfYH807X/Xfu/tY21ydpv156tfbF2vPbVmvfgfx7sPjh48Pn4D+M/jf88/ksJffBO1efXa53f+K9/B2ZO7Rc=</latexit>⌘(s)(X⌘)(s) + 1.

• ⌘(s) =13 0 implies s 2 C

Hasuo (NII, Tokyo)

Additive Ranking Supermartingale[McIver+ PSSE’04] [Chakarov+ CAV’13]

Overapprox.
Exp(Steps to C)Def.
tr
Let (S, S ! DS) be a MC,⌘ : S ! R be a function.

X⌘ : S ! R is defined by Def.
tr
X
Let
(S,0 S ! DS) be a MC, C ✓ S.
0
(X⌘)(s) =tr(s)(s ) · ⌘(s )⌘: S ! R

s0

=

X
s0

<latexit sha1_base64="Y3L9txaOlYwisJtQZ+cZ5JUx8Js=">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</latexit>

0 is a ranking supermartingale for C if

Pr(s ! s0 ) · ⌘(s0 )• 8s 2 S \ C.

⌘(s)

(X⌘)(s) + 1.

= Exp(⌘(s0 ) | s ! s0 ).
<latexit sha1_base64="Bx09C5xaCHJpiLtCsukwLc/M+Uc=">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</latexit>• ⌘(s) = 0 implies s 2 C

Thm.
Let ⌘ be a ranking supermartingale for C. Then:• Pr(ReachC ) = 1 (almost sure reachability)<latexit sha1_base64="W9U6OJV/61xiW/UQ/pJfrh6dqaA=">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
sha1_base64="W9U6OJV/61xiW/UQ/pJfrh6dqaA=">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Exp(Steps from s to C)  ⌘(s)
14

Hasuo (NII, Tokyo)

Towards Automated Reachability Analysis• a pCFG:
finite locations + memory states• a MC:
all the information as explicit states➜ infinite states
15

Hasuo (NII, Tokyo)

The Martingale-Based Approach
Use parametric certificates, i.e. functions

Def.
tr
Let (S, S ! DS) be a MC, C ✓ S.
⌘: S ! R

0 is a ranking supermartingale for C if

• 8s 2 S \ C.

to give approximate answers.
<latexit 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va3tb+v/WPzT5tPNg82D2voT95o+vxhrfPZ/Ppf+iJugQ==</latexit><latexit 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16

⌘(s)

(X⌘)(s) + 1.

• ⌘(s) = 0 implies s 2 C

Hasuo (NII, Tokyo)

The Martingale-Based Approach
Use parametric certificates, i.e. functions

Def.
tr
Let (S, S ! DS) be a MC, C ✓ S.
⌘: S ! R

0 is a ranking supermartingale for C if

• 8s 2 S \ C.

to give approximate answers.
<latexit 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va3tb+v/WPzT5tPNg82D2voT95o+vxhrfPZ/Ppf+iJugQ==</latexit><latexit sha1_base64="Bx09C5xaCHJpiLtCsukwLc/M+Uc=">AAAvDXiczVpJc+TGlaZ6xlt5GWl8mMNcUlPsMNkqlljsUFihFjtaLDalDrPZFEm1GEGwqQSQqEoRGzMTJMto/Ab/FV98c/jq3+B/4/cSKCyZKEke++A6kHj5vny5vTUBNw25VNvbf3vnwX/8549+/JOf/mzw81/88lf/9e57//1aJpnw2FdeEibi3KWShTxmXymuQnaeCkYjN2Rfu9dT5H99y4TkSXymFim7jOgs5gH3qIKmq/fe+aPjshmP84jHPKUzVuQfe1ExcBS7V26QO1nsM4HS830WjIuCOM7gkCmyvnE6IqfOveCzuaJCJHe5E1E1F1GuRFHoZ4+G+X5xurlOXEYoeTkdDdanjsxcyRS7IafrYzJ4uO4wRR1cSQzyVKJ7um5+UlTdgiz2cLJjGHng3MqUeiwff8RgloPVna9yZ8bINgjhEoQ4LErnuaDxNY9nRGYpExEVCggasoIEiSDrU8AGg2o/uGIR/z0rBsTBR7LuAIaGIZEOx7FgCbBlmSTTsXOTUZ/gTDbkJo66sZzGeYGtm9D8wQQXW8sqsbvbMGIEx8wkWddyp+sDh8V+Mzr8dEN9PIOrd4fb4239I/bDpHoYrlW/46v3RvuOn3hZxGLlhVTKi8l2qi5zXL4Hix84mWSwqdcg/gIeYxoxeZlr3SrIQ2jx9f4ESayIbm33yGkk5SJyAYmLliYPG/t4F5kKPr7MeZxmisVeOVCQhUQlBBWV+FwwT4ULeKCe4DBX4s2poJ4Cde6MAsp0D/3J2RyOOsqkQr1Rc0YCLoCoYCgY2sOE+szX+9oSEdDYW7jJ/ahaDf5HCxhVKxgNpAJ9WQh/dEflAiAjsCoXjCgapSJJgtG91suCgEa3V+nf8hQ2M5VKcO9adjebxTMw8fll7lKXhV0eqNplfr8oQNw0iVIespcUZYCqDJz7RZKiSeS3O0WL8jJxy9oNOx4Lw3aDSBRVCDllIezuGUwuB3PPYd6wf0CRjSQg2pzlJvxnBJSSel4ifNghRu64mhOcDxyaIC8T8A3xwPlKss/C8OwuweHgbE5htrFazhd22meBE+IapVqELNfORT8WJU9VXW1u64xmgqZz7t2PBMhB44ApC3aTgZ6UCvoNWJAXZj6rkPI3ZGMBy70n1Icz3xxos6y8Ye4moV90tUDx69+P8M+W55f/b5Rg1TSQDrkrQAtybJV2c7lvIyLnNGXwH/yfNyI0UwkMTEckTSTHYwCvgx4WekL/XPM/HOsVk12Sbz0dPd0F7aOhmv+mWAL11sDcta7vkguPCzDeEUHHEGURwR3Z3R5/5EUjOLMYTgfWtYuGPtC/h//Mb/BQb7BCA5Mp8zCAkKVPGTgxu/OSKKLgp0CxlCzKUADhQ1NFF4F7FhcXjy9zJx9O4MyGO86Tt86T4WPHhHK/qKMKPBtcGoASLgEszB3cExPkJd+HODp6VeT50mUfFZYEqoqLyWUDGU4szLSouVN7BvGsiZLt2UgFiodPA9L8HEG5ZOCLMMbdF/m6I8H9paVFaGHOaL0oVvTZhi4X8OdS/4HOzcjQydpfibtjTLALQdMMAXTSAjzsINIQHR8sC8df34JhwDB1I5mDjTEx6IU3iYWTzmkMNpDHkDeMvqd3vYOYXBybCwpKPvy9qhUn4PHYWrkPqVVL0L7JD3nQ5h9a/b00adR8fwpUD8I1IDApN1EVEgJcTp6C6hWP3v9w6+NUffgsf1poT4zGFiVxQp4BYuupIRk5ILg5MkV5aGBYygt9eHNG/ZWHm6TVBJO03i5oK5w4CXnElTTwd2ymTcFcKgt4EICoJ41KL8UBD9ZbqmnuHIKqhSxQrRk5T2xp7OYHCNu1u0aLL5/vgxpC7z3428MEzlxH6kegLByC43rdad30U0sFkEEulXW8bbav+eW5QWbJEx+8P6blEAIUmyWCYzhgyhsDiI1nYzKcgBPfeszAWw934HGinx7Dk/agxkzmoI3ML32mpLfs4vL9T4eT0XDn6aN8+LhwBJMqEczynhFN9XnpkFr60CvtvrpA2GLU96QLPoGUZGM42ewHFxc7l62eKHVjuGOCfxcyLkNeyl0a0+9g464hcdga74C3ckBEzzBKVF6l3GGsKQxAlkKGoAWHFLWK4BzAE+KzgXXZvBGFhMH/LJw1loqE6Q0Yare8WZYuVzU4uPMt3xIydwXa7UevEr530IMOVgk/2EP0/zfQQz/IjWdQGIaQ5EMCATXXLSSbCnzztcTkORLjawqqGFHElvkAI1gtEMgZeRwwwTBHFFkI2o4YSI80Jokhc5wnErNJSIBgBMgjPJpyRUPEUUlcHkMGRRKwHgqqbBgAFEnozRJjM17o5hcvj9H2zJj7baEVwweHCh4V6paQ+7CeKoONv4UFZpDfsvh2DNNqR9OeH1htKKFgGnuY32LKiiNUAxipQDl0GQ01YL0n+OjZQX3VZeyXjFcnRvuBLj2B89nhocF6fg+RDDOu5+dmXsNQqY9enWHAhqNpaxFO25qjhUpEC5QIax0wT4NPjAOCo6kRcIg2AqfXyIgTZUNwzTWkLMJtEKy+xjC9JTZmrz2W2zcUKCpme7VJWZ7yFeqn30pwlw2mcwIXGro6Sfuu4qfirip/zICk88BVYltZ4grBbUSx1OIolXwW10np09Iz76TK+TbhcX+LleBqO2SxBINCOdXu8DivB4CcoqqQjRxeiJ6u4KWRsaKPLuv7e5WsFf0kA6P1+ztWvBU90Tk1AURTZiaVqTakJA3MF0nUlDNIWJ6DhrMmVIJ2lS19IVuyqIp9IdSe3jXUkTr6iYrCytaI2qpTCSUgWh/8HNKgVs3xOFW65pgnQkUc/EIJ8IQnzGLCiExYRfxLRgk79YgeqdRWN6ECVebZo/zi7cbO5qWVyCCiyDXw7dYjB1Pst4W5F1Kmy71LKbg8CFESAhLUokK0GsA5DIyeflftiF6oUrm/TGg+gnVZqeLXcx5iKqMN28tLsg+jxeunN3osM0uRMm7EaKoHoYXgQ6+ML2BnWnMpyT6MFqOfeuXMF+D4yrK4eJM/Ktob0Jth0rApkU8sduN03fxLkxu3uUcWN4sa7n5ZP1ESMUVvqeDUhcQQc5WTEfmSwP8j05eE7TrvwLYzr6z+31ZWaGjE69rqwVxf2zb/2fP7tAFoykDsdRB7PYhp5DcAJMzzsgBX+UVZZ1sWchCFDRQJ0x/XJaF235V+Y6uJdHuRbg9S1Mi6lgOk6EEqkS1zfkBoyvT9NJQtSEnaFy5SNX4UYN7Shz5cAezzsNc8bTYWhGCDdYESTEMKaW5ZmlVAHkBlCt7kSb2QOYuxaadpYjhvvPMyK2y0/qXMnUambjfE+okWap6glEyopYjWJpQMLcPciO8eFaszJ4fZW4p32sqVkDD4bbbNVa3TxrdGlQuZ9DvRoH3yeKvTgQeWf4KkI6IAv2UenKUmrMuvI9qq4UvSwryIOxgkreRAHUNp0M4PygZzBRCZu8C6xbrg8Z/faJVCvdyFrS+6hgZ9AVMUWPxbtcWMt28mNGkqWeppzbjDNKLPMKS+gKlToxvL9uPmYK0g5ELZ4TbXbzPbc/AOwLMBsgOQOgOwEhoaxjgNfFi+7AM53RLmpKUxJ8zmv2hlZC+shOz0WCReEyxLsg+jg2UH1Rs0gTMTNGqVD3WLgUxnzf6n1nVEOjvlMesgrmpCs6w77gRq32bYkjQDqd9IjK0gE/svKcf9jv1mLN1kIitcB2LMxj+iEWtNp6R75vxDcKcQMk9ajgYj6FWVcvRApzbUurJ3k6SVruxZFh9ljdrwvkH27EG0TNMluIueGIqt1pGL5FvjrhaQutWCJnf1axEUiwmac8f8ma0WXJ40iRlGrpI2FwR5zfP4trWkqsE61g6qJK1r0wyTAZfP5I0Hj9Vld/myl/n4Zh5y6yyyLnR0XbG8EL6ofYcqLvPyamgpCoK64nGWZJJENJVmyMfi4rtldcsPvCMwnKM5F50L5y0x1pwebZmz6lmcSq46QlYtaMV6utM2RZk3Cc2I9ux/6LQPz5dm7on8EHT83DztQ2lCquHOLVU8nJpIyyoPpYWpJj+1X8h9sUh1+lEiNpzpG4zUgVoUm1YGbmL3VmNPTCy6mdXw0zOAtT0O0r2L05y9HqTlfY58zWrZmaYteV2TPes12DPDYvtQB68bZwfPJve0zT61+UdHLf6RVb69pq2C5LXtfERzz6/fEmB5p5vKm9wNkWSxv0l03W7sAJ/FkDO29qBqMHcqWpYnGD0jK8c/jdS8DVDW6wPqq2aNPuSuxhL88u0Ypnst2JscqsWysznirHvCFW2hjoWJq1oM5Nm0U/aVpFk08U5I09s8Z4RHdFZezWcxV4YnnFI/pK03J1MHkplbltPCD+tHK3upe2HCxBX0+qbs8c1q9AttXkuqtjczbxU+E/y2aN2hmpveOaer/APLcch5A5HWSeOXSUrIUOvjcPKmuc47LezLAcycsZwaTq7ySTFy/AQSxBFSkNH3BNluhLVOiPEQvxwpr66QwreIjsAn06N1kxdjWi6Nl4cGucT7K9/2uu5xI+fYukYLuFuWvc213vY2JHvtVynVdbsTUjFjD51D/DcgPwPOep53bqQJXl9SlaTd7stPA7pQ2L3OVxF+chfrsIeszocRKNHoDCWTfXt565WXr/8e68nSf3g11tmgU4N/1TEiNbXygCkevJFWgjKXzZZ1HTTaYN2THQvWuoPSVA8CVaaaVdmhqD4YMqMsa7krJGx+S5RGr5B0mrmNPSNhhifeKmXt27/M1VqOFqeHAgk6CuV9l5wHtFVNImHdXUVdaYBZSqsnv888UC782G/+qnoZ6pzX0ed8FeTzGvL5KshBDbHWKa6bicOzdXmLR6u//Mx7DtYN9TVOydeEBYBMuYVAylTYaKF7lt9ZtUTZuxwtcA5tINLl3SwC8c4/hUP/ZGMy2ppsPnuTv90q6jdiqTucLL+h+ODTnW2SqhH+efoJ1ddcVDyD3nlRXFw2X1V0vxLEr7LBD6O3DKXCL/igWCEEP0LIgL87HWmbdanknjbSXUdEcPY8XJSca7a4S4Rf8dxAMv19iObhITDp0ZSNkNSSiZwnd/gBKQyKL6fkrr4LqzrADJcCd8sLuxHxkxHhwQgjeDzCV9fQFvARSXxoExlSWgLRd5oDu9ZsfbikKdiDq3eHE/MrY/vh9c54sj2efLkzfLZXfYH807X/Xfu/tY21ydpv156tfbF2vPbVmvfgfx7sPjh48Pn4D+M/jf88/ksJffBO1efXa53f+K9/B2ZO7Rc=</latexit>⌘(s)(X⌘)(s) + 1.

• ⌘(s) = 0 implies s 2 C

Template-based synthesis
Fix a template, e.g. “each fl is a linear function”(Coefficients are parameters)
Generate constraints for the parameters, so thatis a supermartingale
Solve the constraints, byLP (linear template, Farkas’ Lemma),
SDP (polynomial template, Positivstellensatz), …
<latexit sha1_base64="ZvzVp2GFqRo/MXbV5YAPAFQ/a3Y=">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</latexit>

16

Hasuo (NII, Tokyo)

… 共通性？
対比の項目は同じ

制約充足問題翻訳
ソフトウェア
検証問題

spec ↦ automaton
model ↦ prop. formula
…

SAT
graph
reachability
…

最適化問題

制御問題

翻訳
17

LMI
SDP
mixed integerstochastic optimization…

… 共通性？
対比の項目は同じ• martingales forprob. programs

制約充足問題翻訳
ソフトウェア
検証問題

spec ↦ automaton
model ↦ prop. formula…

• Lyapunov
functions

制御問題

SAT
graph
reachability
…

最適化問題

翻訳
17

LMI
SDP
mixed integerstochastic optimization…

Outline
制御理論 vs 形式手法・ソフトウェア科学
数学的相似・相互乗り入れ
例1（相似）： 確率的プログラムのマルチンゲールによる解析[Chakarov & Sankaranarayanan, CAVʼ13][Takisaka, Oyabu, Urabe & Hasuo, ATVAʼ18]

例2（乗り入れ）： 近似双模倣による離散化[Girard & Pappas, IEEE TAC ʼ07]

[Kido, Sedwards & Hasuo, IFAC ADHS ʼ18]

実システム応用に向けて
機械学習・AIとの協働
ソフトウェア工学，現実的解決
ERATO MMSD プロジェクトの取り組み
18

Approximate Bisimulation[Girard & Pappas, IEEE Transactions on Automatic Control’07]

Ti = (Qi , L, !, O, Hi ) for i = 1, 2i(O equipped with a metric d)Def.
R" ✓ Q1 ⇥ Q2 is an εapproximate bisimulation
relation between T1 and T2 if
for all (q1 , q2 ) 2 R" ,

T1

T2

R"
l

l
R"

• d(H1 (q1 ), H2 (q2 ))  ✏
l

l
0
0
0
0
8q
!
q
,
9q
!
q
,
s.t.
(q
,
q
•
1
2
1
2
1 2 ) 2 R"
1
2
l
l
0
0
0
0
8q
!
q
,
9q
!
q
,
s.t.
(q
,
q
•
2
1
2
1
1 2 ) 2 R"
2
1
Hasuo (NII, JP) 19

Approximate Bisimulation[Girard & Pappas, IEEE Transactions on Automatic Control’07]

Ti = (Qi , L, !, O, Hi ) for i = 1, 2i(O equipped with a metric d)Def.
R" ✓ Q1 ⇥ Q2 is an εapproximate bisimulation
relation between T1 and T2 if
for all (q1 , q2 ) 2 R" ,

T1

T2

R"
l

l
R"

• d(H1 (q1 ), H2 (q2 ))  ✏
l

l
0
0
0
0
8q
!
q
,
9q
!
q
,
s.t.
(q
,
q
•
1
2
1
2
1 2 ) 2 R"
1
2
l
l
0
0
0
0
8q
!
q
,
9q
!
q
,
s.t.
(q
,
q
•
2
1
2
1
1 2 ) 2 R"
2
1

Once in a relationship,
always henceforthHasuo (NII, JP) 19

Approximate Bisimulation[Girard & Pappas, IEEE Transactions on Automatic Control’07]

Ti = (Qi , L, !, O, Hi ) for i = 1, 2i(O equipped with a metric d)Def.
R" ✓ Q1 ⇥ Q2 is an εapproximate bisimulation
relation between T1 and T2 if
for all (q1 , q2 ) 2 R" ,

T1

T2

R"
l

l
R"

• d(H1 (q1 ), H2 (q2 ))  ✏
l

l
0
0
0
0
8q
!
q
,
9q
!
q
,
s.t.
(q
,
q
•
1
2
1
2
1 2 ) 2 R"
1
2
l
l
0
0
0
0
8q
!
q
,
9q
!
q
,
s.t.
(q
,
q
•
2
1
2
1
1 2 ) 2 R"
2
1

Once in a relationship,
always henceforthHasuo (NII, JP) 19

Incremental Stability (δ-GUAS)
Incrementally Globally Uniformly Asymptotically StableDef.
A dynamics is said to be -GAS if
9 KL function s.t.
x(x, t)x(y, t)



kx

yk, t

where x(x, ) is the trajectory starting at x.
<latexit sha1_base64="+Jgywg36ygy/4moXYL2t6vqxoYY=">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</latexit>

Hasuo (NII, JP) 20

Incremental Stability (δ-GUAS)
Incrementally Globally Uniformly Asymptotically StableDef.
A dynamics is said to be -GAS if
9 KL function s.t.
x(x, t)x(y, t)



kx

GAS

yk, t

where x(x, ) is the trajectory starting at x.
<latexit sha1_base64="+Jgywg36ygy/4moXYL2t6vqxoYY=">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</latexit>

t
δ-GAS

t

Hasuo (NII, JP) 20

• increasing in kx

ykIncremental Stability (δ-GUAS)– 0 if kx

yk = 0– ! 1 if kx• decreasing in t
<latexit sha1_base64="W1f9wnRLwEWTp6zmbeqHVz00R7M=">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</latexit>

yk ! 1

Incrementally Globally Uniformly Asymptotically Stable– ! 0 if t ! 1

Def.
A dynamics is said to be -GAS if
9 KL function s.t.
x(x, t)x(y, t)



kx

GAS

yk, t

where x(x, ) is the trajectory starting at x.
<latexit sha1_base64="+Jgywg36ygy/4moXYL2t6vqxoYY=">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</latexit>

t
δ-GAS

t

Hasuo (NII, JP) 20

• increasing in kx

ykIncremental Stability (δ-GUAS)– 0 if kx

yk = 0– ! 1 if kx• decreasing in t
<latexit sha1_base64="W1f9wnRLwEWTp6zmbeqHVz00R7M=">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</latexit>

yk ! 1

Incrementally Globally Uniformly Asymptotically Stable– ! 0 if t ! 1

Def.
A dynamics is said to be -GAS if
9 KL function s.t.
x(x, t)x(y, t)



kx

GAS

yk, t

where x(x, ) is the trajectory starting at x.
<latexit sha1_base64="+Jgywg36ygy/4moXYL2t6vqxoYY=">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</latexit>Def.
For a dynamics ẋ = f (x) in Rn , a smooth function
V : Rn ⇥ Rn ! R+ is a common -GAS Lyapunov function if
there exist K1 functions ↵, ↵ and  > 0 such that↵(kx
@V<latexit sha1_base64="pUU+6u+ntJzXpLhbmdZ1Cl9XY9U=">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</latexit>@xyk)  V (x, y)  ↵(kx

(x, y)f (x) +@V@y(x, y)f (y) 

t
δ-GAS

yk)
V (x, y)t

Hasuo (NII, JP) 20

• increasing in kx

ykIncremental Stability (δ-GUAS)– 0 if kx

yk = 0– ! 1 if kx• decreasing in t
<latexit sha1_base64="W1f9wnRLwEWTp6zmbeqHVz00R7M=">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</latexit>

yk ! 1

Incrementally Globally Uniformly Asymptotically Stable– ! 0 if t ! 1

Def.
A dynamics is said to be -GAS if
9 KL function s.t.
x(x, t)V(x,y) is more or less|| x - y ||…

x(y, t)



kx

GAS

yk, t

where x(x, ) is the trajectory starting at x.

<latexit sha1_base64="+Jgywg36ygy/4moXYL2t6vqxoYY=">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</latexit>Def.
For a dynamics ẋ = f (x) in Rn , a smooth function
V : Rn ⇥ Rn ! R+ is a common -GAS Lyapunov function if
there exist K1 functions ↵, ↵ and  > 0 such that↵(kx
that decreases
along thedynamics<latexit sha1_base64="pUU+6u+ntJzXpLhbmdZ1Cl9XY9U=">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</latexit>@V@xyk)  V (x, y)  ↵(kx

(x, y)f (x) +@V@y(x, y)f (y) 

t
δ-GAS

yk)
V (x, y)t

Hasuo (NII, JP) 20

• increasing in kx

ykIncremental Stability (δ-GUAS)– 0 if kx

yk = 0– ! 1 if kx• decreasing in t
<latexit sha1_base64="W1f9wnRLwEWTp6zmbeqHVz00R7M=">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</latexit>

yk ! 1

Incrementally Globally Uniformly Asymptotically Stable– ! 0 if t ! 1

Def.
A dynamics is said to be -GAS if
9 KL function s.t.
x(x, t)V(x,y) is more or less|| x - y ||…

x(y, t)



kx

GAS

yk, t

where x(x, ) is the trajectory starting at x.

<latexit sha1_base64="+Jgywg36ygy/4moXYL2t6vqxoYY=">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</latexit>Def.
For a dynamics ẋ = f (x) in Rn , a smooth function
V : Rn ⇥ Rn ! R+ is a common -GAS Lyapunov function if
there exist K1 functions ↵, ↵ and  > 0 such that↵(kx
that decreases
along thedynamics<latexit sha1_base64="pUU+6u+ntJzXpLhbmdZ1Cl9XY9U=">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</latexit>@V@xyk)  V (x, y)  ↵(kx

(x, y)f (x) +@V@y(x, y)f (y) 

t
δ-GAS

yk)
V (x, y)Thm.
There is a -GAS Lyapunov function=) incrementally stable ( -GAS).

t<latexit sha1_base64="oEMlW7WLHpaWjokmXb+5B6tSEZc=">AAACrXicbVFNb9QwEHXCVwlfC1yQuFjsIpUD26QXOKECh/bQQxG7baX1spo4k8SqY6e2U1hF+VP8G+78EJxtJErLSCM9vZl5M3qT1lJYF8e/gvDW7Tt3723djx48fPT4yejps2OrG8NxzrXU5jQFi1IonDvhJJ7WBqFKJZ6kZ5/7+skFGiu0mrl1jcsKCiVywcF5ajX6yVIshGoroUQNBXZtEvOqiyhz+MOlecsalaHp5dtZWU27jp2fN5DRaFaiQSoshYhOWIbSweTt/sev9HANdaP0Bc0bxfstUTRhh1oVRhSlA2P09wnrNSIqFDdYoXIg5ZpaB/5sun1V7c2URgxV9vfAaDUax9N4E/QmSAYwJkMcrUa/WaZ50+/hEqxdJHHtli0YJ7j0gqyxWAM/8+oLDxVUaJftxtuOvvZMRnNtfCpHN+zViRYqa9dV6jsrcKW9XuvJ/9UWjcvfL1uh6sah4peL8kZSp2n/KJoJg9x5WzIB3Ah/K+UlGODOvzNiveDw2DbVMuu8Lcl1E26C491pEk+TL7vjvU+DQVvkJXlFtklC3pE9ckCOyJzw4EXwIdgPDsKdcB6y8NtlaxgMM8/JPxEWfwCKNdIs</latexit>
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Hasuo (NII, JP) 20

• increasing in kx

ykIncremental Stability (δ-GUAS)– 0 if kx

yk = 0– ! 1 if kx• decreasing in t
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yk ! 1

Incrementally Globally Uniformly Asymptotically Stable– ! 0 if t ! 1

Def.
A dynamics is said to be -GAS if
9 KL function s.t.
x(x, t)V(x,y) is more or less|| x - y ||…

x(y, t)



kx

GAS

yk, t

where x(x, ) is the trajectory starting at x.

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For a dynamics ẋ = f (x) in Rn , a smooth function
V : Rn ⇥ Rn ! R+ is a common -GAS Lyapunov function if
there exist K1 functions ↵, ↵ and  > 0 such that↵(kx
that decreases
along thedynamics<latexit sha1_base64="pUU+6u+ntJzXpLhbmdZ1Cl9XY9U=">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</latexit>@V@xyk)  V (x, y)  ↵(kx

(x, y)f (x) +@V@y(x, y)f (y) 

t
δ-GAS

yk)
V (x, y)Thm.
There is a -GAS Lyapunov function=) incrementally stable ( -GAS).

t<latexit sha1_base64="oEMlW7WLHpaWjokmXb+5B6tSEZc=">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</latexit>
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Switched extension: δ-GUAS, common Lyapunov func.
[Girard, Pola, Tabuada, IEEE TAC ’10]

Hasuo (NII, JP) 20

Discrete Abstraction via Incremental Stability[Girard, Pola, Tabuada, IEEE TAC ’10]

For a switched dynamics ẋ = fp (x) …
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Commonwitnesses
δ-GAS
Lyapunov func.

V

Incremental
stability(δ-GUAS)

Hasuo (NII, JP) 21

Discrete Abstraction via Incremental Stability[Girard, Pola, Tabuada, IEEE TAC ’10]

For a switched dynamics ẋ = fp (x) …
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sha1_base64="4LCzMseb0W4aHc9jzDSRRbHgPjA=">AAAB63icbZC7TsMwGIX/lFsJBcrMYlEhMVUJC4xILIxFoheprSrHcVqrjh3Zf5BK1Bdg5QXYEI/EzoPgtBmg5UiWjs7x5fcXZVJYDIIvr7azu7d/UD/0jxr+8clps9GzOjeMd5mW2gwiarkUindRoOSDzHCaRpL3o/l92fefubFCqydcZHyc0qkSiWAUXdSZNFtBO1iJbJuwMi2oNGl+j2LN8pQrZJJaOwyDDMcFNSiY5Et/lFueUTanUz50VtGU23GxGnNJLl0Sk0QbtxSSVfr7REFTaxdp5HamFGd2syvD/7phjsntuBAqy5Ertn4oySVBTco/k1gYzlAunKHMCDcrYTNqKENHxh+VF1aMikjLeOmohJsMtk3vuh0G7fAxgDqcwwVcQQg3cAcP0IEuMIjhFd68F+/d+1jTq3kVxjP4I+/zB6n+kjc=</latexit>
sha1_base64="LH3bPeus+w+du5iygLcEFAPh56Y=">AAACAHicbZDNSsNAFIVv6l+tVaNbN4NFqJuSuNGNILhxWcH+QBvCZDpph04mYWZSWkIewTdwqy/gTnwR9z6IkzYLbT0wcDhn5s7lCxLOlHacL6uytb2zu1fdrx3UD4+O7ZN6V8WpJLRDYh7LfoAV5UzQjmaa034iKY4CTnvB9L7oezMqFYvFk14k1IvwWLCQEaxN5Nv2cBTrbJ7fhn6W5M35pW83nJazFNo0bmkaUKrt299mBEkjKjThWKmB6yTay7DUjHCa14apogkmUzymA2MFjqjysuXmObowyQiFsTRHaLRMf7/IcKTUIgrMzQjriVrvivC/bpDq8MbLmEhSTQVZfRSmHOkYFRjQiElKNF8Yg4lkZldEJlhiog2s2rAYWGLLgpiPcoPFXYewabpXLddpuY8OVOEMzqEJLlzDHTxAGzpAYAYv8Apv1rP1bn2sAFaskuQp/JH1+QMn1pos</latexit>
sha1_base64="o/SdJbZfj+PjZWcm6LvkTzckCEw=">AAACC3icbZC7TsMwGIUdrqXcAowsFhVSWaqEBRakChbGItGL1EaR4zitVceObKdqFeUReANWeAE2xMpDsPMgOG0GaDmSpaPzX/zrCxJGlXacL2ttfWNza7uyU93d2z84tI+OO0qkEpM2FkzIXoAUYZSTtqaakV4iCYoDRrrB+K6odydEKir4o54lxIvRkNOIYqRN5Nv2IBQ6m+Y3kZ8leX164ds1p+HMBVeNW5oaKNXy7W+zAqcx4RozpFTfdRLtZUhqihnJq4NUkQThMRqSvrEcxUR52fzyHJ6bJISRkOZxDefp74kMxUrN4sB0xkiP1HKtCP+r9VMdXXsZ5UmqCceLj6KUQS1ggQGGVBKs2cwYhCU1t0I8QhJhbWBVB8XCElsWCBbmBou7DGHVdC4brtNwH5xa87YEVAGn4AzUgQuuQBPcgxZoAwwm4Bm8gFfryXqz3q2PReuaVc6cgD+yPn8A2Jybng==</latexit>

Commonwitnesses
δ-GAS
Lyapunov func.

V

Incremental
stability(δ-GUAS)

induces8

Approximate bisimulation

Statespace abstraction

ANTOINE GIRARD, GIORDANO
AND PAULO TABUADA(Fig. from POLA,[GirardPT’10])Thm.
Let V be a common -GAS Lyapunov func. lower-bounded by ↵. Then<latexit sha1_base64="Zdv1yJRAdPhY+PP3Zak2zOsE5lU=">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</latexit>

def

()

V (x, y)  ↵(")

yields an "-approximate bisimulation.

ses
ctn eerre ntco aragu(x, y) 2 R

q0

||x

q 0 ||  ⌘x(⌧s , q, p)qHasuo (NII, JP) 21

Discrete Abstraction via Incremental Stability[Girard, Pola, Tabuada, IEEE TAC ’10]

Discrete verification,
synthesis, supervisory control,…

For a switched dynamics ẋ = fp (x) …
<latexit sha1_base64="b1aAoYEKuxppQ5RIx5a34r84bUQ=">AAACC3icbZDLSgMxGIUz9VbrbdSlm2AR6qbMiKAboejGZQV7gXYYMmmmDc0kQ5IpLcM8gm/gVl/Anbj1Idz7IGbaWWjrgcDh/Jf8fEHMqNKO82WV1tY3NrfK25Wd3b39A/vwqK1EIjFpYcGE7AZIEUY5aWmqGenGkqAoYKQTjO/yemdCpKKCP+pZTLwIDTkNKUbaRL5t9wdCp9PsJvTTOKtNz3276tSdueCqcQtTBYWavv1tVuAkIlxjhpTquU6svRRJTTEjWaWfKBIjPEZD0jOWo4goL51fnsEzkwxgKKR5XMN5+nsiRZFSsygwnRHSI7Vcy8P/ar1Eh9deSnmcaMLx4qMwYVALmGOAAyoJ1mxmDMKSmlshHiGJsDawKv18YYEtDQQbZAaLuwxh1bQv6q5Tdx8uq43bAlAZnIBTUAMuuAINcA+aoAUwmIBn8AJerSfrzXq3PhatJauYOQZ/ZH3+ANncm6I=</latexit>
sha1_base64="4LCzMseb0W4aHc9jzDSRRbHgPjA=">AAAB63icbZC7TsMwGIX/lFsJBcrMYlEhMVUJC4xILIxFoheprSrHcVqrjh3Zf5BK1Bdg5QXYEI/EzoPgtBmg5UiWjs7x5fcXZVJYDIIvr7azu7d/UD/0jxr+8clps9GzOjeMd5mW2gwiarkUindRoOSDzHCaRpL3o/l92fefubFCqydcZHyc0qkSiWAUXdSZNFtBO1iJbJuwMi2oNGl+j2LN8pQrZJJaOwyDDMcFNSiY5Et/lFueUTanUz50VtGU23GxGnNJLl0Sk0QbtxSSVfr7REFTaxdp5HamFGd2syvD/7phjsntuBAqy5Ertn4oySVBTco/k1gYzlAunKHMCDcrYTNqKENHxh+VF1aMikjLeOmohJsMtk3vuh0G7fAxgDqcwwVcQQg3cAcP0IEuMIjhFd68F+/d+1jTq3kVxjP4I+/zB6n+kjc=</latexit>
sha1_base64="LH3bPeus+w+du5iygLcEFAPh56Y=">AAACAHicbZDNSsNAFIVv6l+tVaNbN4NFqJuSuNGNILhxWcH+QBvCZDpph04mYWZSWkIewTdwqy/gTnwR9z6IkzYLbT0wcDhn5s7lCxLOlHacL6uytb2zu1fdrx3UD4+O7ZN6V8WpJLRDYh7LfoAV5UzQjmaa034iKY4CTnvB9L7oezMqFYvFk14k1IvwWLCQEaxN5Nv2cBTrbJ7fhn6W5M35pW83nJazFNo0bmkaUKrt299mBEkjKjThWKmB6yTay7DUjHCa14apogkmUzymA2MFjqjysuXmObowyQiFsTRHaLRMf7/IcKTUIgrMzQjriVrvivC/bpDq8MbLmEhSTQVZfRSmHOkYFRjQiElKNF8Yg4lkZldEJlhiog2s2rAYWGLLgpiPcoPFXYewabpXLddpuY8OVOEMzqEJLlzDHTxAGzpAYAYv8Apv1rP1bn2sAFaskuQp/JH1+QMn1pos</latexit>
sha1_base64="o/SdJbZfj+PjZWcm6LvkTzckCEw=">AAACC3icbZC7TsMwGIUdrqXcAowsFhVSWaqEBRakChbGItGL1EaR4zitVceObKdqFeUReANWeAE2xMpDsPMgOG0GaDmSpaPzX/zrCxJGlXacL2ttfWNza7uyU93d2z84tI+OO0qkEpM2FkzIXoAUYZSTtqaakV4iCYoDRrrB+K6odydEKir4o54lxIvRkNOIYqRN5Nv2IBQ6m+Y3kZ8leX164ds1p+HMBVeNW5oaKNXy7W+zAqcx4RozpFTfdRLtZUhqihnJq4NUkQThMRqSvrEcxUR52fzyHJ6bJISRkOZxDefp74kMxUrN4sB0xkiP1HKtCP+r9VMdXXsZ5UmqCceLj6KUQS1ggQGGVBKs2cwYhCU1t0I8QhJhbWBVB8XCElsWCBbmBou7DGHVdC4brtNwH5xa87YEVAGn4AzUgQuuQBPcgxZoAwwm4Bm8gFfryXqz3q2PReuaVc6cgD+yPn8A2Jybng==</latexit>

Commonwitnesses
δ-GAS
Lyapunov func.

V

Incremental
stability(δ-GUAS)

induces8

Approximate bisimulation

Statespace abstraction

ANTOINE GIRARD, GIORDANO
AND PAULO TABUADA(Fig. from POLA,[GirardPT’10])Thm.
Let V be a common -GAS Lyapunov func. lower-bounded by ↵. Then<latexit sha1_base64="Zdv1yJRAdPhY+PP3Zak2zOsE5lU=">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</latexit>

def

()

V (x, y)  ↵(")

yields an "-approximate bisimulation.

ses
ctn eerre ntco aragu(x, y) 2 R

q0

||x

q 0 ||  ⌘x(⌧s , q, p)qHasuo (NII, JP) 21

Bounding Errors by Switching Delays[Kido, Sedwards & Hasuo, IFAC ADHS ’18]ẋ = f1 (x)ẋ = f2 (x)
≤δ0

≤δ0
τ

2τ

≤δ0
3τ

≤δ0
4τ

Ideal, without delayt

Actual, with delay

Hasuo (NII, JP) 22

Bounding Errors by Switching Delays[Kido, Sedwards & Hasuo, IFAC ADHS ’18]ẋ = f1 (x)ẋ = f2 (x)
≤δ0

≤δ0
τ

2τ

≤δ0
3τ

≤δ0
4τ

Ideal, without delayt

Actual, with delayThm. If the switched system has a common δ-GAS
Lyapunov function V, then the gap (L∞ dist.,
✓
◆
⌫ 0
1
↵
1 e (⌧ 0 )) is bounded by<latexit sha1_base64="QtOVjoJ4nsLHd/3PcZICSmWaF54=">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</latexit>

Here α, κ, ν are param’s of the dynamics and V.

Hasuo (NII, JP) 22

Bounding Errors by Switching Delays[Kido, Sedwards & Hasuo, IFAC ADHS ’18]ẋ = f1 (x)ẋ = f2 (x)
≤δ0

≤δ0
τ

2τ

≤δ0
3τ

≤δ0
4τ

Ideal, without delayt

Actual, with delayThm. If the switched system has a common δ-GAS

Lyapunov function V, then the gap (L∞ dist., ) is bounded by
✓
◆
⌫ 0
1
↵
1 e (⌧ 0 )No accumulation<latexit sha1_base64="QtOVjoJ4nsLHd/3PcZICSmWaF54=">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</latexit>

Here α, κ, ν are param’s of the dynamics and V.

of

Hasuo (NII, JP) 22

Outline
制御理論 vs 形式手法・ソフトウェア科学
数学的相似・相互乗り入れ
例1（相似）： 確率的プログラムのマルチンゲールによる解析[Chakarov & Sankaranarayanan, CAVʼ13][Takisaka, Oyabu, Urabe & Hasuo, ATVAʼ18]

例2（乗り入れ）： 近似双模倣による離散化[Girard & Pappas, IEEE TAC ʼ07]

[Kido, Sedwards & Hasuo, IFAC ADHS ʼ18]

実システム応用に向けて
機械学習・AIとの協働
ソフトウェア工学，現実的解決
ERATO MMSD プロジェクトの取り組み
23

実システム応用に向けて自動運転を例に．
センシング ➜ 物体認識 ➜ 経路計画 ➜ 経路トレース

24

Autonomoose Project, U. Waterloo
www.autonomoose.net

実システム応用に向けて自動運転を例に．
センシング ➜ 物体認識 ➜ 経路計画 ➜ 経路トレース

24

Autonomoose Project, U. Waterloo
www.autonomoose.net

実システム応用に向けて自動運転を例に．
センシング ➜ 物体認識 ➜ 経路計画 ➜ 経路トレース

25

Autonomoose Project, U. Waterloo
www.autonomoose.net

実システム応用に向けて自動運転を例に．
センシング ➜ 物体認識 ➜ 経路計画 ➜ 経路トレース

26

Autonomoose Project, U. Waterloo
www.autonomoose.net

実システム応用に向けて自動運転を例に．
センシング ➜ 物体認識 ➜ 経路計画 ➜ 経路トレース

26

Autonomoose Project, U. Waterloo
www.autonomoose.net

実システム応用に向けて自動運転を例に．
センシング ➜ 物体認識 ➜ 経路計画 ➜ 経路トレース

27

Autonomoose Project, U. Waterloo
www.autonomoose.net

実システム応用に向けて自動運転を例に．
センシング ➜ 物体認識 ➜ 経路計画 ➜ 経路トレース

27

Autonomoose Project, U. Waterloo
www.autonomoose.net

機械学習コンポーネントと，
品質保証・説明責任

センシング

物体認識

経路計画

28

gtspirit.com
moralmachine.mit.edu

経路トレース

機械学習コンポーネントと，
品質保証・説明責任

センシング

物体認識

経路計画

28

gtspirit.com
moralmachine.mit.edu

経路トレース

機械学習コンポーネントと，
品質保証・説明責任

センシング

経路トレース

物体認識

経路計画

事故 ＆ 原因解明 & 社会に対する説明
28

gtspirit.com
moralmachine.mit.edu

機械学習コンポーネントと，
品質保証・説明責任

センシング

経路トレース
ニューラルネットは
ほぼブラックボックス…

物体認識

経路計画

事故 ＆ 原因解明 & 社会に対する説明
28

gtspirit.com
moralmachine.mit.edu

統計的機械学習との協働
統計的
機械学習

帰納的
（データから学ぶ）

制御理論・
形式手法

演繹的
（証明）

不確かさ
ブラックボックス（データのノイズ）
数学的・論理的厳密性

29

ホワイトボックス

統計的機械学習との協働
統計的
機械学習

帰納的
（データから学ぶ）

制御理論・
形式手法

演繹的
（証明）

不確かさ
ブラックボックス（データのノイズ）
数学的・論理的厳密性

29

ホワイトボックス

統計的機械学習との協働
統計的
機械学習

帰納的
（データから学ぶ）

制御理論・
形式手法

演繹的
（証明）

不確かさ
ブラックボックス（データのノイズ）
数学的・論理的
厳密性

制御理論・

統計的

形式手法

機械学習

29

ホワイトボックス

統計的機械学習との協働
統計的
機械学習

帰納的
（データから学ぶ）

制御理論・
形式手法

演繹的
（証明）

不確かさ
ブラックボックス（データのノイズ）
数学的・論理的
厳密性

制御理論・

統計的

形式手法

機械学習
不確かな機械学習コンポーネントを包み込む “safety envelope”
例： 自動運転の経路計画．safety
envelope の中で最適化
29

ホワイトボックス

Key: 論理と統計的学習の間で，システムレベルのタスク切り分けと構造化

統計的機械学習との協働
統計的
機械学習

帰納的
（データから学ぶ）

制御理論・
形式手法

演繹的
（証明）

不確かさ
ブラックボックス（データのノイズ）
数学的・論理的
厳密性

制御理論・

統計的

形式手法

機械学習
不確かな機械学習コンポーネントを包み込む “safety envelope”
例： 自動運転の経路計画．safety
envelope の中で最適化
29

ホワイトボックス

Key: 論理と統計的学習の間で，システムレベルのタスク切り分けと構造化

統計的機械学習との協働
統計的
機械学習

帰納的
（データから学ぶ）

制御理論・
形式手法

演繹的
（証明）

不確かさ
ブラックボックス（データのノイズ）
数学的・論理的厳密性

総当たり探索の代わりに，
データから傾向を読み取り高速に証明や補題を探索

制御理論・

統計的

形式手法

機械学習
不確かな機械学習コンポーネントを包み込む “safety envelope”
例： 自動運転の経路計画．safety
envelope の中で最適化
29

ホワイトボックス

制約充足問題

翻訳
ソフトウェア
検証問題

SAT
graph
reachability…

spec ↦ automaton
model ↦ prop. formula
…

最適化問題

制御問題

LMI
SDP
mixed integerstochastic optimization
…

翻訳

30

制約充足問題

翻訳
ソフトウェア
検証問題

SAT
graph
reachability… 確率的最適化

spec ↦ automaton
model ↦ prop. formula
…

機械学習
最適化問題
制御問題

LMI
SDP
mixed integerstochastic optimization
…

翻訳

30

制約充足問題

翻訳
ソフトウェア
検証問題

SAT
graph
reachability… 確率的最適化

spec ↦ automaton
model ↦ prop. formula
…

機械学習
最適化問題
制御問題

LMI
SDP
mixed integerstochastic optimization
…

翻訳

thr

例：

sp

60
30

Simulink モデルと時相論理式仕様に対する
サーチベーステスト，反例生成

0

br

ti

Cf. S-TaLiRo, BREACH, …

10

20

30

0

10

20

30

R

70
35

[Zhang, Ernst, Sedwards, Arcaini & Hasuo, EMSOFTʼ18]0ti

確率的最適化によるブラックボックスアプローチ

0⇤[0,30] (speed < 120)30制御理論・形式手法のソフトウェア工学的応用

困難
スケーラビリティシステムの「定義」（＝ホワイトボックス・モデル）が必要
実システムではむずかしい（自動車部品，NNなど）
産業応用における証明の意義．「労力の割にありがたがられない」
環境への仮定に依存 ➜ 仮定の正しさをどう担保する？

31

制御理論・形式手法のソフトウェア工学的応用

困難
スケーラビリティシステムの「定義」（＝ホワイトボックス・モデル）が必要
実システムではむずかしい（自動車部品，NNなど）
産業応用における証明の意義．「労力の割にありがたがられない」
環境への仮定に依存 ➜ 仮定の正しさをどう担保する？
➜ 現行の帰納的品質保証手段（＝テスト）のサポートに注力
ソフトウェア工学の考え方．テスト，実行時検証
形式手法の演繹的推論が決定的に有効（サーチベーステストのスコア定義，テスト空間の削減，…）
テストの有効性の数学的議論 ➜ 品質保証の業界標準，国際規格，…
31

制御理論・形式手法のソフトウェア工学的応用

困難「〇〇の範囲にバグはない」ことを証明

スケーラビリティシステムの「定義」（＝ホワイトボックス・モデル）が必要➜ 〇〇の範囲をテストする必要はない

実システムではむずかしい（自動車部品，NNなど）
産業応用における証明の意義．「労力の割にありがたがられない」
環境への仮定に依存 ➜ 仮定の正しさをどう担保する？
➜ 現行の帰納的品質保証手段（＝テスト）のサポートに注力
ソフトウェア工学の考え方．テスト，実行時検証
形式手法の演繹的推論が決定的に有効（サーチベーステストのスコア定義，テスト空間の削減，…）
テストの有効性の数学的議論 ➜ 品質保証の業界標準，国際規格，…
31

制御理論・形式手法のソフトウェア工学的応用

困難「〇〇の範囲にバグはない」ことを証明

スケーラビリティシステムの「定義」（＝ホワイトボックス・モデル）が必要➜ 〇〇の範囲をテストする必要はない

実システムではむずかしい（自動車部品，NNなど）
産業応用における証明の意義．「労力の割にありがたがられない」
環境への仮定に依存 ➜ 仮定の正しさをどう担保する？
➜ 現行の帰納的品質保証手段（＝テスト）のサポートに注力
ソフトウェア工学の考え方．テスト，実行時検証「何をチェックすれば十分と認められるのか」

形式手法の演繹的推論が決定的に有効の工学的見極め（サーチベーステストのスコア定義，テスト空間の削減，…）
テストの有効性の数学的議論 ➜ 品質保証の業界標準，国際規格，…
31

Outline
制御理論 vs 形式手法・ソフトウェア科学
数学的相似・相互乗り入れ
例1（相似）： 確率的プログラムのマルチンゲールによる解析[Chakarov & Sankaranarayanan, CAVʼ13][Takisaka, Oyabu, Urabe & Hasuo, ATVAʼ18]

例2（乗り入れ）： 近似双模倣による離散化[Girard & Pappas, IEEE TAC ʼ07]

[Kido, Sedwards & Hasuo, IFAC ADHS ʼ18]

実システム応用に向けて
機械学習・AIとの協働
ソフトウェア工学，現実的解決
ERATO MMSD プロジェクトの取り組み
32

プロジェクト紹介
プロジェクト目標： 工業製品の設計サポート
形式手法をソフトウェアから物理情報システムへ
工業製品の品質保証をサポート，省力化
Challenge: 関連分野の多様性
計算機科学，制御理論，ソフトウェア工学，機械学習，etc.
方法論： 圏論・代数学的メタ理論による統合> 10 名の特任教員・研究員を含む 50 名体制で理論研究を推進．
国内製造業企業（自動車関連），U Waterloo 自動運転プロジェクトと協働，社会応用特任教員・特任研究員募集中 (-2022)．https://group-mmm.org/eratommsd/openpositions/33

Hasuo (NII, Tokyo)

私達の取り組み： ３つの柱連続量を扱う形式手法：
統計的機械学習と協働する形式手法
演繹的手法のソフトウェア工学的応用

34

私達の取り組み： ３つの柱連続量を扱う形式手法：
統計的機械学習と協働する形式手法
演繹的手法のソフトウェア工学的応用
制御理論

形式手法
ソフトウェア
工学

機械学習
34

私達の取り組み： ３つの柱連続量を扱う形式手法：

制御理論との協働．

統計的機械学習と協働する形式手法

潮俊光（阪大），岸田昌子（NII），
脇隼人（九大）演繹的手法のソフトウェア工学的応用
制御理論

形式手法
ソフトウェア
工学

機械学習
34

私達の取り組み： ３つの柱連続量を扱う形式手法：

制御理論との協働．

統計的機械学習と協働する形式手法

潮俊光（阪大），岸田昌子（NII），
脇隼人（九大）演繹的手法のソフトウェア工学的応用

数理的
メタ理論

制御理論

形式手法
ソフトウェア
工学

機械学習
34

35

35

独自の方法論：
メタ理論による移転+ e1 ➜ T1[e1]T2 + e2 ➜ T2[e2]T3 + e3 ➜ T3[e3]T1

新たな関心事ための形式手法

連続量，不確かさ，…

…

…

…
ソフトウェアの物理情報システム
36のための形式手法

独自の方法論：
メタ理論による移転
メタ理論家
論理学，
代数学，圏論，…

+ e1 ➜ T1[e1]T2 + e2 ➜ T2[e2]T3 + e3 ➜ T3[e3]T1

新たな関心事ための形式手法

連続量，不確かさ，…

…

…

…
ソフトウェアの物理情報システム
36のための形式手法

独自の方法論：
メタ理論による移転

理論拡張のプロセス自体を理論的に解析

メタ理論家
論理学，
代数学，圏論，…

+ e1 ➜ T1[e1]T2 + e2 ➜ T2[e2]T3 + e3 ➜ T3[e3]T1

新たな関心事ための形式手法

連続量，不確かさ，…

…

…

…
ソフトウェアの物理情報システム
36のための形式手法

独自の方法論：
メタ理論による移転

理論拡張のプロセス自体を理論的に解析

メタ理論家… T + e ➜ T[e]の包括的な一般論を構築（さまざまな T と e に一挙に適用可能）+ e1 ➜ T1[e1]T2 + e2 ➜ T2[e2]T3 + e3 ➜ T3[e3]T1

新たな関心事ための形式手法

連続量，不確かさ，…

…

…

…
ソフトウェアの物理情報システム
36のための形式手法

論理学，
代数学，圏論，…

Existing Technique

T1

37

Abstract TechniqueT[_]
Identify“mathematicalessense”
Existing Technique

T1

37

Abstract TechniqueT[_]
Identify“mathematicalessense”

Chooseparameter e1

Existing Technique

T1 = T[e1]37

Abstract TechniqueT[_]
Identify“mathematicalessense”

Chooseparameter e1

Existing TechniqueChooseparameter e2
Novel Technique

T1 = T[e1]

T[e2]

37

論理学，
代数学，圏論，✓◆
theory of coalgebras
Defn. T (e) = …
, where Be is a monadin K`(Be )Abstract TechniqueThm. Coalgebraic proof methods (such as (bi)simulations) are v
A

T[_]
Identify“mathematicalessense”

Proof.

::=

true | false | A1 ^ A2 | ¬A | a1 < a2 |8x 2 ⇤N. A | 8x 2 ⇤R. A

By the following characterizations in K`(Be ).
FX
O
c
X

F behc /

F OZ
final
/Z

behc
system behavior

Chooseparameter e1

Existing Technique

FX
O
c
X

Chooseparameter e2

T[e2]

37

w

/ FY
O
d
/Y

f
simulation

Novel Technique

T1 = T[e1]

Ff

論理学，
代数学，圏論，✓◆
theory of coalgebras
Defn. T (e) = …
, where Be is a monadin K`(Be )Abstract TechniqueThm. Coalgebraic proof methods (such as (bi)simulations) are v
A

T[_]
Identify“mathematicalessense”

Proof.

::=

true | false | A1 ^ A2 | ¬A | a1 < a2 |8x 2 ⇤N. A | 8x 2 ⇤R. A

By the following characterizations in K`(Be ).
FX
O
c
X

F behc /

F OZ
final
/Z

behc
system behavior

T1 = T[e1]

w

/ FY
O
d
/Y

f
simulation

Novel Technique
物理情報システムの多様な応用に即応
産業界（製造業）のニー
ズに根ざした理論展開
現代数学の抽象性から生まれる応用37

Ff

Chooseparameter e2

Chooseparameter e1

Existing Technique

FX
O
c
X

T[e2]

まとめ
制御理論 vs 形式手法・ソフトウェア科学
数学的相似・相互乗り入れ
例1（相似）： 確率的プログラムのマルチンゲールによる解析[Chakarov & Sankaranarayanan, CAVʼ13][Takisaka, Oyabu, Urabe & Hasuo, ATVAʼ18]

例2（乗り入れ）： 近似双模倣による離散化[Girard & Pappas, IEEE TAC ʼ07]

[Kido, Sedwards & Hasuo, IFAC ADHS ʼ18]

実システム応用に向けて
機械学習・AIとの協働
ソフトウェア工学，現実的解決
ERATO MMSD プロジェクトの取り組み
38