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Approximating
Reachability Probabilities
by (Super-)Martingales
Ichiro Hasuo
National Institute of Informatics
Tokyo, Japan

First Question
Each of Alice, Bob and Charlie has someapples.
On average one has <= 2.5 apples.
How many can Alice have, at most?

2

Hasuo (NII, Tokyo)

Overview
Reachability in probabilistic programs
Need of “parameters”
Foundation of fixed points, in the non-probabilistic setting
Ranking functions, invariants
Foundation: Knaster-Tarski, Cousot-Cousot
Known supermartingale methods
Roles of concentration lemmas
Something new (from our recent results)
Automated Synthesis
3

Hasuo (NII, Tokyo)

Based on Joint Works:
Natsuki Urabe, Masaki Hara and Ichiro Hasuo.
Categorical Liveness Checking by Corecursive Algebras.
LICS 2017
Toru Takisaka, Yuichiro Oyabu, Natsuki Urabe and Ichiro Hasuo.
Supermartingales for Refuting Reachability of Probabilistic
Programs: Completeness.
In preparation

4

Hasuo (NII, Tokyo)

Probabilistic Programs
Programs withrandom assignment

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

x := Gaussian(0, 0.2)

probabilistic branching
if prob(0.2)(say, m = 16 and p = 0.2)

(also nondet. assignment& branching)<latexit 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zuPRMIjfOphBE898xayR6bcIyKCNpkjZSQQc6c5cGvN1otLhoI9uHpvOLbfMnYfXu5uj3e2x1/tDp8c1G8gv7Px7xv/sbG1Md74r40nG59vnGx8vRHee3jv5N5v7/1u9IfRn0b/PfqfCvqTe3Wff9vofEb/9//2g5Rz</latexit>

Example: right (random walk)
We disregard the Bayesian aspects
observations, conditioning, priors/posteriors, etc.
See e.g. [Gordon+, FOSE’14])
5

Hasuo (NII, Tokyo)

Probabilistic Programs
float skillA, skillB,skillC;
float perfA1,perfB1,perfB2,
x := perfC2,perfA3,perfC3;mwhile= Gaussian(100,10);x > 0 doskillAskillB
Gaussian(100,10);i f =prob(p)doskillC = Gaussian(100,10);1234
x := x 1
5 // firste l s e game:A vs B, A won
Gaussian(skillA,15);
6 perfA1 x= :=x+1
perfB1 = Gaussian(skillB,15);
7 observe(perfA1fi> perfB1);
8 //odsecond game:B vs C, B won<latexit 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zuPRMIjfOphBE898xayR6bcIyKCNpkjZSQQc6c5cGvN1otLhoI9uHpvOLbfMnYfXu5uj3e2x1/tDp8c1G8gv7Px7xv/sbG1Md74r40nG59vnGx8vRHee3jv5N5v7/1u9IfRn0b/PfqfCvqTe3Wff9vofEb/9//2g5Rz</latexit>

perfB2 = Gaussian(skillB,15);(say, m= 16 and p = 0.2)
perfC2= Gaussian(skillC,15);
observe(perfB2 > perfC2);// third game:A vs C, A won
perfA3 = Gaussian(skillA,15);
perfC3 = Gaussian(skillC,15);
observe(perfA3 > perfC3);

Figure 9: Bayesian skill rating (TrueSkill [26]).

each tuple t has an associated indicator random varie Xt which takes a value 1 if the tuple is present in theabase and 0 if the tuple is absent. Each instantiation
values to all of the random variables is called a world.
e probabilistic database is the joint probability distribun over the random variables, which implicitly specifies a
tribution over all the possible worlds. The answer to aery Q on a probabilistic database is the set of tuples T
t are possible answers to Q, along with a probability P (t)every t 2 T denoting the probability that t belongs thewer for query Q. A probabilistic database together with

int goats, tigers;
double c1, c2, c3, curTime;// initialize populations
goats = 100; tigers = 4;// initialize reaction rates
c1 = 1; c2 = 5; c3 = 1;//initialize time
curTime = 0;
while (curTime < TIMELIMIT){
if (goats > 0 && tigers > 0){
double rate1, rate2, rate3,
rate;
rate1 = c1 * goats;rate2 = c2 * goats * tigers;
rate3 = c3 * tigers;
rate = rate1 + rate2 + rate3;

else if (goats > 0){
double rate;
rate = c1 * goats;double dwellTime =Exponential(rate);
curTime += dwellTime;goats++;
double dwellTime =}Exponential(rate);
else if (tigers > 0)
int discrete ={Disc3(rate1/rate,rate2/rate);
double rate;
curTime += dwellTime;rate = c3 * tigers;
switch (discrete)
double dwellTime ={Exponential(rate);case 0: goats++; break;curTime += dwellTime;case 1: goats--; tigers++;
tigers--;break;}
case 2: tigers--; break;}//end while loop}
return(goats,tigers);}}[Gordon+, FOSE’14]
Figure 10: Lotka-Volterra Population Model.

6
Hasuo (NII, Tokyo)
from a Gaussian distribution with mean 100 and variance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

real-valued variables(say, m = 16 and p = 0.2)
➜ infinite states
A state is a pair (program location, memory state)➜ not for automated analysis<latexit 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zuPRMIjfOphBE898xayR6bcIyKCNpkjZSQQc6c5cGvN1otLhoI9uHpvOLbfMnYfXu5uj3e2x1/tDp8c1G8gv7Px7xv/sbG1Md74r40nG59vnGx8vRHee3jv5N5v7/1u9IfRn0b/PfqfCvqTe3Wff9vofEb/9//2g5Rz</latexit>

Transition by small-step operational semantics➜ MDP (Markov decision processes) M

If nondet. isangelic…
<latexit 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ycHzRfIH2z8buM/NjY3xht/2Hiy8dXGyca3G9HGf2/89b1fvPfL7RfbavvH7Xc19P33mj7/vtH52/6vvwEHYn40</latexit>

7

Hasuo (NII, Tokyo)

Probabilistic Control Flow Graph(pCFG) See e.g. [Chatterjee+, POPL’17]
Finite graph,“parametric” in memory states(… justifies my presence here!)

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>

State = (program location)
Transition types:

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)

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)

Our Approach
Use parametric certificates, i.e. functions<latexit sha1_base64="ZvzVp2GFqRo/MXbV5YAPAFQ/a3Y=">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</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)

Why Overapproximation?
Pr(ReachC) <= αif and only if
Pr(NotReachC) >= 1- α
Refutation of reachabilityisverification of safety

12

Hasuo (NII, Tokyo)

Overview
Reachability in probabilistic programs
Need of “parameters”
Foundation of fixed points, in the non-probabilistic setting
Ranking functions, invariants
Foundation: Knaster-Tarski, Cousot-Cousot
Known supermartingale methods
Roles of concentration lemmas
Something new (from our recent results)
Automated Synthesis
13

Hasuo (NII, Tokyo)

In Case of No Probabilities…
pCFG
Reachability question:is there a schedulerthat leads to C?
Verify reachability by ranking functions,refute reachability (= verify safety) by invariants

From now on, our theory is not parametrized for simplicity
Kripke frames instead of CFGs
MDPs (or even MCs) instead of pCFGs
14

Hasuo (NII, Tokyo)

Ranking Functions(S, ! ✓ S ⇥ S): Kripke frame, C ✓ S.
A ranking function is⌘ : S ! N [ {1}Thm.
⌘(s) = 1 implies C isreachable from s.

such that• 8s 2 S \ C. 9s0 2 S.
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SdUX3NR8Qh6F2V5ftG8VdF+SxBfxQY/jN4ykgrf4INihRB8CSEH/t5kqG3Wp5IH2kj3PBHD2fNoUXGu2OImFWHN86eS6fdDNA8PgcmAZmyIpJZM5Dy9wRdIYVD8cUru6buwugPMcClwr7qwG5IwHRI+HWIET4b40zW0TfmQpCG0iRwpLYHoO82eW2saLy5pCvbg8u3+2H7L2H14sTsa74zGX+70H+3XbyD/dON3G7/f2NwYb3y08Wjjs43jja82gju/uLN758GdT0d/Gv1l9NfR3yronbfqPr/daH1Gf/8n4aXqlQ==</latexit>
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SdUX3NR8Qh6F2V5ftG8VdF+SxBfxQY/jN4ykgrf4INihRB8CSEH/t5kqG3Wp5IH2kj3PBHD2fNoUXGu2OImFWHN86eS6fdDNA8PgcmAZmyIpJZM5Dy9wRdIYVD8cUru6buwugPMcClwr7qwG5IwHRI+HWIET4b40zW0TfmQpCG0iRwpLYHoO82eW2saLy5pCvbg8u3+2H7L2H14sTsa74zGX+72H+3XbyD/dON3G7/f2NwYb3y08Wjjs43jja82gju/uLN758GdT0d/Gv1l9NfR3yronbfqPr/daH1Gf/8n4kXqlw==</latexit>⌘(s0 ) + 1• ⌘(s) = 0 implies s 2 Cη overapproximates #(steps to C)
Note: {natural numbers} is well-founded
Discrete analogue of (certain) Lyapunov functions…
15

Hasuo (NII, Tokyo)

Invariants(S, ! ✓ S ⇥ S): Kripke frame, C ✓ S.
An invariant for S \ C isI✓S<latexit sha1_base64="NoQAwh6CrHyzyPHa/BoumC0P+5g=">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sPzna3h9tbwm+21N9f+tPbntY214dpf1h6ufbl2tPZsLVr75623br1364+bZ5t/3fxb9YbyG7fqV5XfW2t9Nv/+L460gGU=</latexit>Thm.
s 2 I implies C is
not reachable from s.

such that• I ✓ 2I, where
2I = {s | 8s0<latexit sha1_base64="nfROi+CB2tpoVorWPbooVe8bbLk=">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</latexit>s. s0 2 I}

• I ✓S\C

Get trapped in I
Discrete analogue of barrier certificates…
16

Hasuo (NII, Tokyo)

Ranking Functions & Invariants(S, ! ✓ S ⇥ S): Kripke frame, C ✓ S.
An invariant for S \ C is

(S, ! ✓ S ⇥ S): Kripke frame, C ✓ S.
A ranking function isI✓S⌘ : S ! N [ {1}such thatsuch that• 8s 2 S \ C. 9s0 2 S.
s ! s0 and ⌘(s)

• I ✓ 2I, where
2I = {s | 8s0⌘(s0 ) + 1• ⌘(s) = 0 implies s 2 C<latexit sha1_base64="nfROi+CB2tpoVorWPbooVe8bbLk=">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</latexit>s. s0 2 I}

• I ✓S\CThm.
⌘(s) = 1 implies C isreachable from s.

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sha1_base64="NoQAwh6CrHyzyPHa/BoumC0P+5g=">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</latexit>Thm.
s 2 I implies C is
not reachable from s.

How come the difference?
NB:

ReachC =µ C [ 3ReachC

NotReachC =⌫ (S \ C) \ 2NotReachC
17

Hasuo (NII, Tokyo)

Lattice-Theoretic Foundation
L: complete lattice,f : L ! L monotone

Thm. (Knaster-Tarski)• µf =
min{l 2 L | f (l) v l}=)f (l) v lµf v l• ⌫f =
max{l 2 L | l v f (l)}=)

l v f (l)l v ⌫fThm. (Cousot-Cousot)? v f (?) v · · · v f ! (?) v · · ·stabilizes, and converges to µf=)f ↵ (?) v µf

(8↵ 2 Ord)

> w f (>) w · · · w f ! (>) w · · ·stabilizes, and converges to ⌫f=)⌫f v f ↵ (>)(8↵ 2 Ord)Sound approx. fromHasuo
below(Tokyo)Overview

KT
lfp

CC

overapprox. underapprox.

gfp underapprox. overapprox.

Reachability in probabilistic programs
Need of “parameters”

Foundation of fixed points, in the non-probabilistic setting
Ranking functions, invariants
Foundation: Knaster-Tarski, Cousot-Cousot
Known supermartingale methods
Roles of concentration lemmas
Something new (from our recent results)
Automated Synthesis
19

Hasuo (NII, Tokyo)

Answers whatquestion?

Additiverankingrankingsupermartingale(lower bd) Multiplicative
rankingsupermartingale[Urabe+ LICS’17]

ε-decreasing
supermartingale(upper bd)

Exp(StepsC) <= ??

(elementary)[McIver+ PSSE’04] [Chakarov+ CAV’13]

supermartingale

repulsing

Pr(ReachC) =? 1

Underlying mathrepulsingsupermartingale• LPcharacterization
Pr(ReachC) >= ??
• (Categoricalconsideration)

Pr(ReachC) <= ??

Azuma’sinequality

Pr(ReachC) <= ??

Markov’s
inequality[Chatterjee+ POPL’17]

Nonnegative
repulsingsupermartingale[Steinhardt+ IJRR’12][Takisaka+, in preparation]

20

for martingale
concentration

for martingale
concentration

Hasuo (NII, Tokyo)

Disclaimer
From now on:
no nondeterminism, no parametrization
Our setting is therefore:tr

A Markov chain (S, S ! DS), and C ✓ S<latexit 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2vHa1+uRWt/Wfvb2t/X/rG1vXW2db51UUPffafp87u1zm8r+SchYHLy</latexit>

21

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)

X⌘ : S ! R is defined by
X(X⌘)(s) =tr(s)(s0 ) · ⌘(s0 )
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 ).

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

• ⌘(s) =22 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 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ytB/DAb/DB6y0hp/IIPihVC8COEDPj705GxWZ8qHhgj3fdkDGfPo2XJuWXLeyHDiufPFDPfhxgeHgJTAU3ZCEkjmaiFuMcPSGFQfDml9s1dWNUBZrgSuF9e2I1IKEaEz0YYwZMRvrqGthkfERFCm8yQMhKIudMcuLVm68MlQ8Ee3Hw8nNhfGbsPb/fGk93x5Ou94YuD6gvkn278YuOXG1sbk41fb7zY+GrjdOObjeDRPzx68ejVo38a//P4X8f/Nv73Evroo6rP3290fsb/8V9mKO+u</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=">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) = 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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</latexit>• Exp(Steps from s to C)  ⌘(s)
23

Hasuo (NII, Tokyo)

Answers whatquestion?

Additiverankingrankingsupermartingale(lower bd) Multiplicative
rankingsupermartingale[Urabe+ LICS’17]

ε-decreasing
supermartingale(upper bd)

Exp(StepsC) <= ??

(elementary)[McIver+ PSSE’04] [Chakarov+ CAV’13]

supermartingale

repulsing

Pr(ReachC) =? 1

Underlying mathrepulsingsupermartingale• LPcharacterization
Pr(ReachC) >= ??
• (Categoricalconsideration)

Pr(ReachC) <= ??

Azuma’sinequality

Pr(ReachC) <= ??

Markov’s
inequality[Chatterjee+ POPL’17]

Nonnegative
repulsingsupermartingale[Steinhardt+ IJRR’12][Takisaka+, in preparation]

24

for martingale
concentration

for martingale
concentration

Hasuo (NII, Tokyo)

ε-Decreasing Repulsing Supermartingale[Chatterjee+ POPL’17]

Def.
tr
Let (S, S ! DS) be a MC, C ✓ S, " > 0,  > 0.
⌘ : S ! R is an "-decreasing repulsing supermartingale for C with -b’dd di↵erence if• 8s 2 S \ C.
• 8c 2 C.

<latexit sha1_base64="YKmmSfnoWqw1LP70qMQvq/1zjTk=">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</latexit>

⌘(s)

⌘(c)

(X⌘)(s) + ".

0.

• 8s 2 S. 8s0 2 supp(tr(s)).
|⌘(s) ⌘(s0 )|  .

Thm.
Let ⌘ be such a repulsing supermartingale for C.
Assume ⌘(s) < 0. Then
Pr(Reachs,C )  ↵<latexit sha1_base64="7nmqtuMihGciaeViP/Yld7UPwlg=">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</latexit>

where ↵ = e25"·⌘(s)(+")2,=ed|⌘s|/e1"22(+")2.

Hasuo (NII, Tokyo)

ε-Decreasing Repulsing Supermartingale[Chatterjee+ POPL’17]

Def.
tr
Let (S, S ! DS) be a MC, C ✓ S, " > 0,  > 0.
⌘ : S ! R is an "-decreasing repulsing supermartingale for C with -b’dd di↵erence if• 8s 2 S \ C.
• 8c 2 C.

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qe4aQSHZ+vyFo9Wf/lZ9BysF+lrnIquGxYAMuUWAlumwsZzPbL6zqrFyt7leI4ytIHYru5mEYh3/hkc+m83xs7WePPFu+L9Vrl8I5Z5w/HiG4pPf7ezTTLl4J+931J9zUXFCxhdlOXlVfNVRfcrQfzWHPwwestIKvyCD4oVQvAjhBzouxNH26xHJfe1ke66Ioaz59G8otyw+X0qgprmhZLp70M0DQ+BSZ9mzMGm5kzkLL3HD0hhUnw5JXf1XVg9ACRcMNytLuwcEqQO4aGDETxx8NU19IXcIWkAfSLHluZA9J3mwK41Wx8u6RbswfVHw7H5lbH98HZnNN4ejb/aGb7Yr79A/unav679Zm1jbbz272sv1r5cO1n7es3/4K8f/urDf/vw10/Z0z8//Y+n/1lBP/hRPeZXa53f0//6G48xNjQ=</latexit>

⌘(s)

⌘(c)

Thm.
Let ⌘ be such a repulsing supermartingale for C.
Assume ⌘(s) < 0. Then

(X⌘)(s) + ".

0.

Pr(Reachs,C )  ↵• 8s 2 S. 8s0 2 supp(tr(s)).
|⌘(s) ⌘(s0 )|  .

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q65qHwCvfOiuLisv6pofyWIH1yDH0ZvKZTGL/igWCEEP0LIgL8/HRqb9anigTHSfU9GcPZcLEvONVveJjKseP5MMfN9iOHhITAV0JQNkTSSiVokt/gBKQyKL6fUvrkLqzrADFcC98sLuyEJkyHhsyFG8HiIr66hbcaHJAmhTWZIGQnE3Gn23Fqz8eGSoWAPrj7sj+2vjN2HV5PReG80/nrSf3JQfYH8061fb/1ma3trvPUfW0+2vtw62fpmK/jg/+79+72P7/1m9Hb0n6P/Gv13Cb33QdXnV1ut3+h//gx44eak</latexit>

26

where ↵ = e"·⌘(s)(+")2,=ed|⌘s|/e1"22(+")2.

Hasuo (NII, Tokyo)

ε-Decreasing Repulsing Supermartingale[Chatterjee+ POPL’17]

Def.
tr
Let (S, S ! DS) be a MC, C ✓ S, " > 0,  > 0.
⌘ : S ! R is an "-decreasing repulsing supermartingale for C with -b’dd di↵erence if• 8s 2 S \ C.
• 8c 2 C.

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qe4aQSHZ+vyFo9Wf/lZ9BysF+lrnIquGxYAMuUWAlumwsZzPbL6zqrFyt7leI4ytIHYru5mEYh3/hkc+m83xs7WePPFu+L9Vrl8I5Z5w/HiG4pPf7ezTTLl4J+931J9zUXFCxhdlOXlVfNVRfcrQfzWHPwwestIKvyCD4oVQvAjhBzouxNH26xHJfe1ke66Ioaz59G8otyw+X0qgprmhZLp70M0DQ+BSZ9mzMGm5kzkLL3HD0hhUnw5JXf1XVg9ACRcMNytLuwcEqQO4aGDETxx8NU19IXcIWkAfSLHluZA9J3mwK41Wx8u6RbswfVHw7H5lbH98HZnNN4ejb/aGb7Yr79A/unav679Zm1jbbz272sv1r5cO1n7es3/4K8f/urDf/vw10/Z0z8//Y+n/1lBP/hRPeZXa53f0//6G48xNjQ=</latexit>

⌘(s)

⌘(c)

Thm.
Let ⌘ be such a repulsing supermartingale for C.
Assume ⌘(s) < 0. Then

(X⌘)(s) + ".

0.

Pr(Reachs,C )  ↵• 8s 2 S. 8s0 2 supp(tr(s)).
|⌘(s) ⌘(s0 )|  .

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q65qHwCvfOiuLisv6pofyWIH1yDH0ZvKZTGL/igWCEEP0LIgL8/HRqb9anigTHSfU9GcPZcLEvONVveJjKseP5MMfN9iOHhITAV0JQNkTSSiVokt/gBKQyKL6fUvrkLqzrADFcC98sLuyEJkyHhsyFG8HiIr66hbcaHJAmhTWZIGQnE3Gn23Fqz8eGSoWAPrj7sj+2vjN2HV5PReG80/nrSf3JQfYH8061fb/1ma3trvPUfW0+2vtw62fpmK/jg/+79+72P7/1m9Hb0n6P/Gv13Cb33QdXnV1ut3+h//gx44eak</latexit>

27

where ↵ = e"·⌘(s)(+")2,=ed|⌘s|/e1"22(+")2.

Hasuo (NII, Tokyo)

ε-Decreasing Repulsing Supermartingale[Chatterjee+ POPL’17]

Def.
tr
Let (S, S ! DS) be a MC, C ✓ S, " > 0,  > 0.
⌘ : S ! R is an "-decreasing repulsing supermartingale for C with -b’dd di↵erence if• 8s 2 S \ C.
• 8c 2 C.

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qe4aQSHZ+vyFo9Wf/lZ9BysF+lrnIquGxYAMuUWAlumwsZzPbL6zqrFyt7leI4ytIHYru5mEYh3/hkc+m83xs7WePPFu+L9Vrl8I5Z5w/HiG4pPf7ezTTLl4J+931J9zUXFCxhdlOXlVfNVRfcrQfzWHPwwestIKvyCD4oVQvAjhBzouxNH26xHJfe1ke66Ioaz59G8otyw+X0qgprmhZLp70M0DQ+BSZ9mzMGm5kzkLL3HD0hhUnw5JXf1XVg9ACRcMNytLuwcEqQO4aGDETxx8NU19IXcIWkAfSLHluZA9J3mwK41Wx8u6RbswfVHw7H5lbH98HZnNN4ejb/aGb7Yr79A/unav679Zm1jbbz272sv1r5cO1n7es3/4K8f/urDf/vw10/Z0z8//Y+n/1lBP/hRPeZXa53f0//6G48xNjQ=</latexit>

⌘(s)

⌘(c)

Thm.
Let ⌘ be such a repulsing supermartingale for C.
Assume ⌘(s) < 0. Then

(X⌘)(s) + ".

0.

Pr(Reachs,C )  ↵• 8s 2 S. 8s0 2 supp(tr(s)).
|⌘(s) ⌘(s0 )|  .

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q65qHwCvfOiuLisv6pofyWIH1yDH0ZvKZTGL/igWCEEP0LIgL8/HRqb9anigTHSfU9GcPZcLEvONVveJjKseP5MMfN9iOHhITAV0JQNkTSSiVokt/gBKQyKL6fUvrkLqzrADFcC98sLuyEJkyHhsyFG8HiIr66hbcaHJAmhTWZIGQnE3Gn23Fqz8eGSoWAPrj7sj+2vjN2HV5PReG80/nrSf3JQfYH8061fb/1ma3trvPUfW0+2vtw62fpmK/jg/+79+72P7/1m9Hb0n6P/Gv13Cb33QdXnV1ut3+h//gx44eak</latexit>

28

where ↵ = e"·⌘(s)(+")2,=ed|⌘s|/e1"22(+")2.

Hasuo (NII, Tokyo)

Concentration of Measure Inequalities

29

Hasuo (NII, Tokyo)

Concentration of Measure Inequalities

30

Hasuo (NII, Tokyo)

Concentration of Measure I:
Azuma’s Inequality Used in [Chatterjee+ POPL’17]tr

If one-step
difference isbounded…

Def. Let (S, S ! DS) be a MC.
A supermartingale is ⌘ : S ! R such that8s 2 S. ⌘(s)(X⌘)(s)Rem. Let s0 ! s1 ! · · · be a run of the MC, with s0 fixed.
Then sn is a rand. var. in S for each n, and ⌘(sn ) is a rand. var.
in R.
Thm. (Azuma)
and n 2 N,

Let ⌘ have -bdd di↵erence. For eachPr ⌘(sn )⌘(s0 )<latexit sha1_base64="AV3d91n/qB0S1TjvPv+bBJd3Flc=">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</latexit>

31

 e

>0

2
2n2

Hasuo (NII, Tokyo)

Concentration of Measure II:
The Chebyshev-Cantelli Inequality
Used e.g.in [Bouissou+ TACAS’16]

Thm. (Chebyshev–Cantelli) Let X be a random variablewith mean µ and variance 2 . Then for each k > 0,
Pr X

µ

k

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

1
1 + k2
If variance is
known…

32

Hasuo (NII, Tokyo)

Concentration of Measure III:
Bernstein’s Inequality
Used e.g.in [Bouissou+ TACAS’16]

Thm. (Bernstein)Let ⌘ be a supermartingale, where• conditional variance is bounded: 9such that

Var ⌘(s0 ) s ! s0

2

• deviation from the mean is bounded:
9M such that, 8i 2 [0, n],⌘(si )

Exp(⌘(si ))⌘(s0 )

⇣

 M

Then
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33

2

2n

2 + 2M3⌘
Hasuo (NII, Tokyo)

Answers whatquestion?

Additiverankingrankingsupermartingale(lower bd) Multiplicative
rankingsupermartingale[Urabe+ LICS’17]

ε-decreasing
supermartingale(upper bd)

Exp(StepsC) <= ??

(elementary)[McIver+ PSSE’04] [Chakarov+ CAV’13]

supermartingale

repulsing

Pr(ReachC) =? 1

Underlying mathrepulsingsupermartingale• LPcharacterization
Pr(ReachC) >= ??
• (Categoricalconsideration)

Pr(ReachC) <= ??

Azuma’sinequality

Pr(ReachC) <= ??

Markov’s
inequality[Chatterjee+ POPL’17]

Nonnegative
repulsingsupermartingale[Steinhardt+ IJRR’12][Takisaka+, in preparation]

34

for martingale
concentration

for martingale
concentration

Hasuo (NII, Tokyo)

Overview
Reachability in probabilistic programs
Need of “parameters”
Foundation of fixed points, in the non-probabilistic setting
Ranking functions, invariants
Foundation: Knaster-Tarski, Cousot-Cousot
Known supermartingale methods
Roles of concentration lemmas
Something new (from our recent results)
Automated Synthesis
35

Hasuo (NII, Tokyo)

Nonnegative Repulsing Supermartingale[Steinhardt+ IJRR’12] [Takisaka, Oyabu, Urabe, IH, in preparation]
trDef. Let (S, S ! DS) be a MC, C ✓ S and M > 0.
⌘ : S ! R is a nonnegative repulsing supermartingale for C at
M if
• 8s 2 S.

⌘(s)• 8s 2 S \ C.
<latexit sha1_base64="TABzeyW+iKvx9PfcjJKI6Et605M=">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</latexit>• 8c 2 C.

0

⌘(s)

⌘(c)

(X⌘)(s)M

36

Hasuo (NII, Tokyo)

Nonnegative Repulsing Supermartingale[Steinhardt+ IJRR’12] [Takisaka, Oyabu, Urabe, IH, in preparation]
trDef. Let (S, S ! DS) be a MC, C ✓ S and M > 0.
⌘ : S ! R is a nonnegative repulsing supermartingale for C at
M if
• 8s 2 S.

⌘(s)• 8s 2 S \ C.
<latexit 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Gf/t1Rfc1HxEnoXZXl13XxV0f1KED/PBj+M3jKSCr/gg2KFEPwIIQf+3tTRNutRyX1tpHuuiOHsebSoOLds8ZCKoOZ5oWT6+xDNw0Ng0qcZc5DUkomcpw/4ASkMii+n5J6+C6s7wAyXAveqCzuHBKlDeOhgBE8cfHUNbSF3SBpAm8iR0hKIvtMc2LVm68MlTcEe3Hw8nJhfGdsP73bHk53x5Kvd4cuD+gvkn278y8a/bmxuTDb+bePlxpcbpxtfb/hPyJOjJ2+fnI7/ffyf4/8a/3cFffJR3eeXG53f+E//A3sm8K8=</latexit>• 8c 2 C.

0

⌘(s)

⌘(c)

(X⌘)(s)M

37

Hasuo (NII, Tokyo)

Nonnegative Repulsing Supermartingale[Steinhardt+ IJRR’12] [Takisaka, Oyabu, Urabe, IH, in preparation]
trDef. Let (S, S ! DS) be a MC, C ✓ S and M > 0.
⌘ : S ! R is a nonnegative repulsing supermartingale for C at
M if
• 8s 2 S.

⌘(s)• 8s 2 S \ C.
<latexit sha1_base64="TABzeyW+iKvx9PfcjJKI6Et605M=">AAAvFniczVpbc+TGdabWiS+TxJHsx7y0MrNlcg2OONxSxeU1WSsOl9aWuVyKpFasIrhUA2jMtIgbuxskx1j8j/yVvOQtlVe/+t/knAZmAHRjZCXxg+eBxOnz9enbuTbgZRGXamfnzx89+dHf/f2Pf/LTnw3+4R//6ef//PEnv3gn01z47Gs/jVJx6VHJIp6wrxVXEbvMBKOxF7FvvNsp8r+5Z0LyNLlQi4xdx3SW8JD7VEHTzScf/YfrsRlPipgnPKMzVhaTHT8uB65ij8oLCzdPAiZQfHHIwnFZEvcup8HgmCky2jx3yLn7KPhsrqgQ6UPhxlTNRVwoUZb62adRcVieb42Ixwglb6YOGU1dmXuSKXZHzkeEJgEZvdnfGY3JYEBGLlPUxYUlIFqlWojnFWfliHAJElwWZ/MiSZOEzWAR94wIluWR5MmMyDxjIqZCAUEjVpIwFTAcjAGTfQMCwkG9XK5YzP/IygFx8RGGBSiNIiJdDgOP9SJxKptyy50xsjNag3RhHbB3uSTTqhNp9dpczv6yxNYtaLbl+Cin09nXnd+MBi5Lgmaqmlqd0+Dm4+HOeEf/iP0wqR+GG/Xv9OYT59ANUj+PWaL8iEp5NdnJ1HWB++VHOEAuWUb9WxB/BY8JjZm8LrSSleQptAR6Q8M0UUS3tnsUNJZyEXuAxEVLk4eNfbyrXIW/uS54kuWKJX41UJhHRKUENZYEXDBfRQt4oL7gMFfiz6mgvgK97owCGvUI/cnFHDQlzqVCnVNzRkIugKhhKBjao5QGLBjAry0ipIm/8NJHp14N/kdLcOoVOAOpQMEWInAeqFwAxAHz8kARYycTaRo6jxJkgeqRwdP2KoN7nsFmZlIJ7t/K7mazZAa2Pr8uPOqxqMsDFbkuHhcliJumccYj9oaiDCbJwH1cpBkacnG/W7YoPxf3rN2w67MoajeIVFGFkHMWwe5ewOQKMPsC5g37BxTZTEOibVpuwX9GQEmp76cigB1i5IGrOcH5wKEJ8iYFH5EM3K8l+yKKLh5SHA7O5hxmm6jlfGGnAxa6Ea5RqkXECu1k9GNZ8VTd1ea2zmgmaDbn/qMjQA5aBkxZsLsc9KRS0G/BovwoD1iNlL8imwtY7iOhAZz51kCbZe0WCy+NgrKrBYrf/tHBP9t+UP2/U4LV00A64p4ALSiwVdrN1b45RM5pxuA/OEHfITRXKQxMHZKlkuMxgJtCTws9oX+h+Z+N9YrJHim29539PdA+Gqn5r8olUG8NzF3r+h658rkA43UIOoY4jwnuyN7O+HM/duDMEjgdWNceGvpA/57+f36Dp3qDFRqYzJiPkYQsfcrATdiDn8YxePQCFEvJsooHEEY0VXYRuGdJefX8unCL4QTObLjrvvjgvhg+d00oD8pVaIFng0tDUMIlgEWFi3tigvz0LyFOTt6WRbF02SelJYGq8mpy3UCGEwszLVfcqT2DZNaEyvZspALFw6cBaX6uoFwy8EXF+HM462LkSnB/WWURWpjrjMpyTZ8d6HIFf671H+jcjAydrP2VuDvGBLsQNM0IQGctwNMOIovQ8cGycPzRNgwDhqkbyRxsjIlBL7xJMNxsThOwgSKB5MH5C71XO4gZxqm5oLDiw9+bleKEPBlbKw8gx2oJOjT5EQ/b/GOrv5+ljZofToHqQXgGBCblpapGQoAryD6oXvns08+2f5Opz14W+6X2xGhscZqk5CUgtvcNycgBwc2RKcojA8MyXurDmzMarD3cNKsnmGar7YK20k3SiMdcSQP/wGbaFMylspCHIYh60aj0UhzwYL2VmhbuMahaxELVmpH7wpbG7n6AsD27a7z46tUhqCH0PoC/PUzgzHWkfgbKwiE4jladRqafWiqADAuprONtswPNr84NUlGeBuD9MT+HEKDYLBUcwwFT/hhAbDwbk+EEnPj2cwbeergLjxP99ByetAc1ZjIHbWRB5TMlvWdX15/+bjhxhrv7z4rh89IVTKpUMMt7xjTT56VDauVDb7T76gJhi1Hf0y74DFKSzeFkqx9cXu1et3qi1M3hrgn+Q8S4jHgld2lMf4CNu4XEYXu8C97KBRE9wyhRe5Vqh7GwMAB5BhmCFhxR1CqCcwBPiM8G1mPzRhQSBv+LaNZYKhKmN2Co3fJuWb/crMDhQ2D5loh5a9BeP3qd8IOjHnS4TvjRAaL/r4Ee+kFuPIMKEUoTrLJ8LLIkxHya3EpMnmMxvqWgijFFbJUPMILVAoGckSchEwxzRJFHoO2IgfRIY9IEMsd5KjGbhAQIRoA8wqcZVzRCHJXE4wlkUCQF66GgyoYB8BgdlUqNzXitm1+/OUXbM2Pud6VWjAAcKnhUqFsiHsB66gw2+Q4WmEN+y5L7MUyrHU17fmC1kYSCaexjfospK45QD2CkAtXQVTTUgFFP8NGzg/qqyzisGG/PjPYjXTIC54vjY4P16hEiGWZcry7NvIahUp+8vcCADUfT1iKctjVHC5WKFigV1jpgngafGAcER7NCwCHaCJxeIyNJlQ3BNa8gVfFsg2D1KwzTW2JjDtpjeX1DgaJitrcyKctTvkX9DFoJ7rLBdE7gQiNPJ2nfV/zU3HXljxmQdB64TmwrS1wjuI0ol1ocZ5LPklVSul955t1Mud+lPOlvsRJcbYcskWBQKKfeHZ4UqwEgp6grZCOHF6KnK3hpZKzpo8v6/l4Va00/ycBog/6ONW9NT3ROTQDRlJlJ5aoNqUgD82UaN+UMEpbnoNGsCZWgXVVLX8iWLK5jXwS1p38LdaSOfqKmsLI1orbqVEIpiNYHP4c0qFVzPM+UrjnmqVAxB79QAXzhC7OYMCITVhF/lVGiTj2iR6q01UupQJV5+ay4+rC5u3VtJTKIKAsN/LD9zMUU+0Np7oWU2XLvMgouD0KUhIAEtagQrQZwDgOjZ9BVO6IXqlQRLBOaz2FdVqr4zZxHmMpow/aLiuzDaPH66b0ey8xSpEwaMZrqQWgh+NAr40vYmdZcKrIPo8Xop1458wU4vqosLt8Xz8r2BvRmmDRqSuQzi904Xa/4yuQmbe6Jxc3jhntY1U+UxEzReyo49SAxxFzlzCFfEfh/YvqSqF3nHdl25lfV/4faCg2NeLeyejDXd7bNf/HqMWsAmjIQBx3EQQ9iGgcNAAnzvCzATXFV1dmWhRzFUQNFwvTHq5JQu+9av7HVRHq9SK8HKVbIVS0HSNGDVCJf5vyA0JTp+2kkW5CKtC9cpGr8KMD8pQ99ugbY52FvedZsLAjBBusCJZxGFNLcqjSrgTyEyhS8yYvVQuYswabdponhvPHOy6yw0fqXMncbmbrdEBukWqh5glIyoZYiWptQMbQMcyO+f1SsztwCZm8p3nkrV0LC4LfZNle1ThtfHdUuZNLvRMP2yeOtTgceWv4Jko6YAvye+XCWmrAuv05oq4avSAvzOulgkLSSA3UKpUE7P6gazBVAZO4CVy3WBU/w6k6rFOrlHmx92TU06AuYssTi36otZrx9M6FJU8kyX2vGA6YRfYYh9QXMKjW6s2w/aQ7WCkIelB1ec/02sz0H7wB8GyA7AKkzACuhoVGC08AHwOLLOpTTLWHOWhpzxmz+61ZG9tpKyM5PReo3wbIi+zA6WHZQvUETODNB41b5sGoxkNms2f/Muo7IZuc8YR3EzYrQLOuOO4Xatxm2Is1AGjQSEyvIJMEbynG/k6AZSzeZyBrXgRizCU5ozFrTqeieOf8Q3DmEzLOWo8EIelOnHD3QqQ21ruy9NG2lKweWxcd5oza8b5ADexAt03QJ3qInhmKrdeQi/c64qwWkbrWg6cPqtQiKxQTNfWDBzFYLLs+axAwjV0WbC4K85lVy31pS3WAdawdVkda1aY7JgMdn8s6Hx/qyu3rZywJ8lQ+5dR5bFzq6rlheCF+tfIcqr4vqamgpCoK64kme5pLENJNmyMfi4vtldcsPvCMwnKM5F50LFy0x1pyebZuz6lmcSm86QtYtaM16utM2RZk3Cc2I9ux/6LSPL5dm7oviGHT80jztY2lC6uEuLVU8nppIyyqPpYWpJz+1X8h9uch0+lEhNt3pe4zUoVqUW1YGbmIP1mPPTCy6mfXw8wuAtT0O0r2L05yDHqTlfU4CzWrZmaYteV2Tveg12AvDYvtQR+8aZwfPJve8zT63+ScnLf6JVb69o62C5J3tfERzz6/fEmB5p5uqm9xNkeZJsEV03W7sAJ8lkDO29qBuMHcqXpYnGD1jK8c/j9W8DVDW6wMaqGaNAeSuxhKC6u0Ypnst2PsCqsWqsznirHvCNW2hToWJq1sM5MW0U/ZVpFk08U5I09s8Z4THdFZdzecJV4YnnNIgoq03J1MXkpl7VtAyiFaPVvay6oUJE1fQ69uqx7fr0a+1eS2plb2ZeasImOD3ZesO1dz0zjndFL+2HIecNxBpnTR+maSEjLQ+Difvm+u889K+HMDMGcup4eSmmJSOG6SQIDpIQUbfE2S7EdY6IcYj/HKkurpCCt8iugKfTI/WTV6MaXk0WR4a5BKfrn3b63mnjZxT6xot5F5V9jbXejs7kOy1X6XU1+1uRMWMPXWP8d+A/Aw4o6Lo3EgTvL6kKs263ZefBnShsHudryKC9CHRYQ9ZnQ8jUKLRGUom+/by3q8uX/821pNn/+vVWGeDTg3+1ceI1NTKA6Z48EZaCcpcNVvWddRog3VPdipY6w5KUz0IVJl6VlWHsv5gyIyyrOWukLD5LVEavUbSee419oyEGZ54q5S1b/9yT2s5WpweCiToKFT0XXIe0VY1iYR1dxV3pQFmKW01+UPmg3Lhx37zt/XLUPdyFX0u10F+v4L8fh3kaAWx1ilum4nDs3V5i0erv/wseg7Wi/Q1TsXXhAWATLmFQMpU2Hihe1bfWbVE2bscL3AObSDS1d0sAvHOP4ND/+3mxNmebL18X3zYLldvxDJvOFl+Q/Hr3+3ukEw5+Gf/t1Rfc1HxEnoXZXl13XxV0f1KED/PBj+M3jKSCr/gg2KFEPwIIQf+3tTRNutRyX1tpHuuiOHsebSoOLds8ZCKoOZ5oWT6+xDNw0Ng0qcZc5DUkomcpw/4ASkMii+n5J6+C6s7wAyXAveqCzuHBKlDeOhgBE8cfHUNbSF3SBpAm8iR0hKIvtMc2LVm68MlTcEe3Hw8nJhfGdsP73bHk53x5Kvd4cuD+gvkn278y8a/bmxuTDb+bePlxpcbpxtfb/hPyJOjJ2+fnI7/ffyf4/8a/3cFffJR3eeXG53f+E//A3sm8K8=</latexit>• 8c 2 C.

0

⌘(s)

⌘(c)

(X⌘)(s)M

38

Hasuo (NII, Tokyo)

First Question, Again
Each of Alice, Bob and Charlie has someapples.
On average one has <= 2.5 apples.
How many can Alice have, at most?

39

Hasuo (NII, Tokyo)

Nonnegative Repulsing Supermartingale[Steinhardt+ IJRR’12] [Takisaka, Oyabu, Urabe, IH, in preparation]
trDef. Let (S, S ! DS) be a MC, C ✓ S and M > 0.
⌘ : S ! R is a nonnegative repulsing supermartingale for C at
M if
• 8s 2 S.

⌘(s)• 8s 2 S \ C.
<latexit 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Gf/t1Rfc1HxEnoXZXl13XxV0f1KED/PBj+M3jKSCr/gg2KFEPwIIQf+3tTRNutRyX1tpHuuiOHsebSoOLds8ZCKoOZ5oWT6+xDNw0Ng0qcZc5DUkomcpw/4ASkMii+n5J6+C6s7wAyXAveqCzuHBKlDeOhgBE8cfHUNbSF3SBpAm8iR0hKIvtMc2LVm68MlTcEe3Hw8nJhfGdsP73bHk53x5Kvd4cuD+gvkn278y8a/bmxuTDb+bePlxpcbpxtfb/hPyJOjJ2+fnI7/ffyf4/8a/3cFffJR3eeXG53f+E//A3sm8K8=</latexit>• 8c 2 C.

0

⌘(s)

⌘(c)

(X⌘)(s)M

Thm. (upper bound)
Pr(Reachs,C ) 

1
M

⌘s

Thm. (completeness) There exists a nonnegative repulsing
supermartingale ⌘ for C at M such that
Pr(Reachs,C ) =<latexit sha1_base64="qGXSQCDpbZ0BL18JLL1jfhbi074=">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</latexit>

1
M

⌘s

Proof. Take M = 1, ⌘(s) = Pr(Reachs,C ).
40

Hasuo (NII, Tokyo)

Answers whatquestion?

Additiverankingrankingsupermartingale(lower bd) Multiplicative
rankingsupermartingale[Urabe+ LICS’17]

ε-decreasing
supermartingale(upper bd)

Exp(StepsC) <= ??

(elementary)[McIver+ PSSE’04] [Chakarov+ CAV’13]

supermartingale

repulsing

Pr(ReachC) =? 1

Underlying mathrepulsingsupermartingale• LPcharacterization
Pr(ReachC) >= ??
• (Categoricalconsideration)

Pr(ReachC) <= ??

[Chatterjee+ POPL’17]

Nonnegative
repulsingsupermartingale

Pr(ReachC) <= ??

[Steinhardt+ IJRR’12][Takisaka+, in preparation]

Azuma’sinequality
for martingale
concentration

Markov’s
Knasterinequality
Tarski!
for martingale
concentration

41

Hasuo (NII, Tokyo)

Concentration of Measure: No Need!
trDef. Let (S, S ! DS) be a MC, C ✓ S.
Let XC : [0, 1]S ! [0, 1]S be defined by0
XC ⌘ : S
! [0,
⇣
⌘
( 1]
XC
B
1
⌘ : S ! [0, 1] 7 ! @s 7 !
(X⌘)(s)1

if s 2 C C
A
if s 62 C

Thm. (upper bound)
Pr(ReachC ) =µ XC Pr(ReachC )(Easy from the Cousot-Cousot characterization of lfp)
Rem.
• The least fixed point<latexit sha1_base64="YYIvX6N32iFBBZ/2+1Y8rYbZlRM=">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</latexit>• In the complete lattice [0, 1], so that a sink state gets 042

Hasuo (NII, Tokyo)

Lattice-Theoretic Foundation
L: complete lattice,f : L ! L monotone

Thm. (Knaster-Tarski)• µf =
min{l 2 L | f (l) v l}=)f (l) v lµf v l• ⌫f =
max{l 2 L | l v f (l)}=)

l v f (l)l v ⌫fThm. (Cousot-Cousot)? v f (?) v · · · v f ! (?) v · · ·stabilizes, and converges to µf=)f ↵ (?) v µf

(8↵ 2 Ord)

> w f (>) w · · · w f ! (>) w · · ·stabilizes, and converges to ⌫f=)⌫f v f ↵ (>)(8↵ 2 Ord)

Hasuo (Tokyo)

Concentration of Measure: No Need!
Thm.

Let ⌘ : S ! [0, 1] be such that
XC (⌘)  ⌘ ,that is,• 8s 2 S \ C.
• 8c 2 C.

⌘(s)

⌘(c)

(X⌘)(s)1Then, for each s 2 S,
Pr(Reachs,C )  ⌘(s)<latexit sha1_base64="ecryPLomDBd/SG0OQr6yQjFqveo=">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</latexit>

Pr(ReachC ) =µ XC Pr(ReachC ) .

Derivation of nonnegative repulsing supermartingale
from the Knaster-Tarski theorem
Soundness and completeness follow easily
44

Hasuo (NII, Tokyo)

Answers whatquestion?

Additiverankingrankingsupermartingale(lower bd) Multiplicative
rankingsupermartingale[Urabe+ LICS’17]

ε-decreasing
supermartingale(upper bd)

Exp(StepsC) <= ??

(elementary)[McIver+ PSSE’04] [Chakarov+ CAV’13]

supermartingale

repulsing

Pr(ReachC) =? 1

Underlying mathrepulsingsupermartingale• LPcharacterization
Pr(ReachC) >= ??
• (Categoricalconsideration)

Pr(ReachC) <= ??

[Chatterjee+ POPL’17]

Nonnegative
repulsingsupermartingale

Pr(ReachC) <= ??

[Steinhardt+ IJRR’12][Takisaka+, in preparation]

Azuma’sinequality
for martingale
concentration

Markov’s
Knasterinequality
Tarski!
for martingale
concentration

45

Hasuo (NII, Tokyo)

eMethodology

FX
O
c
X

F behc /

F OZ
final
/Z

behc
system behavior

FX
O
c
X

Ff
w

/ FY
O
d
/Y

f
simulation“categorical ranking function”&
soundness theorem

generalization

concretization
categoricallygeneralized systemranking function&
soundness theorem
nondeterministic
systemNatsuki Urabe (U. Tokyo)“probabilistic
rankingfunction”?

probabilistic
automaton
46

Categorical Ranking FunctionDef:: F ⌦ ! ⌦ is a triple

A ranking domain wrt.

( r : F R ! R, q : R ! ⌦, vR ) s.t.
1. R is a complete lattice and c,r is monotone
2. q is monotone, ?-preserving and continuous
3. q r v
F q 4. r is corecursive
Def:
An arrow b : X ! R is a ranking arrow wrt. (r, q, vR ) if:

FXOc

Natsuki Urabe (U. Tokyo)

X

Fr

Fb

c

47

/ FR

Fq

v

b vR r

r

v

b

✏
/R

q

/ F⌦

✏
/7 ⌦

47

Corecursive AlgebraDef:

An algebra r : F R ! R is corecursive if for all
coalgebra c : X ! F X , a coalgebra-algebra
homomorphism from c to r uniquely exists.
FX
O
c

X

F LrMc

/ FR

=

r

LrMc

✏
/R

•

It has been used to ensure productivity of general
structured corecursion [Capretta et al., SBMF ‘09]•

We use it to ensure termination

Natsuki Urabe (U. Tokyo)

48

Scaled Multiplicative Ranking Supermartingale
Def:
2 (0, 1) , a function b : X ! [0, 1] is aFormultiplicative ranking supermartingale if:

·

X

0

x0 2X

0

Pr(x ! x ) · b(x )-scaledb(x)By soundness of (categorical) ranking arrows,Thm:b(x)  Pr

an accepting stateis reached from x!

Quantitative reasoning
Natsuki Urabe (U. Tokyo)

49

Some Afterthoughts…
LP characterization of reachability probabilities(e.g. [Baier & Katoen])
Minimize x
subject to x = Ax + b
Here A is the transition matrix, and b is the membership of C
Our scaled multiplicative ranking supermartingale
Scale the constraint to x = γ(Ax + b) with 0 < γ < 1➜ Makes the solution x0 unique. It is an lfp and a gfp
Once we find x such that x ≦ γ(Ax + b), we have x ≦ x0 by
Knaster-Tarski(Still the categorical axiomatics gives a nice clue, I believe)
50

Hasuo (NII, Tokyo)

eMethodology

FX
O
c
X

F behc /

F OZ
final
/Z

behc
system behavior

FX
O
c
X

Ff
w

/ FY
O
d
/Y

f
simulation“categorical ranking function”&
soundness theorem

generalization

concretization
categoricallygeneralized systemranking function&
soundness theorem
nondeterministic
systemNatsuki Urabe (U. Tokyo)“probabilistic
rankingfunction”?

probabilistic
automaton
51

Overview
Reachability in probabilistic programs
Need of “parameters”
Foundation of fixed points, in the non-probabilistic setting
Ranking functions, invariants
Foundation: Knaster-Tarski, Cousot-Cousot
Known supermartingale methods
Roles of concentration lemmas
Something new (from our recent results)
Automated Synthesis
52

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>

Idea: a state l in pCFG standsfor (l, [x 7! r]) r 2 R<latexit 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v1l7eHa07WDtVdrwdpf1/6+9o+1fw5Gg5PB+YBU0Pffq/v8ca31G7z9F/5AdTc=</latexit>

Supermartingales are given by⌘l : R<latexit sha1_base64="hIZV7Z4a/RhG1giriAbsvYbQ1og=">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
sha1_base64="9Tkonw+yH3M8d8hCNgzmw1RdwXE=">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</latexit>

{x}

! R,

i = 1, . . . , 8
Hasuo (NII, Tokyo)

Template-Based Synthesis
Let’s use linear templates, for example
Meaning:⌘l : R{x} ! R,

i = 1, . . . , 8⌘l (r) ⌘ al r + bl<latexit sha1_base64="1HTrirtgJ15QT2ZaNAg4D8Fjkvs=">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</latexit>

al, bl: undetermined parameters
They are determined so that supermartingale
constraints are satisfied, e.g.
Def.
tr
Let (S, S ! DS) be a MC, C ✓ S.

Constraint solving
by LP, SDP, QE,…

⌘: S ! R

0 is a ranking supermartingale for C if

• 8s 2 S \ C.
<latexit sha1_base64="Bx09C5xaCHJpiLtCsukwLc/M+Uc=">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</latexit>⌘(s)(X⌘)(s) + 1.

• ⌘(s)= 0 implies s 2 C
54

Hasuo (NII, Tokyo)

Answers whatquestion?

Additiverankingrankingsupermartingale(lower bd) Multiplicative
rankingsupermartingale[Urabe+ LICS’17]

ε-decreasing
supermartingale(upper bd)

Exp(StepsC) <= ??

(elementary)[McIver+ PSSE’04] [Chakarov+ CAV’13]

supermartingale

repulsing

Pr(ReachC) =? 1

Underlying mathrepulsingsupermartingale• LPcharacterization
Pr(ReachC) >= ??
• (Categoricalconsideration)

Pr(ReachC) <= ??

Azuma’sinequality

Pr(ReachC) <= ??

Markov’s
inequality[Chatterjee+ POPL’17]

Nonnegative
repulsingsupermartingale[Steinhardt+ IJRR’12][Takisaka+, in preparation]55

for martingale
concentration

for martingale
concentration

Hasuo (NII, Tokyo)

Preliminary Experimental Results
1
2
3
4
5
6
7
8
9
10
11
12

x := mw h i l e x > 0 do
i f x < N doi f prob(p) do
x := x 1
else
x := x + 1
fielse
x := x + 1
fiod

Fig. 1. APP corresponding to Example 1
1
2
3
4
5
6
7
8

x := Uniform[0, 1]
w h i l e x < 1 do
i f prob(p) do
x := 2 ⇤ x
else
x := 0.5 ⇤ xfiodFig. 3. APP corresponding to Example 3

1
2
3
4
5
6
7
8
9
10
11

x := Geometric(0.5)
w h i l e x 1 do
x := x 1
od
i f prob(0.5) do
w h i l e x  0 do
x := x 1
od
else
skip
od

Fig. 2. APP corresponding to Example 2

1
2
3
4
5
6
7
8

x := mw h i l e x > 0 do
i f prob(p) do
x := x 1
else
x := x + 1
fiod

Fig. 4. APP corresponding to Example 4
56

Hasuo (NII, Tokyo)

Preliminary Experimental Results

1
2
3
4
5
6
7
8
9
10
11
12

x := mw h i l e x > 0 do
i f x < N doi f prob(p) do
x := x 1
else
x := x + 1
fielse
x := x + 1
fi
od

1
2
3
4
5
6
7
8
9
10
11

Fig. 1. APP corresponding to Example 1
1
2
3
4
5
6
7
8

x := Uniform[0, 1]
w h i l e x < 1 do
i f prob(p) do
x := 2 ⇤ x
else
x := 0.5 ⇤ xfiodFig. 2. APP corresponding to Example 2

1
2
3
4
5
6
7
8

Fig. 3. APP corresponding to Example 3

x := Geometric(0.5)
w h i l e x 1 do
x := x 1
od
i f prob(0.5) do
w h i l e x  0 do
x := x 1
od
else
skip
od

x := mw h i l e x > 0 do
i f prob(p) do
x := x 1
else
x := x + 1
fiod

Fig. 4. APP corresponding to Example 4

example true reachability probability boundby LNRepSM
by 1-LRepSM
Nonnegative(ours) boundε-decreasing– Pr(l3 , l4 ) = p, Pr(l3 , l5 ) = 1 p.
– G(l2 , l3 ) = (x < 1) and G(l2 , l6 ) = (x1).

(0.4/0.6)5 (0.4/0.6)10⇡ 0.116364
Moreover, let C = {(l6 , [x 7! 1r]) | r 2 R} ✓ L ⇥ RV , and let C be a 10
1 (0.4/0.6) linear predicate
map representing C. They correspond to an APP in Figure 3. It is a variant of geometric(binomial) random walk [23], which is often used in the context of finance.
If we define a linear expression map f and a linear predicate map I by f(l1 ) = 12 ,f(l2 ) = f(l3 ) = x, f(l4 ) = 2x, f(l5 ) = 12 x and I(l) = x 0 for each l 2 L, then f is an
LNRepSM for C at 1 supported by I.2
In contrast, no RepSM can refute almost-sure reachability to ~C. Let ⌘ : L ⇥ RV !
R be an ✏-RepSM for C that has -bounded di↵erences
some  > 0. Assume
R
1 for
0.25
f (l1 , 0) = A < 0. Then we have: A = ⌘(l1 , 0)f (l2 , x) + ✏dlog. Hencethere ex2 (1/x)ex2[0,1]
0.75
ists x > 0 such that f (l2 , x)  A < 0.
0N
0.25
Let N 2 N be the minimum number such that 2N x 1. As(l6 , 21x) 2 C, we haveNN0⌘(l2 , 2 x) ⌘(l6 , 2 x) + ✏ 0 , which means that there exists0.75
n 2 {0, . . . , N 1} suchn0
n0 +1that ⌘(l2 , 2 x) ⌘(l2 , 2 x) > 0.

2
2a
2b
3
4

2

R

0.5
0.5
0.5

(
(

)
dx ⇡ 0.2) ⇡ 0.333333
57

0.5054945055
0.5
0.5
0.5
0.5
—

Table 1. experimental results

We note that if we let 13 < p  12 , then the reachability probability to ~C is strictly smaller

<1
—
—
—
—
<1
Hasuo (NII, Tokyo)

Summary
Reachability in probabilistic programs is richMartingale methods for (over-/under-) approximations
Role of concentration lemmas
Knaster-Tarski foundation is still useful
Automated synthesis by constraint solving, numeric
optimization
Also showcased use of categorical/coalgebraic methods
We’re hiring!
Max 4 yrs, PD & senior researchers
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58

Thank yo u for yo ur atten
tion!
Ichiro Hasuo (NII, Tokyo)
http://group-mmm.org/~ichiroHasuo (NII, /Tokyo)