Proving Positive Almost-Sure Termination
Olivier Bournez, Florent Garnier
To cite this version:
Olivier Bournez, Florent Garnier. Proving Positive Almost-Sure Termination. [Intern report]
A04-R-409 —— bournez04h, 2004, 16 p. <inria-00099867>
HAL Id: inria-00099867
https://hal.inria.fr/inria-00099867
Submitted on 26 Sep 2006
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≤ Ei [V (Xn+1 )1n<τ |X0n ]
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³ EF8:¿%BA3 ¶43=1A˝3=8ÃEF8 ¸736µz36·=0H870365:´ E=19Ë63W.
°¸:EF,-8 M ¿70
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06ۑBA¸70™´dBAEWBA0z´9º:E6°
0z3F» M ,-´ˆ¿?0Ï:870m¿ÔE6´
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B3=»@EF*-*1A0HEF*-,-;HEF¹:*G0Z¸7,œ´tBA3=1A,G0m´&3F» A Ç
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´4»¼3=1^¸7,œ´thBA3=1A=³ aa aa ·· ·· ·· aa t <
∆µ V (a) ≤ −.
φ
φ
φ
h,ht
0 1
n
0 1
n
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ph,h = 1
φ
φ
φ
W (a0 a1 · · · an ) = V (an )
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Î 0ˆ¸:Eb.60
∆W (h) = ∆ V (h) ≤ −
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Î 0z°EF8 B9¸70H8ÔEFº:º7*G³ÌÒ&¸70H3=1A0µ Ä368 <“¶^,GB9¸ 0HÁ/5:E=*–BA3™B9¸:0z´90
BZ3F»B90H19µ`,-8:EF*
190mEF*-,G;mEF¹7*-0^¸7,œ´tBA3=1A,G0m´B93°3=8:°*G5‘¿?0^B9¸‘EWBvBA¸70+M¿?0H19,-.WEWBA,G368:C´v´dBAEF19B9,-87·(»¼1A3=µ a ,-8 M 190mE=°¸
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08‘°
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³ φ 190mE=°¸zB90H19µ`,-8:EF*
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BA0+»¼361&Ï:87,GB90H*G³™¹:1AE=8:°¸7,-87·z´9³?´tBA0µ´HÇ
2j#q]S52E%:,$`g8
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A = (A, →)
0 1
n
µ
φ
φ
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a → µm
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a(
µ %
2E%Eb
P Q X Y TT GPN Y V>X N Y V(Y sT TWX ! T " \ ek]_5R2E%G,1`a8
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A=
(A, →)
m m
|
2E%Eb
65U2E5KJ %>b,15%Eb,$'),^,&+2j#&%O#130;h,Oe./5U*1%(24365
V
:
A
→
R
( %
inf
V
(i)
>
−∞
(
#$36;7,
#&/*cb %>b6% e$30'h0`E` #&%:0%:,
e$30'h0`E`
2E%>b i∈A
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> 0(
a ∈ A(
µ %
a → µ(
a
(
"*.*.30')-62E5KJl%G3
-",(]_5U,.-l!$8
µ
∆µ V (a) =
X
µ(i)V (i) − V (a)
i
#$6%(2j#>]f,&#
∆µ V (a) ≤ −.
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Ç Î ¸708
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³6<•¶T0`¶^φ 1A,rBA0 T [h, φ]T [a,»¼361(φ]B9¸7<0+∞
µ`0HE=8ÉB9,-µz0
¹‘0»¼3=1A0Z1A0HE6°¸7,-87· ⊥ EW» BA01¸7,œ´dB93=1A³ h Ç
L3FBA0BA¸:EWB(»¼361(E=8}³º“3=*-,œ°
³
<¶^¸70H8 h ,œ´ZEÀ1A0HEF*-,-;HEF¹:*G0›873=8?ÑSB90H19µ`,-8:EF*¸7,œ´dB93=1A³=<“¶40
φ
¸:Eb.=0
X
Ä
T [h, φ] = 1 +
φ(h)(x)T [hx, φ]
')33 e
x∈A
ÈP»@º“3=*-,-°³ φ ,œ´ =E 19Ë63W.},-E=8•<6¶40(¸:Eb.=0
<7EF8:¿¸70H8:°
0
T [hx, φ] = T [x, φ]
X
φ(h)(x)T [x, φ].
T [h, φ] = 1 +
Í
Ò&¸70(,-¿?0mEz,-´4BA3™°
3=8‘´d,œ¿?014BA¸70 t¶43=1´dB +´dB91EWB90H·=³ Ç:Ò&¸:,-´´dB91EWBA0·6³™°EF8¹“0ˆ¹757,-*rBDE6´
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¿?,-´dB91A,-¹75?B9,-3=8 µ ¶^,rBA¸ a → µ B9¸:EFBµ`EF½?,Gµ`,G;H0H´ sup P µ(x)T [hx, φ] Ç
¯},G8‘°
0ZB93™E=8/³À´tBA1AEFB90·6³ φ 368°HEF8¤E=´A´d3?°
,œEWBA0ZE´dB91EWBA0·6³ φ ¶^,rBA¸
x∈A
0
n
BAE=Ë=0
φ
x∈A
n
0
T [hx, φ] = T [x, φ0 ]
φ0 (h0 ) = φ(hh0 )
sup
»¼3=1E=8}³™190mEF*-,G;mEF¹7*-0Z873=8?ÑSB90H19µ`,-8:EF*¸:,-´dB93619³ h <
0
X
µ(x)T [hx, φ] = sup
X
µ(x)T [x, φ],
EF8:¿À¸708‘°
0 Φ ,-´ E=19Ë63W.},-E=8•Ç
Î 0°*-E=,Gµ BA¸:EWB^BA¸7,œ´^,-´&,-8:¿?0H0H¿ÀB9¸:0ˆ¶T361A´dB&´dB91EWBA0·6³=<},…Ç 0=Ç
φ x∈A
φ x∈A
±
sup T [h, φ] ≤ T [h, Φ]
»¼3=1EF*-*1A0HEF*-,-;HEF¹:*G0Z873=87хBA01Aµz,-8:E=*¸7,-´dB93619³ h Ç
Ò&¸7,-´v»¼3=*-*G3W¶´»¼1936µ BA¸70^»¼3=*-*-3W¶^,G87·E=19·657µ`08/BA´HÛW»¼361TEF8}³›,G8/B90H·=0H1 <=*-0
B ¹‘0B9¸70Z´d0B
3F»^´tBA1AEFB90·6,G0m´+B9¸:EFB›°
36,G8:°,-¿70H´(¶^,GB9¸ Φ 3=8%EF*-*v¸7,œ´tBA3=1A,G0m´(3F»&*G0H87·FBA¸Éi*-0H´A´ZB9Φ¸‘EF8 i ÇD´9,G8:·
190Hº‘0mEWB90m¿?*-³ NÁ/5:EWBA,G368%Ä=<?3=8:0Z·60
B´4»¼3=1EF*-*,-86BA0·601 i <
φ
i
sup T [h, φ] ≤ sup T [h, φ]
»¼3=1EF*-*1A0HEF*-,-;HEF¹:*G0Z873=87хBA01Aµz,-8:E=*¸7,-´dB93619,-0H´ h 3=»@*-08:·FB9¸*-0H´A´4B9¸:E=8 i Ç
L3W¶ˆ<T´9,-8:°
0
,œ´™BA¸70*-,Gµ`,GB3=» sup T [h, φ] ¶^¸708 i 6· 3/0m´™BA3Ù,-8?Ï:87,GBt³=<
NÁ/5:EWBA,G368¤±z¸736*-¿:T´Ç[h, Φ]
L3W¶ˆ<?,-8¤E=8}³873=8?ÑSB90H19µ`,-8:EF* <7¶^,GB9¸
<
a
a→µ
P
P
µ(x)T [x, Φ] ≤ sup P
µ(x)T [x, φ]
Õ
!
≤ sup
Φ(a)(x)T
[x, φ]
P
≤
Φ(a)(x)T [x, Φ].
¶^¸701A0Ï:1´tB›,G870mÁ/5:EF*-,rBt³É°
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´d0m°
3=8‘¿»¼1936µßB9¸:0›¿?0Ï:87,GB9,-3=8Ì3F»
<•EF8:¿BA¸70ˆB9¸:,G1¿»¼1936µÊBA¸70» E6°BBA¸:EWBDB9¸70z´d57º
3F»Ez´957µß,œ´E=*G¶&Eb³?´N*-0H´A´&B9¸‘EWB&B9¸7Φ(a)(x)
0´957µž3F»@B9¸70´957º:´HÇ
»¼3=1E=*G*–´dBAEFB90 <:EF8:¿ <?¶40
Î 0E=190(¿?3=8:0=Û7,-8:¿?0H0H¿<7,G»¶T0ZBAE=Ë=0
Ë/8:3W¶B9¸‘EWB V ,-´873=8?ÑP870H·6EWBA,G.60=<}EF8:¿À»¼36V1(a)EF8}³ =µT¶^[a,,GB9¸ Φ]a → µ <7¶T0(¸:Eb.6a0 = 1
φ
φ∈Φi
φ∈Φi
x∈A
φ
x∈A
φ
x∈A
x∈A
P
∆µ V (a) = Px∈A µ(x)V (x) − V (a)
[x, Φ] − T [a, Φ]P
= x∈A µ(x)T
P
= −1 + ( x∈A µ(x)T [x, Φ] − x∈A Φ(a)(x)T [x, Φ])
≤ −1
¶^¸701A0ZB9¸7,-1A¿À0HÁ/5:E=*G,GBt³™°
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Õ:Ç
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B9¸70H3=1A0µÊ,œ´m µE=8:¿7EWBA3=1A³KÛ?B9¸7,œ´D°HEF8̹“0´9008 E6´E™°3=8:´90HÁ/570H8:°
0ˆ3=» 0µEF1AËÀ±:Ç:Ȓ8:¿?0H0H¿<
°
3=8‘´d,œ¿?01+B9¸:0`°3=578/BA019ÑS0½?E=µ`º7*G0›EW» B90H1 1936º‘3/´d,GB9,-3=8 1 <°
368:´d,œ¿?0H190m¿ E=´ZE"º:1936¹:EF¹7,-*-,-´dB9,œ°
EF¹:´dB91E=°
B190m¿?5:°BA,G368Ì´d³?´dB90µ¤Ç7ÈP»BA¸701A0¶401A0(Ez»¼5:8:°BA,G368 EF8:¿Ì´93=µ`0
E=´^,-8¤B9¸70
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3=8‘°
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368:´dBAEF8/B&,r»V870m°
0H´A´AEF1A³=<?EF8:¿À>µ05:*rBA,Gº7*-³},G8:·
¹/³ 1/ ,G»Z8:0H°
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873=8?ÑSB90H19µ`,-8:EF*K´dBAEWBA0 a ¶^,rBA¸ a → µ <?´d,-8:°
0+¶T0Z´d¸:3=57*œ¿¸:Eb.=0 ∆ V (a) ≤ −1 <}EF8:¿™´9,-8:°
0
,-´4E ,G1E=°B9¸:EFBT,œ´ 0½?°0º?B43=8"´93=µ`0º“3=,-8/B ¶^¸701A0D,GBT¸:E6´v.WEF*-570 </¶T0µ›5:´dBT¸:Eb.60
µ
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368:´d,œ¿?0H1+BA¸70™1 ´dB91EWB90H·=³Ì·63=,-87·
»¼1936µ
BA3 k) < (0, k − 1) <•ÇHÇÇH< (0, 0) Ç V µ5:´dBD¿?0H°190mE=´90+3=»EFB*G0mE=´dB 1 EFB0HE6°¸
B91EF8:´9,r(1,BA,G368•1)Ç:Ò&¸:(0,
EFB+*-0HE6¿7´&B93ÀE™°
3686BA1AE6¿?,œ°B9,-3=8–<:´d,-8:°0´tBEF19B9,-87·`»¼1A3=µ k <:3=8:0°EF8Ì8:3FB+¿?3
,rB k + 1 BA,Gµ`0m´&Ë=00Hº7,-87· V 873=8?ÑP870H·6EWBA,G.60=Ç
Ô©7_ª @ Z® d)® 6¨?®9­ Ù©:®’¨ f=¨ Î 0"E=1908:3W¶Ý,-8/B91A3}¿75:°
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.bEFB9,-3=8Ÿ,œ´B93·60
B›´93=µ`0B9¸7,-87· BA¸:EWB`°
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0m´Ç 36,G8:·`´93:</¶40(°HEF8°
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Y V4XWV4PUY MON0P V (V4TWXWV N0V4X T TWX ! " \ = ~U')3+!c+!$2E`g2j#.%I24*F'), '&2E%G,l'&/`},
%
µ
j2 #05z,1`},1;7,15% 3 e
=
T (Σ, X) × Dist(T (Σ, X)) m
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B•190m¿?5:°
Ñ
B9,-3=8 (T (Σ, X), → T´9³?´tBA0µß3W.601TB9¸70ˆ´90
B^3F»B90H19µ´ T (Σ, X) ¶^¸:01A0 → ,-´^¿70
Ï:870m¿
E=´+»¼3=*-*G3W¶´HÛ Î ¸708 t ∈ T (Σ, X) ,-´ˆEÀB90H19µ¤<*-0
B P os(t) ¹“0›BA¸70´d0Bˆ3F»T,GBA´(º“36´9,GB9,-3=8:´HÇ
>73=1
<}*-0
B ¹‘0+B9¸70(´957¹?B90H19µ 3=» EFBTº“36´9,rBA,G368 <?E=8:¿*-0
B ¿?08:3FB90+B9¸:0
190Hº7*-E6ρ°
∈0HµzP0Hos(t)
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BZ3F»vEFt[s]
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Y V4XWV4PUY
"
R
R
ρ
ρ
9#.#13W*1240%:,.-%>b,e$30`E`}3
%
2E5KJ
=@?BA
(T (Σ, X), →)
30D0,$'d%:,1'&;k#
t →R µ
2 %>b,1'),
2j# s'&/`},
#130;h,h~39#&2E%I24305
(g, M ) ∈ R6( U
#./*.b %>b6%
5 - e$30'k0`E` 0
σ ∈ Sub (
t|p = σ(g) (
t(
(
µ(t0 ) =
X
p ∈ P os(t) (
#130;h,^#./!$#&%(2E%I/%I24305
M (d).
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BA¸Ÿ.WEF*-570 368 EF8:¿
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3 ¶^,rBA¸
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t0 =t[σ(d)]p
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> 0(
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d
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P
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−V (f (s1 , . . . , s, . . . , sn )))
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∆f (s1 ,...,µ,...,sn ) V (f (s1 , . . . , s, . . . , sn )) ≤ −.
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µ
0
P
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a→µ
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P
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= t0 =σ(s0 ) µ0 (s0 )V (σ(s0 )) − V (a)
= ∆σ(µ0 ) V (σ(s))
≤ −.
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<–»¼3=1
<¶T0™¸‘Eb.=0
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0
1
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n
n
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= ∆f (s1 ,...,µ0 ,...,sn ) V (f (s1 , . . . , s, . . . , sn ))
≤ −.
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ÈP»
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m
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σ(M )
d
d
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Ç
1/2 × 10 + 1/2 × 2 − 10 < 0
& 0;@~U`},
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m
a → a : 1/2|b : 1/2
f (x) → x : p1 |f (f (x)) : 1 − p1
f (x) → x : p2 |f (f (x)) : 1 − p2
,-´ZE7Ç ´HÇBA01Aµz,-8:EFB9,-87·+,G» p > 1/2 EF8‘¿ p > 1/2 ÇHȒ8:¿?00m¿<F°
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m
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2
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i
2
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