From f886c947ef97fe0691de30df3dbfc3864f468937 Mon Sep 17 00:00:00 2001 From: Zhengyi Chen Date: Mon, 6 May 2024 02:50:52 +0100 Subject: [PATCH] 6/6 pages completed --- 09-concurrency.tex | 27 ++++++++++++++++++++++----- 99-notes-packed.pdf | Bin 340866 -> 343007 bytes 2 files changed, 22 insertions(+), 5 deletions(-) diff --git a/09-concurrency.tex b/09-concurrency.tex index 7ed8a3a..f8100c0 100644 --- a/09-concurrency.tex +++ b/09-concurrency.tex @@ -83,12 +83,12 @@ Whether safety/liveness properties hold in a system depends often on whether we We hence define the following completeness criteria in increasing strength (or reverse implication chain): \begin{definition}[Progress] - + In a \textbf{closed system} (i.e., without external influence of transition decisions), the \textbf{progress} assumption assumes that we will not stay forever in a state with an outgoing transition -- i.e., a finite path is complete iff it ends in a state without outgoing transitions. + + To show this is weaker than weak fairness, simply show that task $T$ cannot occur whatsoever in last state of a non-progress path (because it is enabled, but trivially not occurred -- because of non-progression past that state). \end{definition} -\begin{definition}[Justness] - -\end{definition} +There is also \textbf{justness} but this was not covered in lecture. \begin{definition}[Weak Fairness] Define a \textbf{task} to be a set of transitions in a Kripke structure (you can think of it as a subprocedure in a program). Append the Kripke structure $(S, \rightarrowtriangle, I)$ with a novel structure $\tau$: @@ -103,9 +103,14 @@ We hence define the following completeness criteria in increasing strength (or r A path $\pi$ is hence \textbf{weakly fair} if it satisfies the following LTL formula: \begin{alignat*}{2} \mathsf{WF}(T) - &\coloneqq &&\ G(\ G(\mathsf{enabled}(T)) \implies F(\mathsf{occurs(T)})\ ) \\ + &\coloneqq &&\ G[G(\mathsf{enabled}(T)) \implies F(\mathsf{occurs(T)})] \\ &\Leftrightarrow &&\ F(G(\mathsf{enabled}(T))) \implies G(F(\mathsf{occurs}(T))) \end{alignat*} + + \begin{quote} + Read it in this way: if, after some $s \in \pi$, part of T can always happen for $s^{'} \in \pi^{'}$ (i.e., perpetually enabled), then at least some part of T \underline{will} happen in $\pi^{'}$. + \end{quote} + where predicates $\mathsf{enabled}$, $\mathsf{occurs}$ follow above definition -- a state $s$ is marked with $\mathsf{enabled}(T)$ iff $T$ is enabled at state $s$ i.e. exists outgoing transition from $s$ that is also in $T$. Hence, define $\models^{WF}$ as: @@ -115,7 +120,19 @@ We hence define the following completeness criteria in increasing strength (or r \end{definition} \begin{definition}[Strong Fairness] + A task $T$ is \textbf{relentlessly enabled} on $\pi$ if for every suffix of $\pi$ there exists a state $s^{'}$ where $T$ is enabled (note the difference between perpetually enabled). + + A path $\pi$ is hence \textbf{strongly fair} if it satisfies the following LTL formula: + \begin{alignat*}{2} + \mathsf{SF}(T) &\coloneqq\ && G[GF(\mathsf{enabled}(T)) \implies F(\mathsf{occurs}(T))] \\ + &\Leftrightarrow\ && GF(\mathsf{enabled}(T)) \implies GF(\mathsf{occurs}(T)) + \end{alignat*} + + \begin{quote} + In other words: if, after some $s \in \pi$, part of T will always be available occasionally for some $s^{'} \in \pi^{'}$ (i.e., relentlessly enabled), then some part of T \underline{will} happen in $\pi^{'}$. + \end{quote} + Likewise, define $\models^{SF}$ in mirror case of $\models^{WF}$. \end{definition} \end{document} \ No newline at end of file diff --git a/99-notes-packed.pdf b/99-notes-packed.pdf index d050d5a83f48d445683dee9300076751b668522b..d947b74da4c64d01b7decf96a3177cc79607bd86 100644 GIT binary patch delta 76109 zcmXt;LvSt(6QyI@=8bLJwsm9Md}G_TZQHhu8{1CioB3;2-Bss2t6p^1sSZ3v+j~W; zzX}H9U}0iULdKy1I3Ka6`rR5F_->4=A&YDe(Z04qGXQC1=SYECUwlA zjn61UrDGMG?(5JW-1y27I$I#nJ$3a5%DHx+&1aVY5{(uJYVSX&b#+rdMOnb1QZP?@ zmvcUplhEa}G0<#37Z4fuz9!fh2v~1X;nu8M^Ug@WB3^o_KLPsx_tF&(`If;2M;F6qyU}@!kK#X zG0#x|j^hG4Pl$f}i|YAElDTM>gk`JKQspT+MEBS(L-pj8>F&ln&VK6GGKh6E^%u}? z+rN0ljZtP@euJb%g%;iS!uXS6_oK>q_SAN`XcoLUqJG${(z(adF+2h0fy>m> zU=?h=z9~uLBcDULb%$}9TMM6$=a+d!a4-5NWDOE2#*r=;g+v5Oyul`t8r zML)Hgs@4$g=m7+A%OK+f2Dh}pt(H(|1`IX=c8B_rAU8FEW}iLppFqCe*97XCRw{6S zlK=687=vJ6fK|0lcbP#{U}V|%T!RB;7JO*kwwh1_6)<@L*6Zp}t(9YOJrl<~!=|9S zR1%6*2Ku`c_;r(hX`N7OzDkGim3QyXWV>)eg-}5d*zS>_D~8R76!)X&9$DP8*NkgZ zH>+(-Ce*)TxRGkcx8e>0KltnBy?&wRTAEgDsg_40$f zD`8`yIG{XBZ|eB6QuN^H{-h9|0MRYVu2PcQyXk<{!mW1{-q{d?D_4xbe1RFjhK+b> zm8N{pSSFN-_mwH@!U0e@sNP*I*C=qR!w@5X10n;316HVlU(XSejmFDBSnlb`_QAJX zDB4KrDAdS7OoOmF45h_(sVd=6GPHNP1DlB*pdUK2LozWl+?Lqu;;Mdq_IYfc3%+$+ zga_D%7#sVbC|LvdOOJ&!nA!uF1=1Vrg;w6g6w&+OD~)r?wFh;_#Z+A@2Z}1wK8J=n z4@dcH>m>xQd*lbr63n%wu=Eo=FZ1HE`s8A$%4-UcIS0VksR9WG69u;)h$(BkKNM|W z>F^5n(IQi@b-_boW^CQItr*EYT6+#E5D}C7hZx|P`eQEDGb{8 zFxih!mszL(8%=3G1UcXwDplDFD?K8x#&bUlo~^T3JoR=Boi9rL`@|oc2>gW-`NH4S#oA@kEL5wuTfwT;sHRj|EK=6g!$KYsx8@v zubtWfnwUi&3KhIzyB6>kTIcGOlXq>-hs9KrWH*rZrT1mmtd+3rg04cyn)6UrYuGY; z0u}ZC6Nc|=1gobh&TVRW5DYxPWsi_JI^DEP$&_PV#P)fkKHS|+YYOT8d z_YAhJ-E%WFs}r!}aBtjWkT3~vP7$0wF?oF)+KiFP~C%rX*z|6j|Z zz}kPE8GQ0C;bvP`GV2(F$ItivW(Cmu!M7M7x zL;U87nNxx znZfh=TI*&4?XZoXVwE3*GUHpw@(Ip7cM`LC#}t9E!!-E6fJ|@kb!2kt`phE2$@vRg zfkcx`JNJ2nH6O00!Nq>;AqK1poxzWz8;B`9a7{pdiPIeiX4(OPtL&)nlkPcC9ev|L zmjb{*&jo1^Ttv<`k_qdCykDHNJC=PMHC%H$gOyz0gJo;`ZDx5!UbuGB&H?|n zFEbpwt$R!dq<;i;j+gVPYx{8EwK({l6L?U?(a?%vaZ;W8?stf<%Mg^*uWtqW>qvJI zH+;gbd?dqVgL0%^3 z?G6bhIFW7me`;881hQU{n-X$S_k-8}4P$rxk;c36ktY!AtEUw-;RkGRe05RXGGwfkj;)&ZKyy@AQt)?H=wP9D)1YlL zRn2j~Li}prcXS&CGE9sTBXXYdNY>Lm;z+>boa%YP76N%4Tgq2KF+KzVD>ym$N9YNB z(5*rln3?o?>`%zU;ki7k6#5}{YLRoY?j&IC--8|evmkg+bQB;`68$ms90JH918I9X zpJuJFf+`CAk?jFZA=EN7MYRR1R)w>1b=HJ*w}(RqTH>0HZC=bKJ?w!P% z@(w#jPFIj_4F}GBK`ciz`>ny1W`XZevOsn6(w&iT>>r%7B*qFSRYjCv#@=%c*8z1PD#xRFEo+}71rr!2{UeI*e6&@5)4N9@|6z%G83)AD(YeGx1=8_& z9m7VmXm^!QVysk{R>I3{Rvv<;qEeIa&9Z~fO@xhs#;cEBp$lCpo-PADu;|$^jRf!D zW!8aU-IA0)d;gPbKK4iy_t0|s3YW?Hjy(KSpAjQw3Qj{L_}5pMh$B-znT;0(kHT|R zjNU8Ri?^K0W5mQSjsbX<@C(RFvJ}+90bafCy=ZrW*FR^a1NssQ?De{rxFe)?+a~PF zC*^!h2K5ioQzaCf=S(sY1Ja>_$agsI+_1@vzgd3wSYh}S=cOR759`VC!(#A){h4k#*p}Px*+PI)DGQ)*T<48n(?#5X_Mw|4 zt7=>0nJ1b)9eR&^zy^XLmBKt6UXP<>s5@OKhED%}WgHsF-zCMBWIbT5{VWA;tJw@Y z=wIaM!ueN4Zw_cK0%QFrCNR?1wKg#ROGc)EMa8MujJ z<{-e6MISQrHr+mv{SRo^Z^g+CZNfz!8ujgwue{3wy7% z_o~FXE_sBu(8XBe$5h@!{(7*qdgk82`s#vtx|615Icg3`ZSdVV0no|R9b4mF%gAvw zXf3n@1YkfX|G04{92Vype~djaHZXX?d{3uy4gP2IooN-ZS!~$~gDyHXD72oSeLt*i zN8OeV05>VC?n4&z7yonc1dtqqEI~|{i)M^dhYbWWg5p+65R)lWBU<(yZ=&xk)C!hO((KaLA|Faev#;F&e7 zpMlF7rsXT}(dxsBzGw`&K$wZ^ov{#{EYOn%yk-~frOTN)UG8~#OgPG?wzE9!Xy{^Oe@*rz|{OY zFkyJQ2vvOXBm>_l-hNRFLmu%y?8C0*RNGrWOCe&Aj$6CyOi+KH{Y)4;m-f9(2~PmT7g+A4M&_l80}jDf2Ws0Z{EarlYOUcu$X2qOEOIUj7Z2ss%Uojy25)^)GgE*kuHZOzt8Q*nCW`WH# z%3?6~&bu!L>|dCyB1mTr~iC=D~M!hs%6b+%=nF7I(smJbgqQMSwu%8cfH{l?Nu2`uO#bUIL<~X?qKW5=btMZn!3YcLNo^bS
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