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Abstraction over Public Interfaces

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(1)Abstraction over Public Interfaces Dilian Gurov, Marieke Huisman. To cite this version: Dilian Gurov, Marieke Huisman. Abstraction over Public Interfaces. [Research Report] RR-5330, INRIA. 2004, pp.21. �inria-00070670�. HAL Id: inria-00070670 https://hal.inria.fr/inria-00070670 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Abstraction over Public Interfaces Dilian Gurov — Marieke Huisman. N° 5330 October 18, 2004 Thème SYM. N 0249-6399. apport de recherche.

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(171) ho(vf;`yNk)¤ rt_’ZoN„Z\JLx’Ztb} σ beXx’z M -°hoxkM]NapYZ\JLNz ν(σ) byXžxOz;_khoM©x’` M .°hx’MONa¤ R6_khoN_&naNhGpA°_khtxkzAV °z;_ah\M©x’4` -°hx’MON σ ¦NSJBxºnkN top (σ) = λ (σ ) ¤ M #Lf;h[ZoJ;NhGpL ¦N'†YN£Lz;N p Pub x’zB† Priv pB†YN9z;_’Zobcz;‚¢Z\J;N'X\NZX)_’‰byzAZ\NhozLxk`.pBw;f;^;`yby€'x’zL†w;hobcn&x’Z\N NG†Y‚kNGX_’ x#MONZ\J;_Y†  y® yInt ¬ ¡x]X\NZ_kw;fL^;`cbe€„MONZoJ;_Y†;X M pYhoN9X\wBNG€¸Z\bynkN9`cVa¤ ( Ë  1 . . +. Meth. 0. 0. 0. 0. 0. 00. 00. Meth. Meth. 0. 00. M. Meth. 0. IntM (m) = {(v, ε, v 0 )|v → − m v 0 ∧ v |= ¬r} m0. PubM (m) = {(v, m0 , v 0 ) | v −−→m v 0 ∧ v |= ¬r ∧ m0 ∈ M } m0. PrivM (m) = {(v, m0 , v 0 ) | v −−→m v 0 ∧ v |= ¬r ∧ m0 6∈ M 0 }. I JLNO†;N£Lz;bcZ\by_kz»_k(ZoJ;N©bczL`cbyz;byz;‚¼x’`y‚k_ah\bcZ\J;MÈfBX[NGX~x’fYÂYby`cbex’hoV6°f;zL€Z\by_kzLX χ xkzL† ζ ¤¢IJ;NO°f;zB€¸Z\by_kz χ €_kzLX\be†YNhXx’`y`NG†Y‚kNGX)hoN`ex&Z\NG†Z\_•x¢M]NZ\J;_Y† ¦bcZžhoNZ\fLh\zLXbczAZ\N9h\zBx’`xkzL†w;f;^L`cbe€

(172) NG†Y‚kNGX) )b}ZoJ¼hoNzLxkM]NG† zL_W†;N9X ' fLX\byz;‚»Z\J;N6w2NzB†YbczL‚ €xk`c`tX[Zoxa€"p¦xkzL†Ä€x’`y`eXOM]NZ\J;_Y† ζ _azÄw;h\byn&x&ZoN6N9†Y‚aN9X9¤#;f;zB€¸Z\by_kz ζ xa†;†;XžN9†Y‚aN9XZ\_¢ZoJ;N]NzAZ\hoVwB_abczAZ9pxkzL†°ho_kMZ\J;N#h\NZ\f;hoz¼w2_kbyzAZž_’ Z\J;N#w;h\byn&x&ZoN'MONZoJ;_Y†pfLX\bcz;‚•Z\J;N w2NzB†YbczL‚]€9x’`y`2X[Zoxa€ Oxkh\‚af;MONzAZ9pWx’zB†¢bcz;N9€N9XoX\xkh\V]z;_ah\M©x’`ybeX[byz;‚~ZoJ;NžhoN9X\f;`cZ °Z\J;beX¦fLX\N9X(Z\J;Nžœxk€Z(ZoJLx&Z ZoJ;NSwBN9zL†Ybyz;‚#€9x’`y`2X[Zoxa€ ]beX¦x’`y ¦xºVYX x'z;_khoM©x’`ybyX\N9† M -°hx’MON ¤%IJ;N9z•bcZ€JLN9€ YXibc¡Z\JLNžw;hobynºx’Z\NS€x’`y`2byX zL_kzY0h\NG€f;hX\bcnaNkpAbcz )J;be€J€9xkX\NSZ\J;N~w;hobynºx’Z\N„MONZoJ;_Y†beX)byz;`ybcz;NG†h\NG€f;hX[bynkN9`cVa¤ 0. 5. '5. 5. 5 '5.  

(173)   

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(175) !" $#% &  ('

(176)  )&    ) * + -, ./  0. 44658739;:;:;<.

(177) Žº. ¦·Y®o¶&µ

(178) ·YªeÀÁ©±&«. - Ë Ð™ - ­Ë ¬ —  ­³¬ ­SA¬°´;­ž: ÀI­¬ ³­¥±’« ±ÅkÅ2½y­¬  ±&« ]­ýy­a¬ ­œÆ M«2­„³¬°­»´;­~± bcz;À`y­by¬¢z;N9¶l†¯©x’Å2w;·Lw;³`y½¾NªœZ² Ë °— Ï Á©­¬ ´L¶&À À¸·L²´¢¬°´;±’¬ M M ∪ (I − I ) ´L­®o­ α (A) = ((V , L , → , A , λ ), E )  beXx]z;_khoM©x’` M .°hx’MON } p V = {w ∈ V | w p L = M ∪ {ε} S ´L­®\­ →= γ(m, )  

(179)

(180) 6 2    '  M ⊆ I+  0. M. 0. 0. 0. 0. 0. 0. −. +. 0. +. 0. 0. 0. m∈M. χ(m, σ). 0 0 = {(v S · σ, `, v · σ) | (v, `, v ) ∈ 0Int0 M (m) ∪ PubM (m)} ∪ (v,m0 ,v 0 )∈Priv(m) ζ(σ, (v, m , v )). ζ(σ, (v, m0 , v 0 )). A0 = M ∪ {r}. p. = {(v · σ, ε, ν(e · v 0 · σ) | e |= m0 ∧ e ∈ E} ∪ {(ν(rt · v 0 · σ), ε, v 0 · σ) | rt |= (m0 ∧ r)} ∪ if ¬∃i.0 ≤ i ≤ |σ| ∧ (v · σ)i |= m0 then χ(m0 , v 0 · σ) else ∅. pBx’zL†. ¤  ™ 0 - Ë ¦N°_khoN•†YbeX\€fLX\X\byz;‚Dw;ho_kw2Nh\Z\byN9X'_’)ZoJ;N•byz;`ybcz;byz;‚§x’`y‚k_khobcZ\J;Mp (N•£LhXlZOX[J;_& xŸX[byMOw;`cN NÂ;x’MOw;`yNk¤¹PYf;w;w2_aX\NŸ ¦NDJLxºnaNDx’z x’w;w;`yNZxkX†YNwLby€Z\N9† bcz4Z\J;N§`cN Z[0JLx’zB† €_a`cf;MOz _k #%bc‚af;hoNÎk¤ q|zL`cbyz;byz;‚Z\J;beX'xkw;w;`yNZ

(181) )bcZ\J»ZoJ;N¢w;f;^L`cbe€©MONZoJ;_Y†¥X[NZ {m} hoN9X\f;`cZoX~byz»Z\J;Nx’w;w;`yNZ'†;Nw;be€¸ZoN9†»byz»Z\J;N hoby‚kJAZ[0JLx’zB†'€_a`cfLM]zO_k#%bc‚af;hoNSŽa¤%rt_’Zoby€NZ\JLx’Z x’`y`YbczAZ\N9h\zBx’`;x’zB†'w;fL^;`cbe€N9†;‚kN9X‰xkh\Nw;hoN9X\NhonkNG†p& )J;by`cN wLh\byn&x&Z\N#MONZ\JL_W†Ÿ€xk`c`eXžxkh\N'h\N9w;`yxa€NG†6^WVZl ¦_¢NG†Y‚kNGX Zo_•ZoJ;N'N9zaZoh\V6xkzL†°ho_kMZ\JLN'hoNZ\fLh\z¼w2_kbyzAZS_k ZoJ;N

(182) €xk`c`yN9†MONZoJ;_Y†p;hoN9X\wBNG€¸Z\bynkN9`cVa¤ .„˜ Ï  Ë ˜º— Ë – ¨§N]X[Zox’Z\N]X[N9nkNhx’`fLX\N°f;` w;ho_kw2Nh\Z\byN9Xt_’iZoJ;N#byz;`ybczLbcz;‚x’`y‚k_ah\bcZ\J;M¤ #%bchXlZS_’(xk`c`-p2Z\J;N byz;`ybczLbcz;‚¢xk`c‚a_khob}ZoJ;M €_aMOw;fYZ\NGX)x’zx’w;wL`cNZ9¤ .„˜ Ï  Ï – œ— Ï ,  ­¬ A : I ³­¢±&«Ã±oÅaÅ2½c­¬S±’«  M ⊆ I ±¢À­¬ž¶l¯Å2·L³½¾ªœ²OÁ©­¬ ´L¶ & À   ´;­Oª «B½¾ª «2­  ±ÅkÅ"½c­¬ α (A) ´;±&Àª «L¬|­®°¯±a²o­ I = (M, M ∪ (I − I ))  b.¤ Nk¤ α (A) : (M, M ∪ (I − I ))  ¦V•g h\_awB_AX[bcZ\by_kz] (NSZ\JWfLX‚aNZ  λ0 = σ 7→ {λMeth (σ0 )} ∪ (if (|σ| = 1 ∧ σ0 |= r) then {r} else ∅) E 0 = {v ∈ E | λMeth (v) ∈ M }

(183). +. M. −. αM (A). +. −. M. +.

(184). bM (αM (A)) = b(αM (A)). PYbczL€N©Z\JLN•byz;`cbyz;byz;‚¼ZohoxkzLX[°_khoMOx’Z\by_kz α _az;`cV»x "N9€ZoX'M]NZ\J;_Y†;X#z;_kZ'bcz M p α beX~Z\JLN•be†YNzAZ\bcZlV _awBN9hox’Z\by_kz¡¤ .„˜ Ï  Ï – œ— Ï  ­¬ A : I ³­#±’«§±oÅaÅ2½c­¬   ´L­« α (A) = A  I+. M. I+. 021 4 043.

(185) ³À¸¬.®o±k²¬.ªœ¶’«Ÿ¶&µ&­®¦·L³½}ªœ²«L¬|­®°¯±a²o­À. ŽGŠ. m m. a. b v4.v3.v1. v0. v2. v4 a. a a. b. v0. v2.v1 v5.v3.v1. v5. v6.v3.v1. v6 m m. v1. v3.v1 v7.v3.v1. v1. v3. v7. #by‚kfLh\NOŽ*‰miÂYxkMOw;`cN~xkw;w;`yNZ #%byzLx’`y`cVapYZ\JLN~bcz;`ybyz;bczL‚¢x’`y‚k_khobcZ\J;M. .„˜ Ï  Ï – œ— Ï &. ¬°´;±’¬ I. − A. −. + IA. ­¬ A : I ±&«  I ⊆M. A − B. B. N9z&jl_&VYX¦Z\J;N„°_a`c`y_& )bczL‚]†YbeX[Z\hobc^;f;Z\bynAbcZlV•wLh\_awBN9h[ZlVa¤ ’± «  B : I ³o­O±ÅkÅ2½y­¬.À  ±&« ¢½y­¬ M ⊆ I ±&«  M ⊆ I ³ ­'À¸·L²o´   ´L­« −I ⊆M + B. B. + A. A. + B. B. A. $  -œ™Y

(186) — Ï •Ë –  -œ—9– utXx’`yh\NGxk†YV M]N9zAZ\by_kz;NG†p)Z\JLNDbyzaZoNh\œxk€N6^BN9JLxºnWbc_af;h¢_’~ZoJ;ND_ah\by‚kbyzLxk` kx w;w;`yNZ¦byX¦w;ho N9X\NhonkNG†O^AV] ZoJ;Nžbyz;`ybczLbcz;‚]x’`y‚k_ah\bcZ\J;Mp2ª  ­ 2N9nkN9h\V]NÂYN9€fYZ\by_kz•_’¡ZoJ;NtbyzAZ\Nh\œxk€Nt^2NJLxºnWby_kf;h _k A beX

(187) x’z§NÂYN9€fYZ\by_kz§_’(Z\J;N©^2NJLxºnWby_kf;h„_k α (A) ¤¢IJ;byX

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