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What are convention and correspondence? What are they for? Why do we bother? What about the rule that kernel operations cannot call other kernel operations?B O w u  H C (Group 6#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK ,$D 1i  LS?Text Box 4/ ,$D 1 3(We were not drunk when we made it up. Seriously!)<4~8 <Z   (  4   A??DCKB00977Picture 5F:\IMAGES\PEOPLE\MEN\CKB00977.WMF H,$D 1 13___PPT10+#D|' = @B D7' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+"4 l(  "   x Z0e0e ?Rectangle 2`  1 VExtension vs. Kernel     x@[0e0e ?Rectangle 3I " 1 Extension Usually layered over a kernel Does not access representation directly Kernel Defines representation Directly accesses representation\ F  8  F8 2  \`\jJ?Oval 4pP /  N   2  \]jJ?Oval 5p  N     R^jJ?Text Box 6` p G  \Kernel     R`jJ?Text Box 7   _ Extension      ZjJ?AutoShape 8m `    ZjJ?AutoShape 98  tB   LjJ?Line 10 tB   LjJ?Line 11p   n d1?Rectangle 12  N      `>1?Text Box 13Mf4  YRep     Z=?jJ?Text Box 14M8! \Op_1     Z-`jJ?Text Box 15X \Op_2     ZyfjJ?Text Box 16 '  \Op_4     Z`ojJ?Text Box 17z N  \Op_3     T !jJ?Text Box 18|t ^Op_Ext   /@  13" :20(      x!0e0e ?Rectangle 2`  1 TThe Correspondence     x#0e0e ?Rectangle 3 "H$1 1 jConsider the implementation of Sorting_Machine_Kernel with Heapsort What s the math model of Sorting_Machine? What s the representation? What s the math model of the representation?*D r Dr  13" Q I P (  =   x$0e0e ?Rectangle 2`  1 q/Math Model of Sorting_Machine0 0 0  F&?Text Box 4@  ,$D 1 xSorting_Machine_Type is modeled by ( inserting: boolean contents: multiset of Item )Zy   y w    C *Group 11#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK  ,$D1   z@(GH:I/JuKyLujJ?AutoShape 12@   c self.contents   " B # `)G{2HIfdJ-KfdLtM8XNtjJ?AutoShape 13 -  dself.inserting   /  135-___PPT10 +AكD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+" D<` (   -   ~@+0e0e ?Rectangle 1026`  1 [Representation Math Model    RW?Text Box 1027 ? >Rep is modeled by ( inserting_rep: boolean queue: string of Item array: ARRAY_MODEL heap_size: integer )p   4  2  RU?Text Box 1028 G  math subtype ARRAY_MODEL is ( lb: integer ub: integer table: INDEXED_TABLE ) constraint & |z    '  z 6  RW?Text Box 1029   math subtype INDEXED_TABLE is finite set of ( i: integer x: Item ) constraint & ||   &  |  { %`N  C .Group 1046#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK%`N,$D1    ~ gGHBI&JPKLPjJ?AutoShape 1030 %` hself.inserting_rep       @kG}HIbJPK,YLPjJ?AutoShape 1031@ ` self.queue        ~gGHBIJPKLPjJ?AutoShape 1032  ` self.array         hG rHI^JPKWLPjJ?AutoShape 1033^N dself.heap_size   z p    C .Group 1044#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|drs/downrev.xmlDA0oVQEd]<ȂUXhmy)M,Yo{SWZVO"ę% x ydeR `M}.] DJdMlMm \7FB,9,XӾ_vYٞ29ŤhV <~Z  7OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|drs/downrev.xmlPK p ,$D1.   3 mG]H! I{JK{LMNjJ?AutoShape 1037p>  fself.array.table   4   #  mGHIeJnKeLMMXNjJ?AutoShape 1042  p rself.array.lb self.array.ub    o`       13 ___PPT10+ DD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +"N6 &%$%(  $F $  ~@X0e0e ?Rectangle 1026`  1 t2Correspondence Continued&    $ R j?Text Box 1027 ,$D 1 $Sorting_Machine_Type is modeled by ( inserting: boolean contents: multiset of Item )Z 1     $ R@.?Text Box 1028h O " (Rep is modeled by ( inserting_rep: boolean queue: string of Item array: ARRAY_MODEL heap_size: integer )Z   B  h{ b\  $C .Group 1043#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPKb\ ,$D1b $ fkjJ?AutoShape 1033b  N    $ XljJ?Text Box 1034@\Z  Y? (  nz  =@] $C .Group 1039#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|drs/downrev.xmlDA0oVQEd]<ȂUXhmy)M,Yo{SWZVO"ę% x ydeR `M}.] DJdMlMm \7FB,9,XӾ_vYٞ29ŤhV <~Z  7OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|drs/downrev.xmlPK= @],$D12 $ \`ijJ?Oval 1029 =@] N    $ XjjJ?Text Box 1035 m jabstract state space   p|  @ $C .Group 1041#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!R&drs/downrev.xmlDOk@&B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.U9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!R&drs/downrev.xmlPK @,$D12 $ \ gjJ?Oval 1030 @ N    $ X@hjJ?Text Box 1036 y  jconcrete state space   {   $C .Group 1040#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!=ك(drs/downrev.xmlDAk@&-)% VzB g̾ 5+zffM ^D K?_>@8L [ϞVj;7 _a.ҕ5t b{>Ⱦ1M+_]l8,Ѧz-TĤ|ʗ I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! _drs/downrev.xmlDA0 okZeEQDT<*GlKib[Y87|ٙR4T²x N.8Sp9o GXZ&/r\h/5'a*ҥ9tC[nk>:6M)GQ4  9V)}Fv57q_nR~ &w%xPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! _drs/downrev.xmlPK. ,$D1,  $  ~@bG"|HDhIVJjJ?AutoShape 1032P  v (true, <2, 1, 3>, (1, 0, {}), 0)! ! ! 2  $ bcjJ?Oval 1038. ^  N   /  $ $ 13___PPT10+@jD' = @B DX' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-o6Bdissolve*<3<* $D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$+"$ ((  (@ (  xH0e0e ?Rectangle 2`  1 t2Correspondence Continued&   $ (  x,0e0e ?Rectangle 3 " 1 XGiven the Sorting_Machine m with value m = (true, {1, 2, 3}), what s the corresponding value for the representation? How do you know? Conversely, given the representation (true, <2, 1, 3>, (1, 0, {}), 0), what Sorting_Machine value does it represent? How do you know?     13" D<,$(  ,@ ,  x@%0e0e ?Rectangle 2`  1 t2Correspondence Continued&    ,  x`B0e0e ?Rectangle 3 " 1 The correspondence provides the mapping between the abstract value (the sorting machine) and the concrete value (the representation)& s  13"0 0(  0C 0  xo0e0e ?Rectangle 2`  1 w5Sorting_Machine with Heapsort: Correspondence6 6 6- 0  x m0e0e ?Rectangle 3 " 1 aWself.inserting = self.inserting_rep and if self.inserting then self.contents = elements (self.queue) else for all x: Item (x is in self.contents iff there exists i: integer (self.array.lb <= i and i <= self.heap_size and (i,x) is in self.array.table))X         X 13"  4(  4 4  x`o0e0e ?Rectangle 2`  1 PThe Convention   4  xp0e0e ?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!(drs/downrev.xmlDOK1G!fmצTE[AhuM$q8x`Zѓ ƣV Dh5ΒX.iw;,!Cu]&e(j2F#ˬt`WR{<ܴ2M[iPcG뚊QM4_z[}NŰiϮ?zܒ1+>e rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!(drs/downrev.xmlPK "H$1 1 6What Sorting Machine is represented by (true, <2, 1, 3>, (1, 0, {}), 5) (true, <2, 1, 3>, (1, 2, {(1, 7), (2, 8)}), 3) (false, <>, (1, 2, {(1, 7), (2, 8)}), 3)? Is the following a correct implementation of Size? What would you need to know to answer this question? { return self[heap_size]; }R' Zz Z Z'zi- 7 13VN___PPT10..+NwnD ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(Dp ' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4'%(D' =-o6Bdissolve*<3<*4'D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4'H%(D' =-o6Bdissolve*<3<*4'HD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4Hw%(D' =-o6Bdissolve*<3<*4HwD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4w%(D' =-o6Bdissolve*<3<*4wD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*47%(D' =-o6Bdissolve*<3<*47+8+0+41 +" 8}(  88 8  xq0e0e ?Rectangle 2`  1 l*Convention Continued&    8  xs0e0e ?Rectangle 3 " 1 9The convention states any implementation-wide assumptions the implementer makes It restricts the space of possible representation values It allows us to reason about the correctness of each operation independently of the others( Z   13"- V#N# <#(  <8 <  x}0e0e ?Rectangle 2`  1 l*Convention Continued&    < L~?Text Box 3 ,$D 1 $Sorting_Machine_Type is modeled by ( inserting: boolean contents: multiset of Item )Z 1     < L@?Text Box 4h O " >Rep is modeled by ( inserting_rep: boolean queue: string of Item array: ARRAY_MODEL heap_size: integer )p   4  1 w  =@ <C *Group 25#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK= @,$D12 < VdjJ?Oval 5 =@] N   2 < VOjJ?Oval 6 @ N   b < ``jJ?AutoShape 9b  N    < TjJ?Text Box 11 m jabstract state space    < TjJ?Text Box 12 @ '  jconcrete state space   d v  S <C *Group 28#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|drs/downrev.xmlDA0oVQEd]<ȂUXhmy)M,Yo{SWZVO"ę% x ydeR `M}.] DJdMlMm \7FB,9,XӾ_vYٞ29ŤhV <~Z  7OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|drs/downrev.xmlPK S,$D12 < ^`IjJ?Oval 14@0 Pp N     < \JjJ?Rectangle 15p`S v (true, <2, 1, 3>, (1, 0, {}), 5)! ! ! |B < LjJ?Line 16 p02 < ^LjJ?Oval 22 .  N    < T tjJ?Text Box 23   ` convention    |B < LjJ?Line 240 @ S x  <C *Group 26#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!R&drs/downrev.xmlDOk@&B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.U9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!R&drs/downrev.xmlPK,$D1 <  xBG{H@WIWJjJ?AutoShape 7  g(true, {1, 2, 3})     < TCjJ?Text Box 13   dcorrespondence     <  ~@ABCDEFQRUVWXjJ?p^lT<<|@  TFreeform 19`P 2  < ^EjJ?Oval 20K{ N   2  < ^FjJ?Oval 21. ^  N   &  <  xGG"|HDhIVJjJ?AutoShape 8P  v (true, <2, 1, 3>, (1, 0, {}), 0)! ! ! / < < 13  ___PPT10 ++B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<+"  @@(  @F @  xu0e0e ?Rectangle 2`  1 z8Sorting_Machine with Heapsort: Convention9 9 9 @  x@{0e0e ?Rectangle 3 " 1 B2if self.inserting_rep then self.heap_size = 0 and self.array = (1, 0, empty_set) else self.array.lb = 1 and 0 <= self.heap_size and self.heap_size <= self.array.ub and SUB_TREE_IS_ORDERED ( self.array, 1, self.heap_size) and self.queue = empty_string3 Z 'P  3 13" `Di(  D( D  x{0e0e ?Rectangle 2`  1 \Convention: One More Thing   D  x0e0e ?Rectangle 3 " 1 5In implementing a kernel component, the implementer assumes that the convention holds at the beginning of each kernel operation It is the responsibility of the implementer to ensure that the convention holds at the end of each kernel operation   13"  pH(  H8 H  x 0e0e ?Rectangle 2`  1 l*Convention Continued&    H  x@0e0e ?Rectangle 3 " 1 yWhat happens if in the middle of a kernel operation (where the convention may not hold) we call another kernel operation?z z z 13"6 wP_(  P P N ?Rectangle 7`   l* C    P  r0e0eL?Rectangle 2   1j P  x`0e0eL?Rectangle 3 @  1 TAt this point, students should have thought about convention and correspondence for the previous closed lab (implementation of Sorting Machine with one of Queue sort algorithms) and for the homework due today (implementation of Sorting Machine with heapsort) These are the terms/questions they will be able to understand by the end of classU U U  ̙33U 6   T(  T T N ?Rectangle 7`   l* C    T  0e0e L?Rectangle 2   1 T  `0e0eL?Rectangle 3 @  1 rExtensions do not introduce a new type, so they don t need to define a representation. They are implemented by calling the kernel operations of the type being extended, and should not access the representation directly (C++ does allow extensions to access the representation that s by design: we could have made the representation private to prevent it, but in some exceptional situations, for efficiency reasons, we may want to allow an extension to be built directly using the representation) Kernels need to implement a type, and that s why they need to define a representation for the type For the rest of the day we ll talk only about issues involved in implementing kernels (not extensions)     ̙33J6  @X(  X X N ?Rectangle 7`   l* C    X  r0e0eL?Rectangle 2   1 X  x0e0eL?Rectangle 3 @  1 5We have a math model for the type we are implementing (Note that self.inserting and self.contents in specs refer to the fields of the math model) We also have a representation. But the representation has its own math model.    ̙33y6 91p\(  \ \ Ny?Rectangle 7`   l* C    \  r0e0eL?Rectangle 2   1$ \  x@x0e0eL?Rectangle 3 @  1 dThe math model of the representation is obtained by replacing each program type in the representation with its corresponding math model Quickly review ARRAY_MODEL (Note that when we write self.size, self.inserting-rep, etc, we refer to the fields of the representation)     ̙33[6 `(  ` ` Nt?Rectangle 7`   l* C    `  r0e0eL?Rectangle 2   1 `  x@0e0eL?Rectangle 3 @  1 FSe we have the math model of sorting machine (abstract state space) and the math model of the representation of sorting machine (concrete state space) Here are typical values in the two spaces The question is how do we map one to the other?    ̙3336 d(  d d N`?Rectangle 7`   l* C    d  r0e0eL?Rectangle 2   1 d  x@0e0eL?Rectangle 3 @  1 Here is the convention from Lab1 explain the various pieces (again, students should have already thought about it in the homework) What happens if we have a representation where some items are in the queue and some are in the array? Or what if there are 6 items in the queue, but size = 5? The correspondence may or may not give us a value for the sorting machine, but it is clear that we don t want these weird values in the representation. That s what the convention is for. "     ̙336 WOh7(  h h N?Rectangle 7`   l* C    h  x0e0eL?Rectangle 1026   1< h  ~@0e0eL?Rectangle 1027 @  1 v Using the correspondence, determine the values of the sorting machine in each of the three examples (true, {1,2,3}), (true, {1,2,3}), (false, {7,8}) Intuitively, are these valid representations? NO What about question 2? If we know that at all times self[size] contains the actual size of the sorting machine, then this is a correct implmentation. However, if that s not the case, then this is probably not a correct implementation (with the given convention, size =0 in insertion phase, so this is NOT a correct implementation)    ̙336 ly(  l l N?Rectangle 7`   l* C    l  x0e0eL?Rectangle 1026   1~ l  ~0e0eL?Rectangle 1027 @  1 bThese are three ways to state the same idea, or three definitions or properties of the convention.c c c  ̙33K6  0p(  p p N?Rectangle 7`   l* C    p  r0e0eL?Rectangle 2   1 p  x0e0eL?Rectangle 3 @  1 6Example of bad value for representation (size does not match the actual size nor 0) Convention can be viewed as a restriction on valid values in the concrete state space Correspondence only needs to be defined on convention     ̙330 6 Pt(  t t N?Rectangle 7`   l* C    t  r0e0eL?Rectangle 2   1 t  x0e0eL?Rectangle 3 @  1 Here is the convention from lab1 It states that the array is empty in insertion phase, the queue is empty in extraction phase, and size is used only in extraction phase (it is 0 in insertion phase)    ̙33C 6 x(  x x N?Rectangle 7`   l* C    x  r0e0eL?Rectangle 2   1 x  x0e0eL?Rectangle 3 @  1 .We cannot guarantee that the invoked kernel operation will do what it says. That s why we have the rule disallowing this. We could have exceptions if we knew the convention holds at the point where the kernel operation is invoked, or if we could argue that the invoked operation is still going to work. 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