Softbound. March 23, Type 2. 2 Env primitives defined functions axioms well-formed environment...

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1 Softbound March 23, 2009 Content 1 Tye 2 2 Env rimitive defined function axiom we-formed environment Syntax 10 4 Semantic 13 5 Proertie 18 A Notationa Convention 21 Lit of Figure 1 Tye Tye Equivaence Tye Converion Tame Pointer Coure We-formed Tye We-formed Environment Syntax We-formed LHS We-formed RHS We-formed CMD Reut Data Cat Aertion Evauation LHS Evauation RHS Evauation RHS - Con Evauation CMD

2 1 Tye Quaifier: Tye: Size of Tye: q ::= afe eq tame Quaifier a ::= atomic tye int int q ointer tye ::= ointer tye a atomic tye anonymou truct tye id named truct tye void void ::= truct tye ni truct ; id:a con truct tab ::= id otion named truct ook-u tabe aize(a) ::= ize() ::= ize() ::= where 1 a = int 1 a = q where aize(a) = a ize() = ize() = n tab n = ome 1 = void where 0 = ize( ) + aize(a) = ; id:a Figure 1: Tye. 2

3 a a a int a int (E-Int) 1 2 q 1 = q 2 1 q 1 a 2 q 2 (E-Pointer) a 1 a a 2 a 1 a 2 (E-Atomic) (E-AStruct) tab n 1 = ome 1 tab n 2 = ome n 1 n 2 (E-NStruct) n n (E-EqName) void void (E-Void) (E-Ni) 1 2 a 1 a a 2 1 ; id 1 :a 1 2 ; id 2 :a 2 (E-Con) Figure 2: Tye Equivaence. 3

4 a a int int (C-Int) q int (C-Ptr-Int) q afe int q (C-Int-Ptr) afe 2 afe (C-SafePtr) afe 2 eq (C-SafeSeqPtr) eq 2 afe (C-SeqSafePtr) eq 2 eq (C-SeqPtr) 1 tame 2 tame (C-TamePtr) Figure 3: Tye Converion. 4

5 a a a int (TC-Int) a tame (TC-Ptr) a a a (TC-Atomic) (TC-AStruct) tab n = ome n (TC-NStruct) void (TC-Void) (TC-Ni) a a ; id:a (TC-Con) Figure 4: Tame Pointer Coure. 5

6 a a a int (WFT-Int) q tame void a q (WFT-NonTamePtr) a tame (WFT-TamePtr) a a a (WFT-Atomic) (WFT-AStruct) tab n = ome n (WFT-NStruct) void (WFT-Void) (WFt-Ni) a a ; id:a (WFT-Con) Figure 5: We-formed Tye. 6

7 2 Env 2.1 rimitive Tabe 1: rimitive Name Function baeaddr N owet uer-acceibe addr maxaddr N max uer-acceibe addr TOP N tack to addr Stack v otion (d a) tack Mem d (b,e) memory TyeInfo a tye information Env (Stack, M em, T yeinf o) environment Vaue N Vaue of Memory Bae N MetaData, Bae of Memory End N MetaData, Bound of Memory Loc N Location of Memory readmem Mem Loc otion V aue read data readmemmeta Mem Loc otion V aue (Bae,End) read data with meta writemem Mem Loc V aue otion Mem write data writememmeta Mem Loc V aue (Bae,End) otion Mem write data with meta maoc Env N otion (Env Loc) memory aocation udateti T yeinfo Loc P trt ye N T yeinfo udating tye information readmem M : read data from the ocation if it i acceibe readmemmeta M : read data with meta from the ocation if it i acceibe writemem M d: write data to the ocation if it i acceibe writememmeta M d (b,e) : write data with meta to the ocation if it i acceibe maoc : memory aocation udateti : udating tye information 2.2 defined function readmembock M ize readmemmetabock M ize writemembock M d ize writememmetabock M d ize coymembock M d ize coymemmetabock M d ize vaidmem M d.readmem M = ome d d. M.writeMem M d = ome M vaidmembock M ize 7

8 2.3 axiom axiom 2.1 (vaidaddrerange) 0 < baeaddr maxaddr axiom 2.2 (vaid memory) 1. (M, ). ( d.readmem M = ome d) ( d. M.writeMem M d). 2. (M, ). ( d (b,e).readmemmeta M = ome d (b,e) ) ( d (b,e). M.writeMemMeta M d (b,e) ). 3. (M, ). ( d.readmem M = ome d) ( d (b,e).readmemmeta M = ome d (b,e) ). 4. (M, ). ( d. M.writeMem M d) ( d (b,e). M.writeMemMeta M d (b,e) ). axiom 2.3 (unique reut) 1. (M, ). ( (d, d ). readmem M = ome d readmem M = ome d d = d ). 2. (M, ). ( (d (b,e), d (b,e ) ). readmemmeta M = ome d (b,e) readmemmeta M = ome d (b,e ) d (b,e) = d (b,e ) ). 3. (M, ). ( (d, d ). M.writeMem M d M.writeMem M d ). 4. (M, ). ( (d (b,e), d (b,e ) ). M.writeMemMeta M d (b,e) M.writeMemMeta M d (b,e ) ). axiom 2.4 (udatetyeinfo inverion) 1. If a q, q tame, ize() > 0 and udatetyeinfo T I ize = T I, then ( [, + ize)). T I ( ) = [( ) mod ize()] t and ( < + ize)). T I( ) = T I ( ). 2. If a tame, ize() > 0 and udatetyeinfo T I ize = T I, then ( [, +ize)). T I ( ) = void tame and ( < + ize)). T I( ) = T I ( ). axiom 2.5 (maoc inverion) If maoc E ize = ome ((M, S, T I ), ), then 1. M, T I.E = (M, S, T I) 2. baeaddr + ize < maxaddr ize > 0 3. (readmemmeta M = ome d (b,e) ). readmemmeta M = ome d (b,e) 4. ( < + ize). readmemmeta M = none readmemmeta M = none 5. ( < + ize). readmemmeta M = none readmemmeta M = ome 0 (0,0) 6. ( < + ize). T I( ) = T I ( ) 7. ( < + ize). T I ( ) = int axiom 2.6 (writemem Inverion) 1. If writememmeta M d (b,e) = ome M, then (a) readmemmeta M = ome d (b,e) (b) If ( ). readmemmeta M = d, then readmemmeta M = d (c) If. readmemmeta M = none, then readmemmeta M = none 2. If writemem M d = ome M, then (a) If readmemmeta M = ome d (b,e), then readmemmeta M = ome d (b,e) (b) If ( ). readmemmeta M = d, then readmemmeta M = d (c) If. readmemmeta M = none, then readmemmeta M = none 2.4 we-formed environment 8

9 M ; T I S S baeaddr T OP maxaddr (v,, a). (v (, a)) S (T OP ) (( + aize(a)) < maxaddr) (v,, a). (v (, a)) S (v,, a ). (v (, a )) S ( ( + aize(a))) ( ( + aize(a ))) M ; T I S S (WF-Stack) M ; T I D d (b,e) : a M ; T I D d (b,e) : int true (WFD-Int) M ; T I D d (b,e) : a afe (d = 0) (WFD-ASafe) ((baeaddr d) (d + 1 < maxaddr) (vaidmem M d) (T I(d) = a)) M ; T I D d (b,e) : afe (d = 0) (WFD-SSafe) ((ize() > 0) (baeaddr d) (d + ize() < maxaddr) ( (i [0, ize())). (vaidmem M d + i (T I(d + i) = [i]))) M ; T I D d (b,e) : n afe (d = 0) (WFD-NSafe) (. tab n = ome (ize() > 0) (baeaddr d) (d + ize() < maxaddr) ( (i [0, ize())). (vaidmem M d + i (T I(d + i) = [i]))) M ; T I D d (b,e) : void afe fae (WFD-VSafe) M ; T I D d (b,e) : eq (b = 0) (WFD-Seq) ((b 0) (baeaddr b e < maxaddr) ( (i [b, e)). (vaidmem M i (T I(i) = [(i d) mod ize()]))) M ; T I D d (b,e) : tame (b = 0) (WFD-Tame) ((b 0) (baeaddr b e < maxaddr) ( (i [b, e)). (vaidmem M i q.t I(i) = q tame)) M M ; T I (, d, b, e). readmemmeta M = d (b,e) M ; T I D M M ; T I d (b,e) : T I() (WF-MemTI) E E M E.M ; E.T I E.M ; E.T I S E.S (v,, a). (v (, a)) E.S vaidmem E.M E.T I() = a E E (WF-Env) Figure 6: We-formed Environment. 9

10 3 Syntax Syntax: h ::= rh ::= c ::= h exreion v variabe h dereference h id truct o h n id name o rh exreion i int contant h h exreion (a q)&h reference rh + rh addition (a)rh cat (izeof)a ize (a q)maoc rh aoc command ki ki c ; c equence h = rh aignment Figure 7: Syntax. 10

11 S h:a (v (, a)) S a a S v :a (WFL-Var) S h:a q S h:a (WFL-Def) S h: q [id] = a S h id:a (WFL-StructPo) S h:n q tab n = ome [id] = a S h n id:a (WFL-NamePo) S!tame h:a (v (, a)) S a a S v :a (WFLNT-Var) S!tame h: q q tame [id] = a S!tame h id:a S!tame h:n q q tame tab n = ome [id] = a S!tame h n id:a (WFLNT-StructPo) (WFLNT-NamePo) S tame h:a (v (, void tame)) S S tame v :void tame (WFLT-Var) S tame h: tame [id] = void tame S tame h id:void tame (WFLT-StructPo) S tame h:n tame tab n = ome [id] = void tame S tame h n id:void tame (WFLT-NamePo) Figure 8: We-formed LHS. 11

12 S r rh:a S r i:int (WFR-Cont) S h:a S r h:a (WFR-Lh) S!tame h:a a a afe S r (a afe)&h:a afe (WFR-RefSafe) S!tame h:a a a eq S r (a eq)&h:a eq S tame h:a a a tame S r (a tame)&h:a tame (WFR-RefSeq) (WFR-RefTame) S r rh 1 :int S r rh 2 :int S r rh 1 + rh 2 :int (WFR-Add) S r rh 1 : q q afe S r rh 2 :int S r rh 1 + rh 2 : q (WFR-AddPtr) S r rh:a a a S r (a )rh:a (WFR-Cat) a a S r izeof(a):int (WFR-Size) S r rh:int a q ize() > 0 S r ( q)maoc rh: q (WFR-Aoc) Figure 9: We-formed RHS. S c c S c ki (WFC-Ski) S c c 1 S c c 2 S c c 1 ; c 2 (WFC-Seq) S h:a S r rh:a r a r a S c h = rh (WFC-Aign) Figure 10: We-formed CMD. 12

13 4 Semantic Annotation USAGE d (b,e) d with meta (b, e) d id (b,e) d with meta (b, e), id i the name of d ub fied [id] off the offet ub fied id [id] t the tye ub fied id Reut: r ::= err ::= reut ok Succ ocation (d (b,e), a) data with meta Abort Abort OutOf M em OutOfMem error Abort Abort OutOf M em OutOfMem Figure 11: Reut. datacat: from to aertion (d (b,e), q) (d (b,e), int) (d (b,e), int) (0 (0,0), afe) d = 0 (d (b,e), int) (d (0,0), eq) (d (b,e), int) (d (0,0), tame) (d (b,e), eq) (d (b,e), afe) (d (b,e), afe) (d (d,d+ize()), eq) (v = 0) (b 0 b d <= (e ize())) (d (b,e), ) (d (b,e), ) Figure 12: Data Cat. Aertion: aert d (b,e) a afe d 0 aert d (b,e) a eq b 0 b d d + aize(a) <= e aert d (b,e) a tame b 0 b d d + aize(a) <= e aert d id (b,e) afe d 0 aert d id (b,e) eq b 0 b d + [id] off + aize([id] t ) <= e aert d id (b,e) tame b 0 b d + [id] off + aize([id] t ) <= e Figure 13: Aertion. 13

14 E h r :a (v (, a)) E.S E v :a (Ev-Var) E h :a q readmem E.M = ome (b,e ) aert (b,e ) a q E h :a (Ev-Def) E h e:a E h e:a (Ev-Def-ErrorPro) E h :a q readmem E.M = ome (b,e ) aert (b,e ) a q E h Abort:a (Ev-Def-Abort) E h : q readmem E.M = ome (b,e ) aert id (b,e ) q E h id + [id] off :[id] t E h e:a E h id e:a (Ev-StructPo-ErrorPro) (Ev-StructPo) E h : q readmem E.M = ome (b,e ) aert id (b,e ) q E h id Abort:[id] t (Ev-StructPo-Abort) E h :n q tab n = ome readmem E.M = ome (b,e ) aert (b id,e ) q E h n id + [id] off :[id] t (Ev-NamePo) E h e:a E h n id e:a (Ev-NamePo-ErrorPro) E h :n q tab n = ome readmem E.M = ome (b,e ) aert (b id,e ) q E h n id Abort:[id] t (Ev-NamePo-Abort) Figure 14: Evauation LHS. 14

15 E r rh r :a r E E r i (i (0,0), int):int r E (Ev-Cont) E h :a readmemmeta E.M = ome d (b,e) E r h d (b,e), E.T I():a r E (Ev-Lh) E h e:a E r h e:a r E (Ev-Lh-ErrorPro) a a afe E.S!tame h:a E h :a E r (a afe)&h ( (0,0), a afe):a afe r E (Ev-RefSafe) a a afe E.S!tame h:a E h :a E r (a eq)&h ( (,+aize(a)), a eq):a eq r E (Ev-RefSeq) a a afe E.S tame h:a E h :a E r (a tame)&h ( (,+aize(a)), a tame):a tame r E (Ev-RefTame) E h e:a E r &h e:a r E (Ev-Ref-ErrorPro) E r rh 1 (d 1(b1,e 1), a 1 ):int r E E r rh 2 (d 2(b2,e 2), a 2 ):int r E E r rh 1 + rh 2 (d 1 + d 2(0,0), int):int r E (Ev-Add) E r rh 1 e:a r E E r rh 1 + rh 2 e:a r E (Ev-Add-ErrorPro1) E r rh 1 (d 1(b1,e 1), a 1 ):int r E E r rh 2 e:a r E E r rh 1 + rh 2 e:a r E (Ev-Add-ErrorPro2) E r rh 1 (d 1(b1,e 1), a 1 ): q r E E r rh 2 (d 2(b2,e 2), a 2 ):int r E E r rh 1 + rh 2 (d 1 + d 2 ize() (b1,e 1), q): q r E (Ev-AddPtr) E r rh 1 e:a r E E r rh 1 + rh 2 e:a r E (Ev-AddPtr-ErrorPro1) E r rh 1 (d 1(b1,e 1), a 1 ): q r E E r rh 2 e:a r E E r rh 1 + rh 2 e:e r E (Ev-AddPtr-ErrorPro2) Figure 15: Evauation RHS. 15

16 E r rh r :a r E E r rh (d (b,e), a 0 ):a r E datacat d (b,e) a a = d (b,e ) E r (a )rh (d (b,e ), a 0):a r E (Ev-Cat) E r rh e:a r E E r (a )rh e:a r E (Ev-Cat-ErrorPro) E r rh (d (b,e), a 0 ):a r E datacat d (b,e) a a = d (b,e ) E r (a )rh Abort:a r E (Ev-Cat-Abort) E r izeof(a) (aize(a) (0,0), int):int r E (Ev-Size) E r rh (d (b,e), a):int r E d ize() maoc E d = ome (E, ) E r ( afe)maoc rh ( (0,0), afe): afe r (E.M, E.S, udatetyeinfo E.T I d) (Ev-AocSafe) E r rh (d (b,e), a):int r E maoc E d = ome (E, ) E r ( eq)maoc rh ( (,+d), eq): eq r (E.M, E.S, udatetyeinfo E.T I d) (Ev-AocSeq) E r rh (d (b,e), a):int r E maoc E d = ome (E, ) E r ( tame)maoc rh ( (,+d), tame): tame r (E.M, E.S, udatetyeinfo E.T I d) (Ev-AocTame) E r rh e:a r E E r ( q)maoc rh e:a r E (Ev-Aoc-ErrorPro) E r rh (d (b,e), a):int r E maoc E d = none E r ( q)maoc rh OutofMem:a r (E (Ev-Aoc-OutofMem) E r rh (d (b,e), a):int r E d < ize() E r ( afe)maoc rh Abort:a r (E ) (Ev-AocSafe-Abort) Figure 16: Evauation RHS - Con. 16

17 E c c r c E E c ki ok c E (Ev-Ski) E c c 1 ok c E E c c 2 ok c E E c c 1 ; c 2 ok c E (Ev-Seq) E c c 1 e c E E c c 1 ; c 2 e c E (Ev-Seq-ErrorPro1) E c c 1 ok c E E c c 2 e c E E c c 1 ; c 2 e c E (Ev-Seq-ErrorPro2) E h : q E r rh (d (b,e), ):a r r E datacat d (b,e) q a r writememmeta E.M d (b,e) = ome M E c h = rh ok c (M, E.S, E.T I) (Ev-Aign-Ptr) E h :int E r rh (d (b,e), ):a r r E datacat d (b,e) int a r writemem E.M d = ome M E c h = rh ok c (M, E.S, E.T I) (Ev-Aign-NPtr) E h e:a E c h = rh e c E (Ev-Aign-ErrorPro1) E h :a E r rh e:a r r E E c h = rh e c E (Ev-Aign-ErrorPro2) E h : q E r rh (d (b,e), ):a r r E datacat d (b,e) q a r E c h = rh Abort c E (Ev-Aign-Ptr-Abort) Figure 17: Evauation CMD. 17

18 5 Proertie Lemma 5.1 (tack invariance) 1. If E E, E.S r rh:a and E r rh r :a r E, then E.S = E.S. 2. If E E, E.S c c and E c c r c E, then E.S = E.S. Proof: Part 1 i by induction on E r rh r :a r E, art 2 i by induction on E c c r c E. Lemma 5.2 (h inverion) If E E and E h : a, then vaidmem E.M and 0 baeaddr + aize(a) < maxaddr. Proof: By induction on E h :a. Lemma 5.3 (h tr inverion) If E E and E h :a, then 1. If a = afe, then E.T I() = afe 2. If a = eq, then E.T I() = eq 3. If a = tame, then.e.t I() = tame Lemma 5.4 (rh tr inverion) If E E and E r rh (d (b,e), a ):a r E, then 1. If a = eq and a = int, then b = e = 0 or d = b = 0 2. If a = afe, then a int or a = int d = b = 0 3. If a = tame, a = q and q tame, then b = e = 0 4. If a = tame, a = int, then b = e = 0 5. If a = afe, a = eq, then d = 0 or b 0 b d < e ize() 6. If a = afe, a = tame, then d = If a = eq, a = tame, then d = 0 or b = e = If a = afe, a = eq, then d = b = e = 0 or d = b e = b + ize(). Lemma 5.5 (rh inverion) If E E and E r rh (d (b,e), a ) : a r E, then E.M ; E.T I d (b,e) : a. D Proof: By h tr inverion 5.3, rh tr inverion 5.4. Theorem 5.1 (rh we-formed environment invariance) If E E, E.S r rh:a and E r rh r :a r E, then E E. Proof: By induction on E r rh r :a r E, other cae are trivia excet Ev-AocSafe, Ev- AocSeq and Ev-AocTame. 1. Ev-AocSafe: E r rh (d (b,e), a):int r E and maoc E d = ome (E, ). t.. W F Env(E.M, E.S, udatetyeinfo E.T I d). It i ufficient to how (a) E.M ; udatetyeinfo E.T I d S inverion 2.4. E.S: By maoc inverion 2.5 and udatetyeinfo 18

19 (b) M E.M ; udatetyeinfo E.T I d: By definition, it i to how (, d, b, e ). readmemmeta E.M = d (b,e ) E.M ; udatetyeinfo E.T I d d (b,e ) : (udatetyeinfo E.T I d)( ). i. +d < : By udatetyeinfo inverion 2.4, E.T I( ) = (udatetyeinfo E.T I d)( ). By detruct E.T I( ), each cae i by maoc inverion 2.5 and udatetyeinfo inverion 2.4. ii. < + d: By maoc inverion 2.5 and udatetyeinfo inverion Ev-AocSeq,Ev-AocTame: imiar to Ev-AocSafe. D Theorem 5.2 (cmd we-formed environment invariance) If E E, E.S c c and E c c r c E, then E E. Proof: By induction on E c c r c E, 1. Ev-Aign-Ptr: By rh inverion other: immediate. Theorem 5.3 (h rogre) If E E and E.S h : a, then.e h : a or E h Abort:a. Proof: By induction on E.S h:a, 1. WFL-Var: Immediate. 2. WFL-Def: E.S h 0 :a 0 q with h = h 0 and a = a 0. t.. E.S h 0 :a 0. By IH, (a) E h 0 : a 0 : By h inverion 5.2, we have readmemmeta E.M = ome (b,e ). The reut foow by that aert (b,e ) a 0 q i decidabe. (b) E h 0 Abort:a 0 : Immediate by Ev-Def-ErrorPro. 3. WFL-StructPo and WFL-NamePo: imiar to Cae WFL-Def. Theorem 5.4 (rh rogre) If E E and E.S r rh:a, then (d (b,e), a ), E.E r rh (d (b,e), a ): a r E or E, a.e r rh OutofMem:a r E or E, a.e r rh Abort:a r E. Proof: By induction on E.S r rh:a, 1. WFR-Cont: Immediate. 2. WFR-Lh: E.S h:a with rh = h and a = a. By h rogre 5.3, (a) E h : a : By h inverion 5.2, we have readmemmeta E.M = ome (b,e ). The reut foow by Ev-Lh. (b) E h Abort:a : Immediate by Ev-Lh-ErrorPro. 3. WFR-Ref-Safe: E!tame h:a with rh = &h and a = a afe. We have E h:a becaue E!tame h:a E h:a. By h rogre WFR-Ref-Seq: imiar to Ev-Ref-Safe. 19

20 5. WFR-Ref-Tame: imiar to Ev-Ref-Safe, but by E tame h:a E h:a. 6. WFR-Add: S r rh 1 : int and S r rh 2 : int with rh = rh 1 + rh 2 and a = int. By IH of rh 1, (a) E r rh 1 (d (b,e), a ) : a r E : By rh we-formed environment invariance 5.1, E E. By tack invariance 5.1, E.S = E.S. The deired reut i by IH of rh 2. (b) ee: By Ev-Add-ErrorPro1. 7. WFR-AddPtr: imiar to Cae WFR-Add. 8. WFR-Cat: S r rh :a with rh = (a)rh and a = a. By IH, (a) E r rh (d (b,e), a ):a r E : If datacat d (b,e) a a hod, then the reut i by Ev-Cat, ee it i by Ev-Cat-Abort. (b) ee: By Ev-Cat-ErrorPro. 9. WFR-Size: Immediate. 10. WFR-Aoc:S r rh :int with rh = ( q)maoc rh and a = q. By IH, (a) E r rh (d (b,e), a):int r E : i. maoc E d = ome (E, ): If q afe, then the reut i by Ev-Aoc-Seq and Ev- Aoc-Tame. Otherwie, if d ize(), the reut i by Ev-Aoc-Safe, ee by Ev-Aoc- SafeAbort. ii. maoc E d = none:by Ev-Aoc-OutofMem. (b) ee: By Ev-Aoc-ErrorPro. Theorem 5.5 (cmd rogre) If E E and E.S c c, then E.E c c ok c E or E.E c c OutofMem c E or E.E c c Abort c E. Proof: By induction on E.S c c, 1. WFC-Ski: Immediate. 2. WFC-Seq: S c c 1 and S c c 2 with c = c 1 ; c 2. By IH of c 1, (a) E c c 1 ok c E : By cmd we-formed environment invariance 5.2, W F EnvE. By tack invariance 5.1, E.S = E.S. The deired reut i by IH of c 2. (b) ee: By Ev-Seq-ErrorPro1. 3. WFC-Aign: S h:a, S r rh:a r and a r a where c = h = rh. By h rogre 5.3, (a) E h :a : By rh rogre 5.4, i. E rh (d (b,e), a r):a r E : A. a = int : By h inverion 5.2, writemem E.M d = ome M. In cae, datacat d (b,e) int a r hod. The reut i by Ev-Aign-NPtr. B. a = q : By h inverion 5.2, writememmeta E.M d(b, e) = ome M. In cae, the reut i by the decidabiity of datacat d (b,e) q a r ii. ee: By Ev-Aign-ErrorPro2. (b) ee: By Ev-Aign-ErrorPro1. 20

21 A Notationa Convention TEXT a b c d e err E i id h rh M n q r S t TI v USAGE atomic tye bae command vaue end error Environment int contant identity ocation eft hand ide exreion right hand ide exreion Memory named truct ointer tye quaifier reut anonymou truct Stack tye Tye Information variabe 21