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COMP4161 Advanced Topics in Software Verification

创建日期：2024-05-15 17:41:48

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标签： COMP4161

Advanced Topics in Software Verification

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COMP4161

Advanced Topics in Software Verification
Assignment 3
This assignment starts on Tue, 8 Nov 2022 and is due on Fri, 18 Nov, 20:00h. We will accept
Isabelle .thy files only. In addition to this PDF document, please refer to the provided Isabelle
template for the definitions and lemma statements.
The assignment is take-home. This does NOT mean you can work in groups.
For all questions, you may prove your own helper lemmas, and you may use lemmas proved
earlier in other questions. You can also use automated tools like sledghammer. If you can’t
finish an earlier proof, use sorry to assume that the result holds so that you can use it if you
wish in a later proof. You won’t be penalised in the later proof for using an earlier true result you
were unable to prove, and you’ll be awarded partial marks for the earlier question in accordance
with the progress you made on it.
1 General recursive function (22 marks)
In this assignment, we continue with the theme of garbage collector that we explored in assign-
ment 2. In Question 1, we look into markF, a general recursive function version of the marking
function which was defined as an inductive relation in the assignment 2. Here we need to use
Isabelle’s function command as below and manually prove that the computation of markF does
terminate. To prove termination, we need to define an appropriate measure that decreases with
the each step of the computation.
Recall that, in the assignment 2, we used the following two types, parametrised by a type variable
′data:
type-synonym ′data block = nat list × ′data
type-synonym ′data heap = (nat, ′data block) fmap
For the marking phase, we introduced a flag for marking by instantiating the type variable with
bool × ′data. The block also contains a list of pointers, nat list, pointing to other blocks (see
Figure 1 in the assignment 2). The function markF is defined using the two functions marked
and mark-block as below:
fun marked :: (bool × ′data) block ⇒ bool where
marked (ptr , (tag,data)) = tag
fun mark-block :: (bool × ′data) block ⇒ (bool × ′data) block where
mark-block (ptr , (tag,data)) = (ptr , (True,data))
function markF where
markF heap [] = heap
| markF heap (root#roots) =
(case fmap.lookup heap root of
None ⇒ markF heap roots
| Some blk ⇒ if marked blk then markF heap roots
else markF (fupd root (mark-block blk) heap) (roots@fst blk))
1
(a) Complete the definition of markF and prove its termination. For the termination proof,
think which values will change in each loop step. (8 marks)
(b) Prove the mark-correct-aux lemma for markF. (7 marks)
markF heap roots =
fmap-keys (λptr (ptrs, tag, data). (ptrs, tag ∨ ptr ∈ ureach heap roots, data)) heap
(c) Prove the correctness of markF. (7 marks)
fpred (λptr block. ¬ marked block) heap =⇒
markF heap roots =
fmap-keys (λptr (ptrs, tag, data). (ptrs, ptr ∈ reach heap roots, data)) heap
Note that the ureach and reach relations defined in the a3 template are identical to the ones in
assignment 2.
2 C verification (78 marks)
In Question 2, we will consider a C version of the marking function, which is defined in the file
gc.c. To keep the assignment manageable, we will concentrate on the properties leading up to
the correctness of mark, but only state, not prove, the final lemma.
2.1 Linked lists
The C code uses the following struct block to model a block (a flat piece of data) in the heap.
For the sake of simplicity, we assume that the actual data it carries is of type int:
struct block {
struct blist *nexts;
int flag;
int data;
struct block *m_nxt;
};
Here the nexts field corresponds to the list of pointers to other blocks that the block is linked
to. The block struct also contains a flag field for marking, as well as the m_nxt (”marked next”)
field which is used for collecting marked blocks in the mark function.
In assignment 2, the list of nexts pointers in the block and the roots for the reachability relation
were modelled using Isabelle’s list datatype. In C, we use the following struct blist for these:
struct blist {
struct block *this;
struct blist *next;
};
which gives a linked list of blocks.
In order to further investigate the behaviour of the mark function, we first need to establish a
correspondence between a blist pointer and an abstract representation of a list in Isabelle/HOL.
We do this by using the following predicate list:
With the semantics given by AutoCorres, The blist pointer is given as a blist-C ptr : a pointer
to a blist-C record, which models the blist struct. This record has two fields, corresponding
to the two fields of the C struct: this-C and next-C. For instance, the selector function this-C
2
has type blist-C ⇒ block-C ptr, i.e. given a blist it returns the block-C pointer this field.
Similarly, the next-C selector functions have type blist-C ⇒ blist-C ptr.
The state type that the AutoCorres-generated functions operate on is called lifted-globals, Au-
toCorres also provides a heap for each type, i.e. here, blist-C heap, heap-blist-C, and block-C
heap, heap-block-C, as well as validity functions, is-valid-blist-C and is-valid-block-C.
The following function, for a given lifted-globals state, tests if a blist-C ptr points to a list whose
pointer-structure corresponds to the given blist-C ptr list. A blist-C ptr list is an abstract list
whose elements are blist-C ptrs. Such a structure gives us an abstract way of talking about the
pointer-structure of a list on the blist-C heap.
primrec list :: lifted-globals ⇒ blist-C ptr ⇒ blist-C ptr list ⇒ bool where
list s p [] = (p = NULL)
| list s p (x#xs) =
(p = x ∧ p 6= NULL ∧ is-valid-blist-C s p ∧ list s (next-C (heap-blist-C s p)) xs)
Prove the following statements:
(a) NULL corresponds to an empty list. (3 marks)
list s NULL xs = (xs = [])
(b) If p is not NULL, p is an element of the list it points to. (3 marks)
[[list s p xs; p 6= NULL]] =⇒ p ∈ set xs
(c) When p is not NULL, the list it points to can be split into head and tail. (3 marks)
p 6= NULL =⇒
list s p xs =
(∃ ys. xs = p # ys ∧ is-valid-blist-C s p ∧ list s (next-C (heap-blist-C s p)) ys)
(d) p points to a unique list. (4 marks)
[[list s p xs; list s p ys]] =⇒ xs = ys
(e) The elements in a blist list are distinct. (6 marks)
list s p xs =⇒ distinct xs
(f) Updating a pointer that is not in a blist list does not affect the list. (4 marks)
q /∈ set xs =⇒
list (heap-blist-C-update (λh. h(q := next-C-update (λ-. v) (h q))) s) p xs = list s p xs
Next, we define a function the-list which, given a state s and a pointer p for which list s p xs
holds for a blist list xs, returns the actual list xs.
definition the-list :: lifted-globals ⇒ blist-C ptr ⇒ blist-C ptr list where
the-list s p = (THE xs. list s p xs)
(g) Prove that updating a pointer that is not in the list does not affect the list in terms of
the-list. (5 marks)
[[q /∈ set xs; list s p xs]]
=⇒ the-list (heap-blist-C-update (λh. h(q := next-C-update (λ-. v) (h q))) s) p =
the-list s p
3
2.2 Correctness of append ′
AutoCorres generates a monadic version of mark, which is named mark ′. Similarly, functions
append ′ and append-step ′, which mark ′ depends on, are also generated. We now prove the
correctness of append-step ′ and append ′, which correspond to the following C functions:
struct blist *append_step (struct blist *l1, struct blist *l2)
{
struct blist *tmp = l1->next;
l1->next = l2;
l2 = l1;
l1 = tmp;
return l1;
}
struct blist *append (struct blist *l1, struct blist *l2)
{
while (l1) {
struct blist *tmp = append_step (l1, l2);
l2 = l1;
l1 = tmp;
}
return l2;
}
append takes two blist pointers representing two disjoint block lists, say l1 and l2, and returns
a pointer that represents a list rev l1 @ l2 (concatenation of the two lists but the first one
reversed). append-step implements each step of append, where the head of the first list is moved
to the head of the second list.
(h) Complete the proof of the correctness of append-step ′. (8 marks)
(i) Provide the precondition for the correctness statement for append ′. (4 marks)
(j) Prove the correctness of append ′ by providing appropriate invariants and a measure and
completing the rest of the proof. (12 marks)
2.3 Marked list
The function mark is defined so that, as it marks the blocks, it collects all the marked block as
another linked list, using the m_nxt field of the block. When it finishes marking, it returns the
pointer that corresponds to the list of all marked blocks. This list is similar to the linked lists
nexts and roots that we already saw, except that this one consists directly of blocks using the
m_nxt field, rather than being wrapped as blist.
We can use this list of marked blocks to retrieve the set of all marked blocks. And together with
the reachability notion on blocks, we will be able to state the conditions we expect to hold for
the function mark.

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 534 Elec4430 MATS3001 MGT6047 FINS5514 MIS602 ATS1279 ECMM445 2F CS262 EFIMM0097 FINM2002 ECMT3150 FINC5090 WACC1000 MECH4620 ECOS 2002 SEESGS79 STAT COMP3007 BUSB200F ACST8040 CAES9920 STAT0020 INFR11205 PHYS2111 FINM7402 STAT 428 SOFT3202 ME220 ACC-ACF2100 EC9410 CSE 120 COMP90042 SEE5211 ECON0048 BUSN2036 MKT2010 3SMFE4 EFIMM0117 ECON3034 MTH5120 EFIMM0120 X0 W19186 MATH 034 COMP2100 COMS4104 CS1210 MENG10005 ACCT 5906 ECMT1020 MGTS7607 MATH3090 CIV4100 CSYS5020 ELE2019 PHS4200 ACS6124 MSIN0225 PHAS0061 6CCS3BIM FINM7403 COMP8410 PY406 COMP2620 CS226 SDEV2004 COMP6080 BFF1001 HR410 MCEN30017 AMS3328 ML3 ELE8059 FINC3017 ACAD001 FTEC2101 ES480 3D ELECTENG 734 MTH731U MATH-UA 251 CSCI-UA 9473 A2 MSDM5058 FIT5047 COM6012 1FIN B386F MPHY202P ACCT6003 ECE 6913 HMB312H1 AC1025 MTH6105 Math6168 FINM 3003 CS24320 PHYS2134 ACCT3011 MECH1400 COMP30024 ENGGEN 121 COMP3721 MCD2080 ECON 7002 MAS439 COMP6216 ECON2011 M23367 CSC597 COMP6247 BRIEF QBUS6180 EENG22000 FN1024 MATH0094 MANG 6554 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