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COMP4161 Advanced Topics in Software Verification

创建日期：2024-08-12 15:49:04

所属分类：计算机computer science

标签： COMP4161

Advanced Topics in Software Verification

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COMP4161

Advanced Topics in Software Verification

Assignment 3

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. Each submission

is personal. For more information, see the plagiarism policy: https://student.unsw.edu.au/plagiarism

Submit using give on a CSE machine: give cs4161 a3 a3.thy

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))

(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

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

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

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

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)

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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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