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COMP1600 Assignment
创建日期：2024-05-13 14:30:35
标签： COMP1600
Assignment

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COMP1600 / COMP6260

Basic Ingredient. Hoare-triples
{P} S {Q}
P and Q are predicates (formulae)
S is a code (fragment)
Example. {x > 0} while (x>0) do x := x-1 {x = 0}
Meaning. If we run S from a state that satisfies precondition P and if S
terminates, then the post-state will satisfy Q.
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Hoare Logic
Idea. Proof Rules that allow us to prove all true triples.
Assignment
{Q(e)} x := e {Q(x)}
Precondition Strengthening / Postcondition Weakening
Ps → Pw {Pw} S {Q}
{Ps} S {Q}
{P} S {Qs} Qs → Qw
{P} S {Qw}
Sequence.
{P} S1 {Q} {Q} S2 {R}
{P} S1; S2 {R}
Conditional.
{P ∧ b} S1 {Q} {P ∧ ¬b} S2 {Q}
{P} if b then S1 else S2 {Q}
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Proof Rule for While Loops (Rule 6/6)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b}
I is called the loop invariant.
I is true before we encounter the while statement, and remains true
each time around the loop (although not necessarily midway during
execution of the loop body).
If the loop terminates the control condition must be false, so ¬b
appears in the postcondition.
For the body of the loop S to execute, b needs to be true, so it
appears in the precondition.
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Soundness of the While Rule w.r.t. the semantics
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b}
Soundness. If the premise is true, then so is the conclusion.
assume that I holds in the initial state.
if b is false, nothing happens, so I ∧ ¬b is true in post-state.
if b is true, then (by premise) I holds at end of each iteration
assuming that the loop terminates, b becomes false (and I still holds)
Q. What about non-termination?
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Applying the While Rule
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
Difficult bit. Finding the right invariant.
This requires intuition and practice. There is no automated way of
doing this.
Easy bit. Establishing the desired postcondition
The postcondition we get after applying our rule has form I ∧ ¬b.
But if I ∧ ¬b → Q, we can use postcondition weakening.
Easy bit. Establishing the desired precondition
The precondition we get after applying our rule has form I . But if
P → I , we can use precondition strengtening.
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Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6. {P} while b do S {Q} (Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2.
{I} while b do S {I ∧ ¬b}
(While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1.
{I ∧ b} S {I}
2.
{I} while b do S {I ∧ ¬b}
(While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2.
{I} while b do S {I ∧ ¬b}
(While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3.
I ∧ ¬b → Q
(Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5.
P → I
(Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6.
{P} while b do S {Q}
(Precondition Strengthening, 4, 6)
6 / 49
Applying the While Rule (schematic proof)
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b} {P} while b do S {Q}
1. {I ∧ b} S {I}
2. {I} while b do S {I ∧ ¬b} (While Loop, 1)
3. I ∧ ¬b → Q (Logic)
4. {I} while b do S {Q} (Postcondition Weakening, 2, 3)
5. P → I (Logic)
6. {P} while b do S {Q} (Precondition Strengthening, 4, 6)
6 / 49
Example
Goal. Find condition I to prove that:
{n > 3} while n>0 do n := n-1 {n = 0}
Observation. Need to prove the above using while-rule, i.e.
{I ∧ b} n := n− 1 {I}
{I} while (n>0) do n:= n-1 {I ∧ n ≤ 0}
Want. Loop invariant I such that
It is implied by the precondition:
n > 3→ I
if the loop terminates (i.e. n > 0 is false), then I should imply the
postcondition:
I ∧ n ≤ 0→ n = 0
If I is true and the body is executed, I is true afterwards:
{I ∧ n > 0} n := n− 1 {I}
Loop Invariant. I ≡ (n ≥ 0)
have {n ≥ 0 ∧ n > 0} n := n− 1 {n ≥ 0}
n ≥ 0 ∧ ¬(n > 0)→ n = 0
7 / 49
Example (cont.)
Goal. Find condition I to prove that:
{n > 3} while n>0 do n := n-1 {n = 0}
Loop Invariant. I ≡ (n ≥ 0), we have
n > 3→ n ≥ 0,
n ≥ 0 ∧ n ≤ 0→ n = 0, and
{n ≥ 0 ∧ n > 0} n := n− 1 {n ≥ 0}.
7 / 49
Example, Formally
1. {n − 1 ≥ 0} n := n− 1 {n ≥ 0} (Assignment)
2. n ≥ 0 ∧ n > 0→ n − 1 > 0 (Logic)
3. {n ≥ 0 ∧ n > 0} n := n− 1 {n ≥ 0} (Prec. Strength., 1, 2)
4. {n ≥ 0} while (n > 0) do n := n− 1 {n ≥ 0∧¬(n > 0)} (While Loop, 3)
5. n > 3→ n ≥ 0 (Logic)
6. {n > 3} while (n > 0) do n := n− 1 {n ≥ 0 ∧ ¬(n > 0)} (Prec.
Strength., 4, 5)
7. n = 0↔ n ≥ 0 ∧ ¬(n > 0) (Logic)
8. {n ≥ 0} while (n > 0) do n := n− 1 {n = 0} (Post. Equiv., 6, 7)
Other Invariants
e.g. true or n = 0
both are invariants, and give n = 0 as postcondition
but n ≥ 0 is better (weaker) as it is more general.
8 / 49
Let’s Prove a Program!
Program (with specification):
{True}
i:=0;
s:=0;
while (i ̸= n) do
i:=i+1;
s:=s+(2*i-1)
{s = n2}
(The sum of the first n odd numbers is n2)
Goal: prove
{True} Program {s = n2}
9 / 49
A Very Informal Analysis
Let’s look at some examples:
1 = 1 = 12
1 + 3 = 4 = 22
1 + 3 + 5 = 9 = 32
1 + 3 + 5 + 7 = 16 = 42 . . .
It looks OK - let’s see if we can prove it!
Goal: prove
{True} Program {s = n2}
10 / 49
How can we prove it?
First Task. Find a loop invariant I . (NB: S and s are different!)
Post condition and loop condition:
{I ∧ b} S {I}
{I} while b do S {I ∧ ¬b}
while (i ̸= n) do
i:=i+1; (1, 2, 3, ...)
s:=s+(2*i-1) (1, 4, 9, ...)
{s = n2}
Want. (I ∧ i = n)→ (s = n2) to apply postcondweak
Loop Body. Each time i increments, s moves to next square number.
Invariant. I ≡ s = i2.
11 / 49
Check I as (s = i2) is an invariant: prove {I}S{I}
{s = i2} i := i + 1 {Q} {Q} s := s + (2 ∗ i − 1) {s = i2}
Seq{s = i2} i := i + 1; s := s + (2 ∗ i − 1) {s = i2}
Using the assignment axiom and the sequence rule:
1. {Q} s:=s+(2*i-1) {s = i2}
2.
3. {s = i2} i:=i+1 {Q}
4. {s = i2} i:=i+1; s:=s+(2*i-1) {s = i2} (Sequence, 3, 1)
12 / 49
Check I as (s = i2) is an invariant: prove {I}S{I}
{s = i2} i := i + 1 {Q} {Q} s := s + (2 ∗ i − 1) {s = i2}
Seq{s = i2} i := i + 1; s := s + (2 ∗ i − 1) {s = i2}
Q is {s + (2 ∗ i − 1) = i2}
Using the assignment axiom and the sequence rule:
1. {s + (2 ∗ i − 1) = i2} s:=s+(2*i-1) {s = i2} (Assignment)
2.
3. {s = i2} i:=i+1 {s + (2 ∗ i − 1) = i2}
4. {s = i2} i:=i+1; s:=s+(2*i-1) {s = i2} (Sequence, 3, 1)
13 / 49
Check I as (s = i2) is an invariant: prove {I}S{I}
{s = i2} i := i + 1 {Q} {Q} s := s + (2 ∗ i − 1) {s = i2}
Seq{s = i2} i := i + 1; s := s + (2 ∗ i − 1) {s = i2}
Q is {s + (2 ∗ i − 1) = i2}
Using the assignment axiom and the sequence rule:
1. {s + (2 ∗ i − 1) = i2} s:=s+(2*i-1) {s = i2} (Assignment)
2. {s + (2 ∗ (i + 1)− 1) = (i + 1)2} i:=i+1 {s + (2 ∗ i − 1) = i2}
(Assignment)
3. {s = i2} i:=i+1 {s + (2 ∗ i − 1) = i2} (Prec. Equiv., 2)
4. {s = i2} i:=i+1; s:=s+(2*i-1) {s = i2} (Sequence, 3, 1)
So far, so good. (I as (s = i2) is an invariant.)
14 / 49
Completing the Proof of {True} Program {s = n2}
6 Strengthen the precondition to match the While rule premise
{I ∧ b} S {I}
{(s = i2) ∧ (i ̸= n)} i:=i+1; s:=s+(2*i-1) {s = i2}
7 Apply the While rule and postcondition equiv:
s = i2 ∧ i = n↔ s = n2
{s = i2} while ... s:=s+(2*i-1) {s = n2}
8 Check that the initialisation establishes the invariant:
{0 = 02} i:=0 {0 = i2} {0 = i2} s:=0 {s = i2}
{0 = 02} i:=0; s:=0 {s = i2}
9 (0 = 02) ↔ True, so putting it all together with Sequencing we have
{True} i := 0 ; s := 0 ; while (i ̸= n) do S {s = n2}
15 / 49
What about Termination?
Hoare Logic (in this form) proves partial correctness.
Example. while 1+1 = 2 do x:=0.
This will loop forever!
can still prove things about it
Exercise.
{True} while 1+1 = 2 do x:=0 {False}
Termination.
remember functional programs? Something must decrease
need loop variant (later)
16 / 49
Are the Rules Complete?
So far. Have a very simple language
new rules for arrays, for-loops, exceptions, . . .
Focus here. Soundness
every provable Hoare triple is (semantically) true
soundness holds but terms and conditions apply
with these assumptions, also have completeness, i.e. every true Hoare
triple is provable.
Completeness. if {P} S {Q} is true then {P} S {Q} is provable
17 / 49
What are these Assumptions?
The language we use for expressions in our programs is the same as
the language we use in our pre- and postconditions (in our case, basic
arithmetic).
We assumed no aliasing of variables. (In most real languages we can
have multiple names for the one piece of memory.)
How is aliasing a problem?
Suppose x and y refer to the same cell of memory.
▶ We get {y +1 = 5 ∧ y = 5} x:=y+1 {x = 5 ∧ y = 5} (Assignment)
▶ i.e. {y = 4 ∧ y = 5} x:=y+1 {x = 5 ∧ y = 5}
▶ i.e. if initial state satisfies False and x:=y+1 terminates then final state
satisfies {x = 5 ∧ y = 5} (but also works for x = 6 ∧ y = 6)
which makes a mockery of our calculus since it proves rubbish!
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MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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FN1024 MATH0094 MANG 6554 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 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