TMPA-2015: The Verification of Functional Programs by Applying Statechart Diagrams Construction Method

Âåðèôèêàöèÿ ôóíêöèîíàëüíûõ ïðîãðàìì
ìåòîäîì ïîñòðîåíèÿ äèàãðàìì ñîñòîÿíèé
Àíäðåé Ìèðîíîâ
amironov66@gmail.com
ÔÈÖ Èíôîðìàòèêà è óïðàâëåíèå ÐÀÍ
Ìîñêâà, 2015 ã.
Àíäðåé Ìèðîíîâ Âåðèôèêàöèÿ ôóíêöèîíàëüíûõ ïðîãðàìì ìåòîäîì ïîñòð

Êðàòêîå ñîäåðæàíèå
Îïèñàíèå ïðîáëåìû
Ðàññìàòðèâàåòñÿ ïðîáëåìà âåðèôèêàöèè ôóíêöèîíàëüíûõ ïðîãðàìì
(ÔÏ) íàä ñèìâîëüíûìè ñòðîêàìè, ãäå
ñïåöèôèêàöèè ñâîéñòâ ÔÏ îïðåäåëÿþòñÿ äðóãèìè ÔÏ, è
ÔÏ Σ1 óäîâëåòâîðÿåò ñïåöèôèêàöèè, îïðåäåëÿåìîé ÔÏ Σ2, åñëè
êîìïîçèöèÿ ôóíêöèé, îïðåäåëÿåìûõ ÔÏ Σ1 è Σ2, ïðèíèìàåò
çíà÷åíèå 1 íà âñåõ àðãóìåíòàõ.
Ìû ââîäèì ïîíÿòèå äèàãðàììû ñîñòîÿíèé ÔÏ, è ñâîäèì ïðîáëåìó
âåðèôèêàöèè ÔÏ ê ïðîáëåìå àíàëèçà äèàãðàìì ñîñòîÿíèé ÔÏ.
Ïðåäëîæåííûé ïîäõîä èëëþñòðèðóåòñÿ ïðèìåðîì âåðèôèêàöèè ÔÏ
ñîðòèðîâêè.

Ïðèìåð âåðèôèêàöèè ñïåöèôèêàöèè ôóíêöèîíàëüíîé
ïðîãðàììû ñîðòèðîâêè ñòðîê
Ïðîãðàììà:
sort(x) = (x = ε)? ε : insert(xh, sort(xt))
insert(a, y) = (y = ε) ? aε
: (a ≤ yh) ? ay
: yh insert(a, yt)
(1)
Ñïåöèôèêàöèÿ:
∀ x ∈ S ord(sort(x)) = 1 (2)
ãäå
ord(x) =
= (x = ε) ? 1
: (xt = ε) ? 1
: (xh ≤ (xt)h) ? ord(xt)
: 0
(3)

Îáû÷íîå ìàòåìàòè÷åñêîå äîêàçàòåëüñòâî êîððåêòíîñòè
ïðîãðàììû ñîðòèðîâêè
Èíäóêöèÿ ïî äëèíå ñòðîêè x.
Åñëè x = ε, òî, ñîãëàñíî ïåðâîìó óðàâíåíèþ ñèñòåìû (1), âåðíî
ðàâåíñòâî sort(x) = ε, è ïîýòîìó
ord(sort(x)) = ord(ε) = 1
Ïóñòü x = ε. Äîêàæåì ðàâåíñòâî (2) äëÿ ýòîãî ñëó÷àÿ ìåòîäîì
ìàòåìàòè÷åñêîé èíäóêöèè.
Ïðåäïîëîæèì, ÷òî âåðíî ðàâåíñòâî, ïîëó÷àåìîå èç ðàâåíñòâà â (2)
çàìåíîé x íà ëþáóþ ñòðîêó, äëèíà êîòîðîé ìåíüøå äëèíû x.
Äîêàæåì, ÷òî â ýòîì ñëó÷àå ðàâåíñòâî â (2) òàêæå áóäåò âåðíûì.

ïðîãðàììû ñîðòèðîâêè (ïðîäîëæåíèå)
Ðàâåíñòâî â (2) ìîæíî ïåðåïèñàòü â âèäå
ord( insert(xh, sort(xt))) = 1 (4)
Ïî èíäóêòèâíîìó ïðåäïîëîæåíèþ, âåðíî ðàâåíñòâî
ord(sort(xt)) = 1
èç êîòîðîãî ñëåäóåò (4) ïî íèæåñëåäóþùåé ëåììå.
Ëåììà.
Èìååò ìåñòî èìïëèêàöèÿ
ord(y) = 1 ⇒ ord(insert(a, y)) = 1 (5)
Äîêàçàòåëüñòâî.
Äîêàçûâàåì ëåììó èíäóêöèåé ïî äëèíå y.

Åñëè y = ε, òî ïðàâàÿ ÷àñòü â (5) èìååò âèä
ord(aε) = 1
÷òî âåðíî ïî îïðåäåëåíèþ ord.
Ïóñòü y = ε, è äëÿ êàæäîé ñòðîêè z, äëèíà êîòîðîé ìåíüøå äëèíû y,
âåðíà èìïëèêàöèÿ
ord(z) = 1 ⇒ ord(insert(a, z)) = 1 (6)
Îáîçíà÷èì c
def
= yh, d
def
= yt.
(5) èìååò âèä
ord(cd) = 1 ⇒ ord(insert(a, cd)) = 1 (7)

Äëÿ äîêàçàòåëüñòâà èìïëèêàöèè (7) íóæíî äîêàçàòü, ÷òî ïðè óñëîâèè
ord(cd) = 1 âåðíû èìïëèêàöèè
(a) a ≤ c ⇒ ord(a(cd)) = 1,
(b) c a ⇒ ord(c insert(a, d)) = 1.
(a) âåðíî ïîòîìó, ÷òî èç a ≤ c ñëåäóåò
ord(a(cd)) = ord(cd) = 1.
Äîêàæåì (b).

d = ε. Â ýòîì ñëó÷àå ïðàâàÿ ÷àñòü â (b) èìååò âèä
ord(c(aε)) = 1 (8)
(8) ñëåäóåò èç c a.
d = ε. Îáîçíà÷èì p
def
= dh, q
def
= dt.
Â ýòîì ñëó÷àå íàäî äîêàçàòü, ÷òî ïðè c a
ord(c insert(a, pq)) = 1 (9)
Åñëè a ≤ p, òî (9) èìååò âèä
ord(c(a(pq))) = 1 (10)
Ò.ê. c a ≤ p, òî (10) ñëåäóåò èç ðàâåíñòâ
ord(c(a(pq))) = ord(a(pq)) = ord(pq) =
= ord(c(pq)) = ord(cd) = 1

Åñëè p a, òî (9) èìååò âèä
ord(c(p insert(a, q))) = 1 (11)
Ïîñêîëüêó ïî ïðåäïîëîæåíèþ
ord(cd) = ord(c(pq)) = 1
òî c ≤ p, è ïîýòîìó (11) ìîæíî ïåðåïèñàòü â âèäå
ord(p insert(a, q)) = 1 (12)
Ïðè p a
insert(a, d) = insert(a, pq) = p insert(a, q)
ïîýòîìó (12) ìîæíî ïåðåïèñàòü â âèäå
ord(insert(a, d)) = 1 (13)

(13) ñëåäóåò ïî èíäóêòèâíîìó ïðåäïîëîæåíèþ äëÿ ëåììû (ò.å. èç
èìïëèêàöèè (6), â êîòîðîé z
def
= d) èç ðàâåíñòâà
ord(d) = 1
êîòîðîå îáîñíîâûâàåòñÿ öåïî÷êîé ðàâåíñòâ
1 = ord(cd) = ord(c(pq)) = (ò.ê. c ≤ p)
= ord(pq) = ord(d).

Èäåÿ íîâîãî ìåòîäà âåðèôèêàöèè ôóíêöèîíàëüíûõ
ïðîãðàìì
Îòêàç îò äîêàçàòåëüñòâà óòâåðæäåíèÿ î êîððåêòíîñòè ïðîãðàììû,
âûðàæàåìîãî ôîðìóëîé èñ÷èñëåíèÿ ïðåäèêàòîâ ïåðâîãî ïîðÿäêà,
ïóòåì ïîñòðîåíèÿ ôîðìàëüíîãî âûâîäà â ëîãèêå ïåðâîãî ïîðÿäêà.
Ìåòîä âåðèôèêàöèè ïîñòðîåíèå
ãðàôîâîé ìîäåëè âåðèôèöèðóåìîé ïðîãðàììû è
ãðàôîâîé ìîäåëè ïðîãðàììû, âûðàæàùåé ïðîâåðÿåìîå ñâîéñòâî,
ïîñëå ÷åãî âû÷èñëÿåòñÿ ãðàôîâàÿ ìîäåëü äëÿ ñóïåðïîçèöèè
àíàëèçèðóåìîé è ïðîâåðÿþùåé ôóíêöèé, è èññëåäóþòñÿ òåðìèíàëüíûå
âåðøèíû ïîëó÷èâøåéñÿ ãðàôîâîé ìîäåëè.

Èäåÿ íîâîãî ìåòîäà âåðèôèêàöèè ôóíêöèîíàëüíûõ
ïðîãðàìì
Òåîðåìà .
Ïóñòü ÔÏ Σ ◦ Σspec èìååò êîíå÷íóþ äèàãðàììó ñîñòîÿíèé (ÄÑ),
ïðè÷åì çíà÷åíèÿ ñîñòîÿíèé, ñîîòâåòñòâóþùèõ òåì òåðìèíàëüíûì
âåðøèíàì ýòîé ÄÑ, êîòîðûå äîñòèæèìû èç íà÷àëüíîãî ñîñòîÿíèÿ,
ðàâíû 1. Òîãäà fΣ◦Σspec ïðèíèìàåò çíà÷åíèå 1 íà âñåõ ñâîèõ
àðãóìåíòàõ.

Ïðèìåð ãðàôîâîé ìîäåëè ôóíêöèîíàëüíîé ïðîãðàììû

y := sort(x)
ε

ε := x
'

$
%
y := a → u
u := sort(b)
ab := x
y
y
aε

aε := x
'

$
%
z := a → d
cd := sort(b)
ab := x
tail
tail
tail
tail
{a ≤ c}.acd
{c a}.cz
' $
z := a → d, d := p → j
cj := sort(q)
apq := x
'

$
%
z := a → ij
ij := sort(q)
acq := x
'

$
%
cd := sort(b)
ab := x
{c a}.caε

acε := x
c
c
E
E
E
c
rrr
rr‰
¨¨
¨¨¨¨B
rr
rrr‰
¨¨¨
¨¨B
d
d
dds

d
d
d
ds

A B
C
D E
G
F
I
H
{c a, c ≤ i}.cz
{c a, c p}.cz

Ðåäóöèðîâàííàÿ ãðàôîâàÿ ìîäåëü äëÿ ÔÏ ñîðòèðîâêè

y := sort(x)
ε

ε := x
'

$
%
y := a → u
u := sort(b)
ab := x
y
y
aε

aε := x
'

$
%
z := a → d
cd := sort(b)
ab := x
tail
tail
{a ≤ c}.acd
{c a}.cz
'

$
%
cd := sort(b)
ab := x c
c
E
Er
rrr
r‰
¨¨¨
¨¨¨B
d
d
dds

A B
C
D E
G

Ãðàôîâàÿ ìîäåëü ÔÏ ïðîâåðêè óïîðÿäî÷åííîñòè ñòðîêè
a b
c d
e f g
s

s := ord(y)
s

s := ord(cz)
cz := y
1

ε := y
1

cε := y
s'

$
%
s := ord(cvw)
cvw := y

cvw := y
'

$
%
s := ord(vw)
cvw := y
c
c
'
{c ≤ v}.s
E
{v c}.0
E
EE


Ãðàôîâàÿ ìîäåëü ñóïåðïîçèöèè
Aa BaCe
Gc Da
Ec
Gf Ge
1'

$
%
. . .
1'

$
%
. . .E
'

$
%
'

$
%
s := ord(y)
y := sort(x)
c'

$
%
s := ord(y)
y := a → u
u := sort(b)
ab := x
d
d
d
d
ds
'
c
EE
'

$
%
s := ord(cz)
z := a → d
cd := sort(b)
ab := x
'

$
%
s := ord(vw)
vw := a → d
cd := sort(b)
ab := x
E
'

$
%
s := ord(cvw)
vw := a → d
cd := sort(b)
ab := x
'

$
%
s := ord(cd)
cd := sort(b)
ab := x
{a ≤ c}.s
{c a, c ≤ v}.s{c a}.s
{c a}.s s
s
d
d
dds


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TMPA-2015: The Verification of Functional Programs by Applying Statechart Diagrams Construction Method

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TMPA-2015: The Verification of Functional Programs by Applying Statechart Diagrams Construction Method