V erication 1 Objectives of this Lecture Induction on ! - - PowerPoint PPT Presentation

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V erication 1 Objectives of this Lecture Induction on ! - - PowerPoint PPT Presentation

Feb 21, 2024 306 likes •485 views

V erication 1 Objectives of this Lecture Induction on ! Induction on 6! F o rmal p ro of of the vecto r clo ck algo rithm Vija c y K. Ga rg Distributed Systems Sp ring 96 V erication 2

slide-1
SLIDE 1 V erication 1 Objectives
  • f
this Lecture
  • Induction
  • n
!
  • Induction
  • n
6!
  • F
  • rmal
p ro
  • f
  • f
the vecto r clo ck algo rithm c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-2
SLIDE 2 V erication 2 Causally p recedes and its complem ent
  • s
1
  • s
2
  • s
3
  • s
4
  • t
1
  • t
2
  • t
3
  • t
4
  • k
! relation used fo r induction
  • n
!. F
  • r
k > 0, s k ! t 4 = ml (s; t) = k Thus s k ! t if and
  • nly
if s ! t and the longest chain from s to t has length k . c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-3
SLIDE 3 V erication 3 Induction
  • n
! Lemma 1 s ! t , (9k : k > : s k ! t) Lemma 2 s 1 ! t ) s
  • 1
t _ s ; t Is Con v erse true ? Pro
  • f:
s 1 ! t ) ml (s; t) = 1 f defn
  • f
k ! g ) 9c : f ir st(c) = s ^ l ast(c) = t ^ l en(c) = 1 ) s
  • 1
t ^ s ; t f defn
  • f
a chain g Lemma 3 (s k ! t) ^ (k > 1) ) (9u :: s k 1 ! u ^ u 1 ! t) Pro
  • f:
(s k ! t) ^ (k > 1) ) (ml (s; t) = k ) ^ (k > 1) f defn
  • f
k ! g ) (9u :: ml (s; u) = k
  • 1
^ ml (u; t) = 1) f chain lemma g ) (9u :: s k 1 ! u ^ u 1 ! t) f defn
  • f
k ! g c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-4
SLIDE 4 V erication 4 The relation 6!
  • s
1
  • s
2
  • s
3
  • s
4
  • t
1
  • t
2
  • t
3
  • t
4
  • Dene
fo r k
  • 0:
s k 6! t 4 = s 6! t ^ ml (I nit; t) = k Thus s k 6! t if and
  • nly
if s 6! t and the longest chain from some initial state to t has length k . c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-5
SLIDE 5 V erication 5 Induction
  • n
6! Lemma 4 s 6! t , (9k : k
  • :
s k 6! t) Pro
  • f:
s 6! t , (s 6! t) ^ (ml (I nit; t)
  • 0)
f b y defn
  • f
ml (I nit; t) g , (9k : k
  • :
s k 6! t) f defn
  • f
k ! g Lemma 5 s 6! t , I nit(t) c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-6
SLIDE 6 V erication 6 Induction
  • n
6! [Contd.] Lemma 6 (k > 0) ^ (s k 6! t) ^ (u ! t) ) (9j :
  • j
< k : s j 6! u) Pro
  • f
: k > ^ s k 6! t ^ u ! t ) k > ^ s 6! u ^ s k 6! t fotherwise s ! tg ) k > ^ s 6! u ^ ml (I nit; t) = k fdefn
  • f
k 6! g ) k > ^ s 6! u ^ ml (I nit; u) < k fotherwise ml (I nit; t) > k g ) (9j :
  • j
< k : s j 6! u) fdefn
  • f
j 6! g c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-7
SLIDE 7 V erication 7 A va riant
  • f
the vecto r clo ck algo rithm
  • vecto
r comp
  • nents
incremented less frequently; it maintains: (8s; t : s:p 6= t:p : s:v < t:v , s ! t) F
  • r
any initial state s: (8i : i 6= s:p : s:v [i] = 0) ^ (s:v [s:p] = 1) Rule fo r a send event (s; snd; t): t:v := s:v ; t:v [t:p] + +; Rule fo r a receive event (s; r cv (u); t): t:v := max (s:v ; u:v ); Rule fo r an internal event (s; int; t): t:v := s:v ; c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-8
SLIDE 8 V erication 8 Pro
  • f
  • (8s;
t : s:p 6= t:p : s:v < t:v , s ! t). accomplished b y s:p 6= t:p ^ s ! t ) s:v < t:v (1) s:p 6= t:p ^ s:v < t:v ) s ! t (2) Lemma 7 s ! t ) s:v
  • t:v
c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-9
SLIDE 9 V erication 9 Pro
  • f
[Contd.] Pro
  • f
: Sucient to sho w that 8 k > : s k ! t ) s:v
  • t:v
Base (k = 1) : s 1 ! t ) s
  • 1
t _ s ; t flemma 2 g ) (s; int; t) _ (s; snd; t) _ (9u :: (s; r cv (u); t)) _(9u :: (u; r cv (s); t)) fexpand s
  • t
and s ; tg ) (s:v = t:v ) _ (s:v < t:v ) _ (s:v
  • t:v
) _ (s:v
  • t:v
) fSnd, Rcv, and Int rulesg ) s:v
  • t:v
f simplify g Induction: (k > 1) s k ! t ^ (k > 1) ) (9u :: s k 1 ! u ^ u 1 ! t) flemma 3g ) (9u :: s:v
  • u:v
^ u:v
  • t:v
) finduction hyp
  • thesisg
) s:v
  • t:v
fsimplifyg c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-10
SLIDE 10 V erication 10 Use
  • f
induction
  • n
k 6! [Base Case] Contrap
  • sitive
  • f
2: 8s; t : s:p 6= t:p : s 6! t ) :(s:v < t:v ): Lemma 8 (8s; t : s:p 6= t:p : s 6! t ) t:v [s:p] < s:v [s:p]) Pro
  • f
Base (k = 0) : s 6! t ^ s:p 6= t:p ) I nit(t) ^ s:p 6= t:p flemma 7g ) I nit(t) ^ s:p 6= t:p ^ flet u b e initial state in s:pg (9u : I nit(u) ^ u:p = s:p : u = s _ u ! s) ) I nit(t) ^ s:p 6= t:p ^ flemma 7g (9u : I nit(u) ^ u:p = s:p : u:v = s:v _ u:v
  • s:v
) ) t:v [s:p] = ^ fInit ruleg (9u : u:v [s:p] = 1 : u:v = s:v _ u:v
  • s:v
) ) t:v [s:p] < s:v [s:p] fsimplifyg c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-11
SLIDE 11 V erication 11 Pro
  • f
[Induction Case] Induction: (k > 0) s k 6! t ^ s:p 6= t:p ^ k > ) f let u satisfy u
  • 1
t, u exists since :I nit(t) g s k 6! t ^ s:p 6= t:p ^ u:p = t:p ^ u
  • 1
t ) f lemma 6 g s j 6! u ^
  • j
< k ^ u:p 6= s:p ^ u
  • 1
t ) f inductive hyp
  • thesis
g u:v [s:p] < s:v [s:p] ^ u
  • 1
t ) f expand u
  • 1
t g u:v [s:p] < s:v [s:p] ^ ((u; int; t) _ (u; snd; t) _ (u; r cv (w ); t)) Consider each disjunct sepa rately . c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-12
SLIDE 12 V erication 12 Pro
  • f
  • f
Inductive Case [Contd.] Case 1: (u; int; t) u:v [s:p] < s:v [s:p] ^ (u; int; t) ) u:v [s:p] < s:v [s:p] ^ t:v = u:v fInt ruleg ) t:v [s:p] < s:v [s:p] fsimplifyg Case 2: (u; snd; t) u:v [s:p] < s:v [s:p] ^ (u; snd; t) ) u:v [s:p] < s:v [s:p] ^ t:v [s:p] = u:v [s:p] fSnd rule, s:p 6= t:pg ) t:v [s:p] < s:v [s:p] fsimplify g Case 3: (u; r cv (w ); t) u:v [s:p] < s:v [s:p] ^ (u; r cv (w ); t) ) u:v [s:p] < s:v [s:p] ^ (u; r cv (w ); t) ^ fRcv ruleg (t:v [s:p] = u:v [s:p] _ t:v [s:p] = w :v [s:p]) ) (t:v [s:p] < s:v [s:p]) _ fsimplifyg ((u; r cv (w ); t) ^ t:v [s:p] = w :v [s:p]) It suces to p rove the t w
  • cases:
w :p = s:p and w :p 6= s:p. c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-13
SLIDE 13 V erication 13 Pro
  • f
  • f
Inductive Case [Contd.] Case 3A: w :p = s:p t:v [s:p] = w :v [s:p] ^ (u; r cv (w ); t) ) t:v [s:p] = w :v [s:p] ^ (w ; snd; x) 8 > > > > > > > > > < > > > > > > > > > : let x satisfy w
  • x,
x exists since w ; t implies :F inal (w ) 9 > > > > > > > > > = > > > > > > > > > ; ) t:v [s:p] = w :v [s:p] ^ (w ; snd; x) fotherwise s ! tg ^ w ! s ) t:v [s:p] = w :v [s:p] ^ (w ; snd; x) fsince w
  • xg
^ (x = s _ x ! s) ) t:v [s:p] = w :v [s:p] ^ w :v [s:p] < x:v [s:p] ^ (x = s _ x ! s) fSnd ruleg ) t:v [s:p] = w :v [s:p] ^ w :v [s:p] < x:v [s:p] ^ (x:v
  • s:v
) flemma 7g ) t:v [s:p] < s:v [s:p] fsimplifyg c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-14
SLIDE 14 V erication 14 Pro
  • f
  • f
Inductive Case [Contd.] Case 3B: w :p 6= s:p t:v [s:p] = w :v [s:p] ^ (u; r cv (w ); t) ^ w :p 6= s:p ) f use s k 6! t, k > 0, and lemma 6 g t:v [s:p] = w :v [s:p] ^ w :p 6= s:p ^ s j 6! w ^
  • j
< k ) f inductive hyp
  • thesis
g t:v [s:p] = w :v [s:p] ^ w :v [s:p] < s:v [s:p] ) f simplify g t:v [s:p] < s:v [s:p] c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-15
SLIDE 15 V erication 15 Converse Eqn 2 : s:p 6= t:p ^ s:v < t:v ) s ! t Lemma 9 (8s; t : s:p 6= t:p : s ! t ) s:v < t:v ) Pro
  • f
Base (k = 1) : s 1 ! t ^ s:p 6= t:p ) s ; t ^ s:p 6= t:p fdefn
  • f
1 ! and lemma 2g ) s:p 6= u:p ^ (u; r cv (s); t) flet u satisfy u
  • tg
) 8 > > > > < > > > > :
  • therwise
t ! s (since there is
  • nly
  • ne
event b et w een u and t) 9 > > > > = > > > > ; u 6! s ^ s:p 6= u:p ^ (u; r cv (s); t) ) s:v [u:p] < u:v [u:p] flemma 8 and rcv ruleg ^ (8i :: t:v [i] = max (u:v [i]; s:v [i])) ) s:v < t:v c Vija y K. Ga rg Distributed Systems Sp ring 96
slide-16
SLIDE 16 V erication 16 Converse [Contd.] Induction (k > 0) : s k ! t ^ k > ^ s:p 6= t:p ) (9u :: s k 1 ! u ^ u 1 ! t ^ s:p 6= t:p) flemma 3g ) (9u :: s k 1 ! u ^ u 1 ! t ^ fu:p can not have t w
  • valuesg
(u:p 6= t:p _ u:p 6= s:p)) ) (9u :: (s k 1 ! u ^ u 1 ! t ^ u:p 6= t:p) _ (s k 1 ! u ^ u 1 ! t ^ u:p 6= s:p)) ) (9u :: (s k 1 ! u ^ u:v < t:v )_ finductive hyp
  • thesisg
(s:v < u:v ^ u 1 ! t)) ) (9u :: (s:v
  • u:v
^ u:v < t:v )_ flemma 7g (s:v < u:v ^ u:v
  • t:v
)) ) s:v < t:v Theorem 1 (8s; t : s:p 6= t:p : s ! t , s:v < t:v ) c Vija y K. Ga rg Distributed Systems Sp ring 96