SLIDE 1
Challenge How to prove safety of industrial size protocols like FLASH for an arbitrary number of processes ?
2
INVARIANTS FOR FINITE INSTANCES AND BEYOND October, 21 st 2013 - - PowerPoint PPT Presentation
INVARIANTS FOR FINITE INSTANCES AND BEYOND October, 21 st 2013 - - PowerPoint PPT Presentation
INVARIANTS FOR FINITE INSTANCES AND BEYOND October, 21 st 2013 Sylvain Conchon, Amit Goel, Sava Kristi c, Alain Mebsout , Fatiha Za di LRI, Universit e Paris-Sud Strategic CAD Labs, Intel Corporation Challenge How to prove safety of
SLIDE 2
SLIDE 3
Challenge How to prove safety of industrial size protocols like FLASH for an arbitrary number of processes ?
◮ automatically
2
SLIDE 4
The FLASH protocol
Stanford FLASH multiprocessor architecture (1994)
◮ Cache-coherence shared memory ◮ High-performance message passing ◮ Industrial size: 67 million states for 4 processes (28,000
states for German)
3
SLIDE 5
The FLASH protocol
Stanford FLASH multiprocessor architecture (1994)
◮ Cache-coherence shared memory ◮ High-performance message passing ◮ Industrial size: 67 million states for 4 processes (28,000
states for German) Who proved the protocol?
◮ Park and Dill, 1996, PVS proof ◮ Das, Dill and Park, 1999, by predicate abstraction ◮ McMillan, 2001, by compositional model checking ◮ Chou, Mannava, Park, 2004, CMP method inspired by
McMillan’s work
◮ Talapur and Tuttle, 2008, message-flows extension of CMP
None of these proofs are purely automatic
3
SLIDE 6
Solutions
◮ Model checking of parameterized systems ◮ Decidable fragment ◮ Cubicle implements backward reachability
4
SLIDE 7
Solutions
◮ Model checking of parameterized systems ◮ Decidable fragment ◮ Cubicle implements backward reachability
Does it work ?
4
SLIDE 8
Some benchmarks
Cubicle CMurphi Szymanski at 0.30s 8.04s (8) 5m12s (10) 2h50m (12) German Baukus 7.03s 0.74s (4) 19m35s (8) 4h49m (10) German.CTC 3m23s 1.83s (4) 43m46s (8) 12h35m (10) German pfs 3m58s 0.99s (4) 22m56s (8) 5h30m (10) Chandra-Toueg 2h01m 5.68s (4) 2m58s (5) 1h36m (6)
5
SLIDE 9
Some benchmarks
Cubicle CMurphi Szymanski at 0.30s 8.04s (8) 5m12s (10) 2h50m (12) German Baukus 7.03s 0.74s (4) 19m35s (8) 4h49m (10) German.CTC 3m23s 1.83s (4) 43m46s (8) 12h35m (10) German pfs 3m58s 0.99s (4) 22m56s (8) 5h30m (10) Chandra-Toueg 2h01m 5.68s (4) 2m58s (5) 1h36m (6) Szymanski na T.O. 0.88s (4) 8m25s (6) 7h08m (8) Flash nodata O.M. 4.86s (3) 3m33s (4) 2h46m (5) Flash O.M. 1m27s (3) 2h15m (4) O.M. (5)
O.M. > 20 GB T.O. > 20 h
5
SLIDE 10
How to scale ?
◮ Reduce the state space to explore ◮ Invariants for parameterized case ◮ Interesting behaviors often observable on small instances
6
SLIDE 11
Invariants inference
Problem: Invariants often harder to prove than original property
7
SLIDE 12
Invariants inference
Problem: Invariants often harder to prove than original property Idea: use finite instances to infer invariants for parametrized case
◮ Insert and check on the fly in backward reachability loop ◮ Backtrack if necessary
BRAB: Backward Reachability with Approximations and Backtracking
7
SLIDE 13
Backward reachability algorithm
I U 8
SLIDE 14
Backward reachability algorithm
V Q I U 8
SLIDE 15
Backward reachability algorithm
V Q I U 8
SLIDE 16
Backward reachability algorithm
V Q I U 8
SLIDE 17
Backward reachability algorithm
V Q I U 8
SLIDE 18
Backward reachability algorithm
V Q I U 8
SLIDE 19
Backward reachability algorithm
V Q I U 8
SLIDE 20
Backward reachability algorithm
V Q I U 8
SLIDE 21
Backward reachability algorithm
V I U 8
SLIDE 22
BRAB: intuition
I U
9
SLIDE 23
BRAB: intuition
I U 2
9
SLIDE 24
BRAB: intuition
I U 2 2
9
SLIDE 25
BRAB: intuition
V Q I U 2 2
9
SLIDE 26
BRAB: intuition
V Q ϕ I U 2 2
9
SLIDE 27
BRAB: intuition
V Q candidate I U 2 2
9
SLIDE 28
BRAB: intuition
V Q I U 2 2
9
SLIDE 29
BRAB: intuition
V Q V Q I U 2 2
9
SLIDE 30
BRAB: intuition
V Q V Q ϕ I U 2 2
9
SLIDE 31
BRAB: intuition
V Q V Q I U 2 2
9
SLIDE 32
BRAB: intuition
V Q I U 2 2
9
SLIDE 33
BRAB: intuition
V Q I U 2 2
9
SLIDE 34
BRAB: intuition
V Q I U 2 2
9
SLIDE 35
BRAB: intuition
V Q I U 2 2
9
SLIDE 36
BRAB: intuition
I U 2 2
9
SLIDE 37
BRAB: intuition
V Q I U 2 2
9
SLIDE 38
BRAB: intuition
V Q I U 2 2
9
SLIDE 39
BRAB: intuition
V Q I U 2 2
9
SLIDE 40
BRAB: intuition
V Q I U 2 2
9
SLIDE 41
BRAB: intuition
V I U 2 2
9
SLIDE 42
Framework
◮ Symbolic framework for parameterized systems ◮ States : formulas in a decidable fragment of FOL ◮ Pre-image effectively computable ◮ Post-image effectively computable for a finite instance
10
SLIDE 43
Framework
◮ Symbolic framework for parameterized systems ◮ States : formulas in a decidable fragment of FOL ◮ Pre-image effectively computable ◮ Post-image effectively computable for a finite instance
In Cubicle → array-based transition systems
10
SLIDE 44
Example: German-ish cache coherence protocol
Client i:
Cache[i] ∈ {E, S, I}
Directory:
Cmd ∈ {rs, re, ǫ} Ptr ∈ proc Shr[i] ∈ {true, false} Exg ∈ {true, false}
E S I Shr[i] := true Exg := true Exg := true Shr[i] := true Shr[i] := false Exg := false Exg := false Shr[i] := false
Initial states:
∀i. Cache[i] = I ∧ ¬Shr[i] ∧ ¬Exg ∧ Cmd = ǫ
Unsafe states:
∃i, j. i = j ∧ Cache[i] = E ∧ Cache[j] = I ?
(cubes)
11
SLIDE 45
Example: German-ish cache coherence protocol
Client i:
Cache[i] ∈ {E, S, I}
Directory:
Cmd ∈ {rs, re, ǫ} Ptr ∈ proc Shr[i] ∈ {true, false} Exg ∈ {true, false}
E S I Shr[i] := true Exg := true Exg := true Shr[i] := true Shr[i] := false Exg := false Exg := false Shr[i] := false
t5 : ∃i. Ptr = i ∧ Cmd = rs ∧ ¬Exg ∧ Cmd′ = ǫ ∧ Shr′[i] ∧ Cache′[i] = S
11
SLIDE 46
BRAB algorithm
I : inital states U : unsafe states (cubes)
T : transitions
BRAB ():
B := ∅; Kind(U) := Orig; From(U) := U; M := FWD(dmax, k) ;
while BWDA() = unsafe do if Kind(F) = Orig then return unsafe
B := B ∪ { From(F) };
return safe
12
SLIDE 47
BRAB algorithm
I : inital states U : unsafe states (cubes)
T : transitions
BWD ():
V := ∅ ;
push(Q, U ) ; while not empty(Q) do
ϕ := pop(Q);
if ϕ ∧ I sat then return unsafe if ¬(ϕ |
=
ψ∈V ψ) then
V := V ∪ {ϕ};
push(Q, preT (ϕ)); return safe
13
SLIDE 48
BRAB algorithm
I : inital states U : unsafe states (cubes)
T : transitions
BWDA ():
V := ∅ ;
push(Q, U ) ; while not empty(Q) do
ϕ := pop(Q);
if ϕ ∧ I sat then return unsafe if ¬(ϕ |
=
ψ∈V ψ) then
V := V ∪ {ϕ};
push(Q, ApproxT (ϕ) ); return safe
13
SLIDE 49
BRAB algorithm
I : inital states U : unsafe states (cubes)
T : transitions
ApproxT (ϕ): foreach ψ in candidates(ϕ) do if ψ ∈ B ∧ M ψ then
Kind(ψ) := Appr ;
. . . return ψ . . . return preT (ϕ)
14
SLIDE 50
Example: BRAB on German-ish
¬Exg Cmd = ǫ ∀i. Cache[i] = I ¬Shr[i] t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1)
∃i = j. Cache[i] = E Cache[j] = I
15
SLIDE 51
Example: BRAB on German-ish
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1)
∃i = j. Cache[i] = E Cache[j] = I
15
SLIDE 52
Example: BRAB on German-ish
. . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
15
SLIDE 53
Example: BRAB on German-ish
. . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j]
pre(t4(j))
15
SLIDE 54
Example: BRAB on German-ish
. . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
pre(t4(j)) pre(t5(j))
15
SLIDE 55
Example: BRAB on German-ish
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
pre(t4(j)) pre(t5(j)) pre(t6(i))
15
SLIDE 56
Example: BRAB on German-ish
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i))
15
SLIDE 57
Example: BRAB on German-ish
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i))
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j]
∃i. Cmd = rs Cache[i] = E
Extracting a candidate (ApproxT )
15
SLIDE 58
Example: BRAB on German-ish
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i))
15
SLIDE 59
Example: BRAB on German-ish
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i))
Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
∃i. Cmd = rs Cache[i] = E
Checking candidate
15
SLIDE 60
Example: BRAB on German-ish
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i))
| =
Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
∃i. Cmd = rs Cache[i] = E
Checking candidate
15
SLIDE 61
Example: BRAB on German-ish
| =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i))
15
SLIDE 62
Example: BRAB on German-ish
| =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Shr[j] Cache[i] = E ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i))
15
SLIDE 63
Example: BRAB on German-ish
| =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Shr[j] Cache[i] = E ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs pre(t4(j)) pre(t5(j)) pre(t6(i))
15
SLIDE 64
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Shr[j] Cache[i] = E ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs pre(t4(j)) pre(t5(j)) pre(t6(i))
15
SLIDE 65
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
Cache[i] = E ∃i. ¬Exg ∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs ∃i = j. Shr[j] Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i))
15
SLIDE 66
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
Cache[i] = E ∃i. ¬Exg ∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs ∃i = j. Shr[j] Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i)) pre(t5(j))
15
SLIDE 67
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
Cache[i] = E ∃i. ¬Exg ∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs ∃i = j. Shr[j] Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i)) pre(t5(j))
15
SLIDE 68
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
Cache[i] = E
∃i = j. Cmd = re Cache[i] = E Shr[j]
∃i. ¬Exg ∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs ∃i = j. Shr[j] Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i)) pre(t5(j)) pre(t3(j))
15
SLIDE 69
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
Cache[i] = E
∃i = j. Cmd = re Cache[i] = E Shr[j]
∃i. ¬Exg ∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs ∃i = j. Shr[j] Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i)) pre(t5(j)) pre(t3(j))
15
SLIDE 70
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
Cache[i] = E
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. Cmd = re Cache[i] = E Shr[j]
∃i. ¬Exg ∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs ∃i = j. Shr[j] Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i)) pre(t5(j)) pre(t4(j)) pre(t3(j))
15
SLIDE 71
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
Cache[i] = E
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. Cmd = re Cache[i] = E Shr[j]
∃i. ¬Exg ∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs ∃i = j. Shr[j] Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i)) pre(t5(j)) pre(t4(j)) pre(t3(j))
15
SLIDE 72
Example: BRAB on German-ish
| = | =
. . . . . .
¬Exg Cmd = ǫ Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = re Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = re Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] ¬Exg Cmd = rs Ptr = #2 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2]
¬Exg Cmd = rs
Ptr = #1 Cache[#1] = I Cache[#2] = I ¬Shr[#1] ¬Shr[#2] Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = E ¬Shr[#1] Shr[#2] Exg Cmd = ǫ Ptr = #1 Cache[#1] = E Cache[#2] = I Shr[#1] ¬Shr[#2] ¬Exg Cmd = ǫ Ptr = #2 Cache[#1] = I Cache[#2] = S ¬Shr[#1] Shr[#2] Exg
Cmd = rs
Ptr = #2
Cache[#1] = E
Cache[#2] = I Shr[#1] ¬Shr[#2]
t2(#2) t2(#1) t1(#2) t1(#1) t6(#2) t6(#1) t5(#2) t5(#1) t2(#1) t1(#1) t2(#2) t1(#2) t2(#2) t2(#1) t1(#1) ∃i = j. Cache[i] = E Cache[j] = I
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E ∃i = j. ¬Exg Cmd = rs Ptr = j Cache[i] = E
Cache[i] = E
∃i = j. Exg Cmd = rs Cache[i] = E Shr[j] ∃i = j. Cmd = re Cache[i] = E Shr[j]
∃i. ¬Exg ∃i. Cmd = rs Cache[i] = E ¬Exg Cmd = rs ∃i = j. Shr[j] Cache[i] = E pre(t4(j)) pre(t5(j)) pre(t6(i)) pre(t5(j)) pre(t4(j)) pre(t3(j))
15
SLIDE 73
Some benchmarks
BRAB Cubicle CMurphi Szymanski at 0.14s 0.30s 8.04s (8) 5m12s (10) 2h50m (12) German Baukus 0.25s 7.03s 0.74s (4) 19m35s (8) 4h49m (10) German.CTC 0.29s 3m23s 1.83s (4) 43m46s (8) 12h35m (10) German pfs 0.34s 3m58s 0.99s (4) 22m56s (8) 5h30m (10) Chandra-Toueg 2m17s 2h01m 5.68s (4) 2m58s (5) 1h36m (6) Szymanski na 0.19s T.O. 0.88s (4) 8m25s (6) 7h08m (8) Flash nodata 0.36s O.M. 4.86s (3) 3m33s (4) 2h46m (5) Flash 5m40s O.M. 1m27s (3) 2h15m (4) O.M. (5)
O.M. > 20 GB T.O. > 20 h
16
SLIDE 74
Remarks
◮ BRAB is complete only if the framework admits a complete
Backward Reachability
◮ Cubicle goes beyond decidable fragment of array-based
systems
◮ FLASH is expressed outside of this fragment ◮ BRAB remains safe
17
SLIDE 75
Future work
Improvements:
◮ Experiment with real size industrial protocols ◮ Improve backtracking ◮ Difficult to discover candidates for numerical invariants
Certification:
◮ Deductive program verification (with Why3 and Alt-Ergo) and
code extraction
◮ Goal: obtain a certified and efficient model checker
18
SLIDE 76
Thank you
Visit our web site
http://cubicle.lri.fr
19