Partial-Order Reduction for General State Exploring Algorithms
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Partial-Order Reduction for General State Exploring Algorithms
reduction function r assigns to a state the set of actions being considered. (reduced action set) .... For any u in S_r there exists at least one action a in r(u) and state v such that u-aâv .... Katz/Peled 92, Peled 94, Holzmann/Peled 94. LTL, stack.
Partial-Order Reduction for General State Exploring Algorithms Dragan Boanački Eindhoven University of Technology Stefan Leue University of Konstanz Alberto Lluch Lafuente Empoli
Main Messages " Partial order reduction for General State Exploring Algorithms Covers Depth First Search, Breadth First Search and different Directed Search Heuristics Crucial novelty: new cycle proviso Implementation in HFS Spin, an extension of Spin for directed model checking, with encouraging results
Model Checking System
S
Model
M
(Modeling Language)
satisfies
p
f
property
Formal property (Temporal Logic) State space explosion - reduction techniques needed
Partial Order Reduction Proc B
Proc A a1
s0
s0r0 a1
a2
r0
s1
b1 r1
s0r1
s1r0 s2
b1
a2
b1
a1
b2 r2 s0r2
s1r1
s2r0 b1
b2
a2
a1
s1r2
s2r1 b2
b2
a2
s2r2 Proc A || B
Partial Order Reduction reduction function r assigns to a state the set of actions being considered example (reduced action set) a1 s9
b0
s0
a0 s1
s12
c0 b0 s11 s4 04
c0
s10
a1
s6
partial-order reduction exploits independence of actions of concurrent processes
c0
independence
s5
b1
a
s7
b b
c1
b a a
s8
restrictions needed to guarantee that reduction preserves properties of interest
Partial-Order Reduction
restrictions on reduced action set
C0: empty iff no actions enabled C1 ( persistency ) c d a reduced action set
b
e
independent
8 deadlock preservation
? 8 assertion (local-property) preservation
Action Ignoring Partial Order Reduction
a1
a3
Proc B
a3
Proc A
s0
s0r0 a1
a2
r0
s1
b1 r1
s0r1
s1r0 s2
b1
a2
b1
a1
b2 r2 s0r2
s1r1
s2r0 b1
b2
a2
a1
s1r2
s2r1 b2
b2
a2
s2r2 Proc A || B
Action Ignoring Partial Order Reduction
a1
a3
Proc B
a3
Proc A
s0
s0r0
r0
b1
a1 a2
s1
r1 s1r0
s2
a2
b2 r2
s2r0 b1
Reduced Proc A || B
Action Ignoring Solution [Valmari 89] Any action which is temporarily ignored in a given state s must be eventually executed in some state reachable from s s0
a s1
a
s2
a
Main Theorem Each execution sequence σ from the original state space S that begins in a state s of the reduced state space S_r has a representative execution sequence σ in the reduced state space which contains a permutation of σ. s
a1
a2
a3
a4
b1 b2
π(a1) π(a2)
π(a3)
π(a4)
π(b1)
π(b2)
Local Properties /Assertions
f
f a
a
b
f
¬f b
a f
b
f
¬f b
a ¬f
Efficient Provisos Because of efficiency reasons we need a locally checkable version of the condition that prevents action ignoring
Stack Proviso C2s: [Godefroid/Wolper 91, Godefroid 96, Holzmann/Godefroid/Pirottin 92] For any s in S_r, there exists at least one action a in r(s) and state s such that s-a→ s and s is not on the DFS stack, i.e., s is not in stack(s ). Otherwise, r(s) = enabled_T(s). s
r(s) a s’
s’ not on DFS stack
Partial Order Reduction with Stack Proviso
a1
a3
Proc B
a3
Proc A
s0
s0r0
r0
b1
a1 a2
s1
r1 s0r1
s1r0 s2
a1
a2
b2
b2 r2 s0r2
s1r1
s2r0 b1
b2
a2
a1
s1r2
s2r1 b2
a2
s2r2 Stack Proviso Reduced Proc A || B
Queue Proviso C2q: [Boanački/Holzmann 05, Alur/Brayton/Henzinger/Qadeer/Rajamani 97] For any s in S_r there exists at least one action a in r(s) and state s such that s-a→ s and s is in the BFS queue, i.e., s is in queue (s). Otherwise, r(s) = enabled_T(s). s
r(s) a s’
s’ in the queue
General State Exploring/Expanding Algorithm
procedure GSEA(s) Closed = ∅; Open = {s} while not Open.empty() do r(u) u ← Open.extract(); Closed.insert(u); If goal(u) then return solution; For each a in enabled_T(u) do v ← τ(u,a); process(v); If reopenOK(v) then Closed.delete(v); If v not in Closed and v not in Open then Open.Insert(v);
A* Search Algorithm
procedure A*(s) Closed = ∅; Open = {s}; s.g ← 0; s.f ← h(s); Open.insert(s); while not Open.empty() do u ← Open.extractmin(); Closed.insert(u); r(u) if goal(u) then return solution; for each a in enabled_T(u) do v ← t(u,a); v.g ← u.g+cost(a); f’ ← v.g+h(v); if v in Open then if (f’
