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Minimization of Large State Spaces using Symbolic Branching - - PowerPoint PPT Presentation
Minimization of Large State Spaces using Symbolic Branching - - PowerPoint PPT Presentation
Minimization of Large State Spaces using Symbolic Branching Bisimulation Ralf Wimmer (joint work with Marc Herbstritt and Bernd Becker) Institute of Computer Science Albert-Ludwigs-University Freiburg Germany April 18th, 2006 Introduction
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Toolflow
Statemate description Safety requirements symbolic LTS Small explicit LTS Transformation Symbolic Minimization Huge
Afterwards: Addition of stochastic information and application of stochastic model checking.
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Overview
1
Introduction Why Symbolic Branching Bisimulation? Labeled Transition Systems Branching Bisimulation
2
Symbolic Computation Symbolic Representation of LTS, Partitions and Signatures Computation of the Signatures Refinement
3
Experimental Results
4
Further Work
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Why Symbolic Branching Bisimulation?
Nominal behaviour is irrelevant. ⇒ LTS with “unobservable” τ-actions. Branching Bisimulation
preserves all interesting properties (CTL*\X) makes use of τ-actions.
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Labelled Transition System
A labelled transition system M is a triple M = (S, A, T):
s1 s2 s3 s4 s5 s6 s7 s8 s9 τ a b τ τ τ τ τ τ τ a a a b
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Branching Bisimulation
An equivalence relation B is a branching bisimulation if for all (s, t) ∈ B s
a
− → s′ implies: Bi
τ s s′ t
- r
a a
τ ∗
Bi Bk
s s′ t t′′ t′
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Example: Branching Bisimulation
s1 s2 s3 s4 s5 s6 s7 s8 s9 τ a b τ τ τ τ τ τ τ a a a b
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Signatures
Signature = set of pairs (action a, block B) meaning: “With the action a you can go to the block B under certain conditions”. (a, Bk) ∈ sig(s) iff
a τ τ
Bk Bj s s′ s′′
a = τ ∨ Bi = Bj
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Refinement
The states are grouped according to their signatures: sigref(π) = {{t ∈ S | sig(t) = sig(s)} | s ∈ S} Iteration to the fixpoint yields the coarsest branching bisimulation.
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Symbolic Representation
Unique numbers are assigned to each block of the current partition π = {B0, . . . , Bm−1}. BDD representation state space: S(s) = 1 iff s ∈ S transition relation: T (s, a, t) = 1 iff s
a
− → t partitions: P(s, k) = 1 iff s ∈ Bk. signatures: σ(s, a, k) = 1 iff (a, Bk) ∈ sig(s).
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Signatures
Formal Definition
sig(s) = {(a, B) | ∃s′, s′′ ∈ S : s
τ ∗
− →
π
s′
a
− → s′′ ∈ B ∧ (a = τ ∨ s ≡π s′′)}
The signatures can be computed using standard BDD operations: Boolean connectives existential quantification reflexive transitive closure Problem Efficient implementation of the refinement operator is not possible using standard BDD operations. How can it be done efficiently?
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Refinement (1)
Observation Assuming the variable order si < aj ∧ si < kl. Then each signature is represented by a unique node of the BDD. Idea Substitute these signature nodes by new block numbers.
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Refinement (2)
s0 a0 node v Signature of all states that lead to node v s0 k0 node v
refine BDD-representation
- f the new block number
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Results
0.01 1 100 10000 1e+06 1e+08 1e+10 1 2 3 4 5 6 7 8
Time [s] / Number Kanban parameter
time number of states number of transitions
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Comparison Sigref ↔ bcg min (1)
100 200 300 400 500 600 1 2 3 4 5
Time [s] Kanban parameter
sigref bcg_min
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Comparison Sigref ↔ bcg min (2)
500 1000 1500 2000 1 2 3 4 5 6 7 8
Memory [MB] Kanban parameter
sigref bcg_min
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Further Work
Extension of the approach to other types of bisimulations: Strong Bisimulation Weak Bisimulation Safety Bisimulation ... and stochastic variants thereof.
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