Post on 28-Dec-2015
A Decidable Recursive Logic for Weighted Transition Systems
Bingtian XueAalborg University, DENMARK
ICTAC, Sep 18, 2014
Joint work with
Kim G. Larsen Radu Mardare
Weighted processes to express quantitative properties of real systems the focus is on the weighted behaviors
weighted bisimulation => identical weighted behaviors
Weighted Modal Logics => characterize weighted bisimilarity
Recursion to encode properties of infinite behaviors: safety,
cost-bound liveness
Motivation
ICTAC, Sep 18, 2014
Properties related to timing and resource constraints
Motivation
Weighted Transition System
”Complete Proof Systems for Weighted Modal Logic” [TCS13](Kim Larsen, Radu Mardare)
ICTAC, Sep 18, 2014
Labeled Weighted System (LWS)
non-empty set of states : finite set of ( types of) resources non-empty set of actions labeled transition function
Example:
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Weighted Bisimulation
s.t. whenever If , then s.t.
and; If , then s.t.
and
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Basic formulas
Extended states: : iff and .
Semantics:
where
Recursive Weighted Logic (RWL)
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Basic formulas
Semantics:
Example:
Recursive Weighted Logic (RWL)
ICTAC, Sep 18, 2014
Basic formulas
Semantics:
Example:
Recursive Weighted Logic (RWL)
ICTAC, Sep 18, 2014
Basic formulas
Semantics:
Recursive Weighted Logic (RWL)
ICTAC, Sep 18, 2014
Recursive Weighted Logic (RWL)
Maximal equation block
Semantics:
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Recursive Weighted Logic (RWL)
Example
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Satisfiability Problem
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Satisfiability problem
Given and , ask whether there exists ?
Satisfiability Problem
Example
Only satisfied by infinite model!
Region theory
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Region Theory
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Region theory
One resource: given iff:
Regions: equivalence classes under
Region Theory
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Region theory
Successor: iff:
Multi resources:
Region Theory
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Symbolic LWS
satisfying: if , then
Region Theory
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Symbolic LWS
Region Theory
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Lemma:
If there exists , then there exists LWS s.t.
Proof:
Satisfiability Problem
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Maximal:
Satisfiability Problem
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Mutually-consistent:
Satisfiability Problem
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Lemma: Theorem:
Theorem: -
Satisfiability Problem
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Example:
Satisfiability Problem
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Example:
Satisfiability Problem
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Example:
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Conclusion
Conclusion RWL does not enjoy the finite model
property Satisfiability problem for RWL is decidable
Future work Extensions of RWL Axiomatization Weighted distance
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Recursive Weighted Logic
Thank you!