a decidable recursive logic for weighted transition systems bingtian xue aalborg university, denmark...
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A Decidable Recursive Logic for Weighted Transition Systems
Bingtian XueAalborg University, DENMARK
ICTAC, Sep 18, 2014
Joint work with
Kim G. Larsen Radu Mardare
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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
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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
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Labeled Weighted System (LWS)
non-empty set of states : finite set of ( types of) resources non-empty set of actions labeled transition function
Example:
ICTAC, Sep 18, 2014
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Weighted Bisimulation
s.t. whenever If , then s.t.
and; If , then s.t.
and
ICTAC, Sep 18, 2014
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Basic formulas
Extended states: : iff and .
Semantics:
where
Recursive Weighted Logic (RWL)
ICTAC, Sep 18, 2014
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Basic formulas
Semantics:
Example:
Recursive Weighted Logic (RWL)
ICTAC, Sep 18, 2014
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Basic formulas
Semantics:
Example:
Recursive Weighted Logic (RWL)
ICTAC, Sep 18, 2014
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Basic formulas
Semantics:
Recursive Weighted Logic (RWL)
ICTAC, Sep 18, 2014
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Recursive Weighted Logic (RWL)
Maximal equation block
Semantics:
ICTAC, Sep 18, 2014
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Recursive Weighted Logic (RWL)
Example
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Satisfiability Problem
ICTAC, Sep 18, 2014
Satisfiability problem
Given and , ask whether there exists ?
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Satisfiability Problem
Example
Only satisfied by infinite model!
Region theory
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Region Theory
ICTAC, Sep 18, 2014
Region theory
One resource: given iff:
Regions: equivalence classes under
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Region Theory
ICTAC, Sep 18, 2014
Region theory
Successor: iff:
Multi resources:
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Region Theory
ICTAC, Sep 18, 2014
Symbolic LWS
satisfying: if , then
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Region Theory
ICTAC, Sep 18, 2014
Symbolic LWS
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Region Theory
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Lemma:
If there exists , then there exists LWS s.t.
Proof:
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Satisfiability Problem
ICTAC, Sep 18, 2014
Maximal:
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Satisfiability Problem
ICTAC, Sep 18, 2014
Mutually-consistent:
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Satisfiability Problem
ICTAC, Sep 18, 2014
Lemma: Theorem:
Theorem: -
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Satisfiability Problem
ICTAC, Sep 18, 2014
Example:
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Satisfiability Problem
ICTAC, Sep 18, 2014
Example:
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Satisfiability Problem
ICTAC, Sep 18, 2014
Example:
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ICTAC, Sep 18, 2014
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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ICTAC, Sep 18, 2014
Recursive Weighted Logic
Thank you!