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:

ICTAC, Sep 18, 2014

Weighted Bisimulation

s.t. whenever If , then s.t.

and; If , then s.t.

and

ICTAC, Sep 18, 2014

Basic formulas

Extended states: : iff and .

Semantics:

where

Recursive Weighted Logic (RWL)

ICTAC, Sep 18, 2014

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:

ICTAC, Sep 18, 2014

Recursive Weighted Logic (RWL)

Example

ICTAC, Sep 18, 2014

Satisfiability Problem

ICTAC, Sep 18, 2014

Satisfiability problem

Given and , ask whether there exists ?

Satisfiability Problem

Example

Only satisfied by infinite model!

Region theory

ICTAC, Sep 18, 2014

Region Theory

ICTAC, Sep 18, 2014

Region theory

One resource: given iff:

Regions: equivalence classes under

Region Theory

ICTAC, Sep 18, 2014

Region theory

Successor: iff:

Multi resources:

Region Theory

ICTAC, Sep 18, 2014

Symbolic LWS

satisfying: if , then

Region Theory

ICTAC, Sep 18, 2014

Symbolic LWS

Region Theory

ICTAC, Sep 18, 2014

Lemma:

If there exists , then there exists LWS s.t.

Proof:

Satisfiability Problem

ICTAC, Sep 18, 2014

Maximal:

Satisfiability Problem

ICTAC, Sep 18, 2014

Mutually-consistent:

Satisfiability Problem

ICTAC, Sep 18, 2014

Lemma: Theorem:

Theorem: -

Satisfiability Problem

ICTAC, Sep 18, 2014

Example:

Satisfiability Problem

ICTAC, Sep 18, 2014

Example:

Satisfiability Problem

ICTAC, Sep 18, 2014

Example:

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

ICTAC, Sep 18, 2014

Recursive Weighted Logic

Thank you!