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A calculus of Untyped Aspect-Oriented Programs

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A calculus of Untyped Aspect-Oriented Programs Powered By Docstoc
					            Approximate reasoning
            for probabilistic real-
            time processes
Radha Jagadeesan DePaul University
Vineet Gupta       Google Inc
Prakash Panangaden McGill University
Outline of talk
 Beyond CTMCs to GSMPs
 The curse of real numbers
 Metrics
 Uniformities
 Approximate reasoning
Real-time probabilistic processes
   Add clocks to Markov processes

       Each clock runs down at fixed rate


       Different clocks can have different rates

   Generalized Semi Markov Processes:
    Probabilistic multi-rate timed automata
    Generalized semi-Markov
    processes.
                                  Each state is labelled
            {c,d}   s             with propositional
                                  Information

                                  Each state has a set
                                  of clocks associated
                                  with it.
u   {d,e}               {c}   t
    Generalized semi-Markov
    processes.
                                  Evolution determined by
            {c,d}   s             generalized states
                                    <state, clock-valuation>

                                  <s,c=2, d=1>

                                  Transition enabled when a clock
                                  becomes zero
u   {d,e}               {c}   t
    Generalized semi-Markov
    processes.
                              <s,c=2, d=1> Transition enabled in
            {c,d}   s              1 time unit

                              <s,c=0.5,d=1> Transition enabled in
                                   0.5 time unit



u   {d,e}               {c}   t

                                                  Clock c

                                                  Clock d
    Generalized semi-Markov
    processes.

            {c,d}   s            Transition determines:

                                a. Probability distribution on next
                                   states
     0.2             0.8
                                b. Probability distribution on
                                   clock values for new clocks
u   {d,e}               {c}   t c. This need not be exponential.

                                                   Clock c

                                                   Clock d
Generalized semi Markov
processes
   If distributions are continuous and states are
    finite:

     Zeno traces have measure 0

   Continuity results. If
    stochastic processes from <s, > converge
    to the stochastic process at <s, >
The traditional reasoning paradigm

 Establishing equality: Coinduction
 Distinguishing states: HM-type logics
 Logic characterizes the equivalence (often
  bisimulation)
 Compositional reasoning: ``bisimulation is
  a congruence’’
Labelled Markov       PCTL
Processes             Bisimulation [Larsen-Skou,
                      Desharnais-Edalat-P]


Markov Decision       Bisimulation [Givan-Dean-Grieg]
Processes

Labelled Concurrent   PCTL [Hansson-Johnsson]
Markov Chains
Labelled Concurrent   PCTLCompleteness: [Desharnais-
Markov chains (with   Gupta-Jagadeesan-P]
tau)                  Weak bisimulation
                      [Philippou-Lee-Sokolsky,
                       Lynch-Segala]
With continuous time
Continuous time Markov CSL [Aziz-Balarin-Brayton-
chains                 Sanwal-Singhal-S.Vincentelli]


                            Bisimulation,Lumpability
                            [Hillston, Baier-Katoen-
                            Hermanns,Desharnais-P]

Generalized Semi-           CSL
Markov processes
                            Bisimulation:?????
Stochastic hybrid
systems                     Composition:?????
The curse of real numbers:
instability
                Vs




                Vs
Problem!
   Numbers viewed as coming with an error
    estimate.
   Reasoning in continuous time and continuous
    space is often via discrete approximations.
   Asking for trouble if we require exact match
Idea: Equivalence              metrics
   Jou-Smolka90, DGJP99, …
    Replace equality of processes by (pseudo)
    metric distances between processes


   Quantitative measurement of the
    distinction between processes.
Criteria on approximate reasoning

 Soundness
 Usability
 Robustness
Criteria on metrics for approximate
reasoning
   Soundness



   Stability of distance under temporal
    evolution: “Nearby states stay close”
    through temporal evolution.
``Usability’’ criteria on metrics
 Establishing closeness of states:
  Coinduction.
 Distinguishing states: Real-valued modal
  logics.
 Equational and logical views coincide:
  Metrics yield same distances as real-
  valued modal logics.
``Robustness’’ criterion on
approximate reasoning
 The actual numerical values of the
  metrics should not matter too much.
 Only the topology matters?
 Our results show that everything is defined
  “up to uniformities.’’
What are uniformities?
 In topology open sets capture an abstract
  notion of “nearness”: continuity,
  convergence, compactness, separation …
 In a uniformity one axiomatises the notion
  of “almost an equivalence relation”:
  uniform continuity, …
 Uniform continuity is not a topological
  invariant.
Uniformities: definition
 A nonempty collection U of subsets of SxS
  such that:
 Every member of U contains
 If X in U then so is
 If X in U, there is a Y s.t. YoY is contained
  in X
 Down closed, intersection closed
 Two apparently different
 Uniformities which are actually the
 same




m(x,y) = |x-y|          m(x,y) = |2x + sinx
                                  -2y – siny|
 Uniformities (different)




m(x,y) = |x-y|
Our results
Our results
 A metric on GSMPs based on
  Wasserstein-Kantorovich and Skorohod
 A real-valued modal logic
 Everything defined up to uniformity
Results for discrete time models
               Bisimulation   Metrics

Logic          (P)CTL(*)      Real-valued
                              modal logic
Compositionality Congruence   Non-
                              expansivity
Proofs         Coinduction    Coinduction
Results for continuous time models
                Bisimulation   Metrics

Logic           CSL            Real-valued
                               modal logic
Compositionality ???           ???

Proofs          Coinduction    Coinduction
Metrics for discrete
time probabilistic
processes
Defining metric: An attempt
Define functional F on metrics.
Metrics on probability measures
   Wasserstein-Kantorovich

   A way to lift distances from states to a
    distances on distributions of states.
Metrics on probability measures
Not up to uniformities
   If the Wasserstein metric is scaled you get
    the same uniformity, but when you
    compute the fixed point you get a different
    uniformity because the lattice of
    uniformities has a different structure (glbs
    are different) then the lattice of metrics.
Variant definition that works up to
uniformities
Fix c<1. Define functional F on metrics




 Desired metric is maximum fixed point of F
Reasoning up to uniformities
 For all c<1 we get same uniformity
[see Breugel/Mislove/Ouaknine/Worrell]
Metrics for real-time
probabilistic
processes
    Generalized semi-Markov
    processes.
                                  Evolution determined by
            {c,d}   s             generalized states
                                    <state, clock-valuation>


                                       : Set of generalized states


u   {d,e}               {c}   t

                                                     Clock c

                                                     Clock d
The role of paths
 In the continuous time case we cannot use
  single actions: there is no notion of
  “primitive step”
 We have to talk about a “timed path” of
  one process matching a “timed path” of
  another process.
    Generalized semi-Markov
    processes.

            {c,d}   s         Path:



                              Traces((s,c)): Probability distribution
                                on a set of paths.

u   {d,e}               {c}   t

                                                    Clock c

                                                    Clock d
Accomodating discontinuities:
cadlag functions
(M,m) a pseudometric space.
                     cadlag if:
Countably many jumps, finitely many jumps
higher than any fixed “h”.
Defining metric: An attempt
Define functional F on metrics. (c <1)




traces((s,c)), traces((t,d)) are distributions on
sets of cadlag functions.

What is a metric on cadlag functions???
Metrics on cadlag functions




            x     y

         are at distance 1 for unequal x,y
 Not separable!
Skorohod’s metrics on cadlag
Skorohod defined 4 metrics on cadlag: J1,J2
M1 and M2 with different convergence
properties.
All these are based on “wiggling” the time.
The M metrics “fill in the jumps”.
The J metrics do not.
Skorohod metric (J2)
(M,m) a pseudometric space. f,g cadlag with
  range M.

Graph(f) = { (t,f(t)) | t \in R+}
Skorohod J2 metric: Hausdorff distance between graphs of f,g



                                                          g
          f
                    (t,f(t))
f(t)
g(t)




                               t
Skorohod J2 metric
(M,m) a pseudometric space. f,g cadlag
Examples of convergence to
Example of convergence




            1/2
Example of convergence




            1/2
Examples of convergence




            1/2
Examples of convergence




            1/2
Non-convergence in J2:
Sequences of continuous functions cannot converge to
a discontinuous function.


In general, the number of jumps can decrease in the limit,
but they cannot increase.
Non-convergence
Non-convergence
Non-convergence
Non-convergence
Summary of Skorohod J2
 A separable metric space on cadlag
  functions
 Allows jumps to be nearby
 Allows jumps to decrease in the limit.
 Not complete.
 Defining metric coinductively
Define functional on 1-bounded pseudometrics (c <1)




    a. s, t agree on all propositions

    b.

  Desired metric: maximum fixpoint of F
Results

   All c<1 yield the same uniformity. Thus,
    construction can be carried out in lattice of
    uniformities.
   Real valued modal logic which gives an
    alternate definition of a metric.
   For each c<1, modal logic yields the same
    uniformity but not the same metric.
Proof steps
 Continuity theorems (Whitt) of GSMPs
  yield separable basis.
 Finite separability arguments yield the
  result that the closure ordinal of the
  functional F is omega.
 Duality theory of LP for calculating metric
  distances.
Summary
 Metric on GSMPs defined up to uniformity.
 Real valued modal logic that gives the
  same uniformity.
 Approximating quantitative observables:
  Expectations of continuous functions are
  continuous.
 Might be worth looking at the M2 metric.
Real-valued modal logic
Real-valued modal logic
Real-valued modal logic
Real-valued modal logic




   h: Lipschitz operator on unit interval
Real-valued modal logic




Base case for path formulas??
Base case for path formulas

 First attempt:

      Evaluate state formula F on state
      at time t
 Problem: Not smooth enough wrt time since
 paths have discontinuities
  Base case for path formulas

    Next attempt:

         ``Time-smooth’’ evaluation of state
         formula F at time t on path



Upper Lipschitz approximation to      evaluated
at t
Real-valued modal logic
Non-convergence
Illustrating Non-convergence




               1/2


                     1/2

				
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posted:4/5/2010
language:English
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