# A calculus of Untyped Aspect-Oriented Programs

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					            Approximate reasoning
for probabilistic real-
time processes
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, >

 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-
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:
(M,m) a pseudometric space.
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

What is a metric on cadlag functions???

x     y

are at distance 1 for unequal x,y
Not separable!
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

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 views: 3 posted: 4/5/2010 language: English pages: 70