Approximate reasoning for probabilistic realtime processes

1
Approximate reasoning for probabilistic real-time
processes

• Radha Jagadeesan DePaul University
• Vineet Gupta Google Inc
• Prakash Panangaden McGill University

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Outline of talk

• Beyond CTMCs to GSMPs
• The curse of real numbers
• Metrics
• Uniformities
• Approximate reasoning

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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

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Generalized semi-Markov processes.
Each state is labelled with propositional
Information Each state has a set of clocks
associated with it.
c,d
s
c
t
u
d,e
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Generalized semi-Markov processes.
Evolution determined by generalized states
ltstate, clock-valuationgt lts,c2,
d1gtTransition enabled when a clockbecomes
zero
c,d
s
c
t
u
d,e
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Generalized semi-Markov processes.
lts,c2, d1gt Transition enabled in 1
time unit lts,c0.5,d1gt Transition enabled in
0.5 time unit
c,d
s
c
t
u
d,e
Clock c
Clock d
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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
c
c. This need not be exponential.
t
u
d,e
Clock c
Clock d
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Generalized semi Markov processes

• If distributions are continuous and states are
  finite Zeno traces have measure 0
• Continuity results. If stochastic processes
  from lts, gt converge to the stochastic process
  at lts, gt

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The traditional reasoning paradigm

• Establishing equality Coinduction
• Distinguishing states HM-type logics
• Logic characterizes the equivalence (often
  bisimulation)
• Compositional reasoning bisimulation is a
  congruence

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With continuous time
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The curse of real numbers instability
Vs
Vs
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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

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Idea Equivalence metrics

• Jou-Smolka90, DGJP99,
• Replace equality of processes by (pseudo)
  metric distances between processes
• Quantitative measurement of the distinction
  between processes.

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Criteria on approximate reasoning

• Soundness
• Usability
• Robustness

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Criteria on metrics for approximate reasoning

• Soundness
• Stability of distance under temporal evolution
  Nearby states stay close through temporal
  evolution.

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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.

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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.

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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.

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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

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Two apparently different Uniformities which are
actually the same
m(x,y) 2x sinx -2y siny
m(x,y) x-y
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Uniformities (different)
m(x,y) x-y
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Our results
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Our results

• A metric on GSMPs based on Wasserstein-Kantorovich
  and Skorohod
• A real-valued modal logic
• Everything defined up to uniformity

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Results for discrete time models
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Results for continuous time models
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Metrics for discrete time probabilistic processes
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Defining metric An attempt

• Define functional F on metrics.

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Metrics on probability measures

• Wasserstein-Kantorovich
• A way to lift distances from states to a
  distances on distributions of states.

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Metrics on probability measures
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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.

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Variant definition that works up to uniformities

• Fix clt1. Define functional F on metrics

Desired metric is maximum fixed point of F
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Reasoning up to uniformities

• For all clt1 we get same uniformity
• see Breugel/Mislove/Ouaknine/Worrell

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Metrics for real-time probabilistic processes
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Generalized semi-Markov processes.
Evolution determined by generalized states
ltstate, clock-valuationgt Set of
generalized states
c,d
s
c
t
u
d,e
Clock c
Clock d
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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.

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Generalized semi-Markov processes.

Path Traces((s,c)) Probability distribution
on a set of paths.
c,d
s
c
t
u
d,e
Clock c
Clock d
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Accomodating discontinuities cadlag functions

• (M,m) a pseudometric space. cadlag if

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Countably many jumps, finitely many jumps higher
than any fixed h.
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Defining metric An attempt

• Define functional F on metrics. (c lt1)

traces((s,c)), traces((t,d)) are distributions on
sets of cadlag functions. What is a metric on
cadlag functions???
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Metrics on cadlag functions
x
y
are at distance 1 for unequal x,y
Not separable!
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Skorohods 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.

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Skorohod metric (J2)

• (M,m) a pseudometric space. f,g cadlag with range
  M.
• Graph(f) (t,f(t)) t \in R

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Skorohod J2 metric Hausdorff distance between
graphs of f,g
g
f
(t,f(t))
f(t) g(t)
t
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Skorohod J2 metric

• (M,m) a pseudometric space. f,g cadlag

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Examples of convergence to
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Example of convergence
1/2
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Example of convergence
1/2
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Examples of convergence
1/2
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Examples of convergence
1/2
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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.
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Non-convergence
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Non-convergence
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Non-convergence
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Non-convergence
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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.

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Defining metric coinductively

• Define functional on 1-bounded pseudometrics (c
  lt1)

a. s, t agree on all propositionsb.
Desired metric maximum fixpoint of F
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Results

• All clt1 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 clt1, modal logic yields the same
  uniformity but not the same metric.

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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.

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Summary

• Metric on GSMPs defined up to uniformity.
• Real valued modal logic that gives the same
  uniformity.
• Approximating quantitative observablesExpectatio
  ns of continuous functions are continuous.
• Might be worth looking at the M2 metric.

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Real-valued modal logic
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Real-valued modal logic
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Real-valued modal logic
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Real-valued modal logic
h Lipschitz operator on unit interval
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Real-valued modal logic
Base case for path formulas??
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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
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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
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Real-valued modal logic
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Non-convergence
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Illustrating Non-convergence
1/2
1/2