Metrics for real time probabilistic processes
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Transcript of Metrics for real time probabilistic processes
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Metrics for real time probabilistic processes
Radha Jagadeesan, DePaul University
Vineet Gupta, Google Inc
Prakash Panangaden, McGill University
Josee Desharnais, Univ Laval
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Outline of talk
Models for real-time probabilistic processes
Approximate reasoning for real-time probabilistic processes
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Discrete Time Probabilistic processes Labelled Markov Processes
For each state sFor each label a K(s, a, U)
Each state labelledwith propositional information
0.50.3
0.2
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Discrete Time Probabilistic processes Markov Decision Processes
For each state sFor each label a K(s, a, U)
Each state labelledwith numerical rewards
0.50.3
0.2
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Discrete time probabilistic proceses
+ nondeterminism : label does not determine probability distribution uniquely.
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Real-time probabilistic processes
Add clocks to Markov processes
Each clock runs down at fixed rate r c(t) = c(0) – r t
Different clocks can have different rates
Generalized SemiMarkov Processes Probabilistic multi-rate timed automata
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Generalized semi-Markov processes.
Each state labelledwith propositional Information
Each state has a setof clocks associated with it.
{c,d}
{d,e} {c}
s
tu
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Generalized semi-Markov processes.
Evolution determined bygeneralized states <state, clock-valuation>
<s,c=2, d=1>
Transition enabled when a clockbecomes zero
{c,d}
{d,e} {c}
s
tu
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Generalized semi-Markov processes.
<s,c=2, d=1> Transition enabled in 1 time unit
<s,c=0.5,d=1> Transition enabled in 0.5 time unit
{c,d}
{d,e} {c}
s
tu
Clock c
Clock d
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Generalized semi-Markov processes.
Transition determines:
a. Probability distribution on next states
b. Probability distribution on clock values for new clocks
{c,d}
{d,e} {c}
s
tu
Clock c
Clock d
0.2 0.8
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Generalized semi Markov proceses
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, >
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Equational reasoning
Establishing equality: Coinduction Distinguishing states: Modal logics Equational and logical views coincide Compositional reasoning: ``bisimulation is
a congruence’’
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Labelled Markov Processes
PCTL Bisimulation [Larsen-Skou,
Desharnais-Panangaden-Edalat]
Markov Decision Processes
Bisimulation [Givan-Dean-Grieg]
Labelled Concurrent Markov Chains
PCTL [Hansson-Johnsson]
Labelled Concurrent Markov chains (with tau)
PCTL [Desharnais-Gupta-
Jagadeesan-Panangaden]
Weak bisimulation [Philippou-Lee-Sokolsky,
Lynch-Segala]
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With continuous time
Continuous time Markov chains
CSL [Aziz-Balarin-Brayton-
Sanwal-Singhal-S.Vincentelli]
Bisimulation,Lumpability
[Hillston, Baier-Katoen-Hermanns]
Generalized Semi-Markov processes
Stochastic hybrid systems
CSL
Bisimulation:?????
Composition:?????
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Alas!
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Instability of exact equivalence
Vs
Vs
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Problem!
Numbers viewed as coming with an error estimate.
(eg) Stochastic noise as abstraction Statistical methods for estimating
numbers
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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.
eg. Monte-Carlo methods to approximate probability distributions by a sample.
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Idea: Equivalence metrics
Jou-Smolka, Lincoln-Scedrov-Mitchell-Mitchell
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 --- ``upto uniformities’’.
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Uniformities (same)
m(x,y) = |x-y| m(x,y) = |2x + sinx -2y – siny|
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Uniformities (different)
m(x,y) = |x-y|
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Our results
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Our results
For Discrete time models: Labelled Markov processes Labelled Concurrent Markov chains Markov decision processes
For continuous time: Generalized semi-Markov processes
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Results for discrete time models
Bisimulation Metrics
Logic (P)CTL(*) Real-valued modal logic
Compositionality Congruence Non-expansivity
Proofs Coinduction Coinduction
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Results for continuous time models
Bisimulation Metrics
Logic CSL Real-valued modal logic
Compositionality ??? ???
Proofs Coinduction Coinduction
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Metrics for discrete time probablistic processes
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Bisimulation
Fix a Markov chain. Define monotone F on equivalence relations:
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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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Metrics on probability measures
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Example 1: Metrics on probability measures
Unit measure concentrated at x
Unit measure concentrated at y
x y
m(x,y)
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Example 1: Metrics on probability measures
Unit measure concentrated at x
Unit measure concentrated at y
x y
m(x,y)
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Example 2: Metrics on probability measures
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Example 2: Metrics on probability measures
THEN:
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Lattice of (pseudo)metrics
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Defining metric coinductively
Define functional F on metrics
Desired metric is maximum fixed point of F
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Real-valued modal logic
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Real-valued modal logic
Tests:
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Real-valued modal logic (Boolean)
q
q
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Real-valued modal logic
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Results
Modal-logic yields the same distance
as the coinductive definition However, not upto uniformities since glbs
in lattice of uniformities is not determined by glbs in lattice of pseudometrics.
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Variant definition that works upto uniformities
Fix c<1. Define functional F on metrics
Desired metric is maximum fixed point of F
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Reasoning upto uniformities
For all c<1, get same uniformity
[see Breugel/Mislove/Ouaknine/Worrell]
Variant of earlier real-valued modal logic incorporating discount factor c characterizes the metrics
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Metrics for real-time probabilistic processes
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Generalized semi-Markov processes.
{c,d}
{d,e} {c}
s
tu
Clock c
Clock d
Evolution determined bygeneralized states <state, clock-valuation>
: Set of generalized states
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Generalized semi-Markov processes.
{c,d}
{d,e} {c}
s
tu
Clock c
Clock d
Path:
Traces((s,c)): Probability distribution on a set of paths.
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Accomodating discontinuities: cadlag functions
(M,m) a pseudometric space. cadlag if:
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Countably many jumps, in general
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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???
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Metrics on cadlag functions
Not separable!
are at distance 1 for unequal x,y
x y
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Skorohod metrics (J2)
(M,m) a pseudometric space. f,g cadlag with range M.
Graph(f) = { (t,f(t)) | t \in R+}
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t
fg
(t,f(t))
Skorohod J2 metric: Hausdorff distance between graphs of f,g
f(t)g(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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Examples of non-convergence
Jumps are detected!
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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
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Defining metric coinductively
Define functional on 1-bounded pseudometrics (c <=1)
Desired metric: maximum fixpoint of F
a. s, t agree on all propositions
b.
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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
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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 stateat time t
Problem: Not smooth enough wrt time sincepaths 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 evaluatedat 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
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Results
For each c<=1, modal-logic yields the same distance as the coinductive definition
All c<1 yield the same uniformity. In this case, construction can be carried out in lattice of uniformities.
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Proof steps
Continuity theorems (Whitt) of GSMPs yield separable basis
Finite separability arguments yield closure ordinal of functional F is omega.
Duality theory of LP for calculating metric distances
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Results
Approximating quantitative observables:
Expectations of continuous functions are continuous
Continuous mapping theorems for establishing continuity of quantitative observables
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Summary
Approximate reasoning for real-time probabilistic processes
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Results for discrete time models
Bisimulation Metrics
Logic (P)CTL(*) Real-valued modal logic
Compositionality Congruence Non-expansivity
Proofs Coinduction Coinduction
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Results for continuous time models
Bisimulation Metrics
Logic CSL Real-valued modal logic
Compositionality ??? ???
Proofs Coinduction Coinduction
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Questions?
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Real-valued modal logic