Approximate reasoning for probabilistic real-time processes
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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 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.
c. This need not be exponential.
{c,d}
{d,e} {c}
s
tu
Clock c
Clock d
0.2 0.8
Transition determines:
a. Probability distribution on next states
b. Probability distribution on clock values for new clocks
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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 <s, > converge to the stochastic process at <s, >
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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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Labelled Markov Processes
PCTL Bisimulation [Larsen-Skou,
Desharnais-Edalat-P]
Markov Decision Processes
Bisimulation [Givan-Dean-Grieg]
Labelled Concurrent Markov Chains
PCTL [Hansson-Johnsson]
Labelled Concurrent Markov chains (with tau)
PCTLCompleteness: [Desharnais-
Gupta-Jagadeesan-P]
Weak bisimulation [Philippou-Lee-Sokolsky,
Lynch-Segala]
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With continuous timeContinuous time Markov chains
CSL [Aziz-Balarin-Brayton-
Sanwal-Singhal-S.Vincentelli]
Bisimulation,Lumpability
[Hillston, Baier-Katoen-
Hermanns,Desharnais-P]
Generalized Semi-Markov processes
Stochastic hybrid systems
CSL
Bisimulation:?????
Composition:?????
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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) = |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
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
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 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 c<1. Define functional F on metrics
Desired metric is maximum fixed point of F
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Reasoning up to uniformities
For all c<1 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.
{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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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.
{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, finitely many jumpshigher than any fixed “h”.
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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’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.
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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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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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Non-convergence in J2:
Sequences of continuous functions cannot converge toa 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 <1)
Desired metric: maximum fixpoint of F
a. s, t agree on all propositions
b.
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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.
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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 observables:
Expectations 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 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