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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.
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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 with propositional Information Each state has a set of clocks associated with it. {c,d} s {c} u t {d,e}
Generalized semi-Markov processes. Evolution determined by generalized states <state, clock-valuation> <s,c=2, d=1>Transition enabled when a clockbecomes zero {c,d} s {c} u t {d,e}
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} s {c} u t {d,e} 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 {c} u t c. This need not be exponential. {d,e} 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’’
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) = |2x + sinx -2y – siny| m(x,y) = |x-y|
Uniformities (different) m(x,y) = |x-y|
Our results • A metric on GSMPs based on Wasserstein-Kantorovich and Skorohod • A real-valued modal logic • Everything defined up to uniformity
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.
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]
Generalized semi-Markov processes. Evolution determined by generalized states <state, clock-valuation> : Set of generalized states {c,d} s {c} u t {d,e} 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. Path: Traces((s,c)): Probability distribution on a set of paths. {c,d} s {c} u t {d,e} 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
