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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 {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, > The traditional reasoning paradigm 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- Markov chains (with Gupta-Jagadeesan-P] 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: 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 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 1/2

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