Towards Trace Metrics via Functor Lifting

Transcription

1 Towards Trace Metrics via Functor Lifting Paolo Baldan 1,Filippo Bonchi 2, Henning Kerstan 3 and Barbara König 3 1 Università di Padova, Italy 2 CNRS, ENS Lyon, Université de Lyon, France 3 Universität Duisburg-Essen, Germany CALCO 2015 June 26, 2015 This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit

2 My Talk Today Introduction Lifting Functors [FSTTCS 14] Behavioral (Pseudo-)Metrics [FSTTCS 14] Trace Metrics via Functor Lifting Final Remarks 2

3 Introduction

4 Behavioral Equivalence vs Behavioral Metric Behavioral Equivalence Two systems behave exactly the same. (bisimilarity, trace/language equivalence, failure equivalence...) Behavioral (Pseudo-)Metrics Two systems behave similarly, i.e. they have a small distance. (bisimilarity pseudometric, trace/language pseudometric...) Our Central Contributions FSTTCS 14: Behavioral pseudometrics for coalgebras of Set-functors. This talk: Trace pseudometrics (using Eilenberg-Moore). 4

5 Examples of Trace Distances NFA L x, L y : A 2 d 2 A (L x, L y ) = c min{n N 0 w A n.l x (w) L y (w)} where c (0, 1). Probabilistic Moore automata p x, p y : A [0, 1] d [0,1] A (p x, p y ) = c 1 w A where c 1, c 2 (0, 1], c 1 + c 2 1. ( ) w c2 p x (w) p y (w). A 5

6 Lifting Functors [FSTTCS 14]

7 Lifting of a Functor [FSTTCS 14] Definition: Lifting of a Functor (to PMet) F is a lifting of F if the diagram commutes. F F PMet Set U In Plain Terms: Our Task Given: F: Set Set and a pseudometric space (X, d) Define: pseudometric d F on FX 7

8 Pseudometrics on Probability Distributions Example Based on Optimal Transportation [Villani, 2009] X = {A, B, C}, d X : X X [0, 1], µ, ν: X [0, 1] 1 3 B Transportation Problem A, B, C: producers/consumers supply: µ, demand: ν (in %) transporting along d costs d A 1 2 minimize total cost of transport Optimal Solution first use local production (cost 0) 1 6 C transport 0.2 from A to B: = 1 15 transport 0.3 from A to C: = 1 20 d(µ, ν) = =

9 From Transportation Plans to Couplings [FSTTCS 14] µ, ν D(X), t D(X X), t: X }{{} from }{{} X [0, 1] }{{} to percentage supply is used up x X. t(x, y) = µ(x) y X Dπ 1 (t) = µ demand is satisfied y X. t(x, y) = ν(x) x X Dπ 2 (t) = ν Definition: Coupling F: Set Set, t 1, t 2 FX. The set of couplings of t 1 and t 2 is { Γ F (t 1, t 2 ) := t F(X X) Fπ } 1(t) = t 1. Fπ 2 (t) = t 2 9

10 From Costs to Evaluation Functions [FSTTCS 14] costs d (t) = t(x, y) d(x, y) (x,y) X X d: X X [0, 1] costs d : D( X X ) [0, 1] Definition: Evaluation Functor & Evaluation Function X Set/[0, ] F Set/[0, ] f [0, ] FX Ff F[0, ] ev F [0, ]. }{{} evaluation function 10

11 Putting It Together: The Wasserstein Lifting [FSTTCS 14] Definition: Wasserstein Distance For t 1, t 2 FX define } d F (t 1, t 2 ) = inf { Fd(t) t ΓF (t 1, t 2 ). Proposition: Wasserstein Pseudometric d F is a pseudometric if F preserves weak pullbacks ev F satisfies some constraints (well-behaved) 11

12 Behavioral (Pseudo-)Metrics [FSTTCS 14]

13 Final Coalgebra and Behavior The Essential Basics in a Nutshell (i.e. on one Slide) Definition: Final Coalgebra ω: Ω FΩ is final, if for all c: X FX there is a unique coalgebra homomorphism [[ ]] c : X Ω.![[ ]] c X Ω c ω FX FΩ F[[ ]] c Definition: Behavioral Equivalence x, y X : x c y [[x]] c = [y]] c Definition: Behavioral Pseudometric [FSTTCS 14] x, y X : d c (x, y) = d ω ( [[x]]c, [[y]] c ) for d ω on Ω turning ω into a final F-coalgebra 13

14 Computing Behavioral Distances [FSTTCS 14] For all coalgebras c: X FX we want to have: ( ) d c (x, y) = d ω [[x]]c, [[y]] c. (1) Especially for c = ω, where ω is an iso and [[ ]] ω = id Ω. Definition: Behavioral Pseudometric Define d c : X X [0, ] as least fixpoint of: d c (x, y) = d F ( ) c c(x), c(y) Theorem: Behavioral Pseudometric 1. c = ω yields final F-coalgebra. 2. If F preserves isometries, (1) holds for all c. 3. If ω is obtained via the final chain construction, F preserves isometries and metrics, then d ω is a proper metric. 14

15 Trace Metrics via Functor Lifting

16 Coalgebraic Traces in Eilenberg-Moore Our coalgebras are of the shape c: X FTX where T is a monad on Set and F = B _ A. A nondeterministic automaton is a Set-coalgebra c: X 2 (P fin X) A. A probabilistic Moore automaton is a Set-coalgebra c: X [0, 1] (DX) A. Known: To obtain trace semantics via final coalgebra, we need to move to the Eilenberg-Moore category of T. 16

17 Eilenberg-Moore & Liftings Definition: Lifting of a Functor to EM(T) F is a lifting of F if the diagram commutes. F EM(T) U T F C Proposition: Liftings and Distributive Laws F: EM(T) EM(T) = λ: TF FT Given λ and an FT -coalgebra c: X FTX define: c := (TX TFTX FTTX FTX ) Tc λ T X Fµ X This F-coalgebra yields a F-coalgebra µ X Fµ X. 17

18 Traces in Eilenberg-Moore via the generalized powerset construction [Silva et al., 2013, Jacobs et al., 2015] X η X TX [[ ]] c Ω c c ω FTX F[[ ]] c FΩ Definition: Trace Equivalence x, y X : x c y [[η(x)]] c = [η(y)]] c Definition: Trace Pseudometric [new! ] x, y X : d c (x, y) = d ω ([[η(x)]] c, [[η(y)]] c ) for d ω on Ω turning ω into a final F-coalgebra 18

19 The Big Picture L T F, T PMet EM(T) F U T U V L T F, T Set EM(T) F U T 19

20 Our Results... Ensure the Validity of the Aforementioned Approach compositionality F G = FG lifting of natural transformations λ: F G λ: F G lifting of monads (T, η, µ) (T, η, µ) we obtain the trace pseudometrics of our initial examples! 20

21 Final Remarks

22 Summary, Future and Related Work Our Contribution Trace pseudometrics via functor lifting and determinization! Future/Related Work What about coalgebras X TFX in Kl(T)? Our lifting approach is clearly inspired by [van Breugel and Worrell, 2005]. What about a logical characterization of distances as in [Desharnais et al., 1999]? Directed metrics (simulation distances) [de Alfaro et al., 2009] Vary metric on [0, ] [Chatzikokolakis et al., 2014] Algorithms (polynomial-time [Chen et al., 2012], on-the-fly [Bacci et al., 2013],... ) A fibrational perspective (work in progress) 22

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24 Bibliography I Bacci, G., Bacci, G., Larsen, K. G., and Mardare, R. (2013). On-the-fly exact computation of bisimilarity distances. In Proc. of TACAS 13, pages Baldan, P., Bonchi, F., Kerstan, H., and König, B. (2014). Behavioral metrics via functor lifting. In FSTTCS 2014, LIPIcs Series. Schloss Dagstuhl Leibniz-Zentrum für Informatik. Baldan, P., Bonchi, F., Kerstan, H., and König, B. (2015). Towards Trace Metrics via Functor Lifting. In Proc. of CALCO 2015, Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl Leibniz-Zentrum für Informatik. Chatzikokolakis, K., Gebler, D., Palamidessi, C., and Xu, L. (2014). Generalized bisimulation metrics. In Baldan, P. and Gorla, D., editors, CONCUR 2014 Concurrency Theory, volume 8704 of LNCS, pages Springer Berlin Heidelberg. Chen, D., van Breugel, F., and Worrell, J. (2012). On the complexity of computing probabilistic bisimilarity. In Proc. of FoSSaCS 12, volume 7213 of LNCS, pages de Alfaro, L., Faella, M., and Stoelinga, M. (2009). Linear and branching system metrics. IEEE Transactions on Software Engineering, 25(2). Desharnais, J., Gupta, V., Jagadeesan, R., and Panangaden, P. (1999). Metrics for labeled Markov systems. In CONCUR 99 Concurrency Theory, volume 1664 of LNCS, pages Springer Berlin Heidelberg. 24

25 Bibliography II Jacobs, B., Silva, A., and Sokolova, A. (2015). Trace semantics via determinization. Journal of Computer and System Sciences, 81(5): th International Workshop on Coalgebraic Methods in Computer Science, CMCS 2012 (Selected Papers). Silva, A., Bonchi, F., Bonsangue, M. M., and Rutten, J. J. M. M. (2013). Generalizing determinization from automata to coalgebras. LMCS, 9(1). van Breugel, F. and Worrell, J. (2005). A behavioural pseudometric for probabilistic transition systems. TCS, 331: Villani, C. (2009). Optimal Transport Old and New, volume 338 of A Series of Comprehensive Studies in Mathematics. Springer. 25