1. Ichiro Hasuo Tracing Anonymity with Coalgebras TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A

2. Ichiro Hasuo Tracing Anonymity with Coalgebras The ultimate aim pervasive, important fail easily … we don’t quite understand them!

3. Ichiro Hasuo Tracing Anonymity with Coalgebras Coalgebras Good balance:Good balance:In this thesis:In this thesis: more applications are found further mathematical theory is developed mathematical simplicity (potential) applicability

4. Ichiro Hasuo Tracing Anonymity with Coalgebras Coalgebras

5. Ichiro Hasuo Tracing Anonymity with Coalgebras Overview Coalgebraic theory of traces and simulations (Ch. 2-3) via coalgebras in a Kleisli category apply to both non-determinism probability case study: probabilistic anonymity (Ch. 4) Concurrency in coalgebras (Ch. 5) the microcosm principle appears

6. Ichiro Hasuo Tracing Anonymity with Coalgebras In Sets : bisimilarity system as coalgebra behavior by final coalgebra what they mean exactly depends on which category they’re in NB they are in the category Sets “behavior” captures bisimilarity standard X, FX, FZ, …sets X  Y function

7. Ichiro Hasuo Tracing Anonymity with Coalgebras Bisimilarity vs. trace semantics When do we decide or ? Bisimilarity Anyway we get  or  Trace semantics a a a b b c c  

8. Ichiro Hasuo Tracing Anonymity with Coalgebras Coalgebraic trace semantics Behavior by final coalgebra captures… bisimilarity bisimilarity (standard) in Sets trace semantics trace semantics (Ch. 2) in Kl(T) Kleisli category “Kleisli category” o a category where branching is implicit branching is implicit o X  Y : “branching function” from X to Y branching-type o T : parameter for branching-type   = Generic Trace Semantics via Coinduction IH, Bart Jacobs & Ana Sokolova Logical Method in Comp. Sci. 3(4:11), 2007

9. Ichiro Hasuo Tracing Anonymity with Coalgebras non-deterministic non-deterministic branching Different “branching-types” in Kl(T) captures trace semantics in Kl(T) captures trace semantics branching-type T : parameter for “branching-type” T = P probabilistic probabilistic branching T = D a b c aa b c a 11 trace semantics: a  b a  c trace semantics: a  b a  c trace semantics: a  b : 1/3 a  c : 2/3 trace semantics: a  b : 1/3 a  c : 2/3

10. Ichiro Hasuo Tracing Anonymity with Coalgebras Coalgebraic simulations (Ch. 3) morphism of coalgebras functional bisimulation (standard) in Sets ?? in Kl(T) lax morphism forward = forward simulation simulation lax morphism forward = forward simulation simulation oplax morphism backward = backward simulation simulation oplax morphism backward = backward simulation simulation 9 fwd/bwd simulation  trace inclusion genericity again genericity again : both for T = P (non-determinism) T = D (probability) genericity again genericity again : both for T = P (non-determinism) T = D (probability) Generic Forward and Backward Simulations IH Proc. CONCUR 2006 LNCS 4137

11. Ichiro Hasuo Tracing Anonymity with Coalgebras Summary so far Ch. 3 Ch. 2 theory of bisimilarity theory of traces and simulations genericity genericity : both for T = P (non-determinism) T = D (probability) genericity genericity : both for T = P (non-determinism) T = D (probability)

12. Ichiro Hasuo Tracing Anonymity with Coalgebras Case study: probabilistic anonymity (Ch. 4) Probabilistic Anonymity via Coalgebraic Simulations IH & Yoshinobu Kawabe Proc. ESOP 2007 LNCS 4421

13. Ichiro Hasuo Tracing Anonymity with Coalgebras Concurrency 2-dimensional, nested algebraic structure2-dimensional, nested algebraic structure “concurrency”, “behavior” final coalgebra category of coalgebras

14. Ichiro Hasuo Tracing Anonymity with Coalgebras Concurrency and the microcosm principle (Ch. 5) concurrency, compositionality, behavior, … formalization of microcosm principle in 2-categories generic compositionality theorem The Microcosm Principle and Concurrency in Coalgebra IH, Bart Jacobs & Ana Sokolova To appear in Proc. FoSSaCS 2008 LNCS

15. Ichiro Hasuo Tracing Anonymity with Coalgebras Summary Coalgebraic theory of traces and simulations (Ch. 2-3) via coalgebras in a Kleisli category apply to both non-determinism probability case study: probabilistic anonymity (Ch. 4) Concurrency in coalgebras (Ch. 5) the microcosm principle appears