Ichiro Hasuo Tracing Anonymity with Coalgebras Tracing Anonymity

Ichiro Hasuo Tracing Anonymity with Coalgebras

Tracing Anonymity with Coalgebras Ichiro Hasuo The ultimate aim Better mathematical understanding of computer systems Computer systems • pervasive, important • fail easily • … • we don’t quite understand them!

Tracing Anonymity with Coalgebras Ichiro Hasuo Coalgebras Our mathematical presentation of systems Good balance: mathematical simplicity (potential) applicability In this thesis: • more applications are found • further mathematical theory is developed

Tracing Anonymity with Coalgebras Ichiro Hasuo Coalgebras coalgebraically system behaviorpreserving map behavior coalgebra morphism of coalgebras by final coalgebra

Tracing Anonymity with Coalgebras Ichiro Hasuo 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

Tracing Anonymity with Coalgebras Ichiro Hasuo In Sets: bisimilarity system as coalgebra category = “universe ” Sets, Top, Stone, Vect, CLat, … behavior by final coalgebra NB • what they mean exactly depends on which category they’re in X, FZ, … sets function standard • they are in the category Sets • “behavior” captures bisimilarity

Tracing Anonymity with Coalgebras Ichiro Hasuo Bisimilarity vs. trace semantics a b Bisimilarity c Also captured by = final coalgebra? a When do we decide or ? Trace semantics a c b Anyway we get or

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

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

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

Tracing Anonymity with Coalgebras Ichiro Hasuo genericity : both for • T = P (non-determinism) • T = D (probability) Summary so far coalgebra morphism of coalgebra in Sets in Kl(T) system functional bisimilarity forward simulation (lax) backward similation (oplax) Ch. 3 Ch. 2 by final coalgebra bisimilarity trace semantics theory of bisimilarity theory of traces and simulations

Tracing Anonymity with Coalgebras Ichiro Hasuo Probabilistic Anonymity via Coalgebraic Simulations IH & Yoshinobu Kawabe Proc. ESOP 2007 LNCS 4421 Case study: probabilistic anonymity (Ch. 4) Simulation-based proof method for non-deterministic anonymity [Kawabe. MST 06] generic, coalgebraic theory of traces and simulations T=P [Ch. 2 -3] T=D Simulation-based proof method for probabilistic anonymity

Tracing Anonymity with Coalgebras Ichiro Hasuo Concurrency “concurrency ” , “behavior” Ck. D running C and D in parallel 2 -dimensional , nested category of algebraic structure coalgebras inner k final coalgebr a outer k the microcosm principle

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

Tracing Anonymity with Coalgebras Ichiro Hasuo 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