Probabilistic systems a place where categories meet probability

Transcription

1 Probabilistic systems a place where categories meet probability Ana Sokolova SOS group, Radboud University Nijmegen University Dortmund, CS Kolloquium, p.1/32

2 Outline Introduction - probabilistic systems and coalgebras University Dortmund, CS Kolloquium, p.2/32

3 Outline Introduction - probabilistic systems and coalgebras Bisimilarity - the strong end of the spectrum University Dortmund, CS Kolloquium, p.2/32

4 Outline Introduction - probabilistic systems and coalgebras Bisimilarity - the strong end of the spectrum Application - expressiveness hierarchy (older result) University Dortmund, CS Kolloquium, p.2/32

5 Outline Introduction - probabilistic systems and coalgebras Bisimilarity - the strong end of the spectrum Application - expressiveness hierarchy (older result) Trace semantics - the weak end of the spectrum (newer result) University Dortmund, CS Kolloquium, p.2/32

6 Systems are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems University Dortmund, CS Kolloquium, p.3/32

7 Systems are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems University Dortmund, CS Kolloquium, p.3/32

8 Systems are formal objects, transition systems (e.g. LTS), that serve as models of real (software, hardware,...) systems a b c c a University Dortmund, CS Kolloquium, p.3/32

9 Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: University Dortmund, CS Kolloquium, p.4/32

10 Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: University Dortmund, CS Kolloquium, p.4/32

11 Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: a[1] a[ 1 3 ] b[ 2 3 ] University Dortmund, CS Kolloquium, p.4/32

12 Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: a University Dortmund, CS Kolloquium, p.4/32

13 Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: 1 a b University Dortmund, CS Kolloquium, p.4/32

14 Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: 1 a 1 b a University Dortmund, CS Kolloquium, p.4/32

15 Probabilistic systems arise by enriching transition systems with (discrete) probabilities as labels on the transitions. Examples: a 1 3 b University Dortmund, CS Kolloquium, p.4/32

16 Coalgebras are an elegant generalization of transition systems with states + transitions University Dortmund, CS Kolloquium, p.5/32

17 Coalgebras are an elegant generalization of transition systems with states + transitions as pairs S, α : S FS, for F a functor University Dortmund, CS Kolloquium, p.5/32

18 Coalgebras are an elegant generalization of transition systems with states + transitions as pairs S, α : S FS, for F a functor based on category theory provide a uniform way of treating transition systems provide general notions and results e.g. a generic notion of bisimulation University Dortmund, CS Kolloquium, p.5/32

19 Examples A TS is a pair S, α : S PS!! coalgebra of the powerset functor P University Dortmund, CS Kolloquium, p.6/32

20 Examples A TS is a pair S, α : S PS!! coalgebra of the powerset functor P An LTS is a pair S, α : S PS A!!! coalgebra of the functor P A Note: P A = P(A ) University Dortmund, CS Kolloquium, p.6/32

21 More examples Thanks to the probability distribution functor D DS = {µ : S [0, 1], µ[s] = 1}, µ[x] = s X µ(x) Df : DS DT, Df(µ)(t) = µ[f 1 ({t})] the probabilistic systems are also coalgebras University Dortmund, CS Kolloquium, p.7/32

22 More examples Thanks to the probability distribution functor D DS = {µ : S [0, 1], µ[s] = 1}, µ[x] = s X µ(x) Df : DS DT, Df(µ)(t) = µ[f 1 ({t})] the probabilistic systems are also coalgebras... of functors built by the following syntax F ::= A P D G + H G H G A G H University Dortmund, CS Kolloquium, p.7/32

23 reactive, generative evolve from LTS - functor P (A ) = P A University Dortmund, CS Kolloquium, p.8/32

24 reactive, generative evolve from LTS - functor P (A ) = P A reactive systems: functor (D + 1) A University Dortmund, CS Kolloquium, p.8/32

25 reactive, generative evolve from LTS - functor P (A ) = P A reactive systems: functor (D + 1) A generative systems: functor (D + 1)(A ) = D(A ) + 1 University Dortmund, CS Kolloquium, p.8/32

26 reactive, generative evolve from LTS - functor P (A ) = P A reactive systems: functor (D + 1) A generative systems: functor (D + 1)(A ) = D(A ) + 1 note: in the probabilistic case (D + 1) A = D(A ) + 1 University Dortmund, CS Kolloquium, p.8/32

27 Probabilistic system types MC D DLTS ( + 1) A LTS P(A ) = P A React (D + 1) A Gen D(A ) + 1 Str D + (A ) + 1 Alt D + P(A ) Var D(A ) + P(A ) SSeg P(A D) Seg PD(A ) University Dortmund, CS Kolloquium, p.9/32

28 Probabilistic system types MC D DLTS ( + 1) A LTS P(A ) = P A React (D + 1) A Gen D(A ) + 1 Str D + (A ) + 1 Alt D + P(A ) Var D(A ) + P(A ) SSeg P(A D) Seg PD(A ) b[1] a[ 1 3 ] a[ 2 3 ] b[1] a[1] University Dortmund, CS Kolloquium, p.9/32

29 Probabilistic system types MC D DLTS ( + 1) A LTS P(A ) = P A React (D + 1) A Gen D(A ) + 1 Str D + (A ) + 1 Alt D + P(A ) Var D(A ) + P(A ) SSeg P(A D) Seg PD(A ) a[ 1 2 ] a[ 1 4 ] c[1] b[ 1 4 ] c[1] University Dortmund, CS Kolloquium, p.9/32

30 Probabilistic system types MC D DLTS ( + 1) A LTS P(A ) = P A React (D + 1) A Gen D(A ) + 1 Str D + (A ) + 1 Alt D + P(A ) Var D(A ) + P(A ) SSeg P(A D) Seg PD(A ) a b University Dortmund, CS Kolloquium, p.9/32

31 Probabilistic system types MC D DLTS ( + 1) A LTS P(A ) = P A React (D + 1) A Gen D(A ) + 1 Str D + (A ) + 1 Alt D + P(A ) Var D(A ) + P(A ) SSeg P(A D) Seg PD(A ) a a b University Dortmund, CS Kolloquium, p.9/32

32 Probabilistic system types MC D DLTS ( + 1) A LTS P(A ) = P A React (D + 1) A Gen D(A ) + 1 Str D + (A ) + 1 Alt D + P(A ) Var D(A ) + P(A ) SSeg P(A D) Seg PD(A ) b[ 3 4 ] a[ 1 3 ] a[ 1 4 ] a[ 2 3 ] University Dortmund, CS Kolloquium, p.9/32

33 Bisimulation - LTS Consider the LTS a s0 t0 a t2 a b b s1 t1 t3 b University Dortmund, CS Kolloquium, p.10/32

34 Bisimulation - LTS Consider the LTS a s0 t0 a t2 a b b s1 t1 t3 b The states s 0 and t 0 are bisimilar since there is a bisimulation R relating them... University Dortmund, CS Kolloquium, p.10/32

35 Bisimulation - LTS Consider the LTS a s0 t0 a t2 a b b s1 t1 t3 Transfer condition: s, t R = s a s ( t ) t a t, s, t R, t a t ( s ) s a s, s, t R b University Dortmund, CS Kolloquium, p.10/32

36 Bisimulation - generative Consider the generative systems a[ 1 2 ] s0 b[ 1 2 ] t0 b[ 1 2 ] a[ 1 2 ] a[ 1 3 ] a[ 1 6 ] t2 s1 t1 t3 b[ 1 2 ] University Dortmund, CS Kolloquium, p.11/32

37 Bisimulation - generative Consider the generative systems a[ 1 2 ] s0 b[ 1 2 ] t0 b[ 1 2 ] a[ 1 2 ] a[ 1 3 ] a[ 1 6 ] t2 s1 t1 t3 b[ 1 2 ] The states s 0 and t 0 are bisimilar, and so are s 0 and t 2, since there is a bisimulation R relating them... University Dortmund, CS Kolloquium, p.11/32

38 Bisimulation - generative Consider the generative systems a[ 1 2 ] s0 b[ 1 2 ] t0 b[ 1 2 ] a[ 1 2 ] a[ 1 3 ] a[ 1 6 ] t2 s1 t1 t3 b[ 1 2 ] Transfer condition: s, t R = s µ ( µ ) t µ, µ R,A µ University Dortmund, CS Kolloquium, p.11/32

39 Bisimulation - simple Segala Consider the simple Segala systems a s0 t a 3 t2 a b b s1 t1 t3 b University Dortmund, CS Kolloquium, p.12/32

40 Bisimulation - simple Segala Consider the simple Segala systems a s0 t a 3 t2 a b b s1 t1 t3 b The states s 0 and t 0 are bisimilar, since there is a bisimulation R relating them... University Dortmund, CS Kolloquium, p.12/32

41 Bisimulation - simple Segala Consider the simple Segala systems a s0 t a 3 t2 a b b s1 t1 t3 b Transfer condition: s, t R = s a µ ( µ ) t a µ, µ R µ University Dortmund, CS Kolloquium, p.12/32

42 Coalgebraic bisimulation A bisimulation between S, α : S FS and T, β : S FS is R S T such that γ: University Dortmund, CS Kolloquium, p.13/32

43 Coalgebraic bisimulation A bisimulation between S, α : S FS and T, β : S FS is R S T such that γ: S π 1 R π 2 T α FS Fπ 1 γ FR Fπ 2 β FT University Dortmund, CS Kolloquium, p.13/32

44 Coalgebraic bisimulation A bisimulation between S, α : S FS and T, β : S FS is R S T such that γ: S π 1 R π 2 T α FS Fπ 1 γ FR Fπ 2 β FT Transfer condition: s, t R = α(s), β(t) Rel(F)(R) University Dortmund, CS Kolloquium, p.13/32

45 Coalgebraic bisimulation A bisimulation between S, α : S FS and T, β : S FS is R S T such that γ: S π 1 R π 2 T α FS Fπ 1 γ FR Fπ 2 β FT Theorem: Coalgebraic and concrete bisimilarity coincide! University Dortmund, CS Kolloquium, p.13/32

46 Expressiveness simple Segala system Segala system a p 1 p 2 p n... University Dortmund, CS Kolloquium, p.14/32

47 Expressiveness simple Segala system Segala system a p 1 p 2 p n... a[p 1 ] a[p n ] a[p 2 ]... University Dortmund, CS Kolloquium, p.14/32

48 Expressiveness simple Segala system Segala system a p 1 p 2 p n... a[p 1 ] a[p n ] a[p 2 ]... When do we consider one type of systems more expressive than another? University Dortmund, CS Kolloquium, p.14/32

49 Comparison criterion Coalg F Coalg G if there is a mapping S, α : S FS T S, α : S GS that preserves and reflects bisimilarity University Dortmund, CS Kolloquium, p.15/32

50 Comparison criterion Coalg F Coalg G if there is a mapping S, α : S FS T S, α : S GS that preserves and reflects bisimilarity s S,α t T,β s T S,α t T T,β University Dortmund, CS Kolloquium, p.15/32

51 Comparison criterion Coalg F Coalg G if there is a mapping S, α : S FS T S, α : S GS that preserves and reflects bisimilarity Theorem: An injective natural transformation F G is sufficient for Coalg F Coalg G University Dortmund, CS Kolloquium, p.15/32

52 Comparison criterion Coalg F Coalg G if there is a mapping S, α : S FS T S, α : S GS that preserves and reflects bisimilarity Theorem: An injective natural transformation F G is sufficient for Coalg F Coalg G proof via cocongruences - behavioral equivalence University Dortmund, CS Kolloquium, p.15/32

53 Example Indeed SSeg Seg since P(A D) Pτ PD(A injective for ) is University Dortmund, CS Kolloquium, p.16/32

54 Example Indeed SSeg Seg since P(A D) Pτ PD(A injective for ) is given by (A D) τ D(A ) University Dortmund, CS Kolloquium, p.16/32

55 Example Indeed SSeg Seg since P(A D) Pτ PD(A injective for ) is (A D) τ D(A ) given by τ X ( a, µ ) = δ a µ, where University Dortmund, CS Kolloquium, p.16/32

56 Example Indeed SSeg Seg since P(A D) Pτ PD(A injective for ) is (A D) τ D(A ) given by τ X ( a, µ ) = δ a µ, where µ µ ( x, x ) = µ(x) µ (x ) University Dortmund, CS Kolloquium, p.16/32

57 Example Indeed SSeg Seg since P(A D) Pτ PD(A injective for ) is (A D) τ D(A ) given by τ X ( a, µ ) = δ a µ, where µ µ ( x, x ) = µ(x) µ (x ) and δ a is Dirac distribution for a University Dortmund, CS Kolloquium, p.16/32

58 The hierarchy... MG PZ Seg Bun SSeg Var Alt React LTS Gen Str DLTS MC University Dortmund, CS Kolloquium, p.17/32

59 The hierarchy... MG PZ Seg Bun SSeg Var Alt React LTS Gen Str DLTS MC * Falk Bartels, AS, Erik de Vink University Dortmund, CS Kolloquium, p.17/32

60 LT/BT spectrum Bisimilarity is not the only semantics... University Dortmund, CS Kolloquium, p.18/32

61 LT/BT spectrum Are these non-deterministic systems equal? x a b c a b y a c University Dortmund, CS Kolloquium, p.18/32

62 LT/BT spectrum Are these non-deterministic systems equal? x a b c a b y a x and y are: different wrt. bisimilarity c University Dortmund, CS Kolloquium, p.18/32

63 LT/BT spectrum Are these non-deterministic systems equal? x a b c a b y a x and y are: different wrt. bisimilarity, but equivalent wrt. trace semantics tr(x) = tr(y) = {ab, ac} c University Dortmund, CS Kolloquium, p.18/32

64 Traces - LTS For LTS with explicit termination (NA) the set of all possible trace = linear behaviors University Dortmund, CS Kolloquium, p.19/32

65 Traces - LTS For LTS with explicit termination (NA) the set of all possible trace = linear behaviors Example: a a y x b tr(y) = b, tr(x) = a + tr(y) = a + b University Dortmund, CS Kolloquium, p.19/32

66 Traces - generative For generative probabilistic systems with ex. termination sub-probability distribution over trace = possible linear behaviors University Dortmund, CS Kolloquium, p.20/32

67 Traces - generative For generative probabilistic systems with ex. termination sub-probability distribution over trace = possible linear behaviors Example: x b[ 1 3 ] z a[1] a[ 1 3 ] 1 3 a[ 1 2 ] y 1 2 tr(x) : 1 3 a a University Dortmund, CS Kolloquium, p.20/32

68 Trace of a coalgebra? University Dortmund, CS Kolloquium, p.21/32

69 Trace of a coalgebra? Power&Turi 99 Jacobs 04 Hasuo& Jacobs 05 Ichiro Hasuo, Bart Jacobs, AS: Generic Trace Theory, CMCS 06 University Dortmund, CS Kolloquium, p.21/32

70 Trace of a coalgebra? Power&Turi 99 Jacobs 04 Hasuo& Jacobs 05 Ichiro Hasuo, Bart Jacobs, AS: Generic Trace Theory, CMCS 06 main idea: coinduction in a Kleisli category University Dortmund, CS Kolloquium, p.21/32

71 Coinduction FX F(beh) FZ α = X system beh Z final coalgebra University Dortmund, CS Kolloquium, p.22/32

72 Coinduction FX F(beh) FZ α = X system beh Z final coalgebra finality =!(morphism for any F- coalgebra) beh gives the behavior of the system this yields final coalgebra semantics University Dortmund, CS Kolloquium, p.22/32

73 Coinduction FX F(beh) FZ α = X system beh Z final coalgebra f.c.s. in Sets = bisimilarity f.c.s. in a Kleisli category = trace semantics University Dortmund, CS Kolloquium, p.22/32

74 Types of systems For trace semantics systems are suitably modelled as coalgebras in Sets X c T F X University Dortmund, CS Kolloquium, p.23/32

75 Types of systems For trace semantics systems are suitably modelled as coalgebras in Sets X c T F X monad - branching type University Dortmund, CS Kolloquium, p.23/32

76 Types of systems For trace semantics systems are suitably modelled as coalgebras in Sets X c T F X monad - branching type functor - linear i/o type University Dortmund, CS Kolloquium, p.23/32

77 Types of systems For trace semantics systems are suitably modelled as coalgebras in Sets X c T F X monad - branching type functor - linear i/o type needed: distributive law FT T F University Dortmund, CS Kolloquium, p.23/32

78 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) University Dortmund, CS Kolloquium, p.24/32

79 Distributive law is needed since branching is irrelevant: a LTS with - PF = P(1 + A ) x a a b b University Dortmund, CS Kolloquium, p.24/32

80 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab University Dortmund, CS Kolloquium, p.24/32

81 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab X c PFX University Dortmund, CS Kolloquium, p.24/32

82 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab X c PFX University Dortmund, CS Kolloquium, p.24/32

83 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab X c PFX PFc PFPFX University Dortmund, CS Kolloquium, p.24/32

84 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab X c PFX PFc PFPFX University Dortmund, CS Kolloquium, p.24/32

85 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab X c PFX PFc PFPFX d.l. PPFFX University Dortmund, CS Kolloquium, p.24/32

86 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab X c PFX PFc PFPFX d.l. PPFFX University Dortmund, CS Kolloquium, p.24/32

87 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab X c PFX PFc PFPFX d.l. PPFFX m.m. PFFX University Dortmund, CS Kolloquium, p.24/32

88 Distributive law is needed since branching is irrelevant: LTS with - PF = P(1 + A ) a x a a b b aa x ab X c PFX PFc PFPFX d.l. PPFFX m.m. PFFX University Dortmund, CS Kolloquium, p.24/32

89 Distributive law is needed for X c T FX to be a coalgebra in the Kleisli category Kl(T ).. University Dortmund, CS Kolloquium, p.25/32

90 Distributive law is needed for X c T FX to be a coalgebra in the Kleisli category Kl(T ).. objects - sets arrows - X f Y are functions f : X T Y University Dortmund, CS Kolloquium, p.25/32

91 Distributive law is needed for X c T FX to be a coalgebra in the Kleisli category Kl(T ).. FT T F : F lifts to F Kl(T ) on Kl(T ). University Dortmund, CS Kolloquium, p.25/32

92 Distributive law is needed for X c T FX to be a coalgebra in the Kleisli category Kl(T ).. FT T F : F lifts to F Kl(T ) on Kl(T ). Hence: coalgebra X c F Kl(T ) X in Kl(T )!!! University Dortmund, CS Kolloquium, p.25/32

93 Distributive law is needed for X c T FX to be a coalgebra in the Kleisli category Kl(T ).. FT T F : F lifts to F Kl(T ) on Kl(T ). Hence: coalgebra X c F Kl(T ) X in Kl(T )!!! in Kl(T ) : X c F Kl(T ) X University Dortmund, CS Kolloquium, p.25/32

94 Distributive law is needed for X c T FX to be a coalgebra in the Kleisli category Kl(T ).. FT T F : F lifts to F Kl(T ) on Kl(T ). Hence: coalgebra X c F Kl(T ) X in Kl(T )!!! in Kl(T ) : X c F Kl(T ) X University Dortmund, CS Kolloquium, p.25/32

95 Distributive law is needed for X c T FX to be a coalgebra in the Kleisli category Kl(T ).. FT T F : F lifts to F Kl(T ) on Kl(T ). Hence: coalgebra X c F Kl(T ) X in Kl(T )!!! in Kl(T ) : X c F Kl(T ) X F Kl(T )c F Kl(T ) F Kl(T ) X University Dortmund, CS Kolloquium, p.25/32

96 Distributive law is needed for X c T FX to be a coalgebra in the Kleisli category Kl(T ).. FT T F : F lifts to F Kl(T ) on Kl(T ). Hence: coalgebra X c F Kl(T ) X in Kl(T )!!! in Kl(T ) : X c F Kl(T ) X F Kl(T )c F Kl(T ) F Kl(T ) X University Dortmund, CS Kolloquium, p.25/32

97 Main theorem - traces If, then F Kl(T ) I F Kl(T ) I η I α = η FI α 1 = I I is initial is final in Kl(T ) University Dortmund, CS Kolloquium, p.26/32

98 Main theorem - traces If, then F Kl(T ) I F Kl(T ) I η I α = η FI α 1 = I I is initial is final in Kl(T ) [α : FI = I denotes the initial F-algebra in Sets] University Dortmund, CS Kolloquium, p.26/32

99 Main theorem - traces If, then F Kl(T ) I F Kl(T ) I η I α = η FI α 1 = I I is initial is final in Kl(T ) [α : FI = I denotes the initial F-algebra in Sets] proof: via limit-colimit coincidence Smyth&Plotkin 82 University Dortmund, CS Kolloquium, p.26/32

100 The assumptions : A monad T s.t. Kl(T ) is DCpo -enriched left-strict composition University Dortmund, CS Kolloquium, p.27/32

101 The assumptions : A monad T s.t. Kl(T ) is DCpo -enriched left-strict composition A functor F that preserves ω-colimits University Dortmund, CS Kolloquium, p.27/32

102 The assumptions : A monad T s.t. Kl(T ) is DCpo -enriched left-strict composition A functor F that preserves ω-colimits A distributive law FT T F: lifting F Kl(T ) University Dortmund, CS Kolloquium, p.27/32

103 The assumptions : A monad T s.t. Kl(T ) is DCpo -enriched left-strict composition A functor F that preserves ω-colimits A distributive law FT T F: lifting F Kl(T ) F Kl(T ) should be locally monotone University Dortmund, CS Kolloquium, p.27/32

104 Corollary ( ) For X c F Kl(T ) X in Kl(T ) University Dortmund, CS Kolloquium, p.28/32

105 Corollary ( ) For X c F Kl(T ) X in Kl(T )... X c T FX in Sets University Dortmund, CS Kolloquium, p.28/32

106 Corollary ( ) For X c F Kl(T ) X in Kl(T )... X c T FX in Sets! finite trace map tr c : X T I in Sets: University Dortmund, CS Kolloquium, p.28/32

107 Corollary ( ) For X c F Kl(T ) X in Kl(T )... X c T FX in Sets! finite trace map tr c : X T I in Sets: in Kl(T ) F Kl(T ) X F Kl(T )(tr c ) c F Kl(T ) I = X tr c I University Dortmund, CS Kolloquium, p.28/32

108 It works for... lift, powerset, sub-distribution monad University Dortmund, CS Kolloquium, p.29/32

109 It works for... lift, powerset, sub-distribution monad shapely functors - almost all polynomial University Dortmund, CS Kolloquium, p.29/32

110 It works for... lift, powerset, sub-distribution monad shapely functors - almost all polynomial Hence: University Dortmund, CS Kolloquium, p.29/32

111 It works for... lift, powerset, sub-distribution monad shapely functors - almost all polynomial Hence: * for LTS with explicit termination P(1 + A ) University Dortmund, CS Kolloquium, p.29/32

112 It works for... lift, powerset, sub-distribution monad shapely functors - almost all polynomial Hence: * for LTS with explicit termination P(1 + A ) * for generative systems with explicit termination D(1 + A ) University Dortmund, CS Kolloquium, p.29/32

113 It works for... lift, powerset, sub-distribution monad shapely functors - almost all polynomial Hence: * for LTS with explicit termination P(1 + A ) * for generative systems with explicit termination D(1 + A ) Note: Initial 1 + A - algebra is A [nil,cons] = 1 + A A University Dortmund, CS Kolloquium, p.29/32

114 Finite traces - LTS with the finality diagram in Kl(P) F Kl(P) X F Kl(P)(tr c ) c F Kl(P) A = X tr c A University Dortmund, CS Kolloquium, p.30/32

115 Finite traces - LTS with the finality diagram in Kl(P) 1 + A X (1+A ) Kl(P)(tr c ) c 1 + A A = X tr c A University Dortmund, CS Kolloquium, p.30/32

116 Finite traces - LTS with the finality diagram in Kl(P) 1 + A X (1+A ) Kl(P)(tr c ) c 1 + A A = X amounts to tr c A tr c (x) c(x) a w tr c (x) ( x ) a, x c(x), w tr c (x ) University Dortmund, CS Kolloquium, p.30/32

117 Finite traces - generative the finality diagram in Kl(D) F Kl(D) X F Kl(D)(tr c ) F Kl(D) A c = X tr c A University Dortmund, CS Kolloquium, p.31/32

118 Finite traces - generative the finality diagram in Kl(D) 1 + A X (1+A ) Kl(D)(tr c ) c 1 + A A = X tr c A University Dortmund, CS Kolloquium, p.31/32

119 Finite traces - generative the finality diagram in Kl(D) 1 + A X (1+A ) Kl(D)(tr c ) c 1 + A A = X amounts to tr c (x) : tr c A c(x)( ) a w y X c(x)(a, y) c(y)(w) University Dortmund, CS Kolloquium, p.31/32

120 Conclusions & future work Coalgebras allow for a unified treatment and expressiveness study of (P)TS University Dortmund, CS Kolloquium, p.32/32

121 Conclusions & future work Coalgebras allow for a unified treatment and expressiveness study of (P)TS Coinduction gives us semantic relations: University Dortmund, CS Kolloquium, p.32/32

122 Conclusions & future work Coalgebras allow for a unified treatment and expressiveness study of (P)TS Coinduction gives us semantic relations: * bisimilarity for F-systems in Sets University Dortmund, CS Kolloquium, p.32/32

123 Conclusions & future work Coalgebras allow for a unified treatment and expressiveness study of (P)TS Coinduction gives us semantic relations: * bisimilarity for F-systems in Sets * trace semantics for T F-systems in Kl(T ) University Dortmund, CS Kolloquium, p.32/32

124 Conclusions & future work Coalgebras allow for a unified treatment and expressiveness study of (P)TS Coinduction gives us semantic relations: * bisimilarity for F-systems in Sets * trace semantics for T F-systems in Kl(T ) Other monads e.g. PD?... suitable monad/order structure yet to be found (Varacca&Winskel) University Dortmund, CS Kolloquium, p.32/32

125 Conclusions & future work Coalgebras allow for a unified treatment and expressiveness study of (P)TS Coinduction gives us semantic relations: * bisimilarity for F-systems in Sets * trace semantics for T F-systems in Kl(T ) Other monads e.g. PD?... suitable monad/order structure yet to be found (Varacca&Winskel) Other semantics?.. between bisimilarity and trace in the spectrum University Dortmund, CS Kolloquium, p.32/32

126 Conclusions & future work Coalgebras allow for a unified treatment and expressiveness study of (P)TS Coinduction gives us semantic relations: * bisimilarity for F-systems in Sets * trace semantics for T F-systems in Kl(T ) Other monads e.g. PD?... suitable monad/order structure yet to be found (Varacca&Winskel) Other semantics?.. between bisimilarity and trace in the spectrum University Dortmund, CS Kolloquium, p.32/32