A Note on Scope and Infinite Behaviour in CCS-like Calculi p.1/32

Transcription

1 A Note on Scope and Infinite Behaviour in CCS-like Calculi GERARDO SCHNEIDER UPPSALA UNIVERSITY DEPARTMENT OF INFORMATION TECHNOLOGY UPPSALA, SWEDEN Joint work with Pablo Giambiagi and Frank Valencia A Note on Scope and Infinite Behaviour in CCS-like Calculi p.1/32

2 Motivation: Scoping Consider µx.p with P = a (a.b X)\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.2/32

3 Motivation: Scoping Consider µx.p with P = a (a.b X)\a Question: Will action b ever be executed? A Note on Scope and Infinite Behaviour in CCS-like Calculi p.2/32

4 Motivation: Scoping Consider µx.p with P = a (a.b X)\a Question: Will action b ever be executed? Answer: It depends... (!?) = Static vs Dynamic Scoping A Note on Scope and Infinite Behaviour in CCS-like Calculi p.2/32

5 Motivation: Infiniteness Parametric vs. Constant definitions 1. CCS-like calculus, with A def = P 2. CCS-like calculus, with A(x 1,..., x n ) def = P A Note on Scope and Infinite Behaviour in CCS-like Calculi p.3/32

6 Motivation: Infiniteness Parametric vs. Constant definitions 1. CCS-like calculus, with A def = P 2. CCS-like calculus, with A(x 1,..., x n ) def = P Can we encode (2) into (1)? A Note on Scope and Infinite Behaviour in CCS-like Calculi p.3/32

7 Motivation: Infiniteness Parametric vs. Constant definitions 1. CCS-like calculus, with A def = P 2. CCS-like calculus, with A(x 1,..., x n ) def = P Can we encode (2) into (1)? Do we need relabelling? A Note on Scope and Infinite Behaviour in CCS-like Calculi p.3/32

8 Motivation: Infiniteness Parametric vs. Constant definitions 1. CCS-like calculus, with A def = P 2. CCS-like calculus, with A(x 1,..., x n ) def = P Can we encode (2) into (1)? Do we need relabelling? What happens with other forms of introducing infinite behaviour? For instance, Replication A Note on Scope and Infinite Behaviour in CCS-like Calculi p.3/32

9 Motivation and Contributions These are important issues when comparing CCS variants Static vs Dynamic Scoping? Parametric vs. Constant definitions? Recursion vs Replication A Note on Scope and Infinite Behaviour in CCS-like Calculi p.4/32

10 Motivation and Contributions These are important issues when comparing CCS variants Static vs Dynamic Scoping? Parametric vs. Constant definitions? Recursion vs Replication We will show that these issues affect Expressiveness Analysis of certain properties A Note on Scope and Infinite Behaviour in CCS-like Calculi p.4/32

11 Overview of the presentation The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks A Note on Scope and Infinite Behaviour in CCS-like Calculi p.5/32

12 Overview of the presentation The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks A Note on Scope and Infinite Behaviour in CCS-like Calculi p.5/32

13 The Finite Core: Syntax Given: A set of names, N (a, b, x, y...) A set of co-names, N = {a a N } A set of actions, Act = N N {τ} (α, β) A Note on Scope and Infinite Behaviour in CCS-like Calculi p.6/32

14 The Finite Core: Syntax Given: A set of names, N (a, b, x, y...) A set of co-names, N = {a a N } A set of actions, Act = N N {τ} (α, β) Processes specifying finite behaviour: P ::= i I α i.p i P \a P P A Note on Scope and Infinite Behaviour in CCS-like Calculi p.6/32

15 The Finite Core: Semantics SUM i I α i.p i α j P j if j I RES P P \a α P α P \a if α {a, a} PAR P α P 1 P Q α P Q Q α Q PAR 2 P Q α P Q COM P l P Q l Q P Q τ P Q A Note on Scope and Infinite Behaviour in CCS-like Calculi p.7/32

16 Overview of the presentation The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks A Note on Scope and Infinite Behaviour in CCS-like Calculi p.8/32

17 Scoping: Example Consider µx.p with P = a (a.b X)\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.9/32

18 Scoping: Example Consider µx.p with P = a (a.b X)\a Consider the following rule: REC P [µx.p/x] α P µx.p α P (without name α-conversion) A Note on Scope and Infinite Behaviour in CCS-like Calculi p.9/32

19 Scoping: Example Consider µx.p with Then, P [µx.p/x] = a (a.b µx.p )\a P = a (a.b X)\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.9/32

20 Scoping: Example Consider µx.p with P = a (a.b X)\a Then, P [µx.p/x] = a (a.b µx.p )\a = a (a.b µx.(a (a.b X)\a))\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.9/32

21 Scoping: Example Consider µx.p with P = a (a.b X)\a Then, P [µx.p/x] = a (a.b µx.p )\a = a (a.b µx.(a (a.b X)\a))\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.9/32

22 Scoping: Example Consider µx.p with P = a (a.b X)\a Then, P [µx.p/x] = a (a.b µx.p )\a = a (a.b µx.(a (a.b X)\a))\a Then b may be executed! A Note on Scope and Infinite Behaviour in CCS-like Calculi p.9/32

23 Scoping: Example 2 Consider again µx.p with P = a (a.b X)\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.10/32

24 Scoping: Example 2 Consider again µx.p with P = a (a.b X)\a Consider now the following rule: REC P [µx.p/x] α P µx.p α P (applying name α-conversion when necessary) A Note on Scope and Infinite Behaviour in CCS-like Calculi p.10/32

25 Scoping: Example 2 Consider again µx.p with Then, P [µx.p/x] = a (ā.b µx.p )\a P = a (a.b X)\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.10/32

26 Scoping: Example 2 Consider again µx.p with Then, P [µx.p/x] = a (ā.b µx.p )\a P = a (a.b X)\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.10/32

27 Scoping: Example 2 Consider again µx.p with Then, P [µx.p/x] = a ( c.b µx.p )\c P = a (a.b X)\a A Note on Scope and Infinite Behaviour in CCS-like Calculi p.10/32

28 Scoping: Example 2 Consider again µx.p with P = a (a.b X)\a Then, P [µx.p/x] = a ( c.b µx.p )\c = a ( c.b µx.(a (ā.b X)\a))\c A Note on Scope and Infinite Behaviour in CCS-like Calculi p.10/32

29 Scoping: Example 2 Consider again µx.p with P = a (a.b X)\a Then, P [µx.p/x] = a ( c.b µx.p )\c = a ( c.b µx.(a (ā.b X)\a))\c A Note on Scope and Infinite Behaviour in CCS-like Calculi p.10/32

30 Scoping: Example 2 Consider again µx.p with P = a (a.b X)\a Then, P [µx.p/x] = a ( c.b µx.p )\c = a ( c.b µx.(a (ā.b X)\a))\c Then b will never be executed! A Note on Scope and Infinite Behaviour in CCS-like Calculi p.10/32

31 Static vs Dynamic Scoping Name α-conversion to avoid name capture = static scoping Otherwise, = dynamic scoping Dynamic scoping: the occurrence of a name may get dynamically (i.e. during execution) captured under the scope of some restriction A Note on Scope and Infinite Behaviour in CCS-like Calculi p.11/32

32 Overview of the presentation The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks A Note on Scope and Infinite Behaviour in CCS-like Calculi p.12/32

33 Infinite Behaviour There are at least four manners of introducing infinite behaviour A Note on Scope and Infinite Behaviour in CCS-like Calculi p.13/32

34 Infinite Behaviour There are at least four manners of introducing infinite behaviour CCS k : Infinite behavior given by a finite set of constant (i.e., parameterless) definitions of the form A def = P. The calculus is essentially CCS (Milner s book 1989) without relabelling nor infinite summations. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.13/32

35 Infinite Behaviour There are at least four manners of introducing infinite behaviour CCS k : A def = P CCS p : Like CCS k but using parametric definitions of the form A(x 1,..., x n ) def = P. The calculus is the variant in Milner s book on the π-calculus A Note on Scope and Infinite Behaviour in CCS-like Calculi p.13/32

36 Infinite Behaviour There are at least four manners of introducing infinite behaviour CCS k : A def = P CCS p : A(x 1,..., x n ) def = P CCS! : Infinite behavior given by replication of the form!p. This variant is presented, e.g. in a paper by Busi, Gabbrielli and Zavattaro. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.13/32

37 Infinite Behaviour There are at least four manners of introducing infinite behaviour CCS k : A def = P CCS p : A(x 1,..., x n ) def = P CCS! :!P CCS µ : Infinite behavior given by recursive expressions of the form µx.p. However, we adopt static scoping of channel names. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.13/32

38 Infinite Behaviour There are at least four manners of introducing infinite behaviour CCS k : A def = P CCS p : A(x 1,..., x n ) def = P CCS! :!P CCS µ : µx.p A Note on Scope and Infinite Behaviour in CCS-like Calculi p.13/32

39 Parametric Definitions: CCS p Syntax: P ::=... A(y 1,..., y n ) where A(x 1,..., x n ) def = P A, fn(p A ) {x 1,..., x n }. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.14/32

40 Parametric Definitions: CCS p Syntax: P ::=... A(y 1,..., y n ) where A(x 1,..., x n ) def = P A, fn(p A ) {x 1,..., x n }. Semantics: CALL P A[y 1,..., y n /x 1,..., x n ] A(y 1,..., y n ) α P if A(x 1,..., x n ) α P (name α-conversion when necessary) def = P A A Note on Scope and Infinite Behaviour in CCS-like Calculi p.14/32

41 Constant Definitions: CCS k Syntax: P ::=... A where A def = P A A Note on Scope and Infinite Behaviour in CCS-like Calculi p.15/32

42 Constant Definitions: CCS k Syntax: P ::=... A where A def = P A Semantics: CONS P A A α P α P if A def = P A A Note on Scope and Infinite Behaviour in CCS-like Calculi p.15/32

43 Constant Definitions: CCS k Syntax: P ::=... A where A def = P A Semantics (alternative): REC P [µx.p/x] α P µx.p α P (without name α-conversion) A Note on Scope and Infinite Behaviour in CCS-like Calculi p.15/32

44 Recursion Expressions: CCS µ Syntax: P ::=... X µx.p A Note on Scope and Infinite Behaviour in CCS-like Calculi p.16/32

45 Recursion Expressions: CCS µ Syntax: Semantics: P ::=... X µx.p REC P [µx.p/x] α P µx.p α P (name α-conversion when necessary) A Note on Scope and Infinite Behaviour in CCS-like Calculi p.16/32

46 Replication: CCS! Syntax: P ::=...!P A Note on Scope and Infinite Behaviour in CCS-like Calculi p.17/32

47 Replication: CCS! Syntax: Semantics: P ::=...!P REP P!P α P!P α P A Note on Scope and Infinite Behaviour in CCS-like Calculi p.17/32

48 Overview of the presentation The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks A Note on Scope and Infinite Behaviour in CCS-like Calculi p.18/32

49 Expressiveness and Classification Criteria Bisimilarity A Note on Scope and Infinite Behaviour in CCS-like Calculi p.19/32

50 Expressiveness and Classification Criteria Bisimilarity CCS σ is as expressive as CCS σ iff for every P Proc σ, there exists Q Proc σ such that P Q A Note on Scope and Infinite Behaviour in CCS-like Calculi p.19/32

51 Expressiveness and Classification Criteria Bisimilarity CCS σ is as expressive as CCS σ iff for every P Proc σ, there exists Q Proc σ such that P Q Divergence A Note on Scope and Infinite Behaviour in CCS-like Calculi p.19/32

52 Expressiveness and Classification Criteria Bisimilarity CCS σ is as expressive as CCS σ iff for every P Proc σ, there exists Q Proc σ such that P Q Divergence P is divergent iff P ( ) τ ω, i.e., there exists τ τ an infinite sequence P = P 0 P A Note on Scope and Infinite Behaviour in CCS-like Calculi p.19/32

53 Expressiveness and Classification Criteria Bisimilarity CCS σ is as expressive as CCS σ iff for every P Proc σ, there exists Q Proc σ such that P Q Divergence We will study: 1. The relative expressiveness w.r.t. weak bisimilarity 2. The decidability of divergence A Note on Scope and Infinite Behaviour in CCS-like Calculi p.19/32

54 Expressiveness Results Encodings: (weak) bisimulation preserving mappings [ ]: Proc σ Proc σ A Note on Scope and Infinite Behaviour in CCS-like Calculi p.20/32

55 Expressiveness Results Encodings: (weak) bisimulation preserving mappings [ ]: Proc σ Proc σ Encoding CCS p into CCS k Encoding CCS k into CCS p Encoding CCS µ into CCS! Encoding CCS! into CCS µ A Note on Scope and Infinite Behaviour in CCS-like Calculi p.20/32

56 Expressiveness Results Encodings: (weak) bisimulation preserving mappings [ ]: Proc σ Proc σ Encoding CCS p into CCS k Encoding CCS k into CCS p Encoding CCS µ into CCS! Encoding CCS! into CCS µ A Note on Scope and Infinite Behaviour in CCS-like Calculi p.20/32

57 Encoding CCS p into CCS k [ ] : Proc p Proc k A Note on Scope and Infinite Behaviour in CCS-like Calculi p.21/32

58 Encoding CCS p into CCS k [ ] : Proc p Proc k Idea: Assume a definition of the form A(x) def = P A Generate as many constants A y as occurrences of A(y) in P A A Note on Scope and Infinite Behaviour in CCS-like Calculi p.21/32

59 Encoding CCS p into CCS k [ ] : Proc p Proc k Idea: Assume a definition of the form A(x) def = P A Generate as many constants A y as occurrences of A(y) in P A Problem: Potentially infinitely many definitions! A Note on Scope and Infinite Behaviour in CCS-like Calculi p.21/32

60 Encoding CCS p into CCS k [ ] : Proc p Proc k Idea: Assume a definition of the form A(x) def = P A Generate as many constants A y as occurrences of A(y) in P A Problem: Potentially infinitely many definitions! - Due to name α-conversion a possible infinite number of fresh names can be generated A Note on Scope and Infinite Behaviour in CCS-like Calculi p.21/32

61 Encoding CCS p into CCS k : Example Let A(x) def = (z.x.0 x.0 A(z))\z A Note on Scope and Infinite Behaviour in CCS-like Calculi p.22/32

62 Encoding CCS p into CCS k : Example Let A(x) def = (z.x.0 x.0 A(z))\z 1. A x def = (z.x.0 x.0 A z )\z A Note on Scope and Infinite Behaviour in CCS-like Calculi p.22/32

63 Encoding CCS p into CCS k : Example Let A(x) def = (z.x.0 x.0 A(z))\z 1. A x def = (z.x.0 x.0 A z )\z 2. A z def = (z.z.0 z.0 A z )\z A Note on Scope and Infinite Behaviour in CCS-like Calculi p.22/32

64 Encoding CCS p into CCS k : Example Let A(x) def = (z.x.0 x.0 A(z))\z 1. A x def = (z.x.0 x.0 A z )\z 2. A z def = (z.z.0 z.0 A z )\z A Note on Scope and Infinite Behaviour in CCS-like Calculi p.22/32

65 Encoding CCS p into CCS k : Example Let A(x) def = (z.x.0 x.0 A(z))\z 1. A x def = (z.x.0 x.0 A z )\z 2. A z def = (z 1.z.0 z.0 A z1 )\z 1 A Note on Scope and Infinite Behaviour in CCS-like Calculi p.22/32

66 Encoding CCS p into CCS k : Example Let A(x) def = (z.x.0 x.0 A(z))\z 1. A x def = (z.x.0 x.0 A z )\z 2. A z def = (z 1.z.0 z.0 A z1 )\z 1 3. A z1 def = (z.z 1.0 z 1.0 A z )\z A Note on Scope and Infinite Behaviour in CCS-like Calculi p.22/32

67 Encoding CCS p into CCS k : Example Let A(x) def = (z.x.0 x.0 A(z))\z 1. A x def = (z.x.0 x.0 A z )\z 2. A z def = (z 1.z.0 z.0 A z1 )\z 1 3. A z1 def = (z.z 1.0 z 1.0 A z )\z Remark: The generation of fresh names could continue forever! A Note on Scope and Infinite Behaviour in CCS-like Calculi p.22/32

68 Encoding CCS p into CCS k Theorem: For any P CCS p with a finite set of definitions, one can effectively construct the associated set of definitions of [P ]. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.23/32

69 Encoding CCS p into CCS k Theorem: For any P CCS p with a finite set of definitions, one can effectively construct the associated set of definitions of [P ]. Theorem: Given a process P CCS p, P [P ]. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.23/32

70 Encoding CCS p into CCS k Theorem: For any P CCS p with a finite set of definitions, one can effectively construct the associated set of definitions of [P ]. Theorem: Given a process P CCS p, P [P ]. Corollary: Injective relabellings are redundant in CCS. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.23/32

71 Encoding CCS µ into CCS! [ ] : Proc µ Proc! Idea: [X i ] = x i.0 [µx i.p ] = (!x i.[p ] x i.0)\x i A Note on Scope and Infinite Behaviour in CCS-like Calculi p.24/32

72 Encoding CCS µ into CCS! : Example Let be the following CCS µ process: P = µx.(a.x) A Note on Scope and Infinite Behaviour in CCS-like Calculi p.25/32

73 Encoding CCS µ into CCS! : Example Let be the following CCS µ process: P = µx.(a.x) Then the corresponding encoding is: [P ] = (!x.a. x x)\x A Note on Scope and Infinite Behaviour in CCS-like Calculi p.25/32

74 Encoding CCS µ into CCS! : Example Let be the following CCS µ process: P = µx.(a.x) Then the corresponding encoding is: [P ] = (!x.a. x x)\x They are clearly not strongly bisimilar: µx.a.x a µ µx.a.x a µ µx.a.x... (!x.a. x x)\x τ! (!x.a. x a. x)\x a! (!x.a. x x)\x... A Note on Scope and Infinite Behaviour in CCS-like Calculi p.25/32

75 Encoding CCS µ into CCS! Theorem: For P Proc µ, P [P ]. Moreover, P diverges iff [P ] diverges. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.26/32

76 Overview of the presentation The finite core Static vs Dynamic scoping Infinite behaviour Expressiveness Concluding Remarks A Note on Scope and Infinite Behaviour in CCS-like Calculi p.27/32

77 Conclusions CCS p CCS k CCS µ CCS! Divergence: Undecidable Divergence: Decidable A Note on Scope and Infinite Behaviour in CCS-like Calculi p.28/32

78 Conclusions CCS p CCS k CCS µ CCS! Divergence: Undecidable Divergence: Decidable Injective relabellings are redundant in CCS Interpretation of Rule REC leads to important differences CCS exhibits dynamic name scope and it does not preserve α-conversion A Note on Scope and Infinite Behaviour in CCS-like Calculi p.28/32

79 Related Work The CCS variant in Milner s book π-calculus uses parametric definitions with static scope Edinburgh Concurrency Workbench tool (CWB) uses dynamic scoping for parametric definitions ECCS advocates the static scoping of names CHOCS uses dynamic name scoping in the context of higher-order CCS A Note on Scope and Infinite Behaviour in CCS-like Calculi p.29/32

80 Auxiliary Slides A Note on Scope and Infinite Behaviour in CCS-like Calculi p.30/32

81 Bisimilarity A relation S Proc Proc is a (strong) simulation if for all (P, Q) S: P S Q α P A Note on Scope and Infinite Behaviour in CCS-like Calculi p.31/32

82 Bisimilarity A relation S Proc Proc is a (strong) simulation if for all (P, Q) S: P S Q α P S α Q A Note on Scope and Infinite Behaviour in CCS-like Calculi p.31/32

83 Bisimilarity A relation S Proc Proc is a (strong) simulation if for all (P, Q) S: P S Q α P S α Q S is a (strong) bisimulation if both S and its converse are (strong) simulations: P Q. A Note on Scope and Infinite Behaviour in CCS-like Calculi p.31/32

84 Bisimilarity A relation S Proc Proc is a weak simulation if for all (P, Q) S: P S Q s = P S = s Q S is a weak bisimulation if both S and its converse are weak simulations: P Q. - = s (where s = α 1.α 2....) is ( ) τ α 1 ( ) τ... ( ) τ α n ( τ ) A Note on Scope and Infinite Behaviour in CCS-like Calculi p.31/32

85 Encoding CCS p into CCS k [ ] : Proc p Proc k Idea: For each P CCS p, let P CCS k replacing in P each occurrence of B(y) with B y For each definition A(x) def = P A, generate a constant definition A x def = P A A Note on Scope and Infinite Behaviour in CCS-like Calculi p.32/32