Understanding Process Semantics

Transcription

1 TU München March, 2017 Understanding Process Semantics David de Frutos Escrig In collaboration with: Carlos Gregorio Rodríguez, Miguel Palomino and Ignacio Fábregas Departamento de Sistemas Informáticos y Programación Universidad Complutense de Madrid Munich, March 2017 David de Frutos Escrig Understanding Process Semantics 1 / 32

2 Outline 1 Introduction and Motivation 2 Observational Unification 3 Axiomatic Unification David de Frutos Escrig Understanding Process Semantics 2 / 32

3 Outline 1 Introduction and Motivation 2 Observational Unification 3 Axiomatic Unification David de Frutos Escrig Understanding Process Semantics 3 / 32

4 Introduction Observational Unification Axiomatic Unification Process Theory agents that act and interact continuously with other similar agents and with their common environment. The agents may be real-world objects (even people), or they may be artefacts, embodied perhaps in computer hardware or software systems C.A.R. Hoare David de Frutos Escrig Understanding Process Semantics 4 / 32

5 Introduction Observational Unification Axiomatic Unification Process Theory agents that act and interact continuously with other similar agents and with their common environment. The agents may be real-world objects (even people), or they may be artefacts, embodied perhaps in computer hardware or software systems C.A.R. Hoare David de Frutos Escrig Understanding Process Semantics 4 / 32

6 Introduction Observational Unification Axiomatic Unification Process Theory agents that act and interact continuously with other similar agents and with their common environment. The agents may be real-world objects (even people), or they may be artefacts, embodied perhaps in computer hardware or software systems C.A.R. Hoare David de Frutos Escrig Understanding Process Semantics 4 / 32

7 Introduction Observational Unification Axiomatic Unification Process Theory agents that act and interact continuously with other similar agents and with their common environment. The agents may be real-world objects (even people), or they may be artefacts, embodied perhaps in computer hardware or software systems C.A.R. Hoare David de Frutos Escrig Understanding Process Semantics 4 / 32

8 Process Theory agents that act and interact continuously with other similar agents and with their common environment. The agents may be real-world objects (even people), or they may be artefacts, embodied perhaps in computer hardware or software systems C.A.R. Hoare David de Frutos Escrig Understanding Process Semantics 5 / 32

9 Process Theory Processes Mathematical abstraction of agent behaviour David de Frutos Escrig Understanding Process Semantics 6 / 32

10 Process Theory Processes Dijkstra David de Frutos Escrig Understanding Process Semantics 6 / 32

11 Process Theory Processes Dijkstra Hoare David de Frutos Escrig Understanding Process Semantics 6 / 32

12 Process Theory Processes Dijkstra Hoare Milner David de Frutos Escrig Understanding Process Semantics 6 / 32

13 Process Theory Processes Operational David de Frutos Escrig Understanding Process Semantics 6 / 32

14 Process Theory Processes Observational Operational David de Frutos Escrig Understanding Process Semantics 6 / 32

15 Process Theory Processes Axiomatic Observational Operational David de Frutos Escrig Understanding Process Semantics 6 / 32

16 Process Theory? p = q Processes Axiomatic Observational Operational David de Frutos Escrig Understanding Process Semantics 6 / 32

17 Process Theory? p = q? p q Processes Axiomatic Observational Operational David de Frutos Escrig Understanding Process Semantics 6 / 32

18 Linear Time-Branching Time spectrum bisimulation 2-nested simulation ready simulation possible worlds possible futures complete simulation ready trace failure trace readiness failure impossible futures simulation complete trace trace David de Frutos Escrig Understanding Process Semantics 7 / 32

19 Linear Time-Branching Time spectrum complete simulation bisimulation traces(0) = { } 2-nested simulation traces(ap) = { } { a t t traces(p)} traces(p + q) = traces(p) traces(q) ready simulation p = T q traces(p) possible worlds = traces(q) possible futures failure trace ready trace traces(p) = {σ p p σ p, σ = a 1... a n } failure readiness p = T q traces(p) = traces(q) impossible futures simulation complete trace trace David de Frutos Escrig Understanding Process Semantics 7 / 32

20 Linear Time-Branching Time spectrum a complete simulation bisimulation 2-nested simulation A relation S is a bisimulation if ready simulation a p p then q such that q q and p Sq a a possible worlds q q then p such that p p and q Sp ready trace possible futures We say p = B q if there is a bisimulation S such that psq failure trace readiness failure impossible futures simulation complete trace trace David de Frutos Escrig Understanding Process Semantics 7 / 32

21 Linear Time-Branching Time spectrum bisimulation 2-nested simulation ready simulation possible worlds possible futures complete simulation ready trace failure trace readiness failure impossible futures simulation complete trace trace David de Frutos Escrig Understanding Process Semantics 7 / 32

22 Examples of Equivalence Relations a p 0 = T a p 1 = F a p 2 = RS a p 3 a b F b b b RS b b B b b b c d c d c d c d c d c ab(c + d) a(bc + bd + b(d + c)) a(bc + bd) a(bc + bd) + abc = T Traces = F Failures = RS Ready Similarity = B Strong Bisimulation David de Frutos Escrig Understanding Process Semantics 8 / 32

23 Looking for Unification It would be interesting to get uniform descriptions of the (quite) different semantics for processes This would shed light into the differences and similarities between those semantics We have identifed, classified and compare groups and families of process semantics Based on them we obtain general results with general proofs applicable to (wide) families of semantics David de Frutos Escrig Understanding Process Semantics 9 / 32

24 Looking for Unification It would be interesting to get uniform descriptions of the (quite) different semantics for processes This would shed light into the differences and similarities between those semantics We have identifed, classified and compare groups and families of process semantics Based on them we obtain general results with general proofs applicable to (wide) families of semantics David de Frutos Escrig Understanding Process Semantics 9 / 32

25 Looking for Unification It would be interesting to get uniform descriptions of the (quite) different semantics for processes This would shed light into the differences and similarities between those semantics We have identifed, classified and compare groups and families of process semantics Based on them we obtain general results with general proofs applicable to (wide) families of semantics David de Frutos Escrig Understanding Process Semantics 9 / 32

26 Looking for Unification It would be interesting to get uniform descriptions of the (quite) different semantics for processes This would shed light into the differences and similarities between those semantics We have identifed, classified and compare groups and families of process semantics Based on them we obtain general results with general proofs applicable to (wide) families of semantics David de Frutos Escrig Understanding Process Semantics 9 / 32

27 BCCSP processes Definition The set BCCSP(Act) of processes is defined by: p ::= 0 ap p + q where a Act; 0 represents the process that performs no action; for every action in Act, there is a prefix operator; and + is a choice operator. Definition The operational semantics for BCCSP terms is defined by ap a p a p p q a q p + q a p p + q a q David de Frutos Escrig Understanding Process Semantics 10 / 32

28 BCCSP processes Definition The set BCCSP(Act) of processes is defined by: p ::= 0 ap p + q where a Act; 0 represents the process that performs no action; for every action in Act, there is a prefix operator; and + is a choice operator. Definition The operational semantics for BCCSP terms is defined by ap a p a p p q a q p + q a p p + q a q David de Frutos Escrig Understanding Process Semantics 10 / 32

29 Observational framework Observations Branching observations for simulation semantics Linear observations for extensional semantics (degenerated branching) Deterministic branching observations for rare semantics Concepts Local observation function at states L N, parameter N Branching general observation (bgo) a partial aggregated view of a collection of computations of a process and observations L N at its states David de Frutos Escrig Understanding Process Semantics 11 / 32

30 Observational framework Observations Branching observations for simulation semantics Linear observations for extensional semantics (degenerated branching) Deterministic branching observations for rare semantics Concepts Local observation function at states L N, parameter N Branching general observation (bgo) a partial aggregated view of a collection of computations of a process and observations L N at its states David de Frutos Escrig Understanding Process Semantics 11 / 32

31 Constrained simulations Definition Given a relation N, an N-constrained simulation is a relation S N such that ps N q implies: For every a, if p a p there exists q, q a q and p S N q, pnq. Notation: p NS q. Constraints for the Simulations The universal relation U relating all processes gives rise to the simulation semantics. Relation I, relating processes with the same initial actions, corresponds to ready simulation. Relation T, that relates processes with the same traces, corresponds to trace simulation. David de Frutos Escrig Understanding Process Semantics 12 / 32

32 Constrained simulations Definition Given a relation N, an N-constrained simulation is a relation S N such that ps N q implies: For every a, if p a p there exists q, q a q and p S N q, pnq. Notation: p NS q. Constraints for the Simulations The universal relation U relating all processes gives rise to the simulation semantics. Relation I, relating processes with the same initial actions, corresponds to ready simulation. Relation T, that relates processes with the same traces, corresponds to trace simulation. David de Frutos Escrig Understanding Process Semantics 12 / 32

33 Local observations L N We observe locally at each state of the process the information needed to decide the corresponding constraint: Observations Plain simulation: L U = { }, L U (p) =. Ready simulation: L I = P(Act), L I (p) = I(p). Complete simulation: L C = Bool, L C (p) is true if p 0 and false otherwise. Trace simulation: L T = P(Act ), L T (p) = T (p), the set of traces of p. 2-nested simulation: L S = {[p] S p BCCSP}, L S (p) = [p] S. David de Frutos Escrig Understanding Process Semantics 13 / 32

34 Branching observations Branching general observations The domain BGO N of branching general observations of p corresponding to the constraint N contains the finite trees doubly labelled, at their nodes by local observations in L N, and at their arcs by actions in Act. BGO N (p) The set BGO N (p) can be compositionally defined as BGO N (p) = { L N (p), S S {(a, bgo) bgo BGO N (p ), p a p }} David de Frutos Escrig Understanding Process Semantics 14 / 32

35 Branching observations Branching general observations The domain BGO N of branching general observations of p corresponding to the constraint N contains the finite trees doubly labelled, at their nodes by local observations in L N, and at their arcs by actions in Act. BGO N (p) The set BGO N (p) can be compositionally defined as BGO N (p) = { L N (p), S S {(a, bgo) bgo BGO N (p ), p a p }} David de Frutos Escrig Understanding Process Semantics 14 / 32

36 Branching observations Interpreting bgo s Any bgo BGO N (p) can be interpreted as a partial aggregated view of a collection of computations of p, by observing L N at their states. Example {a} {a} a {b} b {c, d} c d {b} b {c} c {a} a 2 22 a {b} b {c, d} Some bgo s in BGO I (p) for p = a(b(c + d) + bc + bd) d David de Frutos Escrig Understanding Process Semantics 15 / 32

37 Branching observations Interpreting bgo s Any bgo BGO N (p) can be interpreted as a partial aggregated view of a collection of computations of p, by observing L N at their states. Example {a} {a} a {b} b {c, d} c d {b} b {c} c {a} a 2 22 a {b} b {c, d} Some bgo s in BGO I (p) for p = a(b(c + d) + bc + bd) d David de Frutos Escrig Understanding Process Semantics 15 / 32

38 Characterization of Branching Semantics Observational Characterization Definition: p b N q if BGO N(p) BGO N (q). For all N {U, I, C, T, S} and any two processes p and q, p NS q iff p b N q. Why it works We may include, but we do not must!, in a bgo several partial computations that start executing several transitions. We may duplicate a single transition by including different computations starting with the same transition. David de Frutos Escrig Understanding Process Semantics 16 / 32

39 Characterization of Branching Semantics Observational Characterization Definition: p b N q if BGO N(p) BGO N (q). For all N {U, I, C, T, S} and any two processes p and q, p NS q iff p b N q. Why it works We may include, but we do not must!, in a bgo several partial computations that start executing several transitions. We may duplicate a single transition by including different computations starting with the same transition. David de Frutos Escrig Understanding Process Semantics 16 / 32

40 Characterization of Branching Semantics Observational Characterization Definition: p b N q if BGO N(p) BGO N (q). For all N {U, I, C, T, S} and any two processes p and q, p NS q iff p b N q. Why it works We may include, but we do not must!, in a bgo several partial computations that start executing several transitions. We may duplicate a single transition by including different computations starting with the same transition. David de Frutos Escrig Understanding Process Semantics 16 / 32

41 Linear observations for Linear Semantics Linear observations The set LGO N of linear general observations for a local observer L N is the subset of BGO N defined as: l, LGO N for each l L N. l, {(a, lgo)}, whenever a A and lgo LGO N. The set of linear general observations of a process p with respect to the local observer L N is LGO N (p) = BGO N (p) LGO N. Since lgo s are linear, they can be presented as decorated traces: LGO N (p) ::= { L N (p) } { L N (p), a lgo p a p, lgo LGO N (p )} David de Frutos Escrig Understanding Process Semantics 17 / 32

42 Linear observations for Linear Semantics Linear observations The set LGO N of linear general observations for a local observer L N is the subset of BGO N defined as: l, LGO N for each l L N. l, {(a, lgo)}, whenever a A and lgo LGO N. The set of linear general observations of a process p with respect to the local observer L N is LGO N (p) = BGO N (p) LGO N. Since lgo s are linear, they can be presented as decorated traces: LGO N (p) ::= { L N (p) } { L N (p), a lgo p a p, lgo LGO N (p )} David de Frutos Escrig Understanding Process Semantics 17 / 32

43 Linear observations for Linear Semantics Linear observations The set LGO N of linear general observations for a local observer L N is the subset of BGO N defined as: l, LGO N for each l L N. l, {(a, lgo)}, whenever a A and lgo LGO N. The set of linear general observations of a process p with respect to the local observer L N is LGO N (p) = BGO N (p) LGO N. Since lgo s are linear, they can be presented as decorated traces: LGO N (p) ::= { L N (p) } { L N (p), a lgo p a p, lgo LGO N (p )} David de Frutos Escrig Understanding Process Semantics 17 / 32

44 Characterization of Linear Semantics Characterization of the Finest Linear Semantics Definition: p l N q, if LGO N(p) LGO N (q). (1) l U = T ; (2) l I = RT ; (3) l C = CT The linear semantics corresponding to decorated traces observing L N are the finest linear semantics corresponding to each constraint. David de Frutos Escrig Understanding Process Semantics 18 / 32

45 Characterization of Linear Semantics There are other coarser semantics for which a different treatment of the linear observations is needed. Orders for the rest of linear semantics for N = I For T, T LGO I we define the orders l I, lf I, and lf I : T l I T X 0 a 1 X 1... X n T Y 0 a 1 Y 1... Y n T i 0..n X i Y i T lf I T X 0 a 1 X 1... X n T Y 0 a 1 Y 1... Y n T X n = Y n T lf I T X 0 a 1 X 1... X n T Y 0 a 1 Y 1... Y n T X n Y n Then, we write p lx I q if LGO I (p) lx I LGO I (q). David de Frutos Escrig Understanding Process Semantics 19 / 32

46 Characterization of Linear Semantics There are other coarser semantics for which a different treatment of the linear observations is needed. Orders for the rest of linear semantics for N = I For T, T LGO I we define the orders l I, lf I, and lf I : T l I T X 0 a 1 X 1... X n T Y 0 a 1 Y 1... Y n T i 0..n X i Y i T lf I T X 0 a 1 X 1... X n T Y 0 a 1 Y 1... Y n T X n = Y n T lf I T X 0 a 1 X 1... X n T Y 0 a 1 Y 1... Y n T X n Y n Then, we write p lx I q if LGO I (p) lx I LGO I (q). David de Frutos Escrig Understanding Process Semantics 19 / 32

47 Characterization of Linear Semantics Linear Observations for some other Classical Semantics lf I lf I l I generates the readiness preorder; generates the failures preorder; generates the failures trace preorder. David de Frutos Escrig Understanding Process Semantics 20 / 32

48 Characterization of Linear Semantics Three closure operations can be defined for the sets of linear observations Closures characterising the Linear semantics T = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n Y n T i 0..n X i Y i }. T f = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n X n T }. T f = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n Y n T X n Y n }. For X {, f, f } we define LGO X I (p) = LGO I (p) X. Set characterization of the Linear Semantics p lx I q LGO X I (p) LGO X I (q) David de Frutos Escrig Understanding Process Semantics 21 / 32

49 Characterization of Linear Semantics Three closure operations can be defined for the sets of linear observations Closures characterising the Linear semantics T = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n Y n T i 0..n X i Y i }. T f = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n X n T }. T f = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n Y n T X n Y n }. For X {, f, f } we define LGO X I (p) = LGO I (p) X. Set characterization of the Linear Semantics p lx I q LGO X I (p) LGO X I (q) David de Frutos Escrig Understanding Process Semantics 21 / 32

50 Characterization of Linear Semantics Three closure operations can be defined for the sets of linear observations Closures characterising the Linear semantics T = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n Y n T i 0..n X i Y i }. T f = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n X n T }. T f = {X 0 a 1 X 1... a n X n Y 0 a 1 Y 1... a n Y n T X n Y n }. For X {, f, f } we define LGO X I (p) = LGO I (p) X. Set characterization of the Linear Semantics p lx I q LGO X I (p) LGO X I (q) David de Frutos Escrig Understanding Process Semantics 21 / 32

51 Characterization of Linear Semantics Linear Observations for the rest of Linear Semantics The preorders X I can be generalised simply substituting I by an arbitrary constraint, thus getting X N. In this way we characterize two other semantics more recently added to the spectrum: lf T generates the Possible Futures preorder; generates the Impossible Futures preorder; lf T Many other new linear semantics that however can be introduced in a quite natural way. David de Frutos Escrig Understanding Process Semantics 22 / 32

52 Deterministic Branching semantics Deterministic branching observations A bgo is deterministic if for every node in it, its set of children {(a i, bgo i )} satisfies a i a j whenever i j. Deterministic branching observations of a process The set of deterministic branching observations of a process p is dbgo N (p) = BGO N (p) dbgo N. We write p db N q if dbgo N(p) dbgo N (q). The Possible Worlds semantics Deterministic branching observations capture the possible worlds semantics plus many other (a bit strange) ones. p db I q p PW q David de Frutos Escrig Understanding Process Semantics 23 / 32

53 Deterministic Branching semantics Deterministic branching observations A bgo is deterministic if for every node in it, its set of children {(a i, bgo i )} satisfies a i a j whenever i j. Deterministic branching observations of a process The set of deterministic branching observations of a process p is dbgo N (p) = BGO N (p) dbgo N. We write p db N q if dbgo N(p) dbgo N (q). The Possible Worlds semantics Deterministic branching observations capture the possible worlds semantics plus many other (a bit strange) ones. p db I q p PW q David de Frutos Escrig Understanding Process Semantics 23 / 32

54 Deterministic Branching semantics Deterministic branching observations A bgo is deterministic if for every node in it, its set of children {(a i, bgo i )} satisfies a i a j whenever i j. Deterministic branching observations of a process The set of deterministic branching observations of a process p is dbgo N (p) = BGO N (p) dbgo N. We write p db N q if dbgo N(p) dbgo N (q). The Possible Worlds semantics Deterministic branching observations capture the possible worlds semantics plus many other (a bit strange) ones. p db I q p PW q David de Frutos Escrig Understanding Process Semantics 23 / 32

55 The New Linear Time-Branching Time Spectrum B New 2S New New New New New TS New New IF PF FT RS PW RT F R CS New CT S New T David de Frutos Escrig Understanding Process Semantics 24 / 32

56 Axiomatisations for the semantics in the spectrum BCCSP axiomatisations for the equivalences B RS PW RT FT R F CS CT S T (x + y) + z = x + (y + z) x + y = y + x x + 0 = x x + x = x I(x) = I(y) a(x + y) = a(x + y) + ay + v v v v v v v v v a(bx + by + z) = a(bx + z) + a(by + z) + v v v v v v I(x) = I(y) ax + ay = a(x + y) + + v v v v ax + ay = ax + ay + a(x + y) + v v v a(bx + u) + a(by + v) = a(bx + by + u) + a(by + v) + + v v ax + a(y + z) = ax + a(x + y) + a(y + z) + v v a(x + by + z) = a(x + by + z) + a(by + z) + v v v a(bx + u) + a(cy + v) = a(bx + cy + u + v) + v a(x + y) = a(x + y) + ay + v ax + ay = a(x + y) + David de Frutos Escrig Understanding Process Semantics 25 / 32

57 Axiomatisations for the semantics in the spectrum BCCSP axiomatisations for the preorders B RS PW RT FT R F CS CT S T (x + y) + z = x + (y + z) x + y = y + x x + 0 = x x + x = x ax ax + ay v v v v a(bx + by + z) = a(bx + z) + a(by + z) + v v v v v v I(x) = I(y) ax + ay = a(x + y) + v v v v v ax + ay a(x + y) + v v v a(bx + u) + a(by + v) a(bx + by + u) + v v v ax + a(y + z) a(x + y) + v v ax ax + y + + v v a(bx + u) + a(cy + v) = a(bx + cy + u + v) + v x x + y + + ax + ay = a(x + y) + David de Frutos Escrig Understanding Process Semantics 25 / 32

58 Unifying the Axiomatization A uniform axiomatization for process semantics A first axiom capturing the Branching behaviour: (NS) N(x, y) x x + y. Another axiom identifying certain forms of non-deterministic choices: (ND) M(x, y, w) a(x + y) ax + a(y + w). The condition in the second axiom establishes which identifications are needed. Related with the Unification of the Observational Semantics The Branching axiom characterizes the Simulation Semantics. There is a Single axiom for the Linear Semantics. But we need adequate conditions to capture each of them. David de Frutos Escrig Understanding Process Semantics 26 / 32

59 Unifying the Axiomatization A uniform axiomatization for process semantics A first axiom capturing the Branching behaviour: (NS) N(x, y) x x + y. Another axiom identifying certain forms of non-deterministic choices: (ND) M(x, y, w) a(x + y) ax + a(y + w). The condition in the second axiom establishes which identifications are needed. Related with the Unification of the Observational Semantics The Branching axiom characterizes the Simulation Semantics. There is a Single axiom for the Linear Semantics. But we need adequate conditions to capture each of them. David de Frutos Escrig Understanding Process Semantics 26 / 32

60 Unifying the Axiomatization A uniform axiomatization for process semantics A first axiom capturing the Branching behaviour: (NS) N(x, y) x x + y. Another axiom identifying certain forms of non-deterministic choices: (ND) M(x, y, w) a(x + y) ax + a(y + w). The condition in the second axiom establishes which identifications are needed. Related with the Unification of the Observational Semantics The Branching axiom characterizes the Simulation Semantics. There is a Single axiom for the Linear Semantics. But we need adequate conditions to capture each of them. David de Frutos Escrig Understanding Process Semantics 26 / 32

61 Unifying the Axiomatization A uniform axiomatization for process semantics A first axiom capturing the Branching behaviour: (NS) N(x, y) x x + y. Another axiom identifying certain forms of non-deterministic choices: (ND) M(x, y, w) a(x + y) ax + a(y + w). The condition in the second axiom establishes which identifications are needed. Related with the Unification of the Observational Semantics The Branching axiom characterizes the Simulation Semantics. There is a Single axiom for the Linear Semantics. But we need adequate conditions to capture each of them. David de Frutos Escrig Understanding Process Semantics 26 / 32

62 Axiomatization of simulation semantics The Bisimulation Axioms (B 1 ) x + y y + x (B 3 ) x + x x (B 2 ) (x + y) + z x + (y + z) (B 4 ) x + 0 x Behaviour Preorder A preorder relation over processes is a behavior preorder if it is weaker than bisimilarity, i.e. p B q p q, and it is a precongruence with respect to the prefix and choice operators, i.e. if p q then ap aq and p + r q + r. The Simulation Axiom Whenever N is a behavior preorder, N-similarity can be axiomatically defined by means of the conditional axiom (NS) N(x, y) x x + y, together with B 1 B 4. David de Frutos Escrig Understanding Process Semantics 27 / 32

63 Axiomatization of simulation semantics The Bisimulation Axioms (B 1 ) x + y y + x (B 3 ) x + x x (B 2 ) (x + y) + z x + (y + z) (B 4 ) x + 0 x Behaviour Preorder A preorder relation over processes is a behavior preorder if it is weaker than bisimilarity, i.e. p B q p q, and it is a precongruence with respect to the prefix and choice operators, i.e. if p q then ap aq and p + r q + r. The Simulation Axiom Whenever N is a behavior preorder, N-similarity can be axiomatically defined by means of the conditional axiom (NS) N(x, y) x x + y, together with B 1 B 4. David de Frutos Escrig Understanding Process Semantics 27 / 32

64 Axiomatization of simulation semantics The Bisimulation Axioms (B 1 ) x + y y + x (B 3 ) x + x x (B 2 ) (x + y) + z x + (y + z) (B 4 ) x + 0 x Behaviour Preorder A preorder relation over processes is a behavior preorder if it is weaker than bisimilarity, i.e. p B q p q, and it is a precongruence with respect to the prefix and choice operators, i.e. if p q then ap aq and p + r q + r. The Simulation Axiom Whenever N is a behavior preorder, N-similarity can be axiomatically defined by means of the conditional axiom (NS) N(x, y) x x + y, together with B 1 B 4. David de Frutos Escrig Understanding Process Semantics 27 / 32

65 A uniform characterization of linear semantics The diamond of classic semantics coarser than ready simulation: (RS) ready simulation (RT) ready trace (FT) failure trace (R) readiness (F) failure Classic axiomatizations: Failures: (F) a(x + y) ax + a(y + w) Readiness: (R) a(bx + by + u) a(bx + u) + a(by + v) Failure traces: (FT ) a(x + y) ax + ay Ready traces: (RT ) I(x) = I(y) ax + ay a(x + y) David de Frutos Escrig Understanding Process Semantics 28 / 32

66 A uniform characterization of linear semantics The diamond of classic semantics coarser than ready simulation: (RS) ready simulation (RT) ready trace (FT) failure trace (R) readiness (F) failure Classic axiomatizations: Failures: (F) a(x + y) ax + a(y + w) Readiness: (R) a(bx + by + u) a(bx + u) + a(by + v) Failure traces: (FT ) a(x + y) ax + ay Ready traces: (RT ) I(x) = I(y) ax + ay a(x + y) David de Frutos Escrig Understanding Process Semantics 28 / 32

67 Synthetizing the most general conditions for (ND) The original conditions for each preorder M F (x, y, w) ::= ( I(x) I(y) I(x) ) or ( I(w) I(y) I(x) I(y) ) M R (x, y, w) ::= ( I(y) I(x) ) or ( I(x) I(y) = I(y) I(w) ) M FT (x, y, w) ::= ( I(x) I(y) ) ( I(y) I(w) ) M RT (x, y, w) ::= ( I(y) I(x) ) and ( I(x) = I(y) I(w) ) Simplified forms when used together with (RS) (ND F ) M F (x, y, w) ::= true (ND R ) M R (x, y, w) ::= I(x) I(y) (ND FT ) (ND RT ) M FT (x, y, w) ::= I(w) I(y) M RT (x, y, w) ::= ( I(x) = I(y) ) and ( I(w) I(y) ) David de Frutos Escrig Understanding Process Semantics 29 / 32

68 Synthetizing the most general conditions for (ND) The original conditions for each preorder M F (x, y, w) ::= ( I(x) I(y) I(x) ) or ( I(w) I(y) I(x) I(y) ) M R (x, y, w) ::= ( I(y) I(x) ) or ( I(x) I(y) = I(y) I(w) ) M FT (x, y, w) ::= ( I(x) I(y) ) ( I(y) I(w) ) M RT (x, y, w) ::= ( I(y) I(x) ) and ( I(x) = I(y) I(w) ) Simplified forms when used together with (RS) (ND F ) M F (x, y, w) ::= true (ND R ) M R (x, y, w) ::= I(x) I(y) (ND FT ) (ND RT ) M FT (x, y, w) ::= I(w) I(y) M RT (x, y, w) ::= ( I(x) = I(y) ) and ( I(w) I(y) ) David de Frutos Escrig Understanding Process Semantics 29 / 32

69 The coarsest semantics in the spectrum Plain simulation: U(x, y) ::= true (S) U(x, y) ::= true (ND U ) a(x + y) ax + a(y + w) All the semantics in the diamond define the trace semantics. Complete simulation: C(x, y) ::= (x = 0 y = 0) Taking C(x) ::= (x = 0), (C-ND R ) M CR (x, y, w) ::= ( C(x) C(y) ) (C-ND FT ) M CFT (x, y, w) ::= ( C(y) C(w) ) (C-ND RT ) M CRT (x, y, w) ::= ( C(x) C(y) and C(y) C(w) ) Once again, all four semantics in the diamond correspond to complete trace semantics. David de Frutos Escrig Understanding Process Semantics 30 / 32

70 The coarsest semantics in the spectrum Plain simulation: U(x, y) ::= true (S) U(x, y) ::= true (ND U ) a(x + y) ax + a(y + w) All the semantics in the diamond define the trace semantics. Complete simulation: C(x, y) ::= (x = 0 y = 0) Taking C(x) ::= (x = 0), (C-ND R ) M CR (x, y, w) ::= ( C(x) C(y) ) (C-ND FT ) M CFT (x, y, w) ::= ( C(y) C(w) ) (C-ND RT ) M CRT (x, y, w) ::= ( C(x) C(y) and C(y) C(w) ) Once again, all four semantics in the diamond correspond to complete trace semantics. David de Frutos Escrig Understanding Process Semantics 30 / 32

71 The finest semantics in the spectrum Trace simulation: T (x, y) ::= ( T (x) = T (y) ) Impossible futures: (T-ND F ) MF T (x, y, w) ::= true Possible futures: (T-ND R ) MR T (x, y, w) ::= ( T (x) T (y) ) New: impossible traces, possible traces. (T-ND FT ) M T FT (x, y, w) ::= ( T (w) T (y) ) (T-ND RT ) M T RT (x, y, w) ::= ( T (x) = T (y) and T (w) T (y) ) 2-nested simulation: S(x, y) ::= (x S y) Four new linear semantics; for example: (S-ND R ) y S x a(x + y) ax + a(y + z). David de Frutos Escrig Understanding Process Semantics 31 / 32

72 The finest semantics in the spectrum Trace simulation: T (x, y) ::= ( T (x) = T (y) ) Impossible futures: (T-ND F ) MF T (x, y, w) ::= true Possible futures: (T-ND R ) MR T (x, y, w) ::= ( T (x) T (y) ) New: impossible traces, possible traces. (T-ND FT ) M T FT (x, y, w) ::= ( T (w) T (y) ) (T-ND RT ) M T RT (x, y, w) ::= ( T (x) = T (y) and T (w) T (y) ) 2-nested simulation: S(x, y) ::= (x S y) Four new linear semantics; for example: (S-ND R ) y S x a(x + y) ax + a(y + z). David de Frutos Escrig Understanding Process Semantics 31 / 32

73 Some curious archaeological discoveries The real diamond below ready simulation (RS) (RT) (R FT) (FT) (R) (R FT) (F) (R FT) M R FT ::= ( I(x) I(y) ) and ( I(w) I(y) ). (R FT) M R FT ::= ( I(x) I(y) ) or ( I(w) I(y) ). David de Frutos Escrig Understanding Process Semantics 32 / 32