Structure Preserving Bisimilarity,
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1 Structure Preserving Bisimilarity, Supporting an Operational Petri Net Semantics of CCSP Rob van Glabbeek NICTA, Sydney, Australia University of New South Wales, Sydney, Australia September 2015
2 Milner: CCS [[ ]] op LTS
3 Milner: CCS [[ ]] op LTS Hoare: CSP [[ ]] den failures model
4 Milner: CCS [[ ]] op LTS Hoare: [[ ]] op LTS CSP [[ ]] den failures model [Olderog & Hoare 86]
5 CCS [[ ]] op LTS [[ ]] op LTS CSP
6 Nielsen: Olderog: CCSP [[ ]] op LTS
7 Goltz & Mycroft Winskel van Glabbeek & Vaandrager: Petri nets [[ ]] den CCSP [[ ]] op LTS but no treatment of recursion.
8 Degano, De Nicola and Montanari: [[ ]] op [[ ]] den Petri nets CCSP [[ ]] op LTS including a treatment of recursion.
9 Degano, De Nicola and Montanari: [[ ]] op [[ ]] den Petri nets CCSP [[ ]] op LTS including a treatment of recursion. But initial concurrency is not respected.
10 c [[(ad b)+c]] DDM op = a b d b a d
11 Olderog, 1987: [[ ]] op [[ ]] den Petri nets CCSP [[ ]] op LTS including a treatment of recursion. Concurrency is fully respected.
12 a c b [[(ad b)+c]] Old op = d
13 a c b [[(ad b)+c]] Old op = d This is the same result as for the denotational Petri net found in the literature.
14 How to formalise the statement that Olderog s operational respects concurrency?
15 How to formalise the statement that Olderog s operational respects concurrency? We use the well-known fact that the denotational from the literature respects concurrency, at least for those expressions P for which it is defined (the recursion-free ones).
16 How to formalise the statement that Olderog s operational respects concurrency? We use the well-known fact that the denotational from the literature respects concurrency, at least for those expressions P for which it is defined (the recursion-free ones). So we want to prove: [[P]] op = [[P]] den.
17 How to formalise the statement that Olderog s operational respects concurrency? We use the well-known fact that the denotational from the literature respects concurrency, at least for those expressions P for which it is defined (the recursion-free ones). So we want to prove: [[P]] op = [[P]] den. But this fails in the same way as the operational versus denotational of CCSP in terms of labelled transition systems:
18 How to formalise the statement that Olderog s operational respects concurrency? We use the well-known fact that the denotational from the literature respects concurrency, at least for those expressions P for which it is defined (the recursion-free ones). So we want to prove: [[P]] op = [[P]] den. But this fails in the same way as the operational versus denotational of CCSP in terms of labelled transition systems: [[a+a]] op [[a+a]] den.
19 How to formalise the statement that Olderog s operational respects concurrency? We use the well-known fact that the denotational from the literature respects concurrency, at least for those expressions P for which it is defined (the recursion-free ones). So we want to prove: [[P]] op = [[P]] den. But this fails in the same way as the operational versus denotational of CCSP in terms of labelled transition systems: [[a+a]] op [[a+a]] den. So instead one shows [[P]] op [[P]] den for a suitable relation.
20 Showing agreement between an operational and denotational net of CCSP Aim: propose a relation between nets, and show [[P]] op [[P]] den.
21 Showing agreement between an operational and denotational net of CCSP Aim: propose a relation between nets, and show [[P]] op [[P]] den. Which requirements to impose on?
22 Showing agreement between an operational and denotational net of CCSP Aim: propose a relation between nets, and show [[P]] op [[P]] den. Which requirements to impose on? 1. [[P]] op [[P]] den for any P for which both sides are defined.
23 Showing agreement between an operational and denotational net of CCSP Aim: propose a relation between nets, and show [[P]] op [[P]] den. Which requirements to impose on? 1. [[P]] op [[P]] den for any P for which both sides are defined. 2. respects concurrency.
24 Showing agreement between an operational and denotational net of CCSP Aim: propose a relation between nets, and show [[P]] op [[P]] den. Which requirements to impose on? 1. [[P]] op [[P]] den for any P for which both sides are defined. 2. respects concurrency. 3. is a congruence relation for CCSP. R = P Q PN
25 Showing agreement between an operational and denotational net of CCSP Olderog proposes a relation between nets, called strong bisimulation.
26 Showing agreement between an operational and denotational net of CCSP Olderog proposes a relation between nets, called strong bisimulation. 1. [[P]] op [[P]] den for any P for which both sides are defined.
27 Showing agreement between an operational and denotational net of CCSP Olderog proposes a relation between nets, called strong bisimulation. 1. [[P]] op [[P]] den for any P for which both sides are defined. 2. respects concurrency.
28 Showing agreement between an operational and denotational net of CCSP Olderog proposes a relation between nets, called strong bisimulation. 1. [[P]] op [[P]] den for any P for which both sides are defined. 2. respects concurrency. 3. has the typical congruence properties.
29 Showing agreement between an operational and denotational net of CCSP Olderog proposes a relation between nets, called strong bisimulation. 1. [[P]] op [[P]] den for any P for which both sides are defined. 2. respects concurrency. 3. has the typical congruence properties. But it fails to be transitive!
30 Showing agreement between an operational and denotational net of CCSP Olderog proposes a relation between nets, called strong bisimulation. 1. [[P]] op [[P]] den for any P for which both sides are defined. 2. respects concurrency. 3. has the typical congruence properties. But it fails to be transitive! This is not a problem, because can be seen as just a tool to proof things about causal equivalence caus. N 1 caus N 2 if they have the same processes or causal nets.
31 Showing agreement between an operational and denotational net of CCSP Olderog proposes a relation between nets, called strong bisimulation. 1. [[P]] op [[P]] den for any P for which both sides are defined. 2. respects concurrency. 3. has the typical congruence properties. But it fails to be transitive! This is not a problem, because can be seen as just a tool to proof things about causal equivalence caus. N 1 caus N 2 if they have the same processes or causal nets. N 1 N 2 N 1 caus N 2.
32 Showing agreement between an operational and denotational net of CCSP Olderog proposes a relation between nets, called strong bisimulation. 1. [[P]] op [[P]] den for any P for which both sides are defined. 2. respects concurrency. 3. has the typical congruence properties. But it fails to be transitive! This is not a problem, because can be seen as just a tool to proof things about causal equivalence caus. N 1 caus N 2 if they have the same processes or causal nets. caus respects concurrency. N 1 N 2 N 1 caus N 2.
33 Showing agreement between an operational and denotational net of CCSP caus is a linear time equivalence: [[a(b +c)]] op caus [[ab +ac]]. This is less good for capturing phenomena like deadlock behaviour.
34 Showing agreement between an operational and denotational net of CCSP caus is a linear time equivalence: [[a(b +c)]] op caus [[ab +ac]]. This is less good for capturing phenomena like deadlock behaviour. My contribution today is the proposal of a new branching time equivalence that can play the rôle of caus. I call it structure preserving bisimilarity sp.
35 Structure Preserving Bisimulation N 1 N 2
36 Structure Preserving Bisimulation N 1 N 2 a
37 Structure Preserving Bisimulation N 1 N 2 a a
38 Structure Preserving Bisimulation N 1 N 2 a a
39 Structure Preserving Bisimulation N 1 N 2 a a
40 Criteria for choosing this equivalence 2. It should capture concurrency. a b ab + ba causality 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
41 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
42 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
43 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
44 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
45 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 6. It should be finer than caus. 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
46 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 6. It should be finer than caus. 7. It should not distinguish nets whose behaviours are patently the same, i.e. when differing only in unreachable parts. 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
47 Semantic equivalences on Petri nets branching time linear time abstract from causality/concurrency capture causality/concurrency
48 Semantic equivalences on Petri nets branching time bisimulation ib linear time trace it abstract from causality/concurrency capture causality/concurrency
49 Semantic equivalences on Petri nets branching time tree bisimulation ib linear time trace it abstract from causality/concurrency capture causality/concurrency
50 Semantic equivalences on Petri nets tree branching time bisimulation ib sb 2b STb h linear time trace it interleaving st step 2t split STt ST pt partial order abstract from causality/concurrency capture causality/concurrency
51 Semantic equivalences on Petri nets tree branching time bisimulation ib sb 2b STb h linear time trace it interleaving st step 2t split STt ST pt partial order caus structure preserving abstract from causality/concurrency capture causality/concurrency
52 Semantic equivalences on Petri nets tree branching time bisimulation ib sb 2b STb h pb linear time trace it interleaving st step 2t split STt ST pt partial order caus structure preserving abstract from causality/concurrency capture causality/concurrency
53 Semantic equivalences on Petri nets tree occ branching time bisimulation ib sb 2b STb h pb linear time trace it interleaving st step 2t split STt ST pt partial order caus structure preserving abstract from causality/concurrency capture causality/concurrency
54 Semantic equivalences on Petri nets tree occ branching time bisimulation ib sb 2b STb h pb linear time trace it interleaving st step 2t split STt ST pt partial order caus structure preserving abstract from causality/concurrency capture causality/concurrency
55 Semantic equivalences on Petri nets tree occ branching time bisimulation ib sb 2b STb h sp pb linear time trace it interleaving st step 2t split STt ST pt partial order caus structure preserving abstract from causality/concurrency capture causality/concurrency
56 Semantic equivalences on Petri nets branching time tree occ h (collective) bisimulation ib sb 2b STb h sp pb linear time trace it interleaving st step 2t split STt ST pt partial order caus structure preserving abstract from causality/concurrency capture causality/concurrency
57 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 6. It should be finer than caus. 7. It should not distinguish nets whose behaviours are patently the same, i.e. when differing only in unreachable parts. 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
58 Semantic equivalences on Petri nets branching time tree occ h (collective) bisimulation ib sb 2b STb h sp pb linear time trace it st 2t STt pt caus interleaving step split ST partial order structure preserving abstract from causality/concurrency capture causality/concurrency
59 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 6. It should be finer than caus. 7. It should not distinguish nets whose behaviours are patently the same, i.e. when differing only in unreachable parts. 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
60 Semantic equivalences on Petri nets branching time tree occ h (collective) bisimulation ib sb 2b STb h sp pb linear time trace it st 2t STt pt caus interleaving step split ST partial order structure preserving abstract from causality/concurrency capture causality/concurrency
61 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 6. It should be finer than caus. 7. It should not distinguish nets whose behaviours are patently the same, i.e. when differing only in unreachable parts. 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
62 Semantic equivalences on Petri nets branching time tree occ h (collective) bisimulation ib sb 2b STb h sp pb linear time trace it st 2t STt pt caus interleaving step split ST partial order structure preserving abstract from causality/concurrency capture causality/concurrency
63 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 6. It should be finer than caus. 7. It should not distinguish nets whose behaviours are patently the same, i.e. when differing only in unreachable parts. 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
64 Semantic equivalences on Petri nets branching time tree occ h (collective) bisimulation ib sb 2b STb h sp pb linear time trace it st 2t STt pt caus interleaving step split ST partial order structure preserving abstract from causality/concurrency capture causality/concurrency
65 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 6. It should be finer than caus. 7. It should not distinguish nets whose behaviours are patently the same, i.e. when differing only in unreachable parts. 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
66 Semantic equivalences on Petri nets branching time tree occ h (collective) bisimulation ib sb 2b STb h sp pb linear time trace it st 2t STt pt caus interleaving step split ST partial order structure preserving abstract from causality/concurrency capture causality/concurrency
67 Criteria for choosing this equivalence 1. It should be a branching time equivalence. 2. It should capture concurrency. a b ab + ba causality 3. It should respect inevitability. [Mazurkiewicz] 4. It should be real-time consistent. a and b one minute each 5. It should be preserved under action refinement. a a 1 ;a 2 6. It should be finer than caus. 7. It should not distinguish nets whose behaviours are patently the same, i.e. when differing only in unreachable parts. 8. It should be a congruence for CCSP. 9. [[P]] op [[P]] den for any P for which both sides are defined.
68 Inevitability I will show you Petri nets featuring a transition b. I will ask you by a show of hands whether you hold b to be inevitable.
69 Inevitability I will show you Petri nets featuring a transition b. I will ask you by a show of hands whether you hold b to be inevitable. Question 0: Are you insufficiently familiar with Petri nets to answer such questions, or decline to answer for any other reason? Question 1: b (CCSP expression: b)
70 Inevitability Question 2: a b (CCSP expression: a+b)
71 Inevitability Question 3: a b (CCSP expression: E with E def = a.e +b.)
72 Inevitability Question 4: a c b (CCSP expression: E with E def = a.c.e +b.)
73 Inevitability Question 5: a b (CCSP expression: A b with A def = a.a.)
74 Fairness In the literature I found only 4 meaningful types of fairness assumptions: 1. Progress 2. Justness 3. Weak Fairness 4. Strong Fairness These form a hierarchy, thus creating 5 assumptional states.
75 Fairness Shop with 2 customers. When in the shop, a customer is waiting expectantly to be served. Upon being served, the customer leaves the shop, but usually returns right away to buy something else. A customer may leave the shop anytime, and possibly return later.
76 Fairness Shop with 2 customers. When in the shop, a customer is waiting expectantly to be served. Upon being served, the customer leaves the shop, but usually returns right away to buy something else. A customer may leave the shop anytime, and possibly return later. Failure of weak fairness: Customer A remains forever in the shop, ready to be served, but is never served, because the clerk is always busy serving customer B.
77 Fairness Shop with 2 customers. When in the shop, a customer is waiting expectantly to be served. Upon being served, the customer leaves the shop, but usually returns right away to buy something else. A customer may leave the shop anytime, and possibly return later. Failure of weak fairness: Customer A remains forever in the shop, ready to be served, but is never served, because the clerk is always busy serving customer B. Failure of strong fairness: Customer A occasionally leaves the shop, but always returns in the hope to get served. Yet, this never occurs, because the clerk is always busy serving customer B.
78 Fairness Shop with 2 customers. When in the shop, a customer is waiting expectantly to be served. Upon being served, the customer leaves the shop, but usually returns right away to buy something else. A customer may leave the shop anytime, and possibly return later. Failure of weak fairness: Customer A remains forever in the shop, ready to be served, but is never served, because the clerk is always busy serving customer B. Failure of strong fairness: Customer A occasionally leaves the shop, but always returns in the hope to get served. Yet, this never occurs, because the clerk is always busy serving customer B. Failure of progress: Customer A remains forever in the shop, ready to be served, but no-one is ever served. The clerk stares pathetically at the customer(s) without doing anything.
79 Fairness Shop with 2 customers. When in the shop, a customer is waiting expectantly to be served. Upon being served, the customer leaves the shop, but usually returns right away to buy something else. A customer may leave the shop anytime, and possibly return later. Failure of weak fairness: Customer A remains forever in the shop, ready to be served, but is never served, because the clerk is always busy serving customer B. Failure of strong fairness: Customer A occasionally leaves the shop, but always returns in the hope to get served. Yet, this never occurs, because the clerk is always busy serving customer B. Failure of progress: Customer A remains forever in the shop, ready to be served, but no-one is ever served. The clerk stares pathetically at the customer(s) without doing anything. Failure of justness: There are two counters with a clerk each. Customer A is the only customer at counter 1, yet never is served, while customer B is being served repeatedly at counter 2.
80 Fairness in CCSP Are the following processes guaranteed to do action b eventually? Assuming b a+b E with E def = a.e +b. E with E def = a.c.e +b. A b with A def = a.a nothing progr. justness wk. f. str. fair.
81 Fairness in CCSP Are the following processes guaranteed to do action b eventually? Assuming nothing progr. justness wk. f. str. fair. b a+b E with E def = a.e +b. E with E def = a.c.e +b. A b with A def = a.a
82 Fairness in CCSP Are the following processes guaranteed to do action b eventually? Assuming nothing progr. justness wk. f. str. fair. b a+b E with E def = a.e +b. E with E def = a.c.e +b. A b with A def = a.a
83 Fairness in CCSP Are the following processes guaranteed to do action b eventually? Assuming nothing progr. justness wk. f. str. fair. b a+b E with E def = a.e +b. E with E def = a.c.e +b. A b with A def = a.a
84 Fairness in CCSP Are the following processes guaranteed to do action b eventually? Assuming nothing progr. justness wk. f. str. fair. b a+b E with E def = a.e +b. E with E def = a.c.e +b. A b with A def = a.a
85 Fairness in CCSP Are the following processes guaranteed to do action b eventually? Assuming nothing progr. justness wk. f. str. fair. b a+b E with E def = a.e +b. E with E def = a.c.e +b. A b with A def = a.a
86 Fairness in CCSP Are the following processes guaranteed to do action b eventually? Assuming nothing progr. justness wk. f. str. fair. b a+b E with E def = a.e +b. E with E def = a.c.e +b. A b with A def = a.a
87 Fairness in process algebra Strong or weak fairness can be indispensable for certain applications, such as a correctness proof of the alternating bit protocol. patently wrong when used where not appropriate. E with E def = a.e +b.0.
88 Fairness in process algebra Strong or weak fairness can be indispensable for certain applications, such as a correctness proof of the alternating bit protocol. patently wrong when used where not appropriate. E with E def = a.e +b.0. could be a spec. of a mobile phone b is a successful dialling attempt a is an unsuccessful dialling attempt. Fairness amounts to saying that if you try often enough, dialling will succeed. This is wishful thinking. The real world is not fair.
89 Fairness in process algebra Strong or weak fairness can be indispensable for certain applications, such as a correctness proof of the alternating bit protocol. patently wrong when used where not appropriate. E with E def = a.e +b.0. could be a spec. of a mobile phone b is a successful dialling attempt a is an unsuccessful dialling attempt. Fairness amounts to saying that if you try often enough, dialling will succeed. This is wishful thinking. The real world is not fair. When assuming strong or weak fairness, we loose the ability to finitely specify a system like E above that does allow an infinite sequences of as without a b.
90 Semantic equivalences on Petri nets branching time tree occ h (collective) bisimulation ib sb 2b STb h sp pb linear time trace it st 2t STt pt caus interleaving step split ST partial order structure preserving abstract from causality/concurrency capture causality/concurrency
91 HP bisimilarity does not respect inevitability s 2 2 b s 1 1 a t l 2 t2 r a s 3 1 t b 2 s 2 1 b s 1 0 a t l 1 t1 r a s 3 0 t b 1 s 0 s 4 s 2 0
92 HP bisimilarity does not respect inevitability s 2 2 b s 1 1 a t l 2 t2 r a s 3 1 t b 2 s 2 1 b s 1 0 a t l 1 t1 r a s 3 0 t b 1 s 0 s 4 s 2 0
93 Causal equivalence does not respect inevitability b a a... The causal nets of both systems are the infinite one above, and all its finite prefixes.
94 Conclusion 1. I proposed 9 requirements on semantic equivalences on Petri nets. 2. None of the existing equivalences satisfies all (or almost all) of these requirements. 3. I propose a new equivalence that does. 4. A major motivation of this equivalence is its suitability in establishing agreement between the denotational and operational Petri net of CCSP.
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