Distributed transactions and reversibility

Transcription

1 Distributed transactions and reversibility Pawel Sobocinski, University of Cambridge Southampton, 26 September 2006 based on joint work with Vincent Danos and Jean Krivine

2 Motivation Reversible CCS (RCCS) (Danos, Krivine 2004, 2005) process calculus invented in order to model distributed algorithms without having to worry about including explicit backtracking in the specification syntax & SOS extended: terms include histories How to generalise approach to other models? Petri nets, graph transformations,... Here we illustrate the concepts using a simple graph transformation system

3 Example: dining philosophers attributed nodes of two types adhesive (Lack Sobocinski 04, 05) Type graph nodes: agents, attributes: internal state, edges: physical proximity

4 Productions q1 q2 q3

5 Start graph

6 Example q1 q2 q3

7 Example q1 q2 q3

8 Example q1 q2 q3

9 q1 q2 q3 q 2 Example

10 q1 q2 q3 q 2 Example

11 q1 q2 q3 q 2 q 3 Example

12 q1 q2 q3 q 2 q 3 Example

13 q1 q2 q3 q 2 q 3 q 2 Example

14 Deadlock q1 q2 q3

15 Deadlock q1 q2 q3

16 Deadlock q1 q2 q3

17 Deadlock q1 q2 q3

18 q1 q2 q3 Deadlock

19 q1 q2 q3 Deadlock

20 q1 q2 q3 Deadlock

21 q1 q2 q3 philosophers starve Deadlock

22 Transactions Idea: q1 should be undoable. Divide set of productions into: irreversible - actions which take place once a desired distributed state is reached reversible - actions which may lead to unstable states and deadlock Transaction: a causal sequence of zero or more reversible actions followed by a single irreversible action. Note: causality well-understood in an adhesive setting.

23 Examples Problem: Correctly capture the transactions by correctly reversing the reversible computations correctly = avoiding deadlock + no new behaviour

24 q1 Solution? q2 q3 add reverse production +

25 q2 q3 Deadlock avoided

26 q2 q3 Deadlock

27 q2 q3 Deadlock

28 q2 q3 q 2 Deadlock

29 q2 q3 The greedy philosopher who mastered mind control

30 q2 q3 The greedy philosopher who mastered mind control

31 q2 q3 The greedy philosopher who mastered mind control

32 q2 q3 The greedy philosopher who mastered mind control

33 Ad hoc (correct?) solutions Problem under specified? add extra productions, distinguish between left and right & then add reversed productions still must prove that transactions captured risk of making the model too complex/specialised l r l l r r l l l l r l r r l r

34 Category of computations a mathematical structure which captures the concurrent behaviour of the model objects: graphs; arrows: paths of direct derivations modulo concurrency. Note: for adhesive categories modulo concurrency is well-understood. Problem: Correctly capture the transactions by correctly reversing the reversible computations Our problem can be solved at this level. (Danos, Krivine, Sobocinski 06)

35 Why solve abstractly? One can imagine reversing several formalisms for concurrency Petri nets, graph transformations, process algebras each has its own category of computations It is useful to have the construction and proof of correctness at a level which can be applied to all the models

36 Abstract transactions R = reversible subcategory; the computations which consist only of reversible actions. A p B f g if for arbitrary p, q f C h q D g such that gp = qf there exists unique h such that hf = p & gh = q. Let I = { f C g R. f g }. Example: single thread C = {I + R} R = R,, I = {I + R} I + ɛ

37 Example: single thread C = {I + R} R = R,, I = {I + R} I + ɛ I R R I proof: σi σir 1...r m r 1...r m r 1...r n σr r 1...r m σ id r 1...r n r f f g p, q st gp = qf p A B C h q st!h gh = q hf = p D g

38 Abstract transactions, facts: 1. I is the subcategory of irreversible computations (in examples - computations built up of zero or more transactions) 2. I, R is a factorisation system on the category of computations ie computations can be broken up (essentially uniquely) into an irreversible component followed by a reversible component. h = f g

39 History category Suppose that C has a factorisation system I, R. h(c, R) objects: arrows g in R ; P 1 f arrows: commutative g 1 P 2 g 2 diagrams, with f I. Q 1 h Q 2 ie in our example, states are no longer graphs but (concurrent) reversible computations.

40 Reversible history category h (C, R) objects: arrows g in R ; P 1 f arrows: commutative diagrams, with f I. g 1 P 2 g 2 Q 1 h Q 2 + inverses whenever h R. ie histories can be backtracked Formal definition uses categories of fractions.

41 Main result h(c, R) M N N Ψ I h (C, R) M M : h (C, R) I is an equivalence of categories Intuition: the transactions are captured up to reversing of histories. For CCS, translates to functional weak bisimulation (reversible moves treated as silent actions).

42 Adding reversibility 1. Start with favourite model of concurrent computation 1.1. develop syntax for concurrent computations (eg. Petri net processes, special SOS rules for calculi,...) for Petri nets, can use processes, for graph transformations, there is (ongoing) work on processes (Corradini Sobocinski 06) 1.2. replace states with reversible histories, add reverse rules, capturing the reversible history category 2.Main theorem guarantees correctness

43 Example id

44 Evil philosopher?? No extra behaviour, essentially because only things that have been done can be undone.

45 Conclusions irreversible - reversible factorisation as a factorisation system a construction on the cat of computations which implements the backtracking main result shows correctness of backtracking generalises previous work on RCCS by Danos an Krivine applicable to Petri nets, graph transformations and other formalisms difficulty comes in inventing syntax to describe histories Many models, one construction, one proof of correctness.