statistical physics of communicating processes Vincent Danos U of Edinburgh, CNRS SynThsys Centre

Transcription

1 statistical physics of communicating processes Vincent Danos U of Edinburgh, CNRS SynThsys Centre

2 1 ideas

3 idea I two aspects in solving a distributed problem: - local steps towards a solution - backtracking (deadlock escape) sequential case: can try to always make progress to solution, but NP! // case: one has to!

4 idea I - continued backtrack -> infrastructure (make it a "library") code easier to prove and understand universal backtrack strategy p Γ p i.e., add history to a process

5 results/reversible CCS 1. universal cover ppty: distributed history characterizes traces up to concurrency 2. weak-bisimulation: rev(p) + irreversible actions / causal transition system(p) - only irreversible actions observable 3. syntax-independent history construction (eg works for Petri nets, pi-calculus)

6 Jean Krivine Pawel Sobocinski

7 could relax universal cover ppty: introduce flex-moves (never a choice) not memorized weak memories: forget synch partner... not done anything!

8 idea II -hesitation, efficiency \ probabilization of rev(p) exhaustivity \ probabilistic equilibrium

9 idea III' what prob structure? borrow from stat phys distributed CT Metropolis build a potential energy function drive kinetics (Newtonian style, stochastic version) build a causal/concurrent, and convergent potential energy on the state space of reversible CCS

10 2 the reversible CCS transition system

11 reversible communicating processes memory n-fork Γ (p 1,...,p n ) f Γ1 p 1,...,Γn p n synch on a1,..., am Γ 1 (a 1 p 1 + q 1 ),...,Γ m (a m p m + q m ) s a Γ 1 ( Γ,a 1,q 1 ) p 1,...,Γ m ( Γ,a m,q m ) p m with a unique naming scheme and enough info to reverse uniquely

12 symmetric TS (so strongly connected) "simplicity" of TS: at most one jump slight pb with sums between x and y near acyclic countable state space (recursion)

13 potential/rate ratio constraint CTMC ρ(x, y) =q(y, x)/q(x, y) =p(y)/p(x) =e (V (y) V (x)) X e V (x) < that is convergence by def! NB: lower energy/higher probability

14 explosive growths event horizon nb of comp q f 0 p(a), 1 p(ā) 1, 1 1 fs 0a0 p(a), 0a1 p(a), 1ā0 p(ā), 1ā1 p(ā) 2, 2 2 = 0a0 a(p(a),p(a)), 0a1 a(p(a),p(a)), 1ā0 ā(p(ā),p(ā)), 1ā1 ā(p(ā),p(ā)) fs 0a0a0 p(a), 0a0a1 p(a), 0a1a0 p(a), 0a1a1 p(a), 4, 4 4! 1ā0ā0 p(ā), 1ā0ā1 p(ā), 1ā1ā0 p(ā), 1ā1ā1 p(ā)... fs w 2 0w(a) p(a), k w 2 1w(ā) p(ā) k 2 k, 2 k 2 k! is there a concurrent potential that controls the above? upper bound on the number of such (entropy) lower bound on energy of a deep state

15 3 construction of a potential

16 total stack size potential V 1 (p 1,...,p n )=V 1 (p 1 )+...+ V 1 (p n ) V 1 (Γ p) =V 1 (Γi) =V 1 (Γ) V 1 (Γ( Γ, a, q)) = V 1 (Γ)+ a Γ(p),

17 V1 energy balance for forks and synchs V 1 =(n 1)V 1 (Γ) V 1 = m a n-ary fork with memory Γ synch on a realize the ratio constraint as: k f =1 k a =1 k + f = e (n 1)V 1(Γ) k + a = e m a

18 total synch potential Given a path γ from p 0 to p: V 0 (p) = a A x s a y γ( 1)v(s) a ± ratio constraint: k f = k+ f k a /k+ a =exp( a)

19 V1 is truly concurrent, i.e. sensitive to sequential expansion V0 < or equal to V1 potentially more divergent No matter how costly a synch, V0 diverges what about V1?

20 upper bound on the number of such (entropy) Lemma 5 For large ns, log T (n) β + α 2 O(n log n) lower bound on energy of a deep state Lemma 4 Suppose β > 1, m > 0, p Σ n (p 0 ): m log 4 + log(β + +1) n log n V 1(p)

21 sufficient condition for equilibrium Proposition 1 Suppose 1 < β, and β + α 2 log(4(β + +1))< m, then: Z(p 0 ) := p Ω(p 0 ) e V 1(p) < +

22 4 epilogue

23 simulated annealing with "local" temperatures k f =1 k + f = e (n 1)V 1(Γ) the bounds are rough control growth rate? what with irreversible actions? other potentials? work with general steady states? reactive modules? something else than ccs what kind of problem? obvious connexion with rewriting theory

24 idea III: energy-based programming/distributed Metropolis code = statics/potential + transition/moves + compatible kinetics energy as syntax self-organised energy-based dynamics... stochastic machine learning argmax. p X π(,p)= 1 X dπ

25 Nicolas Oury Giorgio Bacci (Udine) Ohad Kammar David Mark