Robustness Analysis of Networked Systems

Transcription

1 Robustness Analysis of Networked Systems Roopsha Samanta The University of Texas at Austin Joint work with Jyotirmoy V. Deshmukh and Swarat Chaudhuri January, 0 Roopsha Samanta Robustness Analysis of Networked Systems / 5

2 Problem Overview P P P Programs rarely execute in isolation Roopsha Samanta Robustness Analysis of Networked Systems / 5

3 Problem Overview P P P Programs interact with other programs and the physical world Roopsha Samanta Robustness Analysis of Networked Systems / 5

4 Problem Overview P P P Predictability in the presence of uncertainty Roopsha Samanta Robustness Analysis of Networked Systems / 5

5 Problem Overview P P P Small perturbation in input Small perturbation in output Roopsha Samanta Robustness Analysis of Networked Systems / 5

6 Problem Overview P P P Robustness! Roopsha Samanta Robustness Analysis of Networked Systems / 5

7 Synchronous Networked System N : C, P C out Directed graph (P, C) P = {P,..., P n } C = I N O Synchronous P C in C, C, Computation alphabet: Σ P C, Roopsha Samanta Robustness Analysis of Networked Systems / 5

8 Process C, Process P i : (In i, Out i, M i ) Mealy Machine M i : (Σ In i, Σ Out i, Q i, q 0i, R i ) C in P C, Roopsha Samanta Robustness Analysis of Networked Systems / 5

9 Network Semantics N : (Σ I, Σ O, Q, q 0, R) Network state: (q,..., q n, c,..., c N ) N : C, P C out Network transition: (q,..., q n, c,..., c I ) P (a,..., a I ), (a,..., a w ) C in C, C, (q,..., q n, c,..., c O ) P Network execution: ρ(s) Network output: N (s) C, Roopsha Samanta Robustness Analysis of Networked Systems 5 / 5

10 Internal Channel Perturbations Instantaneous deletion or substitution of current symbol τ-transition: (q,..., q n, c,..., c I ) ε, ε (q,..., q n, c,..., c I ), Perturbed network execution: ρ τ (s) Perturbed network output: ρ τ (s) Channel-wise perturbation count in ρ τ (s): ρ τ (s) Roopsha Samanta Robustness Analysis of Networked Systems 6 / 5

11 Robust Networked System Given: a networked system N, max. pertb. count of internal channels, δ = (δ,..., δ N ), max. permissible error in output channel, ɛ, distance metric d over Σ P N : C,, δ C in C,, δ C,, δ P P C out, ɛ C,, δ N is defined to be (δ, ɛ)-robust if: s (Σ I ), ρ τ (s) : ρ τ (s) δ = d( N (s), ρ τ (s) ) ɛ Roopsha Samanta Robustness Analysis of Networked Systems 7 / 5

12 Robust Networked System Given: a networked system N, max. pertb. count of internal channels, δ = (δ,..., δ N ), max. permissible error in output channel, ɛ, distance metric d over Σ P N : C,, δ C in C,, δ C,, δ P P C out, ɛ C,, δ N is defined to be (δ, ɛ)-robust if: s (Σ I ), ρ τ (s) : ρ τ (s) δ = d( N (s), ρ τ (s) ) ɛ Roopsha Samanta Robustness Analysis of Networked Systems 7 / 5

13 Problem Definition Given: a networked system N, max. pertb. count of internal channels, δ = (δ,..., δ N ), max. permissible error in output channel, ɛ, distance metric d over Σ P N : C,, δ C in C,, δ C,, δ P P C out, ɛ C,, δ Check if N is (δ, ɛ)-robust. Roopsha Samanta Robustness Analysis of Networked Systems 8 / 5

14 Solution Strategy Given: a networked system N, max. pertb. count of internal channels, δ = (δ,..., δ N ), max. permissible error in output channel, ɛ, distance metric d over Σ P N : C,, δ C in C,, δ C,, δ P P C out, ɛ C,, δ Construct machine A: N is (δ, ɛ)-robust iff L(A) is empty. Roopsha Samanta Robustness Analysis of Networked Systems 9 / 5

15 Solution Strategy Construct machine A: N is (δ, ɛ)-robust iff L(A) is empty. In a run of A on s, A simultaneously: simulates unperturbed, perturbed executions: ρ(s), ρ τ (s) tracks channel perturbations along ρ τ (s), tracks distance between outputs of N along ρ(s), ρ τ (s). A accepts s iff ρ τ (s): ρ τ (s) δ and d( N (s), ρ τ (s)) > ɛ. Roopsha Samanta Robustness Analysis of Networked Systems 9 / 5

16 Review: Levenshtein Distance, d Lev (s, t) Minimum number of symbol insertions, deletions and substitutions required to transform s into t. d Lev (abc, bcd) = b c d a b c d Lev (acbcd, ccff ) = c c f f or, a c b c d c c f f a c b c d Roopsha Samanta Robustness Analysis of Networked Systems 0 / 5

17 Review: Levenshtein Distance, d Lev (s, t) Minimum number of symbol insertions, deletions and substitutions required to transform s into t. d Lev (abc, bcd) = b c d a b c d Lev (acbcd, ccff ) = c c f f or, a c b c d c c f f a c b c d Roopsha Samanta Robustness Analysis of Networked Systems 0 / 5

18 Review: Levenshtein Distance, d Lev (s, t) Minimum number of symbol insertions, deletions and substitutions required to transform s into t. d Lev (abc, bcd) = b c d a b c d Lev (acbcd, ccff ) = c c f f or, a c b c d c c f f a c b c d Roopsha Samanta Robustness Analysis of Networked Systems 0 / 5

19 Review: Levenshtein Distance Computation t c c f f s a c b d Lev (s[0], t[0]) = 0 d Lev (s[0, i], t[0]) = i d Lev (s[0], t[0, j]) = j c d 5 5 Roopsha Samanta Robustness Analysis of Networked Systems / 5

20 Review: Levenshtein Distance Computation t s a c b c 0 c c f f 0 0 subs del ins? d Lev (s[0, i], t[0, j]) = min(d Lev (s[0, i-], t[0, j-]) + (s[i], t[j]), d Lev (s[0, i-], t[0, j]) +, d Lev (s[0, i], t[0, j-]) + ) (a, b) equals 0 if a = b, else. d 5 5 Roopsha Samanta Robustness Analysis of Networked Systems / 5

21 Review: Levenshtein Distance Computation t s a c b 0 c c f f 0 0 d Lev (s[0, i], t[0, j]) = min(d Lev (s[0, i-], t[0, j-]) + (s[i], t[j]), d Lev (s[0, i-], t[0, j]) +, d Lev (s[0, i], t[0, j-]) + ) c (a, b) equals 0 if a = b, else. d 5 5 Roopsha Samanta Robustness Analysis of Networked Systems / 5

22 Review: Levenshtein Distance Computation t s a c b 0 c c f f 0 0 d Lev (s[0, i], t[0, j]) = min(d Lev (s[0, i-], t[0, j-]) + (s[i], t[j]), d Lev (s[0, i-], t[0, j]) +, d Lev (s[0, i], t[0, j-]) + ) c (a, b) equals 0 if a = b, else. d 5 5 Roopsha Samanta Robustness Analysis of Networked Systems / 5

23 Review: Levenshtein Distance Computation t s a c b 0 c c f f 0 0 d Lev (s[0, i], t[0, j]) = min(d Lev (s[0, i-], t[0, j-]) + (s[i], t[j]), d Lev (s[0, i-], t[0, j]) +, d Lev (s[0, i], t[0, j-]) + ) c (a, b) equals 0 if a = b, else. d 5 5 Roopsha Samanta Robustness Analysis of Networked Systems / 5

24 Review: Levenshtein Distance Computation t 0 c c f f 0 0 To check if d Lev (s, t) > ɛ, focus on ɛ-diagonal. a s c b c Construct DFA D ɛ Lev: runs on a string pair (s, t), and accepts iff d Lev (s, t) > ɛ. d 5 ɛ = Roopsha Samanta Robustness Analysis of Networked Systems / 5

25 Distance-Tracking Automaton, D ɛ Lev c c f f # # (λ, λ,,, 0,, ) (a, c) (c, f ) 0 0 (a, c,,,,, ) (bc, ff,,,,, ) a (c, c) (d, #) c (ac, cc,,,,, ) (cd, f #,,,,, ) b (b, f ) (#, #) c (cb, cf,,,,, ) accept d 5 # 6 ɛ = Roopsha Samanta Robustness Analysis of Networked Systems / 5

26 Distance-Tracking Automaton, D ɛ Lev D ɛ Lev accepts a pair of strings (s, t) iff d Lev(s, t) > ɛ. Roopsha Samanta Robustness Analysis of Networked Systems / 5

27 Review: Counter Machines (One-way, Nondeterministic) h-counter machine A: finite automaton with h integer counters In each step, A may read an input symbol, test counter values change state update each counter by some constant Transition: (x, σ, test, x, c,..., c h ) s is accepted by A iff (x 0, s 0, 0,... 0) A (acc, s j, z,..., z h ) Most interesting questions are undecidable. Roopsha Samanta Robustness Analysis of Networked Systems / 5

28 Review: Counter Machines (One-way, Nondeterministic) h-counter machine A: finite automaton with h integer counters In each step, A may read an input symbol, test counter values change state update each counter by some constant Transition: (x, σ, test, x, c,..., c h ) s is accepted by A iff (x 0, s 0, 0,... 0) A (acc, s j, z,..., z h ) Most interesting questions are undecidable. Roopsha Samanta Robustness Analysis of Networked Systems / 5

29 Review: Counter Machines (One-way, Nondeterministic) h-counter machine A: finite automaton with h integer counters In each step, A may read an input symbol, test counter values change state update each counter by some constant Transition: (x, σ, test, x, c,..., c h ) s is accepted by A iff (x 0, s 0, 0,... 0) A (acc, s j, z,..., z h ) Most interesting questions are undecidable. Roopsha Samanta Robustness Analysis of Networked Systems / 5

30 Review: Counter Machines (One-way, Nondeterministic) h-counter machine A: finite automaton with h integer counters In each step, A may read an input symbol, test counter values change state update each counter by some constant Transition: (x, σ, test, x, c,..., c h ) s is accepted by A iff (x 0, s 0, 0,... 0) A (acc, s j, z,..., z h ) Most interesting questions are undecidable. Roopsha Samanta Robustness Analysis of Networked Systems / 5

31 Review: Reversal-bounded Counter Machines (One-way, Nondeterministic) r-reversal bounded, h-counter machine: each counter alternates between an increasing mode and a decreasing mode at most r times. [GurariIbarra98] Nonemptiness for a r-reversal bounded, h-counter machine A is solvable in time polynomial in the size of A. Roopsha Samanta Robustness Analysis of Networked Systems / 5

32 Review: Reversal-bounded Counter Machines (One-way, Nondeterministic) r-reversal bounded, h-counter machine: each counter alternates between an increasing mode and a decreasing mode at most r times. [GurariIbarra98] Nonemptiness for a r-reversal bounded, h-counter machine A is solvable in time polynomial in the size of A. Roopsha Samanta Robustness Analysis of Networked Systems / 5

33 Robustness Analysis for Levenshtein distance Construct machine A δ,ɛ Lev A: N is (δ, ɛ)-robust iff L(A) is empty. In a run of A on s, A simultaneously: simulates unperturbed, perturbed executions: ρ(s), ρ τ (s) tracks channel perturbations along ρ τ (s), tracks distance between outputs of N along ρ(s), ρ τ (s). A accepts s iff ρ τ (s): ρ τ (s) δ and d( N (s), ρ τ (s)) > ɛ. Roopsha Samanta Robustness Analysis of Networked Systems 5 / 5

34 A: -reversal-bounded N -counter machine A state x: (q, r, q Lev ), q, r Q, q Lev Q Lev Initial state: (q 0, q 0, q 0 Lev ) Set of counters Z = {z,..., z N }, initially 0 Final state: {acc} Roopsha Samanta Robustness Analysis of Networked Systems 6 / 5

35 A: Initialization transition Initialize counters with perturbation bounds: ( (q 0, q 0, q 0 Lev ), ε, ) z k = 0, (q 0, q 0, q 0 Lev ), (+δ,..., +δ N ) k Roopsha Samanta Robustness Analysis of Networked Systems 7 / 5

36 A: Unperturbed network transitions Simulate pair of unperturbed N -transitions, track output distance: ( (q, r, q Lev ), a, ) z k 0, (q, r, q Lev), 0 k Roopsha Samanta Robustness Analysis of Networked Systems 8 / 5

37 A: Perturbed network transitions Simulate τ-transition, decrement counters of perturbed channels by : ( (q, r, q Lev ), ε, ) z k 0, (q, r τ, q Lev ), c k Roopsha Samanta Robustness Analysis of Networked Systems 9 / 5

38 A: Rejecting transitions Reject if any internal channel perturbation exceeds bound: ( (q, r, q Lev ), ε, ) z k < 0, rej, 0 k (rej, a, true, rej, 0) Roopsha Samanta Robustness Analysis of Networked Systems 0 / 5

39 A: Accepting transitions Accept witness to non-robustness: ( (q, r, acc Lev ), ε, ) z k 0, acc, 0 k Roopsha Samanta Robustness Analysis of Networked Systems / 5

40 Main Result(s) Checking if N is (δ, ɛ)-robust w.r.t the Levenshtein distance is polynomial in the size of N and exponential in ɛ. Checking if N is (δ, ɛ)-robust w.r.t the L -norm is polynomial in the size of N. Roopsha Samanta Robustness Analysis of Networked Systems / 5

41 Main Result(s) Checking if N is (δ, ɛ)-robust w.r.t the Levenshtein distance is polynomial in the size of N and exponential in ɛ. Checking if N is (δ, ɛ)-robust w.r.t the L -norm is polynomial in the size of N. Roopsha Samanta Robustness Analysis of Networked Systems / 5

42 Related Work Sequential programs with perturbed inputs [MS09, CGLN] Input-output stability of finite-state transducers [TBCSM] Sequential circuits, common suffix distance metric [DHLN0] Robust control, wireless control networks [ADJPW, Pappas] Reactive systems with ω-regular spec. in uncertain environment [MRT, CHR0, BGHJ09] Roopsha Samanta Robustness Analysis of Networked Systems / 5

43 Future Work Consider more distance-metrics Generalize error model - input noise, process failures etc. Synthesis of robust networked systems Roopsha Samanta Robustness Analysis of Networked Systems / 5

44 Thank you. Roopsha Samanta Robustness Analysis of Networked Systems 5 / 5