Verification of. Probabilistic Programs. Seminar on. Federico Olmedo. Pr[n = k] = 1 2

Transcription

1 Seminar on Verification of n := 0; repeat n := n + 1; c := coin flip(0.5) until (c=heads); return n Probabilistic Programs Pr[n = k] = 1 2 k Federico Olmedo federico.olmedo@cs.rwth-aachen.de 1

2 Seminar Details Speaker: FEDERICO OLMEDO Software Modelling and Verification Group RWTH Aachen University Structure & Schedule: Language: Pre-requisites: Previous knowledge on program logics and semantics is ONLY advised. Webpage: 6 Weekly Presentations 16:30-17:45 Room 9U10 E3 2

3 Agenda Introduction to probabilistic programs The problem of probabilistic program verification Seminar content Summary 3

4 Agenda Introduction to probabilistic programs The problem of probabilistic program verification Seminar content Summary 4

5 Probabilistic Programs Basics What is a probabilistic program? Ordinary Program + Distribution Sampling In this seminar imperative (eg. Probabilistic C) functional (eg. Church) logical (eg. CHRISM) randomly choose a process with which communicate select a random prime in interval [1, n 2 ] flip a (fair/biased) coin; Input/Output behaviour: Input Probabilistic Program Output Distribution Prob Output 5

6 Probabilistic Programs Examples c 1 := coin flip(0.5); c 2 := coin flip(0.5); return (c 1, c 2 ) n := 0; repeat n := n + 1; c := coin flip(0.5) until (c=heads); return n 6

7 Probabilistic Programs Examples c1 c 1 := coin flip(0.5); c 2 := coin flip(0.5); return (c 1, c 2 ) Output Distribution c2 h t h 1/4 1/4 t 1/4 1/4 n := 0; repeat n := n + 1; c := coin flip(0.5) until (c=heads); return n 6

8 Probabilistic Programs Examples c1 c 1 := coin flip(0.5); c 2 := coin flip(0.5); return (c 1, c 2 ) Output Distribution c2 h t h 1/4 1/4 t 1/4 1/4 n := 0; repeat n := n + 1; c := coin flip(0.5) until (c=heads); return n Output Distribution n 1 2 k 1/2 1/4 1/2 k 6

9 Probabilistic Programs Relevance Algorithms speed-up communication. cryptography Applications in several domains biology computer vision data management optimization Relevance of Randomization Solution of problem where deterministic techniques fail Probabilistic programs are simple and intuitive to understand 7

10 Probabilistic Programs Relevance Algorithms speed-up communication. cryptography Applications in several domains biology computer vision data management optimization Relevance of Randomization Solution of problem where deterministic techniques fail Probabilistic programs are simple and intuitive to understand 8

11 Probabilistic Programs Application Domain Probabilistic Programs 9

12 Probabilistic Programs Application Domain Skill Ranking System Probabilistic Programs 9

13 Probabilistic Programs Application Domain Skill Ranking System Probabilistic Programs Predator Prey Population Model 9

14 Probabilistic Programs Application Domain Skill Ranking System Probabilistic Programs Predator Prey Population Model Dining Philosophers 9

15 Probabilistic Programs Application Domain Skill Ranking System Probabilistic Programs Predator Prey Population Model Dining Philosophers Miller Rabin Primality Test 9

16 Probabilistic Programs Application Domain. Skill Ranking System Probabilistic Programs Graph Isomorphism Predator Prey Population Model Public-Key Encryption Dining Philosophers Miller Rabin Primality Test 9

17 Probabilistic Programs Relevance Algorithms speed-up communication. cryptography Applications in several domains biology computer vision data management optimization Relevance of Randomization Solution of problem where deterministic techniques fail Probabilistic programs are simple and intuitive to understand 10

18 Randomization allows speeding up algorithms Quicksort: QS(A), if ( A apple 1) then return (A); i := b A /2c; A < := {a 0 2 A a 0 < A[i]}; A > := {a 0 2 A a 0 > A[i]}; return QS(A < )++A[i]++QS(A > ) 11

19 Randomization allows speeding up algorithms Quicksort: QS(A), if ( A apple 1) then return (A); i := b A /2c; A < := {a 0 2 A a 0 < A[i]}; A > := {a 0 2 A a 0 > A[i]}; return QS(A < )++A[i]++QS(A > ) Problem of Quicksort: In the average case, it performs fairly well: On a random input of size n, it requires on average O(n log(n)) comparisons (which matches information theory lower bound). But in the worst case, it does not: There exist ill-behaved inputs of size n which require O(n 2 ) comparisons. How to narrow the gap between the worst and average case performance? 11

20 Randomization allows speeding up algorithms Quicksort: rqs(a), if ( A apple 1) then return (A); i := rand[1... A ]; A < := {a 0 2 A a 0 < A[i]}; A > := {a 0 2 A a 0 > A[i]}; return QS(A < )++A[i]++QS(A > ) Problem of Quicksort: In the average case, it performs fairly well: On a random input of size n, it requires on average O(n log(n)) comparisons (which matches information theory lower bound). But in the worst case, it does not: There exist ill-behaved inputs of size n which require O(n 2 ) comparisons. How to narrow the gap between the worst and average case performance? Choose the pivot at random! For any input, the expected number of comparisons matches the average case. No ill-behaved input. 11

21 Randomization allows speeding up algorithms Computing the cardinality of the union of sets: S 1 [ S 2 [...[ S n = X S i i X S i \ S j + X S i \ S j \ S k... i<j i<j<k Incl-Excl Principle Problem: Incl-Excl Principle yields an expensive solution, the RHS has 2 n -1 terms. Solution based on randomization: Random Sampling Technique We can approximate some properties of a set from a randomly chosen subset. P P P estimates P 12

22 Randomization allows speeding up algorithms Computing the cardinality S 1 [ S 2 [...[ S n Solution based on randomization: sample an element x 2 S 1 [ S 2 [...[ S n use cov(x ) = {i x 2 S i } to estimate S 1 [ S 2 [...[ S n. and m := S S n ; Draw a set S from S 1,..., S n with probability Pr[S i ]= S i m ; Draw an element x from S with uniform distribution; r := m cov (x ) ; return (r) It can be shown that r is an unbiased estimator of S 1 [ S 2 [...[ S n, ie E[r] = S 1 [ S 2 [...[ S n Expected value of r 13

23 Randomization allows speeding up algorithms Another application of the Random Sampling technique Approximate the area of a circle 14

24 Randomization allows speeding up algorithms Another application of the Random Sampling technique Approximate the area of a circle Sample random points in the enclosing square. The fraction of points lying in the circle approximates its area. Area( ) Area( ) N N (The approximation improves as N Nr of points hitting the circle Total nr of random points grows larger.) 14

25 Randomization allows speeding up algorithms Another application of the Random Sampling technique Approximate the area of a circle Sample random points in the enclosing square. The fraction of points lying in the circle approximates its area. Area( ) Area( ) N N (The approximation improves as N Nr of points hitting the circle Total nr of random points grows larger.) Approximate a definite integral Z b a f (x) dx Area( ) N N 14

26 Randomization allows speeding up algorithms Testing against the null polynomial Assume we have oracle access O(,..., ) to a multivariate polynomial p in R[x 1,..., x n ]. Determine whether p is identically zero. Solution based on randomization: Exploit the fact if p 6 0 and ā =(a 1,..., a n ) is chosen at random, Pr[p(ā ) = 0] is small. Theorem (Schwartz-Zippel): let S R with S = k deg(p). If each component of ā =(a 1,..., a n ) is chosen independently and uniformly from S, then Pr[p(ā )=0 p 6 0] apple 1 /k. Let S R with S = k deg(p) Draw a 1,..., a n independently and uniformly from S; if O(a 1,..., a n )=0then return ( p 0 ) else return ( p 6 0 ) Alg. outputs p 6 0 p 6 0 Alg. outputs p 0 p 0 Pr[p 6 0 output p 0 ] apple 1 k 15

27 Randomization allows speeding up algorithms Testing against the null polynomial Assume we have oracle access O(,..., ) to a multivariate polynomial p in R[x 1,..., x n ]. Determine whether p is identically zero. Solution based on randomization: Exploit the fact if p 6 0 and ā =(a 1,..., a n ) is chosen at random, Pr[p(ā ) = 0] is small. Theorem (Schwartz-Zippel): let S R with S = k deg(p). If each component of ā =(a 1,..., a n ) is chosen independently and uniformly from S, then Pr[p(ā )=0 p 6 0] apple 1 /k. Let S R with S = k deg(p) Alg. outputs p 6 0 p 6 0 For i =1...m do Draw a 1,..., a n independently and uniformly from S; Alg. outputs p 0 p 0 if O(a 1,..., a n )6=0 then return ( p 6 0 ) return ( p 0 ) Pr[p 6 0 output p 0 ] apple 1 k m 15

28 Randomization allows speeding up Algorithms Underlying techniques Abundance of Witnesses Decision problem whose output depends on the presence (resp. absence) of a witness to prove (resp. disprove) a property. Witnesses abound in a given search space. Given a witness, the property is efficiently verified. Amplification by Independent Trials Used in conjunction with the abundance of witnesses technique to reduce the error probability. Given an algorithm with error probability ε, run it n independent times to reduce the error probability to ε n. 16

29 Randomization allows speeding up Algorithms Underlying techniques Abundance of Witnesses Any ā such that p(ā ) 6= 0 Decision problem whose output depends on the presence (resp. absence) of a witness to prove (resp. disprove) a property. Witnesses abound in a given search space. Given a witness, the property is efficiently verified. Other example: primality testing [Rabin 76]. p 6 0 Any S R with S deg(p) Schwartz-Zippel theorem Amplification by Independent Trials Used in conjunction with the abundance of witnesses technique to reduce the error probability. Given an algorithm with error probability ε, run it n independent times to reduce the error probability to ε n. 16

30 Probabilistic Programs Relevance Algorithms speed-up communication. cryptography Applications in several domains biology computer vision data management optimization Relevance of Randomization Solution of problem where deterministic techniques fail Probabilistic programs are simple and intuitive to understand 17

31 Randomization circumvents the Limitations of Determinism The Dinning Philosopher Problem Theorem (Lehmann & Rabin 81) there exists no fully distributed and symmetric deterministic algorithm for the dining philosopher problem. Randomized Algorithm while (true) do (* Thinking Time *) trying := true while (trying) do s := rand{left, right} Wait until fork[s] is available and take it If fork[ s] is available then take it and set trying to false else drop fork[s] (* Eating Time *) Drop both forks Idea: Do not pick always the same fork first. Flip a coin to choose. If the second fork is not available, release the first and flip again the coin. Algorithm is deadlock-free: At any time, if there is a hungry philosopher, with probability one some philosopher will eventually eat. Algorithm can also be adapted to prevent starvation (ie the hungry philosopher will eventually eat). 18

32 Randomization circumvents the Limitations of Determinism Leader Election P1 Aim: to choose a leader node in a network. Network consists of n identical nodes P1,,Pn connected in a ring fashion. Transmission of messages is allowed between consecutive nodes in the ring. At the end of the process all nodes must agree on the election of the leader. Pn-1 Pn P5 P2 P4 P3 Theorem (Angluin 80) there exists no deterministic algorithm for carrying out the election in a ring of identical processes. Randomized Algorithm repeat s := empty list ; name := rand{1,..., K}; For i =1...n do s := s ++ [name]; send(name) to next node ; receive(name) from previous node until (at least one name in s is unique) return max{n 2 s n occus only once in s} Idea: Each nodes chooses a random name from {1,,K} and propagates it around the ring. At the end of the propagation each process has a list of the names of all the nodes. If there is a name that belongs to only one node, then this is the leader (in case of several, choose eg the largest) If there is no unique name, repeat the process. 19

33 Randomization circumvents the Limitations of Determinism Underlying technique Symmetry Breaking in Distributed Systems For many problems on distributed systems, deterministic solutions do not exists when objects are to be treaded identically. Using randomization to choose among identical objects may help solving the symmetry problem. 20

34 Probabilistic Programs Tradeoffs Advantages Reduction of time/space complexity Reduction of communication complexity in the distributed setting Allows tackling problems that have no deterministic solution Probabilistic programs are simple and easy to understand Wide range of application domains 21

35 Probabilistic Programs Tradeoffs Advantages Reduction of time/space complexity Reduction of communication complexity in the distributed setting Allows tackling problems that have no deterministic solution Probabilistic programs are simple and easy to understand Wide range of application domains Disadvantages Absolute correctness is sometimes sacrificed: probabilistic programs are correct with probability 1 Quicksort, dining philosopher, leader election Definite integral, testing against null polynomial =0 > 0 21

36 Probabilistic Programs Tradeoffs Advantages Reduction of time/space complexity Reduction of communication complexity in the distributed setting Allows tackling problems that have no deterministic solution Probabilistic programs are simple and easy to understand Wide range of application domains Disadvantages Absolute correctness is sometimes sacrificed: probabilistic programs are correct with probability 1 Running time is a random variable Quicksort, dining philosopher, leader election Definite integral, testing against null polynomial =0 > 0 LAS VEGAS ALGORITHM MONTE CARLO ALGORITHM Constant running time (always fast) 21

37 Probabilistic Programs Reliability Deterministic Algorithm Probabilistic Algorithm ALWAYS LONG ALWAYS LONG NEVER SHORT Faster, but less correct NEVER SHORT Correctness Running Time Correctness Running Time Is the probabilistic algorithm still reliable? 22

38 Probabilistic Programs Reliability Deterministic Algorithm Probabilistic Algorithm ALWAYS LONG ALWAYS LONG NEVER SHORT Faster, but less correct NEVER SHORT Correctness Running Time Correctness Running Time Is the probabilistic algorithm still reliable? Yes, absolutely! 22

39 Probabilistic Programs remain Reliable Verifying Data Consistency Goal: RI and RII must communicate to verify whether x=y (x, y 2 {0,1} n ). x=x 1...x n communication y=y 1...y n Requirement: minimize the # of bits exchanged. RI RII Theorem: any deterministic protocol requires the exchange of (at least) n bits. Randomized Algorithm Idea: use random fingerprints of x and y. Routine of R I, p := rand{i 2 [2, n 2 ] prime(i)}; s := x mod p; send(p, s) to R II ; #bits exchanged = #bits(p) + #bits(s) apple 2 log 2(n 2 ) Prot. outputs x6=y apple n 2 apple p apple n 2 x6=y Pr[x 6= y output x=y ] apple ln(n2 ) n Routine of R II, receive(p, s) from R I ; t := y mod p; if (s=t) then return ( x=y ) else return ( x6=y ) 23

40 Probabilistic Programs remain Reliable Verifying Data Consistency Goal: RI and RII must communicate to verify whether x=y (x, y 2 {0,1} n ). x=x 1...x n communication y=y 1...y n Requirement: minimize the # of bits exchanged. RI RII Theorem: any deterministic protocol requires the exchange of (at least) n bits. Randomized Algorithm Idea: use random fingerprints of x and y. Routine of R I, p := rand{i 2 [2, n 2 ] prime(i)}; s := x mod p; send(p, s) to R II ; Routine of R II, receive(p, s) from R I ; t := y mod p; if (s=t) then return ( x=y ) else return ( x6=y ) apple n 2 apple p apple n 2 #bits exchanged = #bits(p) + #bits(s) apple 2 log 2(n 2 ) Prot. outputs x6=y x6=y Pr[x 6= y output x=y ] apple ln(n2 ) n # bits exch. prob. error n= x n= x n= x n= x n= x

41 Agenda Introduction to probabilistic programs The problem of probabilistic program verification Seminar content Summary 24

42 Probabilistic Programs The Verification Problem Input/Output behaviour of probabilistic programs: Input Probabilistic Program Output Distribution Prob Output Verification of probabilistic programs: Probabilistic Program VERIFICATION Probabilistic Assertion about the Program Output Property holds with a given probability Pr[error] apple 0.02 Assertion about the expected (ie average) value of a variable E[r] apple 10 25

43 Probabilistic Programs The Verification Problem Examples of Probabilistic Assertions Cardinality of sets union m := S S n ; Draw a set S from S 1,..., S n with probability Pr[S i ]= S i m ; Draw an element x from S with uniform distribution; r := m cov (x) ; return (r) E[r] = S 1 [ S 2 [...[ S n Leader Election repeat s := empty list ; name := rand{1,..., K}; For i =1...n do s := s ++ [name]; send(name) to next node ; receive(name) from previous node until (at least one name in s is unique) return max{n 2 s n occus only once in s} Pr[termination] =1? 26

44 Probabilistic Programs The Verification Problem Let p be probability that after the random choices of the node names, at least one name is unique. We know that 0 < p < 1. Pr[term] =1 Pr[non term] =1 Pr h in all rounds, there is no unique name =1 lim (1 n!1 p)n =1 i 27

45 Probabilistic Programs The Verification Problem Let p be probability that after the random choices of the node names, at least one name is unique. We know that 0 < p < 1. Pr[term] =1 Pr[non term] =1 Pr h in all rounds, there is no unique name =1 lim (1 n!1 p)n =1 Intricacy i Probabilistic programs may terminate with probability 1, and still admit diverging executions. Insight: (set of) diverging executions have probability 0 27

46 Probabilistic Programs The Verification Problem Let p be probability that after the random choices of the node names, at least one name is unique. We know that 0 < p < 1. Pr[term] =1 Pr[non term] =1 Pr h in all rounds, there is no unique name =1 lim (1 n!1 p)n =1 i ALMOST-SURE TERMINATION (AST) Intricacy Probabilistic programs may terminate with probability 1, and still admit diverging executions. Insight: (set of) diverging executions have probability 0 27

47 Probabilistic Programs The Verification Problem Let p be probability that after the random choices of the node names, at least one name is unique. We know that 0 < p < 1. Pr[term] =1 Pr[non term] =1 Pr h in all rounds, there is no unique name =1 lim (1 n!1 p)n =1 i ALMOST-SURE TERMINATION (AST) Intricacy Probabilistic programs may terminate with probability 1, and still admit diverging executions. Another example of AST: repeat b := flip coin(); until (b = heads) Insight: (set of) diverging executions have probability 0 27

48 Probabilistic Programs The Verification Problem How to verify probabilistic assertions? It is possible to extend standard verification techniques of sequential programs: Hoare logic Weakest precondition calculus Assertions of the form {P} c {Q}, where P and Q are predicates over program states Given in terms of predicate transformer wp[c]: P( )! P( ) initial state s c s 0 final state wp[c](q) = set of initial states that lead to a final state satisfying Q {P} c {Q} is valid i P(s) =) Q(s 0 ) Example: wp[x := x+2](x 0) = x 2 Example: {x 0} x := x+2 {x 0} Connection to Hoare logic Deductive system (ie proof rules) to derive valid assertions (one rule per language construction) {P} c {Q} i P =) wp[c](q) {P }skip{p } {P [x e]}x := e{p } Transformer wp[c] the structure of c: is defined by induction on {P }s 1 {Q} {Q}s 2 {R} {P }s 1 ; s 2 {R} Proof objects are derivations (trees) wp[skip](q) =Q wp[x := e](q) =Q[x/E] wp[c 1 ; c 2 ](Q) =(wp[c 1 ] wp[c 2 ])(Q) 28

49 Agenda Introduction to probabilistic programs The problem of probabilistic program verification Seminar content Summary 29

50 Seminar Content Different extension of Hoare logic and weakest precondition calculus for probabilistic programs. Probabilistic predicate transformers [McIver & Morgan 96] Reward function final states. f :! R over the set of wp[c](f )= Expected reward of c wrt f expected value of f wrt distribution of final states Relational Hoare logics [Barthe 09, 12] Relates the executions of a program from two different initial states. Pre- and post-conditions are relations (rather than predicates) over program states. {= L } c {= L } c in non-interferent Hartog s Hoare logic [Hartog 02] {true} c {8i (iapple0) _ Pr[x=i] =( 1 /2) i } variable x is geometrically distributed 30

51 Agenda Introduction to probabilistic programs The problem of probabilistic program verification Seminar content Summary 31

52 Summary What is a probabilistic program? Probabilistic programs are simple and intuitive to understand Ordinary Program + Distribution Sampling Applications in several domains Algorithms speed-up Relevance of Randomization Solution of problem where deterministic techniques fail We can extend traditional program verification techniques to probabilistic programs. 32