Newtonian Program Analysis

Transcription

1 Newtonian Program Analysis Javier Esparza Technische Universität München Joint work with Stefan Kiefer and Michael Luttenberger

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6 From programs to flowgraphs proc X 1 proc X 2 proc X 3 a c d f X 1 b X 2 X 2 e X 1 g X 2 X 3 X 1 X 3

7 From flowgraphs to equations Again a syntactic transformation. X 1 = a X 1 X 2 + b X 2 = c X 2 X 3 + d X 2 X 1 + e X 3 = f X 1 X 3 + g But how should the equations be interpreted mathematically? What kind of objects are a,..., g? What kind of operations are sum and product? It depends. Different interpretations lead to different semantics

8 From flowgraphs to equations Again a syntactic transformation. X 1 = a X 1 X 2 + b X 2 = c X 2 X 3 + d X 2 X 1 + e X 3 = f X 1 X 3 + g But how should the equations be interpreted mathematically? What kind of objects are a,..., g? What kind of operations are sum and product? It depends. Different interpretations lead to different semantics.

9 Input/output relational semantics Interpret a,..., g as assignments or guards over a set of program variables V with set of valuations Val. R(X i ) = (v, v ) Val Val such that X i started at v, may terminate at v.

10 Input/output relational semantics Interpret a,..., g as assignments or guards over a set of program variables V with set of valuations Val. R(X i ) = (v, v ) Val Val such that X i started at v, may terminate at v. ( R(X 1 ), R(X 2 ), R(X 3 ) ) is the least solution of the equations under the following interpretation: Universe: 2 V V (input/output relations) a,..., g are relations for assignment/guards sum is union of relations, product is join of relations: R 1 R 2 = {(a, b) c (a, c) R 1 (c, b) R 2 }

11 Language semantics Interpret the atomic actions as letters of an alphabet A. L(X i ) = words w A such that X i can execute w and terminate.

12 Language semantics Interpret the atomic actions as letters of an alphabet A. L(X i ) = words w A such that X i can execute w and terminate. ( L(X 1 ), L(X 2 ), L(X 3 ) ) is the least solution of the equations under the following interpretation: Universe: 2 A (languages over A). a,..., g are the singleton languages {a},..., {g}. sum is union of languages, product is concatenation:. L 1 L 2 = {w 1 w 2 w 1 L 1 w 2 l 2 }

13 Counting semantics Given a word w, denote by #(w) the vector saying how many times each of a,..., g occurs in w. Define Co(X i ) = {#(w) w L(X i )}.

14 Counting semantics Given a word w, denote by #(w) the vector saying how many times each of a,..., g occurs in w. Define Co(X i ) = {#(w) w L(X i )}. ( Co(X 1 ), Co(X 2 ), Co(X 3 )) is the least solution of the equations under the following interpretation: Universe: sets of vectors of naturals a,..., g are the singleton sets {(1, 0,..., 0)},..., {(0, 0,..., 1)} sum is union of sets, product is given by S 1 S 2 = {v 1 + R v 2 v 1 S 1, v 2 S 2 }

15 Probabilistic termination semantics Interpret a,..., g as probabilities. T (X i ) = probability that X i terminates.

16 Probabilistic termination semantics Interpret a,..., g as probabilities. T (X i ) = probability that X i terminates. ( T (X 1 ), T (X 2 ), T (X 3 ) ) is the least solution of the equations under the following interpretation: Universe: R + a,..., g are the probabilities of taking the transitions sum and product are addition and multiplication of reals

17 Abstract interpretation [Cousot, Cousot 77] determines an interpretation given its universe, and its relation to a reference semantics (the concrete semantics).

18 ω-continuous semirings Underlying mathematical structure: ω-continuous semirings Algebra (C, +,, 0, 1) (C, +, 0) is a commutative monoid distributes over + (C,, 1) is a monoid 0 a = a 0 = 0 a a + b is a partial order -chains have limits System of equations X = f (X) where X = (X 1,..., X n ) vector of variables, f (X) = (f 1 (X),..., f n (X)) vector of terms over C {X 1,..., X n }. Notice: the f i are polynomials!!

19 Static program analysis Static program analysis = computing the least solution of a system of polynomial equations over a suitable ω-continuous semiring Program = system of equations Analysis problem = concrete semiring Algorithmic solution = equation solver Theory of static analysis = generic solution techniques In this talk: generic solution techniques and some consequences.

20 Kleenean program analysis Theorem [Kleene]: The least solution µf is the supremum of {k i } i 0, where k 0 = f (0) k i+1 = f (k i ) Basic algorithm: compute k 0, k 1, k 2,... until either k i = k i+1 or the approximation is considered adequate. Current state-of-the-art: sufficient condition for termination: finite ascending chains if condition does not hold: widening and narrowing.

21 Kleenean program analysis is slow Set interpretations: Kleene iteration never terminates if µf is an infinite set. X = a X + b µf = a b Kleene approximants are finite sets: k i = (ɛ + a a i )b Probabilistic interpretation: convergence can be very slow [EY STACS05]. X = 1 2 X µf = 1 = Logarithmic convergence : k iterations to get log k bits of accuracy. k n 1 1 n + 1 k 2000 =

22 Kleene Iteration for X = f (X) (univariate case) µf f (X)

23 Kleene Iteration for X = f (X) (univariate case) µf f (X)

24 Kleene Iteration for X = f (X) (univariate case) µf f (X)

25 Kleene Iteration for X = f (X) (univariate case) µf f (X)

26 Kleene Iteration for X = f (X) (univariate case) µf f (X)

27 Kleene Iteration for X = f (X) (univariate case) µf f (X)

28 Kleene Iteration for X = f (X) (univariate case) µf f (X)

29 Kleene Iteration for X = f (X) (univariate case) µf f (X)

30 Newton s Method for X = f (X) (univariate case) µf f (X)

31 Newton s Method for X = f (X) (univariate case) µf f (X)

32 Newton s Method for X = f (X) (univariate case) µf f (X)

33 Newton s Method for X = f (X) (univariate case) µf f (X)

34 Newton s Method for X = f (X) (univariate case) µf f (X)

35 Newton s Method for X = f (X) (univariate case) µf f (X)

36 Evaluation of Newton s method Newton s Method is usually very efficient often exponential convergence... but not robust: may not converge, or may converge only locally (in some neighborhood of the least fixed-point), or may converge very slowly.

37 A puzzling mismatch Program analysis: General domain: arbitrary ω-continuous semirings Kleene Iteration is robust and generally applicable... but converges slowly. Numerical mathematics: Particular domain: the real field Newton s Method converges fast... but is not robust

38 Two questions Can Newton s Method be generalized to arbitrary ω-continuous semirings? Is Newton s method robust when restricted to the real semiring?

39 Mathematical formulation of Newton s Method Let ν be some approximation of µf. (We start with ν = f (0).) Compute the function T ν (X) describing the tangent to f (X) at ν Solve X = T ν (X) new approximation (instead of X = f (X)), and take the solution as the Elementary analysis: T ν (X) = Df ν (X) + f (ν) ν where Df x0 (X) is the differential of f at x 0 So: ν 0 = 0 ν i+1 = ν i + i i solution of X = Df νi (X) + f (ν i ) ν i

40 Generalizing Newton s method Key point: generalize X = Df ν (X) + f (ν) ν In an arbitrary ω-continuous semiring neither the differential Df ν (X), nor the difference f (ν) ν are defined.

41 Differentials in semirings Standard solution: take the algebraic definition Df (X) = 0 if f (X) = c X if f (X) = X Dg(X) + Dh(X) if f (X) = g(x) + h(x) Dg(X) h(x) + g(x) Dh(X) if f (X) = g(x) h(x) Df (X) if f (X) = f i (X). i I i I

42 The difference f (ν i ) ν i Solution: Replace f (ν i ) ν i by any δ i such that f (ν i ) = ν i + δ i ν i+1 = ν i + i where i solution of X = Df νi (X) + δ i But does δ i always exist? Proposition: Yes But ν i+i depends on your choice of δ i! Theorem: No, it doesn t Can t you give a closed form for ν i+1? Proposition: Yes The least solution of X = Df νi (X) + δ i is Df ν i (δ i ):= and so: ν i+1 = ν i + Df ν i (δ i ) j=0 Df j ν i (δ i )

43 The difference f (ν i ) ν i Solution: Replace f (ν i ) ν i by any δ i such that f (ν i ) = ν i + δ i ν i+1 = ν i + i where i solution of X = Df νi (X) + δ i But does δ i always exist? Proposition: Yes But ν i+i depends on your choice of δ i! Theorem: No, it doesn t Can t you give a closed form for ν i+1? Proposition: Yes The least solution of X = Df νi (X) + δ i is Df ν i (δ i ):= and so: ν i+1 = ν i + Df ν i (δ i ) j=0 Df j ν i (δ i )

44 The difference f (ν i ) ν i Solution: Replace f (ν i ) ν i by any δ i such that f (ν i ) = ν i + δ i ν i+1 = ν i + i where i solution of X = Df νi (X) + δ i But does δ i always exist? Proposition: Yes But ν i+i depends on your choice of δ i! Theorem: No, it doesn t Can t you give a closed form for ν i+1? Proposition: Yes The least solution of X = Df νi (X) + δ i is Df ν i (δ i ):= and so: ν i+1 = ν i + Df ν i (δ i ) j=0 Df j ν i (δ i )

45 The difference f (ν i ) ν i Solution: Replace f (ν i ) ν i by any δ i such that f (ν i ) = ν i + δ i ν i+1 = ν i + i where i solution of X = Df νi (X) + δ i But does δ i always exist? Proposition: Yes But ν i+i depends on your choice of δ i! Theorem: No, it doesn t Can t you give a closed form for ν i+1? Proposition: Yes The least solution of X = Df νi (X) + δ i is Df ν i (δ i ):= and so: ν i+1 = ν i + Df ν i (δ i ) j=0 Df j ν i (δ i )

46 The difference f (ν i ) ν i Solution: Replace f (ν i ) ν i by any δ i such that f (ν i ) = ν i + δ i ν i+1 = ν i + i where i solution of X = Df νi (X) + δ i But does δ i always exist? Proposition: Yes But ν i+i depends on your choice of δ i! Theorem: No, it doesn t Can t you give a closed form for ν i+1? Proposition: Yes The least solution of X = Df νi (X) + δ i is Df ν i (δ i ):= and so: ν i+1 = ν i + Df ν i (δ i ) j=0 Df j ν i (δ i )

47 The difference f (ν i ) ν i Solution: Replace f (ν i ) ν i by any δ i such that f (ν i ) = ν i + δ i ν i+1 = ν i + i where i solution of X = Df νi (X) + δ i But does δ i always exist? Proposition: Yes But ν i+i depends on your choice of δ i! Theorem: No, it doesn t Can t you give a closed form for ν i+1? Proposition: Yes The least solution of X = Df νi (X) + δ i is Df ν i (δ i ):= and so: ν i+1 = ν i + Df ν i (δ i ) j=0 Df j ν i (δ i )

48 The difference f (ν i ) ν i Solution: Replace f (ν i ) ν i by any δ i such that f (ν i ) = ν i + δ i ν i+1 = ν i + i where i solution of X = Df νi (X) + δ i But does δ i always exist? Proposition: Yes But ν i+i depends on your choice of δ i! Theorem: No, it doesn t Can t you give a closed form for ν i+1? Proposition: Yes The least solution of X = Df νi (X) + δ i is Df ν i (δ i ):= and so: ν i+1 = ν i + Df ν i (δ i ) j=0 Df j ν i (δ i )

49 The difference f (ν i ) ν i Solution: Replace f (ν i ) ν i by any δ i such that f (ν i ) = ν i + δ i ν i+1 = ν i + i where i solution of X = Df νi (X) + δ i But does δ i always exist? Proposition: Yes But ν i+i depends on your choice of δ i! Theorem: No, it doesn t Can t you give a closed form for ν i+1? Proposition: Yes The least solution of X = Df νi (X) + δ i is Df ν i (δ i ):= and so: ν i+1 = ν i + Df ν i (δ i ) j=0 Df j ν i (δ i )

50 Theorem [EKL DLT07]: Let X = f (X) be an equation over an arbitrary ω-continuous semiring. The sequence ν 0 = f (0) ν i+1 = ν i + Df ν i (δ i ) where δ i satisfies f (ν i ) = ν i + δ i exists, is unique and satisfies for every i 0. k i ν i µf

51 Extensions and simplifications Systems of equations: ν i, i, δ i become vectors (elements of S n ) The differential becomes a function S n S n Geometric intuition: Df νi (X 1,..., X n ) is the hyperplane tangent to f at the (n-dimensional) point ν i Commutative semirings (and left-linear equations): One variable: Df ν (X) = f (ν) X, and so ν i+1 = ν i + f (νi ) δ i Many variables: Df ν (X) = J(ν) X, where J(ν) is the Jacobi matrix of partial derivatives evaluated at ν, and so ν i+1 = ν i + J (ν i ) δ i

52 Newton s method for language equations Language semiring: Universe is 2 A, + is union, is concatenation. For left-linear systems of equations, Newton s method terminates after 1 iteration: X 1 = a X 1 + b X 2 X 2 = a X 1 + b X ν 0 = ν 1 = = ( ( ( ) 0 1 ) ( 0 a b + 1 a b (a + bba ) b (b + aa b) ) ( ) 0 1 ) = ( (a + bb a) (a + bba ) b (b + aa b) a (b + aa b) )

53 A nonlinear equation X = a X X + b f (X) = a X X + b Df ν (X) = a ν X + a X ν ν 0 = b ν 0 + δ 0 = f (ν 0 ) = δ 0 := abb ν 1 = ν 0 + Df b (δ 0) = b + Df b (abb) = b + (X + abx + axb + abaxb +...)(abb) = b + abb + ababb + aabbb + abaabbb +... ν 2 = The method does not terminate. Can we characterize the approximants?

54 Finite-index approximations System X = f (X) induces context-free grammar G def = X f (X). [Ginsburg, Spanier, Salomaa, Gruska, Yntema 67-71]: A word w L(G) has index k if there is a derivation S w 1 w 2... w n w such that each of S, w 1,..., w n contains at most k occurrences of non-terminals (and one of them contains k non-terminals). Example: X = a X X + b b has index 1 (ab) i b has index 2 aabbabb has index 3 X b X axx abx (ab) i X (ab) i b X aaxxx aabbabb

55 Theorem [EKL DLT 07]: Let X = f (X) be a system of language equations, and let G be the derived context-free grammar. For every i 0: ν i = L i+1 (G) We can easily construct grammars G i such that L(G i+1 ) = ν i X = a X X + b G = {X axx b} G 0 = {X 0 b} G 1 = G 0 {X 1 ax 1 X 0 ax 0 X 1 ax 0 X 0 + b} G i+1 = G i {X i+1 ax i+1 X i ax i X i+1 ax i X i + b} Newton s method approximates a context-free grammar by context-free grammars of finite index.

56 Visualizing finite index: Secondary structure of RNA (image by Bassi, Costa, Michel;

57 An stochastic context-free grammar [ ]: Model the distribution of secondary structures as the derivation trees of the following stochastic context-free grammar: L CL L C S psp S CL C s C psp Graphical interpretation: sssppsssspsssssspssppsssspssssspss s s s s s s s s s s s p p s s p p p p s s sss s s s s s s p p ss

58 Visualizing the index of a derivation Index = maximal number of branching points from root to leaf + 1

59 Visualizing the index of a derivation Index = maximal number of branching points from root to leaf + 1

60 Visualizing the index of a derivation Index = maximal number of branching points from root to leaf + 1

61 Grammar leads to two equation systems: L = C L + C S = p S p + C L C = s + p S p ˆL = Ĉ ˆL Ĉ Ŝ = Ŝ Ĉ ˆL Ĉ = Ŝ ν 0 (L) = der. of index 1 ν 1 (L) = der. of index 2 ν 2 (L) = der. of index 3 ν 3 (L) = der. of index 4 ν 4 (L) = der. of index 5 ν 5 (L) = der. of index 6 ˆν 0 (L) = ˆν 1 (L) = ˆν 2 (L) = ˆν 3 (L) = ˆν 4 (L) = ˆν 5 (L) =

62 Idempotent and commutative semirings Theorem [Hopkins-Kozen LICS 99]: The least fixed point of a system X = f (X) of n equations over an ω-continuous idempotent and commutative semiring is reached by the sequence ν 0 = f (0) ν i+1 = J(ν i ) f (ν i ) after at most O(3 n ) iterations. Theorem [EKL STACS 07]: This is exactly Newton s sequence. Moreover, the fixed point is reached after at most n iterations.

63 Idempotent and commutative semirings Theorem [Hopkins-Kozen LICS 99]: The least fixed point of a system X = f (X) of n equations over an ω-continuous idempotent and commutative semiring is reached by the sequence ν 0 = f (0) ν i+1 = J(ν i ) f (ν i ) after at most O(3 n ) iterations. Theorem [EKL STACS 07]: This is exactly Newton s sequence. Moreover, the fixed point is reached after at most n iterations.

64 Idempotent and commutative semirings Theorem [Hopkins-Kozen LICS 99]: The least fixed point of a system X = f (X) of n equations over an ω-continuous idempotent and commutative semiring is reached by the sequence ν 0 = f (0) ν i+1 = J(ν i ) f (ν i ) after at most O(3 n ) iterations. Theorem [EKL STACS 07]: This is exactly Newton s sequence. Moreover, the fixed point is reached after at most n iterations.

65 Idempotent and commutative semirings Theorem [Hopkins-Kozen LICS 99]: The least fixed point of a system X = f (X) of n equations over an ω-continuous idempotent and commutative semiring is reached by the sequence ν 0 = f (0) ν i+1 = J(ν i ) f (ν i ) after at most O(3 n ) iterations. Theorem [EKL STACS 07]: This is exactly Newton s sequence. Moreover, the fixed point is reached after at most n iterations.

66 Idempotent and commutative semirings Theorem [Hopkins-Kozen LICS 99]: The least fixed point of a system X = f (X) of n equations over an ω-continuous idempotent and commutative semiring is reached by the sequence ν 0 = f (0) ν i+1 = J(ν i ) f (ν i ) after at most O(3 n ) iterations. Theorem [EKL STACS 07]: This is exactly Newton s sequence. Moreover, the fixed point is reached after at most n iterations.

67 An example The Newton sequence terminates for all idempotent and commutative analyses, the Kleene sequence does not. X = a X X + b f (X) = a X + a X = a X For one equation: µf = ν 1 = f (ν 0 ) ν 0 We obtain: ν 0 = b ν 1 = (ab) b

68 This result provides a computational version of Parikh s theorem: [Hopkins, Kozen LICS 99], [Aceto, Esik, Ingólfsdottir ITA 02] The regular language (a b) b has the same Parikh image ( counting semantics ) as the context-free language generated by the grammar X axx b

69 Our two questions Can Newton s Method be generalized to arbitrary ω-continuous semirings? Is Newton s method robust when restricted to the real semiring?

70 Newton s method on the real semiring On the real field Newton s method may not converge, or converge only locally On the real semiring these problems disappear [EKL TCS 08]: Newton s method always converges [EY STACS 05] It always exhibits linear or exponential convergence [EKL STOC 07] For strongly connected systems there is a threshold k such that after k iterations each subsequent iteration gains at least one bit of accuracy [EKL STACS 08] For important classes the threshold is linear in the size of the system [EKL STACS 08].

71 Conclusions Newton did it all but never saw Iceland... and I did!

72 Conclusions Newton did it all but never saw Iceland... and I did!

73 Conclusions Newton did it all but never saw Iceland... and I did!