Abstract Acceleration of General Linear Loops

Transcription

1 Abstract Acceleration of General Linear Loops Bertrand Jeannet, Peter Schrammel, Sriram Sankaranarayanan Principles of Programming Languages, POPL 14 San Diego, CA

2 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs).

3 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs). Challenge Linear loops generate non-linear trajectories.

4 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs). Challenge Linear loops generate non-linear trajectories. while(3*y+x<=10){ x = x+1; y = y+x; }

5 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs). Challenge Linear loops generate non-linear trajectories. while(y<=3){ x = 1.5*x; y = y+1; }

6 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs). Challenge Linear loops generate non-linear trajectories. while(y<=10){ } x = 1.1*x *y; y = 0.05*x + 1.1*y;

7 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs). Challenge Linear loops generate non-linear trajectories.

8 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs). Challenge Linear loops generate non-linear trajectories.

9 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs). Challenge Linear loops generate non-linear trajectories.

10 Motivation and Challenge Motivation Inferring polyhedral invariants on programs containing non-trivial numerical loops (e.g. control programs). Challenge Linear loops generate non-linear trajectories. Goal Compute convex polyhedra containing all such trajectories.

11 Existing Techniques Standard polyhedral analysis (Cousot&Halbwachs 1978) Compute iteratively (α τ γ) (X) Interproc,... Not precise Thresholds, landmarks, abstract acceleration,... improve precision for linear behavior

12 Existing Techniques Standard polyhedral analysis (Cousot&Halbwachs 1978) Compute iteratively (α τ γ) (X) Interproc,... Not precise Thresholds, landmarks, abstract acceleration,... improve precision for linear behavior Non-linear constraint solving (Colon et al 2003, Sankaranarayanan et al 2004) Solving directly X = τ(x) over convex polyhedra X Relaxation and reduction to non-linear constraint solving problem Sting

13 Existing Techniques Standard polyhedral analysis (Cousot&Halbwachs 1978) Compute iteratively (α τ γ) (X) Interproc,... Not precise Thresholds, landmarks, abstract acceleration,... improve precision for linear behavior Non-linear constraint solving (Colon et al 2003, Sankaranarayanan et al 2004) Solving directly X = τ(x) over convex polyhedra X Relaxation and reduction to non-linear constraint solving problem Sting Ellipsoid methods (Kurzhanski&Varaiya 2000, Feret 2004,...) Restricted to stable loops Astrée

14 Existing Techniques Standard polyhedral analysis (Cousot&Halbwachs 1978) Compute iteratively (α τ γ) (X) Interproc,... Not precise Thresholds, landmarks, abstract acceleration,... improve precision for linear behavior Non-linear constraint solving (Colon et al 2003, Sankaranarayanan et al 2004) Solving directly X = τ(x) over convex polyhedra X Relaxation and reduction to non-linear constraint solving problem Sting Ellipsoid methods (Kurzhanski&Varaiya 2000, Feret 2004,...) Restricted to stable loops Astrée...

15 Thermostat assume(temp ext =14 && 16 temp 17); while(1) { time = 0; while(temp 22.0) { temp = (15.0*temp - temp ext )/16.0; time++; } time = 0; while(temp 18.0) { temp = (15.0*temp - temp ext )/16.0; time++; } } heater on heater off

16 Thermostat temp Interproc time

17 Thermostat temp Astrée Interproc time

18 Thermostat temp Sting Astrée 14 Interproc time

19 Thermostat temp Sting our approach Astrée Interproc time

20 Existing Techniques Standard polyhedral analysis (Cousot&Halbwachs 1978) Compute iteratively (α τ γ) (X) Not precise Interproc,... Thresholds, landmarks, abstract acceleration,... improve precision for linear behavior Non-linear constraint All these solving methods (Colon alcompute 2003, Sankaranarayanan et al 2004) Solving directly X = τ(x) inductive overinvariants convex polyhedra X Relaxation and reduction in abstractodomains non-linear tooconstraint weak solving problemto express precise invariants of linear systems. Sting Ellipsoid methods (Kurzhanski&Varaiya 2000, Feret 2004,...) Restricted to stable loops Astrée...

21 Abstract Acceleration of Linear Loops (Gonnord&Halbwachs 2006) Approach Given a linear loop τ = ( G x h x = A x + b ) provide an operator s.t. τ α τ γ in the convex polyhedra domain.

22 Abstract Acceleration of Linear Loops (Gonnord&Halbwachs 2006) Approach Given a linear loop τ = ( G x h x = A x + b ) provide an operator s.t. τ α τ γ (vs. Kleene: (α τ γ) ) in the convex polyhedra domain.

23 Abstract Acceleration of Linear Loops (Gonnord&Halbwachs 2006) Approach Given a linear loop τ = ( G x h x = A x + b ) provide an operator s.t. τ α τ γ (vs. Kleene: (α τ γ) ) in the convex polyhedra domain. Then, given a polyhedron X τ (X) is a precise over-approximation of τ (X)

24 Contribution Abstract acceleration of any linear loop τ Integrates smoothly into a static analyzer

25 Contribution Abstract acceleration of any linear loop τ Integrates smoothly into a static analyzer

26 Our Approach Loops without guard convex polyhedron X true x = A x template polyhedron A A = {A k k 0} AX }{{} A X = {A k x k 0, x X} convex polyhedron

27 Loop without Guard: Example Example x=x0; y=y0; while true do x=1.5*x; y=y+1; done

28 Loop without Guard: Example Example x=x0; y=y0; while true do x=1.5*x; y=y+1; done Homogeneization: we introduce ξ for constant coefficient x=x0; y=y0; ξ=1; while true do x=1.5*x; y=y+ξ; done

29 Loop without Guard: Example Example x=x0; y=y0; while true do x=1.5*x; y=y+1; done Homogeneization: we introduce ξ for constant coefficient x=x0; y=y0; ξ=1; while true do x=1.5*x; y=y+ξ; done x x Loop: true y = A y with A = ξ ξ 0 0 1

30 Loop without Guard: Example Approximating A 1.5 k 0 0 A k = 0 1 k 0 0 1

31 Loop without Guard: Example Approximating A 1.5 k 0 0 A k = 0 1 k We look for a template polyhedron ϕ A (m 1, m 2 ) such that m A = 0 1 m 2 ϕ A (m 1, m 2 ) A

32 Loop without Guard: Example Approximating A 1.5 k 0 0 A k = 0 1 k We look for a template polyhedron ϕ A (m 1, m 2 ) such that m A = 0 1 m 2 ϕ A (m 1, m 2 ) A Let s consider octagonal templates over m 1 and m 2 : m m ϕ A = m 2 0 ϕ A 1 m 1 m m

33 Loop without Guard: Example Approximating AX with a convex polyhedron Properly multiplying generators (vertices & rays) of A by generators of X

34 Loop without Guard: Example Approximating AX with a convex polyhedron Properly multiplying generators (vertices & rays) of A by generators of X Result on our example: y X x

35 Loop Without Guard: General Case convex polyhedron X true x = A x template polyhedron A A = {A k k 0} AX }{{} A X = {A k x k 0, x X} convex polyhedron

36 Loop Without Guard: General Case convex polyhedron X true x = A x template polyhedron (1) A A = {A k k 0} (2) AX }{{} A X = {A k x k 0, x X} convex polyhedron

37 (1) Over-approximation of A by A (a) Closed-Form Expression of A k real Jordan normal form A = Q 1 JQ

38 (1) Over-approximation of A by A (a) Closed-Form Expression of A k real Jordan normal form A k = Q 1 J k Q

39 (1) Over-approximation of A by A (a) Closed-Form Expression of A k real Jordan normal form A k = Q 1 J k Q coefficients of the form ( k ) p λ k p cos((k p)θ r π 2 )

40 (1) Over-approximation of A by A (a) Closed-Form Expression of A k real Jordan normal form at most n distinct coefficients A k = Q 1 J k Q coefficients of the form ) λ k p cos((k p)θ r π 2 ) ( k p

41 (1) Over-approximation of A by A (a) Closed-Form Expression of A k real Jordan normal form at most n distinct coefficients A k = Q 1 J k Q coefficients of the form ) λ k p cos((k p)θ r π 2 ) ( k p (b) Over-approximation of J by a template polyhedron J vector of coefficients of J k { m l i e i ( m) u i }

42 (1) Over-approximation of A by A (a) Closed-Form Expression of A k real Jordan normal form at most n distinct coefficients A k = Q 1 J k Q coefficients of the form ) λ k p cos((k p)θ r π 2 ) ( k p (b) Over-approximation of J by a template polyhedron J vector of template expressions coefficients of J k (e.g. logahedra) { m l i e i ( m) u i }

43 (1) Over-approximation of A by A (a) Closed-Form Expression of A k real Jordan normal form at most n distinct coefficients A k = Q 1 J k Q coefficients of the form ) λ k p cos((k p)θ r π 2 ) ( k p (b) Over-approximation of J by a template polyhedron J vector of coefficients of J k { m l i e i ( m) u i } template expressions (e.g. logahedra) lower and upper bounds computed by asymptotic analysis techniques

44 (1) Over-approximation of A by A (a) Closed-Form Expression of A k real Jordan normal form at most n distinct coefficients A k = Q 1 J k Q coefficients of the form ) λ k p cos((k p)θ r π 2 ) ( k p (b) Over-approximation of J by a template polyhedron J vector of coefficients of J k (c) A = Q 1 J Q { m l i e i ( m) u i } template expressions (e.g. logahedra) lower and upper bounds computed by asymptotic analysis techniques

45 (2) Computing AX AX = {A x A A x X}

46 (2) Computing AX AX = {A x A A x X} 1 In general: neither convex nor a polyhedron! 0 0 1

47 (2) Computing AX AX = {A x A A x X} 1 In general: neither convex nor a polyhedron! Over-approximate AX AX ConvexSpan(V 1 V }{{ 2, V } 1 R 2 R 1 V 2 R 1 R 2 ) }{{} vertices rays

48 (2) Computing AX AX = {A x A A x X} 1 In general: neither convex nor a polyhedron! Over-approximate AX AX ConvexSpan(V 1 V }{{ 2, V } 1 R 2 R 1 V 2 R 1 R 2 ) }{{} vertices rays Best convex over-approximation if R 1 = R 2 =

49 Loop without Guard: Example 2 τ = true x 0.8 cos π sin π 6 0 = 0.8 sin π cos π 6 0 x m 1 m 2 0 A A = m 2 m 1 0 ϕ A (m 1, m 2 ) τ (X) = AX With octagonal template: 1 m 2 3 y m 1 1 X x

50 Loop with Guard convex polyhedron X G x 0 x = A x maximum number of iterations template polyhedron with N = argmin k 0 GA k x > 0 A [0,N] A [0,N] = {A k 0 k N} A [0,N] X A [0,N] X }{{} convex polyhedron

51 Loop with Guard: Example Program We want to compute a loop invariant of x=x0; y=y0; while y<=3 do x = 1.5*x; y = y+1; done with X(x 0, y 0 ) = x 0 [1, 3] y 0 [0, 2] Computing the maximum number of iterations y 0 [0, 2] y =y+1 y 3 = max nb. of iterations N = 4

52 Loop with Guard: Example Exploiting the maximum number of iterations We compute X τ ( A [0,3] (X G) ) }{{} y m x A [0,3] A [0, ] m 1 y 3

53 Influence of Template for Approximating A Program x=x0; y=y0; while x+2*y <= 100 do x = 1.5*x; y = y+1; done with X(x 0, y 0 ) = x 0 [1, 3] y 0 [0, 2] Result y X x+2y template expressions 8 template expressions N = x

54 Implementation and Experiments Implementation Based on Apron for polyhedral computations Sage for computation of Jordan normal form Loops with conditionals transformed into multiple self-loops Acceleration of inner loops, widening for outer loops Benchmarks Small filters (single loops) Forced oscillator with stable and unstable switching modes (nested loops) Thermostat system (nested loops) Car convoy car2 car1 car0 x 2 x 1 x 0 for all i [1, N 1] : ẍ i = c(ẋ i 1 ẋ i )+d(x i 1 x i 50)

55 Experimental Results: Inferred Bounds Percentage of inferred bounds that are finite 100% 80% 60% 40% * * * * * * * INTERPROC STING Our Approach 20% * * * * * 0% * * * Benchmarks *

56 Experimental Results: Analysis Time Analysis time 10 5 timed out * * * * * * * * * * * INTERPROC STING Our Approach * * * * * Benchmarks

57 Summary and Prospects Contribution Precise polyhedral overapproximation of general linear loops Matrix abstract domain Precise loop bound estimation Integrates smoothly into static analyzer

58 Summary and Prospects Contribution Precise polyhedral overapproximation of general linear loops Matrix abstract domain Precise loop bound estimation Integrates smoothly into static analyzer Rationale of the Approach Use non-linear deductions within polyhedra analysis

59 Summary and Prospects Contribution Precise polyhedral overapproximation of general linear loops Matrix abstract domain Precise loop bound estimation Integrates smoothly into static analyzer Rationale of the Approach Use non-linear deductions within polyhedra analysis Ongoing Work Open systems (with inputs) Applications Hybrid systems with linear differential equations Stability/termination analysis