Basic Algorithms for Periodic-Linear Inequalities and Integer Polyhedra

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1 Basic Algorithms for Periodic-Linear Inequalities and Integer Polyhedra Alain Ketterlin / Camus IMPACT 2018: January, 23, 2018

2 Motivation Periodic-Linear Inequalities The Omicron Test Decomposition

3 Motivation / Polyhedra and Integers 1 Omega s nightmare 3 11x + 13y x 9y 6 LS(N) 2 3y x 5 1 N 2y x 1

4 Motivation / Polyhedra and Integers 1 Omega s nightmare 3 11x + 13y x 9y 6 LS(N) 2 3y x 5 1 N 2y x 1 empty (no integer points) non-empty rational projections holes fuzzy vertices periodic y-span

5 Motivation / Incomplete Variable Elimination 2 O Omega s nightmare 11x y 11x x 6 9y 7x + 8 Fourier-Motzkin variable elimination (of y) 9 ( 11x + 3) 13 (7x + 8) 13 (7x 6) 9 ( 11x + 21) x 267

6 Motivation / Incomplete Variable Elimination 2 O Omega s nightmare 11x y 11x x 6 9y 7x + 8 Fourier-Motzkin variable elimination (of y) 9 ( 11x + 3) 13 (7x + 8) 13 (7x 6) 9 ( 11x + 21) x 267 What went wrong? Loose bounds... A tight bound would be: 9y 7x + 8 (7x + 8) mod 9 Combinations accumulate and amplify the slack 77 + [ 9(12 (2 11x) mod 13) + 13((7x + 8) mod 9) ] 190x } {{ } up to = 212

7 Motivation / Imprecise Bound Comparisons 3? LS(N) x + 2 3y x + 5 x + 1 N 2y x + 1 for 1 <= x <= 3N+7 for max( x+2 x+1 N 3, 2 ) <= y <= min( x+5 x+1 3, 2 ) exec S(x,y)

8 Motivation / Imprecise Bound Comparisons 3? LS(N) x + 2 3y x + 5 x + 1 N 2y x + 1 Which (upper) bound is effective where? 3y = x y = x

9 Motivation / Imprecise Bound Comparisons 3? LS(N) x + 2 3y x + 5 x + 1 N 2y x + 1 Which (upper) bound is effective where? 4 2y = x

10 Motivation / Imprecise Bound Comparisons 3? LS(N) x + 2 3y x + 5 x + 1 N 2y x + 1 Which (upper) bound is effective where? 3y = x

11 Motivation / Imprecise Bound Comparisons 3? LS(N) x + 2 3y x + 5 x + 1 N 2y x + 1 Which (upper) bound is effective where? 3y = x y = x < < < = < = = = = > = > > >

12 Motivation / Imprecise Bound Comparisons 3? LS(N) x + 2 3y x + 5 x + 1 N 2y x + 1 Which (upper) bound is effective where? 3y = x y = x (x + 1 (x + 1) mod 2) 2 (x + 5 (x + 5) mod 3)

13 Motivation / What Problem are we Trying to Solve? 4 1. Tightening bounds find a workable modulo representation define precise combinations 2. The Omicron Test Fourier-Motzkin-like decision procedure correct and complete 3. Polyhedron Decomposition via affine unswitching applications: transformation, projection, optimization Strategy Be Radically Integral no or, no inexact division no loose bound! Focus on Representation, not on Algorithms find a set of elementary operations algorithms should simply repeat while possible

14 Motivation Periodic-Linear Inequalities The Omicron Test Decomposition

15 Periodic-Linear Inequalities / Periodic Numbers 5 A periodic number is a collection of numbers indexed by the congruence class of an expression: v0, v 1,..., v π 1 π x = v 0 v 1. v π 1 if x 0 mod π if x 1 mod π Essentially a notation, with useful operations: Rotation Division Distribution Separation v 0, v 1,... π x+1 = v 1,..., v 0 π x if x (π 1) mod π v 0,... π cx = i..., v (ci mod π ),... π / gcd(π,c) x v 0,... α w 0,...,w β 1 β x = i..., v 0,... α w i,... β v 0,..., v α 1 α x+y = i..., v 0,..., v α 1 α i+y,... α x x

16 Periodic-Linear Inequalities / Modulos 6 Modulos: for any expression X X mod π = 0, 1,..., π 1 X π The maximal multiple of π less than or equal to X X X mod π = X 0, 1,..., π 1 X π Tightened linear bounds have: a sharp linear part a periodic correction 9y 7x + 8 (7x + 8) mod 9 7x + 8 0,..., 8 9 7x+8 7x + 8 8, 6, 4, 2, 0, 7, 5, 3, 1 9 x Works for any number of variables: 9y = 7x + 8 v 0, v 1, v 2 3 2x+6y+5z 1 = v2, v 1, v 0 3 x 1 y, v 1, v 0, v 2 3 x 1 y, v 0, v 2, v 1 3 x 1 y 3 z

17 Periodic-Linear Inequalities / Normal Forms 7 Periodic-linear expressions (PLEs) in normal form over [x 1,..., x n ]: n a i x i +, i=1 π 1 x 1 π πn n 1, x n 1 x n or in simplified normal form a n x n +, a n 1 x n 1 + π n 1 x n 1, π n x n If X and Y are PLEs, n an integer, then: nx, (X + Y ), (X mod π ), X[Y/x k ] are all PLEs Periodic-linear inequalities are PLEs compared to zero: a n x n + X 0,... π n x n 0 with X 0,... PLEs over [x 1,..., x n 1 ]

18 Periodic-Linear Inequalities / Linearity and Periodicity 8 LS(N) over [N, x, y] with tightened inequalities: x + 3, 2, 4 x 3y x + 3, 5, 4 x x N + 2, 1 N, 1, 2 N x 2y x + 0, 1 x and over [N, x] after combination 2, 1 N, 0, 1 N x N 0 0 3, 0, 3 x 0, 3 N, 7, 4 6, 1, 8, 3, 4, 5 x x x 3N + N, 2, 5 N, 3, 0 N, 4, 7 N, 5, 2 N Categories: linear : 3y 1 x + 3, 5, 4 x y periodic : 2, 1 N, 0, 1 N x N 0x mixed : 6, 1, 8, 3, 4, 5 x x 6 x

19 Periodic-Linear Inequalities / Tightening 9 Given a (potentially loose) periodic-linear inequality over [..., x n ]: a n x n X 0,... π n x n or X 0,... π n x n a n x n the following inequality is an equivalent tight bound a n x n i..., X i 0, 1,..., a n π n 1 a n π πn n,... X i ia n x n the rhs is a multiple of a n for all phases of x n modulo π n (and all phases of the other variables)

20 Periodic-Linear Inequalities / Periodicity and Disjunction 10 Mixed tight bounds are fuzzy x + 3, 2, 4 x 3y 6, 1, 8, 3, 4, 5 x x 2y x + 0, 1 x Disjoin turns a mixed bound into a disjunction of linear bounds: it computes a major bound plus outliers 6, 1, 8, 3, 4, 5 x x (x = 1) (3 x) 6 0, 3 N, 7, 4 x 3N + N, 2, 5 N, (x 3N + 5) (x = 3N + 7) 3, 0 N, 4, 7 N, 5, 2 N x Omega s nightmare (left corner) 11x x 13y 9y 7x x 117, 73, 29, 15,..., 29, x 190x 117, 1, 2, 3,..., 115, x x 1 x O 1 117

21 Motivation Periodic-Linear Inequalities The Omicron Test Decomposition

22 The Omicron Test / Correct and Complete Decision 11 Fourier-Motzkin elimination on Q relies on an equivalence: f l (x 1,..., x n 1 ) ax n bx n f u (x 1,..., x n 1 ) Z, Q Q b f l (x 1,..., x n 1 ) a f u (x 1,..., x n 1 ) Restoring completeness on Z by tightening loose bounds tight bounds combination (4 3x) L ax bx U (3x 5) (6 3x) L ax bx U (3x 3) bl au (6 3) keep bounds tight at all times

23 The Omicron Test / Inequality Maintenance 12 Combining mixed bounds may not eliminate the variable 0, 3 N, 7, 4 6, 1, 8, 3, 4, 5 x x x 3N + N, 2, 5 N, 3, 0 N, 4, 7 N, 5, 2 N 6 2, 1 N, 2, 1 N, 2, 1 N, N 0, 1 N, 0, 1 N, 0, 1 N x apply Disjoin, and fork the system (if needed) There is no way to combine periodic bounds { 2, 1 N, 0, 1 N 2 x = 2x x N + 0 2, 1 N N x = 2x + 1 0, 1 N N splinter the system and change variables 6 x

24 The Omicron Test / On Omega s Nightmare x 13y 21 11x 7x 6 9y 7x + 8 tighten+combine 117, 73, 29,..., x 190x 190x 117, 73, , x? tighten+disjoin 1 x 0 combine false

25 The Omicron Test / On Omega s Nightmare x 13y 21 11x 7x 6 9y 7x + 8 tighten+combine 117, 73, 29,..., x 190x 190x 117, 73, , x? tighten+disjoin alternative: splintering by x 0 combine x 117 tighten+combine false false (115 more) x tighten+combine false

26 The Omicron Test / Projection? 14 On Q, Fourier-Motzkin Elimination can be used for projection Omicron can as well, but produces a disjoint union. x + 2 3y x + 5 x N + 1 2y x + 1 (x even) (x odd) 4 2x + 0 3N + 4, 7 N 1 N 1 2x + 1 3N + 7, 4 N 0 N A decomposition is a partition of a polyhedron such that, in each part: each variable has a contiguous non-empty range with elementary bounds (no min / max)

27 Motivation Periodic-Linear Inequalities The Omicron Test Decomposition

28 Decomposition / Polyhedra and ASTs 15 To keep a collection of (disjoint) polyhedra: an AST if condition then statements [else statements] arbitrary logical combinations all inequalities properly tightened no mixed inequalities (thanks to Disjoin) exec label for/when PLE <= scale counter <= PLE scale used only to keep bounds tight, e.g., statements for 2x + [x:6,4,8] <= 6y <= 3x + [x:0,3]... The AST keeps a layer for each variable. On LS(N), start with when _ <= N <= _ for _ <= x <= _ for _ <= y <= _ if 3y <= x+[x:3,5,4] and... then exec S

29 Decomposition / Affine Unswitching 16 Starting from an inequality and its innermost enclosing loop for/when L sx n U... ax n X... Affine unswitching produces: al if sx < al then [ ] for L sx n U do sx... ax n X... // = false else if sx < au then al for al asx n sx do [ ]... ax n X... // = true sx for s(x + a) asx n au do... ax n X... // = false al else // sx au [ for L sx n U do... ax n X... // = true au ] au ] au ] ] sx

30 Decomposition / Hoisting Inequalities 17 Periodic inequalities need special treatment: X 0, X 1,... π n x n 0 is viewed as X 0 0, X 1 0,... π n x n then individual inner inequalities are hoisted, eventually leaving a periodic boolean: b 0, b 1,... π n x n with b i {true, false} At this point, the for-range on x n is unrolled by a factor π n

31 Decomposition / On LS(N) 18 when N = 0 exec S(1,1); exec S(3,2); exec S(5,3); exec S(7,4) when N = 1 [...] when N = 2 [...] when N = 3 [...] when N = 4 [...] when 5 <= N <= _ exec S(1,1) for 3 <= x <= 8 10 for 2x+[x:6,4,8] <= 6y <= 3x+[x:0,3] exec S(x,y) exec S(9,4) for 4 <= y <= 5 exec S(10,y) for 11 <= x <= 3N-3 for x+[x:3,2,4] <= 3y <= x+[x:3,5,4] exec S(x,y) for N <= y <= N+1 exec S(3N-2,y) exec S(3N-1,N+1) for 3N <= x <= 3N+5 for 3x-3N+[x:[N:6,3],[N:3,6]] <= 6y <= 2x+[x:6,10,8] exec S(x,y) exec S(3N+7,N+4) 25

32 Decomposition / Example 19 (0, 0, 0) 0 i P 0 j i 0 k i j Q = i + j + k Q < P Q = P Q > P (P, P, 0) (P, 0, 0) when P = 0 when Q = 0 exec S(0,0,0) when 1 <= P <= _ when 0 <= Q <= P for Q+[Q:0,1] <= 2i <= 2Q for 0 <= j <= -i+q exec S(i,j,-j-i+Q) when P+1 <= Q <= 2P for Q+[Q:0,1] <= 2i <= 2P for 0 <= j <= -i+q exec S(i,j,-j-i+Q) (P, 0, P)

33 Decomposition / Polyhedral Operations 20 Most polyhedral operations can be implemented by hoisting: image (and pre-image): e.g., skewing a rectangle for _ <= x <= _ for _ <= y <= _ if 0 <= x <= 19 and 0 <= y <= 9 then exec S(x,y)

34 Decomposition / Polyhedral Operations 20 Most polyhedral operations can be implemented by hoisting: image (and pre-image): e.g., skewing a rectangle for _ <= x' <= _ for _ <= y' <= _ for _ <= x <= _ for _ <= y <= _ if 0 <= x <= 19 and 0 <= y <= 9 then if x' = x+y and y' = y then exec S(x',y',x,y)

35 Decomposition / Polyhedral Operations 20 Most polyhedral operations can be implemented by hoisting: image (and pre-image): e.g., skewing a rectangle for _ <= x' <= _ for _ <= y' <= _ for _ <= x <= _ for _ <= y <= _ if 0 <= x <= 19 and 0 <= y <= 9 then if x' = x+y and y' = y then exec S(x',y',x,y) After unswitching: for 0 <= x' <= 9 for 0 <= y' <= x' exec S(x',y',-y'+x',y') for 10 <= x' <= 19 for 0 <= y' <= 9 exec S(x',y',-y'+x',y') for 20 <= x' <= 28 for x' - 19 <= y' <= 9 exec S(x',y',-y'+x',y')

36 Decomposition / Polyhedral Operations 20 Most polyhedral operations can be implemented by hoisting: image (and pre-image): e.g., skewing a rectangle for _ <= x' <= _ for _ <= y' <= _ for _ <= x <= _ for _ <= y <= _ if 0 <= x <= 19 and 0 <= y <= 9 then if x' = x+y and y' = y then exec S(x',y',x,y) After unswitching: for 0 <= x' <= 9 for 0 <= y' <= x' exec S(x',y',-y'+x',y') for 10 <= x' <= 19 for 0 <= y' <= 9 exec S(x',y',-y'+x',y') for 20 <= x' <= 28 for x' - 19 <= y' <= 9 exec S(x',y',-y'+x',y')

37 Decomposition / Polyhedral Operations 20 Most polyhedral operations can be implemented by hoisting: image (and pre-image): e.g., skewing a rectangle for _ <= x' <= _ for _ <= y' <= _ for _ <= x <= _ for _ <= y <= _ if 0 <= x <= 19 and 0 <= y <= 9 then if x' = x+y and y' = y then exec S(x',y',x,y) After unswitching: for 0 <= x' <= 9 for 0 <= y' <= x' exec S(x',y',-y'+x',y') for 10 <= x' <= 19 for 0 <= y' <= 9 exec S(x',y',-y'+x',y') for 20 <= x' <= 28 for x' - 19 <= y' <= 9 exec S(x',y',-y'+x',y')

38 Decomposition / Polyhedral Operations 20 Most polyhedral operations can be implemented by hoisting: image (and pre-image): e.g., skewing a rectangle for _ <= x' <= _ for _ <= y' <= _ for _ <= x <= _ for _ <= y <= _ if 0 <= x <= 19 and 0 <= y <= 9 then if x' = x+y and y' = y then exec S(x',y',x,y) After unswitching: for 10 <= x' <= 19 for 0 <= y' <= 9 for x'-y' <= x <= x'-y' for y' <= y <= y' exec S(x',y',-y'+x',y')

39 Decomposition / Lexicographic Extrema 21 After repeated hoisting/unswitching: no if-then-else conditional parts no empty range lexicographic extrema are readily available for 0 <= x' <= 9 for 0 <= y' <= x' exec S(x',y',-y'+x',y') for 10 <= x' <= 18 for 0 <= y' <= 9 exec S(x',y',-y'+x',y') for 19 <= x' <= 28 for x' - 19 <= y' <= 9 exec S(x',y',-y'+x',y')

40 Decomposition / Lexicographic Extrema 22 Linear optimization: min/maximize 15z + 2 = x over LS(N) when _ <= N <= _ for _ <= x <= _ for _ <= y <= _ if... then exec S

41 Decomposition / Lexicographic Extrema 22 Linear optimization: min/maximize 15z + 2 = x over LS(N) when _ <= N <= _ for _ <= z <= _ for _ <= x <= _ for _ <= y <= _ if... then if 15z+2 = x then exec S 18 z = 0 z = 1 z = 2 z = 3 49

42 Decomposition / Lexicographic Extrema 22 Linear optimization: min/maximize 15z + 2 = x over LS(N) when _ <= N <= _ for _ <= z <= _ for _ <= x <= _ for _ <= y <= _ if... then if 15z+2 = x then exec S produces when N = 4 exec S(1,17,7) when 5 <= N <= _ for 5 <= 5z <= N+[N:0,-1,-2,-3,1] exec S(z,15z+2,5z+2) i.e., z min = 1 (at x = 17) and z max = N 0,1,2,3, 1 N 5 = N 3 5 = N+1 5

43 Decomposition / Finite State Machines 23 States: one per (static) exec statement Transitions: given by function Next ( ), defined with First ( ): [...] for 3 <= x <= 10 do for 2x+[x:6,4,8] <= 6y' <= 3x+[x:0,3] do exec S2(x,y') done done for 11 <= x <= 3N-3 do for x+[x:3,2,4] <= 3y' <= x+[x:3,5,4] do exec S3(x,y') [...] Note: the result of Next tests each variable exactly once ( at done)

44 Summary A new representation for inequalities Tightening Precise combination/comparison A correct and complete decision procedure Polyhedron decomposition into simple ranges Essential polyhedral operations reformulated More work needed on: reducing size/complexity of representations and algorithms delay normalization of deeper levels leverage more arithmetic properties strategies & heuristics for simplest decomposition very frequent excessive fragmentation avoidance or correction?

46 Backup Slides / Normal Forms and Space Complexity The modulo of a PLE is a PLE ( an x n + X 0,... π n x n ) mod β = 0, 1,... β a n x n + X 0,... xn πn = k ( ) π n..., an k + X (k mod β ) mod β... where π n = lcm(π n, β/ gcd(a n, β)) The overall size of a corrective term for a n x n X 0,... π n x n x n is n β lcm(π n, gcd(a n, β) ) i=1

47 Backup Slides / Periodicity and Disjunction Mixed tight bounds are fuzzy x + 3, 2, 4 x 3y 2y x + 0, 1 x 6, 1, 8, 3, 4, 5 x x These can be turned into disjunctions of linear bounds Disjoin_1( v 0,... x π ax) let v m = max{v i } let M = v m a(π 1) let O = {v i v i < M} return (M ax) ( d O (x = d)) Provides a major bound (M) plus outliers (O) (x = 1) (3 x) bound [M, v m ]

48 Backup Slides / Periodicity and Disjunction 31y 10x 32y 10x 32y 31y 10x 0, 497, 498,..., 493, 494, x x x = 0 x = 16 x = 19 x = 22 x = 25 x = x 32 x = 35 x = 38 x = 41 x = x x x x x x x x x x x

49 Backup Slides / Periodicity and Disjunction Multidimensional mixed bounds rely on transposition, e.g.: 6 0, 3 N, 7, 4 x 3N + N, 2, 5 N, 3, 0 N, 4, 7 N, 5, 2 N 1. Build the uni-dimensional bound for all phases of all other variables x x 3N + 0, 7, 2, 3, 4, 5 x, 3, 4, 5, 0, 7, 2 x N 2. Apply Disjoin_1 on each sub-bound to obtain the major bound and outliers x 3N + [[M 0 = 5, O 0 = {7}]], [[M 1 = 5, O 1 = {7}]] 3. Collect phase-specific major bounds and outliers into periodic numbers (+ simplify, + other details) (x 3N + 5, 5 N ) (x = 3N + 7, 7 N ) N

50 Backup Slides / Periodicity and Disjunction With multidimensional bounds X 0,... π n x n a n x n transpose to sink x n at the lowest level apply Disjoin_1 (for each phase of each other variable) transpose back the results X 00 X 01. M 0 M 1. π n 1 X 10 π n 1 X 11 x n 1,. x n 1 π n 1 O 01 π n 1 O 11 x n 1,. x n 1 π n,... x n,... (a) (c) [X 00, X 10,...] πn x n [X 01, X 11,...] πn x n. { M0, O 01,... } { M1, O 11,... }. π n 1 x n 1 π n 1 x n 1 (b)

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