Array Dependence Analysis as Integer Constraints. Array Dependence Analysis Example. Array Dependence Analysis as Integer Constraints, cont

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1 Theory of Integers CS389L: Automated Logical Reasoning Omega Test Işıl Dillig Earlier, we talked aout the theory of integers T Z Signature of T Z : Σ Z : {..., 2, 1, 0, 1, 2,..., 3, 2, 2, 3,..., +,, =, >} This theory also called linear arithmetic over integers Since equal in expressive power to Presurger arithmetic, people also refer to it as Presurger arithmetic Today and next lecture: Look at algorithms for deciding satisfiaility in quantifier-free fragment of T Z Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 1/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 2/35 Prolem Description As in previous two lectures, we ll consider T Z formulas without disjunctions Prolem we want to solve: Given an m n matrix A with only integer coefficients and a vector in Z n, does have any integer solutions? A x Prolem very similar to that from last lecture, ut this time we only accept integer solutions; rational solutions not ok! This requirement actually makes prolem much harder Finding solution over rationals is poly-time, ut integer prolem is NP-complete even without disjunctions Geometric Description As efore the system A x defines a polytope Last time we asked the question: Is the polytope empty? This time, we want to know if polytope contains integer points While the polytope is convex, the space formed y all integer points in polytope is not convex Unfortunately, non-convexity makes prolem much harder to solve Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 3/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 4/35 Overview of Techniques Two different techniques for solving linear integer inequalities 1. Elimination-ased techniques: Omega Test, Cooper s method 2. Relaxation-ased techniques: Branch-and-ound, Gomory cuts, Cuts-from-Proofs Elimination-ased techniques eliminate variales one y one until system ecomes trivially solvale Relaxation-ased techniques first drop the integrality requirement and look for a real valued solution Then, they iteratively introduce additional constraints to guide search for integer solution The Omega Test: Historical Perspective Omega Test: invented in early 1990 s for compiler optimizations Particular application: array dependence analysis Array dependence analysis: Can two expressions a[i] and a[j] refer to same element? Can use this information to reorder read and writes from the array and perform operations in parallel Called relaxation-ased ecause they first solve LP-relaxation Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 5/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 6/35 1

2 Array Dependence Analysis Example Array Dependence Analysis as Integer Constraints Consider the following code snippet: for(i=1; i<= 100; i++) { for(j=i; j<= 100; j++) a[i, j+1] = a[100, j] } Can the expressions a[i, j+1] and a[100,j] ever refer to same element (not necessarily in same iteration)? No! Thus, no array element is oth read and written to in the loop Hence, we can optimize code y performing assignments in parallel! for(i=1; i<= 100; i++) { for(j=i; j<= 100; j++) a[i, j+1] = a[100, j] } Can express dependence analysis as linear integer constraints Variales w i and w j denote array indices when write is performed Variales r i and r j denote array indices when read is performed How do we express that same element is oth read and written to? w i = r i w j = r j Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 7/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 8/35 Array Dependence Analysis as Integer Constraints, cont for(i=1; i<= 100; i++) { for(j=i; j<= 100; j++) a[i, j+1] = a[100, j] } Based on loop start/end conditions, what are constraints on w i? 1 w i 100 What are constraints on w j? w i < w j 101 What are constraints on r i? r i = 100 What are constraints on r j? w i r j 100 r j = w j 1 Array Dependence Analysis as Integer Constraints, cont Thus, an array element may e oth read and written in the loop if the conjunction of these constraints is satisfiale: w i = r i w j = r j 1 w i 100 w i < w j 101 r i = 100 w i r j 100 r j = w j 1 Since array indices can t e real numers, only interested in integer solution Since this constraint has no integer solutions, there is no dependence etween array reads and writes Thus, all writes can e done in parallel Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 9/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 10/35 Applications of Theory of Integers Omega Test: Main Idea Array dependence analysis one application of decision procedure for theory of integers Omega Test was initially invented to do etter jo with array dependence analysis Many other applications in software verification, compiler optimizations, operations research,... Main idea: Eliminate variales one y one from the initial system A x Geometrically, eliminating a variale corresponds to computing a projection of a polytope in n-dimensional space to an n 1-dimensional space Since the polytope has one less dimension at each step, resulting prolem is easier to solve than the previous one Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 11/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 12/35 2

3 Projections in Omega Test Omega test computes three kind of projections, called shadows: 1. Real Shadow Overapproximates satisfiaility over integers Omega Test Work Flow If real shadow has no solutions, neither does original prolem 2. Dark Shadow Underapproximates satisfiaility over integers If dark shadow has solution, original prolem has solution 3. Gray Shadows These correspond to areas etween real and dark shadow that might contain integer points Omega test constructs multiple gray shadows Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 13/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 14/35 The Real Shadow When constructing the real shadow, we ignore requirement that solution must e integer Thus, resulting projection overapproximates satisfiaility of original prolem To construct real shadow, we use the Fourier-Motzkin variale elimination technique Fourier-Motzkin Variale Elimination Suppose we want to eliminate variale x n from A x Consider an inequality n i This can e rewritten as x n i n1 If is positive, this yields an upper ound on x n : x n i n1 If is negative, this yields lower ound on x n : x n i n1 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 15/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 16/35 Fourier-Motzkin Variale Elimination, cont. Thus, if we have A x has two rows i and k with positive and negative coefficients for x n, this yields the inequality: n1 k a kj x n i n1 We eliminate x n y removing it from the middle of inequality: n1 k a kj i n1 If we do this for every pair of inequalities with positive and negative coefficients for x n, this yields the real shadow Fourier-Motzkin Example Consider the set of inequalities: x y + 10 y 15 x + 20 y Let s compute real shadow on x-axis using Fourier-Motzkin Isolate y on one side: (1) x 10 y (2) y 15 (3) x + 20 y From (1) and (2), we get x 10 15, i.e., x 25 From (2) and (3), we get x , i.e. x 5 Thus, real shadow on x-axis is 5 x 25 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 17/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 18/35 3

4 Dark Shadow The second projection Omega test constructs is dark shadow Dark shadow underapproximates satisfiaility Math Behind the Dark Shadow As in real shadow, consider a pair of inequalities corresponding to lower and upper ounds on x: Suppose we want to eliminate variale x from A x L ax x U Dark shadow only projects those parts of polytope that are at least one unit thick in the x-dimension If dark shadow has integer solution, original polytope must also have integer solution. Why? Since polytope is at least one unit thick aove the dark shadow in x-dimension, we are guaranteed to have an integer solution for x as well! These imply: L a x U Now, suppose there is no integer etween L a and U Consider first integer i smaller than L a Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 19/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 20/35 Math Behind the Dark Shadow, cont. Math Behind Dark Shadow, cont If we rearrange this equation, we get: I L au + a a I I I I I I I I I I I I I Finally, multiplying oth sides y 1: Thus, we have the following inequalities: au L a a ( ) If we sum these up, we get: L a i 1 a i + 1 U 1 L a U a + 1 Recall: We derived this equation y assuming that there is no integer solution for x That is, we showed If there is no integer solution for x, then (*) must hold Thus, negation of (*) guarantees there exists integer solution for x! Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 21/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 22/35 Math Behind Dark Shadow, cont Dark Shadow Example Thus, negation of (*) au L > a a ( ) guarantees there is an integer value for x! Thus, to construct dark shadow, we remove inequalities containing x and add inequality ( ) Resulting projection is underapproximation ecause only projects those parts that are at least one unit thick, ut there might e an integer solution for x even if it s not unit thick Using (1), (3), we have a = 4, L = x, and = 3, U = 7 x: 4(7 x) 3x > x < 23 7 Using (2), (3), a = 2, L = 6 3x, and = 3, U = 7 x: 2(7 x) 3(6 3x) > x > 5 7 Since 5 7 < x < 23 7 has integer solutions, system is satisfiale Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 23/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 24/35 4

5 Dark Shadow Example, cont. Gray Shadows Recall: Real shadow overapproximates the prolem, and dark shadow underapproximates it. Geometrically: If real shadow has integer solutions, ut dark shadow does not, we still don t know if original prolem has integer solutions. In this case, Omega test constructs projections called gray shadows Gray shadows look for integer solutions outside the dark shadow, ut inside the real shadow. Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 25/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 26/35 Constructing the Gray Shadow Consider again the pair of inequalities: L ax x U By construction, any point in the real shadow satisfies: L ax au (1) Also, y construction, any point outside dark shadow satisfies: au L a a We can rewrite aove as: au L + a a (2) Comining (1) and (2), we have: L ax L + a a Constructing Gray Shadow, cont. Thus, any point inside real shadow ut outside dark shadow must satisfy: L ax a + L a Dividing y, points in the gray shadow must satisfy: L ax L + a a Oserve: If x is an integer, ax must also e integer Furthermore, ax must e equal to L + i for some i in the range [0, aa ] Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 27/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 28/35 Constructing Gray Shadow, cont. Remark aout Gray Shadows Thus, we construct each gray shadow y adding the equality: ax = L + i for each integer i in the range [0, aa ] If any suprolem has integer solution, then so does original prolem If no suprolem has integer solution, original prolem unsatisfiale Oserve: If there are n integers etween 0 and aa, Omega test constructs n gray shadows Thus, Omega test is very sensitive to coefficients in formula The larger a is, the more gray shadows we must consider Nightmare for Omega test: Real shadow has solution, dark shadow has no solution, and coefficient a is very large, and prolem is unsatisfiale In this case, Omega test must solve a very large numer of suprolems Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 29/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 30/35 5

6 Gray Shadow Example Example, cont. Derive gray shadow using constraints (1) and (3): 5y 3x + 3 (LB) 4y 3 (UB) Here, L = 3, a = 4, U = 3x + 3, = 5 Try 4x = 3 + i for i [0, ] (i.e., i [0, 2]) Real shadow has solutions, ut dark shadow does not; so, need to explore gray shadows... Satisfiale for i = 1 Has integer solutions Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 31/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 32/35 Example, cont. Geometrically: An Optimization Omega test uses important optimization to handle equality constraints Equality constraints can e expressed as pair of inequalities, ut handling equalities directly much more efficient Thus, Omega test has special preliminary step where it gets rid of all equality constraints Uses interesting coefficient-reducing technique ased on symmetric modulo Details are in paper posted on class wepage - strongly encouraged to read! Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 33/35 Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 34/35 Omega Test Summary Omega test is an elimination-ased technique for solving linear inequalities over integers Constructs three kinds of projections: real shadow, dark shadow, gray shadow Prolem has no solution if real shadow has no solution Prolem has solution if dark shadow has solution Otherwise, prolem has solution iff one of the dark shadows has solution Omega test handles equalities specially using the symmetric modulo technique Işıl Dillig, CS389L: Automated Logical Reasoning Omega Test 35/35 6