ABC: Analyzing Loop Bounds and Complexities

Transcription

1 ABC: Analyzing Loop Bouns an Coplexities Régis Blanc, Thoas A. Henzinger, Thibau Hottelier, an Laura Kovács EPFL, Switzerlan Abstract. We present ABC, a software tool for autoatically coputing sybolic upper bouns on the nuber of iterations of progra loops. The syste cobines static analysis of progras with sybolic suation techniques to erive loop invariant relations aong progra variables. Iteration bouns are obtaine fro the inferre invariants, by substituting variables with their greatest values. We have successfully applie ABC to a large nuber of exaples. The erive sybolic bouns express non-trivial polynoial relations over loop variables. We also report on results to autoatically infer sybolic expressions over haronic nubers as upper bouns on loop iteration counts. 1 Introuction Establishing tight upper bouns on the execution ties of progras is both ifficult an interesting, see e.g. [6,, 5, 1]. We present ABC, a new software tool for autoatically coputing tight sybolic upper bouns on the nuber of iterations of progra loops. The erive bouns express polynoial relations over loop variables. ABC is fully autoatic, cobines static analysis of progras with sybolic suation techniques, an requires no user-guiance in proviing aitional set of preicates, teplates an assertions. ABC is also able to erive sybolic expressions over haronic nubers as upper bouns on loop iteration counts, which, to the best of our knowlege, is not yet possible by other works. In our approach to boun coputation, we have ientifie a special class of neste loop progras, calle the ABC-loops (Section 3.1). Further, we have built a loop converter to transfor, whenever possible, arbitrary loops into their equivalent ABC-loop forat (Section 3.). Inforally, an ABC-loop is a neste for-loop such that each loop fro the neste loop contains exactly one iteration variable with only one conition an (non-initializing) upate on the iteration variable. For such loops, our etho erives exact bouns on the nuber of loop iterations. We assue that each progra stateent is annotate with the tie units it nees to be execute. For siplicity, we assue that an iteration of an unneste loop takes one unit tie, an all other instructions of the unneste loop nee zero tie. The key steps of our approach to boun coputation are as follows (Section 3.3). (i) First, we instruent the innerost loop boy of an ABC-loop with a new variable that increases at every iteration of the progra. We enote this variable by z. Upper bouns on the value of z thus express upper bouns on the nuber of loop iterations. (ii) Next, the value of z is copute as a polynoial function over the neste loop s iteration variables. We call the relation between z an loop s iteration variables the z-relation. To this en, for each loop of the ABC-loop recurrence equations of z an loop iteration variables are first constructe. Close fors of variables are then erive using our This research was supporte by the Swiss NSF.

2 Loop Sybolic Sus Sybolic Solver Close Fors Loop Converter ABC Loop Boun Coputer z-relation Fig. 1. The ABC tool. sybolic solver which integrates special techniques fro sybolic suation (Section 3.4). The erive close fors express vali relations aong z an loop iteration variables, an thus the z-relations are loop invariant properties. (iii) Further, by substituting loop iteration variables by their greatest values in the copute z-relation, bouns on the value of z are obtaine. These bouns give us tight sybolic upper bouns on the nuber of iterations of the progra. Our etho can be generalize for the tiing analysis of loops whose iteration bouns involve haronic expressions over the loop variables (Section 3.5). Ipleentation. ABC was ipleente in the the Scala prograing language [11], contains altogether 5437 lines of Scala coe, an is available at Inputs to ABC are loops written in the Scala syntax. ABC first rewrites the input loop into an equivalent ABC-loop by using its loop converter, an then coputes bouns on loop iteration counts using its boun coputer. The boun coputer relies on the sybolic solver in orer to erive close fors of sybolic sus an siplify atheatical expressions. The overall workflow of ABC is given in Figure 1. Note that we o not rely on external coputer algebra package for sybolic suation. Experients. We successfully applie ABC on exaples fro [6, 5], as well as on ninety loops extracte fro the JAMA package [8] see the appenix an entione URL. Altogether, we ran ABC on 558 lines of JAMA. ABC copute exact upper bouns on iteration counts for all loops, an inferre the z-relation for eighty-seven loops, all in less than one secon on a achine with a.8 GHz Intel Core Duo processor an GB of RAM. The three loops for which ABC was not able to erive the z-relation were actually sequences of loops. Relate Work. We only present soe of the nuerous ethos that are relate to ABC. Paper [9] infers polynoial loop invariants aong progra variables by using polynoial invariant teplates of boune egree. Unlike [9] where no restrictions on the consiere loops were ae, we require no user guiance in proviing invariant teplates but autoatically erive invariants (z-relations) for a restricte class of loops. The approach presente in [7] infers invariants an boun assertions for loops with neste conitionals, by eploying sybolic coputations techniques. Neste loops cannot be hanle in [7]. In contrast, we infer boun assertions as z-relations for neste loops, but, unlike [7], our assertions are only over loop iteration variables an not arbitrary progra variables. Paper [6] erives iteration bouns of neste loops by pattern atching siple recurrence equations. Contrarily to [6], we solve ore general recurrence equations using the Gosper algorith an ientities over haronic nubers.

3 for (i = 1; i n; i = i + 1) o for (j = 1; j n; j = j + 1) o (a) for (i = 0; i n; i = i + 1) o for (j = 0; j ; j = j + ) o (b) Fig.. Exaples illustrating the power of ABC to (i) copute z-relations as loop invariants, an (ii) infer tight upper bouns on the nuber of iterations of loops. Sybolic upper bouns on iteration counts of ulti-path loops are autoatically erive in [1]. The approach eploys control-flow refineent ethos to eliinate infeasible loop paths an rewrite ulti-path loops into a collection of sipler loops for which boun assertions are inferre using abstract interpretation techniques [1]. The progras hanle by [1] are ore general than the ABC-loops. Unlike [1], we o not rely on abstract interpretation, an are able to infer haronic expressions as upper bouns on loop iterations counts. There has been a great eal of research on estiating the worst case execution tie (WCET) of real-tie systes, see e.g. [, 5]. The cite works autoatically infer loop bouns only for siple loops; bouns for the iteration nubers of ulti-path loops ust be provie as user annotations. The ait tool [] eterines the nuber of loop iterations by relying on a cobination of interval-base abstract interpretation with pattern atching on typical loop patterns. The SWEET tool [5] eterines upper bouns on loop iterations by unrolling loops ynaically an analyzing each loop iteration separately using abstract interpretation. In contrast, our etho is fully autoatic an path-insensitive, but it is restricte to ABC-loops. Motivating Exaples We first give soe exaples illustrating what kin of iteration bouns ABC can autoatically generate. Consier Figure (a) taken fro the JAMA library [8]. ABC first instruents the innerost loop of Figure (a) with a new variable z, initialize to 1, for counting the nuber of iterations of Figure (a). The such obtaine loop is presente in Figure 3(a). Further, by applying ABC on Figure 3(a), we erive the z-relation 1 z = (i 1)n + j as an invariant property of the loop. By substituting i an j with their greatest values (i.e. n) in the z-relation, the nuber of iterations of Figure (a) is boune by n. Consier next Figure (b) with a non-unit increent, an its instruente version in Figure 3(b). We obtain the z-relation: z = 1 + ( j + i ) + 1, yieling: 1 + (1 + n) + n as a tight upper boun on loop iteration counts, where enotes the largest integer not greater than. In the sequel, we illustrate the ain steps of ABC on Figure (b). 1 Actually, the loops of Figure are first translate into their equivalent ABC-forat, an then the z variable is introuce in their innerost loop boy. For siplicity, in Figure 3 we present the instruentation step irectly on the loops of Figure an not on their ABC-loop forats. In our work we i not consier analyzing the relations between the sallest an greatest sybolic values of the loop iteration variables. It ay be however the case that these sybolic values are such that the loops are never execute (e.g. n < 0).

4 z = 1 for (i = 1; i n; i = i + 1) o for (j = 1; j n; j = j + 1) o z = z + 1 (a) z = 1 for (i = 0; i n; i = i + 1) o for (j = 0; j ; j = j + ) o z = z + 1 (b) 3 ABC: Syste Description Fig. 3. Figure instruente by ABC We have ientifie a special class of loops, calle the ABC-loops (Section 3.1), an esigne a loop converter for translating progras into their equivalent ABC-loop shape (Section 3.). Algorithic ethos fro sybolic suation, ipleente in our sybolic solver (Section 3.4), are further eploye in ABC to autoatically erive upper bouns on loop iterations of ABC-loops (Section 3.3). 3.1 ABC-Loops We enote by Z the set of integer nubers, an by Z[x] the ring of polynoial functions in ineterinate x over Z. We consier progras of the following for: for (i 1 = 1; i 1 c; i 1 = i 1 + 1) o for (i = 1; i f 1(i 1); i = i + 1) o for (i = 1; i f 1 (i 1,, i 1 ); i = i + 1) o (1) where i 1,, i are pairwise isjoint scalar variables (calle loop iteration variables) with values fro Z, c is an integer-value sybolic constants, an f k Z[i 1,, i k ] are polynoial functions (k = 1,, 1). In what follows, loops satisfying (1) will be calle ABC-loops. 3. The Loop Converter Converting loops into ABC-loops is one as presente in Algorith 1. The algorith (i) converts loops into equivalent ones such that the sallest values of the loop iteration variables are 1, an (ii) converts loops with arbitrary increents over the iteration variables into equivalent loops with increents of 1. In ore etail, Algorith 1 takes as input a neste for-loop F an an epty list conversion list, an returns, whenever possible, an ABC-loop F that is equivalent to F. The conversion list is use to store the list of changes ae by Algorith 1 on the iteration variables of F. Lines 4-9 of Algorith 1 are require to convert F into an equivalent loop whose outerost loop has the following properties: it iterates over a new variable nvar instea of the iteration variable ovar of the outerost loop of F, where nvar an ovar are polynoially relate; the sallest value of nvar is 1 (instea of the sallest value olboun of ovar); nvar is increent by 1 (instea of the oincr increent value of ovar); the greatest value of nvar is given by the largest integer not greater than the rational expression ouboun olboun oincr + 1, where ouboun is the

5 Algorith 1 Loop Converter Input: For-loop F an conversion list = {} Output: ABC-loop F an conversion list 1: ovar, oincr := outer iteration variable(f ), outer iteration increent(f ) : olboun, ouboun := outer iteration lowerboun(f ),outer iteration upperboun(f ) 3: nvar:= fresh variable() 4: F 0:= loop boy(f ) /. (ovar oincr (nvar + olboun 1 ) ) 5: conversion list:=conversion list {ovar = oincr (nvar + olboun 1 ) } 6: if isloop(f 0) then 7: F := for-loop(nvar, 1, ouboun olboun oincr + 1, 1, Loop Converter(F0)) 8: else 9: F := for-loop(nvar, 1, ouboun olboun oincr + 1, 1, F0) 10: en if greatest value of ovar. The (appropriately oifie 3 ) loop boy F 0 of F is processe in the siilar anner, yieling finally the ABC-loop F that is equivalent to F. The for loop(v, e 1, e, e 3, boy) notation is a short-han notation for the loop for (v = e 1 ; var e ; var = var + e 3 ) o boy. Exaple 1 Consier Figure (b). By applying Algorith 1, the loop iteration variables i 1 an j 1 are introuce with i = i 1 1 an j = (j 1 1) (lines 3-5 of Algorith 1). The sallest values of i 1 an j 1 are 1, their greatest values are respectively n + 1 an + 1, an i 1 an j 1 are increente by 1 (lines 6-9 of Algorith 1). The ABC-loop forat of Figure (b) is given in Figure 4(a). Base on Algorith 1 an keeping the notations of (1), we conclue that the general shape of loops that can be converte into ABC-loops is: for (i 1 = l; i 1 c; i 1 = i 1 + inc 1) o for (i = g 1(i 1); i f 1(i 1); i = i + inc ) o for (i = g 1 (i 1,, i 1 ); i f 1 (i 1,, i 1 ); i = i + inc ) o where l, inc 1,, inc are integer-value sybolic constants, an g k Z[i 1,, i k ]. 3.3 The Boun Coputer We assue that each progra stateent is annotate with the tie units it nees to be execute. For siplicity, we assue that an iteration of an unneste ABC-loop takes one unit tie, an all other instructions of the unneste loop nee zero tie (e.g. assignent stateents take zero tie to be execute). That is we copute a boun on the total nuber of loop iterations of an ABC-loop (1). 3 The expression e 1/.x e is obtaine fro e 1 by substituting each occurrence of the variable x by the expression e. ()

6 for (i 1 = 1; i 1 n + 1; i 1 = i 1 + 1) o for(j 1 = 1; j 1 + 1; j1 = j1 + 1) o (a) z = 1 for (i 1 = 1; i 1 n + 1; i 1 = i 1 + 1) o for(j 1 = 1; j 1 + 1; j1 = j1 + 1) o z := z + 1 (b) Fig. 4. ABC-loop forat of Figure(b) an its instruente version, where i = i 1 1 an j = (j 1 1). Note that / Z., In our approach to boun coputation, we instruent the innerost loop boy of (1) with a new variable that increases at every iteration of the progra, an is initialize to 1 before entering the ABC-loop. We enote this variable by z. We hence obtain: for (i 1 = 1; i 1 c; i 1 = i 1 + 1) o for (i = 1; i f 1 (i 1,, i 1 ); i = i + 1) o z := z + 1 (3) Exaple The instruente loop of Figure 4(a) is given in Figure 4(b). Upper bouns on the value of z give upper bouns on the nuber of iterations of (3). We are hence left with coputing the value of z as a function, calle the z-relation, over i 1,, i. To this en, the value of z at an arbitrary iteration of the outerost loop of (3) is first copute. Coputing the value of z after an arbitrary iteration of the outerost loop of (3). Let us consier a ore general loop than (3): for (i 1 = 1; i 1 c; i 1 = i 1 + 1) o for (i = 1; i f 1(i 1); i = i + 1) o for (i = 1; i f 1 (i 1,, i 1 ); i = i + 1) o z := z + g(i ) (4) where i 1,, i, c, f 1,, f 1 are as in (1), an g Z[i ]. In particular, if g = 1 then (4) becoes (3). Let s 1,, s l be nonnegative integers (l = 1,, ) such that 1 s 1 c, 1 s f ( i 1 ),, an 1 s l f l 1 (i 1,, i l 1 ). In the sequel we consier s 1,, s l arbitrary but fixe. We write x (l, s1,...,sl ) to ean the value of a variable x {i 1,, i, z} in (4) such that the kth loop of (4) is at its s k th iteration (k = 1,, l), We are thus intereste in eriving z (1, s1 ) for s 1 {1,, c}. We procee as follows. For each loop of (4), starting fro the innerost one, we (i) oel the assignent over z as a recurrence equation, (ii) eploy sybolic suation algoriths to copute the close for of z, an (iii) replace the loop by a single assignent over z expressing

7 the relation between the values of z before the first an after the last execution of the loop. Steps (i)-(iii) are recursively applie until all loops of (4) are eliinate. In ore etail, z (1, s1 ) is erive as follows. We start with the innerost loop of (4). The assignent over z is oele by the recurrence equation: z (, s1,...,s +1 ) = z (, s1,...,s ) + g(i (, s1,...,s 1 ) ), (5) yieling z (, s1,...,s ) = ini z + s g(i(, s1,...,k 1 ) ), where ini z = z (, s1,...,0 ) enotes the value of z before entering the innerost loop of (4). The value of i (, s1,...,s ) is copute fro the recurrence equation i (, s1,...,s +1 ) = i (, s1,...,s ) +1. Naely, we have i (, s1,...,s ) = ini + s 1, where ini = 1 enotes the initial value of i (i.e. before the first iteration of the innerost loop of (4)). Hence, i (, s1,...,s ) = s + 1. (6) Note that (6) hols for each iteration variable, that is i (l, s1,...,s l ) l = s l + 1 for every l {1,, }. For this reason, in what follows we write i l instea of i (l, s1,...,s l ) l an use the relation i l = s l + 1 to speak about the value of i l at iteration s l of the lth loop. We thus obtain z (, s1,...,s ) = ini z + s g(i(, s1,...,k 1 ) ) = ini z + i 1 g(k) Since g Z[i ], the close for of i 1 g(k) always exists [3] an can be copute as a polynoial function over i (see Section 3.4). Finally, we consier the last iteration s = i 1 = f 1 (i 1,, i 1 ) of the innerost loop of (4), an write incr = f 1 (i 1,...,i 1 ) g(k). We ake use of incr Z[i 1,, i 1 ] to eliinate the innerost loop of (4) an obtain: for (i 1 = 1; i 1 c; i 1 = i 1 + 1) o for (i 1 = 1; i 1 f 1 (i 1,, i ); i 1 = i 1 + 1) o z := z + incr Inner loops of (7) can be further eliinate by applying recursively the steps escribe above, since close fors of polynoial expressions over i 1,, i yiel polynoial expressions over i 1,, i whenever the suation variables are boune by polynoial expressions. As a result, the total nuber of increents over z in the s 1 th iteration of the outerost loop of (4) is erive. Let us enote this nuber by incr 1. Then: z (1, s1 ) = z 0 + incr 1, where z 0 = 1 is the value of z before (4). Exaple 3 Consier Figure 4(b). We ai at eriving z (1, s1 ), where 1 s 1 n+1 is arbitrary but fixe such that i 1 = s Fro the innerost loop of Figure 4(b) we get z (, s1,s+1 ) = z (, s1,s ) + 1 for an arbitrary but fixe s, where 1 s + 1 an j 1 = s + 1. Hence, z (, s1,s ) = ini + j 1 1, where ini is the initial value of z before entering the innerost loop. Further, after s = j 1 1 = + 1 iterations of the innerost loop, the total nuber of increents over z is incr = +1 1 = + 1. The innerost loop of Figure 4(b) is next eliinate, yieling: (7)

8 Algorith Boun Coputer Input: ABC-loop F, initial value z 0 of z Output: z-relation zrel 1: inner:= loop boy(f ) : incr:= z reuce loop(inner) 3: ovar, ouboun := outer iteration variable(f ),outer iteration upperboun(f ) 4: nvar:= fresh variable() 5: z i:=z 0 + solve su(nvar, 1, ovar 1, incr/.(ovar nvar)) 6: if isloop(inner) then 7: zrel:= z =Boun Coputer(inner, z i) 8: else 9: zrel:=z = z i 10: en if for (i 1 = 1; i 1 n + 1; i 1 = i 1 + 1) o z = z with the recurrence equation of z as z (1, s1 +1) = z (1, s1 ) Solving this recurrence an using that z 0 = 1 is the initial value of z before the outerost loop of Figure 4(b), we obtain: z (1, s1 ) = 1 + i 1 1 ( + 1) = 1 + (i 1 1) ( + 1). Coputing the z-relation aong arbitrary values of z, i 1,, i. We are intereste in eriving the value of z (, s1,...,s ), where i k = s k + 1 (k = 1,, ), fro which the z-relation can be ieiately constructe as z = z (, s1,...,s ). The value z (, s1,...,s ) (an hence the z-relation) is inferre by Algorith as follows. (a) The value incr is copute s.t. z (1, s1 ) = z 0 + incr (line of Algorith ); (b) The outerost loop of (4) is oitte (line 1 of Algorith ), yieling: for (i = 1; i f 1(i 1); i = i + 1) o for (i = 1; i f 1 (i 1,, i 1 ); i = i + 1) o z := z + g(i ) (c) The value of z at the s n iteration of the outerost loop (8) is next copute, where the initial value of z before (8) is consiere to be z (1, s1 ) (line 7 of Algorith ). As a result, z (, s1,s ) in the loop (4) is obtaine. () Steps (b)-(c) are recursively applie on (8) (lines 6-9 of Algorith ). Exaple 4 Consier Figure 4(b). The outerost loop of Figure 4(b) is oitte (line 1 of Algorith ), yieling: for(j 1 = 1; j 1 + 1; j 1 = j 1 + 1) o z = z + 1 (9) The total nuber of increents incr = + 1 over z ae by (9) is copute, as presente in Exaple 3 (line of Algorith ). The value z i = z (1, s1 ) of z at an iteration s 1 = i 1 1 of the outerost loop of Figure 4(b) is further copute as z i = z 0 + i 1 1 ( nvar=1 + 1) = 1 + (i 1 1) ( + 1) (lines 3-5 of Algorith ). (8)

9 for (i = 1; i n; i = i + 1) for (j = 0; j n; j = j + i) for(i 1 = 1; i 1 n; i 1 = i 1 + 1) for(j 1 = 1; j 1 n 1; j 1 = j 1 + 1) z := z + 1 for(i 1 = 1; i 1 n; i 1 = i 1 + 1) for(j 1 = 1; j 1 n 1; j 1 = j 1 + 1) z := z + 1 (a) Not an ABC-loop (b) Converte loop by ABC with (c) Instruente loop by ABC n 1 = n i + 1, an i = i 1 1, j = i j 1 Fig. 5. ABC on a non-abc-loop. Next, Algorith is calle on (9) with the initial value z i to copute the value of z at an iteration s = j 1 1 of (9) (line 7 of Algorith ). As (9) has no inner loops, we have incr = 1 an z i = z i + j 1 1 nvar=1 1, yieling z(, s1,s ) = z i = (i 1 1) ( + 1) + j 1 (lines -5 of Algorith ). The z-relation of Figure 4(b) is finally erive as z = (i 1 1) ( + 1) + j 1 (line 9 of Algorith ). To obtain the z-relation of Figure (b), we ake use of i = i 1 1 an j = (j 1 1) an have: z = i ( ) j + 1. Substituting i an j with n an in the z- relation, the upper boun on loop iteration counts of Figure (b) is: (n + 1) ( ) Sybolic Solver Siplifying arithetical expressions an coputing close fors of sybolic sus is perfore by the sybolic solver engine of ABC. Our sybolic solver supports the close for coputation of the following sus: e x=e 1 c 1 n x 1 x c r n x r x r where e 1, e are integer-value sybolic constants, n i, i are natural nubers, an c i are rational nubers. Close fors of such sus always exists [3]. For coputing the close fors of these sus we rely on a siplifie version of the Gosper algorith [3]. We have also instruente our sybolic solver to hanle sybolic sus whose close fors involve haronic nubers [4], as iscusse in Section Beyon ABC-Loops ABC is coplete for ABC-loops an for loops satisfying (). That is, it always returns the z-relation an loop iteration boun of an ABC-loop. It is worth to be entione that soe loops violating () can still be successfully analyze by ABC. Consier Figure 5(a) violating (), as upates over j epen on i. However, using Algorith 1 we erive the loop given in Figure 5(b), yieling finally the instruente loop fro Figure 5(c). Further, by applying Algorith, we are left n i 1. This su cannot be further siplifie [4]. with fining the close for of i 1 1 However, we can copute an upper-boun on its close for using the relation: i 1 1 n i 1 i1 1 i 1 n = n i 1 1 i 1 1. Note that i i 1 is the haronic nuber H(i 1 1) arising fro the truncation of the haronic series [4]. We ake use of H(i 1 1) an erive an upper boun on the

10 loop iteration count of Figure 5(a) as being a haronic expression. To this en, we have extene our sybolic solver with soe siple ientities over haronic nubers. ABC can be thus successfully applie to loops for which sybolic coputation ethos can be eploye to copute or approxiate close fors of loop variables. 4 Conclusions We escribe the software tool ABC for autoatically eriving tight sybolic upper bouns on loop iteration counts of a special class of loops, calle the ABC-loops. The syste was successfully trie on a large nuber of exaples. The erive sybolic bouns express non-trivial (polynoial an haronic) relations over loop variables. Future work inclues extening ABC to hanle ore coplex sus, such as e.g. fractions of polynoials [10], an incluing ore sophisticate control-flow refineent techniques, such as [1], into ABC. References 1. P. Cousot an R. Cousot. Abstract Interpretation: a Unifie Lattice Moel for Static Analysis of Progras by Construction or Approxiation of Fixpoints. In Proc. of POPL, pages 38 5, C. Ferinan an R. Heckann. ait: Worst Case Execution Tie Preiction by Static Progra Analysis. In IFIP Congress Topical Sessions, pages , R. W. Gosper. Decision Proceures for Inefinite Hypergeoetric Suation. Journal of Sybolic Coputation, 75:40 4, R. L. Graha, D. E. Knuth, an O. Patashnik. Concrete Matheatics. Aison-Wesley Publishing Copany, n eition, J. Gustafsson, A. Ereahl, C. Sanberg, an B. Lisper. Autoatic Derivation of Loop Bouns an Infeasible Paths for WCET Analysis Using Abstract Execution. In RTSS, pages 57 66, C. A. Healy, M. Sjöin, V. Rustagi, D. B. Whalley, an R. van Engelen. Supporting Tiing Analysis by Autoatic Bouning of Loop Iterations. Real-Tie Systes, 18(/3):19 156, T. A. Henzinger, T. Hottelier, an L. Kovacs. Valigator: A Verification Tool with Boun an Invariant Generation. In Proc. of LPAR, pages , J. Hicklin, C. Moler, P. Webb, R. F. Boisvert, B. Miller, R. Pozo, an K. Reington. JAMA: A Java Matrix Package M. Müller-Ol, M. Petter, an H. Seil. Interproceurally Analyzing Polynoial Ientities. In Proc. of STACS 006, I. Nees an M. Petkovsek. RCop: A Matheatica Package for Coputing with Recursive Sequences. Journal of Sybolic Coputation, 0(5-6): , M. Oersky. The Scala Language Specification S. Srivastava an S. Gulwani. Progra Verification using Teplates over Preicate Abstraction. In Proc. of PLDI, pages 3 34, R. A. van Engelen, J. Birch, an K. A. Gallivan. Array Data Depenence Testing with the Chains of Recurrences Algebra. In Proc. of IWIA, pages 70 81, 004.

11 Appenix: Experiental Results with ABC Loop z-relation Iteration boun Tie [s] for(i = a; i b; i = i + 1) for(i = 0; i n 1; i = i + 1) for(j = 0; j i; j = j + 1) for(i = 1; i ; i = i + 1) for(j = 1; j i; j = j + 1) for(k = i + 1; k ; k = k + 1) for(l = 1; l k; l = l + 1) for(i = 0; i ( n n n 1); i = i + 1) for(j = 0; j n 1; j = j + 1) for(k = 0; k j 1; k = k + 1) for(i = 1; i n; i = i + 1) for(j = 1; j i; j = j + 1) for(i = 1; i n; i = i + 1) for(j = i; j n; j = j + 1) for(i = a; i b; i = i + 1) for(j = c; j ; j = j + 1) for(k = i j; k i + j; k = k + 1) for(i = n; i 1; i = i 1) for(j = ; j 1; i = j 1) z = 1 + i a 1 + b a z = 1 + j + i+i z = i i +i i 4 + i i i3 i 1 + j+j +k +ji +ji++k + 1 z = 1 + k + in in+j j z = i i + j z = (i 1)n + j + i i z = 1 a + i a + i + ac ic + j c + j a + k n+n n 4 n 4 n +n n +n 1 + b a c + bc +ac +b a + b a z = (n i+1) j+1 n 3 Table 1. ABC results obtaine on a achine with a.8 GHz Intel Core Duo processor an GB of RAM. The first four progras of Table 1 are exaples taken fro [13], whereas the last four progras of Table 1 are loops taken fro the JAMA package [8]. The first colun of Table 1 presents the loop being fe into ABC, the secon colun shows the z-relation erive by ABC, whereas the thir one presents the nuber of loop iterations copute by ABC. The forth colun gives the tie neee by ABC to infer bouns on loop iteration counts. Note that iteration bouns are integer-value polynoial expressions (e.g. n + n is ivisible by )