A General Approach for Expressing Infeasibility in Implicit Path Enumeration Technique

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1 A Generl Approch for Expressing Infesibility in Implicit Pth Enumertion Technique Pscl Rymond Univ. Grenoble Alpes, VERIMAG, F Grenoble, Frnce CNRS, VERIMAG, F Grenoble, Frnce ABSTRACT Sttic timing nlysis ims t computing gurnteed upper bound to the Worst-Cse Execution Time (WCET) of progrm. It requires both n ccurte modeling of the hrdwre, nd precise nlysis of the progrm in order to reject infesible executions (in prticulr, ll infinite ones). For the ctul computtion of the worst-cse execution, most of the existing tools nd methods re bsed on the Implicit Pth Enumertion Technique (IPET), which consist in encoding this serch into numericl optimiztion problem (Integer Liner Progrmming, ILP). An interest of this pproch is tht it nturlly integrtes the loop bounds. It lso llows to implicitly prune infesible pths, s fr s they cn be expressed using liner constrints. Severl works on the subject re using this bility in order to enhnce the WCET estimtion: they identify specific property ptterns (e.g., implictions, exclusions) nd propose d hoc trnsltion into numericl constrints. The gol of this pper is to go further thn d hoc resoning by proposing generl method for trnslting infesibility in terms of numericl constrints. It does not ddress the problem of finding infesible pths, only the one of chrcterizing them s precisely s possible. Moreover the pper ims t exploring the limits of the method, nd thus, it does not try to enhnce the result using dditionl methods (e.g., grph trnsformtion). Ctegories nd Subject Descriptors D.2.8 [Softwre Engineering]: Metrics complexity mesures, performnce mesures Generl Terms WCET, infesible pth, Integer Liner Progrmming This work is supported by the french reserch fundtion (ANR) s prt of the W-SEPT project (ANR-12-INSE- 0001). Permission to mke digitl or hrd copies of ll or prt of this work for personl or clssroom use is grnted without fee provided tht copies re not mde or distributed for profit or commercil dvntge nd tht copies ber this notice nd the full cittion on the first pge. Copyrights for components of this work owned by others thn ACM must be honored. Abstrcting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission nd/or fee. Request permissions from Permissions@cm.org. ESWEEK 14, October , New Delhi, Indi. Copyright 2014 ACM /14/10...$ INTRODUCTION Sttic timing nlysis ims t computing, given binry code nd the description of the rchitecture, proven upper bound to the Worst-Cse Execution Time (WCET) of the code (see [8] for n overview of methods nd tools). Existing methods nd tools re minly orgnized in 3 steps. (1) Dt-flow Anlysis considers the semntics of the code (often requiring the vilbility of the source code, e.g., C) in order to identify the pths in the CFG tht re ctully fesible; this phse must t lest find ccurte loop bounds, in order to retin only finite executions. (2) Micro-rchitecture Anlysis builds the Control Flow Grph (CFG) of the code, mde of Bsic Blocks (BB) of sequentil instructions connected by edges representing the possible control pths. It then computes, for ech BB nd/or edges, locl WCET estimtion (k locl weight), tking into ccount s precisely s possible hrdwre fetures like memory cche, pipeline, brnch predictor. (3) Worst-Cse Pth Serch comes t lst, to identify /the worst-cse execution pth ccording to the weights ssigned to ech BB nd/or edges. For this purpose, the mostly used methods re bsed on Implicit Pth Enumertion Technique (IPET), introduced in [6]. The key ide of this method is to encode the problem of finding worst-cse pth into numericl optimiztion problem, more precisely, n Integer Liner Progrmming problem (ILP). An integer vrible is ssocited to ech BB nd/or edge, representing the number of time the block/edge is trversed during n execution. The structure of the grph cn then be expressed s set of liner constrints, ccording to the well-know principle tht incoming flow equls outgoing flow, nd tht the entry point of the progrm is executed once. Other constrints, coming from the dtflow nlysis, re dded to the numericl system, comprising t lest the ones tht re necessry to bound ll loops in the grph. The objective function is then to mximize the weighted sum of ll these counters, nd it cn be solved by stte-of-rt Integer Liner Progrmming solver. A min interest of the IPET method, pointed out since its first proposl [6], is its bility to implicitly prune lots of infesible pths by stting simple liner constrints. The study of this bility is the subject of this pper. To complete this introduction, we present now some relted works nd

2 some exmples tht illustrte the problem nd motivte this work. 1.1 Stte of the rt To our knowledge, there is no previous work specificlly dedicted to the expression of infesible pths by mens of ILP constrints. More precisely, lmost ll works bsed on the IPET method re considering this problem, but minly s complement to the one of finding the properties tht mke the pths infesible. Ppers on this subject re numerous, nd generl chrcteristic is tht they serch for prticulr property ptterns leding to infesible pth, tht re trnslted into prticulr constrint ptterns. These ptterns re minly bsed on the notion of conflicting edges: edges tht cnnot be ll trversed during the sme execution. Ppers re considering pirwise conflicts (conflictpir), or more generlly n-ry conflicts (conflict-list). For instnce, [1, 4] re hndling conflicting lists of edges, tht cn be expressed s bounded sums; it is similr to the notion of conflict we develop in this pper, but only in the cse of progrms without loops. Ppers like [2, 3, 4] go further by considering properties holding in loop scopes, in the cse of pirwise reltions between edges (conflict pirs, exclusions, equlities etc.). Moreover, lmost ll works notice (often implicitly) tht purely conjunctive liner progrmming is unble to express ll kind of conflicting properties. The generl solution is to define mix of cse-by-cse nlysis nd pure ILP. This kind of methods cn be referred to s splitting-bsed methods, in the sense tht they consists in splitting the problem to mke pths more nd more explicit. Techniclly, they cn be bsed on control-flow grph trnsformtion, ddition of extr vribles nd/or disjunctive liner systems (in order to express non convex domins [5, 6, 7]). The work presented in this pper is somehow trnsversl to these relted works. In prticulr it does not consider t ll the problem of finding properties, but only focusses on the expression of these properties s ILP constrints. We im t considering in homogeneous mnner progrms with or without loops, nd reltions between two or more edges (conflicting pirs or lists). Moreover, we try to explore the limits of pure ILP constrints, nd thus we do not consider splitting-bsed methods. 1.2 Exmples Progrm 1 shows n exmple of code written in pseudo- C (left). The ctul CFG, hndled by the timing nlyzing tool, is shown in the middle. For resoning bout pth fesibility, we use the simplified grph on the right: sequentil blocks re bstrcted to outline the brnching possibilities. Ech trnsition is identified by letter (,b,c, etc.) tht helps to relte it with the source C code. In the clssicl IPET pproch, n integer vrible is ssocited to ech edge. This vrible is counter, whose vlue is the number of time the edge is trversed during one prticulr execution. We note these counters with the sme letter thn the corresponding edge. Let E = {, b, c,... this set of vribles. The weights (locl WCET) computed during the micro-rchitecture nlysis re denoted w, w b, etc. The IPET gol is to mximize, under set of liner constrints, the objective function: w x x x E The CFG of Progrm 1 cn be literlly trnslted into set of structurl constrints: { 1 = + d + d = g g + k = h + l S = h = b + e b + e = c + f c + f = k With these structurl constrints only, there is trivilly no bound to the objective function: the estimted WCET is infinite. This is why the control-flow nlysis must t lest produce bound for ech loop in the progrm. We suppose tht such constnt bound n is given (e.g. 100), nd new constrint is dded to obtin the bsic set of constrints : B = S {h n If we only consider the constrints set B (structurl nd loop bounds), the interest of the ILP encoding is not obvious. The WCET cn be computed without the help of numericl solver by pplying simple mx/plus inductive lgorithm: 1. The WCET of sequence is the sum. 2. The WCET of choice is the mx. 3. The WCET of loop bounded by n is n times the WCET of one itertion. The ILP method becomes interesting if there exist dditionl informtion bout edges tht re not structurl but rther semnticl. Such n informtion is for instnce tht two (distnt) edges re incomptible (or conflicting), in the sense tht they cnnot be both tken during the sme execution. The purpose of this work is not t ll to find these properties: we just mke the sttement tht they exist, nd our gol is to express them s precisely s possible in the IPET frmework. Let us now precise this notion of incomptibility on the exmple. Conflict within loop. On the exmple progrm, one cn observe tht executing the brnch e sets the vrible cond to true, nd, s consequence, mkes it impossible to tke immeditely fter the brnch f. This kind of property is very esy to express s n ILP constrint if the progrm hs no loop: since ech edge is executed t most once, the constrint e + f 1 precisely express tht t most one edge cn be executed. Here the problem is slightly more complex becuse the edges belong to loop. A simple d hoc resoning shows tht the conflict holds t ech itertion of the for sttement, nd tht the number of itertions is precisely cptured by the vlue of the edge counter h. The conflict cn be expressed by: e + f h Since we know tht h is bounded by the constnt n, we my lso directly write: e + f n Most of the exmples found in the literture re very similr to this cse: uthors serch for prticulr property ptterns tht led to prticulr constrint ptterns, typiclly sum bounded by constnt.

3 i f ( i n i t ) { / / else { / d / for ( i =0; i <n ; i ++){ i f (Y[ i ] ) { cond = not i n i t nd Z [ i ] ; / b / else { cond = t r u e ; / e / /... / i f ( cond ){ / c / else { / f / k d e f g h l b c Progrm 1: simplified C code, ctul binry CFG nd simplified CFG k e f d g h l b c l e b Figure 1: Unfolding the CFG of Progrm 1 to distinguish executions where is tken or not. Conflict cross loop. A less obvious semntic reltion exists between the initil if sttement nd the two sttements inside the loop: if the vrible init is true, then in ech loop, if Y[i] is true then cond is set to flse nd brnch c becomes unrechble. This is yet nother exmple of conflict, but involving three edges insted of two:, b nd c. Moreover, these edges do not belong to the sme loop scope, mking impossible to simply bound their sum. A usul solution to tret this kind of property consists in splitting the problem ccording to severl cses. This split cn be mde explicitly by unfolding the CFG s shown in Figure 1. We obtin new grph, nd thus new ILP system which is equivlent with respect to the WCET computtion. Note tht the conflict previously discovered between e nd f, simply pplies to the new versions of the edges; from now on we use the more visul nme of vtrs to identify the severl versions of n edge: e + f n nd e + f n. Moreover, we cn now precisely express the new conflict, tht holds only for the prime vtrs of b nd f g h c k c (i.e., fter the execution of only): b + c n CFG trnsformtion (unfolding, loops unrolling) is generl solution for expressing ny kind of infesibility: the more detiled is the grph, the more explicit re the pths. However it contrdicts the spirit of the IPET method, nd it is techniclly limited by the combintoril growing of the grphs. A clever solution consists in keeping the splitting implicit, by introducing s few extr vribles s possible. For instnce, we cn strt with the originl ILP system of Progrm 1 nd introduce only the vtrs of b nd c. The following set of extr constrints, clled A, is sufficient to express the property: A = b + c n b n d c n d b + b = b c + c = c In this system, behves like Boolen orcle whose vlue chooses between two different systems. If = 0, then, becuse of the structurl constrints, we hve d = 1 = 1 nd the set A cn be reduced to b n nd c n, which is redundnt with the structurl constrints. If = 1, A cn be reduced to the expected conflicting constrint : b+c n. The solution of introducing extr vribles is still not relly in the spirit of the IPET, since extr vribles re just wy to mke pths more explicit. In fct, for this prticulr cse, d hoc resoning shows tht the property cn be expressed exctly without the help of splitting method. The following constrint, involving only originl vribles, precisely cptures the expected conflict: n + b + c 2n Conflict between itertions. Progrm 2 hs the sme control flow grph s Progrm 1, but the order of the if stte-

4 i f ( i n i t ) { / / else { / d / cond =... ; for ( i =0; i <n ; i ++){ i f ( cond ){ / b / else { / e / /... / i f (Y[ i ] ) { cond = not i n i t nd Z [ i ] ; / c / else { cond = t r u e ; / f / Progrm 2: Sme CFG s Progrm 1, if sttements re reversed ments in the loop is reversed. As consequence, whenever is tken, the conflict between b nd c does not occur during the sme itertion, but between two consecutive itertions: if c is tken, cond is set to flse, nd in the next itertion (if it exists) b cnnot be tken. Even if it is not so different from the previous exmple, this kind of inter-itertion property is hrdly hndled in existing work. Here gin the problem cn be solved by unfolding the grph or introducing extr vribles. However, here gin, it is not necessry to do so. Let n be the loop bound, the following constrint precisely cptures the conflict: (n 1) + b + c 2n Here is n informl proof sketch: if n = 0, then b = c = 0 nd the constrint becomes tutology: 1 0; if n = 1 or = 0, the constrint becomes redundnt: b + c 2n; if n 2 nd = 1 the constrint becomes b + c 2n (n 1) = n + 1. By doing (virtul) unfolding of the n itertions, on cn exhibit n vtrs of b nd c, denoted b 1, b n, c 1,, c n. Both b 1 nd c n re unconstrined, nd then, their sum is bounded by 2. All others pirs (c k, b k+1 ) re conflicting nd thus c k + b k+1 1. Since there re n 1 such pirs, the sum of ll the vtrs is bounded by 2 + (n 1) = n + 1, which is the bound given by the formul. Limits of numericl constrints. Unfortuntely, it is not possible to exctly reflect ny conflict using only the existing vribles. Progrm 3 is very similr to Progrm 1, except tht the tests within the loop re not depending on the current itertion. For this progrm, if init is true, then depending on the vlue of Y, either b = n nd c = 0 or b = 0 nd c = n. This property cnnot be expressed in clssicl (conjunctive) liner progrmming since it requires to express disjunction: (b = n c = 0) (b = 0 c = n). Here gin, splitting oriented methods (grph unfolding, extr vrible, i f ( i n i t ) { / / else { / d / for ( i =0; i <n ; i ++){ i f (Y) { cond = not i n i t nd Z ; / b / else { / e / cond = t r u e ; /... / i f ( cond ){ / c / else { / f / Progrm 3: Sme CFG s Progrm 1, conditions do not depend on the current itertion disjunctive systems) cn solve precisely the problem, but they re not considered in this pper. Nevertheless, for this progrm, the simple liner constrint n + b + c 2n still holds nd, while not perfect, my prune some infesible pths. 1.3 Motivtion nd pper orgniztion The gol of this work to go further thn d hoc resoning by proposing generl method to produce liner constrints such s the ones presented in the exmples (e + f n, n + b + c 2n, etc.). The method should hndle in homogeneous mnner progrms with or without loops, intr or inter loop properties. For this purpose, first requirement is to precise the nture of n infesibility property. Moreover, we wnt to explore the limits of the ILP expressiveness without the help of ny split-oriented method: no grph trnsformtion, no extr vribles, no disjunction. To summrize, first definition of the gol is: given CFG nd informtion on the incomptibility between edges, find one or more liner constrint(s) tht cut, s precisely s possible, the corresponding infesible pth. The pper is orgnized s follow: section 2 gives the necessry forml definitions. We show tht the notion of conflicting sets is tomic with respect to pth pruning in the sense tht ny set of infesible pths cn be described s union of conflicting sets. Section 3.1 gives first generl solution for trnsforming conflicting sets into liner constrints. This solution requires very few informtion bout the incomptibilities, but the counterprt is tht the result my be rther imprecise. Section 3.2 presents solution for cpturing more precisely the incomptibilities.

5 2. DEFINITIONS AND NOTATIONS 2.1 Progrms nd unfoldings A CFG (or simply progrm) is direct grph, possibly cyclic, mde of finite set of vertices (V ), finite set of edges (E V V ), prticulr entry vertex (), nd one or more exit vertex (X ): d1 1 g 1 l 1 h 1 l 4 k 3 c 3 f 3 e 1 b 1 P = (V, E V V, V, X V ) In the sequel we use lowercse letters to identify edges (e.g.,, b, c, etc., s in the CFG of Progrm 1). f 1 c 1 k 1 l 2 l 3 b 2 c 2 h 2 e 2 f2 b 3 k 2 h 3 e 3 A trce of progrm P is sequence of subsequent edges strting in nd ending in x X. The set of trces of progrm P is denoted T (P ). In the WCET frmework, we re only interested in progrms tht terminte in bounded time. It implies tht we lwys suppose the existence of finite set of ctully fesible executions, which is, in generl, strict subset of the progrm trces. We do not clim tht this finite set is precisely known, just tht it exists. We cll it the set of ctul executions nd note it E T (P ). We introduce now the notion of unfolding in order to formlize the notion of more precise progrm grph. Definition 1. An unfolding of progrm P = (V P, E P, P, X P ), with respect to set of executions E T (P ) is pir (U, δ) such tht: U = (V U, E U, U, X U ) is CFG, δ is mpping from U edges to P edges (i.e., decoding ): δ : E U E P the induced trce mpping is denoted: δ : T (U) T (P ) nd the set of decoded trces is denoted: such tht: T δ (U) = {δ (t), t T (U) E T δ (U) T (P ) For instnce, Figure 1 shows n unfolding of Progrm 1, where the mpping consists in ignoring the prime symbols; in this cse the unfolding is not more precise since T δ (U) = T (P ). Note tht the definition of unfolding cn be generlized by considering neutrl edges in U: in this cse the mpping is function δ : E U E P {τ where τ E P is simply ignored when extending δ to sequences (i.e., δ (τ) = ε). Among the unfoldings of P we re prticulrly interested in those tht re cyclic (i.e., DAGs): they structurlly reject the infinite trces of the originl P. In prticulr, n Figure 2: Acyclic unfolding ofprogrm 1 (for n = 3). cyclic unfolding is exctly wht we get (implicitly) in clssicl WCET techniques when bound is ssigned to ech loop of the progrm. Figure 2 shows n cyclic unfolding of Progrm 1, in the cse tht the number of itertions is bounded by n = 3. Note tht it trivilly exists, within the cyclic unfoldings of P, cnonicl progrm tht exctly cptures the ctul executions E (more precisely, clss of miniml grphs, equivlent modulo edge renming). In the sequel, we consider tht we hve n cyclic unfolding of the progrm P : we do not require tht it must be precisely built or known, only tht it virtully exists. 2.2 Avtrs nd implicit pths Let P be progrm, E the set of ctul executions nd (U, δ) n cyclic unfolding. From now on we cll P the concrete progrm (nd its edges the concrete edges). The edges in the reverse imge of the concrete edge re clled the vtrs of. We note m the number of vtrs of (m = δ 1 () ), nd, by convention, we note the vtrs with subscript indices, like in Figure 2 1 : δ 1 () = { 1, 2,, m Let us now come bck to the IPET frmework. In IPET, sets of trces re chrcterized implicitly by giving the number of occurrences of the edges. We cll these numbers the edge counters, nd note, for instnce, t the number of occurrences of the edge in the trce t. In order to simplify the nottions, nd whenever the context clerly concerns trce t, we will simply note for t. The bsic reltion between the counters of n unfolded trce t nd the corresponding concrete trce t = δ (t) is trivilly: m i = Another trivil property of the unfolded counters is tht they re either 0 or 1: 0 i 1 1 This does not men tht we consider nturl ordering of the vtrs, just think bout them s fmily of symbols.

6 2.3 Conflicting sets Let P be progrm, E the set of ctul executions nd (U, δ) n cyclic unfolding. U structurlly rejects ny infinite infesible pths, however there is still precision gp represented by G = T δ (U) \ E which is the set of infesible finite pths not rejected by U. Definition 2. Let C E U be set of edges, we note T δ U (C) T δ (U) the set of trces of U tht pss by ll the edges in C: C is conflicting set if TU δ (C) E = nd TU δ (C), i.e., it contins only infesible pths, mong them t lest one not rejected by U; C is miniml conflicting set iff ny C C is not conflicting set; A set C = {C 1,, C k is conflict covering iff: T δ (U) E T δ U (C i) 1 i k i.e., the conflicting sets of C re sufficient to reject ny infesible pth from U; C is miniml conflict covering iff it contins only miniml conflicting sets, nd ny C C is not conflict covering. Theorem 1. For ll (P, (U, δ), E), there exist miniml conflict covering. Here is sketch of constructive proof, which is indeed relted to the notion of prime implicnt in Boolen lgebr: there exist (t lest) one conflict covering: the one obtined form G = T δ (U)) \ E by interpreting the pths s (unordered) sets of edges, if conflicting set C in covering is not miniml, replce it by some other conflicting set C C, if covering is not miniml, replce it by some covering C C. This theorem is reltively trivil, but nevertheless importnt since it justifies the fct tht the notion of conflicting sets is (implicitly) considered equivlent to the one pruning properties in the literture. In n unfolding, the incomptibility due to conflicting set cn be expressed exctly in terms of ILP constrints. Consider multiset of n concrete edges (we use superscript indexes to void confusion with vtr nottion): x for x = 1 n. Consider set of n vtrs, one per ech concrete edge x : C = { x i x x = 1 n. Note tht we consider multisets of concrete edges for the ske of generlity: it is possible tht different vtrs of sme concrete edge re conflicting; it rises for instnce when edge in loop cnnot be tken in two consecutive itertions. If C is conflicting set, then, for ny execution: n x i x n 1 x=1 For the sme multiset of concrete edges, it is likely tht mny others sets of vtrs re lso conflicting. This leds to the notion of conflicting from time to time : multiset of (concrete) edges {{ x is conflicting s times (i.e., is s- conflicting) if there exists s sets of their vtrs tht re conflicting. 3. COMPLETION OF S-CONFLICTING 3.1 Rough completion Cse of 3 edges In order to mke the presenttion more cler, nd to keep the nottions redble, we consider here the cse of 3 concrete edges, denoted, b nd c. Note tht nothing in the following resoning requires the these edges should be ll different:, b, c must be understood s multi-set contining 3 edge references (e.g. nd b my refer to the sme concrete edge). The resoning cn be esily extended to the generl cse of ny number of edge references. Even if the resoning is bsed on the existence of n cyclic unfolding, we try to keep the informtion bout it s bstrcted s possible; we suppose given: the number of vtrs of ech edge, denoted m, m b nd m c; the number of conflicting sets of vtrs, denoted s. In order to void confusion, we use different index symbols for the vtrs. Avtrs re denoted i for i = 1 m, b j for j = 1 m b, nd c k for k = 1 m c. The corresponding counters properties, holding for ny execution re: m m b m c = i, b = b j, c = j=1 k=1 We know tht {{, b, c is s-conflicting. Thus, there exists set S of s triples (i, j, k) such tht ech { i, b j, c k is conflicting set. The sum of the corresponding liner constrints gives: (i,j,k) S i + b j + c k 2s The ide is now to complete this constrint in order to get rid of the vtr detils nd come up with reltion between the full counters only. Without ny other informtion thn the number s, we cn hrdly do better thn complete the reltion with ll the triples tht do not belong to S. For these triples, such tht (i, j, k) / S, the following trivil reltion holds: i + b j + c k 3 Moreover, there re m m b m c s such triples. By summing ll these trivil constrints with the conflicting ones, we obtin, in the left hnd side, m b m c times the sum of ll vtrs, tht is, the complete counter. Similrly, the full b ppers m m c times, nd the full c m m b times. Finlly, we obtin reltion free of ny vtr detils: m b m c + m m cb + m m b c 2s + 3(m m b m c s) or equivlently: c k

7 Formul 1. m m + m m b b + m m c c 3m s where m = m m b m c Generl cse The 3-edges formul cn be esily generlized to the incomptibility between ny number of (possibly redundnt) edges: let X be multiset of X concrete edges. Ech edge occurrence x X is chrcterized by its number of vtrs m x, nd the incomptibility is chrcterized globlly by the number of conflicting sets of vtrs s: Formul 2. x X m m x x X m s where m = x X m x Inefficiency of rough completion Let us consider the left hnd prt of Formul 2: ech edge counter x is, by definition, bounded by m x, thus the whole left hnd sum is intrinsiclly bounded by m X. The gin in precision of the Formul is then only s. Except for very specil cses (big s, smll number of edges nd/or smll number of vtrs), the formul is unlikely to give useful informtion. One of the rre cse where the Formul gives n exct informtion is when m = m b = m c = s, (i.e. cycle-free progrm with 3 incomptible tests): the Formul + b + c 2 precisely cuts the infesible pths. As soon s the m s re greter, the formul cuts very few infesible pths, nd thus gives very imprecise results. Consider 3 edges within the sme loop (executed n times), conflicting t ech itertion. The prmeters re m = m b = m c = s = n nd the formul gives: + b + c (3n 2 1)/n which is equivlent (since, b, nd c re integers) to: + b + c 3n 1 The inefficiency is clerly drwbck of the completion, nd not of the ILP pproch, since we know tht there exist, for this prticulr exmple, constrint tht precisely cuts the infesible pths: + b + c 2n. The gol of the next section is to propose method for finding, when it exists, precise ILP formultion of the conflict. 3.2 Precise completion Multiplicity nd lck The problem of the rough completion is tht it introduces huge mount of useless informtion of the type i 1. We will try here to introduce the miniml number of useless informtion. Consider the incomptibility reltion: i + b j + c k 2s (i,j,k) s nd focus for instnce on the term involving the i vtrs. This term is of the form: m α i i with m α i = s The mximum of the α i is clled the multiplicity of nd denoted p = MAX m (α i). For ech α i, we define α i, its complement to p : α i+α i = p. Intuitively, the α i describe the vtrs of tht re missing in the conflict constrint: m m α i i + α i i = p Moreover, the detils of the α i hve no importnce, only their sum is importnt: m α i = p m s We cll it the lck of in the conflict, nd note l = p m s. Consider for instnce Progrm 1 in the cse n = 3, nd the corresponding cyclic unfolding in Figure 2. The numbers of vtrs re m = 1 nd m b = m c = 3. There re s = n = 3 conflicting sets { 1, b 1, c 1, { 1, b 2, c 2 nd { 1, b 3, c 3. The multiplicity of is 3 ( 1 ppers 3 times in the conflicts), while the multiplicity of b nd c is 1 (ech vtr ppers exctly once in the conflicts). It follows tht the lck of is l = = 0, nd the lck of b nd c is l b = l c = = 0. As consequence, there is no lck t ll in the constrints: no vtr is missing in the conflicts sets. Consider now the exmple Progrm 2, with n = 3, for which the CFG in Figure 2 is lso n cyclic unfolding. We still hve m = 1 nd m b = m c = 3, but there re now only s = 2 conflicting sets: { 1, b 2, c 1 nd { 1, b 3, c 2. The multiplicity of is 2, nd thus the lck is l = = 0. The multiplicity of both b nd c is 1, nd thus the lck is l b = l c = = 1. Intuitively, there is lck in the conflict sets: one vtr of b nd one vtr of c re missing. Note tht we do not cre bout wht prticulr vtr is missing or not: only their numbers mtter Lck completion For ech missing vtr, we cn dd trivil constrint stting tht is it less thn 1, nd by summing ll these trivil constrints, we obtin: m m α i i α i = l The sme resoning holds for the terms in b nd c, nd finlly we cn build globl sum of the conflict constrints nd the three lck constrints tht erses the vtr detils:

8 m m α i i + m b j=1 β jb j + mc γ k c k 2s k=1 α i i l m b j=1 β jb j l b m c k=1 γ kc k l c p + p b b + p cc 2s + l + l b + l c Precise completion (3 edges) To summrize, in order to obtin precise trnsltion in ILP of the incomptibility, we need: the numbers of vtrs m, m b nd m c, the number of times the incomptibility holds s (i.e., the number of vtr conflicting sets mong the m m b m c possible ones), for ech edge, its multiplicity in the conflict, tht is, the mximum occurrence of prticulr vtr i in the set of conflicting sets: p, p b nd p c, from these informtion, we compute the reltive lcks of ech edge, e.g., l = p m s, nd then we cn stte tht the (ternry) s-conflicting formul holds: Formul 3. p + p b b + p cc 2s + l + l b + l c Precise completion (generl cse) This result cn be esily generlized to the incomptibility between ny number of (possibly redundnt) edges. Let X be multiset of X concrete edges. Ech edge occurrence x X is chrcterized by its number of vtrs m x. The incomptibility is chrcterized globlly by the number of conflicting sets of vtrs s, nd, for ech edge occurrence x, its multiplicity p x, nd the corresponding lck l x = p xm x s. The generlized n-ry s-conflicting formul is: Formul Exmples x X p xx ( X 1)s + x X Across-loop conflict. In Progrm 1, the conflict between, b nd c holds for ech of the n itertions, thus: s = m b = m c = n nd m = 1, p = n nd thus l = p m s = 0, p b = p c = 1 nd thus l b = l c = p b m b s = 0, nd finlly: n + b + c 2n which is the precise trnsltion of the incomptibility, s it ws obtined by d hoc resoning in the introduction. l x for ( i =0; i <n1 ; i ++){ red ( x ) ; i f ( x ) { / / for ( j =0; j<n2 ; j ++){ red ( y ) ; i f ( not x or y ) { / b / for ( k=1;k<n3 ; k++){ red ( z ) ; i f ( not ( x nd y ) nd z ) { / c / Progrm 4: loops. Exmple of ternry conflict in nested An even simpler exmple is when the 3 edges belong to the sme loop, nd the incomptibility holds for ech of the n itertions: n = s = m = m b = m c, p = p b = p c = 1 nd thus l = l b = l c = 0, nd finlly: + b + c 2n Note tht the sme resoning works for edges tht re in distnt loops: only the number of conflict mtters, not the precise structure of the grph. Nested-loops conflict. Progrm 4 is nother exmple of ternry conflict, but where the conflict propgtes within nested loops. The edges, b nd c re ppering (respectively) in nested loops executed loclly n 1, n 2 nd n 3 times, thus: m = n 1, m b = n 1n 2, m c = n 1n 2n 3, The conflict propgtes to the whole nested loop, i.e.: the first is incomptible with the n 2 first b nd the n 2n 3 first c, nd so on. For this exmple, the vtrs re numbered form 0 to m 1 in order to simplify the nottions. The incomptibility holds for ll the triples: (i, n 2i + j, n 3(n 2i + j) + k) with 0 i < n 1, 0 j < n 2, 0 k < n 3. It follows tht: s = n 1n 2n 3, p = n 2n 3, p b = n 3, p c = 1 nd l = l b = l c = 0 nd finlly: n 2n 3 + n 3b + c 2n 1n 2n 3 b c

9 cond = red ( ) ; for ( i =0; i <n ; i ++){ i f ( cond ){ / / cond = 0 ; else { cond = red ( ) ; Progrm 5: Edge is uto-conflicting between two consecutive itertions. Conflict between different itertions. Consider the Progrm 2, where the conflict holds from one itertion to the following (i.e., if b is tken t loop i, then c cnnot be tken t loop i + 1). In this cse: n = m b = m c nd m = 1, s = n 1, since ll { 1, b i, c i+1 is conflicting, p = n 1, nd thus l = p m s = 0, p b = p c = 1 nd thus l b = l c = p b m b s = 1; the globl lck is then 2s + l + l b + l c = 2(n 1) + 2 = 2n nd finlly: (n 1) + b + c 2n Auto-conflict. This exmple illustrtes the fct tht conflicting concrete edges do not hve to be different. Consider Progrm 5: (1) the loop is bounded by the constnt n, (2) whenever is executed, it becomes unrechble for the next itertion. This is n exmple of pirwise conflict, covered by the generl formul: p + p b b s + l + l b where, indeed, one hs to keep in mind tht = b. The prmeters re m = n, s = n 1, p = 1 nd thus l = 1, nd finlly: + n n CONCLUSION This pper presents generl method for trnslting infesibility properties into Integer Liner Progrmming (ILP) constrints, suitble for the use in IPET method. The trnsltion of infesibility in terms of ILP constrints is fr from new, nd numerous exmples cn be found in the literture. But the gol of this work is not compete on precision or ccurcy with existing pproches. It is theoreticl study tht proposes generl formultion tht in some sense en- compss the existing ones nd outlines the fundmentl limits of the method. In prticulr, it does not consider the problem of finding pruning properties, but only the one of reflecting them s precisely s possible, without the help of ny complementry method (e.g., grph trnsformtion). The resoning is bsed on the existence of n cyclic unfolding of the progrm. However this unfolding is kept minly bstrct: rough solution only requires to identify (1) the number of times ech prticulr edge is unfolded, (2) the number of conflicting sets of edges in the unfolding. In order to provide finer solution, more precise informtion is necessry, tht intuitively gives the number of time prticulr edge is involved in the infesibility property. Even with this finer solution, the formultion is sometimes not perfect, in the sense tht it does not reject ll the infesible pths. This is not drwbck of the proposed method, but generl limittion of ILP, tht rises whenever the exct formultion requires to express disjunction. In this cse, the generl solution is to combine ILP with by-cse resoning, but this somehow orthogonl problem is not considered here since the ide ws to explore the limits of the strict conjunctive ILP formultion. 5. REFERENCES [1] B. Blckhm, M. Liffiton, nd G. Heiser. Trickle: utomted infesible pth detection using ll miniml unstisfible subsets. In Rel Time nd Embedded Technology Applictions Symposium, Berlin, Germny, April [2] J. Engblom nd A. Ermedhl. Modeling complex flows for worst-cse execution time nlysis. In RTSS, pges , [3] J. Gustfsson, A. Ermedhl, C. Sndberg, nd B. Lisper. Automtic derivtion of loop bounds nd infesible pths for WCET nlysis using bstrct execution. In RTSS, [4] C. Hely nd D. Whlley. Automtic detection nd exploittion of brnch constrints for timing nlysis. IEEE Trns. on Softwre Engineering, 28(8), August [5] T. H. Kim, H. Bng, nd S. D. Ch. A systemtic representtion of pth constrints for implicit pth enumertion technique. Softw. Test., Verif. Relib., 20(1):39 61, [6] Y.-T. S. Li nd S. Mlik. Performnce nlysis of embedded softwre using implicit pth enumertion. IEEE Trns. on Computer-Aided Design of Integrted Circuits nd Systems, 16(12), [7] Y.-T. S. Li, S. Mlik, nd A. Wolfe. Efficient microrchitecture modeling nd pth nlysis for rel-time softwre. In Proceedings of the 16th IEEE Rel-Time Systems Symposium, RTSS 95, pges 298, Wshington, DC, USA, IEEE Computer Society. [8] R. Wilhelm, J. Engblom, A. Ermedhl, N. Holsti, S. Thesing, D. Whlley, G. Bernt, C. Ferdinnd, R. Heckmnn, T. Mitr, F. Mueller, I. Puut, P. Puschner, J. Stschult, nd P. Stenström. The worst-cse execution-time problem - overview of methods nd survey of tools. ACM Trns. Embedded Comput. Syst. (TECS), 7(3), 2008.