Decomposing Loops: The S Decomposition

Starting with section 5.5, our goal has been to constructBS, for each shapeS.BSsummarizes

all strings of the formrpiqpqi`1,qi`1q˚, and with shapeS. In section 5.6 we restricted the space of shapes to only those that are upward-flowing, and in section 5.7, we defined a partial orderĎover shapes which constrained the possible shapes of sub-paths. In this section and in section 5.9, we will constructBS. Our construction ofBS is inductive: we assume thatBS1 _{is known for all strictly smaller shapes}S1 Ĺ S_{. Furthermore, since each string in} rpiq

pq_{i`1</sub>,q<sub>i`1}q is non-empty, R Jr

piq

pq_{i`1</sub>,q<sub>i`1}q`K. We therefore separately handle the

case ofand we will now construct an expression vectorB`S which summarizes strings (all

necessarily non-empty) inrpiqpqi`1,qi`1q`and with shapeS. Define the expression vector B_{Sas follows:}

B_S,v,i“

#

ÞÑ _ifS“ txÞÑx,yÞÑy,zÞÑzu, _and bot otherwise.

qi`1 qi`1 qi`1 qi`1 qi`1 qi`1 qi`1 qi`1 S1„S w1 S2„S w2 ¨ ¨ ¨ Sj„S wj ¨ ¨ ¨ Sl„S wl SfĂS wf qi`1 qi`1 qi`1 SpreĂS w<sub>pre</sub>PJr piq<sub>pq</sub> i`1,qi`1q˚K S<sub>suff</sub> w<sub>suff</sub>PJr piq<sub>pq</sub> i`1,qi`1qK

Figure 5.5: Decomposing paths in rpiqpqi`1,qi`1q` whose shape is S. Each w P tw₁,w₂, . . . ,w_luhas shapeSw„S, and each of its proper prefixeswprehas shapeSpreĹS. EquivalentlywPL_firstpS_wq. Each stringw P tw₁,w₂, . . . ,w_lucan itself be unambiguously written as w “ w_pre¨w_{suff, with} w_suff P Jr

piq_pq

i`1,qi`1qKbeing a single iteration of the

q_{i`1</sub> Ñq<sub>i`1 loop.}

Then we can write:

B_S “B_SelseB`_S. _(5.8.1)

S-decompositions. Consider any pathwPJr

piq

pq_{i`1</sub>,q<sub>i`1}q`Kwith shapeS. From proposi-

tions 5.15 and 5.16, we can unambiguously decomposew“w₁w₂¨ ¨ ¨w_lw_{f, where:}

1. each substringw1 P tw1,w2, . . . ,wl,wfuis a self-loop atqi`1:w1 PJr

piq_pq

i`1,qi`1q`K,

2. each sub-path qi`1 Ñw

1

q_{i`1 for} w1 _{P tw}

1,w2, . . . ,wlu has shape Sw1 _{such that} Sw „S, and for each proper looping prefix w2 ofw1 (i.e. such that|w2

| ň |w1

|and

q_{i`1</sub> Ñw2 q<sub>i`1),}S_w2 ĹS_{, and}

3. the final sub-path wf is also a self-loop: wf P Jr

piq

pq_{i`1</sub>,q<sub>i`1}q˚K, but has a strictly

smaller shapeSf ĹS. (Note that the final sub-pathwf is possibly even empty.) We call this splitw“w1w2¨ ¨ ¨wlwf theS-decomposition ofw. See figure 5.5.

Summarizing the segments of theS-decomposition. _{We will now construct expression}

vectorsAS1_{which summarize these minimal sub-paths}w1 P tw₁,w₂, . . . ,w_lu_{. For each shape} S1

„ S_{, let}L_firstpS1

q be the set of non-empty stringsw1 P Jr

piq

pq_{i`1</sub>,q<sub>i`1}q`Ksuch that all

proper prefixeswpreofw1have shapeSpre ĹS1. Observe that for eachw1 P tw1,w2, . . . ,wlu,

w1

PL_firstpS_w1q_{. The string}w1 _{can itself be unambiguously written as}w1 “w_prew_{suff, where} w_suff Prpiq

pq_{i`1</sub>,q<sub>i`1}qis a single iteration of theqi`1 Ñqi`1 loop. Ifwpre has shapeSpre, and wsuff has shape Ssuff, then they are summarized by the expression vectors BSpre and Rp_Siq

suffp

q_{i`1</sub>,q<sub>i`1}qrespectively. We have already constructed both of these expression vectors by the induction hypothesis. Therefore, we can define:

A_S1 “ ÿ S1¨S2“S1,S1ĹS1 BS1¨R piq S2pqi`1,qi`1q. (5.8.2) By construction, we have:

Proposition 5.18. For each shape S1

„ S, the expression vectorA_S1 summarizes all paths in L_firstpS1

q.

Short and longS-decompositions. Pick a small constantl0, and consider stringswsuch

thatqi`1 Ñw qi`1 with shape Sand with shortS-decompositionw “w₁w₂¨ ¨ ¨w_lw_{f, i.e.} lďl0. All such strings are easy to summarize: for eachlP t1, 2, . . . ,l0u, and for each shape

S1 „S_{, we define:} A1_S1 “A_S1, _and Al`1 S1 “ ÿ S1¨S2“S1 Ai_S 1 ¨AS2, so that Aă_Sl1 “ l´1 ÿ j“1 Aj_S1. _(5.8.3)

This still leaves the problem of strings with arbitrarily longS-decompositions. The following result, which is an immediate consequence of proposition 5.17, is an important first step. Recall that we defined a stable shapeSas one with the property that ifuÑSv_thenvÑSv_. Proposition 5.19. Let there bekregisters in the SSTM. Pick a stringwPJr

piq

pq_{i`1</sub>,q<sub>i`1}q`K

with shape S. Let w “ w₁w₂¨ ¨ ¨w_lw_f be its S-decomposition. Then for each sequence of k

consecutive segments,w1

“wjwj`1¨ ¨ ¨wj`k´1,Sw1 is stable. There are two main consequences of this result:

1. For non-stable shapes,lăk_{, and they are therefore easy to summarize.}

2. If theS-decomposition is divided into “k-segments”, each composed ofkconsecutive segments,w“ pw₁w₂¨ ¨ ¨w_kq ¨ pw_{k`1</sub>w<sub>k`2}¨ ¨ ¨w_2kq ¨ ¨ ¨ pw_l1qw_{f, then each}k_-segment

w_s,m“w_{mk`1</sub>w<sub>mk`2}¨ ¨ ¨w_pm`1qk

attains a stable shape. See figure 5.6.

(a) Every non-support register u R SupppSq is reset while processing ws,m. The ultimate value ofuis therefore a function of the suffixwl´k`1wl´k`2¨ ¨ ¨wlwf.

(b) If v P SupppSq, then every incoming register while processing ws,m was reset while processing the previous segmentws,m´1. The value appended tovwhile processingws,m is thereforeentirely determinedbyws,m´1¨ws,m.

We will write an expressionfwhich mapsws,m´1ws,mto the value appended to

v_{after processing}ws,m. The expression of interest will then be chainpf,rq, with

qi`1 qi`1 qi`1 u u u v v v w w Ss,m´1„S, and is stable ws,m´1 “ wpm´1qk`1wpm´1qk`2 ¨ ¨ ¨wmk Ss,m„S, and is stable ws,m“ wmk`1wmk`2¨ ¨ ¨wpm`1qk

Figure 5.6: Analyzing data flows while iteratingk-segments. The entire pathqi`1Ñwqi`1

has shapeS, which is stable. The S-decompositionw “ w₁w₂¨ ¨ ¨w_lw_{f is divided into}k_-

segmentsws,m “wmk`1wmk`2¨ ¨ ¨wpm`1qk ofkconsecutive iterations. In this figure, we

show two consecutivek-segments,ws,m´1ws,m. By proposition 5.19, eachk-segment itself has a stable shape.