SÍNTESIS ORGÁNICA DE COMPUESTOS QUINOLÍNICOS POR MICROONDAS

Definition 5.2. A shapeS:V ÑV˚

is a copyless function over the set of registersV. Given an initial stateqand an input stringwPΣ˚

, letσ“qÑwq1

be the corresponding path through

M_{. The shape of the path}σ_{is the function}S_σ :VÑV˚

such that for all registersvPV_,S_σpvq

is the projection ontoVof the register update expression,µpq,w,vq: Sσpvq “πVpµpq,w,vqq. Because the register setVis finite, there are only a finite number of copyless functions,

S:VÑV˚

x y z x y z (a)q1Ñaq1,SK. x y z x y z 1 2 (b)q2Ñbq3. x y z x y z 1 2 3 (c)q2Ñbbq3,SJ. x y z x y z 1 2

(d)Shape of the update x:“yz,y:“x,x:“. x y z x y z 1 2

(e)ShapeS1of the up-

datex :“ x, y :“ yz, z:“. x y z x y z 1 2

(f)ShapeS2of the up-

date x :“ xz, y :“ y, z:“.

Figure 5.3: _{Visualizing shapes as bipartite graphs. Figures 5.3a–5.3c describe the shapes}

of some paths in the SSTM2from figure 3.2. With the orderxăyăz_{, only the shape in}

figure 5.3d is not upward-flowing. We will refer to the shapes in figures 5.3a, 5.3c, 5.3e, and 5.3f later in this chapter. For convenience, we name themSK,SJ,S1, andS2 respectively. The namesSKandSJare motivated by their position in the pre-orderĎdefined in section 5.7.

Example 5.3. It is helpful to visualize shapes as bipartite graphs, as in figure 5.3. When multiple edges lead to the same vertex, such asx Ñ x_,y Ñ x_{, and}zÑ x_{in figure 5.3c,}

the numbers on the edges disambiguate the order: soSJpxq “xyz. An edgeuÑvcan be informally read as “The value ofuflows intov”. Because of the copylessness restriction, every node on the left is connected to at most one node on the right. 4 When two paths are concatenated, their shapes are combined. We define the concatenation

S₁¨S_{2 of two shapes}S_{1 and}S_{2 as follows. Given a register}vPV_{, let}S₂pvq “v₁v₂¨ ¨ ¨v_{k. For}

eachiP t1, 2, . . . ,ku, definesi“S1pviq. Notice thatsiPV˚is a sequence of register names. Our definition ofpS₁¨S₂qpvq PV˚

should also be a sequence of register names. We define

pS1¨S2qpvq “s1s2¨ ¨ ¨sk,

i.e. the concatenation of the individual sequencessi. Informally, the concatenation of shapes corresponds to the composition of their bipartite graph visualizations. By definition, therefore:

Proposition 5.4. Letσ “q Ñw q1 _andσ1

“q1

Ñw1 q2 _{be two paths through an SSTM,} such that the final stateq1 _{ofσis the same as the initial state ofσ}1_{. Then, for all registersv,}

S_σ¨σ1pvq “ pS_σ¨S_σ1qpvq.

Let S be the shape of a path σ “ q Ñw q1

through an SST. Consider a register v so that Spvq “ v₁v₂¨ ¨ ¨v_{k. The summary update for register} v _{is therefore of the form} µpq,w,vq “s1v1s2v2¨ ¨ ¨vksk`1. We call each positionpv,mq, formP t1, 2, . . . ,|Spvq| `1ua patch in the corresponding shape.

Definition 5.5. _{An expression vector} A _{for a shape}S _{is a map from patches}pv,mqto consistent DReX expressions,Av,m:Σ˚ ùD, such that all the expressions have the same domain: DompAv,mq “DompA_v1_,m1q_{for all patches}pv,mq_andpv1,m1q_.

Pick a languageLĎΣ˚

, and a stateqPQ_{such that all strings}w_{when processed starting}

fromqhave the same shapeS. An expression vectorAsummarizesLif (a) for each string

wPL_{each patch}pv,mqofS,

JAv,mKpwq “sv,m,

wheresv,mis the constant appearing at positionminµpq,w,vq, and (b) for each component

expression,Av,m, DompA_v,mq “L_.

Example 5.6. Consider the loopa˚ at the stateq2ofM2. Consider the concrete stringak. The effect of this string is to updatex:“x¨ak_,y:“y_{, and}z:“z¨bk_{. The shape of this set}

of paths is therefore the identity functionS“ txÞÑx,yÞÑy,zÞÑzu_{. Define the expression}

vectorAas follows:

A_x,1“A_y,1 “A_y,2“A_z,1“a˚

ÞÑ, A_x,2“iterpaÞÑaq, _and

A_z,2“iterpaÞÑbq.

ThenAsummarizes the set of pathsa˚ starting from the stateq2. 4 We now restate the desired invariant (informally described in the proof outline in sec- tion 5.1):

Invariant 5.7. In step iof the SST-to-DReX expression translation, the expression vector Rp_Siqpq,q1_q

summarizes all paths inrpiqpq,q1qwith shapeS.