The lattice of flow diagrams

- THE LATTICE OF FLOW DIAGRAMS by Dana Scott Oxford University Computing Laboratory Programming Research Group ｾＯＮ｜ＩＧＺ ｉｾＢ , I ... . ....,""",..,..:4" ｾＧＢ Ｎ Ｌ . "!, '> I '''''I It11 J C", THE LATTICE OF FLOW DIAGRAMS by Dana Scott Princeton University Technical Monograph PRG-3 November 1970 (l'epl'illttod ｕ｣ｴｯ｢ｾｬＧ 1Y7l:;) Oxford University Computing Laboratory. Programming Research Group, 4S Banbury Road, Oxford. ';l-..., UA, ::'UQ ＨＢＧｾ｣ｲＺｏｦｽｬ © j970 Dana Scotr Depar;::;;Jent of Philosophy, 1879 r:211, Prir.ceton Uni versi ty, Princeton, New Jersey 08540. This pa.lJer also appears in Symposiu.m on Semantics ＱｮｾｅＺ leI' (ed.), Lecture Notes in Ma.thematics Volume 188, ｓ ｰ ｲ ｩ ｮ ｾ ･ ｲ Ｍ Ｂ ･ ｉ Ｇ ｬ ､ Ｆ Ｌ lleidelberg, 1971, and appeilrs as a TechniCdl Monograph by special arrangement with the publishers. of A Zgol'i thmic Langu.agQs, 1t'W'in References in tile lirerature sbe ld be ;:jade to t"he Springer Series, as rhe texts are dentical and the Lecture Notes are generally ｡ ｶ ｩ ｬ ｡ ｢ ｾ ･ n libraries. ABSTRACT This paper represents a continuation of the program sketched in Outline of a Mathematical Theory of Computation (PRG The language under consideration is the elementary 2), ｬ｡ｮｧｵ｡ｧｾ flow diagrams where the level of analysis concerns the of ｦｬｯＧｾ of control but not any questions of storage, assignment. block structure or the use of parameters. A new feature of the approach of this paper is the treatment of both syntax dnd semantics with t-he aid of complete lattices. This provides considerable unification of methods (especially in the use tf recursive definitions) which can be applied to other languages. The main emphasis of the paper is on the method of semanticcl definition, and though the notion of equivalence of diagrams is touched upon a full algebraic formulation remains to be done. CO:-iTIl\TS Page o. Introduction 1. Flow 8iagrar::s 2• ｃｯｮｳｴｲｵ｣ｴｩｮｾ 3. Completing the 4 • The s. Loops alld Other Infinite 6. The Semant ics of Flow Diagrams 7. The ,'1eaning of a Lattices ａｾｧ･｢ｲ｡ of 8. Equivdlence of 9. Conclusion References 12 ｌｾｴ ｩ｣･ 19 ｄｩ｡ｧｲｾｭｳ ?\ ｾｩ｡＾［ｲ｡ｭｳ ｾＺ［ｌｩｬ･ ｌ｣ｾＩ＿ ｾｩ｡ｧｲ｡ｮｊｳ i1 38 4 ｾＬ 48 55 57 TilE LATT! CE OF FLOi', DlAGR.!I.MS O. INTRODUCTION This pLlper represents an initial chaptu in a development of a mathemaTical theory of computation bdsed ))1 la1:tice theory and especially on 1:he use "f con1:inuous functior,s defined on complete la1:tices. may consult Scott Le1: D he a For a general ｯｲｩ･ｮｴ｡ ｩＨｾｲ •• the ｲｾ｡､･ｲ (1970). complete lattice. l;, .1, We use the symbols: T,u,n,U,n to denote respectively the partial Opde1'7:ng, the "Least element, the greateEit element, the joi7l of two element;,;, the meet of two elenents, the join of a set of elements, and the meet of the se1: of elements. The definitions and mathematical properties of these notions ca: he found in many places, for example Birkhoff (1957). Our notatiOll is a bi1: altered from the standard notation to avoid confusion wit e the differently employed notations of set theory and logic. 2 The main reason for attempting to use lattices systematically throughout the discussion relates to following well-known result ｾｨ･ of Tarski: THE FIXED-POINT THEOREM. Let l:O .... 0 be a monotonic fu.nction 0 defined on the complete lattice and taking values also in D. Then f has a minimal fixed point p = f(p) and in fact = p nCr E xl . D:f(x) !; For references and a proof see Bir-khaff (1970), p. lIS, and Bekic A function is called monotonic if whenever x. y E D and (1970). :r!; y, then f(x)!; fCy). Clearly, from the definition of p the element is !; all the fixed points of f (if any). The only trick is to use the monotonic property of f to prove that p is indeed a fixed point. In the case of continuous functions we can be rather more specific. Continuous functions preserve limits. It turns out that in complete lattices the most useful notion of limit is that of forming A subset X the join of a directed subset. ｾ 0 is called ､ｩｾ･｡ｴ･､ if every finite subset of X has an upper bound (in the sense of !;) belonging to empty. x. This applies to the empty subset, so X must be non- This also applies to any pair x. y ex. so there must exist an element B e X with x U Y ｾ B. An obvious example of a directed set is a chain: X = LrO,xl, •••• xn, .•. l where :1: 0 ｾ xl!; .,. !; :1: n ｾ ..•. The limit of the directed set is the element Ux. In the case of a chain (or any sequence for that matter) we write the limit (join) as: O:1: n n=O , A function f: D - 0 is called contin<.<ol.<s if whenever xeD is directed, then f<UX) ｾ Ulf(x)" E xl It is easy to show that continuous functions are monotonic. Note, too, that the definition also applies to functions !:D --. 0' between two different lattices; in which case we read the right-hand side of the above equation as the join-operation in the second lattice 0', In the case of continuous f: 0 ... D, the fixed point turns out to be: p Of"<» 11=0 where fO(J:) = J: and [''1+1(J:) :: !(/n(;r;)). This all seems very abstract, but there is a large variety of quite usefUl complete lattices, and the fixed-point theorem is exactly the right way in which. to introduce functions defined by recursion. This has been known for a long time, but the novelty of the present study centers around the choice of the lattices to which this idea may be applied. In particular, we are going to show that the familiar flow diagrams can be embedded in a useful way in an interesting complete lattice, and ｴｨ･ｾ that the semantics of flow diagrams can be obtained from a continuous function defined with the aid of fixed points. Of course. t::his is only one small application of the method, but it should be inst::ructive. 1 ,flOW DIAGRAMS, like Figure 1. Intuitively, a flow diagram looks very roughly There is a distinguished input information "flows" and an s.:dt or output will (hopefUlly) come. goes on inside the "black box·'. tive ｯｰ･ｲ｡ｴｩｯｾ ｰｯｩｾｴ ･ｾｴｬＧｙ ｰｯｩｾｴ into which the out of which the result The main question. then, is what Now, the box may represent a primi- which we do not analyze further, or the box may be compounded from other diagrams. A trivial example of compounding may be the combination of diagrams whatsoever. ｾ｣ The result is the "straight arrow" of Figure Z. 4 ｟Ｎｾ ｾ Fioure I Fioure THE A FLOW DIAGRAM 2 IDENTITY r----------..., -.,..;·1 d H I;· d' JI L Fioure 3 Fioure A PRODUCT r--------, I I I 1 I I 1 -7"'1ｾ I I I I I 1 _ I I I 1 I J L Fioure 5 A SUM ･ｲｵｾＨｩｆ A 4 A SWITCH 6 LARGE DIAGRAM 5 The in1oI'::Jdtion flowing a:ong SLch Oi.:::'hanr.el exits untransformed; and so that diagram represents the identity function. trivial compound is a ｰｲｯ､ｾ｣ｴＬ shown in Figure 3. ｾｮ A non- this combination, called ｩｾﾭ the output of the first box is fed directly into the put of the second with the obvious result. 'tJith products mation flows, alone not much useful could be done. As infcr- it must be tested and switched into proper channels according to the outcomes of the tests. For these switches we shall aSS'-.lme ::'or simplici ty in this paper that a fixed stock of primitive ones are given. This is not a serious restriction, and the method can just as well be applied when various forms of compounding of switches are allowed. We shall assume, by the way, that information flowing through a switch, though tested, exits untransformed. In diagrams ｲ･ｾ a switch is represented as in Figure 4. In case the suIt of the test is pQsitive, the information flows out of the top; if negative, from the bottom. gram because it has two exits, A switch by itself is not a flow diaIf these "wires" are attached to the inputs of the two boxes, and then if the outputs of the two bOxes are brought together, we have a proper flow diagram. in Figure 5. It is shown We call this construction a sum (of the two boxes) :or short, but it is also called a oonditional because the outcome is conditional on the test. Sums and products are the basic compounding operations for flow diagrams; iterating them leads to large diagrams such as the one shown in Figure 6. Here, the primitive boxes and switches have been labeled for reference and to distinguish them. The attentive reader will notice that we have cheated in the diagram in that the (-) and (+) leads from b l and b 2 have been brought together. for doing this was to avoid duplicating box [4' such shortcuts are not allowed: The reason Strictly speaking, all repetitions must be written vut. The diagrams will thus have a "tree" structure with switches at the 6 ｾｩｴｨ branch points and along tl'€ strings of boxes Ｈｾ･ branches. ､ｲ｡ｾ of the tree all the lea;Js Ｈ｡ｾｹ these trees nur.ber including zero) ｳｩ､･ｾ｡ｹｳＮＩ :Oro'.1ght together fo:- the output. d.c€ What is wrong in figure 57 ｾｳ That is to say, what :::cr.o, Obviously, the ans'.·wr is that there are no grams permit feedback around loops. ｾｳ At the "top" The proper all good ::'101.' ,jia- to allOW looping lHy discussed in the next section; first, we must connect grams in the intuitive sense '..,lith the mathemac.ical Some notation (.Jil1 help. lacking? ｴｾｌ･ｯｲＯ T",'e. helve already used ｾｬｯｷ diaｾｴＢ lattices. the notation D ,D ,b , · · · 2 O 1 Ｚｾｯ｟Ｌｦｊ lor the switches and ｢ｯｸｾｳＬ or binary; while the "F" ｏｾ tions ';'-"2"" information.) recalls ｂｾｯｬ･｡ｮ is used because the boxes represent .r"uncFor the identity (or "d'J;nmy") diagram we may Suppcse J use tne notation I. (The "]" res?ectively. ｡ｾ､ d' are two diagrams, then the product is denoted by: U.;d' ) where :he order is the saIT.e left-right order as ill rigure 3. The sum is written: U'J ..... :i,d') whi::,; is the familiar "C8nditional expression" used ｔｾ form for diagrams. OlO'-> (fO;(I1;(b here in an adapted djagram of Figure 6 may now be written as: l ..... .r'3,i 4 ) ) ) , ( f 2 ;(b 2 .... ,.f'4,t S ) ) ':"his expression has molLY too :;lany parentheses, bU1: we shall have to ｰｲｯ｢ｬ･ｾｳ of case, it is clear ｴｾ｡ discuss ･ｾ ｩｶ｡ｾ･ｮＲ｣ before we can eliminate any. 5U1";'_ a of diagrams we may talk of expressions ｩｾｳ ･｡､ gene:ated from the .-I'i and i ,md product operations. b:t ｾｯｲ･ In any by repeated applications of the various The expressions may get long, but it is obvious what we are talking about. 7 The totality of all expressions obtained in the way described above is a natural and well-determined whole, but just the same, "e are going to embed i t in a much larger complete lattice by a methcd similar to the expansion of the rationals to the reals. The firs1 step is to introduce a sense of approximation, and the second ster is to introduce limits. In our particular case, a very convenient way to achieve the desired goal is to introduce approximate (or: partiaL) expressions which interact with the "perfect" expressions we already know in useful ways not directly analogous to the common notion of approximation in the reals. way to treat reals, however.) (There is an exactly parelJel Existing between approximate expresｾ sions is a partial ordering relation which provides the requirec sense of approximation of one expression by another. the details of setting up We now turn to this relation. If the relationship d i; d' between partial diagrams is to mean that d approximates d', then it seems very likely that in a large number of cases d can approximate many different; d'. In particular, we may as well also assullie the existence of the worst (or most incomplete) diagram ｾ which approximates everything; that is, .1 will hold for all d'. i; d' In pictures we may draw whose contents are undetermined. .J. as a "vague" box Now, these incomplete boxes may occur as parts of other diagrams, as has been indicated in Figure 7. The expression for Figure 7 of course would be written as: (b O ..... (fO;.J.).(f1;(b l --> f 2 ,I))) If we are going to allow incomplete parts of diagrams, then we must also allow ourselves the option of filling in the missing parts. 5 r I - --, I 1 IL Figure I ..J 7 r----, I I AN INCOMPlE,E DIAGRAM I + r---' I I I IL.. ｟ＱｾＭ ..J Figure 8 1 L I ..J r---' r IL I I I oJ r---, A PRODUCT OF TWO INCOMPlETES I I 1 I I ..J L.. Figure 9 A SUM OF TWO INCOM PlETES ｾ Figure 10 THE OVERDETERMINED DIAGRAM 9 Thus, if d is incomplete, then a more precise reading of the reld- tionship d is that d t is like d .1 !;;;" d' d' except that some of the parts left vague in ,1 have been filled in. ship ｾ rhat reading is quite correct for the relation- that must always hold. the desired results In compound cases we can assure by assuming that sum and product formations dre monotonic in the following precise sense: i f dO ｾ d 1 and Ｑｾ (dO;d O) ｾ (oJ. ｾ then (d 1 ;di) and -+ ell,a:!) ｡ｳ ｾｭ･､ must be Ｈｾ itive, and antisymmetroic di. ,1 ,d ) !;;: (b j 0 O ---> Besides this, the relation ｾ to be reflexive, tro,zrrB- is a partial ordering). As an illustration we could fill in the box of figure 7 anj prove by the above assumptions that: (b O - (fO,.l),(f1;(b l ---> f 2 ,I))) ｾ (b O .... (fO;(!l;fO»,(fl;(b l ..... f 2 ,I») In working out these relationships it: seems reasonable to assume in addi tion that: ;.L) (.1 but: riot to assume that: (b U ..... .1,.1) ｾＮＱ as may be appreciated from the pictures in figures 8 and 9. for the sake of mathematical symmetry (and to avoid making exceptions in cert:ain definitions) we also introduce an except:ion21 diagram denoted by T about which we assume: d [;: for all d. and T We can think of .1 as being overdetermi'ied. T as being the The diagram ｵｲｬ､･ｲ､･ｴ ｲｭｩｔｬｾｊ diagram, T is something like a short circuit -- we will make its "meaning" quite precise in the section on semantics. We assume that 10 ( T ; T ) T, b'..lt no t :hat (bj"'T,T)"'T, Other equa- again fc;, reasons that will be semantically motivated. tions tr1at might seem reasonable (say, (d;T) '" T) are postponed to the disclJssion of equivalence. Taking stock of where are now, we can say that we begin with WE certain "atomic" symbols (representing elementary diagrams); namely: fo,f 1'" .1, • ,fn" •• ,1 • T T:'len, we form all combinations generated from these using: and (b Ｈ､［ ｾｊＭ j -. d,d'). These ex:Jressions are partially ordered by a relation ｾ about which we demand first that 1.1;dr;T for all di and then which we subject to the reflexive, transitive, ｡ｾ､ monotonic laws (the so-generated relation will automatically be d:1tisymmetric) . This is the "symbolic" method which is quite reasonable and is well motivated by the pictures. We could even pursue it further and make the totality of expressions into a lattice in the following way. ｔｾ･ join and meet operations must satisfy these laws: dUd' d n d' d' U d = d' n d = n d d d d d U .1 = d d n , , d T - T d n , d dUd U In addition for the atomic expressions other than st ipulate: fi f., U U f I ., j = , f·1, n f·J • .f., n I '" , , .1 and T we 11 where i f. For the case of products we have: j. = (.L ,1-) (T; T) .1 ::: 1 and in the following assume that the pair J,d' exceptional pairs f. i (b f i or T,T: 1- ,.1 = U (d;J') .... J do,d U (b ) O dO -+ 3 f i n (d;d ' ) ::: T U (J;d') ,dO) = (b r.i T where do,d O -+ j , dO ,do) is arbitrary. ｾ j dO,d n (b n I (OJ .L o) n .... do,d j In Cd;d') IUCd;d')=T I U (b is not either of the -+ Cd;d'l ::: O ) = 1- 1 -=.L do Moreover, for any two pairs J ,do) L u ,d6 and dl,di we assume: (dO;d DU d l ) (do n dl;d Dn di) O) U Cd l ;di) (dO U dl;d (dO;db) n Cd l ;di) = Cb Cb J J -+ dO,d o) u ,<'?J .... dO ,dO) n (b j ,J l) = Ｈ｢ｾｬ --> d -+ dl,di) = (b 1 .... dO U dl,d O U j .... dO dU n dl,d O n di) Finally, it" m"ight seem reasonable "to assume; , (b . -dO,db)U (b k .... d1,dU J (b. J when J f. k; but we ---.. dO ,dO) n (I:;/< --> dl,dU ｾ ｾ , postpone this decision. This large number of rules allows us to compute joins and meets for any two expressions (in a recursive way running from the lor.ger to the shorter expression), and it could be shown that in this ｾ the expressions do indeed fcrm a lattice with the parcial ordering. The proof would be long and boring, always the case with symbolic methods. relation as the however, as is The reasons one must exercise care in chis approach are in the main these two: that all cases are ｾ｡ｮ ･ｲ one must be sure covered, and one must be certain that different orders in oarrying out symbolic operations do not lead to inconsistenT 12 Now, it would be quite possible to do .:ill this for our con- results. struction of the la.ttice of diagrams. but it is quiTe unneoessary because s better method is available. The ideil. of the better approach is to work wi th a.re knowM to be lattices from the very start; hence haVe to :heck th", lat:-i::e laws except in SOIT.e trivial structures that we shall never cases. Next, some operations on structures art:' carried out wr:ich are known to transforrl lattices int<:;1 lattices (in our case this '.. i l l correspond TO the fermat ion of compound expressions). finally, main vir:ue of the approach) the extensiun tc a cO"lplete Ｎｊｾｳ｣ｲｩ Ｇ･､ f:lay be ｾｨ･ in a neat way. extent :,) which it has been ｡､ｪｵｮ｣ｴｩｾｔＢｩ ｡ｰ ｬＧｯＮｾｨ･ｮ､･ lattice lattice of expressh·ns to the up "':0 t:Jis point is not com- plete; ､ｾ the "are: :1€ structural 0.pp::ooach will make the exercise of limi t8 requires more or 1<:::55 dutomat ic. this is the (ane I t must tie stressed, J. ｮ ｩ ｡ ｴ ｲ ･ ｾ ar::OG.nt cof of this care that after ｨｯｷ･ｶｾｲ＿ the desired structures are creclted as lat"tices a certain amount of drgUr:lent is required to see thaT the STrUC'·'.;res con:orr:' to our ir!t.Jitiv€ ide'!.s about eXFressions. Though necessary, this will n0t be difficult, as we demonstrate in the next section. ＲＮｾｏｎｓｔｒｕｃｔｉｎｇ are ｴｲｙｾＺＱｧ and I. ate I LATTICES. to construct The initial part c:f the lattice we ｣ｾＩｲ ･ｳｰｯｮ､ｳ to the atomic symbols. 0': 1" . 3ince these symbols play slightly different roles, we separfrom the others. ;Jow, all we really knew abOlt t":1ey are pairwise distin::T; 'lenc", them by elements 0: ｾ it will be ｳｵｦｾｩ｣ ･ｮｴ is that to represent lattice illustrated in figure l l . tures of lattices the partial ordering is represented Ii the In s·.Jch pic- by the ascend- ing lines, the ·..."'aker (smaller) element.:; are below and the stronger (larger) elements are above. (By the way, a lattice is not a flow diagram; the two kinds of pi:='tures should not be confused. trying to make flo·... Jid8rams eLements of a lattice.) We are What the m rl ,..-. ...... .......... w U 1-0 0 0--< i= N ｾ I;i --' ｾ I- w ｾ ｾ ii: i= VI w u i= •• • lL ... ｦＭｾ 0 w i= --< j !, ｾ ii: w J: I- ...J 0 ｾ l- U I- I;i II ｾ + :::E :::> ｾ VI ｾ ｾ ｾ ii: w F 14 pict'-.lre :£ ':ne ｯｲ､･ｲｩｮｾ la""[<::i::e F :..., ?ig'.lr'€ 11 sno.... s is ｾｨ｡ｴ the 'J::1y par1:ial relations allOWed are: ｾ _'"' !;;: T for '" 11 IrJ figurL' 12 '.,;e ｾ･ｳｩｪｅＺ :k 1 anc T have a representalion of '[he ｬ｡ｴ ｩＬｾ･ [I} eler.lents has only one main eler:1er.t which i-o: should I. be menti"ned in setting up these partial orderings that to check that ｦｏｾＧｬ they comp12"te lattices means that every subset of the partially ordered 5et must have a least upper bound (its join) ｾｮ far t'he reS'.Jlt In the two cases we have so of the p:lrtial ordering. the sense is abv le..ls. SU;Jpose now that partial orderings ｾ D and 0' are two given complete lattices with and i;:: T. respectively. Inasmuch as structur2 that is important. we Day assume as sets :;f o and 0' are disJ"oint. unified lattice: it is only elellents that '"Ie wish to comt-ine 0 and 0' together in une it '..... ill be called the sum :Jf the :wo lattices and will be jenoted 1y 0+0' Ｚ ｳ ･ｮｴｾ］ＮｾｬｹＬ it is just the ＺＬＰＱｾ Ｇ ＢＡ of the two sets s:ruct;Jred by the "union";:Jf the two parrial order'ing relations. is This partial ordering not a lattice, however, hecause there is no largest and no small- es t e le;r,ent. These could be adjoined from the outside, but a more convenient and more "economical" procedure is as follows. Let T,T' and 1,1' be the largest and smallest elements of 0 and 0' respec- tively. o 'we have been regarding them as distinct (ai? the elements of were to be distinct [rom the elements of D'), but now j'Jst these two pa:r5 will be r.la::'.e eq'-.lal. ｡ｮｾ r=T' ｲ･ｳｵｬｴｩｾｧ The 13 . ihat is, '.....e s)-,all de::ree for 0+0' 1=1'; though all the other elements are i<:e;Jt separate. that The partial orcering is easily see!! to te a. c::Jmplete lattice. ｰｲ｣ ･ｾＵ of forrr.ing ｴＩｾｩｳ SUI':"L of lattices is ｩｾｬｌＮｳｴｲ｡ｴ･､ in figure lS The initial lattice of atomic expressions (diagrams) we wish to consider, then, is the lattice: F+{I} It will be noted that the notion of sum just introduced could easily be extended to infinitely many factors. Thus, if we considered Lat- tices {fi} that structurally were isomorphic to {I) (but with different elements), then the lattice F could be defined by: F = {fo}+{f1)+ ... +{f )+ ••• n Though they are not by themselves atomic expressions, the symbols b, will diso be thought of as elements of a lattice B defined by: B = {bO}+{ol}+' .. +{b The lattices F, B, different because n }+ •.. and F+{IJ are all isomorphic as lattices but are they have different elements. These, however, are very trivial lattices, and we need much more complicated structures. Suppose D and D' are lattices whose elements represent "dia_ grams" we wish to consider. If we want to form products of diagrams. then according to the intuitive discussion in the last section, the partial ordering on products should be defined (dO;d O ) ｾ that (dl;di) if a'1d o'1ly if dO!; dl'and dO!;' for all dO' d l E D and all dO' di ED'. < dO,d ｾｯ di Abstractly. we usually write > as an ordered pa-£r in place of (dO;d O)' and then write: O D::.:D' for the set of all ordered pairs < d,d' > with d E D and d' E Dr. The above biconditional defines a partial ordering on D::.:D' called the (cartesian) product ordering. and, as is well known. the result is again a complete lattice. The largest and smallest elements of D::.:D' are the pairs < T,T' > and < ｾＬＧ Let the lattice DO = F+{I} >, respectively. 16 b€ the 1attic:e of atomic expressions. Then the lattice °o+(OOXO o ) c8uld ｾ･ regarded ｡Ｎｾ 1::­te lattice ｷｨｩｾｨ ｡､ｇｩｴｾ｣ｮ in 1:) the aTa:;,i::: expressicns has compound expressions which can be thought of as prQducts of t·.... CI atomic expressions. ｴｨ･ｾ In fact, is no rea::,']n tc use the abst:"act notation < d ,1' >; '..Je can ｣ｄｾｰ･ｬ ｩｮｧ use 'the more suggestive (d;d') remembering that lattice­theoretically this is ｾ｣ｴｩ｣･ just an ordered ?air. in this regard thaT: b;: our ｣･ｦｩｮ ｴｩｾＩ［ Ｚ of sums -wd products of lattices we have the equations ( .1 ;.1.) = ( T ; T) .1 = T .'3.utOr.Lat.:._a::'ly. Wh:l.t about diagra.ms? \o,Te11 , even though we wrote (b,-i --.. d,d') ｡｢ｳｴｲ｡｣Ｚｾｹ all we have is an ordered triple < is just an element of the ｔｨｩｾ > . ｨｾ［Ｌ､Ｇ ｩ｡ｴ ｩ｣ｾ BxDxO . (if the reader wants to be especiallY pedanti,:: he to B,(OxO) and < b"d,d' > ｲｾ･ b.;'< d,d' an independent notion of ordered triple. give isomorphic lattices.) sider ＧｾＢｊｕｬ､ or he take BxO"'O car. introduce Structurally, all approaches Hence, the next ｬ｡ｴｾｩ｣･ we wish to con- be 01 ｾ 0o+COOxOO)+(BxDOxDo) Again, :here is no reason to Ｈ｢ｾ », Cd.O ｾＳ･ the abs1:ra:::t noc:ation so that .... ,::',i') can just as "'ell stand for an :)rd.ered triple. ;lotice t),o]t we have in this way introduced SO.Tle eler.,ents noT c0:1sidered as d ｾ＼ｬｧｲ｡ＺＭＮＳ bef:)"e: (T'" d,J') c:.r.d (.1 .... d,d') but we shall fir,d that iT is e"1Sy tc interpret the:\ ｳ･ｭ｡ｲＮｴｩ｣ｾ ｬｹＬ s',: tha: this extra generality costs us no special effort. I f we 17 liy.e, we can also use the more suggestive notation for the lattices themsel yes and wri te: 01 = 00+(OO;Oo)+(B 0 0 ,0 0 ) 4 but for the time being it may be better to retain the abstract notation to emphasize the fact that we know all these structures as lattices. Clearly, 01 contains as elements only very short diagrams. To obtain the larger diagrams we must proceed recursively, iterating our compounding of expressions. Abstractly, this means forming ever more complex lattices: 0n+l DO+{OnxDn)+(BxDnxDn) The way we are construing the elements of these lattices, 00 is ｾ Bubset of each 0 : " DO C On and, in fact, DO is a 8ltbZatti<:€. on This means that partial ordering DO is the restriction of the intended partial ordering on On (res- tricted to the subset). And r..esicJes I the join of any SUI.H.t=t or" 00 formed within the lattice 0D is exactly the same as the join formed within goes °. (This last is very important to remember.) " for meets, The same but this fact is not so important. Consider that DO c 0 1 ' and that this implies that O'oxOo c 0 1 xOl both as a subset and as a sublattice. Similarly, we have; BxOOxOD C BxOlxOl . It then follows that 00+(ODxDO)+(BxODxOO) C 00+(OlxOl)+(BxOlxOl) both as a subset and as a sublattice. By definition we have: 18 D1 C 02 ' and continuing in this way, we prove: D>J ｾ D'1 + 1 • Therefore, c­ D'!'l D 11 whenever n .; m . What we have just done is to take advantage of general properties of t:te sum lattices, ｾｮ､ preduct constructions on lattices as regards sub- These general properties atollt "the comparisons of the par- tial orderings and thf' joins and_ ｭｾ･ｴ｟ｳ are_ very simple to prove ab- stractly, and the reader is urged to work out the details for him- As a result of self inc:uding the asserti'7ns of the last paragraph. these considerations it '.... ill be seen that the union set ｾｄｮ ｃｏｾ･ｲｮｴ has a partial ordering. complete lattice (we shall see why, later). joins and meets do C'n the other hand, many exist; in particular, the join of every finite (The reason is that any finite subset is subset exists in the union. wholly contained in one of the D .) n is a It is not a Is this a lattice? So, the union of the lattices finitely complete lattice (a kind of struoture that is ordiJJarily ｾｵｳｴ called ｗｾ｡ｴ a lattice). are the elements of this union lattice? ｾ･ all the finite combinations They are exactly desired generated from grams by means of the two 1':10des of compositi:)n. the atomic dia- Furt hermor€, the abstract lattice structure obtained in this way provides perfectly all the laws of computation we listed in the last section. abstract approach gives us 2. structure which we know is a Thus, the (finiTely complete) lattice on the basis of simple, general principles. Then, by reference to the construction, the laws of computation are worked 19 oat. Having worked them out in this case, we can see by inspecti:m of cases that we have all we need beca..lse there are only a limi'.:El number of types of ste;J is to cOJ.lplete elements formed in an iterative fashion. The next the lattice and the:1 "to figure out what is obtained. 3.CDNPLETING in birkJlOft (19£7). flow diagrams in a THE LATTICE. p126), Every lattice can be completed (as but we shall want to complete the ｣ｾ special way that allows "..IS to apprehend the nature of the limit elements very clearly. approximation will ｬ｡ＭＺ ｾｩ｣･ In particular, the notion 0: be made quite precise. Roughly speaking, the­elements of the lattice On are diagrams of "length" at most n. generators by nesting the most n. ｾ imation, t\.w modes of composition to Consider DO and D or it may not. l Dl If . a E 01' .1. at then it may belong If d E DO, then it is iTS own best ｡ｰｾｲｯｸﾭ If d If- DO, then since the elements of DO are not compounds (except in a trivial sense) the best we can do in DO is to d by 0: a level This suggests that the elements of 0n+l might be approximable by elements of On' to DO ｾＡｊ･ More exactly, they can be obtained from ｡ｰ ｲｯｾｩｭ｡ｴ･ In other words, wc have defined a mapping lfO:Dl --> DO ' where for d E D1 we have: 'fO(d) {: if d E DO if not. As can easily be established this mapping is ｣ｯｾｴｩｮｵｯ ｳ (in fact, a more general theorem relating to sum formation of lattices is provable), and this is important as all the fll.lf,pin£G we Ciliploy ought to be con:i<1uo'..ls. Now, consider 0n+2 and 0n+l' We wish to define f,!+1:On+2 .... 0"+1 . 20 for d E 0>;+2' the eler.tent oJ + (d) \0,1::11 be the best ap;oroxima­::ilJn n 1 Recall that tc, d by ar, element in 0>:+1' oｾＫｬ 0 +(0 "0 Dr.+:::' 00+(0r.+l"Dn+l)+(BxDn+lxDn+'.) 0 ｾ n )+(8)(0 n ｾｄ ) ｾ and Inductively, we may assume that we have already defined the marring v" ·0 0:' Clearly, what is called for is ..... 0 r;+ 1 ｴｾｩｳ '" definiti0n: ­LldED d ＨｾＧ､ＭＩｊ｢ III r.+ l Cd) 1 n n (b ..... ｾ Now, Cd")) (d')'iJ; n 'n (d"» O ; if d Ci' ',d") ,"r (b ... d' ,d") d these three Cdses are strictly speaking rict mutua,lly exclusive, but on the only possibilities of IJverlap we fine: agreement because l/J n ::: rand (T) iJ; (.1) ::: 3y a proof that need not detain us here, we .1. rt Note aIsQ that we may prove inductively show that iJ.'n+l is continuous. for all n that for dE 0n+l we have: dE 0 ｔｾ･ if n mapping ｾ ｮ Ｚ ｏ ｮ Ｋ Ｑ ｾ 'J,J n (d) ::; d ""S . Dr. is easily illustrated. the first belongs to 0 two diagrams are given: the reslJI t of applying a'Jd only -Lf to the first. 5 In figure 14 and the Second is I1: will be not:ed trlat drawn diagr.J.::1' are slightly ambiguous; this d.rohiguit:y is removed when one chases an expression for the Ｚ ｬｩｾｧｲ｡ｭＮ In this example we chose to associate to the right and to .interpret a long arrow without boxes as a single occurrence of I and not as a product of severa.l ['5. The lJHer diagral'7l is c.:Jmplete; 'dhile I.'h'"'­t we !':light call its pl'oj;;ctiQn from 06 into 05 is necessarily ｩｮ｣ｯｭｰｬ･ｾ Ｎ Clearly, we can recapture the upper figure by removing the vagueness of the IOl.'er figure. a very simple idea. in one position This is the way a:;::.proximation works. I t is --, I ­< I I __ J :::Ii c<: Q: '"Cc<: c<: '"z "i= ｾ f lrl .., '" Q: lL ｾ ;;: 0 22 Suppose d' E 0"1+1 and dE Dr. and d ｾ c')ntinLlOUS, but also monotonic. ";'r;CdJ ｾ d vie have therefore showr. thiit 3.'. Now if", is not only Thus, If"Cd') !; d' ｾ (d') is the z.argest elerr.ent of 0 " which approximates d'. n This reinforces our conception of " n a, a projection of 0n+1 upon On; the idea will now be carried a step further. Assume that we already knew how to complete the union ｾｄｮ to a ｣ｯｭｰｾ･ｴ lattice Om' If we were to be able to preserve relation- ships, we ought to be able to project DID successively ante each On' say by a mapping " "'n ,DO ｾOD n But these projections really should fit hand in glove with the projections we already have. One way of expressing the goodness of fit is by the functional equation ifron :: lbn°'foo(n+l) ｾ･｡ｮｳ which ｰｲｯｪ･｣ｴｾｯｮ that the projection from from o 00 onto 0 n . ｾｯｷＬ Dn + l onto ｏｾ onto D + n l followed by the On ought to be exactly the projection from Suppose this is so. let d E Om be any element of this ultimate lattice. Define a sequence of elements in the known lattices by the equation: dn for all n. "Cd) mn By what we have conjectured 1f',/d n + l ) = dn holds for all n, and so ､ｏｾ､ｬＡ［ !; d n !; d n + l ｾ 23 How does the limit element d fit into the picture? Since If",," Well. since Do<> is n ｭｩｾｬ｣ n= 0 n is a projection. we at least have d the limit of the d We Od ｾ d Easy. must also approximate d. n !;;: d for all n; thus, But why the equality? to be the completion of the union, each element of D", is determined as the directed join <limit) of all elements of the union ｾ it. (All elements of 0", must be approximable as closely as we please by elements from the union lattice.) If d 1 belongs to the union and d' !; d, then since d'E On for some n we must have d' !; d , n ilence ttle ..:quality. We have seen that each element d E D", determines a sequence < d n >n='" 0 such that Ibn (d"n) holds for all n. Furthermore, distinct elements of distinct sequences. (Because each d its corresponding sequence.) of elements d n = dn D.o" determine is determined as the limit of Suppose conversely that such a sequence E On is given and we define d E 0", by the equaticn d Ud n= 0 n We are going to prove that for all n: d n = """'nCd) In the first place. sinc:e these projections are continuous we have: l/I"'n For m ..; n. since d m (d) = Utn(d m) m=O E On' it follows that l/I"'n U m ) = d m 24 For 1"1 > Ｇｾ ･ >1, are going to prove thaT I/;",,"l 7his is L:'':= for",: \o,'e .ugue by inducticn on t!1e .::jClanti*:.y "'! ｩｾ Having JUST: checked Cd",) :: i n :or the value 0, suppose '(he value is [Josiei';e and that \o,'e know the result for ':::;'e previous value. :Jse the ＢＬ ［ＺＧｾ｡ｴｩｯｮ ("1-n). ｲ･ｬ｡Ｑｾｩｮｧ > Thus, m n. :""8 the \/d['ioI15 projections d'ld compute: V -n (l) 1,!;n("vooCn+l) (d"1) m vn(d n + 1 ) d Trv.on " :oince the required eq'.ldtio:1 is proved ',.Ie see: >J; oon ｾ［ｪ (ei) U ell U •.. U d U d n U n c­:>j U •.• 1 'i That is d >' to say, in the infiniTe join all the ｡ｮｾ but the previous ones r;;; d n after n ｴ･ｲｾｳ ｯｮ･ＭｾＢＧｬ which ｳＳｾｩｳｦｹ cot't'es00 Do;<> d.nd the sequences < dJ'J>';::O ＬＢＩｾ pondence he Tween "Lie ele:nents dI'E' anyway. In Ocher words, we have shown that there is a ｾ Ｌｾ the equations ¢'"Cd"q) :: d'1 Ｌｶｬ ､｣ｩｴ｡ｭ･ｮ ｾ this is ｶ･ｾｹ satisfactory ｢･｣｡ｾｳ･ it means that instead of a.ssll..,i"g t:'at we knol>.' D"" we can ｣ｯＢＮｾｴ Ｇｵ｣ｴ being :he set of these sequences. ｡ｾ･ In this sho'"rn "'[0 be mathematically consistent. it as actually ｾ｡ｹＬ ｯｵｾ intuitive ideas This construction is par- ticularly pleasant hecallse the partial ordering on Dc<> has this easy sequential definition: d The ｯｴｾ･ｲ sequences. we ｾ d' i f a.nd ody if d n ｾ ､Ｌｾ fop aU "l. lattice operations also have easy definitions based on Hie But for the moment; the details need not deto'lin us. really need to know is that the desired lattice exist and tho'lt the projections behdve nicely. ｄｾ All does indeed In fact, it can be 25 ｰＬ･ｮ ｲ｡ｊＮｾＧＯ proved qllite ･ＳｃｾＮ U­.at is additive in the ,jl=r! for all X ｴｾ･ C r.OL ＡｾｬｮＩＧ ｾＩｵｴ con'CinClous ｾ ｓｲＧ sense that l./!"'nrUX) '" of "0; ｽＮｾ U{I;",r;iX);:: E' Hence. we Ci'ln ottai.rl a reas(mably clear­ picture 000' lattice structure of C",_ But 000 has Ｂｾｬ ･｢ｲ｡ｩ｣Ｂ structGre as well, and we no\o,' turn to its examination. 4. il THE ALGEBRA OF special '.·.. ay, the Ｎｓｾｍｒｇａｉｄ Because tre 0 n were cor.s1:ructeG in complete lattice 0", is much more than just a l:lt­rlce. Since we want to interpret the elerr.ents of 0"" as diavral'1s, the !:1(Jre abstr<Jct notation of the previous section by our­ earlier al- notation. ｧ･ｴｲｾｩ｣ ｲ･ｾＱ ｡｣ｅＧ \..'€ Thus. by construction, if d. d' E On and if ,­ E B. then bot;' (,l;d ' ) and (b ... d,d') are elements (j f 0 nn . Fhat if d, Will these d' ,:; Den? ｡ｬｾ･｢ｲ｡ｩ｣ cor- binations; of elements Il'Clye sense! In order to answer this in"terestinv question, we shall employ for elements d ED"" this abbreviated notation for project jon: dn = Femember that we regard each D n W""n(dl c;: D"" and so d is the largest element n If dE D + , then' n 1 Of On l,)hich approximates d. d n = 1i-'n(d) also, Using these convenient SUbscripts, we may then define for d. d' ED",,: (d'd') = , lJ"" (d n'-d') n ond (b ­+­ n=O d ,d') = Q(b n=O ­+­ Ｉｾ､ＬＮｲ The idea is that the new elements of 0"" will have the following projections: (d;e' )n-tl = Ｈ､ｮ［ｾＩ and (b ­+­ d.d' )n-tl (b ­+­ Ｉｾ､Ｂ 26 (The projections onto DO behave differently in view of the special nature of ｾｏ as defined in Section 3.) It can be shown that these operations (d;d') and (b - d,d') defiDed on 0"" are not Dnly contin(This answers the question at the end of Section uous but acditive. 2.) Hence, 0", is a lattice enriched with algebraic operations (called products and Bums and not to be confused with products and sums of IJflOle 'Lattices.) Let (0",;0",,) be the totality of all elements (d;d') with ｄｾＮ d, d' E This is a sublattice of 0"", a sub lattice of 0", Similarly, (6 ­ 0"",0",,) is In view of the cOnstruction of 0n+l from On we can show that in faeL: ｾ 0"" Because if d E 0 d' d" "' " E 0 " ｾ and ｂＨｾＩＬ Ｂｄ［Ｌ ＨＫｯＰ 0""0,,,) 4 if d €I DO, then we can find elements such that either for all n: d n " l = Ｈ､ｾ［Ｉ or there is some b E B such that for all d "'1 ｾ (b ­­+ Ｉ［ ､Ｌｾ Setting d' Ud ' "0 and d" n::O 7l . ｕ､ｾ n::O we find that either d:: (d';d") or d = (b ..... d',d ll (One ｭｾｳｴ also check that the T and 1 ) elements match.) Since there can obviously not be any partial ordering relationships holding between the three different types of elements, we thus see why Om decomposes into the sum of three of its sublattices. Inasmuch as Om is our ultimate lattice of expressions for diagrams, it will look neater if we call it E from now on. Having ob- tained the algebra and the decomposition, we shall find very little need to refer back to the projections. Thus, we can write: 27 E an equation which F+([)+(E;E'+(B ca.n be read very Ev;:cr!i erpressio01. or is the E;E) s/lloothly in words: eir;;het' a funcr;ion symbol, {$ ｩ､･Ｇｬｴｩ ｾ ｾ 8;.-:t::OZ, of 01' is the produ.ct 01' is the sum of two erpt'essions. tUG e:rpt'essions, These words are very suggestive but in a way are a bit vague. lie sho\ol ne>1t how to specify additional structure on E thdt will turn the above sentence into To carry out lattice: 15. o d mathematical equation. this last program we need to use a very important the lattice T of truth values. Aside from the ubiquitous (false) and 1 (true). operations fI (and). II .1 It is illustrated in Figure and T it has two special elemerts Defined on this lattice are the Boolear (or), I (not) give.n by the tables of Figur­e 15. For our present purposes, these operations are not too important however, and we discuss them no further. What is much more impor- tant is the conditional. Given an arbi tra['y lattice 0, the conditional is a functicn :l:TxDxD ..... 0 such that :l(t,d,d') , r" i ,r' T , ·if d' if t , if t 0 .= , The reason for the choice of this definition is to make J an add£.- tive function on TxOxD. ing us to test t. of the conditional. result of the test tional. Intuitively, we can read ｾＨｴＬ､ ＧＩ as tell- If the result is 1 (true), we take d as the value If the result is 0 (fa 1 se) we take d'. I f the is underdetermined, so is the value of the condi- If the result is overdetermined, we take the join of the """",-ot- > Ot-Ot- t- 0000 0 """",-ot"""","""",00 " J> .0. en ｾ ｾ !l:: ." -l .0. I '" -l "ｾ I """",-Ot- ｾ III :;; -l 0 0 0> < '" --I-t- t- In -1- Ot- 0 ." 0 -l- -l- -I r r 0 0 J> Z ." '"J> ;0 :::l 0 zen ｾ """",O-t- '" r ｾ ｾ 0 '" ­­I :!1 '"ｾ 0 ｾ u; t-<>-l 29 values we would have obtained in the self­determined cases. This last is conventional, but it seems to be the most convenient convention. It will be easier to read if we write (t :J d,d') :J{t,d,d'), and say in words if t It is common to write then d else d' in place of but we have chosen the latter :J, to avoid confusion with the conditional e:r:pres8io'fi. in E. Returning now to our lattice E there are four fundamental functions: func:E --> T idty:E .... T prod:E -+ T sum:E .. T All of ' these func-:ions map with d 1- T and d # .J. T to T and to .1 .1. For elements we have i.f d E F ''''Cd) 0 idty(d) = prod Cd) = sum (d) These functions ｡ｾ･ = { : {: : {: { if d e F if E en d ifdfi{J} if d E (E;E) e (E;E) if d i f d E ( B ..... E,£) if d II: (B .... E, E) all continuous (even: addi tive). They are the functions that correspond to the decomposition of E into four kinds of expressions. 30 Besides these there are five other fundamenTal functions: first:E E secnd:E E left:E E right:£ ­­>­ E bool:E ­+ B In case d E (E; E), we helve: d'" (firstC:D;secnd(d» otherwise: secndU) first(d) L In case dE (8 ..... E,U, we have: :i = (bool(d) -+ leftCd),rightCc:.'» otherwise: left(,l) r;ghtU) " .1 (:in E) and bool (d) 1 (in B) These functions are all continuous. These nine functions together with the notions of products and sums of elements of E give a complete analysis of the structure of E. In fact, we can now rewrite the informal statement mentioned pre- viously as the following equation which holds for all dEE: :i = (func(d) .J Ii (idtyCd) :J I CprodCd) :) (firstL!);secndCd») (sum(d) ::J (boo1(ci) .... leftC­­:),rightCd»,.L)) ａｾｯｴｨ･ｲ be this: way to say what the result of our construction is would the lattice E ｲ･ｾｬ｡｣ｳ the usual notions of syntax. This lattice is constructed "synthetically", but what we have just verified is the basic equation of "dndlytic" syntax. All we really need to knoloi about E is thdt it is a complete lattice that decomposes into 31 a sum of its algebraic parts. These algebraic parts are either gen- erators or products' and sums. The complete analysis of an element (down on.e level) is provided by the above equation which shows that the algebraic terms out of which an element is formed are uniquely determined as continuous functions of the element itself. Except for stressing the lattice-theoretic completeness and the continuity of certain functions. this sounds just like ordinary syntax. The parallel was intended. But our syntax is n.ot ordinary; it is an essential generalization of the ordinary notions as \Ie now show. 5.LOOPS AND OTHER INFINITE DIAGRAMS. In Figure 17 we have the most well-known ccnstruction of a flow diagram which allows tle information to flow in circles; the so-called while-loop. It represents, as everyone knows, one of the very basic ideas in programming languages. Intuitively, the notion is one of the simplest: enters and is tested (by b )' O tion is transformed (by f O information If the test is positive, the informa- ) and is channeled back to the test in pre- paration for recirCUlation around the loop. positive, the circulation continues. While tests turn out Eventually, the cumulatlve effects of the repeated transformations will produce a ｮ･ｧ｡ｴｩｾ test result ( i f the procedure is to allow output), and then the information exits. None of our finite diagrams in E the On lattices) has this form. overlooked something. ing E complete. (that is, diagrams in any of It might then appear that we had But we did not, and that was the point of mak- To appreciate this, ask whether the diagram lnvolv- ing a loop in Figure 17 is not an abbreviation for a more ordinary diagram. There are many shortcuts one can take in the drawing of diagrams to avoid tiresome repetitions; we have noted several previously. Loops may just be an extreme case of abbreviation. Indeed, J> 0 0 . :!I ｾ . '" 0. ;;! ;;: ;;: z C '" '"r r 0 0 " :!I ｾ ." "' 0; Z =i + '"< '" + ;0 (J) 6 z 0 ." --i :I: '" 50 " + 33 instead of bendi ng the channel bad<- around to the front of the dia- gram, l.Je could write the test again. And, after the next transfor- :nation, l.Je could And again. write it out again. And again, and The beginning of the infinite diagram that will again, and thereby be produced is shown in Figure 18. Obviously, the infinite diagram will produce the same results as the loop. (Actually, this assertion requires proof.) Does what we have just said make any sense? '''''ill help to see it dDes. tllat fa. the transformation Some symbolization We have symbols for the test hi) and Let the diagram we seek be called d. again at Figure 18. diagram repeats i-rself. Look After the first test and transformation the This simple pattern can easily be eX,rressed in symbols thus: d In other words, O .... Cfa;d),I) we have a test on the other hand, ｦｯｬ ｯｾｩｮｧＮ (b ｾ･ ｾｩｴｨ an exit on negative. compound fa ｾｩｴｨ the same procedure If Jositive, ｩｭ ･､ｩ｡ｾ･ｬｹ Therefore, the diagram contains it8etf as a part, That is all very pretty, but does this diagram d really exist in [? To see that it does, recall that all our algebraic operations are Du"l tinuou8 on E. Consider the function ｾＺｅ .... E defined by the equation: (l(x) The function $ ::: (b O .... (fO;x),I) is evidently a continuous mapping of diagrams. Every auntinuuus fU"lDtion on a DQ.mp'Lete lattiD€ into itself has a fixed po i n t . In this case, we of course want d to be the least fixed point: d ::: <J>(d) because the diagram should have no other quality aside from the endless repetition. The infinite diagram d dues exist. ｮｯｾ (It can:lOt be finite, as is Obvious.) We can see why we did not introduce loops in the beginning: their existence follows from completeness 34 and coo"tinui:y. In any case, they are very special and only Cln€ among many diverse types of infinite diagrams, Figure 19 shows a slightly more complex example l.1i th a double We shall not attempt to draw the infinite picture. since that loop. exercise is unnecessary. The figure \.lith loops is explicit enough to allow us to pass directly to the correct symbolization. plish this, label the two re­entry points d and d t • To accom- Following the flow of the diagram we can write these equations: d = (IO;d ' ) and d' = (b o ­­+ f Ｈｦｬ［､ｾＩＬＨ｢ｬＭ 2 ,(f , d ) ) 3 SUbstituting the first equation in the second we find: d t = (b O (f1;d'),(b --+ .... [2'([3;([0;d'»» 1 Now, the "polynomial" = (b 'l'(x) O -+ (f1;x),(b 1 f2 .... is a bit more complex than the previous ,(f ;(f ;x»» 3 o ＴｬＨｾＩＬ but just the same it is continu:Jus and has its least fixed point d'. Thus, d t , and there- fore d, does exist in E. Sometimes, the simple elimination procedure we have just illustrated does not work. A case in point is shown in Figure 20. The loops (whose entry points are marked d and d') are so nested in one another that each fully involves the other. drawing ｴｾ･ (By now, an attempt at infinite diagram is quite hopeless.) The symbolization is easy, however; d = (b O .... fo,(b l .... (f 1 ;d),(f ; i ' ) ) ) 2 and d t = (b 2 .... f 3 ,0'3 ­ Cf 4 ;d),(f 5 ;d'»). In this situation any substitution of either equation in the other leaves us with an expression still containing both letters d and d'. 35 That is 1:0 say. tne t;.;o dia>,;rams C'alled d and d strilcted ainl<4ltaneoJaZy. Is this possible? fact that E"E is also a complete lattice. e: [x[ ­ r h'j',l'" It is, ':0 l)e C'an_ Consider the Introduce the function E>([ defined as follows: 8C<.x,y» = ....:CbO--+f'O,CbJ .... Cfl;x),Cf2;y»),Cb2-f3.(b3 ...... Cf4;;;::).<fs;y»)> Now, this function e is continuous and has a least fixed ｰｯｩｮｾ［ < J,d'> = e« d,d'» and this rair is exactly the pair of diagrams we \o,·anted. This method can now be seen to be flexible and of wide applicability, For €xanple, if using our algebra on E, we write dewn any system of pOlynomials in several variables: nOexO,x 1 ｾｸＢ ... ) ,n 1 CXa,I l ,J'":;" .•. ) ,fl (x 'x2.'x , •• ) 2 O 2 I'" then on a sui table product lattice: [x[xEx ••• we can solve faT' :ixed points: dO "n o (d o .d l ,d 2 ,···) d l =n l U O,d l ,d 2 •· .. ) d2 "n 2 (d O ,dl'd 2 ,···) Diagrams constructed in this way may be called aZ-gebJ'aic eler:ents of E. The fini te diagrams in the union of the On may be called J'ationaZ-. of E; This classification does not by far exhaust the elements there are besides a continuum numbe'r of transcendental elements. (The reader may construct one from Figure 18 by replacing the sequence of boxes fa. f ' fa, ... by the sequence fa, f l , f 2 , o or by some other nonrepeating sequence.) Whether these other elements of E are of any earthly good remains to be seen. there, in any case. not do so. They are If you do not care to look at them. you need It will be your loss not theirs. }> c ." ::!! (") c: c:j ..,"' ｾ • '" I c I I }> 3: ." .c. c c '"0 '" ;;l j;; Cl :ll Z l/) \ .- ­" 50 \ Q. ­ " hI ｾ / / " l/) ; + 37 It is not elements of E. too easy to draw pictures of some of the algebraic Tcke, for example, this defining equation: d = (to ..... <j"o;(d;!l)),I) A first and an unsatisfactory attempt to draw this as a diagram is The question is what to fill in the middle. shown in figure 21. ',ve need another copy of d itself; but this involves still another copy of d. And" so on. There seem to be no shortcuts available. Any attempt to introdu:::e loops will not make it clear that in anyone tour of the chann.ds the same number of visited. f 0 boxes as f 1 boxes But this is a failure of the picture language. must be The alge- braic language i s unambiguous (Ilenct:. better!). Nevertheless, this example does suggest that there is a classification of the algebraic elements of E that needs adiit:ional thought. Now that we see something of the scope of E, we can organize the study of its elements \oOith the aid of further notations. For example, the while-loop is so fundamental that it deserves its own notation: (b*d) which stands for the least fixed point of the function (b It can be eas ily sho\oOn that E. (d;x) ,Il is a continuous function on BxE into There are many others. This is the place to clear up a continuing notational confusion. Since, in order to comllunicate mathematical facts, we need tc wri te formulas involving symbols, \oOe have to be clear about the distinction between a symbol and what it denotes. This distinction becomes par- ticularly critical when we study the theory of syntax, as we have been doing here. So, let us be very pedantic about the nature of the constructions we have been discussing. of E actually? What are the elements Either they are elements of F or of {I} or they are 38 pairs or triples of pairs or trip:es of ... of elements of Band E. Or they are limit points of these, wtlich finite Ｕｴｾ ｣ｴｬｹ speaking, are in- ("convergent") sequences of rational elements of E. Alas, E contains no symboZs, only mathematical constructs. But defined on E is a whole array of functions and constants: fo,f"-, ,T,(x;yJ,(b func(x), --> X,i!) ,first(.T.), ... ,;· •.Y' , , etc. ｬ･ｴｾ･ｲｳＬ Thus, such things as subscripts, capital parer.theses, semi- colons, arroW's, corrunas, bold-face letters, and stars do not actually occur as pdrts of any of the elements· of -our "expression" space E. Rather, E is La be regarded as a ｾｯＮＱｴｨ･ｭ｡Ｚ［ｩ｣ｬ "lode!. of a It is only one of many similar Do::lels. expr'eseio'll'. Or, E is a model lut' a. theory of [J6ometl'io d£.aUl'am6, ｾｮＬｊ tory theor",", at thu.t. ｴｾ you care make ture to guide yoUt' The it. Jf ｬ｡ｴ ｩ｣Ｚｾ E ｾｯ if yO'..! like, "'" quite Satisfac- E does not care what a.pplications E is abstrac:t. ｴｨｯｵｧｾﾷｈＺｳＮ gives you a fixed struc- It is the same with the theory of the real numbers and analytic geometry. it is up theoT'Y of These structures are "pure": us to supply the plot and to write excit:ing stories about them using a careful choice of language (that is, functions, relations, etc.>' In the Case of E, however, we can ask not only what it is, ar.d what its elements do, but also wha"t rjo they mean. 6.1HE SEMANTICS OF FLO", DIAGRAMS. ･ＧＮﾷｾ the flow of information through a diagram. have spoken all along of It is intuitively clear whaT is meant, but ev.:'ntually one must introduce tions S;}ffie if he €'/er hopes to get any defini te resul ts. precise definiIn other words, it is no.. time to present in detail a rr,athematical model of the concept of fZowing. Up to this point, everything is static: the ele- ments 0; E do not move; they do not light up, make noise, or otherwise show signs of life. We have sketched many pictures of elements 39 of E and on the paper, an these diagrams, we can move shift our eyes back anc. forth. OUT' fingers or The abstract elements of E I'e,':lain impassive, however, and must remain so, frozen in the eternal realm of ideas. But they neither expect or want our pity. free to study then" And, we are to talk about them as we do of works of art. Clearly, the first requirement i.n the study of the meaning of E is a theory of information. the artifacts in enough, in this Disappointingly pdper we shall not make a very deep study of this We shall take it as axiomatic that the qt.<a'lta of essential notion. information form ｬ｡ｴｩｾ･ a called; S If you prefer, you can also consider the lattice S as being the lattice of states (states cf "nature"). we do not say. Where the lattice comes from, We shall give some examples, by and by, but shall not be able to discuss lattices in general here. sonably evident It should be rea- from the sucoess we have had in constructing lattices with useful properties, that this assumption is no loss of gerlerality. Indeed, it can be argued that the requirement is a gain of gerlerality. In order to specify the meanings of the elements E, we must begin with the Ji E F. Here, we have great freedom: their mean- ings can be determined at will -- within certain limits. are set by this transformed. meaning of f each f i If i reasoning: The limits as information passes through a box it is the box is labeled with the symbol Ii' then the is this transformation. That is, corresponding to is a fu.nction 'J<fi)'S ｾ S which provides the means of transforming S. Note that the transfor- mation depends only on the label and not on the context of occurrence of a box, because we intend like labeled boxes to perform the same transformat ililn. Since we have gone to the trouble of saying that '0 S is a complete lattice, we will also require each func­cion 3<fi.) to be C'onti'lIWuB. Think for a ｾｯｭ･ｮｴ of the collection of all continuous func- 'tions from S into S. If u and v dre such, there is a most natural way of defining what it means for u to approximate v: lj if u(o) v if and onLy ｾ ｾ v(a) fol' all a 5 E It can easily be established that the set of continuous becomes in this way a complete lattice itself. lattice by (5 .... SJ. functions We denote this do not confuse this notation wit:-t (Ca:.tt"ior:: the earlier (B ..... E,E), which is a certain subl<l.ttice of E.) In a highly useful short-hand way we can say that a,F-rS 4 SJ We even require ';1 , as a mapping, to be continuous. 'JoE [F --jo Thus, [5 .... 5)] . In this manner. we indicate succinctly what is called the logical type of ;} as a mapping. Attention paid to logical types is atten- t ion well spen t. The next project is to attach meanings to elements of B. If bE B it designates some test that may be applied to elements of 5. The outcome of a test is a truth value. ment of the lattice T. For us, that means an ele- Hence, to have meanings is to have a (con- tinuous) function ｉＱｬＬｂＴｲｓｾｔｊ Both [5 ... 5] and [S - TJ have largest elements (both are la'ttices). In [5 - 5] it is the constant function T (obviously, a func'tion). We should write T[5-"'SJ E [S - where for all 0 SJ E 5: T[5_SJ(O) = T5 . continuous 41 But we drop the 1(0) :: 1. subscripts and write Tea) :: T. Similarly, for The sanle slightly ambiguous notation is used for [S ­­.. TJ. faT simplicity, '3 we require both and CB ｾＨｔＩ :: T 'J(ll :: (6(T) :: T @(l) to have the property that 1 w!1er€ it is left TO the reader to deterr:line to which lattices each of the S belong. T'S and -L' The functions 71 <B and may be chosen freely wi thin their respective logical types -- but that is all the freedom we have. The meanings of relative to this all the other elements of E are uniquely determined ':J. choice of and ca . To show how this works out, we shall determine a functicn "" Cagain;continuousl such that ＢＬｅｾ｛ｓｾ ｊ If dEE, then ""Cd) is the "value" of d (given ｾ and CI3 ). The intention is that if a E S is the initial state of the information entering the flow diagram d, then "J(d) is the final state upon exiting. (a) We thus do not teach you hc\ol to swiJ:l through the channels of the flow diagram, but content ourselves with telling you \oIhat you will. look like when you come out as a function of what you looked like when you jumped in. is, of course, continuous, (over all d and The transformation And, merely knowing this transformation all a) is sufficient for a mathematical theory of fZowi >i.g. The precise definition of out an equation V is obtained by simply writing that corresponds to what you yourself would do in swimming through a diagram. We write it first and then read it: "' VCd)(o) ,,(func(dP Cidty(d):.l (prod CdP ';J(d)(o) 0, "Ltc secnd (d» (U'Cfi rs lCd» (sumCd):Je03 (bool (d» (One small point: because ';J(d) ;;: ";Jed} .l (a» (0):1'11(1 eftCd» (0) ,t1'( r Igh ted» (cr» ,.d») we may regard"J. as being of type ·:1­:E ..... [5 .... SJ is a good value if de. F. Or, we should replace by "1<ld) Idl ; : where (dl = d if d E F, and .l E F if d e. The Translation of the above equation runs as follows: To compute the outcome of the passage 0; a through d J tion symbol. ';1ld) (0). ｦｾＢ｣ﾭ first ask whether d is a If it is, tne Dutcoml!J is If it is not. aek whether d is If it is, the identity symbol. ou.tcome is a. it is not, ask o.1hethel' If d is a product. then the If it is, [iTld the first and second terms of d. PaSB a through tile first term of d obtaining the propel' outcome. the Take this ｳ･ｾｯｮ､ final ｯｾｴ｣ｯｭ･ and pass it through term of d. ｯｵｴｾｑｭ･Ｎ That gives the desired If d is not a product, ask whether it is a sum. If it is, find the boolean part of d and test cr by it. Depend- ing on the result vf the test, pass a through either the left or the right branch of d, taining the desired ｯｾｴ｣ｯｾ･Ｎ sum (this ｾｷｴ is cose will ab- If d is not a arise), the outcome .L. One soon learns to appreciate equations. And, the equations are more precise as well as being more perspicuous -times they become so involved as to be unreadable, pIe, how our equation for 'tt though some- Note, for exam- tells us exactly what to do in case F.) 43 (B(booHd)) (0) J. 01"' ｾ T This would be rather TiresolTie to put in words. The question we need to ask now, however, is whether this equation really defines" . Obviously, iT is net an expl-[c-[t both sides of the ""­=luation. ",.. exists. definition because V occurs on Hence, we cannot claim straight off thaT: To prove that it does, some fixed points must be found in some rather sophisticated lattices. ｾｴ was not :ust an idle remark [5 is a complete la ttiee. ｾ to point out that 5J Knowing this, we have by the same token that ｛ｅｾ｛ＵｾＵｊ｝ is also complete­. And this lattice gives the logical type of" : 1J" E To find this 1.1" ｾ 2 [E ｾ [5 ｾ 5J] . then, as a fixed point, we would need a function [[E E ｾ [5 ｾ 5]] ｛ｅｾ｛ＵｾＵ｝ｊ｝ which is a lattice somewhat removed from everyday experience. that does not mat­cer: But we know all the general definitions. nere is the specific principle we need. pression the var·iables E, and 5 respectively. 3' , In the following ex- d. and 0 occur of types [E ... [5'" 5J]. The expression is: (fundd):) "J-{d) (0) ( idty(dPa (prod(d)) ｾ Ｈ ｳ ･ ｣ ｮ ､ Ｈ Ｉ (X (fi rst(d)) (a)) (sumCd):)(ca (bool (d)) (a):) 3(1 eft(d) )(a), JE(right(d))(o)) ,.d))) This is a function of three variables. at it, -chat it .is ｾｏＢｬｴｩＢｬｕｏ Ｖ We can prove, just by lookinc in its three variables. Forget about the exact form of The above €'Xpression ane imagine any such continuous €'Xpression: ( ••• 3£. •.• d ... o) Holding 1: and d fixed we have a function of 0.' The logical type of 44 the value of the expression is also S. ::: '( jf ,d) ､･ｾｊＢＬｮ､ｩ ｧ Dn Ｌｾｮ･ｶｩｧ :'hus, thet"'e is a :=:'( X,d)(a) ( ... X .. . d ..• 0) The logical L:ype of ::':'( 1,d) is [$ .... SJ. ｾ･ｨｴｯ '::n words, ::: '( is an "expression" whose value depends on l:. and 1.. thaL: ::: 'e 3( ,n function d such that is continuous in l: dnd d. ｾＬ､Ｉ .... t:' can sho\oJ Going around again, "[here must :'02 a function (a uniquely determined function) :::(:l-) such that E(3! )(d) ;E' (J( ,d) , so that E(}:) E [E • [S ­ But this correspondence is continuous in E E [[E - All ｃｏｮｴｩｮﾷｾｯｵｳ [S • S]] ­ [E - 5]] ｾ . So, really . [S ­ S]]] . functions have fixed points (when they map a lattice into i'tseif), and so our 1.1 is given hy ＱＡｾ ElV") with the u:lderstanding that ""e take the least such ­1J ( as an element of the lattice (E .... [$ .... SJJ). Yes, the argument is abstract, but then it is very general. The easie,t thing to do is simply to accept the existence of a con- tinuous (ninimal) ",. and to carryon from thel'e. worry about the ｬ｡ｴｾｩ｣･ theory -- as long as he is sure functions that he defines are continuous. take care of themselves. (Jne need not Generally, ｴｨ｡ｾ all they seem to Intuitively, the definition of "tJ'is nothing more than a recursive definition which gives the meaning of one diagram in terms of "smaller" diagrams. common ard are well understood. Such ,jefinitions are In the present context, we might only begin to .....orry ..,hen .....e remember th3.t a portion of diagram is not really "smaller" (it may even be eq!<:lZ inal). an infinite to the orig- :t is this little worry which the method of fixed points lays 45 "to rest. Let us examine what happens with the while­loop. 7.THE MEANING OF A WHILE­LOOP. Let b E Band dEE. Recall the defini tion of Ch .d) . It is the least ･ｬ ｾ･ｮｴ of E satisfying the equation: (b :r ::: We see that (b*d) E ­to (d;x) ,I) E,E) and (B .... bool «b*d» left ((b*d» = b (d;(b*d)) right (Cb_d» Hence, by the definition of 'l.t«b*d»)(o) ::: But Cd;(b*d» E ([;E) ( ::: I for '" , (J <8(b)(o):::) E S we have: l1«d;(bd»)(o),o) and first«d;(b*d») d secnd«d;(b*d») (b*d) So we find that; 'lJCChod»)(o) 0 C COCh)Co) 0 VC(h.d»)( '\JCd)(o»,o) The equation is too hard to read with comfort. . Let w ::: (b*d) und Ii 0 03Cb) E [S ｾ TJ J 0 '\JCd) E [S ｾ S] and . We may suppose that band d are "known" functions. The diagram is the while­loop formed fpom band d, and the semantical equation above now reads: 1Jcw)(o) (5(0'):::) "(w)(a(o)),O) The equation is still too fussy, because of all those o's. Let IE(s .... sJ 46 be such that for all Cl E S ; ICo) ::: ｾＬ For any two functions Cl • v E [5 .... SJ, let ;':'vE[S­S] be such that for all cr E S: (u·v)(o) ::: v{u(a) (This is fU:lctional composition, but note the order.) For dny P E [5 .... S J, and u, () E [5 .... S J. let (p .Y u,v) E [S ..... S] be such that for all Cl E S: (p u,v)(cr) -;> ::: (pCo) J u(er),v(o» Now, we have enough notation to suppress all the o's and to write: V(w) 0 (6 (d·!J(w)) ,I) at: last an almost: readable equation. It is impGrtant to notice that: tions (0: • and;" are eontinuo!./s func- rs several variables) on the lattices ｾ TJ and [5 ... SJ. Injeed, we have a certain varallelism: E [5 5] B [5 TJ fi [1: I 7 Cj E (x ;y) (li • U ) J (b:>x,y) (p with '.I prodOict of inter"'.st. also (l.I°v) (T > the obvioU5 functions}. to be a continuOUB ｡ｉｧ･ｄｲｊ［ｾ U ,v) S] is an algebra with constants f We could even say that [5 T, ;> and with sums u,v) and (1 The function ｨｊｭｯｾ ｲｰｨｩｳｾＮ (5 '..t,C') (and, u,v) wbere T, ''':E ｾ 1 [5 .. S] It is nJt a lattice homomorphism since it is continuous and not ing. E i anJ if they are [5 ｾ T] are tben proves in general join preserv- 47 IT does, however. preserve all products and sums ­ as we hav!'. illustrated in one case. Let us make use of this observation. I f we 11'. t ¢l: E ..... E be such that \0'" ¢lex) trlen our while­loGp w (b*a"l (d;x),J) is giv".n by w '" U4>l'1(.l) 71"'0 ｏｾ The function 4> is an algebraic operation i:rs . . . ｾ S] ... lS E; we shall let SJ be the corresponding operati0n such that (Ii . . . ((i·u),I) i(u) From what we have said about the continuity and algebraic properties of"', it follows that ""(1.1) Oin(l.) = ncoQ This proves that V'(lJ) is tIle least solution u E [5 -+ S] of the equation u Thus, on [$ 'lJ' preserve s ->- T]x[S ... shown that t1 is Actually, ,Il also a homomorphism with respect to this operation. tr.e solution to the equation (b -+ (d;x) ,I) It is not so in the algebra has only one solution. also. (;l·u) SJ analogous to the * operation on BxE, and we have u least solution. (6)- whi 1e; more precisely, there is an operation x = is unique in E. = == (E >­ [5 -+ 5J that thE equation «Iou) ,I) But we have shown that 1J" picks out the This observation could be applied more generally 48 In any case, we can now state definitely that the quantity 'I)(w)(o) is computed by the following iterative scheme: 21«1) = If the result is 1 (tro:e), the:'l puted. first 0(0) is comis compute,1 and '1' the whole procedure is started over on '1I(U)(O') If b(o) L 0, ;,. the result I::; [J at once. = .L, the :--e'3d:, iG ( I f h(<J) If bCo) = T, t!1e result is 1J(w)(a') U 0, ""'hich generally is not too ｩｮｾｦＧｲ･ＮＧＺ［ｴｩｮｧＮＩ The J:lin':'mality of the solution to the equa- tion in [S .... SJ means :hat we get ｩＱＺＯｴｨｾｮＮ ＧＱ "'ore than what is strictly implied by this computation scheme. This result is hardly sUf'prising; it was not meant to be. ｾｯｲＧ ･｣ｴＮ What it shows is that Jur definition is Everyone cOl1lputes a whi1e in the way indicated and the function what was expected: no more, no less. We can say that semantic function which maps the diagram ing" VC!<'). 10' " is the to its "value" or "mean- AId, we have just shown that the neaning of a whlle- loop is exactly a function in [S ..... sJ to be while-manner, process. '\YCw) gives us just The' meaning ｯｾ ｾｵｮｰｵｴ･､ in the uS'Jdl the diag:,amatic while is the wh41e- No one wnuld want it any other way. It is to be hopec. that the reader can extend this style of argument to othe.r' ('onfi gurations that may interest him. a.EQUIVALENCE OF DIAGRANS. Strictly speaking, the semantical interpretation defined nnn illustrated in the last two sections depends not only on the choice of S but on that of Indeed. we should ｷｲｾｴ･ ( ｾ S and take the logical -:ype of Of course, E [[F ｾ [S ｾ and 18 . more fully: 'lt 'll's ') S]J ｾ ) ( 8 )(d)(o) 'tt [[B to be: [S C'1 ) «8) is contir.uous in T]] ｾ [E ｾ '"3' and in lB . [S - S]]]] . 49 If we like, we can call the set S the set state.s of a "lachine. 'do The functions and give the behavior of the "hardware" of a (8 Thus, the lattice machine. [ F ｾ [S ｾ ｾ Sll. [B [S ｾ Tll may be called the lattice of ma",hi>;e8 (relative to the giver: S). This is obvious ly a very superficial analysis of the nature of machines: we have not discussed the "content" nor have we explained how a function values. Thus. 'Jet} of the states in S, manages to produce its for example, the functions have no estimates of "cost" of execution attached to them (e.g. the time required for computation or the like), The level of detail, however, is that generally common in studies in automata theory (cf. Arbib (1969) as a recent reference), but it is sufficient to draw some distinctions. Certainly, lattices are capable of providing the structure of finite state machil'l€S with partial functions (as in (19£17» I dlld much m0rp: ｴｨｾ Scott uses of aontinuoua functions on cert:ain infinite lattices S are more subtle than ordinary employmen: of point-to-point functions. The demonstration that the present gener- alization is really fruitful will have to wait for future publications, though. Whenever (Cf. als0 l3ek.ic, dnd Park (19b9)) one has some semantical construction that assigns "meaning" to "syntactical" objects, it is always possible to introduce a relationship of "synonymity" of expressions. the relation simply: We shall call equivalence, and for z, if E E write: x ;;c to mean that foT' all S and a l l ' ) y and £B relative to this S we have: "'S' "J )( Ql )(r) ｾ 'lJ S ' '3' H Ql Hy) This relationship obviously has all the properties of an eqJivalence relation, but i t will be the "algebraic" properties that will be of so more interest. In this connection, there is one algebraic relation- l;; Y J: to mean that '"3 all S and all :01' E we write: for x, y E ship that suggests itself at once. <:B and relative to this S we have: 'lJS (';J)(<!3)(y) 'lJ's('))«(J3)(X)S These relationships are very strong -- but not as strong as equal- ity. as we 51',03.11 see. to show, r; ｾ The ｾ and are related; for, as it is easy is refle;dlJo? and tl"ansitive. and further J: Y if and only if ｾ Z ｾ r; Y and y But these are only the simplest properties of ｾ x . and r;. For additional properties we must refer back to the exact def ini tioD 0: U' In the first place, the definition of in Section 6. 1Jwas tied very closely to the al-gebroa of E involving products and sums of diagrams. The meaning of a product turned out to be aOM- position of functions; and that of a sum, a eQnditional "join" of functions. ｾ The meanings of in the funcrion space [S and ｾ are equality and approximation SJ, respectively. Hence, it follows from the monotonic character of compos i tions and conditionals x r;; x' and y and (b r;; y ｾ implies (x;y) S x,y) ｾｯｲ ･ｳｰｯｮ､ｩｮｧ (x' ;y') x' ,y') and S noted in the last para- principle with r;; replaced by In view of the connection between graph, the (b r;; that: ｾ ｾ also follows. A somewhat more abstract way to state the fact just noted can be obtained by passing to equivalence classes. For x E E, we write ｸＯｾ '" {x' E E:x for the equivalen"e alaB8 of x under ｾ ｾ x'} We also write 51 to denote the set of all such equivalence classes, the so­called ｡ｚｧ･｢ｾ｡Ｎ quotient ｅＯｾｩｂ And the point is that an algebra in the sense that products and sums are well defined on the equivalence classes, as we We shall have just seen. " be able to make E seem even more like an algebra, if we write: xtY for all :c, y E those c::B E (6 ..... E. = (T ­+ x,y) No",..', in Section 6 we restricted consideration to [5'" T]] such that (8CT) = T Thus "'s( ｾＩＨｃｂＩＨｸｴｹＩＨ｡ ｾ that is the meaning of ｸｴｾ ＢｓＨｾＩ ｑｉＩＨｸ ｡Ｉ u VS(':})(tB)(y)(a) is the lattice­theoretic join (or full sum) of the functions assigned to x and to y. This, of course, As a diagram we would draw xty as in Figure 21. seems very special. The intended interpretation is that flow of information is directed through both x and y and is "joined" at the output. The sense of "join" being used is that of the join in the lattice S. Pushing the algebraic analogy a bit further we can write certain conditionals as scalar products: b·x = (b ..... x.J.) anc Cl-b)'Y = (b ..... J.,y) These two compounds are ､ ｩ ｾ ｧ ｲ ｡ ｭ ･ ､ information through provided r. is false. ｾ in Figure 22. The first passes provided b is true; the second, through Y Now, our first really "algebraic" result is this equivalence: (b ..... x,y) That is. up to "" (b·x)t(l-b)·y . equivalence, the conditional sum can be "defined" by the full sum ""it}­1, the aid of scalar mUltiples. A fact that can 52 b·X I . ＬＭｾ ----...11 ill ] L_ .J Figure 21 A(FULL) SUM ｾｬｬ r­, ... , ｾ ｾＭ (I-b)'Y FiQure 22 TWO SCALAR PRODUCTS ,, FiQure 23 AN INFINITE SUM , '\ 53 be easily appreciated from the diagrams. This would not be very interesting if we did not have further algebraic ｾｱｵｩｶ｡ｬ･ｮ｣･ｳＬ but note the following: b'(xty) ｾ b'xtb'y b'(o'x) '" b'x ｾ b'(o'x) o'(b'x) r:;. x b·x (If we had introduced some algebra into S, These results would be even more regular. must take C,lre But we chose here not to alBebr·,.lici ... ｾ to remember that T .is not d 11.) One Buolean alGebra; thus I while it 11:1 correct that: ｾ b-xt(1-0)'% x , it is not correct that: b· (1-0)'x The reason being that ;;<; .1 • = T is possible. (BCb)(o) Having sufficient illustration of the properties of scalar multiples, we turn now to products. First, there is one distributive law: x;(ytz) :: (x;y)1ex;z) that is correct; but the opposite (ytz);x ｾ is not correct. of these two (y;x)t(z;x) The I'eader may ;::arry out the semantical analysis ｰｲｯｾ ｳ･､ laws. the fact that for The correctness of the first turns on f, f' E (S .... s] and 0 E S we have (f U f") (0) = l(o) U f" (0) • The incorrectness of the second is a consequence of the failure in geneI'al of the equation l(o U for l E [S -+ 0') S) and 0, cr' E S. = leo) U l(o') Similarly, one must take care to note that neither of the following are correct: ,. (O':c};y C'(x;y), J(' x;(b·y};;:; However, the ｡ｳ ｯｾｩ｡ｴｩｶ･ b'(x;y) law for; is valid; (,:C;Y);2 "" %;(Y;'<:) Returning to consideration of the operation 1 , we remark that it: is as sociat i ve up to equivalence a Iso; and since it is a join operation, we can prove :ciY £ B if and only if :c £ z and y £ z This means that E/ii; is algebraically a se77li-Iattiae with i join. as the Whether E/>e is a lattice, the author does not know at the moment of I,./:riting. However, we can define countably infinite joins in the partially ordered set E/:o:: as follows of elements of E, ｾ Given a sequence :en there is a unique element we shall call L>n n=Q which is characterized by the equation IX n X 01 L xl'! n=l n.=O In pictorial form the diagram is illustrated in Figure 23. This type of combination is clearly only of theoretical interest, but it does show why E/;;;: is countably complete. I t may be possible that E/:;e is a complete lattice. but the author doubts it. As examples of equivalences involving infinite sums we have: (Xi In case Y rt LY n ) n=Q Ie LJx;y n ) n=D i;; Yn+l holds for all n, we would also have: ＨｻＲＺｙｮＩ［ｘｾ ­ n=O as a consequf'nce of ｾｯｮｴｩｮｵＮｩｴｹＮ - L(Yn;,r) 1"1=0 There are many other similar laws. 55 9. CONCLU S ION. Starting with very simple­minded ideas about flow diagrams as actual diagrams, we introduced the idea of approx- imation which led to a pal'tiatty o··dered set of diagrams. A rigor- OllS, mathematical construction of this set produced what proved to be a complete lattice -- the lattice of flow diagrams. Defined on this lattice were several atgebJ'aic ope ratione and the lattice as a whole satisfied an equation that connected it with the approach of analytic synta.x. But the limits available in a complete lattice introduced something new into the picture: infinite diagpama. In particular, these infinite diagrams provided solutions to algebra i.e equations (the solutions were atgebJ'aio elements) which could be identified with the intuitive concepts of loops and other "reaursive" diagrams (i.e. with feedback). So much for synta.r. Semantios of flow diagrams entered when the mapping of evaluation was defined from the algebra of diagrams into the algebra of functions on a state spacB (which was also a lattice). A bit of argument was required to see that evaluation captured the intuitive idea of flow in a diagram. but it became clearer in the example of a wh11 e-loop. From there. it was safe to introduce the notion of eqwivalenae of diagrams and to study the resulting algebra. The reason for working out the equivalence algebra is. of course, to formalise some general facts about semantics of flow diagrams. Much remains to be done before we have a perfect understanding even of this elementary area of the theory of computation. For one thing, only a start on the systematizcltion of the algebra under equi" valence was made in Section 8. It may be that equivalence is "ot at all the most important notion, for there may be too few equations between diagrams holding as equivalences. the oonditional equation. A more useful notion is That is. we might write :r: = :r:' I- y ; Y I 56 x', for x, if, all a E S, y' E E to r:>€an that for aIlS and all (,].5 a Ｌｾ and if it is the case thdt V"S' ';1"d! ".r)(O> thetl '3', ｾ｣ｮｳ･ｱｪＬＨＩ P€ ' "'5' 'J" B".r' "0> have 't1 s (3)1(f3)(Y)(Ol = t"S(":1)(rfI)(y'){O) iJote that this is not an 1l1'Dlication between two €(]uivalences, but from each in$tal'lce of one equation to the corresponding :i nstC'lnce cf the other. SiMilarly, ｾＧ･ CDule. '­"rite: x E;: .r' f- y !; '-, Also it is Lj3eful to be able to write: ｸｾＬ ｘｏｾｘ Ｌｘｬ " . I- y y' to mean that in each instance in which all the hypotheses left are true. the conclusion on the rip,ht follows. on the Many iMporTant algebr'aic IdWS Cdn be given such a form. Thus:, in order to have a really systematic and useful algebra of flow diagrams, one should study the consequence relation I- dnd attempt to dxiomatize all of its laws. An effective ｡ｸｩｯｾ､ｴｩｺ｡ｴｩｯｮ may only be possible for equdtions involving d restr1:cted portion of E. beoause E contains so many infinite diagrams. an ｩｮｴ･ｲ ｳｴｩｮｾ question tD ｾｯｮ､･ｲＮ But it is 57 References [lJ M. Arbib, Theories of Eall Series in [2J H. Ｌｾｩｫ･ｂ ａｵｴｏｾ､ｬｩ｣ ａ｢ｳｴｲｾ｣ｴ ａｵｴｯｾ｡ｴ Ｎ Prentice- Computdtion (1969) Defi"able Dperatio"s in Gs"eral Algebras, and the Theory of and Flowcharts, Private Ｇｾｴｯｾ｡ｴ Communication. [3J G. lattice Theory, American Mathematical Ｎｦｾｯｨｫｲｩｂ Society Colloquium Publications, Vol. 25, Third Ole\.,') Edition (1967). [4J ｾＮ ｾ､ｲｫＬ Fixpoint Induction an,", Proofs or Program Properties, in Machine Intell1gence 5 (1969). Edinburgh University Press, pp. [5J D. Scott, Some Theory, ｄｾｦｩｮ ｴ ｯｮ｡ｚ 59-78. Suggestions for Automata Journal of Comnuter and System Sciences. Vol. 1, No.2 (967). Academic Press, pp. 187-212. [6J D. Scott, OutZine of a Mathematical Theory of Computation in Proceedir.rs of the fourth Annual Princet0n Conference on Information Sciences and Systooms (1970). pp. 169-176. Pr'uI;r'd.JMIing Rese,J.I',_11 Gr.:."up lechrdcal Monag} dplis August 19"/8 This is d ser"ies ut tl', ＱｾｉＮｩ｣､ｬ monographs on topics in tile li' I,] 01 c:omputati011. ｲｬ ｐｴｨｾｲ copies may be obt<.line,j from tI,." Progranunirog Hel:lel3r, 'i GJ"()UP, (Technical MO[]ogrdpkl) 1 45 Hanbury Road, Oxford, JX2 f,PL. ｅｮｾｬ｡ｮ､Ｎ (The cost indicated ｩ ｮ ｣ ｬ ｵ ､ ･ ｾ ｩ［ｵｲｴ ｬＨｾ Ｇ postage. If faster ,.1eliver'y is re'1uir'ed it should l,· indjc':.Jred and all a.ddltional 30 per tellt sent.) f'llG­l PRG-2 (Out or 1'r'irtT) Vand SC'.Jtt. ｡ｷｴｌｩｾＸ PHG-J of Nathemaeioal J ｔｨＮ､ｲｾ of Cdmputation (£0.50) VaJld Scutt. The !,attidOl ot' Flow Diagrams (£1. 00) lled) ｐｒＨｾＭｉ Ｈｃ｡Ｎｮ｣ｾ PRG­' I 'Ina Scot t. Data T!tPflB all ＡＬ｡ｴ ｩｻ＿ｰｾ (o.OO) PRG-6 Dana Scott and Christopher' ;j'tJ'Bcl,oo>y. TOIJard a Matl,9matidaZ ｓｾｭ｡ｾｴｩＨＢＱｊ for CompwtQr Langwag8B (£.0.60) PRG-7 Dana Scott. ｃｯｮｴｩｮｵｯｷｾ ｌ｡ｴ ｩｑ･ｾ (£o.,O) ｐｒｇＭｾ Joseph Stoy and Christopher Str"l.c"hey. aS8 ­ An ｏｲｑ ｡ｴｩｾｧ Sy,t8m for a SMall Ccmput.r (£.1. 00) I [(:'-:, Lid lSlu;:JJt"l ［Ｂｴｉ Ｎｾ＼ ｉ ｌﾷＱ "'11'1 'I'll" 1'.:.1: ｾ ",j J ,. ","'1"1, ｏｾＢｨｬＮｩＢ ＨｾｌＡｊｵＩ Ａ［ｲＺｾ - I iJ ｬＧＭ ｨ＾ｪＢｴｾｩＧｬ ｊ［ :...t['­1chey. ｖｾｰｩｾｴｩ､ｾ oj ｾｲｯｧｲ｡ｭ ｩＢｾ ｌｾｮｕ ｾ ､ ＨｾｉｊＮｾ［ !'P:(;- i ; I ｾＮｬ l"L""l,ll<:r !::ILl:'<J.(;/h:':j .1111.1 CJll·jut,... ＱＧ ｪＬｾ ｃｯｮｴｩｮｵｾｇｩｯｮＹＺ A Muthsmatietll Jid' HunJlinfj r'u.ll ｊ ｷ Ｂ ･ ｾ l'. 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