Review of Control Flow Semantics

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1 Review of Control Flow Semntics Riccrdo Pucell Deprtment of Computer Science Cornell University April 19, 2001 I hve to dmit, I ws looking forwrd to reviewing this book. It nswered wht ws for me 6-yer old question. Six yers go, I ws pursuing Mster s degree t McGill University. working on progrmming lnguge semntics. My trining in semntics ws rther trditionl: lmbd clculus, functionl progrmming lnguges, denottionl semntics bsed on complete prtil orders, etc. At the time, Frnck vn Breugel ws visiting McGill, nd I cme cross the fct tht Frnck ws lso working on semntics of progrmming lnguges, but on semntics bsed on metric spces. For someone with n undergrdute bckground in mthemtics, this ws seriously intriguing. Unfortuntely, I never got round to sking Frnck bout his work. This book is n nswer to tht question tht never ws. The purpose of the book is to describe n pproch to provide semntics to impertive lnguges with vrious types of control flow models, including concurrency. The pproch hndles both opertionl semntics nd denottionl semntics, ll in topologicl setting. (We will come bck to this lter.) The issue of relting the two semntics for ny given lnguge is centrl theme of the pproch. I will provide in the next section n introduction to topologicl semntics. For now, let me sy word on the pplicbility of the pproch. As stted, the min interest is in providing semntics to impertive lnguges. Impertive progrms cn be thought of, for our purposes, s sequences of ctions performed on stte. Typiclly, sttes re sets of vribles, nd ctions include modifying the vlue of vrible in stte. An importnt chrcteristic of impertive progrms is tht they embody the notion of computtion step: progrm being sequence of ctions, it forces sequence of intermedites sttes. Typiclly, the intermedite sttes re observble, mening tht one cn observe something bout tht intermedite stte, either by looking up the vlue of vrible, by witnessing n output opertion, etc. When the intermedite sttes of computtion re observble, it becomes resonble to tlk bout infinite (nonterminting) computtions. (The clssicl exmple of this is of course n operting system, which t lest theoreticlly is n infinite process; the min motivtion for the topologicl pproch, s we shll see, is to mke sense of such infinite computtions.) Contrst this with functionl lnguges, which re often used s motivting exmples for the study of semntics. In pure functionl lnguge, ll infinite progrms re equivlent, nd in trditionl denottionl semntics bsed on prtil orders (à l Scott nd Strchey [6]), every nonterminting progrm is mpped to, the lest element of the pproprite prtil order. This is not helpful in setting where we wnt to discuss observbly different infinite behviors. We distinguish two kinds of impertive lnguges. The distinction between them is kin to the distinction between propositionl nd first-order logic. 1 Uniform lnguges re bsed on primitive, bstrct notion of ction, combined with pproprite opertors. For instnce, progrm my hve the form ; b; c; (d + ) where ; represents sequentil composition nd + nondeterministic choice opertor. The primitive ctions,b,c,d re uninterpreted. The stte is implicitly defined by the ctions tht hve been performed. On the other hnd, nonuniform lnguges hve n interprettion ssocited with the ctions; typiclly, s we mentioned, the stte is set of vribles, nd ctions include modifying the vlue of vrible in stte. To showcse the verstility of their pproch, the uthors study different lnguges. The min difference between the vrious lnguges, side from the question of uniformity, is the progrm composition opertors considered. The J. de Bkker, E. de Vink, Control Flow Semntics, MIT Press, 1996, 564pp, ISBN This nlogy cn be mde precise when looking t dynmic logics, fmily of logics for resoning bout progrms in such lnguges [1]. 1

2 following groups of relted opertors re studied: The first group consists of opertors including sequentil composition nd choice. The ltter introduces nondeterminism in the frmework, with suitble complictions. Mny versions of choice re investigted, including bcktrcking choice. The second group of opertors consists of recursion nd itertion. Such opertors re required to get universlity (in the computbility theory sense). The third group of opertors includes prllel composition opertors. Modeling such opertors forces one to del with issues such s dedlock, synchroniztion nd communiction. Lnguges with such opertors include CSP [2] nd CCS [3]. Relted to the lst group of opertors, one my distinguish between sttic nd dynmic configurtion of processes. Finlly, we cn investigte issues of loclity nd scope of vribles. All in ll, 27 lnguges re studied in the book, encompssing vrious fetures described bove (nd others, such s the kernel of logic progrmming lnguge). For ech lnguge, n opertionl semntics is given, long with denottionl semntics bsed on topologicl spces. The reltionship between ech semntics is investigted. Here re the chpter titles, to give n ide of the brekdown of content: 1. Recursion nd Itertion, 2. Nondetermincy, 3. Vritions, 4. Uniform Prllelism, 5. Unbounded Nondeterminism, 6. Loclity, 7. Nonuniform prllelism, 8. Recursion Revisited, 9. Nested Resumptions, 10., Domin Equtions nd Bisimultion, 11. Brnching Domins t Work, 12. Extensions of Nonuniform Prllelism, 13. Concurrent Object-oriented Progrmming, 14. Atomiztion, Commit, nd Action Refinement, 15. The Control Flow Kernel of Logic Progrmming, 16. True Concurrency, 17. Full Abstrctness, 18. Second-order Assignment. In the next section, I summrize the first chpter of the book, to give feel for the pproch. Overview of topologicl semntics Given L collection of progrms in lnguge, semntics for L is mpping M : L P tking progrm p to n element M(p) from domin of menings P. The domin P should hve enough mthemticl structure to cpture wht we wnt to model. The study of semntics centers round the development of methods to specify M nd ssocited P for rnge of lnguges L. We cn distinguish essentilly two wys of specifying M: Opertionl O : L P O, which cptures the opertionl intuition bout progrms by using trnsition system (xioms nd rules) describing the ctions of n bstrct mchine. This is the structurl pproch to opertionl semntics (SOS) dvocted by Plotkin [4]. Denottionl D : L P D, which is compositionl; the mening of composite progrm is given by the mening of its prts. This is helpful to derive progrm logics, to reson bout correctness, termintion nd equivlence. Also, in generl, denottionl semntics re less sensitive to chnges in the presenttion of lnguge. Consider the following simple exmple, to highlight the difference between the two styles of semntics. Let A be n lphbet, nd W the set of structured words over A, given by the following BNF grmmr: w ::= (w 1 w 2 ) where is n identifier rnging over the elements of A. If A = {, b, c}, then ( (b )), (( b) (c b)), ((( b) ) b) re structured words over A. We choose to ssign, s the mening of n element of W, its length. We derive both n opertionl nd denottionl semntics to ssign such mening to elements of W. We tke P O = P D = N (where N is the set of nturl numbers). To define the opertionl semntics, we consider the slightly extended lnguge V = W {E}, where intuitively E stnds for the empty word. We define trnsition system with trnsitions of the 2

3 form (v, n) (v, n ) where v, v V nd n, n N. (Such trnsition counts one letter of the word v.) Let be the lest reltion stisfying the following inference rules: We cn define the opertionl semntics O by: (, n) (E, n + 1) (v 1, n) (v 1, n ) ((v 1 v 2 ), n) ((v 1 v 2 ), n ) (v 1, n) (E, n ) ((v 1 v 2 ), n) (v 2, n ) O(w) = n if nd only if (w, 0) (v 1, 1) (E, n) The denottionl semntics D is much esier to define: D() = 1 D(w 1 w 2 ) = D(w 1 ) + D(w 2 ) It is strightforwrd to show, by structurl induction, tht in this cse the opertionl nd denottionl semntics gree (tht is, they give the sme result for every word w W ). Let us now turn to somewht more relistic exmple. Recll tht there re two kinds of impertive lnguges we consider, uniform nd nonuniform. Let s define simple uniform lnguge with recursive opertor. This exmple is tken stright from Chpter 1 of the book. The lnguge, L rec, is defined over n lphbet A of primitive ctions. We ssume set of progrm vribles PVr. (Stt) s ::= x (s 1 ; s 2 ) (GStt) g ::= (g; s) A sttement s is simply sequence of ctions; vribles re bound to gurded sttements g, which re simply sttements tht re forced to initilly perform n ction. When vrible is encountered during execution, the corresponding gurded sttement is executed. A declrtion is binding of vribles to gurded sttements, nd the spce of ll declrtions is defined s Decl = PVr GStt. The lnguge L rec is defined s L rec = Decl Stt. We write n element of L rec s (x 1 g 1,..., x n g n s), representing the sttement s in context where x 1,..., x n re bound to g 1,..., g n, respectively. The opertionl semntics is defined by trnsition system over Decl Res, where Res = Stt {E}; the intuition is tht E denotes sttement tht hs finished executing. We nottionlly identify the sequence E; s with the sttement s. This will simplify the presenttion of the reduction rules. The trnsitions of the system tke the form s D r where s Stt, r Res, A, nd D Decl ; this trnsition should be interpreted s the progrm sttement s rewriting into the sttement r, long with computtionl effect. (For simplicity the computtionl effect is tken to be the ction performed.) Agin, the D reltion is the lest reltion stisfying the following inference rules: g D r x D r D E if D(x) = g s 1 D r 1 s 1 ; s 2 D r 1 ; s 2 3

4 We tke the domin P O of opertionl menings to be the set of finite nd infinite sequences of ctions, P O = A = A A ω. We define the opertionl semntics O : Decl Res P O s: { O(D r) = 1 2 n if r 1 D r 2 1 D n D r n = E 1 2 if r 1 D r 2 1 D For instnce, we hve O(D 1 ; ( 2 ; 3 )) = 1 2 3, nd O(x (; y), y (b; x) x) = (b) ω. Deriving denottionl semntics is slightly more complicted. A progrm in L rec my describe infinite computtions. To mke sense of those, we need the notion of the limit of computtion. In mthemticl nlysis, limits re usully studied in the context of metric spces [5]. This is the setting in which we will derive our semntics. A metric spce is pir (M, d) with M nonempty set nd d : M M R 0 (where R 0 is the set of nonnegtive rel numbers) stisfying: d(x, y) = 0 iff x = y, d(x, y) = d(y, x), nd d(x, y) d(x, z) + d(z, y). A metric spce (M, d) is α-bounded (for α < ) if d(x, y) α for ll x nd y in M. We cn define metric on A s follows. For ny w A, let w[n] be the prefix of w of length t most n. The Bire-distnce metric d B : A A R 0 is defined by { 0 if v = w d B = 2 n if v w nd n = mx{k : v[k] = w[k]} We sy sequence {x n } n=1 is Cuchy if for ll ɛ > 0 there exists n i such tht for ll j, k i, d(x j, x k ) ɛ. In other words, the elements of Cuchy sequence get rbitrry close with respect to the metric. A metric spce (M, d) is complete if every Cuchy sequence converges in M. It is esy to check tht the metric spce (A, d B ) is complete. If (M, d) is α-bounded for some α, nd X is ny set, let (X M, d F ) be the function spce metric spce defined s follows: X M is the set of ll functions from X to M, nd d F (f, g) = sup{d(f(x), g(x)) : x X} (α-boundedness on M gurntees tht this is well-defined). One cn check tht if (M, d) is complete, then so is (X M, d F ). A centrl theorem of the theory of metric spces, which is used hevily in the book, is Bnch s fixed point theorem. Essentilly, this theorem sys tht every function f from metric spce to itself tht decreses the distnce between ny two points must hve fixed point ( point x such tht f(x) = x). We need more definitions to mke this precise. Define function f : (M 1, d 1 ) (M 2, d 2 ) to be contrctive if there exists n α between 0 nd 1 such tht d 2 (f(x), f(y)) αd 1 (x, y). For exmple, the function f : (A, d B ) (A, d B ) defined by f(x) = x is 1 2 -contrctive. Theorem (Bnch): Let (M, d) be complete metric spce, f : (M, d) (M, d) contrctive function. Then 1. there exists n x in M such tht f(x) = x, 2. this x is unique (written fix(f)), nd 3. fix(f) = lim f n (x 0 ) for n rbitrry x 0 M, where f n+1 (x 0 ) = f(f n (x 0 )). This is the bsic metric spce mchinery needed to get simple denottionl semntics going. Returning to our smple lnguge L rec, we tke s trget of our denottionl semntics the domin P D = A {ɛ}. (We do not llow the empty string for technicl resons. Notice tht the empty string cnnot be expressed by the lnguge in ny cse.) Wht we wnt is function D defined s follows: D(D ) = D(D x) = D(D D(x)) D(D s 1 ; s 2 ) = ;(D(D s 1 ), D(D s 2 )) (for some function ; defined over A, ment to represent sequentil composition, nd to be defined shortly.) Notice tht this definition of D is not inductive. We will use Bnch s theorem to define the function ; over A, nd to define the function D. 4

5 Let us concentrte on ;. Intuitively, we wnt ; to be function A A A stisfying ;(, p) = p ;( p, p) = ;(p, p) Note tht the bove properties do not form n inductive definition of ; due to the presence of infinite words in A. Insted, we will define ; s the fixed point of the pproprite higher-order opertor. Let Op = A A A be the complete metric spce of functions. 2 Define the following opertor Ω ; : Op Op: Ω ; (φ)(, p) = p Ω ; (φ)( p, p) = φ(p, p) Note tht the bove equtions do define function Ω ;. One cn check tht Ω ; is in fct 1 2-contrctive mpping from Op to Op. Therefore, by Bnch s theorem, there exists unique fixed point (cll it ;) such tht Ω ; (;) = ;. It is esy to see tht this ; stisfies the originl equtions we were iming for. Now tht we hve such function ;, let us turn to the problem of ctully defining D. We proceed similrly, by defining D s the fixed point of the pproprite higher-order opertor, through n ppliction of Bnch s theorem. Consider the metric spce Sem D = L rec A, which is complete since A is complete. Define the following function Ψ : Sem D Sem D by: Ψ(S)(D ) = Ψ(S)(D x) = Ψ(S)(D D(x)) Ψ(S)(D s 1 ; s 2 ) = ;(Ψ(S)(D s 1 ), S(D s 2 )) (There is some subtlety in coming up with the lst eqution; s you ll notice from looking t the righthnd side, there is recursion over Ψ in only one of the two cses. The book explins this.) Once gin, we cn show tht Ψ is 1 2-contrctive function (in S), nd thus by Bnch s theorem there is unique fixed point of Ψ (cll if D) such tht Ψ(D) = D. It is strightforwrd to check tht this D stisfies our requirements for the denottionl semntics function. A finl result of interest fter ll of these developments is the reltionship between O, the opertionl semntics bsed on n intuitive notion of computtion, nd D, the denottionl semntics with its compositionl properties. It turns out tht in this cse, O = D, nd moreover this result cn be derived from third ppliction of Bnch s theorem. The detils cn be found in the book. Opinion As technicl book, imed t describing n pproch to provide semntics to wide vriety of impertive lnguge control flow structures, this book is complete. All the exmples re worked out with enough detils to grsp the subtleties rising. Let the reder be wrned, however, tht the book is dense both in terms of the technicl mteril, nd in terms of the presenttion. The first few chpters should be red slowly nd with pencil in hnd. The book does not require s much bckground knowledge of topology s one my expect. Prior exposure is of course beneficil, but in fct, only the bsics of topology nd metric spces re ctully used, nd whtever is needed is presented in the first few chpters. On the other hnd, the presenttion does ssume wht my best be clled mthemticl mturity. In the grnd scheme of things, problem with this book is one of motivtion nd followup. This is hrdly new in the field of semntics. Specificlly, the use of denottionl semntics is hrdly motivted, considering tht most of the mchinery in the book is imed t coming up with denottionl semntics nd showing tht it grees with the intuitive opertionl semntics. There is throw-wy line bout the fct tht denottionl semntics cn help in developing logics for resoning bout progrms, but most of the interesting developments re buried in the bibliogrphicl notes t the end of the chpters. This will not deter the hrdcore semnticist, but my hve other reders go: so wht?. Sd, since denottionl semntics is useful. And for the curious, Frnck s ctul work cn be found in [7]. 2 Strictly speking, we need to consider the spce of bounded functions to ensure tht the spce is complete. This will be irrelevnt t our level of discussion. 5

6 References [1] D. Hrel, D. Kozen, nd J. Tiuryn. Dynmic Logic. The MIT Press, Cmbridge, Msschusetts, [2] C. Hore. Communicting Sequentil Processes. Prentice-Hll, [3] R. Milner. Communiction nd Concurrency. Prentice-Hll, [4] G. D. Plotkin. A structurl pproch to opertionl semntics. Technicl Report DAIMI FN-19, University of Arhus, [5] W. Rudin. Principles of Mthemticl Anlysis. McGrw-Hill, third edition, [6] D. S. Scott nd C. Strchey. Towrd mthemticl semntics for computer lnguges. In J. Fox, editor, Proceedings of the Symposium on Computers nd Automt, New York, Polytechnic Institute of Brooklyn Press. [7] F. vn Breugel. Comprtive Metric Semntics of Progrmming Lnguges: Nondeterminism nd Recursion. Birkhäuser,