The second hope is that by tanslating between inteaction nets and tem ewiting systems we can chaacteise new classes of each fomalism with good popetie

Transcription

1 Inteaction Nets and Tem Rewiting Systems Maibel Fenandez DMI - LIENS (CNRS URA 1327) Ecole Nomale Supeieue 45 Rue d'ulm, Pais, Fance maibeldmi.ens.f Ian Mackie LIX (CNRS URA 1439) Ecole Polytechnique Palaiseau Cede, Fance mackieli.polytechnique.f Abstact Tem ewiting systems povide a famewok in which it is possible to specify and pogam in a taditional synta (oiented equations). Inteaction nets, on the othe hand, povide a gaphical synta fo the same pupose, but can be egaded as being close to an implementation since the eduction pocess is local and asynchonous, and all the opeations ae made eplicit, including discading and copying of data. Ou aim is to bidge the gap between the above fomalisms by showing how to undestand inteaction nets in a tem ewiting famewok. This allows us to tansfe esults fom one paadigm to the othe, deiving syntactical popeties of inteaction nets fom the (well-studied) popeties of tem ewiting systems; in paticula concening temination and modulaity. Keywods: tem ewiting, inteaction nets, temination, modulaity. 1 Intoduction Tem ewiting systems povide a geneal famewok fo specifying and easoning about computation. They can be egaded as a univesal pogamming language whee dieent paadigms (functional, logical, paallel, etc.) can be epessed, o as an abstact model of computation (abstact in the sense that they specify actions but not contol, fo instance they ae fee fom stategies thee is no intentional behaviou implied by the ewite ules). Recently, inteaction nets have been poposed by Lafont [22] as a new paadigm in ewiting, based on ewiting of netwoks athe than tems; hence a gaphical synta. Inteaction nets ae a genealisation of poof nets [12]. Because of the linea logic foundation, they give a moe ened view of computation, which is eemplied by thei successful use in the study of the dynamics of computation (shaing in the -calculus fo eample [14]). Inteaction nets ae close to an implementation than tem ewiting systems, since the inteaction ules ae non-ambiguous and conuent, and the eduction pocess is local and asynchonous. The two fomalisms in ewiting outlined above have been developed sepaately, isolating each paadigm fom pogess in the othe. The aim of ou wok is two-fold. Fist, and pimay, the hope is to bidge the gap between the two fomalisms. This would then allow us to: eason about inteaction nets in a taditional tem ewiting famewok. Tem ewiting is now a vey ich eld, with well established theoies and esults such as type systems, modulaity and temination poof techniques. An encoding of inteaction nets into tem ewiting systems should allow all this knowledge to be hanessed. This is an impotant point if we see inteaction nets as a pogamming paadigm (as pesented by Lafont [22]). eason about tem ewiting systems in a gaphical synta, thus allowing the use of popeties and gaphical intuitions of inteaction nets to deduce popeties of tem ewiting systems. Thee is also the possibility of applying some of the ecent developments in semantics of inteaction nets to tem ewiting, and, in paticula, undestanding tem ewiting systems in the inteaction net famewok allows us to apply to tem ewiting languages the implementation techniques of inteaction nets. 1

2 The second hope is that by tanslating between inteaction nets and tem ewiting systems we can chaacteise new classes of each fomalism with good popeties (obtained as images of the tanslations). Moe specically, when compaing inteaction nets and tem ewiting systems a numbe of questions aise natually: Is it possible to tanslate between the two fomalisms in a faithful way What popeties ae well behaved unde these tanslations Thee ae a numbe of dieent classes of inteaction nets. What classes of tem ewiting systems coespond to these The same question also applies the othe way aound. Some classes of tem ewiting systems do not coespond to any paticula class of inteaction nets. Can this lead to the denition of new and inteesting etensions Fo eample inteaction nets only captue conuent and sequential computations, so thee is no notion of paallel function. Since tem ewiting systems can code such functions (e.g. paallel-o), can we geneate a new notion of inteaction net whee these ae captued but still etain the salient featues In this pape we begin this wok by showing how to undestand inteaction nets in the tem ewiting wold. The epession \tem ewiting system" is nomally used fo ewite systems that deal with st-ode tems. As we will see in the following sections, a natual tanslation of the gaphical synta of inteaction nets to tems involves the use of bound vaiables, which takes us away fom the wold of standad tem ewiting systems. We will conside a genealisation of the st-ode systems, intoduced by Klop [18] unde the name of combinatoy eduction systems, that combines st-ode ewiting with the pesence of bound vaiables. We will use the epession tem ewiting systems in a boad sense, including st-ode systems and etensions like combinatoy eduction systems and shaed-ewiting. Afte pesenting two dieent styles of tanslations fom inteaction nets to tem ewiting systems, we will show that useful popeties like conuence and temination ae peseved unde the tanslations. As a consequence, we can apply the techniques developed fo tem ewiting systems in poofs of temination of inteaction nets (conuence holds by constuction). Moeove, we will show that many of the modulaity esults fo temination of tem ewiting can be efomulated in a simple way fo unions of inteaction nets. The study of the evese tanslations, encoding tem ewiting systems in the inteaction famewok, was stated in [9] as a st step towads the development of an inteaction-net based implementation of tem ewiting systems. In ode to encode inteesting systems like paallelo (moe geneally, non-sequential o non-conuent tem ewiting systems), a genealisation of inteaction nets is equied. This led to the denition of paallel inteaction nets with state which is also epoted in [9]. Inteaction nets have been used as a tool to study optimal implementations of the -calculus [14]. Using these ideas, Laneve [24] etended the notion of optimality to inteaction systems (a subclass of combinatoy eduction systems). It tuned out that inteaction systems then coesponded to (a class of) inteaction nets. Also, inteaction nets and othe elated models that ae founded on linea logic [12], like the Geomety of Inteaction [13], have been successfully used fo the implementation of vaious -calculi [25]. Ou wok can be egaded as a continuation of this last eseach line. On one hand, we study the elations between inteaction nets and tem ewiting systems with the hope of making these semantic esults and implementation techniques applicable also to tem ewiting systems and to languages that combine tem ewiting and -calculus. On the othe hand, seeing inteaction nets as a pogamming paadigm, ou study allows us to apply the pogamming techniques developed fo tem ewiting languages, in paticula concening modulaity, to inteaction nets. The latte point is the main subject of this pape. The pape is oganised as follows. In the net section we eview cetain classes of tem ewiting system and inteaction net that we will use in the sequel. In Section 3 we povide tanslations of inteaction nets into vaious classes of tem ewiting systems. In Section 4 we study some applications of the tanslations, in paticula modulaity of temination. Finally, in Section 5 we 2

3 conclude ou ideas and suggest futhe diections. This pape is a evised and etended vesion of the pape [10] pesented at CAAP'96. 2 Basic Concepts In this section we ecall the fomalisms that we will use thoughout this pape. We efe the eade to the suveys [6, 19] fo futhe eamples of ewite systems, and to [22] fo inteaction nets. 2.1 Tem Rewiting Systems Tem ewiting systems can be seen as pogamming o specication languages, o as fomulae manipulating systems that can be used in vaious applications such as pogam optimisation o automated theoem poving. We ecall biey the denition of st-ode tem ewiting systems, and then descibe two etensions: shaed-ewiting and combinatoy eduction systems Fist-ode Tem Rewiting Systems A signatue F is a nite set of function symbols togethe with thei (ed) aity. X denotes a denumeable set of vaiables, and T (F; X ) denotes the set of tems built up fom F and X. Tems ae identied with nite labeled tees, as usual. The symbol at the oot of t is denoted by oot(t). Positions ae stings of positive integes. The subtem of t at position p is denoted by tj p and the esult of eplacing tj p with u at position p in t is denoted by t[u] p. This notation is also used to indicate that u is a subtem of t. The stict subtem elationship is denoted by (then denotes the supetem odeing). We use to denote syntactic equivalence of objects. Va(t) denotes the set of vaiables appeaing in t. A tem is linea if vaiables in Va(t) occu at most once in t. Substitutions ae witten as in f 1 7! t 1 ; : : : ; n 7! t n g whee t i is assumed dieent fom i. We use Geek lettes fo substitutions and post notation fo thei application. Denition 2.1 Given a signatue F, a tem ewiting system on F is a set of ewite ules R = fl i! i g i2i, whee l i ; i 2 T (F; X ), l i 62 X, and Va( i ) Va(l i ). A tem t ewites to a tem u at position p with the ule l! and the substitution, witten t! p u, o simply t! R u, l! if tj p = l and u = t[] p. Such a tem t is called educible. Ieducible tems ae said to be in nomal fom. We denote by! R (esp.! R ) the tansitive (esp. tansitive and eeive) closue of the ewite elation! R. The subinde R will be omitted when it is clea fom the contet. The signatue F of a tem ewiting system is patitioned into a set D of dened symbols: D = ff j oot(l) = f fo some l! 2 Rg, and a set C of constuctos: C = F D. In most pogamming languages based on ewiting, the constucto discipline is assumed, that is pogams ae constucto systems: Denition 2.2 A constucto system is a tem ewiting system ove a signatue F = C [ D (whee C is the set of constuctos and D the set of dened symbols) with the popety that evey left-hand side f(l 1 ; : : : ; l n ) of a ule in R satises f 2 D and l 1 ; : : : ; l n 2 T (C; X ). A constucto system is then specied by a tiple (D; C; R). Let l! and s! t be two ewite ules (we assume that the vaiables of s! t wee enamed so that thee is no common vaiable with l! ), p the position of a non-vaiable subtem of s, and a most geneal unie of sj p and l. Then (t; s[] p ) is a citical pai fomed fom those ules. Note that s! t may be a enamed vesion of l!. In this case a supeposition at the oot position is not consideed a citical pai. A tem ewiting system R is conuent if t! u and t! v implies u! s and v! s fo some s, 3

4 teminating (o stongly nomalising) if all eduction sequences ae nite, left-linea if all left-hand sides of ules in R ae linea, non-ovelapping if thee ae no citical pais, othogonal if it is left-linea and non-ovelapping, non-duplicating if fo all l! 2 R and fo all 2 Va(l), the numbe of occuences of in is less than o equal to the numbe of occuences of in l Shaed Rewiting Many implementations of tem ewiting systems use diected acyclic gaphs (dags) athe than tees fo eciency easons. Common subtems ae stuctually shaed in a dag. In this way, multiple occuences of a subtem may be simultaneously educed to a common tem. This eduction elation is called shaed-ewiting; it is a paticula case of tem gaph ewiting (see [3, 16]) whee gaphs ae acyclic. In a shaed-eduction step, subtems that coespond to the same vaiable in the left-hand side of the ule ae not copied but shaed in the esulting dag, even if the ight-hand side of the ule has multiple occuences of this vaiable. Fomally, in ode to dene shaed-eductions we use maked tems to epesent dags, and we dene a ewite elation on maked tems that coesponds to dag ewiting (fo moe details see [21]). Denition 2.3 Conside a countably innite set M of objects called maks (M will usually be the set of integes). Let F = ff m j f 2 F and m 2 Mg be the set of maked function symbols. Similaly, the set of maked vaiables is denoted X. We dene mak(x m ) = m fo X m 2 F [ X. The elements of T (F ; X ) ae called maked tems. A tem t 2 T (F ; X ) is well-maked (o a dag) if fo evey pai of subtems t 1, t 2 of t, mak(oot(t 1 )) = mak(oot(t 2 )) implies t 1 t 2. The subset of well-maked tems of T (F ; X ) is denoted by D(F ; X ). Well-maked tems coespond to dags as follows: A maked symbol m in a maked tem t coesponds to a node m labeled by in the dag. If two subtems t 1, t 2 of t have the same mak at the oot they must be identical, because thee is only one subgaph in the dag fo t 1 and t 2. In a dag t, fo each mak m occuing in t thee is a unique subtem s of t that satises mak(oot(s)) = m, which is denoted by tm. Note also that t is well-maked if and only if all its subtems ae well-maked. Two occuences s 1, s 2 of a subtem of a well-maked tem t ae shaed in t if and only if mak(oot(s 1 )) = mak(oot(s 2 )). Eample 2.4 Let f; g; a be function symbols of aity 2; 1; 0 espectively. The maked tem t f 0 (g 3 (g 1 (a 2 )); g 4 (g 1 (a 2 ))) is well-maked, and both occuences of the subtem t1 g 1 (a 2 ) of t ae shaed. Also shaed ae the occuences of t2 a 2. The tem t 0 f 0 (g 3 (g 5 (a 6 )); g 4 (g 1 (a 7 ))) is also well-maked, but thee ae no shaed subtems in it. The gaphical epesentations of t and t 0 ae shown in the following diagam: t t 0 f 0 f 0 g 3 g 4 g 3 g 4 g 1 g 5 g 1 a 2 a 6 a 7 4

5 In ode to dene shaed-ewiting we need the notion of maked substitution and maked ewite ule. Let e be a function that eases all the maks in a maked tem (we denote by e(t) the unmaked tem obtained fom t by easing all the maks), and let us say that t e t 0 (fo t; t 0 2 T (F ; X )) if e(t) = e(t 0 ). A maked-substitution is a substitution in T (F ; X ) such that fo all 1 ; 2 2 X, if e( 1 ) = e( 2 ) then 1 e 2. A maked ewite ule is a ewite ule l! such that l and ae maked tems with the usual convention: l 62 X, and Va( ) Va(l ). This implies that the numbe of copies of a (maked) vaiable in the gaph coesponding to is less than o equal to the numbe of copies of that vaiable in the gaph coesponding to l. A maked ewite ule l! is a maked vesion of a ewite ule l! if e(l ) = l and e( ) =. Now we can dene the shaed-eduction elation! s R induced by a ewiting system R on the set T (F ; X ) of maked tems: Denition 2.5 Let t be a maked tem, l! a maked vesion of the ule l! 2 R and a maked substitution. Let p 1 ; : : : ; p n be the set of positions in t such that tj pi = l, and let t 0 be t[ ] p1 : : : [ ] pn. Then t! s l! t0. Remak that all occuences of l in t ae educed simultaneously. Eample 2.6 Conside the ewite ule: f(1; ; )! f(; ; ). The tem f 1 (1 2 ; 0 3 ; 0 4 ) educes to f 1 (0 4 ; 0 4 ; 0 4 ) using the maked vesion f 1 (1 2 ; 3 ; 4 )! f 1 ( 4 ; 4 ; 4 ), and to f 1 (0 3 ; 0 3 ; 0 4 ) using f 1 (1 2 ; 3 ; 4 )! f 1 ( 3 ; 3 ; 4 ). The coesponding dag eductions ae: f f f f =jn! jjj and =jn! ==n This eample shows that we can choose the degee of shaing of vaiables at each eduction step, within cetain limits: the numbe of copies of a vaiable in a ule cannot incease. Howeve, in the following when efeing to the shaed-ewite elation! s R induced by a ewite system R, we will assume maimal shaing of vaiables, that is, all the occuences of the same vaiable in the ight-hand side ae shaed. It is inteesting to note that well-makedness may be lost if abitay makings ae used in ules (fo instance, conside a ule a 1! b 2 and a ewite step f 2 (a 1 )! f 2 (b 2 )). A simple sucient condition fo peseving well-makedness [21] is to assume that in t! s l! t0 t[ ] p1 : : : [ ] pn is fesh with espect to t, that is, all the maks in ae dieent fom those in t. It is easy to see that evey shaed-ewiting sequence coesponds to a tem ewiting sequence, but the convese does not hold in geneal. Howeve, fo othogonal systems evey tem eduction sequence can be etended to a sequence which does coespond to a shaed-ewiting sequence. This is a consequence of a geneal theoem by Kennaway et al. [16] showing the adequacy of gaph ewiting fo simulating tem ewiting in the case of othogonal systems. Popety 2.7 If R is an othogonal tem ewiting system then fo any ewiting sequence thee eists a shaed-ewiting sequence t 0! t 1! : : :! t n t 0 0!s t 0 1!s : : :! s t 0 n such that t 0 = e(t 0 0) and fo all i, t i! e(t 0 i). Poof: In [16] it is shown that if an othogonal gaph-ewiting system G is nitay (i.e. contains only nite gaphs) and acyclic (i.e. gaphs in ewite ules do not contain cycles), and a tem 5

6 ewiting system R is obtained by unavelling of G (unavelling, in the case of dags, is just the function that eases all the maks) then the unavelling mapping fom G to R is adequate. A shaed-ewite system dened by an othogonal tem ewiting system R satises these conditions. Moeove, adequacy (see [16]) implies, in paticula, that if e(t 0 0)! t i, then thee eists t 0 i such that t 0 0! s t 0 i and t i! e(t 0 i) Combinatoy Reduction Systems Combinatoy eduction systems wee designed by Klop [18] with the aim of combining the usual st-ode tem ewiting systems with the pesence of bound vaiables as in the -calculus. A combinatoy eduction system is a pai consisting of an alphabet and a set of ewite ules. The alphabet consists of 1. vaiables ; y; z; : : :, 2. metavaiables Z k i each with a ed aity, whee k is the aity of Z k i (indees ae often omitted), 3. function symbols f; g; : : : ; F; G; : : :, each with a ed aity, 4. a binay opeato [] fo abstaction ove vaiables, 5. impope symbols such as (; ). In combinatoy eduction systems a distinction is made between metatems and tems. Metatems ae the epessions built fom the symbols in the alphabet, in the usual way: Denition 2.8 The set MTems of metatems ove an alphabet is dened as the smallest set such that: 2 MT ems fo evey vaiable, if is a vaiable and s 2 MT ems then []s 2 MT ems, if s 1 ; : : : ; s n 2 MT ems and f is a function symbol of aity n then f(s 1 ; : : : ; s n ) 2 MT ems, if s 1 ; : : : ; s n 2 MT ems and Z n is a metavaiable, then Z n (s 1 ; : : : ; s n ) 2 MT ems. In the last two cases n 0, and if n = 0 we omit the backets as usual. Tems ae metatems that do not contain metavaiables. Vaiables that ae in the scope of the abstaction opeato ae bound, and fee othewise. A (meta)tem is closed if evey vaiable occuence is bound. As in the -calculus, naming poblems can aise. We adopt the usual convention: all bound vaiables ae chosen to be dieent fom the fee vaiables. We dene now the ewite ules of combinatoy eduction systems, which ae pais of metatems (but they induce a eduction elation on tems, by assigning tems to metavaiables as eplained below). Denition 2.9 A ewite ule is a pai of metatems, witten l!, whee l; ae closed metatems, l has the fom f(s 1 ; : : : ; s n ), the metavaiables that occu in occu also in l, and the metavaiables Zi k that occu in l occu only in the fom Zi k( 1; : : : ; k ), whee 1 ; : : : ; k ae paiwise distinct vaiables. Eample 2.10 The -eduction ule fo the -calculus is witten in the synta of combinatoy eduction systems as: app(lambda([]z()); Z 0 )! Z(Z 0 ) whee the binay function symbol app epesents application and the unay function symbol lambda epesents -abstaction. Z is a unay metavaiable, and Z 0 a nullay metavaiable. 6

7 The metavaiables in metatems can be thought of as holes that must be instantiated by tems. In othe wods, ules act as schemes dening a eduction elation on tems. To etact the actual ewite elation on tems fom the ewite ules, each metavaiable is eplaced by a special kind of -tem, and in the obtained tem all -edees and the esiduals of these -edees ae educed (i.e. a development is pefomed). This opeation is well-dened in the -calculus since all developments ae nite. Fomally, to dene the ewite elation we have to conside a notion of substitution using substitutes and valuations. Denition 2.11 An n-ay substitute is an epession of the fom 1 : : : n :t, whee t is a tem and 1 ; : : : ; n ae dieent vaiables (n 0). It can be applied to an n-tuple s 1 ; : : : ; s n of tems, and the esult is the tem t whee 1 ; : : : ; n ae simultaneously eplaced by s 1 ; : : : ; s n. A valuation is a map that assigns an n-ay substitute to each n-ay metavaiable. This is etended to a mapping fom metatems to tems: given a valuation and a metatem t, st we eplace all metavaiables in t by thei images in and then we pefom the developments of the -edees ceated by this eplacement. When making a substitution, we must take cae of bound vaiables as usual. We can now dene the ewite elation on tems: Denition 2.12 A contet is a tem with an occuence of a special symbol [ ] called a hole. A ewite step is dened as follows: if l! is a ewite ule, a valuation, and C[ ] a contet, then C[l]! C[]. As an eample, we show the ewite step that coesponds to -eduction accoding to the ule given in Eample 2.10: Eample 2.13 Let be the valuation that maps Z to z:f(z; g(z)) and Z 0 to the tem y. We apply now to the left-hand side of the ule given in Eample 2.10: app(lambda([]z()); Z 0 ) = app(lambda([](z:f(z; g(z)))()); y) The application of to the ight-hand side gives: = app(lambda([]f(; g()); y)) Z(Z 0 ) = (z:f(z; g(z)))(y) = f(y; g(y)) Hence, accoding to the pevious denition, thee is a ewite step app(lambda([]f(; g()); y))! f(y; g(y)): A combinatoy eduction system is left-linea if it does not contain a left-hand side in which some metavaiable has multiple occuences. It is non-ovelapping if wheneve an instance t of a left-hand side l contains a educible (stict) subtem u, then u is contained in one of the instantiated metavaiables of l. It is othogonal if it is left-linea and non-ovelapping. 2.2 Inteaction Nets The inteaction net paadigm was intoduced by Lafont in [22] as a new ewiting famewok fo pogamming, founded on poof nets of linea logic [12]. These nets ae vey appealing fom a computational point of view. On one hand they ae a vey simple, gaphical ewiting system which enjoys popeties such as conuence, and on the othe hand they bing out the paallelism in the ewiting pocess making them well-suited as a basis fo paallel implementations. Hee we will biey eview the paadigm. The eade will nd additional eamples in the aticles by Lafont [22] and Gay [11]. 7

8 Denition 2.14 An Inteaction Net (; IR) is specied by the following data: A set of symbols (o agents), each chaacteised by a label, and an aity n 2 N (n 1) which is the numbe of pots it has. Each agent has a distinguished pot, called the pincipal pot, whee inteaction can take place. All the othe pots ae called auiliay pots. Agents ae epesented gaphically as follows, whee we indicate the pincipal pot by an aow in confomity with Lafont; note that pots ae odeed. : : : Auiliay Pots Pincipal Pot A net on is an undiected gaph whose vetices ae agents in, and whose edges join dieent pots in the same o in dieent agents. The wods agent and node ae often used as synonyms. A net may be empty, o consist just of edges without agents. Pots that ae not connected to othe pots in the net ae called fee. Fee pots ae maked with edges that have a fee eteme, as in the diagam above. Then each node has as many incident edges as the aity of the agent. The inteface of a net is the (odeed) set of fee etemes of edges (in paticula, fo a net consisting only of an edge, the inteface contains the two etemes of the edge). A set IR of inteaction ules which ae net ewiting ules whee the left-hand side is a net consisting of two agents connected on thei pincipal pots, and the ight-hand side is an abitay net with the only constaint that it must have the same inteface as the left-hand side. Thee is at most one ule fo each pai of agents. The following diagam shows the geneal fom of an inteaction ule, using agents and of aity 3 and 4 espectively. The ight-hand side N is any net, which may contain occuences of the agents in the left-hand side (we epesent nets with dashed lines). Note that the inteface is peseved; thee ae equal numbes of fee pots befoe and afte the inteaction. We use names (a; b; c; d; e) to indicate the coespondence between the fee pots in the leftand ight-hand sides of the ule, but we will often omit them when thee is no ambiguity. d e a - b c d N e A net ewite step on a net W, called an inteaction, eplaces in W a pai of agents connected on thei pincipal pots (i.e. an occuence of a left-hand side of an inteaction ule) by the coesponding ight-hand side, plugging the edges in the inteface of the ight-hand side to the coesponding pots in W. We wite inteactions as W W 0. To give an eample of inteaction nets, we show the inteaction ules of two ubiquitous agents, namely the ease (), of aity 1, which deletes eveything it inteacts with, and the duplicato (), of aity 3, which copies eveything. These ae epesented by the following diagams, whee is any node. a b c 8

9 The st ule shows that the inteaction deletes the node and places ease nodes on all the fee edges of the node. Fo the second ule, we see that the node is copied, and all its fee edges ae too. Futhe eamples of inteaction nets ae given thoughout the pape, making use of the above agents and ules. A net is in nomal fom when no inteactions ae possible. We say that an inteaction net (; IR) is teminating if all sequences of inteactions in nets on ae nite. As an almost immediate consequence of this denition of net ewiting we have the following featues: Conuence. The estiction of inteaction only on the pincipal pot of an agent, and the constaint that thee is at most one ule fo each pai of agents, suce to give the stongest notion of conuence: if N N 1 and N N 2 (N 1 dieent fom N 2 ) then thee eists a net N 3 such that N 1 N 3 and N 2 N 3. Local implementation. The local inteface is peseved duing an inteaction two agents inteacting do so in thei own \space" and do not aect any othe pat of the netwok. Asynchony. As a diect consequence of the above points, we have the possibility of a paallel implementation no ode on the inteactions is equied since any two agents eady to inteact can do so in any ode. Inteaction nets can be egaded as a genealisation of poof nets fo multiplicative linea logic, and indeed, this is thei oigin. Roughly, the elationship is given by setting the set of symbols to be the logical symbols; the pincipal pot of each symbol is the conclusion and the auiliay pots ae the pemises of the ule fo that symbol; and the ewite ules ae specied by the ules fo cut-elimination fo multiplicative linea logic. We efe the eade to [23] fo a complete pesentation; see also [25] fo anothe appoach. In the following we will use inteaction nets o inteaction net systems as synonyms Classication of Nets Lafont [22] intoduced a type discipline fo inteaction nets, using a set of constant types (atom, nat, list-nat,... ). Fo each agent, pots ae classied between input and output. An input pot will be assigned a type and an output pot a type. A net is well-typed if input pots ae connected to output pots of the same type. In the following, fo the sake of simplicity, we will conside only one type. In othe wods we will only distinguish between input pots (maked with a sign) and output pots (maked with ). Fo eample, fo the agents and we conside the following typings: Assuming has a positive pincipal pot, the nets in the inteaction ules above can be typed as follows: 9

10 Denition 2.15 Agents can be divided into constuctos and destuctos: if the pincipal pot of an agent is an output pot, the agent is a constucto, othewise it is a destucto. In the pevious eample, and ae destuctos, wheeas is a constucto. The division between constuctos and destuctos oiginates in the logical system that inspied the fomalism of inteaction nets: destuctos and constuctos ae espectively associated with left and ight intoduction ules of logical opeatos. Fo each agent, the auiliay pots ae divided into patitions (the notion of patition has also its oigins in the sequent calculus that inspied the fomalism). Denition 2.16 Each agent 2 has a pincipal pot and a (possibly empty) set of auiliay pots which ae divided into one o seveal classes, each of them called a patition. A patition mapping establishes, fo each agent in, the way its auiliay pots ae gouped into patitions. The patitions given by Lafont fo and ae as follows:, which does not have any auiliay pot, has one patition, which is empty (see [2] fo a detailed discussion of the meaning of empty patitions); fo both auiliay pots ae in the same patition ( has one patition containing two output pots). The notion of patition was intoduced in [22] with the pupose of dening a class of inteaction nets, called semi-simple nets, that ae deadlock fee, that is, a class of nets such that vicious cicles of pincipal pots, as depicted in the diagam below, cannot be ceated duing computation Denition 2.17 A net is called semi-simple if it can be constucted using only the following opeations: 1. LINK, which builds an edge: As mentioned befoe, an edge is a paticula case of net. It can be typed by assigning opposite signs to the etemes. 2. CUT, that connects two disjoint nets using a single edge: A B In paticula, when A and B ae just an edge, we obtain an edge. 3. GRAFT, that adds an agent to a set of nets accoding to its patitions. The pincipal pot emains fee, and all the pots belonging to the same patition of the agent ae connected to the same net, but each patition is connected to a dieent net, as shown in the following diagam: 10

11 Agent A1 An We assume that the agent has n patitions, whee n 0. A 1 ; : : : ; A n ae semi-simple nets, with intefaces such that each auiliay pot in the ith-patition of the agent can be connected to the coesponding net A i (in paticula these nets may be just edges, and may contain moe fee pots in the inteface, that will emain fee afte the GRAFT is made). Types have to be espected in ode to obtain a well-typed net. 4. MIX, jutaposing two nets: A B 5. and EMPTY, which constucts an empty net. A semi-simple net is then dened by a sequence of opeations, and in the following we assume that they ae well-typed. Fo eample, the nets in the inteaction ule fo ae semi-simple: assume that the patitions of ae unitay (i.e. contain only one pot), then the left-hand side is obtained by making a CUT of the nets GRAFT(, EMPTY) and GRAFT(, LINK, : : :, LINK), and the ight-hand side is obtained by making MIX of GRAFT(,EMPTY), : : :, GRAFT(,EMPTY). In fact, the constuction of the left-hand side of an inteaction ule can always be done in this way, as the following lemma shows. Lemma 2.18 Left-hand sides of inteaction ules ae always built by a CUT of two GRAFTS on LINKS (possibly combined by MIX) o EMPTY nets. Poof: By denition of inteaction ule, the left-hand side is a net consisting of a pai of agents connected on thei pincipal pots. Hence it is a net built by a CUT of two nets that consist just of one node each. Then these subnets ae built by GRAFTS made on LINKS, combinations of LINKS by MIX, o EMPTY nets, accoding to the aities and patitions of the agents. Inteaction nets whee the patitions of all agents ae unitay ae called discete. As emaked by Lafont [22], in the discete case a net is semi-simple if it is a gaph without cycles; and if the opeations of MIX and EMPTY ae not used, then it is a connected gaph without cycles (i.e. a tee). In the geneal case, a semi-simple net contains no vicious cicle. A ule is semi-simple if when fee pots have been gouped accoding to the patitions in the left membe, the ight membe becomes semi-simple. Fo eample, the ules fo and ae semi-simple. Semi-simple nets ae closed unde eduction by semi-simple ules. Following Laneve [24], if a negative pot (i.e. an input pot) eists in a patition we will call it an input patition, othewise it will be called an output patition. Hence, an input patition may contain some output pots, wheeas an output patition contains only output pots. Accoding to this, thee ae two classes of inteaction nets: 1. dependent inteaction nets: if a positive pot appeas in an input patition of some agent, 2. non-dependent inteaction nets: if evey agent has only negative pots in input patitions. 11

12 3 Fom Inteaction Nets to Tem Rewiting Systems In this section we study the encodings of inteaction nets into tem ewiting systems. Fist we conside semi-simple nets, and then the geneal case of inteaction nets. 3.1 Tanslation of Semi-simple Nets It is known that discete semi-simple inteaction nets coespond to (a esticted class of) stode tem ewiting systems (see [22, 24]). We will dene a tanslation function that tansfoms a semi-simple inteaction net into a combinatoy eduction system. As a paticula case, we will see that non-dependent semi-simple nets (which include the class of discete nets) ae mapped to st-ode tem ewiting systems. In the discete case we obtain a linea st-ode tem ewiting system. As in [24], we will assume that constuctos do not have any output patition (the pincipal pot is an output pot, and they may have output pots in input patitions). Destuctos may have one output patition like in the case of, o none, like in the case of. Summaising, we assume that evey agent has at most one output patition and a numbe (maybe 0) of input patitions, that may o may not contain positive pots. These assumptions allow us to give a smoothe tanslation of semi-simple nets (dependent o non-dependent) into tem ewiting systems. Of couse this is a estiction on the class of nets unde study, but this class is suciently ich in that it can captue all computable functions (and includes the whole class of inteaction systems [24]). We conside st the case of non-dependent semi-simple nets and then genealise the tanslation to deal with output pots in input patitions (dependent nets) Non-dependent nets We stat by dening a mapping : Nets! T (F; X ) which takes a non-dependent semi-simple net and gives a tem. Moe pecisely, since a semi-simple net is dened by a sequence of opeations of LINK, CUT, GRAFT, MIX, EMPTY, takes as input a sequence of opeations that build a net, and gives a tem. With the help of this function we will tanslate inteaction ules into ewite ules: the tanslation of an inteaction ule will simply be obtained by tanslating each membe. The tanslation of a net with an inteface consisting of inputs 1 ; : : : ; n and outputs y 1 ; : : : ; y m, will be a tem t[ 1 ; : : : ; n ] epesenting an m-tuple. The tanslation function is dened by induction on the denition of the semi-simple net. Denition 3.1 Let (; IR) be an inteaction net, and F be a set of function symbols containing the agents in, the constant empty, a binay symbol P fo pai fomation, and unay symbols i fo pojections (we assume that does not contain agents called P, i, empty). P is assumed to be associative, so we use the at notation P ( 1 ; : : : ; n ) fo any n. To simplify the notation we will use some abbeviations: given a tem t P (t 1 ; : : : ; t n ), i (t) = t i and t i (t) = P (t 1 ; : : : ; t i1 ; t i1 ; : : : ; t n ). In othe wods, the epessions of the fom i (P (t 1 ; : : : ; t n )) and P (t 1 ; : : : ; t n ) i (P (t 1 ; : : : ; t n )) used in the metalanguage have to be eplaced by the coesponding denition (the tems t i and P (t 1 ; : : : ; t i1 ; t i1 ; : : : ; t n ) espectively). Hee we assume that when only one element emains, P (t 1 ) = t 1. A sequence u 1 ; : : : ; u n will be abbeviated by ~u. 1. The tanslation of an empty net is simply the constant empty: (EMPTY) = empty. 2. The tanslation of a link is a vaiable: (LINK) =. We assume that fesh vaiables ae obtained on demand. 3. The tanslation of a net constucted fom two nets A, B by a MIX opeation is the pai fomed by the tanslations of A and B: (MIX(A; B)) = P ((A); (B)): 12

13 Note that A and B may in tun be tanslated as pais, which means that we can obtain tuples of abitay length. When (N) is a tuple, we assume (without loss of geneality) that i ((N)) coesponds to the ith output in the inteface of N. 4. The tanslation PSfag of aeplacements net constucted fom two nets A, B by a CUT opeation connecting y and A y B is: (CUT(A; B)) = P ((B)f 7! i ((A))g; (A) i ((A))); whee we assume that y is the ith output pot in the inteface of A, and is the input pot in B connected to y. Note that in this fomula we used a metalanguage with substitution and abbeviations like (A) i ((A)). In fact, (CUT(A; B)) is the tem that we obtain fom the epession P ((B)f 7! i ((A))g; (A) i ((A))) afte making the opeations in the metalanguage (substitution and eplacement of abbeviations by thei denitions). In the fomula above, we have taken into account the fact that A can be a net with multiple output pots. In that case the tanslation of A will be a tuple, and to obtain the tanslation of the CUT we have to select the coesponding element i ((A)) to \plug" in the tanslation of B. The est of the outputs of A (i.e. (A) i ((A))) ae still outputs of the net esulting fom the CUT (ecall that even if A has only one output pot, we use the notation i ((A)) with i = 1, identifying a tuple of length one with its only element). In the discete case we have: (CUT(A; B)) = (B)f 7! (A)g as paticula case of the pevious fomula. 5. The tanslation of a GRAFT depends on whethe the agent that is added is a constucto o a destucto (to simplify the fomulas, we will assume that the agent has two patitions; the genealisation is staightfowad). The esult of gafting the agent is a net of one of these foms: Const Dest A B A B To dene the tanslation of a GRAFT, we conside these two cases sepaately. (a) If the agent is a constucto: (GRAFT(Const; A; B)) = P (Const(! i ((A));! i ((B))); (A)! i ((A)); (B)! i ((B))): In this fomula we have taken into account the fact that, accoding to ou assumptions, in the case of a non-dependent net the only output pot of a constucto is the pincipal pot, and that A and B can be nets with multiple output pots. In that case the tanslations of A; B will be tuples, and to obtain the tanslation of the GRAFT we have to select the coesponding elements of the tuples. This is abbeviated by! i ((A)),! i ((B)). The est of the outputs of A and B (i.e. (A)! i ((A)); (B)! i ((B))) ae still outputs of the net esulting fom the GRAFT, as we can see in the fomula above. When A and B have only one output pot each, the fomula educes to: 13

14 (GRAFT(Const; A; B)) = Const((A); (B)): If the constucto has aity one, then we simply obtain (GRAFT(Const)) = Const: (b) If the agent is a destucto: (GRAFT(Dest; A; B)) =! P ((A)fy j 7! i (Dest(;! i ((B))))g; (B)! i ((B))) In this fomula epesents the pincipal pot of Dest, and ~y ae the input pots of A connected with the destucto. Again we can see that in the simplest case (discete net) this fomula gives the natual tanslation: (GRAFT(Dest; A; B)) = (A)fy 7! Dest(; (B))g In the degeneate case of a destucto without output pots (see Eample 3.2, pat 2 below) we apply the same fomula with a dummy output, in othe wods, we take: (GRAFT(Dest; B)) = P (Dest(;! i ((B))); (B)! i ((B))) The tanslation function etends to inteaction ules in the natual way, i.e. by applying to each side of the ule (giving consistent vaiable names to the edges in both sides). The tanslation of an inteaction net system (; IR) is a tem ewiting system on the signatue F dened above, with a set (IR) of ewite ules that contains the tanslations of the ules in IR and the additional ules = f i (P ( 1 ; : : : ; n ))! i g dening the pojections. In the denition of the tanslation function we did not conside the case of a CUT of two LINKS (since it gives again a LINK). As we aleady mentioned, ou tanslation function takes as input a sequence of opeations that constuct a semi-simple net and gives a tem as esult. The sequence of opeations needed to build a given semi-simple net is not uniquely dened. It is possible in geneal to change the ode in which the opeations of CUT ae done, and the same fo GRAFTS. A change in the ode of CUTS does not aect the esult of the tanslation, but a change in the ode of GRAFTS can change it. This, howeve, does not cause any poblem since the popeties of the tanslation function that we will show below do not depend on the paticula sequence of opeations used to build a net. We now give some eamples to illustate the denition of. Eample Conside an inteaction net fo lists, specied by the constuctos Cons and Nil and the destucto Append. The net z u Cons Append v y is a semi-simple net obtained by st ceating the LINKS z and v, and the GRAFT of Append on them (which gives the tem zfz 7! Append(u; v)g Append(u; v)), and then making a CUT with the net epesented by Cons(; y), which gives: Append(u; v)fu 7! Cons(; y)g Append(Cons(; y); v). 2. Fo an eample with a non-discete net, conside the system of inteaction ules to add and multiply natual numbes given in [22]. The agents S and 0 ae constuctos, and Add and Mul ae destuctos. In the standad denition of multiplication using addition, the second agument is not used when the st is 0, and it is used twice othewise. In the inteaction net pesentation this equies easing and copying (duplication), which is done with the agents and. We show only the inteaction ules fo multiplication: 14

15 z z Mul 0 z 0 S Mul z y Mul Add y The tanslations of the tems in the left-hand sides of the ules ae simila to the pevious eample. Moe inteesting is the tanslation of the ight-hand sides. Fo the st ule, the net in the ight-hand side is the esult of a MIX opeation on the nets obtained by gafting 0 and. Fomally: (MIX(GRAF T (0); GRAF T (; EMP T Y ))) = P (0; ()) Note that since is a destucto without output, we used the special case of the fomula in Denition 3.1, pat 5b. Fo the second ule, the net in the ight-hand side is constucted by thee GRAFTS (and LINK opeations), as shown in the following pictue: z Add N 1 u Mul v 1 N 2 N 3 v 2 y Accoding to the denition of : (N 1 ) = Add(u; v 2 ) (N 2 ) = (N 1 )fu 7! Mul(; v 1 )g = Add(Mul(; v 1 ); v 2 ) (N 3 ) = (N 2 )fv 1 7! 1 ((y)); v 2 7! 2 ((y))g = Add(Mul(; 1 ((y))); 2 ((y))) In this eample we can see that if the inteaction net is not discete then a vaiable can occu moe than once in the image of. In this case shaed-ewiting mimics eductions on the net (see emak below). Theoem 3.3 Let (; IR) be a non-dependent semi-simple inteaction net. The tem ewiting system (IR) on F is a constucto system, and it is othogonal. Poof: It is easy to see that, by Lemma 2.18, the tanslation of the left-hand side of an inteaction ule is a tem of the fom (( 1 ; : : : ; n ); y 1 ; : : : ; y m ) (ecall that P and i in the denition of ae pat of the metalanguage). The system (IR) contains the tanslations of the inteaction ules IR, and the pojection ules. Since the tanslations of left-hand sides of inteaction ules ae linea tems, the system is left-linea. Moeove, since they do not contain P and i, and since each agent is eithe a constucto o a destucto, and thee is at most one inteaction ule fo each pai of agents, we obtain a constucto system without citical pais. 15

16 Remak 3.4 We can dene in the same way a tanslation function fom nets to maked tems (dags), and conside the tanslation of an inteaction net (; IR) as a shaed-ewiting system (IR). The denition of is simila to that of, but in the cases of CUT and GRAFT we have to add maks so that all occuences of (A) and (B) ae shaed. Net we conside the geneal case of semi-simple nets, which includes dependent nets Dependent nets We will use tems with bound vaiables to encode nets whee agents have positive pots in input patitions. By abuse of the language, we still call the tanslation function, although its image now will be MTems, the set of metatems associated to the signatue F of Denition 3.1, and inteaction ules will be tanslated as combinatoy eduction ules. The tanslation function is again dened by induction. The cases of LINK, EMPTY, MIX, and CUT ae simila to the pevious denition, so we will only pesent the case of a GRAFT. Denition 3.5 The tanslation of a GRAFT depends as befoe on whethe the agent that is being added is a destucto o a constucto. Again, to simplify the fomulas we will assume that the agent has two patitions. Thee ae two cases: Const Dest A B A B 1. If it is a constucto: (GRAFT(Const; A; B)) = P (Const(! i ([~](A));! i ([~y](b))); [~](A)! i ([~](A)); [~y](b)! i ([~y](b))): In this fomula we have taken into account the fact that the input patition of Const connecting to A (esp. B) can have some positive pots. The tanslation of A (esp. B) is used in the negative pot of the patition of Const, the positive pots of the same patition ae not epesented as aguments of Const, since they coespond to bound vaiables that will typically appea in (A) (esp. (B)). In case no infomation about the net A (esp. B) is available, the tanslation of A (esp. B) in the fomula above is just X(~) (esp. Y (~y)). Also, as in the pevious denition, A and B can be nets with multiple output pots. Then the tanslations of A; B ae tuples, and to obtain the tanslation of the GRAFT we select the coesponding elements of the tuples. 2. If it is a destucto, we have to conside that the input patition of Dest connecting with B can contain positive pots. Then the tanslation of the GRAFT is:! (GRAFT(Dest; A; B)) = P ((A)fy j 7! i (Dest(;! i ([~z](b))))g; [~z](b)! i ([~z](b))) whee, as in Denition 3.1, epesents the pincipal pot of Dest, and ~y ae the input pots of A connected with the destucto. The vaiables ~z, epesenting the negative pots in B that ae connected to the destucto, will typically appea in (B). Note that the fomulas fo the tanslation of non-dependent nets can be obtained as a paticula case of these when no positive pot appeas in an input patition (hence no bound vaiables). Net we give an eample to illustate the denition of in the case of dependent nets. 16

17 Eample 3.6 Let us ecall the inteaction ules fo appending dieence lists given by Lafont [22]. The agents ae Cons, Di, Dappend, and Open, whee Cons and Di ae the constuctos and Dappend and Open the destuctos. Di has two auiliay pots in the same patition: one is positive and the othe negative (hence the system is dependent). The inteaction ules dening Dappend ae: w w A B Dappend u v Diff y X(y) Diff X(y) y Open v 0 0 y 0 Open u Diff y X(y) y 0 y X(y) Note that in these ules we have emphasised the fact that the pots and y of Di ae in the same patition, connected to a net fom which we do not have any infomation (hence its tanslation is X(y)). Open has two patitions, with one pot each. Let us tanslate the st inteaction ule. The net in the left-hand side is obtained by a CUT of two GRAFTS, one tanslated as Dappend(u; v) and the othe as Di([y]X(y)). The net in the ight-hand side is the esult of the composition of two GRAFTS. The st one is tanslated as X(Open(v; z)), and its composition with the second, using the constucto case of Denition 3.5, gives Di([z]X(Open(v; z))). Then the tanslation of the st ule accoding to Denition 3.5 is the combinatoy eduction ule Dappend(Di([y]X(y)); V )! Di([z]X(Open(V; z))) In the same way the tanslation of the second ule is: Open(Di([y]X(y)); Y 0 )! X(Y 0 ) Thee is a coespondence between sequences of inteactions in a net and sequences of ewite steps in its tanslation. Fo eample, the sequence 17

18 Diff Dappend Diff y Cons Cons T Cons S C Diff Cons T Cons C S Open Diff y Cons Diff Cons T Cons C Cons S is tanslated as: Dappend(Di([]Cons(T; Cons(C; ))); Di([y]Cons(S; y)))! Di([]Cons(T; Cons(C; Open(Di([y]Cons(S; y)); ))))! Di([]Cons(T; Cons(C; Cons(S; )))) which is a ewite sequence using the combinatoy eduction ules above. Note again that the tanslation of an inteaction net system is an othogonal system. It is easy to pove that the tanslation function peseves ewite sequences (and nomal foms). Moe pecisely: Theoem 3.7 Let N; N 0 be nets in a semi-simple inteaction net system. If N N 0 then (N)! (N 0 ). If N is a nomal fom, then so is (N). Poof: By induction on the denition of the semi-simple net N. 1. If N = EMPTY o N = LINK then N is a nomal fom, and so is (N). 2. If N = MIX(A,B) then (N) = P ((A); (B)), and the popeties follow by induction. 3. If N = CUT(A,B) then (N) = P ((B)f 7! i ((A))g; (A) i ((A))). The eductions inside A and B ae peseved by induction. If thee is an inteaction between the agent in A and the agent in B connected by the CUT (in this case these agents will be eplaced by a new net M accoding to the ight-hand side of the inteaction ule), then (N) contains a ede: the tanslation of the inteaction ule can be applied to (B)f 7! i ((A))g, and the ede will be eplaced by (M) with the coesponding substitution. This esults in a tem t that coincides with (N 0 ) modulo!, moe pecisely: (N)! t! (N 0 ). Hence (N)! (N 0 ). If N is a nomal fom, so ae A and B, and hence (A) and (B) ae in nomal fom by induction. Moeove, if the CUT does not ceate a ede in the net N, then no ewite ule applies to (N) eithe (because all the ewite ules that coespond to inteaction ules cannot be applied, and the pojections do not apply because they wee aleady applied in the tanslation function: they ae in the metalanguage). 4. If N = GRAFT(Const; A; B) o N = GRAFT(Dest; A; B) then the popety follows by induction, since thee is no CUT on the pincipal pots (the agent being gafted has its pincipal pot fee). Note that in the case of a CUT o a GRAFT the tanslation of a pat of N can appea seveal times in (N), and then an inteaction step in N may coespond to seveal ewite steps in (N). 18

19 Remak 3.8 Fo shaed-ewiting systems we can pove in the same way, using the function of Remak 3.4: If N N 0 then (N)! s (N 0 ). If N is a nomal fom, then so is (N). The poof is simila to the pevious one: in the case of a CUT we also have (N)! s t! (N 0 ), and the same fo a GRAFT (only one step! s suces because using epeated subtems ae shaed). As a consequence of this theoem: Coollay 3.9 Let (; IR) be an inteaction net such that the ewite system (IR) on F teminates. Then (; IR) is teminating. The convese also holds, as shown below, because the ewite systems ae left-linea and nonovelapping (which implies that shaed-ewiting can mimic standad ewiting). Hence temination of (; IR) implies temination of (IR). This popety will be used in the net section to deive modulaity esults fo inteaction nets fom the modulaity esults of tem ewiting systems. Theoem 3.10 Let (; IR) be a teminating semi-simple inteaction net. Then (IR) teminates. Poof: Since modulaity of temination of tem ewiting systems has been studied mostly fo st-ode tem ewiting systems, we pove the theoem fo st-ode systems (but the poof genealises to combinatoy eduction systems as well). Fist we show that thee is a simple coespondence between a tem (o a dag epesenting a tem) and a net. Recall that each function symbol in F coesponds to an agent in ecept fo P, i and empty. Given a tem t 2 T (F; X ) we build a net with a node fo each occuence of a function symbol in t ecept fo P, which is not visible in the net, the pojections i which select edges in the net, and empty, which coesponds to the empty net. To ensue that the net is well-fomed in the case of a non-linea tem, if a non-linea vaiable occus in a subtem which is not of the fom (), we add duplicato nodes to join the coesponding edges. We denote by N t the net associated to t. The tem ewiting system (IR) is othogonal by Theoem 3.3, hence, by Popety 2.7 we can mimic eductions sequences by shaed-eduction ones. We pove by contadiction that (IR) is teminating: Let t 0 be a minimal non-teminating tem (i.e. all its stict subtems ae teminating), and t 0! t 1! t 2 : : : an innite eduction sequence stating fom t 0. We conside the coesponding shaed-ewiting sequence, and the inteaction sequence stating fom the net N t0 that coesponds to t 0. This is depicted in the following diagam, whee t 0! s t 0 1! s t 0 2 : : : is the shaed-ewiting sequence that mimics the sequence t 0! t 1! t 2 : : :. p 1 t 0! t 0 N t0! p 2 p 3 t 1 t 2 l 1! 1 l 2! 2 # #! s t 0 2! : : : l 3! 3! s : : :! s t 0 1 N t 0 1 N t 0 2 : : : Tem ewiting steps using the pojection ules ae not visible in the inteaction sequence: if t 0 1!s t0 2 then N t 0 1 N t 0 2. Howeve, since t is minimal non-teminating, any innite eduction sequence stating fom t contains innite ewite steps at the oot position, which coespond to an innite numbe of inteactions in the sequence that stats fom N t, and this contadicts the assumption of temination of IR. Hence, temination of IR implies temination of (IR). 19