Soft Session Types (Long Version)

Soft Session T yp es Ugo Dal Lago ∗ P aolo D i Giam b erardino † June 24, 2018 Abstract W e sho w how systems of session types can enforce interactions to b e b ound ed for all typable processes. The type system w e p rop ose is based on Lafon t’s soft linear logic and is strongly inspired b y recent w orks ab out session typ es as in tuitionistic linear logic formula s. Our mai n result is the existence, for every typable process, of a p olynomial boun d on the le ngth of reduction sequences starting from it and on th e size of its reducts. 1 In tro duc tion Session t yp es are one of the m os t succes sful paradigms around which communication can be disciplined in a concurren t or ob ject-ba sed environmen t. They can come in many differen t flav ors, depe nding on the under lying progra mming language and on the degr ee of flexibility they a llow when defining the s tructure of sessions . As an example, sy stems of session types for m ulti-par ty int era ction hav e b een r ecently int ro duced [8], while a for m o f higher-or der session has b een shown to b e definable [11]. Recur sive t yp e s, on the other hand, ar e part of the standar d to olset of session t yp e theories since their inceptio n [7]. The key property induced b y sys tems of session types is the following: if tw o (or more) pro cesse s can b e typed w ith “dual” ses sion type s , then they can in terac t with each o ther without “g oing wrong”, i.e. av oiding situa tions whe r e one party needs some data with a c e rtain t yp e and the other(s) offer so mething of a different, incompatible type. Sometimes, one w ould like to g o b eyond that and design a type system which g uarantees strong er pr op erties, including quantitativ e ones . An example of a prop erty that w e find particularly interesting is the following: suppo se that t wo pro cesses P and Q interact by crea ting a sessio n having t yp e A throug h which they co mm unicate. Is this in teraction guar anteed to b e finite? How long would it la st? Moreover, P and Q may be force d to interact w ith other proces ses in order to b e able to offer A . The question could then b ecome: can the global amount of in tera ction be kept under control? In other words, one could b e interested in pr oving the int er action ind uc e d by sessions to b e b ounde d . This problem has b een a lmost negle c ted by the rese a rch communit y in the ar e a of s ession t yp es, altho ug h it is the manifesto of the so-called implicit computational co mplexity (ICC), where one aims a t giving machine-free characterizations of complexity classes ba s ed on pr ogramming lang ua ges and logica l systems. Linear logic ( LL in the following) has been introduced tw ent y-five years ago by Jean-Yv es Girard [6]. One of its grea test merits has b een to allow a finer analys is of the computational conten t o f bo th int uitionistic and classical logic . In turn, this is made pos sible by distinguishing m ultiplicative as well as additive connectives, by an inv olutive notion of negation, a nd by giving a new status to structura l rules allowing them to b e a pplicable o nly to mo dal formulas. One of the many co nsequences of this new, r e fined wa y of lo o king at pro of theory has b een the introduction of natural characterizations of complexity classes b y frag ments of linear logic. This is p o ssible bec ause linear lo gic so mehow “isolates” complexit y in the mo dal fragment of the logic (which ∗ Unive rs i t` a di Bologna & INRIA Sophia An tip olis, dallago@c s.unibo.it † Dipartiment o di Matematica e Informatica, Universit` a di Cagliari , digiambe @unica.it 1 is so lely r esp onsible for the hyperexp o nent ial complexity of cut elimination in, say intuitionistic logic), which can then be re stricted so a s to get exactly the expressive power needed to capture small complexity classes. O ne of the simplest and most elegant of those systems is Lafont’s so ft linear log ic ( SLL in the following), which ha s b een shown to co rresp ond to p olynomia l time in the realm of class ical [9], quantum [5] and higher-o rder concur rent computation [4]. Recently , Caires a nd Pfenning [1] ha ve shown how a s ystem of sessio n t yp es can be built around int uitionistic linear logic, by introducing π D ILL , a t yp e system for the π -calculus where types and rules are derived from the ones of intuitionistic linear lo gic. In their sy s tem, multip lica tive connectives like ⊗ and ⊸ allow to mo del sequentialit y in sessions, while the additive connectives & and ⊕ model e xternal and internal choice, r esp ectively . The moda l connective !, on the other hand, allows to model a ser ver of t yp e ! A which can o ffer the functionality ex pressed by A man y times. In this pap er, we study a restr iction of π DILL , c alled π DSLL , which can b e thought of as b eing derived fro m π DIL L in the s ame w ay as SLL is obtained from LL . In o ther words, the op erator ! behaves in π DSL L in the sa me wa y as in SL L . The main res ult we prov e ab out π DSLL is prec is ely ab out bo unded in terac tio n: whenever P ca n be typed in π DSLL and P → n Q , then b o th n a nd | Q | (the size of the pr o cess Q , to b e defined later) ar e poly nomially related to | P | . This ensures an a bs tract but quite strong form of bo unded in teractio n. Another, p erha ps more “interactiv e” formulation of the same result is the fo llowing: if P and Q interact via a channel o f type A , then the “ complexity” of this interaction is bo unded by a p olynomial on | P | + | Q | , whos e degree o nly depe nds o n A . The pr o of o f b o unded interaction for π DSLL is structurally similar to the one o f po lynomial time soundness for SL L , but there a re a few p eculiarities which makes the argument more complicated (see Section 5 for mo re details). W e see this pap er as the first successful attempt to bring techniques from implicit co mputa- tional complexity into the realm of s e ssion t yp es. Although pr oving bo unded int era ction has b een techn ica lly nontrivial, due to the p eculiarities of the π -ca lculus, we think the main contribution of this work lies in showing that b ounded termination can be enforced by a natural adaptation o f known systems o f session types. 2 An Informal Accoun t on π DI LL In this sectio n, we will outline the main prop erties of π D ILL , a session type system recently int ro duced by Cair e s and Pfenning [1, 2]. F or more informa tion, please consult the t wo cited pap ers. In π DILL , sessio n types a r e nothing more than fo r mulas of (prop ositional) intuitionistic line a r logic without a toms but with (multiplicativ e) constants: A ::= 1 | A ⊗ A | A ⊸ A | A ⊕ A | A & A | ! A. These types are assig ned to channels (names ) by a for mal system der iving judgments in the form Γ; ∆ ⊢ P :: x : A, where Γ and ∆ are contexts assig ning types to channels, and P is a pro c e ss of the name-pa ssing π -calculus. The judgment ab ov e can be read as follows: the pr o cess P a cts on the channel x according to the sessio n type A whenever comp ose d with pro cesses be having according to Γ and ∆ (eac h on a s pe c ific c hannel). Inf or mally , the v ario us constructions on sessio n t yp es can b e explained as follows: • 1 is the type o f an empty sessio n c hannel. A pro c e ss offering to communicate v ia a session channel t yp ed this way simply synchronizes with another pro c e s s thr o ugh it without exchanging anything. This is meant to be an abstraction for all ground ses s ion types, e.g. natura l num b ers, lists, etc. In linea r logic, this is the unit for ⊗ . • A ⊗ B is the type of a s ession ch annel x through which a message carrying another c hannel with t yp e A is sent. After per forming this actio n, the underlying pro cess behaves accor ding to B on the same channel x . 2 • A ⊸ B is the adjoint to A ⊗ B : on a channel with this type, a pr o cess comm unicate by first per forming an input and receiv ing a channel with type A , then acting a ccording to B , again on x . • A ⊕ B is the type of a channel on which a pro c ess either sends a s pe cial messag e inl and per forms according to A or sends a sp ecial message inr and p er forms accor ding to B . This corres p o nds to internal choice. • The type A & B ca n be assigned to a channel x on which the underlying pro ces s offer s the po ssibility o f choos ing betw een pro ceeding a ccording to A or to B , b oth on x . So, in a sense, & mo dels externa l choice. • Finally , the type ! A is a ttributed to a channel x only if a pro cess r ep eatedly receive a channel y through x , then b ehaving on y acco rding to A . In o ther words, ! A is the type of a pro ces s which offers to op en new sessio n of t yp e A . The ass ig nments in Γ and ∆ are o f tw o different natures: • An assignment of a type A to a channel x in ∆ sig nals the need by P o f a pro ces s offering a session of type A on the channel x ; for this reason, ∆ is ca lle d the line ar c ontext ; • An as signment of a t yp e A to a c hannel x in Γ, on the other ha nd, represents the need by P of a pro cess offering a session of type ! A on the c hannel x ; th us, Γ is the exp onential c ontext . Typing rules π DILL are very similar to the ones of DILL , itself one o f the many p oss ible formulations of linear logic as a sequent calculus. In particular, there are tw o cut rules, each corre sp o nding to a different p or tion of the cont ext: Γ; ∆ 1 ⊢ P :: x : A Γ; ∆ 2 , x : A ⊢ Q :: T Γ; ∆ 1 , ∆ 2 ⊢ ( ν x )( P | Q ) :: T Γ; ∅ ⊢ P :: y : A Γ , x : A ; ∆ ⊢ Q :: T Γ; ∆ ⊢ ( ν x )(! x ( y ) .P | Q ) :: T Please observe how cutting a pro cess P ag ainst an a ssumption in the exp o nential context requires to “wrap” P ins ide a replicated input: this allows to turn P int o a server . In or de r to illustrate the intuitions a bove, we now give an example. Suppose that a pro cess P mo dels a s ervice which acts on x as follows: it receives tw o natural num b er s, to b e interpreted as the num b er and secr et co de of a cr edit car d a nd, if they c orresp o nd to a v alid account, r eturns an MP3 file and a rece ipt co de to the client. O therwise, the session terminates . T o do s o , P needs to interact with a nother ser vice (e.g. a banking service ) Q through a channel y . The ba nking service, a mong o thers, provides a w ay to v er ify whether a given num b er and co de co rresp ond to a v alid credit card. In π DILL , the pr o cess P would receive the type ∅ ; y : ( N ⊸ N ⊸ 1 ⊕ 1 )& A ⊢ P :: x : N ⊸ N ⊸ ( S ⊗ N ) ⊕ 1 , where N and S a re pseudo-types for natura l num b er s and MP 3s, resp ectively . A is the type o f all the other functionalities Q provides. As an example, P could b e the following pro ce s s: x ( nm 1 ) .x ( c d 1 ) .y . inl ; ( ν nm 2 ) y h nm 2 i . ( ν c d 2 ) y h c d 2 i . y . cas e ( x . in l ; ( ν mp ) x h mp i . ( ν rp ) x h rp i , x. inr ; 0) Observe how the credit card num b er and secret co de forwarded to Q are not the o nes sent by the client: the flow of infor mation ha pp ening inside a pro c e ss is abstra cted awa y in π DILL . Similarly , one can write a pro cess Q and assig n it a type as follows: ∅ ; ∅ ⊢ Q :: y : ( N ⊸ N ⊸ 1 ⊕ 1 )& A . Putting the t wo deriv ations together , we obtain ∅ ; ∅ ⊢ ( ν x )( P | Q ) :: x : N ⊸ N ⊸ ( S ⊗ N ) ⊕ 1 . Let us now make an observ a tion whic h will probably b e a ppreciated b y the reader familiar with linear logic. The pro cesses P and Q can be typed in π DIL L without the use of any exp onential rule, nor of cut. What allows to t yp e the par allel comp osition ( ν x )( P | Q ), on the o ther hand, is precisely the cut rule. The interaction b etw een P and Q corresp onds to the elimination of that cut. Since there isn’t any expo nent ial around, this pr o cess must be finite, since the size of the underlying pro cess shr inks at every s ingle r eduction step. F rom a pr o cess-a lg ebraic point of view, on the other hand, the finiteness of the interaction is an immedia te consequence of the absence o f any replicatio n in P and Q . 3 The banking ser vice Q can only serve one single sessio n and would v anish a t the end o f it. T o make it into a p ersistent server offering the same kind o f session to po ssibly many differen t clients, Q must b e put into a replica tion, o btaining R =! z ( y ) .Q . In R , the channel z can b e given t yp e !(( N ⊸ N ⊸ 1 ⊕ 1 )& A ) in the empty context. The pro cess P should b e somehow adapted to b e able to interact with R : befo r e per forming the tw o outputs on y , it’s neces sary to “spawn” R by per forming an o utput on z a nd passing y to it. This way we obtain a pr o cess S such that ∅ ; z :!(( N ⊸ N ⊸ 1 ⊕ 1 )& A ) ⊢ S :: x : N ⊸ N ⊸ ( S ⊗ N ) ⊕ 1 , and the comp o sition ( ν z )( S | R ) ca n be g iven the sa me t yp e as ( ν x )( P | Q ). Of c ourse, S could hav e used the c hannel z more than once, initiating distinct sessions. This is mean t to mo de l a situation in which the sa me c lie n t interacts with the same s e rver b y cr eating more than one session with the same type , itse lf do ne by p erforming m or e than one output o n the same channel. Of course, ser vers can themselves dep end o n other servers. And these dep endencies are naturally mo deled by the expo nential mo dality of linear logic. 3 On Bounded Int eraction In π DILL , the p oss ibilit y of mo deling p ersistent servers which in turn dep end on other servers makes it p os sible to type pro ce sses which ex hibit a very co mplex and combinatorially heavy interactive behavior. Consider the following pro cesses , the fir st one para meter ized o n a natural num b er i ∈ N : dupser i . = ! x i ( y ) . ( ν z ) x i +1 h z i . ( ν w ) x i +1 h w i . ; dup client . = ( ν y ) x 0 h y i ; ser . =! x ( y ) . 0 In π DI LL , these pro cesse s can b e t yp ed as follows: ∅ ; x i +1 :! 1 ⊢ dupser i :: x i :! 1 ; ∅ ; x 0 :! 1 ⊢ dup client :: z : 1 ; ∅ ; ∅ ⊢ s er :: x :! 1 . Then, for every n ∈ N o ne can type the parallel co mpo sition mulser n +1 . = ( ν x 1 . . . x n )( dupser n || . . . || dupser 0 ) as follows ∅ ; x n :! 1 ⊢ mu lser n :: x 0 :! 1 . Informally , mulser n is a per sistent se r ver which offers a session t yp e 1 on a channel x 0 , provided a ser ver with the same functionality is av ailable on x n . The pro ce ss mu lser n is the paralle l comp osition o f n s ervers in the form dupser i , each s pawning tw o different sess ions provided by dupser i +1 on the same channel x i +1 . The pro cess mulser n cannot be further reduced. But notice that, once ser , mulser n and dup client are co mpo sed, the following exp o ne ntial blowup is b o und to happ en: ( ν x 0 )( ser | mulser n | dup client ) ≡ ( ν x 0 . . . x n )( ser | dupser n || . . . || dupser 0 | dup client ) → ( ν x 0 . . . x n )( ser | dupser n || . . . || dupser 1 | P 1 ) → 2 ( ν x 1 . . . x n )( ser | dupser n || . . . || dupser 2 | P 2 | P 2 ) → 4 ( ν x 2 . . . x n )( ser | dupser n || . . . || dupser 3 | P 3 || . . . || P 3 | {z } 4 times ) → ∗ ( ν x n )( ser | dupser n | P n || . . . || P n | {z } 2 n times ) → 2 n 0 . 4 Here, for every i ∈ N the pro ces s P i is s imply ( ν y ) x i h y i . ( ν z ) x i h z i . Notice that b oth the num b er or reduction steps and the size of intermediate pro cesses are exp onential in n , while the size of the initial pro ces s is linear in n . This is a p erfectly legal pro ces s in π DIL L . Mo reov er the type ! 1 of the channel x 0 through w hich dup client and mulser n communicate do es not contain a ny informa tion ab out the “complexity” of the interaction: it is the sa me for every n . The deep reas ons why this phenomenon can happen lie in the very general (and “gener ous”) rules governing the behavior of the exp onential mo dality ! in linear lo gic. It is this generality that allows the embedding of prop os itio nal intuitionistic logic into linear logic. Since the c omplexity of normalizatio n for the former [12 , 10] is nonele men tar y , the exp onential blowup describ ed ab ov e is not a sur prise. It would b e desir able, on the o ther hand, to b e s ure that the interaction caused b y any pro cess P is b o unded: whenev er P → n Q , then there’s a r e asonably low upper b ound to bo th n a nd | Q | . This is prec isely what we achiev e by restricting π D ILL into π DSL L . 4 π DS LL : Syn tax and Main Prop erties In this section, the syntax of π DSL L will b e introduced. Mo reov er, some basic op erationa l prop erties will b e s ta ted a nd proved. 4.1 The Pro cess Algebra π DSLL is a t yp e sys tem for a fair ly standard π -ca lculus, exactly the one on top of which π DIL L is defined: Definition 1 (Pro cess es) Given an infinite set of na mes or channels x, y , z , . . . , t he set of pro- cesses is define d as fol lows: P ::= 0 | P | Q | ( ν x ) P | x ( y ) .P | x h y i .P | ! x ( y ) .P | x. inl ; P | x. inr ; P | x. case ( P , Q ) The only non- standard constructs are the la st three, whic h allow to define a c hoice mechanism: the pro cess x. case ( P , Q ) can evolv e as P or as Q after having r eceived a signal in the form inl o inr throug h x . Pro cesse s sending such a signal through the c hannel x , then contin uing like P are, res pe c tively , x. inl ; P and x. inr ; P . The set of names o c curring free in the pro ce ss P (hereb y denoted fn ( P )) is defined as usual. The same holds for the capture av oiding substitution of a name x for y in a pro cess P (denoted P { x/y } ), and for α -equiv alence b etw een pro cess es (deno ted ≡ α ). Structural congr uence is an equiv alence relation identifying those pro cesses which are syn tac- tically different but can b e considered eq ual for very simple structural r easons: Definition 2 (Structural Co ngruence) The r elation ≡ , c al le d str uctural co ngruence , is the le ast c ongruenc e on pr o c esses satisfying t he fol lowing seven axioms: P ≡ Q whenever P ≡ α Q ; ( ν x )0 ≡ 0; P | 0 ≡ P ; ( ν x )( ν y ) P ≡ ( ν y )( ν x ) P ; P | Q ≡ Q | P ; (( ν x ) P ) | Q ≡ ( ν x )( P | Q ) whenever x / ∈ fn ( Q ); P | ( Q | R ) ≡ ( P | Q ) | R. F o r mal sy s tems for reduction and la be lle d s e mantics can b e defined in a standard wa y . W e refer the reader to [1] for more details. A quantit ative a ttr ibute of pro cesses whic h is delica te to mo del in pro cess alg ebras is their size : how can we measure the size of a pro cess? In particular , it is not stra ightf or ward to define a measure which b oth reflects the “ n umber of s ymbols” in the pro cess and is in v ariant under structural cong ruence (this wa y facilitating a ll pro ofs). A go o d co mpromise is the following: 5 Definition 3 (Pro cess Si ze) The size | P | of a pr o c ess P is define d by induction on the structure of P as fol lows: | 0 | = 0; | x ( y ) .P | = | P | + 1; | x. inl ; P | = | P | + 1; | P | Q | = | P | + | Q | ; | x h y i .P | = | P | + 1; | x. inr ; P | = | P | + 1; | ( ν x ) P | = | P | ; | ! x ( y ) .P | = | P | + 1; | x. case ( P, Q ) | = | P | + | Q | + 1 . According to the definition above, the empty pro cess 0 has null size, while r estriction do es no t increase the size of the underlying pro cess. This allo ws for a definition of size whic h remains inv a riant under structural cong ruence. The price to pay is the fo llowing: the “num b er of symbols ” of a pr o cess P can be arbitr arily big ger than | P | (e.g. for every n ∈ N , | ( ν x ) n P | = | P | ). How e ver, we hav e the following: Lemma 1 F or every P, Q , | P | = | Q | whenever P ≡ Q . Mor e over, ther e is a p olynomial p : N → N such that for every P , t her e is Q with P ≡ Q and the num b er of symb ols in Q is at most p ( | Q | ) . Pro of. The fact P ≡ Q implies | P | = | Q | can b e proved by a simple insp ection of Definition 1 . The second part o f the lemma c a n b e prov ed by induction on P once the p olyno mial p is fixed a s p ( x ) = x 2 . ✷ 4.2 The T yp e System The language of types of π D SLL is exactly the same a s the one of π DILL , a nd the interpretation of t yp e cons tructs do es not change (see Section 2 for s o me infor ma l details). T yping judgments and t yping rules, how ever, are significa nt ly different, in particular, in the tr eatment of the exp onential connective !. More sp ecifically , π DILL allows to give type to the following pro ces ses: • F or every t yp e A , there is a pro ces s D ER A such that ∅ ; x :! A ⊢ DER A :: y : A . As an ex ample, DER 1 is ( ν z ) x h z i . Intuitiv ely , DER A is a pro ces s op ening a new s ession of type A b y calling a server of type ! A . • F or every t yp e A , there is a pr o cess CONT A such that ∅ ; x :! A ⊢ CONT A :: y :! A ⊗ ! A . Int uitively , CONT A is a pro cess offering firs t a sessio n o f type ! A a nd then pro ceeding as ! A along the channel y . All this with the need o f only a server of type ! A fro m x . As an exa mple, CONT 1 is ( ν w ) ( s h w i . ((! w ( y ) . ( ν z ) x h z i ) | (! s ( y ) . ( ν z ) x h z i ))) . • F or every type A , there is also a pro cess DIG A such that ∅ ; x :! A ⊢ DIG A :: y :!! A , whic h turns a server into a server of servers. The r eader is invited to define DIG 1 as an exer c ise. As we will see at the end of this section, o nly DER A can b e given a type in π DSLL , while CONT A and DIG A cannot. In π DSLL , typing judgments b ecome s y nt actica l expressio ns in the for m Γ; ∆; Θ ⊢ P :: x : A. First of all, obse r ve how the context is divided into thr e e ch unks now: Γ and ∆ hav e to b e inter- preted as exp onential cont exts, while Θ is the usua l linear context from π DIL L . The necessity of having two exp o nential con texts is a cons e quence of the finer, less canonical expo nential discipline of SL L compared to the one of LL . W e use the following terminolog y: Γ is said to be the aux iliary context, while ∆ is the multiplexor con text. Typing r ule s are in Figure 1. The rules gov erning the typing constant 1 , the m ultiplicatives ( ⊗ and ⊸ ) and the a dditives ( ⊕ a nd &) are exact analo g ues o f the ones fro m π DILL . The only differences come from the presence of tw o exp onential contexts: in binary multiplicativ e rules ( ⊗ R and ⊸ L ) the auxiliar y context is treated multiplicativ ely , while the multiplexor context is treated 6 Γ; ∆; Θ ⊢ P :: T Γ; ∆; Θ , x : 1 ⊢ P :: T 1 L Γ; ∆; ∅ ⊢ 0 :: x : 1 1 R Γ; ∆; Θ , y : A, x : B ⊢ P :: T Γ; ∆; Θ , x : A ⊗ B ⊢ x ( y ) .P :: T ⊗ L Γ 1 ; ∆; Θ 1 ⊢ P :: y : A Γ 2 ; ∆; Θ 2 ⊢ Q :: x : B Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ ( ν y ) x h y i . ( P | Q ) :: x : A ⊗ B ⊗ R Γ 1 ; ∆; Θ 1 , y : A ⊢ P :: T Γ 2 ; ∆; Θ 2 , x : B ⊢ Q :: T Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 , x : A ⊸ B ⊢ ( ν y ) x h y i . ( P | Q ) :: T ⊸ L Γ; ∆; Θ , y : A ⊢ P :: x : B Γ; ∆; Θ ⊢ x ( y ) .P :: x : A ⊸ B ⊸ R Γ; ∆; Θ , x : A ⊢ P :: T Γ; ∆; Θ , x : B ⊢ P :: T Γ; ∆; x : A ⊕ B , Θ ⊢ y . cas e ( P, Q ) :: T ⊕ L Γ; ∆; Θ ⊢ P :: x : A Γ; ∆; Θ ⊢ x. inl ; P :: x : A ⊕ B ⊕ R 1 Γ; ∆; Θ ⊢ P :: x : B Γ; ∆; Θ ⊢ x. inr ; P :: x : A ⊕ B ⊕ R 2 Γ; ∆; Θ , x : A ⊢ P :: T Γ; ∆; Θ , x : A & B ⊢ x. inl ; P :: T & L 1 Γ; ∆; Θ , x : B ⊢ P :: T Γ; ∆; Θ , x : A & B ⊢ x. inr ; P :: T & L 2 Γ; ∆; Θ ⊢ P :: x : A Γ; ∆; Θ ⊢ P :: x : B Γ; ∆; Θ ⊢ y . case ( P, Q ) :: x : A & B & R Γ; ∆ , x : A ; Θ , y : A ⊢ P :: T Γ; ∆ , x : A ; Θ ⊢ ( ν y ) x h y i .P :: T ♭ # Γ; ∆; Θ , y : A ⊢ P :: T Γ , x : A ; ∆; Θ ⊢ ( ν y ) x h y i .P :: T ♭ ! Γ; ∆ , x : A ; Θ ⊢ P :: T Γ; ∆; Θ , x :! A ⊢ P :: T ! L # Γ , x : A ; ∆; Θ ⊢ P :: T Γ; ∆; Θ , x :! A ⊢ P :: T ! L ! Γ; ∅ ; ∅ ⊢ Q :: y : A ∅ ; ∆; !Γ ⊢ ! x ( y ) .Q :: x :! A ! R Γ 1 ; ∆; Θ 1 ⊢ P :: x : A Γ 2 ; ∆; Θ 2 , x : A ⊢ Q :: T Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ ( ν x )( P | Q ) :: T cut ∆; ∅ ; ∅ ⊢ P :: y : A Γ; ∆ , x : A ; Θ ⊢ Q :: T Γ; ∆; Θ ⊢ ( ν x )(! x ( y ) .P | Q ) :: T cut # Γ 1 ; ∅ ; ∅ ⊢ P :: y : A Γ 2 , x : A ; ∆; Θ ⊢ Q :: T Γ 1 , Γ 2 ; ∆; Θ ⊢ ( ν x )(! x ( y ) .P | Q ) :: T cut ! Figure 1: T yping rules for π DSLL . 7 additively , as in π DI LL 1 . Now, co nsider the rules gov erning the e x po nential connective !, which are ♭ ! , ♭ # , ! L ! , ! L # and ! R : • The rules ♭ ! and ♭ # bo th a llow to spawn a ser ver. This co rresp onds to turning an ass umption x : A in the linear context in to one y : A in one of the exp onential contexts; in ♭ # , x : A could be already prese nt in the m ultiplexor context, while in ♭ ! this cannot happ en; • The rules ! L ! and ! L # lift a n assumption in the exp onential contexts to the linear context; this requires changing its t yp e fr om A to ! A ; • The rule ! R allows to turn an or dinary pro cess into a server, by pack aging it int o a r e plicated input and mo difying its type. Finally there a re thr e e cut rules in the system, namely cut , cut ! and cut # : • cut is the usual linear cut r ule, i.e. the natur a l generaliz a tion of the one from π DI LL . • cut ! and cu t # allow to eliminate an assumption in one of the the tw o ex po nent ial contexts. In b oth ca ses, the pro cess which a llows to do that m ust b e t y pable with empt y linear and m ultiplexor contexts. Observe how b oth CONT A and DIG A are not typable in π DSLL . T ake, as an example, CONT 1 : the tw o o ccurr ences o f x ar e in the sc op e of a r eplicated input, and this pattern is not allowed in the restricted setting of soft linear logic. On the other hand, DER A is indeed t ypable. Actually , a genera liz ation o f it called MUL T n A (where n ≥ 0) can b e typed as follows ∅ ; ∅ ; x :! A ⊢ MUL T n A :: y : A ⊗ . . . ⊗ A | {z } n + 2 times . F o r example, MUL T 2 1 is the following pro cess: ( ν x ) y h x i . ( ν x 1 ) y h x 1 i . ( ν x 2 ) y h x 2 i . 4.3 Bac k to Our Example Let us now reco nsider the example pro cesses introduced in Sectio n 3. The basic building blo ck ov er whic h everything is built was the proc ess dupser i =! x i ( y ) . ( ν z ) x i +1 h z i . ( ν w ) x i +1 h w i . . W e claim that for every i , the pro ces s dupser i is not t ypable in π DSL L . T o understand wh y , observe that the only way to type a r e plic a ted input like dupser i is by the typing rule ! R , and that its premise requires the bo dy of the replicated input to b e typable with empty line a r and multiplexor contexts. A quick insp ection on the typing rules reveals that every name in the aux iliary cont ext o ccurs (free ) exactly once in the underlying pro cess (provided we count t wo o ccurrences in the branches of a cas e as just a single o ccur rence). How ever, the na me x i +1 app ears twic e in the bo dy of dupser i . A slight v ariation on the example ab ove, o n the other hand, c an b e t yp ed in π DSLL , but this requires changing its type. 4.4 Sub ject Reduction A basic prop erty most type sys tems for functional langua ges s atisfy is sub ject reduction: typing is preser ved along r eduction. F or pro cesses , this is o ften true for internal r eduction: if P → Q and ⊢ P : A , then ⊢ Q : A . In this section, a sub ject reduction result for π DSL L will be given and some ide a s o n the underlying pro of will be describ ed. Some co ncepts outlined here will b ecome necessary ingredients in the pro o f o f b o unded interaction, to b e done in Section 5 b elow. Sub ject reduction is proved by closely following the pa th tra ced by Caires a nd Pfenning; as a cons equence, we pro ceed quite quickly , concentrating our a ttent ion on the differences with their pro o f. When proving sub ject reduction, o ne constantly w or k with t yp e deriv ations . This is partic- ularly true here, where (internal) r eduction corr esp onds to the cut-elimination pr o cess. A linear 1 The reader familiar with linear logic and pro of nets wi ll r ecognize in the different treatmen t of the auxiliary and multiplexor context s, one of the basic pri ncipl es of SLL : c ontr action is forbidd en on the auxiliary do ors of exp onenti al b oxes . The c hannel names conta ined in the auxiliary con text corresp ond to the auxiliary doors of exponent ial b oxes, so we treat them multiplicativ ely . The cont raction effect induced by the additiv e tr eatmen t of the channel names in the multiplexor conte xt corr esponds to the multiplexing r ul e of SLL . 8 1 L ( x, D ) b D z 1 R 0 ⊗ L ( x, y .z . E ) x ( y ) . b E z ⊗ R ( D , E ) ( ν y ) x h y i . ( b D y | b E x ) ⊸ L ( x, D , y. E ) ( ν y ) x h y i . ( b D y | b E z ) ⊸ R ( x. D ) x ( y ) . b E x cut ( D , x. E ) ( ν x )( b D x | b E z ) cut ! ( D , x. E ) ( ν x )(! x ( y ) . b D y | b E z ) cut # ( D , x. E ) ( ν x )(! x ( y ) . b D y | b E z ) ♭ ! ( x, y . E ) ( ν y ) x h y i . b E z ♭ # ( x, y . E ) ( ν y ) x h y i . b E z ! R ( D , x 1 , . . . , x n ) ! x ( y ) . b D y ! L ! ( x. D ) b D z ! L # ( x. D ) b D z ⊕ L ( x, y . D , z . E ) y . cas e ( b D x , b E z ) ⊕ R 1 ( D ) x. inl ; b D x ⊕ R 2 ( D ) y . inr ; b D y & L 1 ( x, y . E ) x. inl ; b D z & L 2 ( x, y . D ) y . inr ; b D z & R ( D , E ) z . case ( b D z , b E z ) Figure 2: Extraction of pro cesses from pro of ter ms . notation for pro ofs in the fo rm of pr o of t erms can be eas ily defined, allowing for more compact descriptions. As a n e x ample, a pro o f in the form π : Γ 1 ; ∆; Θ 1 ⊢ P :: x : A ρ : Γ 2 ; ∆; Θ 2 , x : A ⊢ Q :: T Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ ( ν x )( P | Q ) :: T cut corres p o nds to the pr o of ter m cut ( D , x. E ), where D is the pro o f term for π and E is the pro o f term for ρ . If D is a pr o of ter m corresp onding to a t yp e deriv ation for the pro cess P , w e write b D = P . F r om now on, pr o of terms will often take the place of pro cesses: Γ; ∆; Θ ⊢ D :: T stands for the existence o f a type der iv atio n D with conclus ion Γ; ∆; Θ ⊢ b D :: T . The notation Γ; ∆; Θ ⊢ D P :: T stands for the ex istence of a t yp e deriv a tion D s uch that Γ; ∆; Θ ⊢ D :: T and b D = P . A pro of term D is said to be nor mal if it do es not contain any instance s of cut rules. In Figure 2 we show in detail how pro cesses ar e asso cia ted with pro of terms. Sub ject reduction will be proved b y s howing that if P is typable b y a t yp e deriv a tio n D and P → Q , then a t yp e deriv ation E for Q exists. Actually , E can be obtained by manipulating D us ing techniques der ived fr om cut-eliminatio n. Noticeably , not ev ery cut-elimination rule is necessary to prove sub ject reduction. In other words, we a re in presence of a weak corres po ndence betw een pro of terms and pro cess e s, and remain fa r from a ge nuine Curry- How ard corresp ondence. Those manipulations of pro o f-terms which are nece s sary to prov e sub ject reduction can be classified as follows: • First of all, a binary relatio n = ⇒ on pro of terms called c omputational r e duct ion can be defined. A t the logical level, this cor resp onds to prop er cut-elimination steps, i.e. those cut-elimination steps in which tw o rules intro ducing the sa me c onnective interact. At the pro cess level, com- putational reduction co rresp ond to in ternal reductio n. = ⇒ is not symmetric. Computational reduction rules are g iven in Figure 3. W e str ess that Lemma 3 b elow is needed in order to prop erly define so me cases of computatio nal reduction. • A binar y relation 7− → on pro of terms called shift r e duction , distinct fro m = ⇒ , must b e in tro - 9 ( cut / ⊗ R / ⊗ L ) : cut (( ⊗ R ( D , E )) , x. ⊗ L ( x, y .x. F )) = ⇒ cut ( D , y . cut ( E , x. F )) ( cut / ⊸ L / ⊸ R ) : cut ( ⊸ R ( y . D ) , x. ⊸ L ( x, E , x. F )) = ⇒ cut ( cut ( E , y . D ) , x. F ) ( cut / & R / & L 1 ) : cut (& R ( D , E ) , x. & L 1 ( x, y . F )) = ⇒ c ut ( D , x. F ) ( cut / & R / & L 2 ) : cut (& R ( D , E ) , x. & L 2 ( x, y . F )) = ⇒ c ut ( E , x. F ) ( cut / ⊕ R 1 / ⊕ L ) : cut ( ⊕ R 1 ( D ) , x. ⊕ L ( x, y . E , z . F )) = ⇒ cut ( D , x. E ) ( cut / ⊕ R 2 / ⊕ L ) : cut ( ⊕ R 2 ( D ) , x. ⊕ L ( x, y . E , z . F )) = ⇒ cut ( D , x. F ) ( cut ! / − /♭ ! ) : cut ! ( D , x.♭ ! ( x, y . E )) = ⇒ cut ( D ⇓ , y . cut # ( D , x. E ⇓ )) ( cut # / − /♭ # ) : cut # ( D , x.♭ # ( x, y . E )) = ⇒ cut ( D ⇓ , y . cut # ( D , x. E )) Figure 3: Computational r eduction rules ( cut / ! R / ! L ! ) : cut (! R ( D , x 1 , . . . , x n ) , x. ! L ! ( x. E )) 7− → ! L ! ( x 1 . ! L ! ( x 2 . . . . ! L ! ( x n . cut # ( D , y . E )) . . . )) ( cut / ! R / ! L # ) : cut (! R ( D , x 1 , . . . , x n ) , x. ! L # ( x. E )) 7− → ! L ! ( x 1 . ! L ! ( x 2 . . . . ! L ! ( x n . cut # ( D , y . E )) . . . )) Figure 4: Shift r eduction rules duced. A t the pro ce s s level, it corres po nds to structural congr ue nc e . As = ⇒ , 7− → is not a symmetric relatio n. Shift reduction rules ar e given in Figure 4. • Finally , an equiv alence relation ≡ on pro of terms called pr o of e quivalenc e is necessar y . At the logical level, this co rresp onds to the so-called commuting conv ersions, while at the pro cess level, the induced pro cesse s ar e either structurally congruent or strongly bisimilar. Equiv alence rules are given in Figure 5. The reflexive and tra nsitive closur e of 7− → ∪ ≡ is denoted with ֒ → , i.e. ֒ → = ( 7− → ∪ ≡ ) ∗ . T o help the reader unders ta nd the rules defining = ⇒ , 7− → and ≡ , let us give some relev ant examples: • Let us consider the pro of term D = cut (( ⊗ R ( F , G )) , x. ⊗ L ( x, y .x. H )) which cor r esp onds to the ⊗ -case of cut elimination. B y a c omputational reduction rule, D = ⇒ E = cut ( F , y . c ut ( G , x. H )). F r o m the pro ces s side, b D = ( ν x )((( ν y ) x h y i . ( b F | b G )) | x ( y ) . b H ) and b E = ( ν x )( ν y )(( b F | b G ) | b H ), where b E is the pro cess obtained from b D by internal passing the channel y throug h the channel x . • Let D = cut (! R ( F , x 1 , . . . , x n ) , x. ! L ! ( x. G )) be the pro of obtained by comp osing a pro of F (whose last rule is ! R ) with a pro of G (whose la st rule is ! L ! ) thro ug h a cut rule. A shift r eduction rule tells us that D 7− → E = ! L ! ( x 1 . ! L ! ( x 2 . . . . ! L ! ( x n . cut ! ( F , y . G )) . . . )), which corresp o nds to the op ening of a box in S LL . The shift reduction do es not have a cor resp onding reduction step at pro cess level, since b D ≡ b E ; nevertheless, it is defined a s an asymmetric rela tion, for technical reasons connected to the pro of of b o unded int er a ction. • Let D = cut # ( F , x. cut ( G , y. H )). A defining rule for pro of equiv alence ≡ , s tates that in D the cut # rule can be p ermuted over the cu t rule, by duplicating F ; namely D ≡ E = cut ( cut # ( F , x. G ) , y . cu t # ( F , x. H )). This is po ssible beca us e the channel x b elongs to the m ul- tiplexor co nt exts of bo th G , H , such cont exts b eing treated additively . At the pr o cess level, b D = ( ν x )((! x ( y ) . b F ) | ( ν y )( b G | b H )) , while b E = ( ν y )((( ν x )(! x ( y ) . b F ) | b G )) | (( ν x )(! x ( y ) . b F ) | b H ))), b D and b E b eing str ongly bisimilar. The rest o f this section is dev oted to pr oving the following result: Theorem 1 (Sub ject Reduction) L et Γ; ∆; Θ ⊢ D :: T . Supp ose that b D = P → Q . Then ther e is E such that b E = Q , D ֒ → = ⇒ ֒ → E and Φ; Ψ ; Θ ⊢ E :: T , wher e Γ , ∆ = Φ , Ψ . The str ucture of the pro of o f Theor em 1 is divided into three steps, each of them consisting in o ne or more auxilia r y results: 1. First, given a pro ces s P and a t yping deriv atio n D of P , we establish a connection b etw een t yping and lab elled semantics, showing that the visible actions of P behav e acco rding to the t yp es a ssigned to the channels in P by D (Lemma 4). 2. Second, we take t wo pro cesses P and Q communicating with each other on the same channel 10 Structural Con v ersions ( cut / − / cu t 1 ) : cut ( D , x. cut ( E x , y . F y )) ≡ cu t ( cut ( D , x. E x ) , y . F y ) ( cut / − / cu t 2 ) : cut ( D , x. cut ( E , y . F xy )) ≡ cu t ( E , x. cut ( D , y. F xy )) ( cut / − / cu t ! ) : cut ( D , x. cut ! ( E , y . F xy )) ≡ cu t ! ( E , y . cut ( D , x. F xy )) ( cut / cut ! / − ) : c ut ( cut ! ( D , y . E y ) , x. F x ) ≡ cut ! ( D , y . c ut ( E y , x. F x )) ( cut / − / cu t # ) : cut ( D , x. cut # ( E , y . F xy )) ≡ cu t # ( E , y . cut ( D , x. F xy )) ( cut / cut # / − ) : cut ( cut # ( D , y . E y ) , x. F x ) ≡ cut # ( D , y . c ut ( E y , x. F x )) ( cut / 1 R / 1 L ) : cut ( 1 R , x. 1 L ( x, D )) ≡ D Strong Bisimi larities ( cut # / − / cu t ) : cut # ( D , x. cut ( E x , y . F xy )) ≡ cu t ( cut # ( D , x. E x ) , y . c ut # ( D , x. F xy )) ( cut # / − / cu t # ) : cut # ( D , x. cut # ( E x , y . F xy )) ≡ cu t # ( D , x. cut # ( E x , y . cut # ( D , x. F xy ))) ( cut # / − / cu t ! ) : cut # ( D , x. cut ! ( E x , y . F xy )) ≡ cu t ! ( E x , y . cut # ( D , x. F xy )) ( cut ! / − / cu t 1 ) : cut ! ( D , x. cut ( E x , y . F y )) ≡ cu t ( cut ! ( D , x. E x ) , y . F y ) ( cut ! / − / cu t 2 ) : cut ! ( D , x. cut ( E , y. F xy )) ≡ cu t ( E , y . cut ! ( D , x. F xy )) ( cut ! / − / cu t ! ) 1 : cut ! ( D , x. cut ! ( E x , y . F y )) ≡ cu t ! ( cut ! ( D , x. E x ) , y . F y ) ( cut ! / − / cu t ! ) 2 : c ut ! ( D , x. cut ! ( E , y . F xy )) ≡ cu t ! ( E , x. cut ! ( D , y . F xy )) ( cut ! / − / cu t # ) : cut ! ( D , x. cut # ( E x , y . F xy )) ≡ cu t # ( E x , y . cut ! ( D , x. F xy )) ( cut # / − / cu t # ) 0 : cut # ( D , x. cut # ( E x , y . F xy )) ≡ cu t # ( E x , y . cut # ( D , x. F xy )) (if y / ∈ F V ( b F )) ( cut # / − / − 0 ) : cut # ( D , x. E ) ≡ E (if x / ∈ F N ( b E )) Commuting Co n v ersio n s ( cut / − / 1 L ) : cut ( D , x. 1 L ( y , E x )) ≡ 1 L ( y , c ut ( D , x. E x )) ( cut / − / ! L ! ) : cut ( D , x. ! L ! ( y . E xz )) ≡ ! L ! ( y . cut ( D , x. E xz )) ( cut / − / ! L # ) : cut ( D , x. ! L # ( y . E xz )) ≡ ! L # ( y . cut ( D , x. E xz )) ( cut / 1 L / − ) : cut ( 1 L ( y , D ) , x. E x ) ≡ 1 L ( y , cut ( D , x. E x )) ( cut / ! L ! / − ) : cut (! L ! ( y . D z ) , x. E x ) ≡ ! L ! ( y . cut ( D z , x. E xz )) ( cut / ! L # / − ) : cut (! L # ( y . D z ) , x. E x ) ≡ ! L # ( y . cut ( D z , x. E xz )) ( cut ! / − / 1 L ) : cut ! ( D , x. 1 L ( y , E x )) ≡ 1 L ( y , c ut ! ( D , x. E x )) ( cut ! / − / ! L ! ) : c ut ! ( D , x. ! L ! ( y . E xz )) ≡ ! L ! ( y . cut ! ( D , x. E xz )) ( cut ! / − / ! L # ) : cut ! ( D , x. ! L # ( y . E xz )) ≡ ! L # ( y . cut ! ( D , x. E xz )) ( cut # / − / 1 L ) : cut # ( D , x. 1 L ( y , E x )) ≡ 1 L ( y , c ut # ( D , x. E x )) ( cut # / − / ! L ! ) : cut # ( D , x. ! L ! ( y . E xz )) ≡ ! L ! ( y . cut # ( D , x. E xz )) ( cut # / − / ! L # ) : cut # ( D , x. ! L # ( y . E xz )) ≡ ! L # ( y . cut # ( D , x. E xz )) Figure 5: Equiv alence rules 11 x , and the cor resp onding typing der iv ations D , E , resp ectively . F or a ll p ossible t yp e assigne - men t of x , we show that b y compo sing D and E with a cut rule and per forming some pro of manipulation we can obtain a pro of F such that F is a typing deriv ation for the pro cess R obtained by p erforming the communication of P and Q (lemmas 5 , 6, 7, 8, 9, 10, 11). 3. Finally , we show that if a pro cess P is typable by a type deriv ation D and P → Q , then a t yp e deriv a tion E for Q exists. This is done by showing that the int erna l re ductio n which brings from P to Q is a consequence o f the c o mmun icatio n o f tw o subpro cesses of P . This communication can o nly happ en in pr esence of a cut on the corr esp onding pr o of terms, so we conclude using the previous lemmas. The following prop ositio ns state the co rresp ondence s b etw een the pr o of ter ms manipulation rules describ ed ab ov e and r elations ov er pro c esses: we omit the pro ofs, le aving to the reader the verifi- cation of ea ch c a se. Prop ositi o n 1 L et Γ; ∆; Θ ⊢ D :: T and Φ; Ψ; Σ ⊢ E :: S . If D = ⇒ E , t hen b D → b E . Prop ositi o n 2 L et Γ; ∆; Θ ⊢ D :: T and Φ; Ψ; Σ ⊢ E :: S . If D 7− → E , t hen b D is e quivalent to b E mo dulo stru ctur al c ongruenc e. Prop ositi o n 3 L et Γ; ∆; Θ ⊢ D :: T and Φ; Ψ; Σ ⊢ E :: S . If D ≡ E , then b D is e quivalent t o b E mo dulo stru ctur al c ongruenc e or str ong bisimilarity. Before pro c eeding to Sub ject Reduction, we give the following tw o lemmas, co ncerning structur al prop erties of the t yp e s ystem: the first o ne s tates that in a pro of der iv ation the multiplexor context can b e weak ened. The sec o nd says that in a pro o f deriv a tio n a s sumptions in the auxiliar y context can b e “lifted” to the multiplexor context, while the underlying pro ces s stays the same. Lemma 2 (W eak e ning lemm a) If Γ; ∆; Θ ⊢ D :: T and whenever ∆ ⊆ Φ , it holds that Γ; Φ; Θ ⊢ D :: T . Pro of. By a simple induction on the structure of D . ✷ Lemma 3 (Lifting l e mma) If Γ; ∆; Θ ⊢ D :: T then ther e exists an E such that ∅ ; Γ , ∆; Θ ⊢ E :: T wher e b E = b D . We denote E by D ⇓ . Pro of. Again, a simple induction on the structure of the pro of term D . ✷ The following is sort of a ge ne r ation lemma ( s ( α ) denotes the sub ject of the actio n α ): Lemma 4 L et Γ; ∆; Θ ⊢ D P :: x : T . 1. If P α − → Q and T = 1 then s ( α ) 6 = x . 2. If P α − → Q and y : 1 ∈ Θ then s ( α ) 6 = y . 3. If P α − → Q and s ( α ) = x and T = A ⊗ B then α = ( ν y ) x h y i . 4. If P α − → Q and s ( α ) = y and y : A ⊗ B ∈ Θ then α = y ( z ) . 5. If P α − → Q and s ( α ) = x and T = A ⊸ B then α = x ( y ) . 6. If P α − → Q and s ( α ) = y and y : A ⊸ B ∈ Θ then α = ( ν z ) y h z i . 7. If P α − → Q and s ( α ) = x and T = A & B then α = x. inl ; or α = x. inr ; . 8. If P α − → Q and s ( α ) = y and y : A & B ∈ Θ then α = y . inl ; or α = y . inr ; . 9. If P α − → Q and s ( α ) = x and T = A ⊕ B then α = x. inl ; or α = x. inr ; . 10. If P α − → Q and s ( α ) = y and y : A ⊕ B ∈ Θ then α = y . inl ; or α = y . inr ; 11. If P α − → Q and s ( α ) = x and T = ! A then α = x ( y ) . 12. If P α − → Q and s ( α ) = y and y :! A or y ∈ Γ or y ∈ ∆ or y ∈ Φ then α = ( ν z ) y h z i . Pro of. T rivial from definitions. ✷ Crucial to the pro of of the Sub ject Reduction Theo rem is a n analysis of how pro cesses interacting with their en viro nment s p er forming dual action can co mm unicate when comp osed by a cut rule. 12 Lemma 5 Assume that: 1. Γ 1 ; ∆; Θ 1 ⊢ D :: x : A ⊗ B with b D = P ( ν y ) x h y i − − − − − → Q ; 2. Γ 2 ; ∆; Θ 2 , x : A ⊗ B ⊢ E :: z : C with b E = R x ( y ) − − − → S . Then: 1. cut ( D , x. E ) ֒ → = ⇒ ֒ → F for some F ; 2. Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ F :: z : C , wher e b F ≡ ( ν x )( Q | S ) . Pro of. By simultaneous inductio n on D 1 , D 2 . The pr op erty stated in the lemma holds als o fo r the system π DILL (see [1]); since the pro o f technique is essentially the sa me mo dulo some minor details, we o mit the pro of. ✷ Lemma 6 Assume 1. Γ 1 ; ∆; Θ 1 ⊢ D 1 P 1 :: x : A ⊸ B with P 1 x ( y ) − − − → Q 1 2. Γ 2 ; ∆; Θ 2 , x : A ⊸ B ⊢ D 2 P 2 :: z : C with P 2 ( ν y ) x h y i − − − − − → Q 2 Then 1. cut ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → D for some D ; 2. Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ D R :: z : C for some R ≡ ( ν x )( ν y )( Q 1 | Q 2 ) . Pro of. See the pro of of Lemma 5 . ✷ Lemma 7 Assume 1. Γ 1 ; ∆; Θ 1 ⊢ D 1 P 1 :: x :! A with P 1 x ( y ) − − − → Q 1 2. Γ 2 ; ∆; Θ 2 , x :! A ⊢ D 2 P 2 :: z : C wi th P 2 ( ν y ) x h y i − − − − − → Q 2 Then 1. cut ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → D for some D ; 2. Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ D R :: z : C for some R ≡ ( ν x )( ν y )( Q 1 | Q 2 ) . Pro of. See the pro of of Lemma 5 . ✷ Lemma 8 Assume 1. Γ 1 ; ∆; Θ 1 ⊢ D 1 P 1 :: x : A & B with P 1 x. inl ; − − − − → Q 1 2. Γ 2 ; ∆; Θ 2 , x : A & B ⊢ D 2 P 2 :: z : C with P 2 x. inl ; − − − − → Q 2 Then 1. cut ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → D for some D ; 2. Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ D R :: z : C for some R ≡ ( ν x )( Q 1 | Q 2 ) . Pro of. See the pro of of Lemma 5 . ✷ Lemma 9 Assume 1. Γ 1 ; ∆; Θ 1 ⊢ D 1 P 1 :: x : A ⊕ B with P 1 x. inl ; − − − − → Q 1 . 2. Γ 2 ; ∆; Θ 2 , x : A ⊕ B ⊢ D 2 P 2 :: z : C with P 2 x. inl ; − − − − → Q 2 . Then 1. cut ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → D for some D ; 2. Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ D R :: z : C for some R ≡ ( ν x )( Q 1 | Q 2 ) . Pro of. See the pro of of Lemma 5 . ✷ Lemma 10 Assume 1. Γ 1 ; ∅ ; ∅ ⊢ D 1 P 1 :: x : A 2. Γ 2 , x : A ; ∆; Θ ⊢ D 2 P 2 :: z : C with P 2 ( ν y ) x h y i − − − − − → Q 2 Then 1. cut ! ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → c ut # ( D 1 , x. D ) for some D wher e x / ∈ F V ( b D ) ; 2. Γ; Φ; Θ ⊢ D R :: z : C fo r some R ≡ ( ν y )( P 1 | Q 2 ) , wher e Γ , Φ = Γ 1 , Γ 2 , x : A, ∆ . 13 Pro of. By induction on D 2 . W e have different cases, dep ending fro m the la st rules of D 2 . Let us just write do wn so me r elev a n t cas e : • Suppos e D 2 = ♭ ! ( x, y . E ); then P 2 ≡ ( ν y ) x h y i .Q 2 and Γ 2 , x : A ; ∆; Θ ⊢ E Q 2 :: z : C by inv ersio n. Now cut ! ( D 1 , x.♭ ! ( x, y . E )) = ⇒ cut ( D 1 ⇓ , y . cut # ( D 1 , x. E ⇓ )) by ( cut ! / − /♭ ! ) ≡ cut # ( D 1 , x. cut ( D 1 ⇓ , y . E ⇓ )) by ( cut / − / cut # ). W e pick D = cut ( D 1 ⇓ , y . E ⇓ ); then Γ; Φ; Θ ⊢ D Q 2 :: z : C for some Q 2 ≡ ( ν y )( P 1 | Q 2 ), where Γ , Φ = Γ 1 , Γ 2 , x : A, ∆. • Suppos e D 2 = cut # ( E 1 , y . E 2 ); then ∆; ∅ ; ∅ ⊢ E 1 R 1 :: w : C a nd Γ 2 , x : A ; ∆; Θ ⊢ E 2 R 2 :: z : B with P 2 ( ν y ) x h y i − − − − − → R 1 | R ′ 2 , by inv ersio n. Now by induction h yp othesis, cut ! ( D 1 , x. E 2 ) ֒ → = ⇒ ֒ → cut # ( D 1 , x. F ) for some F (where x / ∈ F V ( b F )), and Γ; Φ; Θ 2 ⊢ F S :: z : B for some S = ( ν y )( P 1 | R ′ 2 ). cut ! ( D 1 , x. cut # ( E 1 , y . E 2 )) ≡ cut # ( E 1 , y . cut ! ( D 1 , x. E 2 )) by ( cut ! / − / cut # ), ֒ → = ⇒ ֒ → c ut # ( E 1 , y . cut # ( D 1 , x. F )) by co ngruence, ≡ cut # ( D 1 , x. cut # ( E 1 , y . F )) by ( cut # / − / cut # ) 0 . Pick D = cut # ( E 1 , y . F ). Then R = ( ν y ) R 1 | S by cut, a nd Γ; Φ; Θ ⊢ D R :: z : C for some R ≡ ( ν y )( P 1 | Q 2 ). This concludes the pro of. ✷ Corollary 1 Assume 1. Γ 1 ; ∅ ; ∅ ⊢ D 1 P 1 :: x : A 2. Γ 2 , x : A ; ∆; Θ ⊢ D 2 P 2 :: z : C with P 2 ( ν y ) x h y i − − − − − → Q 2 Then 1. cut ! ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → D for some D ; 2. Γ; Φ; Θ ⊢ D R :: z : C for some R ≡ ( ν x )(! x ( y ) .P 1 | ( ν y )( P 1 | Q 2 )) , wher e Γ , Φ = Γ 1 , Γ 2 , ∆ Pro of. F ollows from Lemma 1 0. ✷ Lemma 11 Assume 1. ∆; ∅ ; ∅ ⊢ D 1 P 1 :: x : A 2. Γ; ∆ , x : A ; Θ ⊢ D 2 P 2 :: z : C with P 2 ( ν y ) x h y i − − − − − → Q 2 Then : 1. cut # ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → c ut # ( D 1 , x. D ) for some D ; 2. Φ; Ψ , x : A ; Θ ⊢ D R :: z : C for some R ≡ ( ν x )( ν y )( P 1 | Q 2 ) , wher e Φ , Ψ = Γ , ∆ . Pro of. By induction on D 2 . W e have different cases, dep ending fro m the la st rules of D 2 . Let us just write do wn so me r elev a n t cas e s: • D 2 = cut ( E 1 , y . E 2 ). Assume Γ = Γ 1 , Γ 2 and Θ = Θ 1 , Θ 2 . Now Γ 1 ; ∆ , x : A ; Θ 1 ⊢ E 1 R 1 :: w : B and Γ 2 ; ∆ , x : A ; Θ , w : B ⊢ E 2 R 2 :: z : C by in version. W e have t wo cas es:either P 2 ( ν y ) x h y i − − − − − → R ′ 1 | R 2 or P 2 ( ν y ) x h y i − − − − − → R 1 | R ′ 2 . First ca se: cut # ( D 1 , x. E 1 ) ֒ → = ⇒ ֒ → c ut # ( D 1 , x. F ) for some F ; then Γ 1 ; ∆ , x : A ; Θ 1 ⊢ F S :: w : B for so me S = ( ν y )( P 1 | R ′ 1 ) b y induc- tion hypothesis; cut # ( D 1 , x. cut ( E 1 , y . E 2 )) ≡ cut ( cut # ( D 1 , x. E 1 ) , y . c ut # ( D 1 , x. E 2 )) by ( cut # / − / cut ), ֒ → = ⇒ ֒ → c ut ( cut # ( D 1 , x. F ) , y . cut # ( D 1 , x. E 2 )) by cong ruence ≡ cut # ( D 1 , x. cut ( F , y . E 2 )) by ( cut # / − / cut ). P ick D = cu t ( F , y . E 2 ); then R = ( ν y ) S | R 2 by cut. Then Γ; ∆ , x : A ; Θ ⊢ D R :: z : C for some R ≡ ( ν y )( P 1 | Q 2 ). Second case: cu t # ( D 1 , x. E 2 ) ֒ → = ⇒ ֒ → cut # ( D 1 , x. F ) for some F ; then Γ 2 ; ∆ , x : A ; Θ 2 ⊢ F S :: w : B for so me S = ( ν y )( P 1 | R ′ 2 ) by induction hypothesis ; cut # ( D 1 , x. cut ( E 1 , y . E 2 )) ≡ cu t ( cut # ( D 1 , x. E 1 ) , y . c ut # ( D 1 , x. E 2 )) by ( cut # / − / cut ), ֒ → = ⇒ ֒ → cut ( cut # ( D 1 , x. E 1 ) , y . c ut # ( D 1 , x. F )) b y co ngruence, ≡ cut # ( D 1 , x. cut # ( E 1 , y . F )) by ( cut # / − / cut ) . Pick D = cut # ( E 1 , y . F ); then R = ( ν y ) R 1 | S by cut. Then Γ; ∆ , x : A ; Θ ⊢ D R :: z : C for some R ≡ ( ν y )( P 1 | Q 2 ). • D 2 = cut # ( E 1 , y . E 2 ). ∆; ∅ ; ∅ ⊢ E 1 R 1 :: w : B Γ; ∆ , x : A, w : B ; Θ ⊢ E 2 R 2 :: z : C b y inv e rsion. Now P 2 ( ν y ) x h y i − − − − − → R 1 | R ′ 2 ; cut # ( D 1 , x. E 2 ) ֒ → = ⇒ ֒ → cu t # ( D 1 , x. F ) for some F and Γ; ∆ , x : A, w : B ; Θ ⊢ F S :: w : B fo r some S = ( ν y )( P 1 | R ′ 2 ) by induction h yp oth- esis. cut # ( D 1 , x. cut # ( E 1 , y . E 2 )) ≡ cut # ( D 1 , x. cut # ( E 1 , y . cut # ( D 1 , x. E 2 ))) b y ( cut # / − / c ut # ) ֒ → = ⇒ ֒ → cut # ( D 1 , x. cut # ( E 1 , y . cut # ( D 1 , x. F ))) by cong ruence, ≡ cut # ( D 1 , x. cut # ( E 1 , y . F )) by ( cut # / − / cut # ) . P ick D = cut # ( E 1 , y . F ); then P 2 = ( ν y ) R 1 | S by cut. Then Γ; ∆ , x : A ; Θ ⊢ D R :: z : C for some R ≡ ( ν y )( P 1 | Q 2 ). 14 This concludes the pro of. ✷ Corollary 2 Assume 1. ∆; ∅ ; ∅ ⊢ D 1 P 1 :: x : A 2. Γ; x : A, ∆; Θ ⊢ D 2 Q 1 :: z : C wi th Q 1 ( ν y ) x h y i − − − − − → Q ′ 1 Then 1. cut # ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → D for some D ; 2. Φ; Ψ; Θ ⊢ D Q 2 :: z : C for some Q 2 ≡ ( ν x )(! x ( y ) .P 1 | ( ν y )( P 1 | Q ′ 1 )) , wher e Φ , Ψ = Γ , ∆ . Pro of. This follows fr o m Lemma 11. ✷ W e are finally able to give a pr o of of Sub ject Reductio n for π DS LL : Pro of. (of T he o rem 1) W e rea son by induction o n the s tructure of D . Since b D = P → Q the only po ssible last rules o f D can b e: 1 L , ! L ! , ! L # , a linear cut ( cut ) o r a n e x po nential cut ( cut ! or cut # ). In all the other cases , the underlying pro ces s ca n only per form a visible action, as can be e a sily verified by insp ecting the rules from Figur e 1 . W ith this o bserv ation in mind, let us insp e ct the op era tional semantics deriv ation proving that P → Q . At some p oint we will find tw o subpro cesses o f P , call them R a nd S , whic h communicate, ca us ing an internal reduction. W e here claim that this can o nly happe n in presence of a cut, and only the communication b etw een R and S m ust o ccur along the channel inv olved in the cut. No w, it’s only a matter of showing that the just describ ed situation can b e “resolved” preserv ing types, and this can b e done using the previo us lemmas. Some relev ant case: • D = cut ! ( D 1 , x. D 2 ); assume Γ = Γ 1 , Γ 2 and P ≡ ( ν x )! x ( w ) .P 1 | P 2 . Now Γ 1 ; ∅ ; ∅ ⊢ D 1 P 1 :: x : C and Γ 2 , x : A ; ∆; Θ ⊢ D 2 P 2 :: z : A , by inv ersio n; from P → Q either P 2 → Q 2 and Q = ( ν x )! x ( w ) .P 1 | Q 2 or P 2 ( ν y ) x h y i − − − − − → Q 2 and Q = ( ν x )! x ( w ) .P 1 | ( ν y ) P 1 | Q 2 . First case: Γ 2 , x : A ; ∆; Θ ⊢ E 2 Q 2 :: z : A for some E 2 with D 2 ֒ → = ⇒ ֒ → E 2 by i.h.; cut ! ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → cut ! ( D 1 , x. E 2 ) by cong r uence. Pick E = cut ! ( D 1 , x. E 2 ); then Γ; ∆; Θ ⊢ E Q :: z : A b y cut ! . Second case: cut ! ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → E for some E ; then Γ; ∆; Θ ⊢ E R :: z : A for s ome R ≡ Q b y Corollar y 1. • D = cut # ( D 1 , x. D 2 ). No w, P ≡ ( ν x )! x ( w ) .P 1 | P 2 and ∆; ∅ ; ∅ ⊢ D 1 P 1 :: x : C , Γ; ∆ , x : A ; Θ ⊢ D 2 P 2 :: z : A , b y in version; from P → Q either P 2 → Q 2 and Q = ( ν x )! x ( w ) .P 1 | Q 2 or P 2 ( ν y ) x h y i − − − − − → Q 2 and Q = ( ν x )! x ( w ) .P 1 | ( ν y ) P 1 | Q 2 First case: Γ; ∆ , x : A ; Θ ⊢ E 2 Q 2 :: z : A for some E 2 with D 2 ֒ → = ⇒ ֒ → E 2 by i.h. and cut # ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → cut # ( D 1 , x. E 2 ) by congruence. Pick E = cut # ( D 1 , x. E 2 ); then Γ; ∆; Θ ⊢ E Q :: z : A by cut # Second case: cut # ( D 1 , x. D 2 ) ֒ → = ⇒ ֒ → E for some E ; then Γ; ∆; Θ ⊢ E R :: z : A for some R ≡ Q by Corollar y 2. This concludes the pro of. ✷ 5 Pro v ing P olynomial B ounds In this section, we prove the main res ult of this pap er , namely some poly nomial b ounds on the length o f internal r eduction sequences and on the size of intermediate results for pro c esses typable in π DILL . In other w or ds, interaction will b e shown to be b ounded. The simplest formulation of this result is the following: Theorem 2 F or every typ e A , t her e is a p olynomial p A such that whenever ∅ ; ∅ ; x : A ⊢ D :: y : 1 and ∅ ; ∅ ; ∅ ⊢ E :: x : A wher e D and E ar e normal and ( ν x )( b D | b E ) → n P , it holds that n, | P | ≤ p A ( | b D | + | b E | ) 15 Int uitively , what Theo rem 2 says is that the complexity of the interaction b etw een tw o pro cesses t ypable w itho ut cuts a nd c o mmun icating through a channel with session t yp e A is p olynomial in their sizes, where the sp ecific po lynomial inv olved o nly depends on A itself. In o ther words, the complexity of the interaction is no t only b ounded, but can b e somehow “read o ff ” fro m the t yp es of the co mmun icating par ties. How do es the pro of of Theorem 2 look like? Conce ptua lly , it can be thought of a s being structured into four steps: 1. First of all, a natural n umber W ( D ) is attributed to any pro of term D . W ( D ) is said to b e the weight of D . 2. Secondly , the weigh t of any pro of term is shown to strictly decrease along co mputational reduction, not to increas e along shifting r eduction and to stay the same for equiv a le n t pro of terms. 3. Thirdly , W ( D ) is shown to b e bounded by a p olyno mial on | b D | , where the exp onent only depe nds on the nesting depth of b oxes of D , denoted B ( D ). 4. Finally , the b ox depth B ( D ) o f any pro of term D is shown to b e “readable” from its type int erfa c e. This is exactly what we a re going to do in the re st of this section. Please o bserve how po int s 1 – 3 ab ov e allow to prov e the following s tr onger result, from which Theorem 2 easily follows, given po int 4: Prop ositi o n 4 F or every n ∈ N , ther e is a p olynomial p n such that for every pr o c ess P with Γ; ∆; Θ ⊢ P :: T , if P → m Q , then m, | Q | ≤ p B ( P ) ( | P | ) . 5.1 Preliminary Definitions Some conce pts hav e to b e given b efore w e can em bar k in the pro of of Pro p o sition 4. First of all, we ne e d to define what the b ox-depth of a pro ces s and of a pr o of term are . Simply , given a proc e ss P , its b ox- depth B ( P ) is the nesting-level of replicatio ns 2 in P . As an example, the b ox-depth of ! x ( y ) . ! z ( w ) . 0 is 2, while the one o f ( ν x ) y ( z ) is 0. F o r mally , given a pro of term D its b ox depth B ( D ) is defined as follows, by induction on the structure of D : B ( 1 L ( x, D )) = B ( D ) B ( ⊕ R 1 ( D )) = B ( D ) B ( 1 R ) = 0 B ( ⊕ R 2 ( D )) = B ( D ) B ( ⊗ L ( x, y .z . D )) = B ( D ) B ( ♭ ! ( x, y . D )) = B ( D ) B ( ⊗ R ( D , E )) = max { B ( D ) , B ( E ) } B ( ♭ # ( x, y . D )) = B ( D ) B ( ⊸ L ( x, D , y . E )) = ma x { B ( D ) , B ( E ) } B (! L ! ( x. D )) = B ( D ) B ( ⊸ R ( x. D )) = B ( D ) B (! L # ( x. D )) = B ( D ) B (& L 1 ( x, y . D )) = B ( D ) B (! R ( x 1 , . . . , x n , D )) = 1 + B ( D ) B (& L 2 ( x, y . D )) = B ( D ) B ( cut ( D , x. E )) = max { B ( D ) , B ( E ) } B (& R ( D , E )) = max { B ( D ) , B ( E ) } B ( cut ! ( D , x. E )) = max { B ( D ) + 1 , B ( E ) } B ( ⊕ L ( x, y . D , z . E )) = max { B ( D ) , B ( E ) } B ( cut # ( D , x. E )) = max { B ( D ) + 1 , B ( E ) } Analogously , the b ox-depth of a pro of term D is simply B ( b D ). Now, suppo se that Γ; ∆; Θ ⊢ D :: T a nd that x : A b elongs to either Γ or ∆, i.e. that x is an “exp onential” channel in D . A key parameter is the virtual numb er of o c curr enc es of x in D , which is denoted a s FO ( x, D ). This para meter , as its name sugge s ts, is not simply the num b er o f literal oc currences of x in D , but takes into a ccount pos sible duplications derived from cuts. So, for example, FO ( w , cu t ! ( D , x. E )) = F O ( x, E ) · FO ( w, D ) + FO ( w , E ), while F O ( w, ⊗ R ( D , E )) is merely 2 This terminology is derived from l i near logic, where pro ofs obtained b y the promotion rule are usually called boxes 16 FO ( w, D ) + FO ( w , E ). Obviously , FO ( w , ♭ ! ( x, w . D )) = 1 and FO ( w , ♭ # ( x, w . D )) = 1. F ormally: FO ( w , 1 L ( x, D )) = F O ( w, D ) FO ( w, 1 R ) = 0 FO ( w, ⊗ L ( x, y .z . D )) = FO ( w, D ) FO ( w, ⊗ R ( D , E )) = FO ( w, D ) + FO ( w , E ) FO ( w, ⊸ L ( x, D , y . E )) = FO ( w , D ) + FO ( w, E ) FO ( w, ⊸ R ( x. D )) = FO ( w, D ) FO ( w, cut ( D , x. E )) = FO ( w, D ) + FO ( w , E ) FO ( w, cut ! ( D , x. E )) = FO ( x, E ) · FO ( w, D ) + FO ( w , E ) FO ( w, cut # ( D , x. E )) = FO ( x, E ) · FO ( w, D ) + FO ( w , E ) FO ( w , ♭ ! ( x, w . D )) = 1 FO ( w , ♭ # ( x, w . D )) = 1 FO ( w , ♭ ! ( x, y . D )) = 0 FO ( w , ♭ # ( x, y . D )) = 0 FO ( w, ! L ! ( x. D )) = FO ( w, D ) FO ( w , ! L # ( x. D )) = FO ( w, D ) FO ( w, ! R ( x 1 , . . . , x n , D )) = 0 FO ( w, ⊕ L ( x, y . D , z . E )) = FO ( w, D ) + FO ( w, E ) FO ( w, ⊕ R 1 ( D )) = FO ( w , D ) FO ( w, ⊕ R 2 ( D )) = FO ( w , D ) FO ( w, & L 1 ( x, y . D )) = FO ( w , D ) FO ( w, & L 2 ( x, y . D )) = FO ( w , D ) FO ( w, & R ( D , E )) = FO ( w, D ) + FO ( w , E ) A channel in either the auxilia ry or the exp onential context c an “floa t” to the linear co nt ext as an effect of r ules ! L ! or ! L # . F rom that moment on, it can only b e treated as a linear channel. As a consequence, it makes sense to define the duplic ability factor of a pr o of term D , written D ( D ), simply as the maximum of FO ( x, D ) o ver all instances of the rules ! L ! or ! L # in D , wher e x is the inv olved channel. F or example, D (! L ! ( x. D )) = max { D ( D ) , FO ( y , D ) } and D ( ⊸ L ( x, D , y . E )) = max { D ( D ) , D ( E ) } . F ormally , the duplicability facto r D ( D ) of D is defined a s follows: D ( 1 L ( x, D )) = D ( D ) D ( ⊕ R 1 ( D )) = D ( D ) D ( 1 R ) = 0 D ( ⊕ R 2 ( D )) = D ( D ) D ( ⊗ L ( x, y .z . D )) = D ( D ) D ( ♭ ! ( x, y . D )) = D ( D ) D ( ⊗ R ( D , E )) = max { D ( D ) , D ( E ) } D ( ♭ # ( x, y . D )) = D ( D ) D ( ⊸ L ( x, D , y . E )) = max { D ( D ) , D ( E ) } D (! L ! ( x. D )) = ma x { D ( D ) , FO ( y , D ) } D ( ⊸ R ( x. D )) = D ( D ) D (! L # ( x. D )) = ma x { D ( D ) , FO ( y , D ) } D (& L 1 ( x, y . D )) = D ( D ) D (! R ( x 1 , . . . , x n , D )) = D ( D ) D (& L 2 ( x, y . D )) = D ( D ) D ( cut ( D , x. E )) = max { D ( D ) , D ( E ) } D (& R ( D , E )) = max { D ( D ) , D ( E ) } D ( cut ! ( D , x. E )) = max { D ( D ) , D ( E ) } D ( ⊕ L ( x, y . D , z . E )) = max { D ( D ) , D ( E ) } D ( cut # ( D , x. E )) = max { D ( D ) , D ( E ) } It’s now po ssible to give the definition of W ( D ), namely the weight of the pro of ter m D . Befor e doing that, ho wev er, it is neces s ary to give a parameter ized no tion of weigh t, denoted W n ( D ). Int uitively , W n ( D ) is defined s imilarly to | b D | . How ever, every input and output action in b D can po ssibly count for more than one: 17 • Everything inside D in ! R ( x 1 , . . . , x n , D ) counts for n ; • Everything inside D in either cut ! ( D , x. E ) o r cut # ( D , x. E ) co unt s for FO ( x, E ). F o r example, W n ( cut # ( D , x. E )) = FO ( x, E ) · W n ( D ) + W n ( E ), while W n (& L 2 ( x, y . D )) = 1 + W n ( D ). F o r mally: W n ( 1 L ( x, D )) = W n ( D ) W n ( 1 R ) = 0 W n ( ⊗ L ( x, y .z . D )) = 1 + W n ( D ) W n ( ⊗ R ( D , E )) = 1 + W n ( D ) + W n ( E ) W n ( ⊸ L ( x, D , y . E )) = 1 + W n ( D ) + W n ( E ) W n ( ⊸ R ( x. D )) = 1 + W n ( D ) W n ( cut ( D , x. E )) = W n ( D ) + W n ( E ) W n ( cut ! ( D , x. E )) = FO ( x, E ) · W n ( D ) + W n ( E ) W n ( cut # ( D , x. E )) = FO ( x, E ) · W n ( D ) + W n ( E ) W n ( ♭ ! ( x, y . D )) = 1 + W n ( D ) W n ( ♭ # ( x, y . D )) = 1 + W n ( D ) W n (! L ! ( x. D )) = W n ( D ) W n (! L # ( x. D )) = W n ( D ) W n (! R ( x 1 , . . . , x n , D )) = n · ( W n ( D ) + 1) W n ( ⊕ L ( x, y . D , z . E )) = 1 + W n ( D ) + W n ( E ) W n ( ⊕ R 1 ( D )) = 1 + W n ( D ) W n ( ⊕ R 2 ( D )) = 1 + W n ( D ) W n (& L 1 ( x, y . D )) = 1 + W n ( D ) W n (& L 2 ( x, y . D )) = 1 + W n ( D ) W n (& R ( D , E )) = 1 + W n ( D ) + W n ( E ) Now, W ( D ) is simply W D ( D ) ( D ). 5.2 Monotonicit y Results The crucia l ingredient for proving p olynomial b ounds are a series of res ults ab out how the weigh t D evolves when D is put in r elation with ano ther pro of term E by wa y of either = ⇒ , 7− → or ≡ . Lemma 12 F or every D , D ( D ) = D ( D ⇓ ) and for every n , W n ( D ) = W n ( D ⇓ ) . Whenever a pro of ter m D computationally r educes to E , the underlying weigh t is guara nteed to strictly decrea s e: Prop ositi o n 5 If Γ; ∆; Θ ⊢ D :: T and D = ⇒ E , then Φ; Ψ; Θ ⊢ E :: T (wher e Γ , ∆ = Φ , Ψ ) , D ( E ) ≤ D ( D ) and W ( E ) < W ( D ) . Pro of. By induction o n the pro o f that D = ⇒ E . So me int er e sting cases: • Suppos e that D = cut ( ⊸ R ( y . F ) , x. ⊸ L ( x, G , x. H )) = ⇒ cu t ( cut ( G , y . F ) , x. H ) = E . Then, D ( D ) = max { D ( F ) , D ( G ) , D ( H ) } = D ( E ); W ( D ) = W D ( D ) ( D ) = 3 + W D ( D ) ( F ) + W D ( D ) ( G ) + W D ( D ) ( H ) > 2 + W D ( E ) ( F ) + W D ( E ) ( G ) + W D ( E ) ( H ) = W D ( E ) ( E ) = W ( E ) . • Suppos e that D = cut (& R ( F , G ) , x. & L 1 ( x, y . H )) = ⇒ cut ( F , x. H ) = E . Then, D ( D ) = max { D ( F ) , D ( G ) , D ( H ) } = D ( E ); W ( D ) = W D ( D ) ( D ) = 3 + W D ( D ) ( F ) + W D ( D ) ( G ) + W D ( D ) ( H ) > 2 + W D ( E ) ( F ) + W D ( E ) ( G ) + W D ( E ) ( H ) = W D ( E ) ( E ) = W ( E ) . 18 • Suppos e that D = cut ! ( F , x.♭ ! ( x, y . G )) = ⇒ cut ( F ⇓ , y . cut # ( F , x. G ⇓ )) = E . Then, D ( D ) = max { D ( F ⇓ ) , D ( G ⇓ ) } = max { D ( F ) , D ( F ) , D ( G ) } = D ( E ); W ( D ) = W D ( D ) ( D ) = FO ( x, ♭ ! ( x, y . G )) · W D ( D ) ( F ⇓ ) + W D ( D ) ( ♭ ! ( x, y . G )) = W D ( D ) ( F ) + W D ( D ) ( ♭ ! ( x, y . G )) = W D ( D ) ( F ) + 1 + W D ( D ) ( G ) ≥ W D ( E ) ( F ) + 1 + W D ( E ) ( G ) > W D ( E ) ( F ) + W D ( E ) ( G ) = W D ( E ) ( F ) + 0 · W D ( E ) ( F ) + W D ( E ) ( G ) = W D ( E ) ( F ) + FO ( x, G ) · W D ( E ) ( F ) + W D ( E ) ( G ) = W D ( E ) ( E ) = W ( E ) . • Suppos e that D = cu t # ( F , x.♭ # ( x, y . G )) = ⇒ cut ( F ⇓ , y . cut # ( F , x. G )) = E . Then we can pr o ceed exactly a s in the previous cas e. This concludes the pro of. ✷ Shift r eduction, on the other hand, is not guar anteed to induce a strict decrea se on the underlying weigh t which, how ever, cannot increa se: Prop ositi o n 6 If Γ; ∆; Θ ⊢ D :: T and D 7− → E , then Γ; ∆; Θ ⊢ E :: T , D ( E ) ≤ D ( D ) and W ( E ) ≤ W ( D ) . Pro of. By induction o n the pro o f that D 7− → E . So me int er e sting cases: • Suppos e that D = cut (! R ( x 1 , . . . , x n , F ) , x. ! L ! ( x. G )) 7− → ! L ! ( x 1 . ! L ! ( x 2 . . . . ! L ! ( x n . cut ! ( F , y . G )))) = E . Then, D ( D ) = max { D ( F ) , D ( G ) } = D ( E ) W ( D ) = W D ( D ) ( D ) = D ( D ) · W D ( D ) ( F ) + W D ( D ) ( G ) ≥ F O ( y , G ) · W D ( D ) ( F ) + W D ( D ) ( G ) = FO ( y , G ) · W D ( E ) ( F ) + W D ( E ) ( G ) = W D ( E ) ( E ) = W ( E ) . • Suppos e that D = cu t (! R ( x 1 , . . . , x n , F ) , x. ! L # ( x. G )) 7− → ! L # ( x 1 . ! L # ( x 2 . . . . ! L # ( x n . cut # ( F , y . G )))) = E . Then we can pr o ceed as in the pr evious cas e. This concludes the pro of. ✷ Finally , equiv a lence leav es the weigh t unchanged: Prop ositi o n 7 If Γ; ∆; Θ ⊢ D :: T and D ≡ E , then Γ; ∆; Θ ⊢ E :: T , D ( E ) = D ( D ) and W ( E ) = W ( D ) . Pro of. By induction o n the pro o f that D ≡ E . Some interesting c ases: • Suppos e that D = cut ( F , x. cut ( G x , y . H y )) ≡ cut ( cut ( F , x. G x ) , y . H y ) = E . Then: D ( D ) = max { D ( F ) , D ( G x ) , D ( H y ) } = D ( E ) W ( D ) = W D ( D ) ( D ) = W D ( D ) ( F ) + W D ( D ) ( G x ) + W D ( D ) ( H y ) = W D ( E ) ( F ) + W D ( E ) ( G x ) + W D ( E ) ( H y ) = W D ( E ) ( E ) = W ( E ) . 19 • Suppos e that D = cut ( F , x. cut ( G , y . H xy )) ≡ cut ( G , x. cut ( F , y . H xy )) = E . Then we can pr o ceed as in the pr evious cas e. • Suppos e that D = cut ( F , x. cut ! ( G , y . H xy )) ≡ cut ! ( G , y . cut ( F , x. H xy )) = E . Then, since FO ( y , F ) = 0, D ( D ) = max { D ( F ) , D ( G ) , D ( H xy ) } = D ( E ) W ( D ) = W D ( D ) ( D ) = W D ( D ) ( F ) + FO ( y , H xy ) · W D ( D ) ( G ) + W D ( D ) ( H xy ) = W D ( D ) ( F ) + FO ( y , cut ( F , x. H xy )) · W D ( D ) ( G ) + W D ( D ) ( H xy ) = W D ( E ) ( F ) + FO ( y , cut ( F , x. H xy )) · W D ( E ) ( G ) + W D ( E ) ( H xy ) = W D ( E ) ( E ) = W ( E ) . • Suppos e that D = cu t # ( F , x. cut ( G x , y . H xy )) ≡ cut ( cut # ( F , x. G x ) , y . c ut # ( F , x. H xy )) = E . Then, D ( D ) = max { D ( F ) , D ( G x ) , D ( H xy ) } = D ( E ) W ( D ) = FO ( x, cut ( G x , y . H xy )) · W D ( D ) ( F ) + W D ( D ) ( G x ) + W D ( D ) ( H xy ) = ( FO ( x, G x ) + FO ( x, H xy )) · W D ( D ) ( F ) + W D ( D ) ( G x ) + W D ( D ) ( H xy ) = ( FO ( x, G x ) · W D ( D ) ( F ) + FO ( x, H xy )) · W D ( D ) ( F ) + W D ( D ) ( G x ) + W D ( D ) ( H xy ) = W D ( D ) ( cut # ( F , x. G x )) + W D ( D ) ( cut # ( F , x. H xy )) = W D ( D ) ( E ) = W D ( E ) ( E ) = W ( E ) . This concludes the pro of. ✷ Now, consider again the sub ject reduction theorem (Theorem 1): what it guar antees is that whenever P → Q and b D = P , there is E with b E = Q and D ֒ → = ⇒ ֒ → E . In view of the three prop ositions w e hav e just stated and prov ed, it’s clear that W ( D ) > E . Altogether , this implies that W ( D ) is a n upp er b o und on the num b er or in terna l reductio n steps b D can p erform. But is W ( D ) itself b ounded? 5.3 Bounding the W eigh t What kind o f b ounds can w e exp ect to prov e for W ( D )? More sp ecifically , how r elated are W ( D ) and | b D | ? Lemma 13 Supp ose Γ; ∆; Θ ⊢ D :: T . Th en 1. If x ∈ Γ , then FO ( x, D ) ≤ 1 ; 2. If x ∈ ∆ , then FO ( x, D ) ≤ | D | ; 3. If x ∈ Θ , then FO ( x, D ) = 0 ; Pro of. By induction on the structure of a type deriv atio n π for Γ; ∆; Θ ⊢ D :: T . Some interesting cases: • If π is ρ 1 : Γ 1 ; ∆; Θ 1 ⊢ D 1 :: z : A ρ 2 : Γ 2 ; ∆; Θ 2 ⊢ D 2 :: y : B Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ ⊗ R ( D 1 , D 2 ) :: y : A ⊗ B ⊗ R 20 then FO ( x, ⊗ R ( D 1 , D 2 )) = FO ( x, D 1 ) ≤ 1 if x ∈ Γ 1 FO ( x, ⊗ R ( D 1 , D 2 )) = FO ( x, D 2 ) ≤ 1 if x ∈ Γ 2 FO ( x, ⊗ R ( D 1 , D 2 )) = FO ( x, D 1 ) + FO ( x, D 1 ) ≤ | D 1 | + | D 2 | ≤ | ⊗ R ( D 1 , D 2 ) | if x ∈ ∆ FO ( x, ⊗ R ( D 1 , D 2 )) = FO ( x, D 1 ) = 0 if x ∈ Θ 1 FO ( x, ⊗ R ( D 1 , D 2 )) = FO ( x, D 2 ) = 0 if x ∈ Θ 2 • If π is Γ 1 ; ∅ ; ∅ ⊢ ∅ :: D 1 z : A Γ 2 ; Γ 1 , y : A ; Θ ⊢ D 2 :: T Γ 2 ; Γ 1 ; Θ ⊢ cut # ( D 1 , y . D 2 ) :: T cut # then: FO ( x, cut # ( D 1 , y . D 2 )) = FO ( y , D 2 ) · FO ( x, D 1 ) + FO ( x, D 2 ) ≤ | D 2 | · 1 + | D 1 | ≤ | cut # ( D 1 , y . E 2 ) | if x ∈ Γ 1 FO ( x, cut # ( D 1 , y . D 2 )) = FO ( y , D 2 ) · FO ( x, D 1 ) + FO ( x, D 2 ) ≤ | D 2 | · 0 + 1 = 1 if x ∈ Γ 2 FO ( x, cut # ( D 1 , y . E 2 )) = FO ( y , D 2 ) · FO ( x, D 1 ) + FO ( x, D 2 ) ≤ | D 2 | · 0 + 1 = 1 if x ∈ Θ • If π is Γ 1 ; ∅ ; ∅ ⊢ ∅ :: D 1 z : A Γ 2 ; ∆; Θ ⊢ D 2 :: T Γ 2 ; ∆; Θ ⊢ cut w ( D 1 , y . D 2 ) :: T cut w then: FO ( x, cut w ( D 1 , y . D 2 )) = FO ( y , D 2 ) · FO ( x, D 1 ) + FO ( x, D 2 ) ≤ 0 · 1 + 0 = 0 if x ∈ Γ 1 FO ( x, cut w ( D 1 , y . D 2 )) = FO ( y , D 2 ) · FO ( x, D 1 ) + FO ( x, D 2 ) ≤ 0 · 0 + 1 = 1 if x ∈ Γ 2 FO ( x, cut w ( D 1 , y . E 2 )) = FO ( y , D 2 ) · FO ( x, D 1 ) + FO ( x, D 2 ) ≤ 0 · 0 + | D 2 | ≤ | cut # ( D 1 , y . E 2 ) | if x ∈ ∆ FO ( x, cut w ( D 1 , y . E 2 )) = FO ( y , D 2 ) · FO ( x, D 1 ) + FO ( x, D 2 ) ≤ 0 · 0 + 1 = 1 if x ∈ Θ This concludes the pro of. ✷ Lemma 14 Supp ose Γ; ∆; Θ ⊢ D :: T . Th en D ( D ) ≤ | D | . Pro of. An easy induction on the structure of a t yp e der iv ation π for Γ; ∆; Θ ⊢ D :: T . Some int ere s ting cases: • If π is Γ 1 ; ∅ ; ∅ ⊢ ∅ :: D 1 z : A Γ 2 ; ∆ , y : A ; Θ ⊢ D 2 :: T Γ 2 ; ∆ , Γ 1 ; Θ ⊢ cut # ( D 1 , y . D 2 ) :: T cut # then, by Lemma 13 and b y induction hypothesis: D ( cut # ( D 1 , y . D 2 )) = max { D ( D 1 ) , D ( D 2 ) } ≤ max {| D 1 | , | D 2 |} ≤ | cut # ( D 1 , y . D 2 ) | 21 This concludes the pro of. ✷ Lemma 15 If Γ; ∆; Θ ⊢ D :: T , then for every n ≥ D ( D ) , W n ( D ) ≤ | b D | · n B ( b D )+1 . Pro of. By induction o n the structure of D . Some interesting cases: • If D = ⊗ R ( E , F ), then: W n ( ⊗ R ( E , F )) = 1 + W n ( E ) + W n ( F ) ≤ 1 + | E | · n B ( E )+1 + | F | · n B ( F )+1 ≤ 1 + ( | E | + | F | ) · n max { B ( E )+1 , B ( F )+1 } ≤ (1 + | E | + | F | ) · n max { B ( E )+1 , B ( F )+1 } ≤ | ⊗ R ( E , F ) | · n B ( ⊗ R ( E , F ))+1 • If D = cut ! ( D , x. E ), then: W n ( cut ! ( D , x. E )) = FO ( x, E ) · ( W n ( D ) + 1) + W n ( E ) ≤ FO ( x, E ) · ( | D | · n B ( D )+1 + 1 ) + | E | · n B ( E )+1 ≤ n · | D | · n B ( D )+1 + n + | E | · n B ( E )+1 ≤ | D | · n B ( D )+2 + n B ( E )+1 + | E | · n B ( E )+1 ≤ ( | D | + | E | + 1) · n max { B ( D )+2 , B ( E )+1 } = | cut ! ( D , x. E ) | · n B ( cut ! ( D ,x. E )) . • If D = ! R ( x 1 , . . . , x n , E ), then: W n (! R ( x 1 , . . . , x n , E )) = n · ( W n ( E ) + 1) ≤ n · | E | · n B ( E )+1 + n ≤ | E | · n B ( E )+2 + n B ( E )+2 = (1 + | E | ) · n B (! R ( x 1 ,...,x n , E ))+1 = | ! R ( x 1 , . . . , x n , E ) | · n B (! R ( x 1 ,...,x n , E ))+1 . This concludes the pro of. ✷ 5.4 Putting Ev erything T ogether W e now hav e almost all the necessar y ingr e die nts to obtain a pro of o f Pr op osition 4: the only missing tales ar e the b ounds on the siz e of any reducts, since the p olynomial bounds on the length of int er na l reductions ar e ex a ctly the ones from Lemma 1 5. Obs erve, how ever, that the latter induces the former: Lemma 16 Supp ose that P → n Q . Then | Q | ≤ n · | P | . Pro of. By induction on n , enriching the statement as follows: whenever P → n Q , b o th | Q | ≤ n · | P | and | R | ≤ | P | for every s ubpro cess R of Q in the form ! x ( y ) .S . ✷ Lemma 17 F or every D , B ( D ) = B ( b D ) and | D | = | b D | . Finally: 22 Pro of. [Pro p o sition 4] Let { q n } n ∈ N the polyno mials coming from Lemma 1 5. The p olynomia ls we are lo o king for are defined a s follows: p n ( x ) = q n ( x ) + x · q n ( x ) . Now, supp ose that P → m Q . By Theore m 1 , there are pro o f ter ms D , E such that P = b D , Q = b E and D ( ֒ → = ⇒ ֒ → ) m E . Now, from prop os itions 5, 6 and 7, it follows that W ( D ) ≥ m + W ( E ) ≥ m. As a cons equence, by Lemma 15 and L e mma 17, m ≤ q B ( D ) ( | D | ) ≤ q B ( P ) ( | P | ) ≤ p B ( P ) ( | P | ) . By Lemma 1 6, it follows that | Q | ≤ m · | P | ≤ q B ( P ) ( | P | ) · | P | ≤ p B ( P ) ( | P | ) . This concludes the pro of. ✷ Let us now consider Theorem 2: how can we deduce it from Pr op osition 4? Everything bo ils down to show that for no rmal pro cesses, the b ox-depth can b e read off fr o m their t yp e. In the following lemma, B ( A ) and B (Γ) are the nesting depths of ! inside the type A and inside the types app earing in Γ (for every type A a nd context Γ). Lemma 18 Supp ose that Γ; ∆; Θ ⊢ D :: x : A and that D is normal. Then B ( b D ) = max { B (Γ) , B (∆) , B (Θ) , B ( A ) } . Pro of. An easy induction on D . ✷ The pro o f of b ounded int era ction is s imila r in str uc tur e to the one o f p olynomia l time soundness for SLL (see [9]). How ever, the p eculiar ities of dual systems and of pro cess algebr as mak e it slightly more complicated. As an example, some of the stro ng bisimilarities o n pro o f terms which are nece s sary to simulate pro ces s reduction (e.g . ( cut # / − / cut ), see Figure 5 ) exhibit complicated combinatorial b ehaviors, which need to be taken into acco unt here. 6 Conclusions In this paper, we introduced a v a riation on Caires and Pfenning’s π D ILL , called π DSL L , being inspired by Lafont’s soft linear logic. The key feature of π DSLL is the fact that the amount of interaction induced b y allowing tw o pro ces ses to in teract with each other is bounded by a po lynomial whose degr ee ca n be “read off ” from the type of the session channel through which they communicate. What we co nsider the main achievemen t of this pa p er is the “trans fer of technology” from the functional w or ld of implicit computationa l complexity to the concurrent framework of π -calc ulus and ses sion t yp es, ra ther than the pro of of the p olynomial b o unds itself, which can b e o btained by a dapting the ones in [5] or in [4] (although this an ywa y presents s ome technical difficulties due to the low-lev el nature of the π -calculus compared to the lambda ca lculus or to hig her -order π -calculus). Another asp ect that w e find in teresting is the fo llowing: it seems that the constr a ints on pro cesses induced by the adoption of the more s tringent t yping discipline π D SLL , as opp ose d to π DILL , are quite natural and do not rule out too many interesting examples. In particular , the way sessions can b e defined rema ins essentially untouc hed: what changes is the wa y sessions can b e offered, i.e. the disc ipline governing the offer ing of multiple sessions by servers. All the ex amples in [1] and the one from Sectio n 2 ar e indeed t ypable in π DSLL . T o pics for future work include the acco mmo da tion of recursive type s in to π DSLL . This co uld be easier than exp ected, due to the robustness of light logics to the presence of r ecursive types [3]. 23 References [1] Lu ´ ıs Caires & F ra nk P fenning (2 0 10): Session T yp es as Intuitionistic Li ne ar Pr op ositions . In: CO NCUR 2010 , LNCS 62 69. Springer, pp. 222–23 6. [2] Lu ´ ıs Ca ires, Ber nardo T oninho & F rank Pfenning (20 11): Dep endent Session T yp es via Intu- itionistic Line ar T yp e The ory . In: PPDP 20 11 . A CM Press, pp. 161 –172 . T o app ear . [3] Ugo Dal La go & Patric k Baillo t (2006): On light lo gics, uniform enc o dings and p olynomial time . 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