Proof nets for additive linear logic with units

Transcription

1 Proo nets or additive linear loic with units Willem Heijltjes LFCS, School o Inormatics, University o Edinburh bstract dditive linear loic, the rament o linear loic concernin linear implication between strictly additive ormulae, coincides with sum-product loic, the internal lanuae o cateories with ree inite products and coproducts. Decidin equality o its proo terms, as imposed by the cateorical laws, is complicated by the presence o the units (the initial and terminal objects o the cateory) and the act that in a ree settin products and coproducts do not distribute. The best known desicion alorithm, due to Cockett and Santocanale (CSL 2009), is hihly involved, requirin an intricate case analysis on the syntax o terms. This paper provides canonical, raphical representations o the cateorical morphisms, yieldin a novel solution to this decision problem. Startin with (a modiication o) existin proo nets, due to Huhes and Van Glabbeek, or additive linear loic without units, canonical orms are obtained by raph rewritin. The rewritin alorithm is remarkably simple. s a decision procedure or term equality it matches the known complexity o the problem. main technical contribution o the paper is the substantial correctness proo o the alorithm. I. INTRODUCTION Proo nets, introduced by Girard in the seminal [6], are a beautiul, eometric description o linear loic proo. They aim to reproduce the qualities o the conjunction implication rament o intuitionistic natural deduction, that have made it into a prominent model o computation, via the Curry Howard correspondence. The speciics o this motivation have been described in subtly dierent ways: proo nets are to remove the uninterestin, bureaucratic permutations rom the cut-elimination procedure in the sequent calculus; to identiy proos that are morally the same; or to obtain conluent cut-elimination, to name a ew. natural interpretation that subsumes those above, is that proo nets aim to be a canonical representation o proo in linear loic, with respect to its cateorical semantics (see [5] and []). This means there is a correspondence between proo nets and cateorical morphisms, in the way that normal proos in neative intuitionistic natural deduction uniquely describe morphisms in a ree Cartesian closed cateory (see, e.., [2]). The oriinal nets or linear loic were canonical, in this sense, only or the multiplicative connectives. It has proven exceedinly diicult to extend them to larer raments, and in particular, to include the units, which are the neutral elements or the our binary connectives. widely hailed success, ater a partial result in [7], are the canonical proo nets or the combined multiplicative additive rament o linear loic, without the our units, in [9]. Successive approaches to includin the multiplicative units are [2] and [3], and more recently [6] and [8]. lthouh some o the proposed representations have conluent normalisation, a treatment o the multiplicative units that is canonical with respect to the semantics provided by -autonomous cateories has remained out o reach. This paper presents a new notion o proo net, or additive linear loic, the rament o linear implication between additive ormulae, includin a canonical treatment o the two additive units, which have thus ar not appeared in proo nets. To quote Girard, in [7, ppendix.3]: There is still no satisactory approach to additive neutrals [... ]. The only way o handlin is by means o a box or, i one preers, by means o a second order translation: on this Kamtchatka o linear loic, the old problems o sequent calculus are not ixed. The absence o a satisactory treatment o calls or another notion o proo-net... Sum product loic dditive linear loic is also known as sum product loic: it is the internal lanuae o sum product cateories, cateories that have all inite products and coproducts. Free sum product cateories are the inite, discrete versions o Joyal s ree bicomplete cateories [], which are ree completions with all limits and colimits. s such, ree sum product cateories are characterised by the sotness property described in that paper (and expanded on in [0]), a property related to cutelimination and the subormula property in loic. Cut-elimination or sum product loic was investiated by Cockett and Seely in [4]. They show that it reduces the word problem decidin equality o (proo) terms or sum product loic to a small set o simple permutations, makin it decidable. However, this observation does not ive a tractable alorithm, as the equivalence classes involved are exponentially sized. One actor complicatin the search or an eicient decision procedure is the presence o the units, the cateorical initial and terminal objects; omittin the units, a simple notion o proo identity via raphs is described in [5]. nother is the absence o distributivity; products distribute over coproducts in most situations where both occur, simpliyin the problem considerably. In particular, no solution is provided by orientatin the permutations as rewrites, a standard technique. Recently, Cockett and Santocanale [3] have presented an intricate, polynomial-time decision alorithm or the word problem or sum product loic. The oriinal text reads,... which are ortunately extremely uninterestin in practice. One can only uess at the reasons or questionin the siniicance o the additive units; ater all, they are an interal part o linear loic, and in the opinion o the author, and probably that o others who have worked on them, pose a demandin challene with interestin technical consequences.

2 a hom(, B) a B 0? X X! X idx X Sum product nets X t0 Y 0 X t Y X t0,t Y 0 Y t X 0 t 0 Y X Y X 0 X [ t 0,t ] Y Id Fi.. X t Y Y s Z X s t Z Sum product loic X i t Y t π X 0 X i Y X t Y i X ιi t Y 0 Y Cut The proo nets presented in this paper provide raphical representations o proos in additive linear loic, that are canonical with respect to the cateorical semantics o ree sum product cateories, constitutin a novel solution to their word problem. First, in Section II, the proo terms o sum product loic are translated to sum product nets, proo nets similar to the unit-ree MLL-nets in [9]. These are canonical or the unitree rament, and actor out the permutations o that do not involve the units. This sharply isolates the challene posed by the additive units, in the orm o an equational theory over sum product nets, induced by the remainin permutations. This equational theory is addressed by rewritin to canonical orms. The rewrite alorithm, called saturation, is extremely simple, and is the irst main contribution o the paper, presented in Section III. s a decision procedure consistin o translatin terms to proo nets, saturation, and testin or syntactic equality it shares the polynomial time complexity o the decision alorithm by Cockett and Santocanale [3]. The correctness proo or the saturation alorithm is hihly involved. This is the second main contribution o the paper, discussed in Section IV. The nets obtained by saturation orm a syntactic characterisation o ree sum product cateories. Section V explores composition and identity morphisms in this context. II. SUM PRODUCT NETS For the remainder, ix a cateory C and denote by ΣΠ(C) its ree completion with products and coproducts (or an ι i (t π j ) = (ι i t) π j ι i [ t, s] = [ ι i t, ι i s] t π i, s π i = t, s π i [ t 0, t ],[ s 0, s ] = [ t 0, s 0, t, s ] Fi. 2. Permutations in sum product loic! =! π i! = [!,!] ι i? =??,? =?! 0 =? introduction to products and coproducts, see [4]). For the purposes o the present paper, ΣΠ(C) can be understood as a syntactic cateory enerated by sum product loic [4], which will be described below. The objects in ΣΠ(C), raned over by variables X, Y, Z, are iven by the rammar X ::= C 0 X X X X. The sequent calculus presentation o sum product loic, illustrated in Fiure, provides a term calculus or the morphisms in ΣΠ(C), in which the cut-rule and identity-rule are admissible. The permutations in Fiure 2 orm an equational theory over proos in sum product loic. In the syntactic description o the cateory, the morphisms o ΣΠ(C) are the equivalence classes o (cut-ree, identity-ree) proos [4]. The notation or the connectives was chosen to aree with that o [4], and is natural iven the intended interpretation as cateorical products and coproducts. To retrieve the notation o linear loic, interpret the unit as, and the connectives and respectively as and &. The case o additive linear loic is iven by choosin a discrete base cateory C (one with only identity morphisms). sum product net will consist o a source object and a taret object rom the cateory ΣΠ(C), and a collection o links connectin vertices in the syntax tree o the ormer to vertices in that o the latter. Two example nets are drawn below, toether with the terms they represent. id B id B B [ (ι 0 id ), (ι id B )] [?,!,! ] Interpreted inormally, nets are to be read rom let to riht. Solid edes in the object trees correspond to projections and injections, while dashed edes correspond, rouhly, to (the application o) the inerence rules, and [, ]. Links correspond to axioms, and are distinuished rom solid edes in the object trees by bein slihtly detached rom vertices. For the ormal deinition, the vertices (or positions) in the syntax tree o an object X are iven as binary words, with ε the empty word and ( ) the standard preix orderin, and collected in the set pos(x). The subormula o X at a vertex v is denoted X v, and v is Y will mean X v = Y when X is understood. In this deinition, i v is a product or coproduct it has children v0 and v, and none otherwise. Deinition (Pre-nets). ΣΠ(C)-pre-net (X, Y, R) consists o a source object X, a taret object Y and a relation R pos(x) ( hom(c) { } ) pos(y ) such that or any v,l, w R, i l = then X v = 0 or Y w =, and otherwise l C(X v, Y w ). Variables,, h and k are used or pre-nets. The links in a pre-net are the elements v,l, w o R, which may be

3 rendered v, w when the label l is understood or irrelevant. link v,, w (the label will be omitted rom diarams) is a unit link; i v is 0 it is an initial link, i w is a terminal link. link labelled with a C-morphism is atomic. switchin ς o an object X is a unction choosin one branch o each product vertex: ς(v) {0, } i X v is a product, while otherwise ς(v) is undeined. The dual notion o a co-switchin is a unction choosin branches on coproduct vertices. vertex w is switched on by a [co-]switchin ς, written ς w, i or any ancestor (i.e. preix) o w that is a [co]product, ς selects the branch containin w; ς w ( vi w v dom(ς) ) ς(v) = i. switchin or a pre-net (X, Y, R) is a pair (ς, τ) o a coswitchin ς o X and a switchin τ o Y. link v, w is switched on by (ς, τ) i ς v and τ w. Deinition 2 (Nets). ΣΠ(C)-net is a pre-net that satisies the ollowin correctness criterion (the switchin condition). ny switchin (ς, τ) or switches on precisely one link. Let NET denote the set o all ΣΠ(C)-nets. Sum product nets without units orm the purely additive rament o the MLL-nets by Huhes and Van Glabbeek [9, Section 4.0]. The addition o unit links has only minor technical consequences, which are due to the act that unlike atomic links, unit links may connect to non-lea nodes. How diarams and deinitions relate is illustrated below. ε 0 id ε ( 0, ( ), { 0,id, 0,,, } ) The edes o nodes subject to (co-)switchins (in the switchin condition) are dashed. The switchin condition can be separated into an at least one and an at most one part. pre-net satisyin the latter, i.e. one or which any switchin (ς, τ) switches on at most one link, is called a partial net. (n insihtul way o ainin amiliarity with the switchin condition is by convincin onesel that there are no natural nets rom (B C) to ( B) ( C), showin that sum and product do not distribute.) The proo terms o sum product loic suest an inductive construction method or sum product nets. Usin the abbreviation (X, Y,l) or (X, Y, { ε,l, ε }), there are basic nets a B? Y = (0, Y, )! X = (X,, ) (, B, a) or each X, Y ΣΠ(C) and a C(, B), correspondin to the axioms note the upriht ont, to contrast with terms. Inerence rules coincide with the our constructors, (π i (X);) [, ], (;ι j (Y )), π 0 ;, [, ] ;ι 0 Fi. 3. Net constructors illustrated in Fiure 3. Usin the notation u R = { uv,l, w v,l, w R} R u = { v,l, uw v,l, w R}, the constructors are deined, on pre-nets, by π i (X 0 X );(X i, Y, R) = (X 0 X, Y, i R) [(X, Z, R), (Y, Z, S)] = (X Y, Z, (0 R) ( S)) (X, Y, R), (X, Z, S) = (X, Y Z, (R 0) (S )) (X, Y i, R);ι i (Y 0 Y ) = (X, Y 0 Y, R 0). The translation rom (cut-ree) proo terms to nets, implicit in the namin o constructors, is made explicit as below.? Y =? Y! X =! X a : B = (, B, a) t π i = π i ; t [ t, s] = [ t, s ] t, s = t, s ι j t = t ;ι j pplyin a constructor is called construction, the reverse deconstruction. pre-net o the orm π i ; or [, ] is letconstructible; one o the orm, or ;ι i riht-constructible; one that is either, constructible; and one that is both, biconstructible. Both construction and deconstruction preserve the switchin condition, and moreover, all nets are basic or constructible. This ives the sequentialisation result below, which states that all nets correspond to some term. Proposition 3. NET is the smallest set containin all basic nets, closed under construction. (This is a minor variant on the analoous result by Huhes and Van Glabbeek in [9].) For bi-constructible pre-nets the ollowin equations are immediate rom the deinitions, as illustrated in Fiure 4. Proposition 4. Construction o pre-nets satisies: (π i ;);ι j = π i ;(;ι j ) [, h], [, k] = [,, h, k ] [, ];ι j = [(;ι j ), (;ι j )] (π i ;), (π i ;) = π i ;, These equations correspond to the our equations on the let in Fiure 2; precisely those not involvin the units.

4 h k The notation {} v,w denotes a pre-net with the sub-prenet v,w replaced by a parallel pre-net. Formally, or pre-nets = (X, Y, R) and = (X v, Y w, S), deine the ollowin. R{S} v,w = { v,l, w R v v w w } (v S w) {} v,w = (X, Y, R{S} v,w ) The eneral orm o rewritin in context is iven by the ollowin relation. {} v,w =[ h ] v,w {h} v,w Fi. 4. Bi-constructible nets! =! π 0! = [!,!] The relation =[ h ] v,w replaces the pre-net between vertices v and w, which is required to be, with the parallel prenet h, leavin the context intact. n equivalent ormulation would be =[ v,w h ] v,w {h} v,w. Droppin the subscript v, w indicates the union over all v and w. Deinition 6 (Equivalence). The equational theory (equivalence) on ΣΠ-nets is the equivalence relation enerated by the ollowin our relations. =[! π i ;! ] =[! [!,!] ] =[??,? ] =[??;ι j ] 0 0? = ι 0? 0 0? =?,? Fi. 5. The unit laws orce an equational theory over nets O the remainin equations in Fiure 2, those involvin the units, only? =! 0 is actored out (by labellin initial and terminal links uniormly). The other our will orm an equational theory over nets, equivalence ( ), illustrated in Fiure 5. The natural way o deinin it is via raph-rewritin, as rewrite rules that replace one subnet with another which irstly requires a notion o subnet. In the eneral notion a subpre-net o (X, Y, R) will mean a pre-net between subormulae o X and Y, with a subcollection o the links between them: a pre-net (X v, Y w, S) such that v S w R. Call two pre-nets parallel i they have the same source objects and the same taret objects, and deine, on parallel pre-nets, (X, Y, S) (X, Y, R) and, or a pre-net = (X, Y, R), R v,w v,w S R, = { v,l, w vv,l, ww R} = (X v, Y w, R v,w ). Deinition 5 (Subnets). sub-pre-net o a pre-net is a prenet v,w. I is a net, it is a subnet o. These our rewrite rules are the equivalences illustrated in Fiure 5, interpreted as rewrite steps rom let to riht, on subnets. With some eort, it then ollows rom the description o ree sum product cateories by Cockett and Seely in [4], that ΣΠ-nets up to equivalence, too, characterise ree sum product cateories. Proposition 7. For cut-ree proo terms t and s o sum product loic, ΣΠ(C) = t = s t s. III. DECIDING EQUIVLENCE OF NETS The equivalence relation over nets will be decided by rewritin equivalent nets to a common canonical orm. natural irst question is whether a suitable, conluent rewrite relation can be obtained by orientatin the equivalence rewrites, i.e. by restrictin them to one direction. Two straihtorward candidates are to rewrite towards the leaves or towards the the roots o the trees. In act, neither option is conluent. For the irst, an example o non-conluence is illustrated in Fiure 6. For the second option, the situation is more delicate. The non-conluent example in Fiure 7 could in principle be resolved by introducin a novel kind o link connectin both root nodes, while preservin the switchin condition as the correctness criterion or nets. However, the non-conluence o the example in Fiure 8 has no solution alon these lines. I conluent rewritin is impossible without breakin the switchin condition, the obvious next step is to break it. Then when two nets rewrite into each other, the easiest way to obtain conluence is to combine the links o both, as in the example o Fiure 9. This ives a simple rewrite relation called saturation. To deine the saturation relation a dierent orm o rewritin is required, whereby links are added to a net, rather than

5 ? Fi. 6. Rewritin towards the leaves is non-conluent Fi. 9. Saturation v and w. In order to provide saturation with a standard notion o termination, the irrelexive variant is deined. Both and will be reerred to as saturation, with the distinction only made when necessary. Proposition 9. The saturation relation ( stronly normalisin. ) is conluent and? Fi. 7. Rewritin towards the roots is non-conluent () (π 0 ;!!) id (! π 0 ;!) id id ([!,!]!)? Fi. 8. Rewritin towards the roots is non-conluent (2) (! [!,!]) replaced. Let the union o two parallel pre-nets be the union o their collections o links, (X, Y, R) (X, Y, S) = (X, Y, R S). statement = { } v,w then expresses the condition that a pre-net must have as a subnet, v,w (where holds the other links in v,w ). Deine a second rewrite relation: ( h) v,w { v,w h} v,w i v,w. Deinition 8. The saturation relation union o the ollowin eiht relations. on pre-nets is the (π i ;!!) ([!,!]!) (?,??) (?;ι j?) (! π i ;!) (! [!,!]) (??,? ) (??;ι j ) The relation is the irrelexive restriction o. The eiht saturation steps in Deinition 8 are illustrated in Fiure 0. In eneral, the relation ( h) v,w is relexive or nets that already have h (and ) as a subnet between vertices 0 (?,??) 0 0 (??,? ) 0 0 (?;ι 0?) 0 0 (??;ι 0 ) 0 Fi. 0. Saturation steps

6 Proo: For stron normalisation it is suicient to observe that each step in adds one or two unit links to a prenet, while the number o unit links in a pre-net (X, Y, R) is bounded by the size o pos(x) pos(y ). For conluence, let = (X, Y, R), let = (X v, Y w, S), and let h = (X x, Y y, T ). Observe that the result o applyin a saturation step ( ) v,w to is just { v,w } v,w = (X, Y, R v S w). The ollowin diaram shows local conluence or. (X, Y, R) proposed way. The irst is that a saturated net σ is the union o all nets equivalent to a act that will ollow rom aspects o the eventual soundness proo, and will have a separate use in characterisin composition o saturated nets. Despite bein naturally suested by the proo idea above, the statement is diicult to prove in the way suested. In particular, that every link in σ belons to some net equivalent to would ollow by induction on the saturation path, i not or the saturation step below (and its dual). ( ) v,w (h h ) x,y (X, Y, R v S w) (h h ) x,y (X, Y, R x T y) ( ) v,w (X, Y, R v S w x T y) Then also is locally conluent, and in the context o stron normalisation this implies is conluent. The normal orm o a pre-net w.r.t. is denoted σ, and, i is a net, is called a saturated net. The idea is that saturation provides a decision procedure by comparin saturated nets, i.e. i and only i σ = σ. Theorem 0 (Completeness). For nets and, i then σ = σ. The completeness proo o this decision procedure is straihtorward: two nets that are equivalent by a sinle rewrite step have saturation steps h and h with a common taret h; the statement then ollows rom conluence. The soundness theorem is stated below; its elaborate proo will be the subject o the next section. Theorem (Soundness). For nets and, i σ = σ then. IV. THE SOUNDNESS PROOF natural approach to provin the soundness theorem would be by induction on the saturation paths o σ and σ; e.. or this is a sequence 2... σ. One could imaine a proo to proceed as ollows. Each stae k in the saturation path would be taken to represent a collection o equivalent nets, by their union, with representin just itsel. saturation step k ( h) v,w k would then create the collection o k by closin that o k under the correspondin rewrite step, =[ h ] v,w. To complete the arument, it would suice to show that i σ = σ then the collections o equivalent nets that both saturations represent, which contain and respectively, share at least one net. proo alon these lines aces several obstacles, mainly in the orm o statements that are true, but hard to prove in the The let-hand side o this step contains two links; even i each occurs in some net equivalent to, or the correspondin equivalence step to apply, both links must occur toether in a sinle net, and it is not obvious how to show this is the case. One approach to the above problem would be to characterise, or a saturated net, the nets whose saturation it is; call such nets representatives o the saturated net. simple potential answer, that any subnet o a saturated net would be a representative, turns out to be alse: the saturated net below let (the saturation o!;ι ) has that on the riht as a subnet, but the latter is already saturated In act, no representative o the saturated net on the let contains both links rom the net on the riht. The main diiculty o this proo idea, however, is the inal step it suests, o showin that the equivalence classes represented by σ and σ overlap. Without a characterisation o representatives, there are not many immediate indications let on how this should be approached. The previous served to illustrate how saturation is diicult to characterise, and prove properties o, via saturation paths alone. Instead, thereore, the soundness proo will proceed by induction on the source and taret objects o nets, and rely on a dierent description o saturation, which ollows the construction o a net. For space reasons many technical details are necessarily omitted. s a irst overview, there will be three cases, or nets and with the same saturation (X, Y, R): one o X and Y is an atom or unit, X is a coproduct or Y is a product, and X is a product and Y a coproduct. The irst two cases are relatively straihtorward, and will be treated in the next subsection. The main body o the proo is concerned with the third case, which is that o nets ;ι j and

7 π i ; as illustrated below. Lemma 3. The saturation o initial and terminal nets is ull. The second case o the soundness proo concerns nets whose source is a coproduct or whose taret is a product; call these coproduct nets and product nets, respectively. There are three main obstacles to overcome. ) Inductive saturation: To apply the induction hypothesis it must be possible to relate, e.., a saturated net σ(;ι 0 ), to the saturation o its component net, σ. This will be addressed by Lemma 8, which describes saturated nets σ inductively, on the construction o. Subsection IV-B presents supportin material or the lemma, whose main content will be discussed in Subsection IV-C. 2) Representatives: The second obstacle is that nets constructed over dierent projections and injections, e.. ;ι 0 and π 0 ;, but also ;ι 0 and ;ι, may have the same saturation; naturally, in such a case the induction hypothesis cannot be applied to σ and σ. This will be solved by Lemma 9, which, or a saturated net σ, inds a representative equivalent to containin iven initial links v, ε, or terminal links ε, w, rom σ. From the presence o these links it can then be deduced that is let-constructible or riht-constructible, respectively, and over which projection or injection it is constructed. This is described in Subsection IV-D. 3) Deconstruction alone may not suice: The third obstacle is that nets constructed over the same projection or injection, e.. ;ι 0 and ;ι 0, may have the same saturation, while their components, and, do not. By isolatin the exact cause o this discrepancy it will be possible to transorm the net ;ι 0 into an equivalent net h;ι 0, such that h does have the same saturation as. This will be treated in Subsection IV-E.. The irst two cases The irst case o the soundness proo concerns parallel nets whose source or taret is an atom or unit. For nets with source X and taret Y, this ives six, pairwise dual, possibilities. Four are immediate: i X is an atom or, or dually i Y is an atom or 0, illustrated below, it is easily observed that no rewrite or saturation steps apply. 0 For such nets and, it ollows that i σ = σ then =. For the remainin two cases, nets with source object 0 will be called initial, and with taret, terminal. The links in an initial net (0, Y, R) can move up and down the syntax tree o Y essentially without hindrance. From this Lemma 2 and Lemma 3, below, ollow. Lemma 2. ll parallel initial nets are equivalent, as are all parallel terminal nets. This proves soundness o saturation or initial and terminal nets, and althouh it need not reer to saturated nets, their characterisation will be useul. Call a pre-net ull i it contains all possible unit links (but no atomic links), i.e. one o the orm (X, Y, { v,, w X v = 0 or Y w = } ). product net is riht-constructible unless it contains links v, ε connectin to the root o its taret. Such a link must be an initial link, and can be moved away rom the root by the ollowin equivalence step. This ives the lemma below. 0 0 Lemma 4. product net is equivalent to a net 0,. coproduct net is equivalent to a net [ 0, ]. similar result holds or the saturation o (co)product nets. For a net,, i irst the saturation steps in and are applied, the remainin steps to be applied to σ, σ are o the ollowin kind. 0 0 pplyin these steps completes the saturation o, : ater a step o the kind above, to the newly added link no saturation steps apply, as the only possible step would be the reverse. Moreover, the links added by these steps are all o the orm v, ε and thus, in σ,, illustrated below riht, separate rom links in σ and σ. σ σ Lemma 5. Saturation o (co)product nets satisies: (σ[ 0, ]) i,ε = σ i (σ 0, ) ε,i = σ i. Usin the two lemmata on (co)product nets, the present case in the soundness proo will be completed. For parallel product nets and with the same saturation, Lemma 4 ives equivalent nets 0, and 0, respectively. By Lemma 5 σ i = (σ 0, ) ε,i = (σ 0, ) ε,i = σ i or i {0, }. The induction hypothesis o the soundness proo ives i i, and the equivalences below ollow. σ σ 0, 0,

8 B. Pointed and copointed nets point is a map out o a terminal object, a copoint one into an initial object. n object that has a [co]point is [co]pointed. In ree sum product cateories the pointed and copointed objects are iven by the ollowin rammars, respectively. P := P X X P P P Q := 0 Q Q Q X X Q Here, X may be any object: pointed, copointed, or neither. Note that an object is never both pointed and copointed, and that in ΣΠ( ), the ree sum product completion o the empty cateory, where atoms are absent, every object is either pointed or copointed. cateorical map that actors throuh a point, i.e. one o the orm p!, where p is a point, is called pointed; one that actors throuh a copoint,? q, copointed. Pointed and copointed nets are deined slihtly more narrowly, by the ollowin rammars over the net constructors. p :=! p;ι j p, p q :=? [q, q] π i ;q. Up to equivalence, the deinition corresponds to the cateorical one, but it requires pointed and copointed nets to have the ollowin syntactic orm. Call a terminal link ε,, w and an initial link v,, ε rooted; pointed nets are those consistin entirely o rooted terminal links, and copointed nets those consistin o rooted initial links. map that is both pointed and copointed will be called bipointed. Bipointed maps eature heavily in the decision procedure o Cockett and Santocanale [3] where they are called disconnects because o the ollowin property: there is precisely one bipointed map rom a copointed object Q to a pointed object P, and none between other objects. The uniqueness property is easily observed rom the act that in the diaram below the copoint q and the point p are arbitrary. Q q! 0!? The correspondin notion or nets will aain be restricted to a useul syntactic orm: let a bipointed net be a net rom a copointed to a pointed object that is itsel pointed or copointed. The properties o bipointed maps carry over, by the ollowin two lemmata. Lemma 6. ny two parallel bipointed nets are equivalent. Lemma 7. The saturation o a bipointed net is ull. C. Saturation via construction Previous lemmata showed that the saturation o initial, terminal, and bipointed nets is ull (Lemma 3 and 7). The next result will be that saturation is completely described by this dynamic, i.e. by the illin o initial, terminal, and bipointed subnets. This enables, and is proved via, a characterisation o saturation by induction on the construction o a net. For the? p P base cases, saturation o the basic nets (0, Y, ) and (X,, ) is ull, by Lemma 3, while no saturation steps apply to a net (, B, a). The saturation o nets, h and [, h] was described inormally in Subsection IV-, leavin the cases π i ; and ;ι j (illustrated below or i = j = 0). s an example, the saturation o ;ι 0 = (X, Y, R) rom σ can be described as ollows. Firstly, any rooted initial link v, ε in σ orms an initial subnet between v and the root o Y in (σ);ι 0, which will be illed. Secondly, since copointed nets consist o initial links, i σ contains a copointed subnet q (σ) v,ε, the duplication o initial links will produce a copointed subnet, in the saturation o ;ι 0, between v and any w in Y. Then i w is pointed, such a copointed subnet is bipointed. Lemma 8, below, states that illin these bipointed subnets completes the saturation o ;ι 0. In a pre-net = (X, Y, R), say that a vertex v in X has a rooted copointed subnet i there is a copointed net q v,ε. I v is minimal amon the vertices in X that have rooted copointed subnets in, then v is said to have a maximal copointed subnet; let MXCP() denote the set o such variables in. Dually, let MXP() be the set o variables in Y that have maximal pointed subnets, i.e. are minimal amon the vertices that have rooted pointed subnets. Lemma 8. For a net ;ι j let σ = (X, Y j, R) and let σ(;ι j ) = (X, Y, S). Then S = (R j) Γ, where Γ = { v,, w X v = 0, v,, ε R} = { v,, w X v = 0 or Y w =, v v. v MXCP(σ), w w. Y w is pointed } The case π i ; is dual. The lemma is proved by showin that its description o a saturated net, as (X, Y, (R j) Γ ), is closed under. D. Deconstruction o saturated nets The previous lemma (Lemma 8) illustrates that retrievin the saturation o rom that o ;ι 0 is easy in some cases, but not in others. simple case that ollows immediately, or example, is that σ(;ι 0 ) = (σ);ι 0 i and only i σ contains no rooted initial links. For the remainin part o the soundness proo, on nets rom products into coproducts, this solves the case or saturated nets that are constructible. However, this need not be the case; and moreover, parallel nets constructed over dierent projections and injections, as illustrated in Fiure, may have the same saturation (lower riht). For an inductive proo this is clearly problematic: or nets ;ι j and π i ;, with the same saturation, equivalent nets must be ound that are constructed over the same projection or injection. In other words, equivalent nets must be ound that

9 Fi.. Dierently constructed nets with the same saturation allow the deconstruction o a saturated net alon a certain projection or injection. Fortunately the examples in Fiure also suest a solution. The two nets on the let are both bipointed, and thus equivalent (by Lemma 6). Then or the one top riht, it needs to be shown that rom the act that its saturation is ull, it ollows that it is equivalent to a bipointed net. Since not all saturated nets are between pointed and copointed objects, a eneralisation is needed. Recall that a partial net is a pre-net that has at most one link or each switchin. Call a partial net [co]pointed i it consists entirely o rooted terminal [initial] links note that in this deinition the taret o a partial pointed net need not be pointed. Lemma 9. I is a net and q σ is a partial pointed or copointed net, then there is a net s.t. q and. This solves the deconstruction problem, or nets ;ι j and π i ;, as ollows. Suppose the saturation o ;ι j is nonconstructible (i.e. not o the orm π i ;h or h; ι j, nor h, k or [h, k], or any pre-nets h and k). This can only be the case i the saturation o contains rooted initial links, which rewrite rom v, j in σ(;ι j ) to v, ε. Then the above lemma provides a net h equivalent to ;ι j containin v, ε, since this link constitutes, on its own, a partial copointed subnet o σ(;ι j ). Now h, containin v, ε, is not riht-constructible, and so must be let-constructible; moreover, i 0 v then h is constructed over π 0, and i v then over π. Since the saturation o π i ; contains the same link v, ε, it has an equivalent k constructed over the same projection as h. E. Completin the proo The inal case in the soundness proo, or nets and between a product and a coproduct, is nearly complete. It was shown that i their (common) saturation is constructible, the induction hypothesis can be applied immediately, and that i it is not, there are equivalent nets ;ι j and ;ι j constructed over the same injection (or projection). inal obstacle is the act that their components and need not have the same saturation, and indeed need not be equivalent. The simple example below illustrates the idea These two nets, π 0 ;?;ι 0 and π ;?;ι 0, have the same saturation, which is ull. However, their components π 0 ;? and π ;? do not: they are already saturated. That this is a eneral problem can be observed rom Lemma 8. Consider the saturation o a net (;ι 0 ), with a pointed taret Y, described by (σ);ι 0 Γ (abusin notation). With Y pointed, i a vertex v has a maximal copointed subnet q in σ, the subnet between v and ε is illed, by. Now suppose is identical to, except that v has a dierent maximal copointed subnet k in the saturation σ. Then ;ι 0 and ;ι 0 have the same saturation, but and miht not. The solution is illustrated below. v v q k ε ε v q ε v k ε The subnets q and k need not be equivalent, but q;ι 0 and k;ι 0 are. They are equivalent to q and k by movin their links up to the root (each u, 0 becomes u, ε ). Because Y is pointed, q and k are bipointed, and hence equivalent by Lemma 6. The eneralised application o this idea is as ollows. I nets ;ι 0 and ;ι 0 have the same saturation, it can be shown that the same vertices v have maximal copointed subnets in σ as in σ. It may be assumed, by Lemma 9, that a maximal copointed subnet o σ is also a subnet o. Then a net h is ormed rom as ollows: or every vertex v that has a maximal copointed subnet q in and one k in, replace q in with k. Then h;ι 0 is equivalent to ;ι 0 by the above reasonin (the equivalence o q;ι 0 and k;ι 0 ), while h and have the same saturation, allowin the induction hypothesis to be applied. V. THE CTEGORY OF STURTED NETS The soundness result, toether with completeness, means that saturated nets are in one-to-one correspondence with morphisms in ΣΠ(C). complete description o this cateory requires also composition and identities to be deined. useul result in this respect is the characterisation o saturated nets as unions over equivalence classes o nets. Proposition 20. The saturation o a net is { }. Identity nets are the translation and saturation o identity proos in sum product loic: nets σ(id X ) where id XY id XY = [(id X ;ι 0 ), (id Y ;ι )] id 0 =? 0 = (π 0 ;id X ), (π ;id Y ) id =!. Huhes and Van Glabbeek established composition or unitree nets as relational composition [9]. Deine, or pre-nets, (X, Y, R) (Y, Z, S) = (X, Z, S R)

10 (denotin relational composition by ( ); labels l and k may be composed as k l i both are morphisms in C, and otherwise). In the presence o the units, this does not work immediately: the ollowin composition would be empty. 0 s is clear rom this simple example, relational composition does work or nets whose links only connect to leaves call these composable and, by movin unit links up towards the leaves, any net is equivalent to a composable one. Furthermore, that composition preserves equivalence ollows by a comparison with the cut-elimination procedure or sum product loic in [4], which it closely resembles. Lemma 2. For composable nets, i and then ( ) ( ). For non-composable nets and the composition o equivalent, composable nets and may be used; this does not deine a unique result, but by the above lemma the possible outcomes are equivalent. Consequently, in the cateory o saturated nets, the composition o σ and σ must be the saturation σ( ). It is obtained rom σ and σ as ollows. Proposition 22. Composition o saturated nets is relational composition ollowed by saturation. Since saturated nets are unions over equivalence classes, by Proposition 20, their relational composition is iven by σ σ = {h k h, k }. It is easily shown that this is a sub-pre-net o the desired solution σ( ) mentioned above in particular, the inclusion o pre-nets h k or non-composable h and k is harmless. In many cases, it will be strictly smaller: σ( ) is the union over all nets equivalent to, which are not necessarily o the orm h k with h and k. Fortunately, the remainin nets can be captured by saturatin σ σ. VI. NOTES, CONCLUSIONS ND FURTHER WORK The saturated nets presented in this paper are canonical representations o proos in additive linear loic with units, or equivalently o morphisms in ree sum product cateories. The saturation alorithm, by which they are obtained rom sum product nets, is simple and tractable, which makes comparin saturated nets an eective decision procedure. Finally, some smaller issues will be picked up here. Time complexity: The time complexity o saturation is as ollows. Usin an appropriate representation o nets, a saturation step may be perormed in constant time. Bounded by the maximum number o unit links in a net (X, Y, R), saturation has a time complexity bound o O( X Y ) (where X denotes the number o vertices in X). By comparison, the alorithm by Cockett and Santocanale [3] has a time complexity bound o O((ht(X) ht(y )) X Y ) (where ht(x) is the heiht o the syntax tree o X). Correctness and sequentialisation: tractable alorithm to ind representatives o saturated nets, or sequentialisation, can be obtained rom the soundness proo, usin, in particular, the inductive characterisation o saturated nets. Such an alorithm also constitutes a correctness criterion, separatin saturated nets rom arbitrary pre-nets. useul addition would be an eleant combinatorial correctness criterion, such as, possibly, a modiication o the switchin condition. Games semantics: ruitul branch o research into loic, linear or otherwise, is that o ame-theoretic semantics, which interprets ormulae as two-player ames and proos as (winnin) strateies. Sum product nets admit a simple ameinterpretation, where two ames are played in parallel, one on the source object and one on the taret object. It appears that saturated nets, viewed as strateies, exhibit interestin ametheoretic properties concernin this parallelism, openin an intriuin anle or uture work. CKNOWLEDGEMENTS Many thanks to lex Simpson or uidance and support. Thanks also to the anonymous reerees or helpul, constructive comments. The author was supported by a Ph.D. studentship on EPSRC research rant EP/F042043/. REFERENCES [] M. Barr: -utonomous cateories and linear loic. Mathematical Structures in Computer Science :59 78 (99) [2] R.F. Blute, J.R.B. Cockett, R..G. Seely, T.H. Trimble: Natural deduction and coherence or weakly distributive cateories. 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