Resource modalities in game semantics

Resour ce modalities in game semantics Paul-André Melliès Nicolas T abareau ∗ Abstract The description of r esour ces in game semantics has never achieved th e simplicity and precision of linear logic, because of a misleading co nception: the belief that lin- ear logic is mor e primitive than game semantics. W e ad- vocate th e contrary here: that game sema ntics is conc ep- tually mor e primitive than linear logic. Starting fr om this r evis ed po int of view , we design a cate gorical mo del of r e- sour ces in game semantics, and construct an ar ena game model wher e the usual notio n of brac keting is extended to multi-brack eting in or d er to capture various r esour ce poli- cies: linear , af fine and e xponential. 1 Intr oduction Game semantics and linear logic. Game semantics is the young er siblin g of linear logic: born (or reborn) at the b e- ginning of the 19 90s, in the turm oil p roduced by the re- cent discovery o f linear logic by Jean-Yves Girard [9], it remained und er its spiritual influ ence for a very lo ng time. This ascendan cy of linear log ic was extraordina rily he althy and pr ofitable in the ear ly day s. Pro perly gu ided, g ame se- mantics developed steadily , fo llowing the idea that e very formula o f linea r logic describes a game ; and tha t every pr o of of the form ula describes a strate gy for playing on that game. This corresponden ce between formulas of linear logic and games is supported by a series of elegant and striking analogies. One basic principle of linea r lo gic is that e very formu la beha ves as a reso urce, which d isappears once co n- sumed. In particular, a p roof of the fo rmula A ⊸ B is required to de duce the con clusion B by using (o r consum- ing) its hypo thesis A exactly once. This pr inciple is nice ly reflected in game semantics, by th e idea that playing a game is just like consuming a resource, the game itself. Another b asic principle of linear logic is that n egation A 7→ ¬ A is in voluti ve. This means that every formula A is ∗ This work has been supported by the ANR Inv ariants algébriques des systèmes informatique s (INV AL). Postal address: Equipe PPS, Unive rsité Paris VII, 2 pl ace Jussie u, Case 7014, 75 251 Pa ris Ce dex 05, FRANCE. Email addresses: mellies@pps.jussieu.fr and tabareau@pps.jussieu.fr equal (or at least isomorp hic) to the formula negated twi ce: A ∼ = ¬¬ A. (1) Again, this pr inciple is nicely reflected in game semantics by the ide a that negating a g ame A con sists in permu ting the r ôles of the two p layers. Hence, negating a game twice amounts to p ermuting the rô le o f Prop onent an d Oppon ent twice, which is just like doing nothing. The connectives of linear lo gic are also nicely r eflected in game semantics. For instance, the ten sor product A ⊗ B of tw o f ormulas A an d B is suitably inter preted as th e game (or fo rmula) A played in parallel with the g ame (or for- mula) B , where only Op ponent may switch f rom a com- ponen t to the other o ne. Similar ly , the sum A ⊕ B of two formu las A an d B is suitably interpreted as the game wh ere Propon ent plays the first move, wh ich co nsists in choo sing between the game A and the game B , before carrying on in the selected compon ent. Fina lly , the expon ential modality of linear logic ! A ap plied to the formula A is suitably in- terpreted as the game where se veral cop ies of the ga me A are played in parallel, and only Opp onent is allowed (1) to switch from a co py to an other o ne and (2) to o pen a fresh copy of the game A . What we d escribe here is in essence the game semantics of linear log ic defined by An dreas Blass in [ 6]. Simp le and elegant, th e model re flects the full flav our of the resource policy of linea r logic. It is also remar kable th at this game semantics is an early predecessor to linear logic [5]. A schism with linear logic. The destiny of g ame seman- tics has been to emancipa te itself fro m linea r logic in the mid–19 90s, in order to comply with its own designs, inher- ited from denotationa l sema ntics: 1. the desire to inter pret pr ograms written in prog ram- ming langu ages with effects (recursion, states, etc. ) and to ch aracterise exactly their interacti ve b ehaviour inside fully abstract models; 2. the d esire to und erstand the algebraic princip les of pro- grammin g languages and effects, using the languag e of category theory . So, a ne w ge neration o f gam e semantics arose, propelled by (at least) two dif ferent lines of research: 1. Samson Abramsky an d Radh a Jag adeesan [2] noticed that th e (alter nating variant of th e) Blass mode l does not define a categorical m odel of linear logic. W orse: it do es no t even define a category , f or lack of asso- ciativity . Abram sky du bs this p henome non the B lass pr o blem and describes it in [1]. 2. Martin H yland and L uke On g [16] in troduced the n o- tion of a r ena ga me , and charac terised th e inter acti ve behaviour of pro grams written in th e fu nctional lan- guage PCF — the simply-typed λ -calculu s with con- ditional test, arithmetic and recursion. So, the Blass p roblem indicates that it is difficult to con - struct a (sequential) gam e model of linear log ic; and at about the same tim e, are na games become mainstream al- though they do no t define a mo del of linear logic. These two r easons (at least) ope ned a schism between game se- mantics and linear log ic: it suddenly beca me accepted that categories of ( sequential) ga mes an d strategies would only capture fragments of lin ear log ic (intuitionistic o r polarised) but not the whole thing. On the oth er hand, defin ing the resour ce mo dalities of linear lo gic for gam e seman tics re quires to reunify the two schismatic su bjects. Since th e d isagreement s tarted with category theory , this reun ification should occ ur at the cat- egorical level. W e explain (in §2) ho w to achieve this by r e- laxing the in voluti ve ne gation of linear log ic into a less con- strained tensorial n egation. This negation indu ces in tur n a linear continua tion m onad, whose unit A − → ¬¬ A (2) refines the isomorp hism (1) of linear logic. Moving from an in voluti ve to a ten sorial negation means that we r eplace linear logic by a more gene ral and p rimitive logic – which we call tensorial logic . As we will see, th is shift to ten- sorial log ic clarifies the Blass pro blem, an d describes th e structure of are na game s. It also enab les the expressions of resour ce m odalities in game semantics, ju st as it is u su- ally done in linear logic. Howe ver , because the presentation of modalities may ap pear difficult to read ers not familiar with categorica l sema ntics, w e p refer to recall first th e n o- tion o f well-brack eting in arena gam es — an d explain h ow it can be reunderstood as a resource policy , and extended to multi-brac keting. Arena g ames. Recall that an arena is defined as a for est of rooted trees, whose nodes are called the moves of th e game. One writes m ⊢ n and says that the move m ena bles the move n when th e move m is the immed iate an cestor of the move n in the arena. E very m ove m is assigne d a polarity λ OP ( m ) ∈ {− 1 , +1 } . By co n vention, λ OP ( m ) = +1 when the move is Pro ponen t, and λ OP ( m ) = − 1 when it is Opp onent. Fi- nally , one requir es that the arena is alternating: m ⊢ n = ⇒ λ OP ( m ) = − λ OP ( n ) and that all roo ts ( called o pening m oves) of th e a rena have the same polarity . A typical example o f arena is the boolean arena B : q x x x x x H H H H H true false (3) where the Oppon ent mov e q justifi es the two Propon ent moves true and f alse . Every arena game A indu ces a set of justified plays , which are essentially sequences of moves (we will a v oid d iscussing pointers he re.) T y pically , the PCF type ( B 3 ⇒ B 2 ) ⇒ B 1 defines the arena q 1 y y H H f f f f f f f f f f f f q 2 y y H H f f f f f f f f f f f f true false q 3 x x I I true false true false where the indices 1 , 2 , 3 distinguish th e thr ee instances o f the boolean arena B . This arena contains the justified play q 1 · q 2 · q 3 · true 3 · true 2 · true 1 (4) also depicted using the con vention b elow: ( B ⇒ B ) ⇒ B q q q true true true (5) Note that the play (4-5 ) belong s to the strategy implem ented by the PCF program λf .f ( true ) . W ell-bracketing. Hyland and Ong dem onstrate in their work [16] that a (finite) strategy can b e implemented in PCF if and only if it satisfies two fu ndamental conditions, called inn ocence and well-brac keting . W e will focus h ere on the well-b racketing co ndition, which is very similar to a stac k d iscipline . The cond ition is usu ally expr essed in th e follo wing way . Arenas are refin ed by attaching a mode λ QA ( m ) ∈ { Q , A } to every move m of the ar ena. A move m is called a question when λ QA ( m ) = Q , and an an swer when λ QA ( m ) = A . One th en req uires that no answer move m ju stifies another answer move n : m ⊢ n = ⇒ λ QA ( m ) = Q or λ QA ( n ) = Q. The intu ition inde ed is that a n answer n respon ds to the question m wh ich justifies it in th e p lay . Note that alter na- tion ensures that P ropon ent an swers the questions raised by Oppone nt, and vice versa: hen ce, a p layer never answers h is own qu estions. For instance, the arena g ame B is refined by declaring that the Opponent m ove q is a question , and that the two Proponent mov es true and fals e are answers. Now , a justified play s is called well-brac keted when e v - ery an swer n appea ring in the play respon ds to the “pen d- ing” question m . The terminology is supported by the intu- ition t hat (1) ev ery question “opens” a bracket and (2) every answer “c loses” a brac ket, which sh ould match the b racket opened by the answered question. T y pically , th e p lay (4-5) is well-bracketed , beca use every an swer respond s pro perly to the last unan swered qu estion, thu s lead ing to the well- bracketed sequence: q 1 · q 2 · q 3 · true 3 · true 2 · true 1 ( 1 1 ) ( 2 2 ) ( 3 3 ) On the other hand, the play ( B ⇒ B ) ⇒ B q q q true (6) is no t well-bracketed, because the move tr ue answers the first question of the p lay , whereas it sho uld have an swered the third ( and pending) question. This may be d epicted in the following way: q 1 · q 2 · q 3 · true 1 ( 1 1 ) ( 2 ( 3 (7) In fact, the play (6-7) b elongs to a strategy which tests whether the fun ction f : B ⇒ B is strict, that is, interro- gates its argument: this test cannot be implemen ted in the languag e PCF – although it can b e implemented in PCF ex- tended with the control operator call-cc , see [7, 22]. Counting resources. W e would like to understand well- bracketing as a resource discipline, ra ther than simply as a stack discipline. One key step in this dir ection is the o bser- vation that a well-bracketed play may be d etected simp ly by counting two specific numbers on a path: • the num ber κ + of Propon ent questions opened but left unanswered , • the numb er κ − of Opp onent questions o pened but left unanswered . Of cou rse, it is not su ffi cient to co unt th e two nu mbers κ + and κ − of a play s to detec t wh ether the p lay is well- bracketed. T yp ically , th e well-bracketed p lay ( a ) and the non well-b racketed play ( b ) introd uced in (6-7) indu ce th e same numbers κ + and κ − : ( a ) q 1 · q 2 · q 3 · true 3 7− → κ + = 1 , κ − = 1 ( b ) q 1 · q 2 · q 3 · true 1 7− → κ + = 1 , κ − = 1 In order to detect well-bracketing , on e n eeds to apply the count to the subpaths ( c ) and ( d ) of these play s. T his reveals a key difference: ( c ) q 3 · true 3 7− → κ + = 0 , κ − = 0 ( d ) q 3 · true 1 7− → κ + = 0 , κ − = 1 The elementary but ke y characterisation follows: Proposition 1 A play s is well-brac keted if and only if ev- ery subpath m · t · n of the play s satisfies κ + ( m · t · n ) = 0 = ⇒ κ − ( m · t · n ) = 0 when m is Oppone nt and n is Pr opo nent; and dually κ − ( m · t · n ) = 0 = ⇒ κ + ( m · t · n ) = 0 when m is Pr op onent and n is Opp onent. Let us explain this briefly . Suppose that m · t · n is a sub- path of a well-bracketed play s , whe re m is Oppon ent an d n is Propo nent. The first con dition says that if there is an Oppone nt question unan swered in m · t , th en either Player answers it – in which ca se κ − ( m · t · n ) = 0 – o r there is a Player question unanswered in m · t · n – in which case κ + ( m · t · n ) 6 = 0 . The other condition is dual. A resource policy . Reformulated in this way , the well- bracketing look s very much like a re source po licy . Th e ba- sic intu ition is that every question m emits a query fo r a linear session . This q uery is noted b y a op ening b racket ( i and cou nted by κ ± where ± is the p olarity of the m ove m . The q uery is then complied with by a r esponse emitted by an answer mov e n , and noted b y a closing bracket i ) . I n our example, the move q 3 emits a que ry ( 3 which is later com - plied with in the play (4-5) by the response 3 ) emitted b y the move t rue wh ereas it remains unanswered in the play (6- 7). Hence, a play like (6-7) is not well-bracketed because it breaks the linear ity policy implemented by the queries. Our game mod el will relate this linearity policy to the fact that the boolean formu la is d efined as B = O ¬ P ¬ (1 ⊕ 1) (8) in tensorial logic. Here, the tags O and P are mnemon ics to indicate that the external negation ¬ O is inter preted as an Opponen t mov e, whereas the internal negation ¬ P is in- terpreted as a Proponen t move. The story told by (8) goes like this: Opponent plays the external negation, fo llowed by Propon ent, who play s the internal negation and at the same time reso lves the choice 1 ⊕ 1 between tru e and false . This refines the picture con veyed b y the boolean arena (3) by d ecompo sing the Player m oves true and fal se in two compou nd stages: negation an d c hoice – whe re n egation thus encapsulates the two moves t rue and false . This en- ables to relax the well-bracketing policy by interpreting the boolean formula as B = O ¬ ! • w P ¬ (1 ⊕ 1) (9) where the affine mo dality ! • w of tensorial lo gic is inserted between the two n egations. The intuition istic hierarch y on the boolean form ula (8) coincides with the well-br acketed arena game m odel of PCF d escribed by Hyland an d Ong in [16] whereas the intuitionistic hierarchy on the boolean formu la (9) – where the affine modality ! • w is replaced b y the exponential modality ! • e – coinc ides with the non-w ell- bracketed arena game model of PCF with control described by Jim Laird in [22] and Oli vier Laurent in [24]. Multi-bracketing. This analysis leads u s to the no tion of multi-brack eting in ar ena games. I n linear logic, every pro of of the formula ( B ⊗ B ) ⊸ B asks the value of its two bo olean arguments, and we would like to understand this as a kind of well-bracketing condi- tion. So , the play ( B ⊗ B ) ⊸ B q q true q true true (10) would be “we ll-bracketed” in the new setting, whereas the play ( B ⊗ B ) ⊸ B q q true true (11) would not be “well-bracketed ”, because it does n ot explore the second argu ment of the fu nction. This extended well- bracketing is captured by t he idea that the first question emits thr ee querie s ( 1 and ( a and ( b at the s ame time. Th en, the play (10) ap pears to b e “well-bracketed” if one depicts the situation in the following way: q 1 · q 2 · true 2 · q 3 · true 3 · true 1 ( 1 1 ) ( a a )( 2 2 ) ( b b )( 3 3 ) whereas the play (11) is not “ well-bracketed” because the query ( a is never complied w ith, as can be gue ssed from the picture below: q 1 · q 3 · true 3 · true 1 ( 1 1 ) ( a ( b b )( 3 3 ) W e explain in § 3 a nd § 4 how we ap ply th e well-b racketing criterion devised in Propo sition 1 in ord er to gen eralise well-bracketing to a multi-bracketed framework. Plan of the paper . W e describe ( § 2) a categorical seman- tics of resour ces in gam e semantics, and explain in wh at sense the r esulting topogr aphy refines both linear logic and polarized logic. After th at, we constru ct ( § 3) a compact- closed (that is, self-dual) category o f multi-br acketed Con- way games and well-br acketed strategies, where the re- source p olicy is enfor ced by multi-bracketing . From this, we d erive ( § 4) a mod el of our categorical semantics of re- sources, using a family construction, and conclude ( § 5). Acknowledgements. W e would like to thank M artin Hy - land tog ether with Masahito Hasegawa, Oli vier Laurent, Laurent Regnier and Peter Selinger for stimulating discus- sions at various stages of this work. 2 Categorical models of r esour ces W e introdu ce now the n otion of tensorial ne gation on a symmetric mon oidal category; and the n explain how such a category with negation may b e equip ped with additives and various r esource modalities. The first author d escribes in [2 7] how to extract a syntax of pro ofs from a categorical semantics, using string d iagrams and fu nctorial boxes. The recipe may b e applied here to extract th e sy ntax o f a logic, called ten sorial logic. Ho we ver , we provide in Appe ndix a sequent calculus for tensorial lo gic, in order to compare it to linear logic [9] or polarized linear logic [23]. T e nsorial negation. A tensorial ne gation o n a symmetric monoid al c ategory ( A , ⊗ , 1) is defined as a func tor ¬ : A − → A op together with a family of bijections ϕ A,B ,C : A ( A ⊗ B , ¬ C ) ∼ = A ( A, ¬ ( B ⊗ C )) natural in A, B and C . Given a negation, it is cu stomary to define the formula false as the object ⊥ def = ¬ 1 obtained b y “negating ” the u nit object 1 of the mon oidal category . Note that we use the notation 1 (in stead of I o r e ) in order to rem ain consistent with the notation s of linear logic. Note also that the bijection ϕ A,B , 1 provides then the category A with a one- to-one correspo ndence ϕ A,B , 1 : A ( A ⊗ B , ⊥ ) ∼ = A ( A, ¬ B ) for all obje cts A and B . For that rea son, the definition of a n egation ¬ is often r eplaced by the — somewhat too in- formal — statemen t th at “the ob ject ⊥ is expo nentiable” in the symmetr ic m onoidal category A , with negation ¬ A noted ⊥ A . Self-adjunction. In his Ph D the sis, Hayo Thielecke [35] observes fo r the first time a fundamental “self- adjunction ” pheno menon, rela ted to negation. This observation plays then a key rôle in an unpub lished work b y Peter Selinger and the first autho r [30] on polar categories, a categori- cal sem antics o f p olarized linear log ic, co ntinuation s and games. The same id ea rea ppears rec ently in a nice, co m- prehen si ve study on po larized categories (=d istributors) by Robin Cockett an d Robert Seely [8]. In our situation, the “self-adjun ction” pheno menon amou nts to the fact th at ev- ery tensorial negation is left adjoint to the opp osite functor ¬ : A op − → A (12) because of the natural bijection A op ( ¬ A, B ) ∼ = A ( A, ¬ B ) . Continuation monad. Every ten sorial n egation ¬ in- duces an adjunction , and thus a monad ¬¬ : A − → A This mon ad is called th e continu ation mo nad of the negation. One fu ndamental fact observed by Eugenio Moggi [31] is that the co ntinuation monad is str o ng but not commutative in g eneral. By strong monad , we mean that the monad ¬¬ is eq uipped with a family of morphisms: t A,B : A ⊗ ¬¬ B − → ¬¬ ( A ⊗ B ) natural in A and B , an d satisfy ing a series of cohere nce proper ties. By comm utativ e mona d, we mean a strong monad making the two canonical morphisms ¬¬ A ⊗ ¬¬ B ⇒ ¬¬ ( A ⊗ B ) (13) coincide. A tensorial negation ¬ is ca lled co mmutative when the continu ation monad induce d in A is commutative — or equiv alently , a monoid al monad in the lax sense. Linear implication. A symmetric mon oidal category A with a tenso rial n egation ¬ is not very far from bein g monoid al closed. It is p ossible indeed t o defin e a l inear implication ⊸ when its target ¬ B is a n egated object: A ⊸ ¬ B def = ¬ ( A ⊗ B ) . In th is way , the functo r (12) d efines what we call an expo- nential idea l in th e ca tegory A . When the fun ctor is faith- ful on objects and morp hisms, we may identif y th is expo- nential ideal with the sub category of negated objects in the category A . The expo nential ideal discussed in Guy Mc- Cusker’ s PhD thesis [ 26] arises p recisely in th is way . This enables in particu lar to define the linea r and intuitionistic hierarchies on the arena games (8) and (9). Continuation catego ry . Every symmetric monoida l cate- gory A equipped with a negation ¬ induces a cate gory of continua tions A ¬ with the sam e objects as A , and mor- phisms defined as A ¬ ( A, B ) def = A ( ¬ A, ¬ B ) . Note that the category A ¬ is the k leisli category associated to the c omonad in A op induced by th e adju nction; and that it is at the same time the opposite of the kleisli category as- sociated to the continu ation monad in A . Because the con- tinuation m onad is strong, th e ca tegory A ¬ is p r emonoidal in the sense of Jo hn Power and Ed mund Robinson [ 32]. It should be noted that string diagrams in p remono idal cate- gories are inhere ntly related to co ntrol flow ch arts in soft- ware engineering, as noticed by Alan Jef frey [18]. Semantics of resour ces. A r esou r ce moda lity o n a sym - metric monoidal categor y ( A , ⊗ , e ) is defined as an adjunc- tion: M U & & ⊥ A F g g (14) where • ( M , • , u ) is a symmetr ic monoidal category , • U is a symm etric monoidal functor . Recall that a symmetric mon oidal fun ctor U is a func- tor which transpo rts the symmetric monoida l stru c- ture of ( M , • , u ) to the symmetric monoidal structure of ( A , ⊗ , e ) , up to isom orphisms satisfying suitable cohe r- ence pro perties. A nother mo re conce ptual definition of a re- source modality is possible: it is an adju nction defined in the 2-category of symm etric mono idal categories, lax symm et- ric monoidal functors, an d monoidal transformations. Now , the resource modality is called • affine when the unit u is the terminal object of the cat- egory M , • exponential when the tensor prod uct • is a cartesian produ ct, and the unit u is th e ter minal object of the category M . This defin ition of resource mo dality is inspired by the ca t- egorical semantics o f linear log ic, an d more sp ecifically by Nick Benton’ s notion of Linear-Non-Line ar model [4] — which m ay be ref ormulated now as a symmetric monoidal closed c ategory A equipped with an exponential m odality in our sense. V ery o ften, we will ide ntify the r esource modal- ity and the indu ced co monad ! = U ◦ F o n the category A . T e nsorial logic. In our ph ilosophy , tensor ial logic is en- tirely described b y its categorical sema ntics — which is d e- fined in the follo wing way . First, ev ery symmetric mo noidal category A equipp ed with a tensorial n egation ¬ defines a model of multiplicative tensorial logic. Such a category de- fines a model of multiplicative additive tensor ial logic when the category A has finite coproduc ts (noted ⊕ ) which dis- tribute over the te nsor p roduct: this mean s that the canon i- cal morphisms ( A ⊗ B ) ⊕ ( A ⊗ C ) − → A ⊗ ( B ⊕ C ) 0 − → A ⊗ 0 are isomorph isms. Th en, a model of ( full) tensorial logic is defined as a m odel of mu ltiplicativ e add iti ve tensorial l ogic, equippe d w ith an affine resource modality (w ith comon ad noted ! • w ) as well as an expon ential resource modality ( with comona d noted ! • e ). The diagramm atic syntax of tensor ial logic will be read- ily extracted f rom its categorical definition, using the recipe explained in [2 7]. Howe ver , the reader will find a se- quent calculus o f tensorial logic in Appendix, wr itten in th e more familiar fashion of proof theory . Seen from that p oint of view , the mod ality-free f ragment of ten sorial logic de- scribes a linear v ariant of Girard’ s LC [1 0] thus akin to lu - dics [11] and more precisely to what Lau rent calls MALLP in his PhD thesis [23]. This convergence simply expr esses the fact that these systems are all based o n tensors, sum s and linear continuation s. Arena games and classical lo gic. Starting f rom Thi- elecke’ s work, Seling er [33] designs the notion of con tr ol cate gory in or der to axiomatize the categorical sem antics of classical logic. Th en, pr ompted by a completene ss re- sult established b y Martin H ofmann and T homas Streicher in [15], he proves a b eautiful structure theorem, stating that ev ery contro l category C is the co ntinuation category A ¬ of a r espo nse category A . Now , a response category A — where the m onic r equiremen t on the u nits ( 2) is relaxed — is exactly the sam e thing as a model of multiplicative addi- ti ve tensorial logic, wh ere the tenso r ⊗ is ca rtesian and the tensor unit 1 is te rminal. A purely proo f-theoretic analysis of classical logic leads exactly to th e same conclusion . Starting from Gira rd’ s work on polarities in LC [10] a nd ludics [11], L aurent dev el- oped a c omprehe nsi ve analysis of polar ities in logic, incor- porating classical log ic, co ntrol categories and (non -well- bracketed) arena gam es [23, 2 4]. Now , it appears th at Lau- rent’ s polarized logic LLP coincides with multiplicative ad- ditiv e tensorial logic — where the monoidal s tructure is cartesian . T his is m anifest in the mono lateral for mulation of tensorial log ic, see Appen dix. W e sum u p below the dif- ference between tensorial l ogic and classical logic in a v ery schematic table: T ensorial logic ⊗ is monoid al ¬ is tensorial Classical logic ⊗ is cartesian ¬ is ten sorial Note that e very resou rce modality (14) on a category A equippe d with a tensorial negation ¬ induces a ten sorial negation F op ◦ ¬ ◦ U on th e ca tegory M . This p rovides a model of polarized linear logic, an d thus o f classical logic, whenever M is cartesian. This phenom enon under lies the construction of a control category in [25], see also [ 12] for another constructio n. Linear logic. The co ntinuation mo nad A 7→ O ¬ P ¬ A of game seman tics lifts an Oppo nent-starting game A with an Oppone nt move ¬ O followed b y a Player move ¬ P . Now , it appears that the Blass problem men tioned in § 1 a rises precisely f rom the f act th at the monad is strong, but no t commutative [3 0, 28]. Ind eed, one ob tains a g ame model of (full) pro positional linear logic b y id entifying th e tw o canonical strategies (13) — this leading to a fully co mplete model of linear lo gic expressed in the languag e of asyn- chrono us games [29]. This constructio n in g ame sem antics has a nice c ategor- ical co unterpar t. W e alrea dy men tioned that the continu- ation category A ¬ inherits a premonoid al structure from the symm etric mon oidal structure o f A . Now , Hasegawa Masahito sho ws ( priv ate co mmunicatio n) that the continua- tion c ategory A ¬ equippe d with this prem onoidal structure is ∗ -auto nomou s if and on ly if the co ntinuation monad is commutative. The specialist will r ecognize here a categori- fication of Gir ard’ s phase space sem antics [9]. Anyway , this shows that linear logic is essentially tensorial logic in which the tensorial negation is commutativ e. Linear logic ⊗ is monoid al ¬ is co mmutative In that situatio n, e very resour ce modality on th e cate gory A induces a r esource mod ality on the ∗ -autono mous cate- gory A ¬ , and thus a model of full linear logic. 3 Multi-brack eted Conway games W e define her e and in § 4 a game semantics with resource modalities and fixp oints, in order to inter pret recu rsion in progr amming languag es. W e ach ie ve this by co nstruct- ing first a com pact-closed cate gory B of multi-bracketed Conway games, inspired from And ré Joyal’ s pioneerin g work [19]. The comp act-closed structur e of B induces a trace operator [20] which, in turn , provides en ough fixpoints in the category constructed in § 4 in order to interpr et the languag e PCF enriched with resource modalities. Conway games. A Conway g ame is an o riented roo ted graph ( V , E , λ ) co nsisting of a set V of vertices called the positions of the game, a set E ⊂ V × V of edges called the moves of the game, a fun ction λ : E → { − 1 , +1 } in dicat- ing wheth er a move belong s to Opponent ( − 1 ) or Proponen t ( +1 ). W e no te ⋆ the roo t of the underlying graph. Path and play . A play is a p ath starting f rom the roo t ⋆ A of the multi-bracketed game: ⋆ A m 1 − − → x 1 m 2 − − → . . . m k − 1 − − − → x k − 1 m k − − → x k (15) T wo paths are p arallel when they have the same initial an d final positions. A p lay (15) is alternating when: ∀ i ∈ { 1 , . . . , k − 1 } , λ A ( m i +1 ) = − λ A ( m i ) . Strategy . A strategy σ o f a Con way game is defin ed as a set of alternating plays of e ven length such that: • σ con tains the empty play , • every n onempty play starts with an Oppon ent move, • σ is closed by e ven- length pr efix: for e very play s , and for all moves m , n , s · m · n ∈ σ = ⇒ s ∈ σ, • σ is d eterministic: f or ev ery play s , an d for all moves m, n, n ′ , s · m · n ∈ σ and s · m · n ′ ∈ σ = ⇒ n = n ′ . W e write σ : A when σ is a strategy of A . Note tha t a play in a Conway game is gen erally non-alternatin g, but that alternation is required on the plays of a strategy . Multi-bracketed g ames. A multi-bracketed ga me is a Conway game equipped with • a finite set Q A ( x ) o f queries for each position x ∈ V of the game, • a fu nction λ ( x ) : Q A ( x ) − → {− 1 , +1 } which as- signs to every query in Q A ( x ) a polarity which indi- cates wheth er the query is made by Op ponent ( − 1 ) or Propon ent ( +1 ), • for each move x m − → y , a r esidual relation [ m ] ⊂ Q A ( x ) × Q A ( y ) satisfying: r [ m ] r 1 and r [ m ] r 2 = ⇒ r 1 = r 2 r 1 [ m ] r an d r 2 [ m ] r = ⇒ r 1 = r 2 The d efinition of residuals is then extended to paths s : x ։ y in th e usual way: by co mposition of relations. W e then define r [ s ] def = { r ′ | r [ s ] r ′ } and [ s ] r def = { r ′ | r ′ [ s ] r } . W e say th at a path s : x ։ y : • comp lies with a query r ∈ Q A ( x ) when r has n o re sid- ual after s — that is, r [ s ] = ∅ , • initiates a query r ∈ Q A ( y ) wh en r has no ancestor before s — th at is, [ s ] r = ∅ . W e require that a move m only initiates queries of its own polarity , and only complies with queries of the opposite po- larity . In order to fo rmalise that a residua l of a query is intuitively th e query itself, we also re quire that two parallel paths s and t in duce the same residu al relation: [ s ] = [ t ] . Finally , we require that there ar e no queries at the roo t: Q A ( ⋆ ) = ∅ . Resource function. Ex tending Conway g ames with queries enables the definition of a resource function κ = ( κ + , κ − ) which cou nts, fo r every path s : x ։ y , the n umber κ + ( s ) (respectively κ − ( s ) ) o f Propon ent (respec ti vely Oppone nt) queries in r ∈ Q A ( y ) in itiated by the path s — that is, such that [ s ] r = ∅ . The definition of multi-br acketed games induces thr ee card inal p roperties o f κ ± , which will r eplace the very de finition of κ , and will play th e rô le of ax ioms in all our proo fs – in particular, in the proof that the composite of two well-bracketed strategies is also well-bracketed. Property 1: accuracy. For all path s s : x ։ y an d Propo- nent move m : y → z , κ − ( m ) = 0 and κ + ( s · m ) = κ + ( s ) + κ + ( m ) , as well as the dual equalities for Oppone nt moves. Property 2: suffix dominatio n. For all paths s : x ։ y and t : y ։ z , κ ( t ) ≤ κ ( s · t ) . Property 3: sub- additivity. For all paths s : x ։ y and t : y ։ z , κ ( s · t ) ≤ κ ( s ) + κ ( t ) . Accuracy hold s beca use Play er does no t in itiate Op ponent queries, and does not comply with Player queries. Suffix dominatio n say s that a qu ery cannot already h a ve been com- plied with. Sub-add iti vity expre sses that composing two paths does not increase the numb er of queries. W ell-bracketed plays a nd strateg ies. Once the reso urce function κ is defined o n paths, it b ecomes po ssible to de- fine a well-brack eted pla y a s a play whic h satisfies th e two condition s stated in Prop osition 1 of § 1. So, the prop erty becomes a d efinition here. A strategy σ is then d eclared well-brac keted when, for e very play s · m · t · n of the strat- egy σ wh ere m is an Opponen t mov e and n is (nec essarily) a Proponen t m ove: κ + A ( m · t · n ) = 0 = ⇒ κ − A ( m · t · n ) = 0 . Every well-bracketed strategy σ then preserves well- bracketing in the following sense: Lemma 1 Suppo se s · m · n ∈ σ a nd tha t s · m is well- brac keted. Then, s · m · n is well-bracketed. Hence, when Op ponent and Propo nent play ac cording to well-br acketed strategies, the resulting play is well- bracketed. Dual. Every multi-brac keted game A induc es a dual game A ∗ obtained by reversing the polarity of moves and queries. Thus, ( κ + A ∗ , κ − A ∗ ) = ( κ − A , κ + A ) . T e nsor product. The tensor produc t A ⊗ B of two m ulti- bracketed games A and B is defined as: - its p ositions a re the pairs ( x, y ) noted x ⊗ y , ie. V A ⊗ B = V A × V B with ⋆ A ⊗ B = ( ⋆ A , ⋆ B ) . - its moves are of two kinds: x ⊗ y →  z ⊗ y if x → z in the game A, x ⊗ z if y → z in the game B . - its quer ies at p osition x ⊗ y are th e queries at p osi- tion x in the game A and the q ueries at position y in the game B : Q A ⊗ B ( x ⊗ y ) = Q A ( x ) ⊎ Q B ( y ) . The polarities o f moves an d queries in the g ame A ⊗ B are in herited fr om the games A an d B , and th e r esidual re- lation of a move m in the game A ⊗ B is d efined just in the expected (pointwise) way . The u nique multi-bra cketed game 1 with { ⋆ } as underlyin g Con way game is the neutral element o f the tensor prod uct. As u sual in game seman- tics, every p lay s in the g ame A ⊗ B may be seen as the interleaving o f a play s | A in the game A and a play s | B in the game B . Mo re interestingly , the resource f unction κ is “tensorial” in the following sense: κ A ⊗ B ( s ) = κ A ( s | A ) + κ B ( s | B ) . Composition. W e procee d as in [26, 13], and say th at u is an interaction o n thre e games A, B , C , th is no ted u ∈ int AB C , when the projection of u on each game A ∗ ⊗ B , B ∗ ⊗ C an d A ∗ ⊗ C is a pla y . Given two strategies σ : A ∗ ⊗ B , τ : B ∗ ⊗ C , we define the com position of these strategies as follows : σ ; τ = { u | A ∗ ⊗ C | u ∈ i nt AB C , u | A ∗ ⊗ B ∈ σ, u | B ∗ ⊗ C ∈ τ } As usually , the comp osition of two strategies is a strat- egy . More interestingly , we show th at our no tion of well- bracketing is preserved by composition : Proposition 2 The str ate gy σ ; τ : A ∗ ⊗ C is well-bracketed when the two strate gies σ : A ∗ ⊗ B and τ : B ∗ ⊗ C ar e well-brac keted. Proof: The proof is entirely based o n th e thr ee ca rdinal proper ties of κ mentio ned earlier . Th e proof appears in the Master’ s thesis of the secon d author [34]. The catego ry B of multi-bra cketed games. The cate- gory B h as multi-b racketed games as objects, and well- bracketed strategies σ of A ∗ ⊗ B as morph isms σ : A → B . The identity strategy is the usual copyc at strategy , defined by An dré Joyal in Conway game [19]. Th e resulting cate- gory B is compact-closed in the sense of [21] and thus ad- mits a canonica l trace operator, unique up to eq uiv alence, see [20] for details. Negative and positive ga mes. A multi-bra cketed g ame A is called negative when all the moves star ting f rom the root ⋆ A are Opp onent moves; and po sitive when its dual game A ∗ is negative. The full sub category of negativ e ( resp. positive) multi-b racketed games is no ted B − (resp. B + ). For a multi- bracketed game A , we write A − for the nega- ti ve game obtain ed by re moving all the Player moves from the root. The exponential modality . Every multi-bracketed game A in duces an e xponen tial game ! A as fo llows: - its positions ar e the words w = x 1 · · · x k whose let- ters a re position s x i of th e game A different f rom the root ⋆ A ; the intuitio n is that the letter x i describes th e current position of the i th copy of the game, - its root ⋆ ! A is the empty word, - its moves w → w ′ are either moves play ed in one copy: w 1 x w 2 → w 1 y w 2 where x → y is a m ove in the game A ; o r moves where Oppone nt opens a new copy: w → w x where ⋆ A → x is an Opponen t move in A . - its q ueries at position w = x 1 · · · x n are pair s ( i , q ) consisting of an ind ex 1 ≤ i ≤ n and a qu ery q at position x i in the game A . The p olarities of moves and queries are inherited f rom th e game A i n the expected way , and the resid ual r elation is defined as for the tensor prod uct. Interesting ly , the result- ing m ulti-bracketed game ! A d efines the free co mmutative comono id associated to the well-bracketed g ame A in th e category B . Hen ce, the c ategory B defines a model of multi- plicative expon ential linear logic. This model is de gener ate in the sense that the tensor product is eq ual to its dual, i.e. ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ . Fixpoints. Th e expone ntial modality tog ether with the traced symm etric mono idal structure on B defines a fixpoint operator in B as shown by Hase gawa Masah ito in [14]. Re- mark that th is c onstruction does n ot require that the ca te- gory B is cartesian. 4 A game model with re sour ces W e would like to con struct a m odel of tensorial log ic based on negati ve multi-br acketed game s. Howe ver it is meaningle ss to construc t an af fine m odality on the cate- gory B − itself because its unit 1 is already a term inal ob- ject in the category . So we need to in troduce the notion o f pointed game. Pointed games. A pointed game may be seen in two different ways: (1) as a po siti ve m ulti-bracketed Conway game, with a un ique initial Player move, ( 2) as a negativ e multi-brac keted Con way game, except that the h ypothesis that there are no qu eries at the ro ot ∗ is n ow relaxed fo r Player queries. From now o n, we ado pt the first point of view , and thus see a pointed g ame as a p ositi ve game with a u nique in itial move. Now , a mo rphism σ : A − → B in the categor y B is called transverse wh en, for every play mn of leng th 2 in the strategy σ : A ∗ ⊗ B , the Op ponent move m is in A and the Player move n is in B . W e note B • the subcategory of B with pointed g ames as objects, and well-bracketed transverse strategies as morphisms. Coalesced tensor . Giv en A, B ∈ B • , the co alesced tensor A ⊙ B is the pointed gam e obtained fro m A ⊗ B by syn chro- nising the two in itial Player moves of A and B . Rema rk that the coalesced tensor product pr eserves a ffine gam es, and coincides there with th e tensor product of B − . The cat- egory B • equippe d with ⊙ is symmetr ic monoidal. It is not monoid al closed, but ad mits a tensorial negation . Besides, it inh erits a trace opera tor from the categor y B , which is partial, b ut sufficient to interpret a l inear PCF wit h resource modalities. T e nsorial neg ation. The negation ¬ A o f a p ointed game A is the po inted game obtain ed by lifting the dual game A ∗ with a Propon ent mo ve m which initiates one query . T hen, every in itial Op ponent move in A ∗ complies with this query . Affine modality . A pointed gam e A is called a ffine when its u nique initial Player move does not initiate any query . Note that B − is isomorphic to the full subcategory of affine games in the c ategory B • . The affine game ! • w A associated to a p ointed gam e A is defined b y r emoving all the querie s initiated by the first move — as well a s their residuals. T his defines an affine r esource modality on B • . Exponential modality . Th e expon ential modality ! • e on pointed games is ob tained by com posing the two adju nc- tions underly ing th e comona ds ! • w and ! ( defined in § 3). M ⊆ ' ' ⊥ B − ! g g ⊆ ' ' ⊥ B • ! • w g g In particular, given a pointed game A , ! • e A is defined as ! • e A def = ! ( ! • w A ) Free coproducts. The category B • lacks coprod ucts to b e a mode l of (full) ten sorial logic. W e adjust this by con- structing its free comp letion, no ted F am ( B • ) , und er small coprod ucts [3]. Given a c ategory C , the objects of F am ( C ) are families { A i | i ∈ I } o f objects of the c ategory . A mo r- phism from { A i | i ∈ I } to { B j | j ∈ J } consists of a reindex- ing function f : I → J together with a family of mor phisms { f i : A i → B f ( i ) | i ∈ I } of the category C . F am is a pseudo-co mmutative monad on C at [17]. Hence, the 2 -monad fo r symmetric mon oidal categories dis- tributes over F am . Consequently , (1) the category F am ( C ) inherits the symmetr ic mo noidal stru cture of a symmetric monoid al category C , and (2) the co produ ct of F am ( C ) dis- tributes over that ten sor p roduct, and (3) F am p reserves monoid al ad junctions. Besides, F am preserves categories with fin ite pro ducts and catego ries with a terminal ob- ject. The construction thu s preserves affine and exponential modalities in the sense of § 2. Gathering all those r emarks, we obtain that: Proposition 3 F am ( B • ) is a model of tensorial logic. Moreover , the category F am ( B • ) has a fixpo int operator restricted on its singleton objects — that is, objects { A i | i ∈ I } wh ere I is sin gleton. T his is sufficient to interpre t a lin- ear v ariant of th e language PCF equ ipped with af fine a nd exponential resource modalities, in the cate gory F am ( B • ) . 5 Conclusion In this p aper, w e integrate resource mo dalities in game semantics, in just the same way as they are integrated in linear logic. This is achiev ed by r eunderstand ing the very topogr aphy of the field. More specifically , linear lo gic is re- laxed into tensorial log ic, wh ere the in v olutive negation of linear log ic is rep laced by a tensorial n egation. Onc e this perfor med, it is possible to keep the best of linear logic: re- source m odalities, etc. but transpo rted in the languag e of games and co ntinuations. Then, linear log ic coincide s with tensorial log ic with the ad ditional axiom that the continu- ation monad is com mutative. 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A sequent calculus f or tensorial logic In the bilateral formula tion o f tensorial log ic, the se- quents are of two forms: Γ ⊢ A where Γ is a context, and A is a for mula; Γ ⊢ where Γ is a context (the notation [ A ] expresses the unessential presence of A in the sequen t). Γ ⊢ A ∆ ⊢ B ⊗ -Right Γ , ∆ ⊢ A ⊗ B Γ 1 , A, B , Γ 2 ⊢ [ C ] ⊗ -Left Γ 1 , A ⊗ B , Γ 2 ⊢ [ C ] Unit-Right ⊢ 1 Γ ⊢ [ A ] Unit-Left Γ , 1 ⊢ [ A ] Γ , A ⊢ ¬ -Right Γ ⊢ ¬ A Γ ⊢ A ¬ -Left Γ , ¬ A ⊢ Axiom A ⊢ A Γ ⊢ A A, ∆ ⊢ [ B ] Cut Γ , ∆ ⊢ [ B ] Γ ⊢ A ⊕ -Right-1 Γ ⊢ A ⊕ B Γ ⊢ B ⊕ -Right-2 Γ ⊢ A ⊕ B Γ , A ⊢ C Γ , B ⊢ C ⊕ -Left Γ , A ⊕ B ⊢ C No right introduction rule for 0 0 -Left Γ , 0 ⊢ A ! • e Γ ⊢ A Strengthening ! • e Γ ⊢ ! • e A Γ , A ⊢ [ B ] Dereliction Γ , ! • e A ⊢ [ B ] Γ ⊢ [ B ] W eakening Γ , ! • e A ⊢ [ B ] Γ , ! • e A, ! • e A ⊢ [ B ] Contraction Γ , ! • e A ⊢ [ B ] ! • w Γ ⊢ A Strengthening ! • w Γ ⊢ ! • w A Γ , A ⊢ [ B ] Dereliction Γ , ! • w A ⊢ [ B ] Γ ⊢ [ B ] W eakening Γ , ! • w A ⊢ [ B ] The m onolateral for mulation requires to p olarize fo rmulas, and to clone each construct into a negativ e c ounterpa rt. Positi ves 0 | 1 | ↓ L | P ⊗ Q | P ⊕ Q | ! • w P | ! • e P Negati ves ⊥ | ⊤ | ↑ P | L Γ M | L & M | ? • w L | ? • e L It is then po ssible to reform ulate all the sequent above, as illustrated below by the right and left introdu ction of ⊗ . ⊢ Γ , P ⊢ ∆ , Q ⊗ -Right ⊢ Γ , ∆ , P ⊗ Q ⊢ Γ 1 , L, M , Γ 2 , [ P ] Γ -Left ⊢ Γ 1 , L Γ M , Γ 2 , [ P ]