Deterministic Concurrent Strategies

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1 Dtrministic Concurrnt Stratgis Glynn Winskl Computr Laboratory, Univrsity of Cambridg, UK Abstract. Nondtrministic concurrnt stratgis thos stratgis compatibl with copy-cat bhaving as idntity w.r.t. composition hav bn charactrizd as crtain maps of vnt structurs. This lads to a bicatgory of gnral concurrnt gams in which th maps ar nondtrministic concurrnt stratgis. This papr xplors th important sub-bicatgory of dtrministic concurrnt stratgis. It is shown that dtrministic stratgis in a gam can b idntifid with crtain subgams, with th bnfit that th bicatgory of dtrministic gams bcoms quivalnt to a tchnically-simplr ordr-nrichd catgory. Via a charactrization, dtrministic stratgis ar shown to coincid with th rcptiv ingnuous stratgis of Mlliès and Mimram, providing an xtrnal justification for thir approach, as yilding th most gnral dtrministic concurrnt stratgis for which copy-cat bhavs as idntity. Dtrministic stratgis dtrmin closur oprators, following Abramsky and Mlliès. Known subcatgoris appar as spcial cass: Brry s ordr-nrichd catgory of di-domains and stabl functions ariss as a full subcatgory in which th gams compris solly of Playr movs; th simpl gams of Hyland t al., a basis for much of gam smantics, form a subcatgory in which th gams prmit no concurrncy, Playr-Opponnt movs altrnat and Opponnt always movs first. 1 Finally, winning stratgis ar considrd. 1 Introduction This articl brings th xprinc of concurrncy (vnt structurs, stabl familis, thir tchniqus and constructions originally usd in th smantics of procss languags [2]) to bar on th thory of gams. It considrs a vry gnral dfinition of 2-party concurrnt gams in which Playr (mor accuratly thought of as a tam of playrs) compts against Opponnt (a tam of opponnts) in a potntially highly-distributd fashion, without for instanc insisting on th altrnation of Playr and Opponnt movs. Two-party gams and stratgis ar rprsntd as vnt structurs with polarity, in which polaritis distinguish th movs of Playr and Opponnt cf. [3]. A total map of vnt structurs with polarity with codomain A can b undrstood as a pr-stratgy in a gam A th map nsurs that Playr and Opponnt rspct th constraints of th gam. Following Joyal, a pr-stratgy from a gam A to a gam B is undrstood as a pr-stratgy in a composit gam got by stting th dual gam of A, rvrsing th rols of Playr and Opponnt, in paralll 1 Th rsults on dtrministic stratgis ar rportd without proofs in [1].

2 2 Glynn Winskl with B. From this gnral schm concurrnt stratgis pr-stratgis for which copy-cat stratgis bhav as idntitis w.r.t. composition of pr-stratgis hav rcntly bn charactrizd as thos pr-stratgis which satisfy th two conditions of rcptivity and innocnc [1]. Th major contribution of this papr is a thorough xploration, with full proofs, of this rsult s consquncs for th cas of dtrministic stratgis of which thr is alrady a significant history within smantics. It givs proofs of ncssary and sufficint conditions for copy-cat stratgis to b dtrministic shows why thy ar not in gnral provs dtrministic stratgis compos, and that thy ar ncssarily mono as maps into gams. Th latr implis that w can quivalntly viw dtrministic stratgis in a gam A as crtain subfamilis of configurations of A. A charactrization of prcisly which subfamilis aris from dtrministic stratgis rcovrs th rcptiv ingnuous stratgis of Mlliès and Mimram [4]; thir rcptiv ingnuous stratgis ar rvald as prcisly thos dtrministic pr-stratgis for which copy-cat stratgis bhav as idntitis a satisfying convrgnc with Mlliès programm of asynchronous gams [3]. 2 Evnt structurs and stabl familis An vnt structur compriss (E, Con, ), consisting of a st E, of vnts which ar partially ordrd by, th causal dpndncy rlation, and a nonmpty consistncy rlation Con consisting of finit substs of E, which satisfy { } is finit for all E, {} Con for all E, Y X Con Y Con, and X Con & X X {} Con. Th (finit) configurations, C(E), of an vnt structur E consist of thos finit substs x E which ar Consistnt: x Con, and Down-closd:,. x x. Two vnts which ar both consistnt and incomparabl w.r.t. causal dpndncy in an vnt structur ar rgardd as concurrnt. In gams th rlation of immdiat dpndncy, maning and ar distinct with and no vnt in btwn, will play a vry important rol. For X E w writ [X] for { E X. }, th down-closur of X; not if X Con, thn [X] Con. Oprations such as synchronizd paralll composition ar awkward to dfin dirctly on th simpl vnt structurs abov. It is usful to broadn vnt structurs to stabl familis, whr oprations ar oftn carrid out mor asily, and thn turnd into vnt structurs by th opration Pr blow. A stabl family compriss F, a nonmpty family of finit substs, calld configurations, which satisfy: Compltnss: Z F. Z Z F;

3 Dtrministic Concurrnt Stratgis 3 Coincidnc-frnss: For all x F,, x with /=, y F. y x & ( y y) ; Stability: x, y F. x y x y F. Abov, Z mans x F z Z. z x, and xprsss th compatibility of Z in F; w us x y for {x, y}. W call lmnts of F vnts of F. 2 Proposition 1. Lt x b a configuration of a stabl family F. For, x dfin x iff y F. y x & y y. Whn x dfin th prim configuration [] x = {y F y x & y}. Thn x is a partial ordr and [] x is a configuration such that [] x = { x x }. Morovr th configurations y x ar xactly th down-closd substs of x. Proposition 2. Lt F b a stabl family. Thn, Pr(F) = df (P, Con, ) is an vnt structur whr: P = {[] x x & x F}, Z Con iff Z P & Z F p p iff p, p P & p p. and, A (partial) map of stabl familis f F G is a partial function f from th vnts of F to th vnts of G such that for all configurations x F, fx G & ( 1, 2 x. f( 1 ) = f( 2 ) 1 = 2 ). Maps of vnt structurs ar maps of thir stabl familis of configurations. Maps compos as functions. W say a map is total whn it is total as a function. Say a total map of vnt structurs is rigid whn it prsrvs causal dpndncy. Pr is th right adjoint of th inclusion functor, taking an vnt structur E to th stabl family C(E). Th unit of th adjunction E Pr(C(E)) taks an vnt to th prim configuration [] = df { E }. Th counit max C(Pr(F)) F taks prim configuration [] x to. Dfinition 1. Lt F b a stabl family. W us x y to man y covrs x in F, i.. x y in F with nothing in btwn, and xy to man x {} = y for x, y F and vnt x. W somtims us x, xprssing that vnt is nabld at configuration x, whn xy for som y. W.r.t. x F, writ [) x = df { E x & /= }, so, for xampl, [) x [] x. Th rlation of immdiat dpndnc of vnt structurs gnralizs: with rspct to x F, th rlation x mans x with /= and no vnt in btwn. 2 W can xtnd familis of finit configurations, F, to familis whr configurations may b infinit, F, taking x F iff thr is a dirctd subst S F s.t. x = S.

4 4 Glynn Winskl 3 Procss oprations 3.1 Products Lt A and B b stabl familis with vnts A and B, rspctivly. Thir product, th stabl family A B, has vnts comprising pairs in A B = df {(a, ) a A} {(a, b) a A & b B} {(, b) b B}, th product of sts with partial functions, with (partial) projctions π 1 and π 2 trating as undfind with configurations x A B iff x is a finit subst of A B s.t. π 1 x A & π 2 x B,, x. π 1 () = π 1 ( ) or π 2 () = π 2 ( ) =, &, x. /= y x. π 1 y A & π 2 y B & ( y y). Right adjoints prsrv products. Consquntly w obtain a product of vnt structurs A and B by first rgarding thm as stabl familis C(A) and C(B), forming thir product C(A) C(B), π 1, π 2, and thn constructing th vnt structur A B = df Pr(C(A) C(B)) and its projctions as Π 1 = df π 1 max and Π 2 = df π 2 max. 3.2 Rstriction Th rstriction of F to a subst of vnts R is th stabl family F R = df {x F x R}. Dfining E R, th rstriction of an vnt structur E to a subst of vnts R, to hav vnts E = { E [] R} with causal dpndncy and consistncy inducd by E, w obtain C(E R) = C(E) R. Proposition 3. Lt F b a stabl family and R a subst of its vnts. Thn, Pr(F R) = Pr(F) max 1 R. 3.3 Synchronizd compositions Synchronizd paralll compositions ar obtaind as rstrictions of products to thos vnts which ar allowd to synchroniz or occur asynchronously. For xampl, th synchronizd composition of Milnr s CCS on stabl familis A and B (with lablld vnts) is dfind as A B R whr R compriss vnts which ar pairs (a, ), (, b) and (a, b), whr in th lattr cas th vnts a of A and b of B carry complmntary labls. Similarly, synchronizd compositions of vnt structurs A and B ar obtaind as rstrictions A B R. By Proposition 3, w can quivalntly form a synchronizd composition of vnt structurs by forming th synchronizd composition of thir stabl familis of configurations, and thn obtaining th rsulting vnt structur this has th advantag of liminating suprfluous vnts arlir.

5 Dtrministic Concurrnt Stratgis Projction Evnt structurs support a simpl form of hiding. Lt (E,, Con) b an vnt structur. Lt V E b a subst of visibl vnts. Dfin th projction of E on V, to b E V = df (V, V, Con V ), whr v V v iff v v & v, v V and X Con V iff X Con & X V. 4 Evnt structurs with polaritis W shall rprsnt both a gam and a stratgy in a gam as an vnt structur with polarity, which compriss (E, pol) whr E is an vnt structur with a polarity function pol E {+, } ascribing a polarity + (Playr) or (Opponnt) to its vnts. Th vnts corrspond to (occurrncs of) movs. Maps of vnt structurs with polarity ar maps of vnt structurs which prsrv polarity. 4.1 Oprations Dual Th dual, E, of an vnt structur with polarity E compriss a copy of th vnt structur E but with a rvrsal of polaritis. It obviously xtnds to a functor. Writ E for th vnt complmntary to E and vic vrsa. Simpl paralll composition This opration simply juxtaposs two vnt structurs with polarity. Lt (A, A, Con A, pol A ) and (B, B, Con B, pol B ) b vnt structurs with polarity. Th vnts of A B ar ({1} A) ({2} B), thir polaritis unchangd, with: th only rlations of causal dpndncy givn by (1, a) (1, a ) iff a A a and (2, b) (2, b ) iff b B b ; a subst of vnts C is consistnt in A B iff {a (1, a) C} Con A and {b (2, b) C} Con B. Th opration xtnds to a functor put th two maps in paralll. Th mpty vnt structur with polarity is th unit w.r.t.. 5 Pr-stratgis Lt A b an vnt structur with polarity, thought of as a gam; its vnts stand for th possibl occurrncs of movs of Playr and Opponnt and its causal dpndncy and consistncy rlations th constraints imposd by th gam. A pr-stratgy in A is a total map σ S A from an vnt structur with polarity S. A pr-stratgy rprsnts a nondtrministic play of th gam all its movs ar movs allowd by th gam and oby th constraints of th gam; th concpt will latr b rfind to that of stratgy (and winning stratgy in Sction 8.1). Lt A and B b vnt structurs with polarity. Following Joyal [5], a prstratgy from A to B is a pr-stratgy in A B, so a total map σ S A B. It thus dtrmins a span S σ 1 σ 2 A B,

6 6 Glynn Winskl of vnt structurs with polarity whr σ 1, σ 2 ar partial maps. In fact, a prstratgy from A to B corrsponds to such spans whr for all s S ithr, but not both, σ 1 (s) or σ 2 (s) is dfind. Two pr-stratgis will b ssntially th sam whn thy ar isomorphic as spans. W writ σ τ, for pr-stratgis σ and τ from A to B whn thir spans ar isomorphic. W writ σ A + B to xprss that σ is a pr-stratgy from A to B. Not a pr-stratgy in a gam A coincids with a pr-stratgy from th mpty gam σ + A. 5.1 Composing pr-stratgis Considr two pr-stratgis σ A + B and τ B + C as spans: A σ 1 S σ 2 B T τ 1 τ 2 B C. W show how to dfin thir composition τ σ A + C as th rsult of a synchronizd composition, followd by projction to hid intrnal synchronization vnts. W first form th synchronizd composition of S and T by rstricting th product S T, with projctions Π 1 S T S and Π 2 S T T, to allow only thos synchronizations associatd with complmntary vnts, of diffrnt polaritis, in B and B. Spcifically, th synchronizd composition is S T R 0 whr R 0 = {p S T σ 1 Π 1 (p) is dfind & Π 2 (p) is undfind} {p S T τ 2 Π 2 (p) is dfind & Π 1 (p) is undfind} {p S T σ 2 Π 1 (p) = τ 1 Π 2 (p) with both dfind}. W dfin T S = df (S T R 0 ) V whr V = {p S T R 0 σ 1 Π 1 (p) is dfind} {p S T R 0 τ 2 Π 2 (p) is dfind}. Finally, th composition τ σ is dfind to b th span σ 1Π 1 T S τ 2Π 2 A C. As rmarkd in Sction 3.3, th sam construction is achivd by first forming th synchronizd composition of th stabl familis C(S) and C(T ) (it is this dscription w shall us in proofs): Proposition 4. Th composition T S = Pr(C(S) C(T ) R) V, whr R = {(s, ) s S & σ 1 (s) is dfind} {(, t) t T & τ 2 (t) is dfind} {(s, t) s S & t T & σ 2 (s) = τ 1 (t) with both dfind}.

7 Dtrministic Concurrnt Stratgis 7 Th span τ σ compriss maps υ 1 T S A and υ 2 T S C, which on vnts p of T S act so υ 1 (p) = σ 1 (s) whn max(p) = (s, ) and υ 2 (p) = τ 2 (t) whn max(p) = (, t), and ar undfind lswhr. Th natural isomorphism S (T U) (S T ) U, associatd with th product of vnt structurs S, T, U, rstricts to th rquird isomorphism of spans as th synchronizations involvd in succssiv compositions ar disjoint: Proposition 5. Lt σ A + B, τ B + C and υ C + D b pr-stratgis. Th two compositions υ (τ σ) and (υ τ) σ ar isomorphic. 5.2 Concurrnt copy-cat Idntitis w.r.t. composition ar givn by copy-cat stratgis. Lt A b an vnt structur with polarity. Th copy-cat stratgy from A to A is an instanc of a pr-stratgy, so a total map γ A CC A A A. It dscribs a concurrnt, or distributd, stratgy basd on th ida that Playr movs, of +v polarity, always copy prvious corrsponding movs of Opponnt, of v polarity. For c A A w us c to man th corrsponding copy of c, of opposit polarity, in th altrnativ componnt, i.. (1, a) = (2, a) and (2, a) = (1, a). Dfin CC A to compris th vnt structur with polarity A A togthr with xtra causal dpndncis c CCA c for all vnts c with pol A A(c) = +. Proposition 6. Lt A b an vnt structur with polarity. Thn vnt structur with polarity CC A is an vnt structur. Morovr, x C(CC A ) iff x C(A A) & c x. pol A A(c) = + c x. Th copy-cat pr-stratgy γ A A + A is dfind to b th map γ A CC A A A whr γ A is th idntity on th common st of vnts. 6 Stratgis Th main rsult of [1], prsntd summarily, is that two conditions on prstratgis, rcptivity and innocnc, ar ncssary and sufficint for copy-cat to bhav as idntity w.r.t. th composition of pr-stratgis. Rcptivity nsurs an opnnss to all possibl movs of Opponnt. Innocnc rstricts th bhaviour of Playr; Playr may only introduc nw rlations of immdiat causality of th form byond thos imposd by th gam. Rcptivity. A pr-stratgy σ is rcptiv iff σx a & pol A (a) =!s S. x s & σ(s) = a. Innocnc. A pr-stratgy σ is innocnt whn it is both +-innocnt: if s s & pol(s) = + thn σ(s) σ(s ), and -innocnt: if s s & pol(s ) = thn σ(s) σ(s ). Thorm 1. Lt σ A + B b pr-stratgy. Copy-cat bhavs as idntity w.r.t. composition, i.. σ γ A σ and γ B σ σ, iff σ is rcptiv and innocnt.

8 8 Glynn Winskl 6.1 Th bicatgory of concurrnt gams and stratgis Thorm 1 motivats th dfinition of a stratgy as a pr-stratgy which is rcptiv and innocnt. In fact, w obtain a bicatgory, Gams, in which th objcts ar vnt structurs with polarity th gams, th arrows from A to B ar stratgis σ A + B and th 2-clls ar maps of spans. Th vrtical composition of 2-clls is th usual composition of maps of spans. Horizontal composition is givn by th composition of stratgis (which xtnds to a functor on 2-clls via th functoriality of synchronizd composition). 7 Dtrministic stratgis 7.1 Dfinition W say an vnt structur with polarity S is dtrministic iff X fin S. Ng[X] Con S X Con S, whr Ng[X] = df {s S pol(s ) = & s X. s s}. In othr words, S is dtrministic iff any finit st of movs is consistnt whn it causally dpnds only on a consistnt st of opponnt movs. Say a stratgy σ S A is dtrministic if S is dtrministic. Lmma 1. An vnt structur with polarity S is dtrministic iff s, s S, x C(S). x s & x s & pol(s) = + x {s, s } C(S). Proof. Only if : Assum S is dtrministic, x, s x s and pol(s) = +. Tak X = df x {s, s }. Thn Ng[X] x {s} so Ng[X] Con S. As S is dtrministic, X Con S and bing down-closd X = x {s, s } C(S). If : Assum S satisfis th proprty statd abov in th proposition. Lt X fin S with Ng[X] Con S. Thn th down-closur [Ng[X]] C(S). Clarly [Ng[X]] [X] whr all vnts in [X] [Ng[X]] ar ncssarily +v. Suppos, to obtain a contradiction, that X Con S. Thn thr is a maximal z C(S) such that [Ng[X]] z [X] and som [X] z, ncssarily +v, for which [) z. Tak a covring chain [) z s1 s 2 s k 1 zk = z. As [) [] with +v, by rpatd us of th proprty of th lmma illustratd blow w obtain zz in C(S) with [Ng[X]] z [X], which contradicts th maximality of z. s 1 [] z s 2 s 1 k z = k z [) s 1 z1 s 2 s k zk = z

9 Dtrministic Concurrnt Stratgis 9 So, abov, an vnt structur with polarity can fail to b dtrministic in two ways, ithr with pol(s) = pol(s ) = + or with pol(s) = + & pol(s ) =. In gnral for an vnt structur with polarity A th copy-cat stratgy can fail to b dtrministic in ithr way, illustratd in th xampls blow. Exampl 1. (i) Tak A to consist of two +v vnts and on v vnt, with any two but not all thr vnts consistnt. Th construction of CC A is picturd: A A Hr γ A is not dtrministic: tak x to b th st of all thr v vnts in CC A and s, s to b th two +v vnts in th A componnt. (ii) Tak A to consist of two vnts, on +v and on v vnt, inconsistnt with ach othr. Th construction CC A : A A To s CC A is not dtrministic, tak x to b th singlton st consisting.g. of th v vnt on th lft and s, s to b th +v and v vnts on th right. 7.2 Th bicatgory of dtrministic stratgis W first charactriz thos gams for which copy-cat is dtrministic; thy only allow immdiat conflict btwn vnts of th sam polarity. Lmma 2. Lt A b an vnt structur with polarity. Th copy-cat stratgy γ A is dtrministic iff A satisfis x C(A). x a & x a & pol(a) = + & pol(a ) = x {a, a } C(A). ( ) Proof. Only if : Suppos x C(A) with x a and x a whr pol(a) = + and pol(a ) =. Construct y = df {(1, b) b x} {(1, a)} {(2, b) b x}. Thn y C(CC A ) with y (2,a) and y (2,a ), by Proposition 6(ii). Assuming CC A is dtrministic, w obtain y {(2, a), (2, a )} C(CC A ), so y {(2, a), (2, a )} C(A A). This ntails x {a, a } C(A), as rquird to show ( ). If : Assum A satisfis ( ). It suffics to show for X fin CC A, with X downclosd, that Ng[X] Con CCA implis X Con CCA. Rcall Z Con CCA iff Z Con A A. Lt X fin CC A with X down-closd. Assum Ng[X] Con CCA. Obsrv (i) {c c X & pol(c) = } Ng[X] and (ii) {c c X & pol(c) = +} Ng[X] as by Proposition 6, X bing down-closd must contain c if it contains c with pol(c) = +.

10 10 Glynn Winskl Considr X 2 = df {a (2, a) X}. Thn X 2 is a finit down-closd subst of A. From (i), X 2 = df {a X 2 pol(a) = } Con A. From (ii), X + 2 = df {a X 2 pol(a) = +} Con A. W show ( ) implis X 2 Con A. Dfin z = df [X 2 ] and z + = df [X + 2 ]. Bing down-closurs of consistnt sts, z, z + C(A). W show z z + in C(A). First not z z + C(A). If a z z z + thn pol(a) = ; othrwis, if pol(a) = + thn a z + a wll as a z making a z z +, a contradiction. Similarly, if a z + z z + thn pol(a) = +. W can form covring chains z z + p1 x 1 p 2 p k x k = z and z z + n1 y 1 n 2 n l yl = z + whr ach p i is +v and ach n j is v. Consquntly, by rpatd us of ( ), w obtain x k y l C(A), i.. z + z C(A), as is illustratd blow. But X 2 z + z, so X 2 Con A. A similar argumnt shows X 1 = df {a A (1, a) X} Con A. It follows that X Con A A, so X Con CCA as rquird. n l y l p 1 p x1 y 2 l n l p x2 y 3 p l k x k y l n l n l n 2 n 2 n 2 y 1 p 1 x1 y 1 p 2 n 2 p x2 y 3 1 p k x k y 1 n 1 n 1 n 1 z z + p 1 x1 n 1 p 2 p 3 x2 p k xk Via th nxt lmma, whn gams satisfy ( ) w can simplify th condition for a stratgy to b dtrministic. Lmma 3. Lt σ S A b a stratgy. Suppos xy s & xy s & pol S (s) =. Thn, σy σy in C(A) y y in C(S). Proof. Assum σy σy in C(A), so σy σy σ(s) σy in C(A). As σ(s) is v, by rcptivity, thr is a uniqu s S, ncssarily v, such that σ(s ) = σ(s) and y s x {s, s } in C(S). In particular, x {s, s } C(S). By -innocnc, w cannot hav s s, so x {s } C(S). But now x s and x s with σ(s) = σ(s ) and both s, s v and hnc s = s by th uniqunss part of rcptivity. W conclud that x {s, s} C(S) so y y.

11 Dtrministic Concurrnt Stratgis 11 Corollary 1. Assum A satisfis ( ) of Lmma 2. A stratgy σ S A is dtrministic iff for all +v vnts s, s S and configurations x C(S), x s & x s x {s, s } C(S). Proof. Only if : clar. If : Lt x s and x s whr pol S (s) = +. For S to b dtrministic w rquir x {s, s } C(S). Th abov assumption nsurs this whn pol S (s ) = +. Othrwis pol S (s ) = with σx σ(s) and σx σ(s ). As A satisfis ( ), σx σ(s), σ(s ) C(A). Now by Lmma 3, x {s, s } C(S). Lmma 4. Th composition τ σ of dtrministic stratgis σ and τ is dtrministic. Proof. Lt σ S A B and τ T B C b dtrministic stratgis. Th composition T S is constructd as Pr(C(T ) C(S)) V, a synchronizd composition of vnt structurs S and T projctd to visibl vnts V whr max() has th form (s, ) or (, t). W first not a fact about th ffct of intrnal, or invisibl, vnts not in V on configurations of C(T ) C(S). If within C(T ) C(S), thn ithr within C(S), or z (s,t) w & z (s,t ) w & w w (1) π 1 z s π 1 w & π 1 z s π 1 w & π 1 w π 1 w, (2) π 2 z t π 2 w & π 2 z t π 2 w & π 2 w π 2 w, (3) within C(T ). Assum (1). If t = t thn σ(s) = τ(t) = τ(t ) = σ(s ) and w obtain (2) as σ is a map of vnt structurs. Similarly if s = s thn (3). Supposing s /= s and t /= t thn if both (2) and (3) faild w could construct a configuration z = df z {(s, t), (s, t)} of C(T ) C(S), contradicting (1); it is asy to chck that z is a configuration of th product C(S) C(T ) and its vnts ar clarly within th rstriction usd in dfining th synchronizd composition. W now show th impossibility of (2) and (3), and so (1). Assum (2) (cas (3) is similar). On of s or s bing +v would contradict S bing dtrministic. Suppos othrwis, that both s and s ar v. Thn, bcaus σ is a stratgy, by Lmma 3, w hav σ 2 π 1 w σ 2 π 1 w in C(B). Also, thn both t and t ar +v nsuring π 2 w π 2 w in C(T ), as T is dtrministic. This ntails τ 1 π 2 w τ 1 π 2 w

12 12 Glynn Winskl in C(B ). But σ 2 π 1 w and τ 1 π 2 w, rspctivly σ 2 π 1 w and τ 1 π 2 w, ar th sam configurations on th common vnt structur undrlying B and B, of which w hav obtaind contradictory statmnts of compatibility. As (1) is impossibl, it follows that within C(T ) C(S). z (s,t) w & z (s,t ) w w w (4) Finally, w can show that τ σ is dtrministic. Suppos xy p and xy p in C(T S) with pol(p) = +. Thn, xz 1 2 k 1 zk = y and x 1 z 1 l z l = y in C(T ) C(S), whr k = max(p) and l = max(p ), and th vnts i and j othrwis hav th form i = (s i, t i ), whn 1 i < k, and j = (s j, t j ), whn 1 j < l. By rpatd us of (4) w obtain z k 1 z l 1. (Th argumnt is lik that nding th proof of Lmma 2, though with th minor diffrnc that now w may hav i = j.) W obtain w = df z k 1 z l 1 C(T ) C(S) with w k and w l and pol( k ) = +. Now, w { k, l } C(T ) C(S) providd w { k, l } C(S) C(T ). Inspct th dfinition of configurations of th product of stabl familis in Sction 3.1. If k and l hav th form (s, ) and (s, ) rspctivly, thn dtrminacy of S nsurs that th projction π 1 w {s, s } C(S) whnc w { k, l } mts th conditions ndd to b in C(S) C(T ). Similarly, w { k, l } C(S) C(T ) if k and l hav th form (, t) and (, t ). Othrwis on of k and l has th form (s, ) and th othr (, t). In this cas again an inspction of th dfinition of configurations of th product yilds w { k, l } C(S) C(T ). Forming th st of prims of w { k, l } in V w obtain x {p, p } C(T S). This stablishs that T S is dtrministic. W thus obtain a sub-bicatgory DGams of Gams; its objcts satisfy ( ) of Lmma 2 and its maps ar dtrministic stratgis A catgory of dtrministic stratgis In fact, DGams is quivalnt to an ordr-nrichd catgory via th following lmma. It says dtrministic stratgis in a gam A ar ssntially crtain subfamilis of configurations C(A), for which w giv a charactrization. Lmma 5. A dtrministic stratgy is injctiv on configurations (i.., is mono as a map of vnt structurs). Proof. Lt σ S A b a dtrministic stratgy. W show x z y & σy σx y x, for x, y, z C(S), by induction on x z.

13 Dtrministic Concurrnt Stratgis 13 Suppos x zy and σy σx. Thr ar x 1 and vnt 1 S such that zx 1 1 x. If σ( 1 ) = σ() thn 1, hav th sam polarity; if v 1 =, by rcptivity; if +v 1 =, by dtrminacy with th local injctivity of σ. Eithr way y x. Suppos σ( 1 ) /= σ(). Cas pol( 1 ) = pol() = +: As σ is dtrministic, 1 and ar concurrnt giving x 1 y { 1 }. By induction w obtain y { 1 } x. Cas pol() = : As both σz σ(1) σx and σzσy σ() and σy σx, w hav σ( 1 ) σ() and σ() concurrnt in A. In particular, σx 1 whr pol(σ()) =. By rcptivity, thr is such that x 1 with σ( ) = σ(). Bcaus of innocnc, w cannot hav 1 in S, so 1 and ar concurrnt. Consquntly z. But thn = by th uniqunss part of rcptivity. W dduc that 1 and ar concurrnt yilding x 1 y { 1 }, and by induction y { 1 } x. Th cas whr pol( 1 ) = is vry similar. Anothr, simplr induction on y z now yilds x z y & σy σx y x, for x, y, z C(S), from which th rsult follows. A dtrministic stratgy σ S A dtrmins, as th imag of th configurations C(S), a subfamily F = df σc(s) of configurations of C(A), satisfying: rachability: F and if x F thr is a covring chain x a1 a 2 a k 1 xk = x within F ; dtrminacy: If x a and x a in F with pol A (a) = +, thn x {a, a } F ; rcptivity: If x F and x a in C(A) and pol A (a) =, thn x {a} F ; +-innocnc: If xx a a 1 & pol A (a) = + in F and x a in C(A), thn x a in F (hr rcptivity implis -innocnc); cub: In F, x 1 y 1 implis x 1 y 1 a b a a b b x y z x w a a z b a b b x 2 y 2 Thorm 2. A subfamily F C(A) satisfis th axioms abov iff thr is a dtrministic stratgy σ S A s.t. F = σc(s), th imag of C(S) undr σ. Proof. (Sktch) It is routin to chck that F, th imag σc(s) of a dtrministic stratgy, satisfis th axioms. Convrsly, suppos a subfamily F C(A) satisfis th axioms. W show F is a stabl family. First not that from th axioms of dtrminacy and rcptivity w can dduc: if x a and x a in F with x {a, a } C(A), thn x {a, a } F. x 2 y 2

14 14 Glynn Winskl By rpatd us of this proprty, using thir rachability, if x, y F and x y in C(A) thn x y F ; th proof also yilds a covring chain from x to x y and from y to x y. (In particular, if x y in F, thn thr is a covring chain from x to y a fact w shall us shortly.) Thus, if x y in F thn x y F. As also F, w obtain Compltnss, rquird of a stabl family. Coincidnc-frnss is a dirct consqunc of rachability. Rpatd us of th cub axiom yilds Cub: In F, x 1 y 1 implis x 1 x 2 x 1 x 2 y 1 y 2 x 1 x 2. y 2 x 2 W us Cub to show stability. Assum v w in F. Lt z F b maximal s.t. z v, w. W show z = v w. Suppos not. Thn, forming covring chains in F, zv c1 c 2 c k 1 vk = v and zw d1 d 2 d l 1 wl = w, thr ar c i and d j s.t. c i = d j, whr w may assum c i is th arlist vnt to b rpatd as som d j. Writ = df c i = d j. Now, v i 1 w j 1 = z. Also, bing boundd abov v i 1 w j 1 F and v i w j F. W hav an instanc of Cub: tak x 1 = v i 1, x 2 = w j 1, y 1 = v i and y 2 = w j. Hnc z and z {} x, y contradicting th maximality of z. Thrfor z = v w, as rquird for stability. Now w can form an vnt structur S = df Pr(F ). Th inclusion F C(A) inducs a total map σ S A for which F = σc(s). Not that -innocnc (viz. if xx a a 1 & pol A (a ) = in F and x a in C(A), thn x a in F ) is a dirct consqunc of rcptivity. That S is dtrministic follows from dtrminacy, that σ is a stratgy from th axioms of rcptivity and +-innocnc. W can thus idntify dtrministic stratgis from A to B with subfamilis of C(A B) satisfying th axioms abov. Through this idntification w obtain an ordr-nrichd catgory of dtrministic stratgis (prsntd as subfamilis) quivalnt to DGams. 8 Rlatd work Ingnuous stratgis [4] Via Thorm 2, dtrministic concurrnt stratgis coincid with th rcptiv ingnuous stratgis of Mlliès and Mimram. Stabl spans and stabl functions [6, 7] Th sub-bicatgory of Gams whr th vnts of gams ar purly +v is quivalnt to th bicatgory of stabl spans. In this cas, stratgis corrspond to stabl spans: A σ 1 S σ 2 B S + σ 1 σ + 2 A B,

15 Dtrministic Concurrnt Stratgis 15 whr S + is th projction of S to its +v vnts; σ + 2 is th rstriction of σ 2 to S +, ncssarily a rigid map by innocnc; σ 2 is a dmand map taking x C(S + ) to σ 1 (x) = σ 1 [x] ; hr [x] is th down-closur of x in S. If w furthr rstrict stratgis to b dtrministic (and, strictly, vnt structurs to b countabl) w obtain a bicatgory quivalnt to Brry s di-domains and stabl functions. Closur oprators [8, 4] A dtrministic stratgy σ S A dtrmins a closur oprator ϕ on possibly infinit configurations C(S) : for x C(S), ϕ(x) = x {s S pol(s) = + & Ng[{s}] x}. Clarly ϕ prsrvs intrsctions of configurations and is continuous. Th closur oprator ϕ on C(S) inducs a partial closur oprator ϕ p on C(A). This in turn dtrmins a closur oprator ϕ p on C(A), whr configurations ar xtndd with a top, cf. [8]: tak y C(A) to th last, fixd point of ϕ p abov y, if such xists, and othrwis. Simpl gams [9, 10] Th subcatgory of simpl gams ariss whn w rstrict DGams to objcts and dtrministic stratgis whos configurations tak th form of a tr and polarity altrnats on th vnts of branchs which always bgin with Opponnt. Extnsions Gams, such as thos of [11, 12], allowing copying ar bing systmatizd through th us of monads and comonads [10], work now fasibl on vnt structurs with symmtry [7]. 8.1 Winning stratgis A gam with losing configurations compriss G = (A, L) whr A is an vnt structur with polarity and L C(A) consists of th losing configurations for Playr. A stratgy in G is a stratgy in A. It is rgardd as winning if it always prscribs Playr movs to avoid nding up in a losing configuration, no mattr what th activity or inactivity of Opponnt. Formally, a stratgy σ S A in G is a winning (for Playr) if σx L for all +-maximal configurations x C(S) a configuration x is +-maximal if whnvr x s thn th vnt s has v polarity. In th spcial cas of a dtrministic stratgy σ in G it is winning iff σϕ(x) L for all x C(S), whr ϕ is th closur oprator ϕ C(S) C(S) dtrmind by σ or, quivalntly, th imags undr σ of fixd points of ϕ li outsid L s arlir in this sction for th dfinition of th closur oprator ϕ. Not all gams with losing configurations hav winning stratgis. Considr th gam A consisting of on playr mov and on opponnt mov inconsistnt with ach othr, with losing configurations L = {, { }}. This gam has no winning stratgy; any stratgy σ S A, bing rcptiv, will hav an vnt s S with σ(s) =, and so th losing {s} as a +-maximal configuration. Thr is an obvious dual of a gam with losing configurations: (A, L) is ssntially (A, L c ) whr L c = C(A) L strictly, w should tak th copy of L c in C(A ). With som xprimntation, on rasonabl way to form th paralll composition of two gams with losing configurations is to tak (A, L A ) (B, L B ) = (A B, L A L B )

16 16 Glynn Winskl whr X Y = {{1} x {2} y x X & y Y } whn X and Y ar substs of configurations; in G H to los is to los in both gams G and H. 3 W can again follow Joyal and dfin stratgis btwn gams now with losing configurations: a (winning) stratgy from G to H is a (winning) stratgy in G H. W compos stratgis as bfor. Copy-cat stratgis ar winning. Th composition of winning stratgis is winning. (Th proof rlis on th obsrvation that a configuration x in (C(S) C(T ) R), usd in Proposition 4 to obtain th composition of stratgis T and S, is +-maximal iff its projctions π 1 x and π 2 x ar +-maximal configurations of S and T rspctivly.) Dfining G H = df (G H ) w obtain a gam whr to los is to los in ithr gam G or H. Acknowldgmnt Thanks to Nathan Bowlr, Pirr Clairambault, Pirr-Louis Curin, Marclo Fior, Julian Gutirrz, Jonathan Hayman, Martin Hyland, Paul-André Mlliès, Samul Mimram, Gordon Plotkin, Silvain Ridau and Sam Staton for hlpful rmarks. Th support of a Royal Socity Lvrhulm Trust Snior Fllowship and Advancd Grant ECSYM of th ERC ar acknowldgd with gratitud. Rfrncs 1. Ridau, S., Winskl, G.: Concurrnt stratgis. In: LICS 2011, IEEE Computr Socity (2011) 2. Winskl, G.: Evnt structur smantics for CCS and rlatd languags. In: ICALP 82. Volum 140 of LNCS., Springr (1982) 3. Mlliès, P.A.: Asynchronous gams 2: Th tru concurrncy of innocnc. Thor. Comput. Sci. 358(2-3): (2006) 4. Mlliès, P.A., Mimram, S.: Asynchronous gams : innocnc without altrnation. In: CONCUR 07. Volum 4703 of LNCS., Springr (2007) 5. Joyal, A.: Rmarqus sur la théori ds jux à dux prsonns. Gaztt ds scincs mathématiqus du Québc, 1(4) (1997) 6. Saundrs-Evans, L., Winskl, G.: Evnt structur spans for nondtrministic dataflow. Elctr. Nots Thor. Comput. Sci. 175(3): (2007) 7. Winskl, G.: Evnt structurs with symmtry. Elctr. Nots Thor. Comput. Sci. 172: (2007) 8. Abramsky, S., Mlliès, P.A.: Concurrnt gams and full compltnss. In: LICS 99, IEEE Computr Socity (1999) 9. Hyland, M.: Gam smantics. In Pitts, A., Dybjr, P., ds.: Smantics and Logics of Computation. Publications of th Nwton Institut (1997) 10. Harmr, R., Hyland, M., Mlliès, P.A.: Catgorical combinatorics for innocnt stratgis. In: LICS 07, IEEE Computr Socity (2007) 11. Hyland, J.M.E., Ong, C.H.L.: On full abstraction for PCF: I, II, and III. Inf. Comput. 163(2): (2000) 12. Abramsky, S., Jagadsan, R., Malacaria, P.: Full abstraction for PCF. Inf. Comput. 163(2): (2000) 3 I m gratful to Nathan Bowlr, Pirr Clairambault and Julian Gutirrz for guidanc in th dfinition of paralll composition of gams with losing configurations.