aa r X i v : . [ c s . L O ] J un Learning to Count up to Symmetry
Pierre ClairambaultUniv Lyon, EnsL, UCBL, CNRS, LIP, F-69342, LYON Cedex 07, France
Abstract
In this paper we develop the theory of how to count , in thin concurrent games, theconfigurations of a strategy witnessing that it reaches a certain configuration of thegame. This plays a central role in many recent developments in concurrent games,whenever one aims to relate concurrent strategies with weighted relational models.The difficulty, of course, is symmetry: in the presence of symmetry many config-urations of the strategy are, morally, different instances of the same , only differingon the inessential choice of copy indices. How do we know which ones to count?The purpose of the paper is to clarify that, uncovering many strange phenomena andfascinating pathological examples along the way.To illustrate the results, we show that a collapse operation to a simple weightedrelational model simply counting witnesses is preserved under composition, providedthe strategies involved do not deadlock.
Thin concurrent games [5] are a complex but powerful setting for truly concurrent gamesemantics; one of the latest iterations of a long line of work [1, 11, 13] on game semanticsquestioning the premise that a play should be a total chronological ordering. They arevery expressive, able to express various languages both pure [4] and stateful [5]; includingwith various quantitative aspects [3, 7]. One strength of concurrent games in general isthe clean link they offer with relational-like semantics: a strategy may (slightly naively)be seen as a collection of points of the web (in the sense of relational semantics) enrichedwith causal information. This enables a clean connection with the relational model, whichserved as basis e.g. for Melli`es’ fully complete model of linear logic [10] (see also [6]).Now, relational semantics as well can be enriched with quantitative information; this isthe basis for probabilistic coherence spaces [8]. Probabilistic coherence spaces are obtainedvia a biorthogonality construction on top of the relational model weighted by elements of R + , the completion of non-negative reals R + with a point at infinity. Instead of merelyrelations, morphisms from set A to set B are then matrices( α a,b ) ( a,b ) ∈ A × B ∈ R + A × B β ◦ α ) a,c = X b ∈ B α a,b · β b,c . (1)Beyond real scalars, more generally one can construct a weighted relational model parametrized by certain semirings [9]. Adding typing information one goes beyond semir-ings, for instance the adequate model for the quantum λ -calculus of [12] uses weights fromthe category of finite dimensional Hilbert spaces and completely positive maps.Above, we mentioned a collapse from concurrent games to the relational model. Doesit hold with quantitative information? Such results appear in the literature [3, 7] – thoughwe shall see in this paper that the definition of this collapse in [3] is not quite right. Thisseemingly simple question holds some surprises. This is the question that this paper solves;detailing the basis for part of [7], and identifying and correcting the mistake in [3].As a matter of fact, the difficulty is not in handling the weights , but in listing the rightwitnesses : if (1) originates in a bijection between witnesses, then provided this bijectionpreserves the weights (and it will be generated in such a way that it does), it follows thatadding weights is relatively painless. On the other hand, coming up with the right notion ofwitnesses is really hard. Indeed, in the presence of replication of resources, configurationsin strategies are countably duplicated, so it is meaningless to sum over all of those asone does without symmetry. What are, then, the right witnesses? Symmetry classes ofconfigurations? Something else? In this paper we give the answer, and illustrate it with aproof of a formula like (1) for a simple weighted relational model simply counting witnesses.We shall see that the appealingly simple idea of [3] to use symmetry classes of config-urations as witnesses is, in general, wrong. We give a more refined notion of witnesses,taking advantage of the split of the symmetry into positive and negative reindexings offeredby thin concurrent games [5]. This lets us solve the problem, but with the cost of adding anew condition to thin concurrent games called representability , which states the existence,for every symmetry class, of a canonical representative on which the symmetry decomposesneatly into a positive and negative parts. Outline.
The structure of the paper is as follows. In Section 2 we fix the notations forthin concurrent games used in this paper and recall a few notions. In Section 3 we give atechnical explanation of the problem and its difficulties. In Section 4 we introduce the newnotions of canonicity and representability . In Section 5 we give the central contribution ofthe paper, the proof of (1). Finally, in Section 6 we give a few ending remarks.
In this paper, we assume some familiarity with concurrent games, and more precisely with thin concurrent games [5]. Let us fix a few conventions for notations and terminology.2y strategy we will always mean ∼ -strategy in the sense of [5]. We will sometimes referto pre- ∼ -strategies , which must be understood as in [5]. If σ : S → A ⊥ k B is a strategyfrom A to B , we write σ : A S → B . We often use x S , y S , . . . to range over configurations of S , with S as a superscript. If x S ∈ C ( S ), we take the convention that σx S = x SA k x SB , in the paper we will use x SA ∈ C ( A ) and x SB ∈ C ( B ) without further introduction.If A is a tcg, we write ∼ = A for its symmetry, and θ : x ∼ = A y if the bijection θ : x ≃ y is in ∼ = A – in which case we say that θ is a symmetry . For x, y ∈ C ( A ), we write x ∼ = A y for theinduced equivalence relation. We use similar notations for the positive and negative sub-symmetries, with ∼ = + A for the positive and ∼ = − A for the negative. We use for symmetries onstrategies similar notations as for configurations. For σ : A S → B , we often tag symmetriesin S with S , as in ϕ S : x S ∼ = S y S . Then, we write ϕ SA : x SA ∼ = A y SA and ϕ SB : x SB ∼ = B y SB .In diagrams, dotted lines signify immediate causal links in the game, whereas _ meansimmediate causality in the strategy. If the direction of causal links is unspecified ( e.g. withdotted lines with no arrow head), then it must be read from top to bottom. Consider two strategies σ : A S → B and τ : B T → C .Recall that their interaction τ ⊛ σ : T ⊛ S → A k B k C has set C ( T ⊛ S ) isomorphic to pairs ( x S , x T ) ∈ C ( S ) × C ( T ) such that x SB = x TB = x B ,and which are causally compatible , in the sense that the induced bijection x S k x TC ≃ x SA k x B k x TC ≃ x SA k x T is secured [5]. We write x T ⊛ x S ∈ C ( T ⊛ S ) for the corresponding configuration; then:( τ ⊛ σ )( x T ⊛ x S ) = x SA k x B k x TC . The composition τ ⊙ σ : A T ⊙ S → C is obtained from the interaction through a hiding operation [5]. We recall: Proposition 1.
The set C ( T ⊙ S ) is isomorphic to the set of pairs ( x S , x T ) ∈ C ( S ) × C ( T ) such that x SB = x TB = x B , which are causally compatible and minimal , in the sense that if y S ⊆ x S and y T ⊆ x T are matching and causally compatible, and x SA k x TC = y SA k y TC , then x S = y S and x T = y T . If x S and x T are matching, causally compatible, and minimal,we write x T ⊙ x S ∈ C ( T ⊙ S ) for the corresponding configuration. We then have ( τ ⊙ σ )( x T ⊙ x S ) = x SA k x TC . roof. Direct from the definition. If a pair ( x S , x T ) is matching and causally compatible,then it is minimal iff x T ⊛ x S has all its maximal events visible ( i.e. in A or C ); and thoseare in one-to-one correspondence with configurations of T ⊙ S .Interaction behaves like a cartesian product (restricted to the matching causally com-patible configurations), while composition has this additional minimality assumption. Wewish to get rid of minimality, since we wish to link to weighted relational models, where(intuitively) a witness of the composition is a pair of witnesses. This can be achieved: Definition 2.
Let σ : S → A be a strategy.A configuration x ∈ C ( S ) is +-covered iff all its maximal events have positive polarity.We write C + ( S ) for the set of + -covered configurations of σ . By extension, we say that x T ⊛ x S ∈ C ( T ⊛ S ) is +-covered iff its maximal events arepositive and write x T ⊛ x S ∈ C + ( T ⊛ S ). This notion is useful, because we have: Lemma 3.
Consider σ : A S → B and τ : B T → C two strategies. Then, there is a bijection φ : C + ( T ⊛ S ) ≃ C + ( T ⊙ S ) x T ⊛ x S x T ⊙ x S such that if ( τ ⊛ σ )( x T ⊛ x S ) = x A k x B k x C , then ( τ ⊙ σ )( φ ( x T ⊛ x S )) = x A k x C .Proof. If x T ⊛ x S ∈ C + ( T ⊛ S ), then the pair ( x S , x T ) is automatically minimal: if not,then one can remove an event in B . But it must be negative for either σ or τ , contradiction.So we may simply set φ ( x T ⊛ x S ) = x T ⊙ x S ∈ C + ( T ⊙ S ).We have one last ingredient to introduce. One crucial difference between strategy com-position and composition in weighted relational models, is that strategies may deadlock.This question is fairly well-explored; in particular in settings where we have performedsuch a collapse [3, 7, 6], we have done so under the assumption that strategies satisfied acondition called visibility , which prevents deadlocks [2]. Describing visibility is beyond thescope of this paper, but many of the results given here will be under the assumption thatcertain strategies do not deadlock. Accordingly, we define: Definition 4.
Strategies σ : A S → B and τ : B T → C do not deadlock iff for all x S ∈ C ( S ) , x T ∈ C ( T ) and θ B : x SB ∼ = B x TB , the composite bijection x S k x TC σ k x TC ≃ x SA k x SB k x TC x SA k θ B k x TC ≃ x SA k x TB k x TC x SA k τ − ≃ x SA k x T is secured. This is, in particular, always the case when σ and τ are visible. If σ and τ do notdeadlock then we may forget the causal compatibility condition in their interaction: con-figurations of the interaction correspond to arbitrary matching pairs.We do not assume that all strategies considered do not deadlock. Throughout thepaper, we make it explicit when we consider this hypothesis.4 Towards a Quantitative Collapse
A game A has a natural associated notion of position , given by the set of configurations C ( A ). Configurations inform the relationship with relational-like semantics: if A is a gamearising from a type in a linear type system , then the web (a set) interpreting this type inrelational semantics may be identified with a subset of C ( A ) . Likewise, a strategy σ : A S → B induces a relation ∫ σ = { ( x A , x B ) | ∃ x S ∈ C ( S ) , σx S = x A k x B } ∈ Rel ( C ( A ) , C ( B )).With this definition, for any σ : A S → B and τ : B T → C we automatically have that ∫ ( τ ⊙ σ ) ⊆ ( ∫ τ ) ◦ ( ∫ σ )and the other inclusion holds if σ and τ do not deadlock.This picture above is of course much simplified thanks to linearity. Without the linearityassumption, the games considered need to carry a symmetry . If A arises from a type, thenthe corresponding web is no longer (a subset of) C ( A ), but (a subset of) C ∼ = ( A ), the set of equivalence classes of configurations under symmetry . In particular, we have Lemma 5.
Consider N a negative tcg. Then, C ∼ = (! N ) ∼ = M f ( C ∼ = ( N )) . where M f ( X ) is the set of finite multisets of elements of set X .Proof. Straightforward.We use x , y , . . . as metavariables ranging over symmetry classes.Above, ! stands for the AJM-style exponential described in Section 3.3.4 in [5]. Like-wise, the reader familiar with relational semantics will recognize in M f ( X ) the familiarexponential modality. This traces the path to extend the links between game and relationalsemantics beyond the linear case: simply correct the definition of ∫ σ by setting: ∫ σ = { ( x A , x B ) ∈ C ∼ = ( A ) × C ∼ = ( B ) | ∃ x S ∈ C ( S ) , x SA ∈ x A & x SB ∈ x B } , where σx S = x SA k x SB , a naming convention that we shall adopt. If x S ∈ C ( S ) is such that x SA ∈ x A and x SB ∈ x B , we say that x S is a witness for ( x A , x B ) in ∫ σ .With this definition, it is immediate by definition of composition of strategies that weretain ∫ ( τ ⊙ σ ) ⊆ ∫ ( τ ) ◦ ∫ ( σ ) for any strategies σ : A S → B and τ : B T → C . Typically, in the presence of Question/Answer labeling, those are the complete configurations whereevery question is answered – but details do not matter for this paper. .2 Synchronization up to symmetry More interesting is the reverse inclusion. Of course, the deadlock issue mentioned abovestill applies. But something else is also going on: consider σ : A S → B and τ : B T → C , and( x A , x B ) ∈ ∫ σ ( x B , x C ) ∈ ∫ τ . By definition, this means that there are x S ∈ C ( S ) and x T ∈ C ( T ) such that x SA ∈ x A , x SB ∈ x B , x TB ∈ x B , x TC ∈ x C . In particular, since we have x SB ∈ x B and x TB ∈ x B it follows that there is a (non-unique) θ : x SB ∼ = B x TB , a symmetry on B . So the witnesses x S ∈ C ( S ) and x T ∈ C ( T ) might not quite reachthe same configuration of the game: typically, they might involve completely distinct copyindices, and θ carries a reindexing from one to the other. Independently of the deadlocks,if we wish to provide a witness y ∈ C ( T ⊙ S ) for ( x A , x C ) in τ ⊙ σ , we must in particularfind some y S ∈ C ( S ) and y T ∈ C ( T ) such that y SA ∈ x A , y SB = y TB , y TC ∈ x C , matching on B on the nose . So starting from x S ∈ C ( S ) and x T ∈ C ( T ), we must reindex them until they match on B on the nose. Of course, this issue already arises in theprocess of constructing a game semantics based on copy incides, to show that equivalenceof (uniform) strategies up to the choice of copy indices is stable under composition.In thin concurrent games, the main tool to deal with it is the weak bipullback property : Lemma 6 (Weak bipullback property) . Let σ : S → A and τ : T → A ⊥ be pre- ∼ -strategies.Let x S ∈ C ( S ) and x T ∈ C ( T ) and θ : σx S ∼ = A τ x T , such that the composite bijection x S σ ≃ σx S θ ∼ = A τ x T τ ≃ x T is secured. Then, there are y S ∈ C ( S ) and y T ∈ C ( T ) causally compatible, θ S : x S ∼ = S y S and θ T : y T ∼ = T x T , such that τ θ T ◦ σθ S = θ . Moreover, y S , y T are unique up to symmetry. This appears as Lemma 3.23 in [5]. The intuition is that σ and τ play against each other,each replacing Player copy indices with one they are prepared to play. By ∼ -receptivity, τ must be receptive to a change in copy indices made by Player, and reciprocally; so y S and y T may be constructed by induction on the causal structure induced by the securednessassumption. If σ : A S → B and τ : B T → C , and we have x S ∈ C ( S ) and x T ∈ C ( T ) with θ : x SB ∼ = B x TB ,
6e may apply the lemma above for σ k C ⊥ → A ⊥ k B k C ⊥ and A k τ : A k T → A k B ⊥ k C . Provided some other argument ensures the securedness assumption, then we obtain y S k y TC ∈ C ( S k C ) y SA k y T ∈ C ( A k T )matching on B ; and so we have found an interaction y T ⊛ y S ∈ C ( T ⊛ S )with ( τ ⊛ σ )( y T ⊛ y S ) = y SA k y B k y TC , satisfying y SA ∈ x A and y TC ∈ x C thus providingthrough hiding the desired witness for ( x A , x C ) ∈ ∫ ( τ ⊙ σ ). But the above is purely qualitative: if σ : A S → B then the collapse above lets us definewhich pairs ( x A , x B ) are “inhabited” by σ . This is sufficient in order to link game semanticswith relational semantics. But this is not sufficient if we want to reproduce this feat in thepresence of quantitative information, such as probabilities or quantum valuations.For the purposes of this paper, let us say that we are now interested not in the mereexistence of a witness x S ∈ C ( S ) such that x SA ∈ x A and x SB ∈ x B , but in counting suchwitnesses. For reasons explained in Section 2.2, from now on we consider witnesses for( x A , x B ) not merely those configurations x S ∈ C ( S ) such that x SA ∈ x A and x SB ∈ x B ; butthose that are additionally +-covered, i.e. we have x S ∈ C + ( S ).From a strategy σ : A S → B , we want a N -weighted relation, i.e. a function ∫ σ : C ∼ = ( A ) × C ∼ = ( B ) → N , where N = N ∪ { + ∞} , counting the number of distinct witnesses for ( x A , x B ). In that case,for x A ∈ C ∼ = ( A ) and x B ∈ C ∼ = ( B ), write ( ∫ σ ) x A , x B ∈ N for the corresponding coefficient.In the spirit of weighted relations [9], we then want to prove that for all σ : A S → B and τ : B T → C that do not deadlock, we have that for all x A ∈ C ∼ = ( A ) and x C ∈ C ∼ = ( C ),( ∫ ( τ ⊙ σ )) x A , x C = X x B ∈ C ∼ = ( B ) ( ∫ σ ) x A , x B × ( ∫ τ ) x B , x C . (2)The convergence of the sum on the right hand side is ensured by the fact that weconsider the completed natural numbers N ∪ { + ∞} as in the weighted relational model.How might we, from σ : A S → B , extract the weighted relation ∫ σ ? Intuitively, we need( ∫ σ ) x A , x B = | wit σ ( x A , x B ) | where wit σ ( x A , x B ) captures the witnesses in σ for symmetry classes x A ∈ C ∼ = ( A ) and x B ∈ C ∼ = ( B ), and where | X | simply computes the cardinal, taken to be + ∞ for X infinite.7ituations where strategies carry additional weights, say probabilities or quantum valua-tions, would be dealt with similarly. In any case, the first obstacle to overcome is then togive a satisfactory definition of wit σ ( x A , x B ).Of course counting all x S ∈ C + ( S ) such that x SA ∈ x A and x SB ∈ x B makes no sense:there are almost always infinitely many of them since e.g. the construction ! N introducescountably many copy indices. The definition of witnesses must take symmetry into account. Symmetry classes.
The obvious candidate for witnesses, chosen in [3], is: wit σ ( x A , x B ) = { x S ∈ C + ∼ = ( S ) | ∀ x S ∈ x S , x SA ∈ x A & x SB ∈ x B } , i.e. the symmetry classes of +-covered configurations mapping to x A k x B . This convinc-ingly simple definition in fact hides a major subtlety. Indeed, (2) hints at a bijection wit τ ⊙ σ ( x A , x C ) ∼ = X x B ∈ C ∼ = ( B ) wit σ ( x A , x B ) × wit τ ( x B , x C ) . This seems straightforward. Firstly, if z ∈ wit τ ⊙ σ ( x A , x C ), then any choice z ∈ z is z = z T ⊙ z S ∈ C ( T ⊙ S ) and the symmetry classes of its projections yield z S ∈ wit σ ( x A , x B ) , z T ∈ wit τ ( x B , x C ) , for some x B ∈ C ∼ = ( B ). These data are easily shown to be invariant under the choice of z .Reciprocally, if x S ∈ wit σ ( x A , x B ) and x T ∈ wit τ ( x B , x C ), we may take arbitrary x S ∈ x S , x T ∈ x T , and via Lemma 6 find symmetric y S ∈ x S and y T ∈ x T agreeing on B on thenose. We may then form y T ⊙ y S ∈ C + ( T ⊙ S ) and take its symmetry class in wit τ ⊙ σ ( x A , x C ).But one should not skip the details : we must show that this construction only dependson the symmetry classes x S and x T , not on the specific choices x S ∈ x S and x T ∈ x T andthe symmetry θ B : x SB ∼ = B x TB used to link them. But surely, that must be true, right?Well, about that. . . It certainly was a surprise to us that the symmetry class obtainedthrough synchronization does depend on the symmetry θ B . Example . Consider the following games. Firstly, A = ∅ is the empty game. Secondly, C = (! ⊖ ) ⊥ which has countably many Player moves written X i for all i ∈ N , all symmetric– we adopt here a convention followed throughout the paper: copy indices appear in grey,to distinguish them from other indices.Thirdly, consider the game B = ! HO ( ⊖ _ ⊕ ), where ! HO is the “HO exponential”defined in Definition 2.24 with symmetries in Definition 2.27 in [5] (see also Proposition3.3). This game has events, polarities and causal dependency those pictured in: ⊖ ⊖ . . . ⊖ i . . . ⊕ , ⊕ , ⊕ , . . . ⊕ ,j . . . ⊕ , ⊕ , ⊕ , . . . ⊕ ,j . . . ⊕ i, ⊕ i, ⊕ i, . . . ⊕ i,j . . . We were guilty of that in [3]. S → B ⊖ i ❀ x x(cid:3) ④④④ ✄ (cid:27) (cid:27)& ❈❈❈ ⊕ f ( i ) /o/o/o ⊕ g ( i ) Figure 1: σ : A S → B B T → C ⊕ ❴ (cid:12) (cid:12)(cid:18) ⊕ h ( i ) ❴ (cid:12) (cid:12)(cid:18) X k ( i,j ) ⊖ i ✹ tttt ⊖ j ✳ ♥♥♥♥♥♥♥ Figure 2: τ : B T → C with all finite sets consistent. Its symmetry comprises all order-isomorphisms betweenconfigurations. Its positive symmetry comprises all order-isomorphisms that preserve theinitial (negative) move. Its negative symmetry comprises all order-isomorphisms such that θ ( ⊕ i,j ) = ⊕ i ′ ,j for some i ′ ∈ N , i.e. they preserve the j component of the positive move.In practice, we will omit the first copy index for the event in the second row, which isredundant with the immediate causal antecedent of the event.We now introduce two strategies σ : A S → B and τ : B T → C that we wish to compose,represented on Figures 1 and 2 where the functions f, g, h , and k are assumed injective,and 0 is not in the codomain of h . Note that the representation is symbolic: the diagramsmust be understood by stating that every positive move has one copy for each instantiationof the metavariables i, j ∈ N , with dependencies as indicated in the diagram. These copiesare compatible with each other. Finally, the symmetries comprise order-isomorphisms thatdiffer only by the value of the metavariables i, j ∈ N . In particular, the two moves in theconflicting branches of σ are not symmetric (that would anyway contradict thinness ).First, we compute the composition τ ⊙ σ , and observe that it is: X k ( f (0) ,f ( h ( f (0)))) /o/o/o3s 2r 2r 2r 1q 1q 1q 0p 0p 0p /o /o /o .n .n .n -m -m -m ,l ,l ,l +k4t 4t 4t 3s 3s 3s 3s 3s 3s 2r 2r 2r 2r 2r 1q 1q 1q 1q 1q 0p 0p 0p 0p 0p 0p /o /o /o /o /o .n .n .n .n .n .n -m -m -m -m -m ,l ,l ,l ,l ,l +k +k +k +k +k +k *j *j *j X k ( f (0) ,g ( h ( f (0)))) /o/o/o3s 2r 2r 2r 1q 1q 1q 0p 0p 0p /o /o /o .n .n .n -m -m -m ,l ,l ,l +k X k ( g (0) ,f ( h ( g (0)))) /o/o/o X k ( g (0) ,g ( h ( g (0)))) There are four events, pairwise conflicting, reflecting the two non-deterministic choicesarising from the two calls to σ – one can read back which non-deterministic choice gaverise to which result from the copy indices, but that is another story. None of these eventsare symmetric: again, this would contradict thinness.Now, let us define two configurations x S ∈ C ( S ) and x T ∈ C ( T ) as x S = ⊖ ⊖ h ( f (0)) ⊕ f (0) ⊕ g ( h ( f (0))) x T = ⊕ ⊕ h ( f (0)) ⊖ f (0) ⊖ g ( h ( f (0))) k X k ( f (0) ,g ( h ( f (0)))) These two configurations match on B (and are causally compatible); and their compo-sition yields the configuration { X k ( f (0) ,g ( h ( f (0)))) } . We of course obtain the same result ifwe synchonize them through the trivial symmetry on their common interface:id : x TB ∼ = B x TB . x B = x SB = x TB , namely sw : x TB ∼ = B x TB , exhanging the two copies. Synchronizing x S and x T through sw via Lemma 6 instead gives: { X k ( g (0) ,f ( h ( g (0)))) } which is not symmetric to { X k ( f (0) ,g ( h ( f (0)))) } in T ⊙ S . Indeed, intuitively, in x S we only havethe information that there were two calls to σ , with distinct non-deterministic resolutions.We do not know, just by looking at x S , which one is the “first call” and which one isthe “second call”. The symmetry θ : x SB ∼ = B x TB “plugs” the two calls in x T to their twonon-deterministic resolutions in x S . With id the first call selected ⊕ f ( i ) and the second call ⊕ g ( i ) , and the other way around for sw ; leading to non-symmetric outcomes.Well, this is puzzling. If the obvious candidate for a bijection between witnesses z ∈ wit τ ⊙ σ ( x A , x C ) and pairs of witnesses z S ∈ wit σ ( x A , x B ) and z T ∈ wit τ ( x B , x C ) for some x B does not work, how can we hope to obtain (2)? This makes one wonder by what miraclethe weighted relational model works at all – what does it really count? Concrete witnesses.
To investigate this issue we introduce an alternative, more con-crete choice for witnesses. It is rooted in the following fact (Lemma 3.28 in [5]):
Lemma 8.
Let σ : S → A be a pre- ∼ -strategy on A , and let θ : x ∼ = S y such that σθ ∈ ∼ = + A .Then, x = y and θ = id x . For this, the condition thin plays a crucial role. Intuitively, thinness means that thestrategy has a canonical choice of copy indices for its moves, once Opponent fixes theirchoice of copy indices. Accordingly, the lemma above may be interpreted as saying thatprovided we remain in the positive symmetry ( i.e. we do not change Opponent’s copyindices), then the choice of the concrete configuration x ∈ C ( S ) is unique. This suggeststhat we might take wit σ ( x A , x B ) to range over concrete configurations of S matching with thegame up to positive symmetry – of course, for that we need reference concrete configurationsof the game rather than symmetry classes. So let us fix a choice, for any tcg A and anysymmetry class x A ∈ C ∼ = ( A ), of a concrete representative written x A ∈ x A .Our alternative definition of witnesses is, for σ : A S → B , x A ∈ C ∼ = ( A ) and x B ∈ C ∼ = ( B ): wit + σ ( x A , x B ) = { x S ∈ C + ( S ) | x SA ∼ = − A x A & x SB ∼ = + B x B } It will turn out (see Section 6) that (even assuming representability) these two notionsof witnesses are not equivalent: the weighted relational model counts not symmetry classes,but concrete witnesses up to positive symmetry. In the rest of this paper, we aim to prove(as mentioned above, modulo one additional condition on games) that wit + , unlike wit ,does the trick . This will be quite the ride, so switch off your phone, fasten your seat belt,as we must now embark on a journey into the darkest corners of thin concurrent games. An early sign that wit + is better behaved is that unlike wit , it does not depend on the choice of thesymmetry for σ – recall from Section A.1.2 in [5] that the symmetry is not unique. Canonical configurations and representable games
To motivate the development of this section, let us look at the definition just above: wit + σ ( x A , x B ) = { x S ∈ C + ( S ) | x SA ∼ = − A x A & x SB ∼ = + B x B } . This definition depends on a choice of a representative x A , once and for all, for everysymmetry class x A . Of course, the set of witnesses we obtain this way depends on thischoice: a different choice of representatives yields configurations of S where Opponent usesdifferent copy indices. But what we really need for this definition to be of any use, is thatthe cardinal of wit + σ ( x A , x B ) should not depend on the representatives x A , x B .Bad news: it does. Example . Remember the game B = ! HO ( ⊖ _ ⊕ ) of Example 7. Consider the strategy ⊕ ❴ (cid:12) (cid:12)(cid:18) ⊕ h ( i ) ❴ (cid:12) (cid:12)(cid:18) ⊖ i ✼ ✇✇✇ ⊖ j written σ : S → B ⊥ , which is τ of Example 7 without the last move.Now, imagine that we fix as representative for a symmetry class in B ⊥ the configuration: x B = ⊕ ⊕ ⊖ ⊖ . Let us consider the configurations of S matching x B up to positive symmetry. First, aconfiguration x ∈ C ( S ) matching our requirements has four moves, and each Player movehas exactly one successor. So it must have the following form, for some i, j ∈ N , ⊕ ❴ (cid:12) (cid:12)(cid:18) ⊕ h ( i ) ❴ (cid:12) (cid:12)(cid:18) ⊖ i ✼ ✇✇✇ ⊖ j (3)and finding the witnesses for x B boils down to figuring out all possible positive symmetries θ : ⊕ ⊕ h ( i ) ⊖ i ⊖ j ∼ = + B ⊥ ⊕ ⊕ ⊖ ⊖ The positive symmetry of B ⊥ is the negative symmetry of B : it lets us change theindices of minimal events, but the second component for positive events must be leftunchanged. We may freely associate the minimal events either as ⊕ ↔ ⊕ and ⊕ h ( i ) ↔ ⊕ ;or as ⊕ ↔ ⊕ and ⊕ h ( i ) ↔ ⊕ . But if we do the former, as the symmetry is positive it11orces i = 1 and j = 2. Likewise, if we do the latter, it forces i = 2 and j = 1. So overall,there are exactly two configurations of S matching x B up to positive symmetry: ⊕ ❴ (cid:12) (cid:12)(cid:18) ⊕ h (1) ❴ (cid:12) (cid:12)(cid:18) ⊖ ✻ ✈✈✈ ⊖ ⊕ ❴ (cid:12) (cid:12)(cid:18) ⊕ h (2) ❴ (cid:12) (cid:12)(cid:18) ⊖ ✻ ✈✈✈ ⊖ In particular, there are two witnesses for x B . This is confusing, because these twoconfigurations are symmetric in S , so we seem to be counting the same symmetry class of S twice – and we shall see indeed that this is a pathological example.In contrast, assume we pick as representative for x B the following configuration: x ′ B = ⊕ ⊕ ⊖ ⊖ . Now, there is only exactly one configuration of S matching x ′ B up to positive symmetry: ⊕ ❴ (cid:12) (cid:12)(cid:18) ⊕ h (1) ❴ (cid:12) (cid:12)(cid:18) ⊖ ✻ ✈✈✈ ⊖ Indeed, starting from (3), the positive symmetry forces i and j to be both 1; and weobtain the unique configuration above. So the choice of x B affects the number of witnesses.What is the moral of the story? This is subtle. Notice that while there is indeed exactly one configuration x ∈ C ( S ) matching x ′ B up to positive symmetry, there are still two symmetries θ : σx ∼ = + B ⊥ x ′ B , corresponding to {⊕ ↔ ⊕ , ⊕ h (1) ↔ ⊕ } and {⊕ ↔⊕ , ⊕ h (1) ↔ ⊕ } . So for x B we get two witnesses, and each has one positive symmetryto x B ; while for x ′ B we get one witness, with two positive symmetries. So the mismatchbetween the representatives is explained if one factors in the number of positive symmetries.To comment further: there are two positive endo-symmetries x ′ B ∼ = + B ⊥ x ′ B : the identity,and the swap between positive events. In contrast, in x B , swapping the positive events ⊕ ⊕ ⊖ ⊖ ∼ = + B ⊥ ⊕ ⊕ ⊖ ⊖ while preserving Opponent indices cannot be achieved via an endosymmetry, this requireschanging the configuration. To avoid such pathological cases, we must select x B such thatthe positive symmetry whose effect is, intuitively, merely to swap (the copy indices of) twoPlayer events, still has x B as codomain. We do not have a definition capturing exactlythis, as it is not clear how to formalize this idea of the minimal symmetry “swapping twoPlayer events”. However, for our purposes the following definition does the job. Definition 10.
Consider A a tcg, and x ∈ C ( A ) . e say that x is canonical iff any θ : x ∼ = A x factors uniquely as x θ − ∼ = − A x θ + ∼ = + A x , with in particular x in the middle. So endo-symmetries of canonical configurations decompose as endo-symmetries, positiveand negative. Of course we already know that all endosymmetries (like all symmetries)decompose as the composite of a positive and a negative symmetries (see Lemma 3.19of [5]). But there is a priori no reason why the decomposition should have the sameconfiguration in the middle. This is in fact not always the case: for instance, picking theproblematic configuration x B of the example above, we have the decomposition ⊕ ⊕ ⊖ ⊖ ∼ = − B ⊥ ⊕ ⊕ ⊖ ⊖ ∼ = + B ⊥ ⊕ ⊕ ⊖ ⊖ where rather than drawing the symmetries, we suggest them by considering that theypreserve the position of events in the diagrams. If we wish to avoid the problem mentionedabove, we must project strategies only on canonical representatives of symmetry classes.But for that, we need to be sure that such canonical representatives always exist.Of course, there is no free lunch: in the full generality of tcgs, that is not the case . Example . Consider the tcg A , with events, polarities, and causality and follows: ⊖ ⊖ ⊕ ⊕ Its symmetry comprises all order-isomorphisms between configurations. The negativesymmetry has all order-isomorphisms included in one of the two maximal bijections ⊖ ⊖ ⊕ ⊕ ∼ = − A ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ∼ = − A ⊖ ⊖ ⊕ ⊕ where again, the bijection matches those events in the corresponding position of the dia-gram. Likewise, the positive symmetry has all order-isomorphisms included in one of: ⊖ ⊖ ⊕ ⊕ ∼ = + A ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ∼ = + A ⊖ ⊖ ⊕ ⊕ forming, altogether, a tcg. Then, the endosymmetry ⊖ ⊖ ⊕ ∼ = A ⊖ ⊖ ⊕ The following example is due to Marc de Visme. ⊖ ⊖ ⊕ ∼ = − A ⊖ ⊖ ⊕ ∼ = + A ⊖ ⊖ ⊕ which is not formed of endosymmetries. So this configuration is not canonical, but its onlysymmetric {⊖ , ⊖ , ⊕ } is not canonical either, for the same reason.Fortunately, no such pathological example arises in the games that (to our knowledge)have found a use in semantics of logics and programming languages. Next we shall proposethe existence of a canonical representative as a new axiom for tcgs, and show that it ispreserved by all useful constructions on games. The axiom of representability simply requires the existence of canonical representatives.
Definition 12.
Consider A a tcg.We say that A is representable iff for all x ∈ C ∼ = ( A ) , there is x ∈ x canonical. If A is representable we may consider fixed in advance a choice, for every symmetry class x ∈ C ∼ = ( A ), of a canonical representative x ∈ C ( A ). For this to be a reasonable conditionon tcgs, we must check that all the common game constructions preserve representability. Basic constructions.
First, we review the common game constructions that have fewinteractions with the symmetry. Clearly, the empty game is representable. We have:
Lemma 13.
Consider
A, B representable tcgs. Then, (1) A ⊥ is representable, (2) A k B is representable.Proof. (1) the dual exchanges ∼ = + A and ∼ = − A and the definition of canonical is symmetric. (2) If x A k x B ∈ C ∼ = ( A k B ), we simply set x A k x B = x A k x B . Canonicity followsdirectly from that of x A and x B , exploiting the fact that any endosymmetry θ : x A k x B ∼ = A k B x A k x B must have the form θ = θ A k θ B for endosymmetries θ A : x A ∼ = A x A and θ B : x B ∼ = B x B .The above are the game constructions used in the compact closed structure of thinconcurrent games. With similarly direct proofs, we cover all the frequent constructions ontcgs that are essentially independent of symmetry: the shifts ↑ A (resp. ↓ A ) which prefixthe game A with a new negative (resp. positive) move (see e.g. [7]), the sum P i ∈ I A i having all A i in pairwise conflict (see e.g. [7]), the linear arrow M ⊸ N of negative M, N – for all those, preservation of representability is direct. What requires more care is thefact that the constructions that introduce symmetry do indeed preserve representability.14
O exponential.
We start with the Hyland-Ong style exponential. Recall that it takesan arena in the usual Hyland-Ong sense, i.e. a forestial partial order, without symmetry.We refer to [5] for the definition of ! HO A for A an arena and the associated notations.We have the proposition: Proposition 14.
For A any arena, ! HO A is a representable thin concurrent game.Proof. Within this proof (and only), by ! A we mean ! HO A . Those configurations x ∈ C (! HO A ) with exactly one initial move are entirely determined by: (1) their label lbl(min( x )), (2) their copy index ind(min( x )), (3) for each min( x ) _ a , the sub-configuration starting with a .Any x ∈ C (! A ) with index i , label a and sub-configurations x , . . . , x n may be written x = i · ( { x , . . . , x n } ⊸ a ) ∈ C (! A )where each x i is written similarly, with a notation inspired from intersection types. Butthen, using that similarly any x j is written i j · ( X j ⊸ a j ), we may rewrite x as i · (( i · x ′ , . . . , i n · x ′ n ) ⊸ a )where each x ′ j = X j ⊸ a j . Going one step further, write x = i · (( i · x , . . . , i p · x p ) ⊸ . . . ⊸ ( i m · x m , . . . , i mp m · x mp m ) ⊸ a ) , regrouping sub-trees by symmetry classes. If x is to be canonical, then for any 1 ≤ k ≤ m ,any i kl and i kl ′ should be swapped by an endosymmetry; implying x kl = x kl ′ . So we set x ′ = i · (( i · x , . . . , i p · x ) ⊸ . . . ⊸ ( i m · x m , . . . , i mp m · x m ) ⊸ a ) , which is symmetric to x by construction; moreover if for all 1 ≤ k ≤ m , x k is assumedcanonical by induction hypothesis, then one may verify that x ′ is canonical.We omit the details on that last verification, as it is the exact same reasoning as for theAJM exponential, which we give more formally below. Unsuprisingly, the proof for AJMbears much in common with the one above. We started with HO as we believe that themore concrete nature of games obtained through the HO exponential makes the reasoningslightly more transparent: we wish to construct a configuration where any two moves withswappable copy indices have the exact same sub-trees below, so that the two copy indicesmay be simply swapped leaving the remainder of the configuration unchanged.15 JM exponential.
The AJM exponential is our main source of non-trivial symmetries.
Lemma 15.
Consider N a representable negative thin concurrent game, i.e. all its mini-mal events are negative. Then, the thin concurrent game ! N is representable.Proof. Let x ∈ C (! N ), of the form x = k i ∈ I x i , where x i ∈ C ( N ). Let us partition I as I = ] k ∈ K I k such that for all i, j ∈ I , x i ∼ = N x j iff there is some k ∈ N such that i, j ∈ K . For each i ∈ I ,write f ( i ) ∈ K for the corresponding component. For each k ∈ K , fix some g ( k ) ∈ I k .Now, fix k ∈ K . Since N is representable, there is x g ( k ) ∼ = N canon ( x g ( k ) ) with canon ( x g ( k ) )canonical. Then for each j ∈ I k we replace x j with canon ( x g ( k ) ); or more formally we set x ′ = k i ∈ I canon ( x g ( f ( i )) ) ∈ C (! N ) . We clearly have x ∼ = ! N x ′ ; indeed, for each i ∈ I , we have x i ∼ = N x g ( f ( i )) ∼ = N canon ( x g ( f ( i )) ).Furthermore, x ′ is canonical. Indeed, writing x ′ i = canon ( x g ( f ( i )) ), consider now any symetry θ : k i ∈ I x ′ i ∼ = ! N k i ∈ I x ′ i . By definition, there is π : I → I a permutation, and for all i ∈ I a symmetry θ i : x ′ i ∼ = N x ′ π ( i ) . But by construction, this means that we had x i ∼ = N x π ( i ) as well, so i, π ( i ) belong tothe same component of the partition and g ( f ( i )) = g ( f ( π ( i ))). Therefore, by construction, x ′ i = x ′ π ( i ) . But x ′ i is canonical, so θ i decomposes as x ′ i θ − i ∼ = − N x ′ i θ + i ∼ = + N x ′ i . Setting θ − ( i, e ) = ( π ( i ) , θ − i ( e )) and θ + ( i, e ) = ( i, θ + i ( e )), we have the required decom-position of θ , showing that x ′ is canonical, as required.In the rest of this paper, we aim to make it explicit whenever this condition is required. By now we have added a new condition on games which eliminates some pathologicalexamples, and we have proved that this condition is preserved by all sensible constructionson games. It remains to be seen whether this condition does solve the problem at hand.16 .1 Actions of negative symmetries on strategies
Before we start, recall that for
A, B tcgs (which from now on will always be assumed tobe representable), σ : A S → B a strategy and x A ∈ C ∼ = ( A ), x B ∈ C ∼ = ( B ), we have set wit + σ ( x A , x B ) = { x S ∈ C + ( S ) | x SA ∼ = − A x A & x SB ∼ = + B x B } , where x A and x B are the canonical representatives given by representability of A and B .Our next step will be to investigate how negative symmetries act on witnesses. Ourstarting point for that is the following lemma, Lemma B.4 in [5]. Lemma 16.
Consider σ : S → A a pre- ∼ -strategy, x S ∈ C ( S ) and θ − : x SA ∼ = − A y A . Then,there is a unique ϕ : x S ∼ = S y S s.t. σϕ = θ + ◦ θ − : x SA ∼ = A y SA for some θ + : y A ∼ = + A y SA . This is our main tool to have negative symmetries act on strategies. If x S ∈ C ( S ) and θ − : x SA ∼ = − A y A presents a change in Opponent’s copy indices, we can make θ − “act on” x S :Player adapts to the change of Opponent copy indices and presents some ϕ : x S ∼ = S y S .It is tempting to invoke some group theory here. For any x ∈ C ( A ), we have threegroups: the group S ( x ) of endosymmetries θ : x ∼ = A x , the group S + ( x ) of positive endosymmetries, and the group S − ( x ) of negative endosymmetries. When applied tosymmetry classes, as in S ( x ) for x ∈ C ∼ = ( A ), these operations mean S ( x ). Of course,if x ∼ = A y then any θ : x ∼ = A y provides an iso between S ( x ) and S ( y ) by conjugation.Warning: if x ∼ = A y we do not necessarily have S − ( x ) and S − ( y ) isomorphic (and of course,likewise for S + ( − )), so the notation S − ( x ) is borderline – we insist that it means S − ( x )and depends on the chosen representative. This shall hopefully cause no confusion.Now, for σ : S → A and x A ∈ C ( A ), it is tempting to make S − ( x A ) act on the set X = { x S ∈ C ( S ) | σx S = x A } , but for θ − ∈ S − ( x A ) and x S ∈ X , there is no reason why the ϕ : x S ∼ = S y S obtained viaLemma 16 would satisfy σy S = x A and hence remain in X .So we add a bit of wiggling room. For x A ∈ C ∼ = ( A ), we define the set ∼ + - wit + ( x A ) = { ( x S , θ + ) | x S ∈ C + ( S ) , θ + : x SA ∼ = + A x A } of witnesses for x A along with a specific choice of positive symmetry. Then we indeed have: Proposition 17.
Consider A a tcg and x A ∈ C ∼ = ( A ) . There is a group action ( y ) : S − ( x A ) × ∼ + - wit + ( x A ) → ∼ + - wit + ( x A ) , such that for all ( y S , ψ + ) = ϕ − y ( x S , θ + ) , there is φ S : x S ∼ = S y S making the diagram x SA θ + / / φ SA (cid:15) (cid:15) x Aϕ − (cid:15) (cid:15) y SA ψ + / / x A commute. roof. Consider ( x S , θ + ) ∈ ∼ + - wit + ( x A ) and ϕ − ∈ S − ( x A ). We show that there is unique φ S : x S ∼ = S y S and ψ + : y SA ∼ = + A x A making the following diagram commute: x SA θ + / / φ SA (cid:15) (cid:15) x Aϕ − (cid:15) (cid:15) y SA ψ + / / x A For existence, by Lemma 3.19 of [5], ϕ − ◦ θ + : x SA ∼ = A x A factors uniquely asΞ + ◦ Ξ − : x SA ∼ = A x A . Next, by Lemma 16, there is φ S : x S ∼ = S y S such that we have φ SA = Ω + ◦ Ξ − : x SA ∼ = A y SA for some Ω + : y A ∼ = + A y SA . We then form ψ + = Ξ + ◦ Ω − to conclude.For uniqueness, if we have ϕ : x S ∼ = S y S and ϕ : x S ∼ = S z S satisfying the requirements, y SA ( σϕ ) − / / + (cid:15) (cid:15) x SA + (cid:15) (cid:15) σϕ / / z SA + (cid:15) (cid:15) x A ϕ − − / / x A ϕ − / / x A commutes, so ( σϕ ) ◦ ( σϕ ) − = σ ( ϕ ◦ ϕ − ) is positive, so by Lemma 3.28 of [5] we have ϕ ◦ ϕ − = id, so ϕ = ϕ .Note that in the proof, we have actually not used the representability assumption.However, it will come in to deduce a property useful for elaborate forms of the collapse(namely, in the quantum case). For that, we need the following intermediate lemma. Lemma 18.
Consider A a representable tcg, x A ∈ C ∼ = ( A ) , and x ∈ C ( A ) s.t. x ∼ = + A x A .Then, any θ : x ∼ = A x A factors uniquely as θ − ◦ θ + , where θ + : x ∼ = + A x A and θ − ∈ S − ( x A ) .Proof. Fix some ϕ : x ∼ = + A x A . Now, take θ : x ∼ = A x A . By Lemma 3.19 of [5], θ factorsuniquely as θ − ◦ θ + , where θ + : x ∼ = + A z and θ − : z ∼ = − A x A for some z ∈ C ( A ). But then, ϕ ◦ θ − : x A ∼ = A x A factors via ( ϕ ◦ θ − ) : z ∼ = + A x A and θ − − : x A ∼ = − A z , so x A = z follows since x A is canonical.For A a tcg and x A ∈ C ∼ = ( A ), we have previously defined ∼ + - wit + ( x A ) = { ( x S , θ + ) | x S ∈ C + ( S ) , θ + : x SA ∼ = + A x A } the set of witnesses for x A up to positive symmetry, along with a specific choice of positivesymmetry θ + : x SA ∼ = + A x A . We shall now also consider the variation ∼ - wit + ( x A ) = { ( x S , θ ) | x SA ∼ = + A x A & θ : x SA ∼ = A x A } where we know that x SA ∼ = + A x A , but θ : x SA ∼ = A x A may not be positive.18 orollary 19. Consider A a representable tcg and x A ∈ C ∼ = ( A ) . Then, the function F : ∼ - wit + ( x A ) → ∼ + - wit + ( x A )( x S , θ − ◦ θ + ) θ − y ( x S , θ + ) is such that any X ∈ ∼ + - wit + ( x A ) has exactly | S − ( x A ) | antecedents.Proof. The definition of F makes use of the decomposition of all symmetries θ : x SA ∼ = A x A offered by Lemma 18, using canonicity of x A . The statement on the number of antecedentsis an immediate consequence of the group action of Proposition 17.The reader might not immediately see the point; in fact we will not use this to establish(2), but it fits in this paper as it is required for more elaborate versions of this construction,in particular in the presence of quantum valuations [7]. Let us fix for this section two strategies σ : A S → B and τ : B T → C . Witnessing strategies and interactions.
We write elements of ∼ + - wit + σ ( x A , x B ) astriples ( θ A − , x S , θ B + ); as an alias for ( x S , θ A − k θ B + ) ∈ ∼ + - wit + σ ( x A k x B ). Two witnesses( θ A − , x S , θ B + ) ∈ ∼ + - wit + σ ( x A , x B ) , (Ω B − , x T , Ω C + ) ∈ ∼ + - wit + τ ( x B , x C ) , are causally compatible iff the composite bijection (see Definition 4) is secured. We write ∼ + - wit + σ ( x A , x B ) • ∼ + - wit + τ ( x B , x C )for the set of causally compatible pairs ( w σ , w τ ) ∈ ∼ + - wit + σ ( x A , x B ) × ∼ + - wit + τ ( x B , x C ).To accompany our notions of witnesses for strategies we shall need to provide witnessesfor interactions. If x A ∈ C ∼ = ( A ), x B ∈ C ∼ = ( B ) and x C ∈ C ∼ = ( C ), we write int + τ ⊛ σ ( x A , x B , x C ) = { x T ⊛ x S ∈ C + ( T ⊛ S ) | x SA ∼ = − A x A , x SB = x TB ∼ = B x B , & x TC ∼ = + C x C } . Like for strategies, we also write ∼ + - int + τ ⊛ σ ( x A , x B , x C ) for the set { ( θ A − , x T ⊛ x S , θ C + ) | θ A − : x SA ∼ = − A x A , x T ⊛ x S ∈ int + τ ⊛ σ ( x A , x B , x C ) , & θ C + : x TC ∼ = + C x C } . interaction witnesses along with specific symmetries to the game. Finally, we write: int + τ ⊛ σ ( x A , x C ) = { x T ⊛ x S ∈ C + ( T ⊛ S ) | x SA ∼ = − A x A , & x TC ∼ = + C x C } for the variant of int + τ ⊛ σ ( x A , x B , x C ) with no constraint in B . Clearly, we have: Lemma 20.
Consider σ : A S → B and τ : B T → C , x A ∈ C ∼ = ( A ) and x C ∈ C ∼ = ( C ) . Then: int + τ ⊛ σ ( x A , x C ) = ] x B ∈ C ∼ = ( B ) int + τ ⊛ σ ( x A , x B , x C ) where the notation ⊎ means the plain set-theoretic union when it is disjoint.Proof. Simply partition interactions according to the symmetry class reached in B .19 nteractions up to symmetry. We start with a more explicit variant of Lemma 6.
Lemma 21.
For any pair of causally compatible witnesses ( θ A − , x S , θ B + ) ∈ ∼ + - wit + σ ( x A , x B ) , (Ω B − , x T , Ω C + ) ∈ ∼ + - wit + τ ( x B , x C ) , there are unique symmetries ω S : x S ∼ = S y S , ν T : x T ∼ = T y T , Θ B : x B ∼ = B y B and witness ( ψ A − , y T ⊛ y S , ψ C + ) ∈ ∼ + - int + τ ⊛ σ ( x A , x B , x C ) with y SB = y TB = y B , such that the following diagrams commute: x SAθ A − y y ssssss ω SA (cid:15) (cid:15) x SB θ B + / / ω SB (cid:15) (cid:15) x B Θ B (cid:15) (cid:15) x TBν TB (cid:15) (cid:15) Ω B − o o x TC Ω C + % % ❑❑❑❑❑❑ ν TC (cid:15) (cid:15) x A x C y SAψ A − e e ❑❑❑❑❑❑ y SB y B y TB y TC ψ C + ssssss Proof.
By Lemma 6, there are unique symmetries ω S : x S ∼ = S y S , ν T : x T ∼ = T y T , and( ψ A − , y T ⊛ y S , ψ C + ) ∈ ∼ + - int + τ ⊛ σ ( x A , x B , x C )with y SB = y TB = y B , such that the following diagrams commute: x SAθ A − y y ssssss ω SA (cid:15) (cid:15) x SB θ B + / / ω SB (cid:15) (cid:15) x B x TBν TB (cid:15) (cid:15) / / (Ω B − ) − x TC Ω C + % % ❑❑❑❑❑❑ ν TC (cid:15) (cid:15) x A x C y SAψ A − e e ❑❑❑❑❑❑ y SB y B y TB y TC ψ C + ssssss We simply set Θ B : x B → y B as either path around the center diagram.Thanks to the previous section we may reverse this operation, as shown below. Lemma 22.
For any symmetry Θ B : x B ∼ = B y B and any witness ( ψ A − , y T ⊛ y S , ψ C + ) ∈ ∼ + - int + τ ⊛ σ ( x A , x B , x C ) with y SB = y TB = y B , there are unique symmetries ω S : x S ∼ = S y S , ν T : x T ∼ = T y T and ( θ A − , x S , θ B + ) ∈ ∼ + - wit + σ ( x A , x B ) , (Ω B − , x T , Ω C + ) ∈ ∼ + - wit + τ ( x B , x C ) , a pair of causally compatible witnesses, such that the following diagrams commute: x SAθ A − y y ssssss ω SA (cid:15) (cid:15) x SB θ B + / / ω SB (cid:15) (cid:15) x B Θ B (cid:15) (cid:15) x TBν TB (cid:15) (cid:15) Ω B − o o x TC Ω C + % % ❑❑❑❑❑❑ ν TC (cid:15) (cid:15) x A x C y SAψ A − e e ❑❑❑❑❑❑ y SB y B y TB y TC ψ C + ssssss roof. The first step is to factor Θ − B in two ways, as in the diagram z B Φ B + / / x B z B Ψ B − o o x A x C y SAψ A − e e ❑❑❑❑❑❑ y SB y B Φ B − \ \ ✾✾✾✾✾✾✾✾✾✾ Θ − B O O Ψ B + B B ✆✆✆✆✆✆✆✆✆✆ y TB y TC ψ C + ssssss following Lemma 3.19 of [5]. By Lemma 16 we can make Φ B − act on σ . This yields λ A − : x SA ∼ = − A y SA , ω S : x S ∼ = S y S , ∆ B + : x SB ∼ = + B z B , unique such that the following diagram commutes: x SAλ A − z z tttttt ω SA (cid:15) (cid:15) x SBω SB (cid:15) (cid:15) ∆ B + / / z B Φ B + / / x B z B Ψ B − o o x A y SAψ A − o o ❏❏❏❏❏❏ ❏❏❏❏❏❏ x C y SA y SB y B Φ B − [ [ ✼✼✼✼✼✼✼✼✼✼✼ Θ − B O O Ψ B + C C ✞✞✞✞✞✞✞✞✞✞✞ y TB y TC ψ C + : : ✉✉✉✉✉✉ leaving in grey the irrelevant parts of the full diagram for context. Setting θ A − = ψ A − ◦ λ A − and θ B + = Φ B + ◦ ∆ B + , we have found data making the following diagram commute: x SAθ A − | | ①①①①① ω SA (cid:15) (cid:15) x SBω SB (cid:15) (cid:15) θ B + / / x B Θ B (cid:15) (cid:15) z B Ψ B − o o x A x C y SAψ A − b b ❋❋❋❋ y SB y B Ψ B + E E ☛☛☛☛☛☛☛☛☛☛ y TC y TC ψ C + < < ①①①① We shall now prove uniqueness of this data. Assume that we have other symmetries γ A − : u SA ∼ = − A x A , ̟ S : u S ∼ = S y S , γ B + : u SB ∼ = + B x B , making the following diagram commute: u SAγ A − | | ①①①①① ̟ SA (cid:15) (cid:15) u SB̟ SB (cid:15) (cid:15) γ B + / / x B Θ B (cid:15) (cid:15) z B Ψ B − o o x A x C y SAψ A − b b ❋❋❋❋ y SB y B Ψ B + E E ✡✡✡✡✡✡✡✡✡ y TC y TC ψ C + < < ①①①① Then, it follows that the following diagram also commutes: u SA ( ψ A − ) − ◦ γ A − z z tttttt ̟ SA (cid:15) (cid:15) u SB̟ SB (cid:15) (cid:15) (Φ B + ) − ◦ γ B + / / z B Φ B + / / x B z B Ψ B − o o x A y SAψ A − o o ❏❏❏❏❏❏ ❏❏❏❏❏❏ x C y SA y SB y B Φ B − [ [ ✼✼✼✼✼✼✼✼✼✼✼ Θ − B O O Ψ B + C C ✞✞✞✞✞✞✞✞✞✞✞ y TB y TC ψ C + : : ✉✉✉✉✉✉
21y uniqueness for Lemma 16, it follows that u S = x S , ω S = ̟ S , λ A − = ( ψ A − ) − ◦ γ A − so γ A − = θ A − , and (Φ B + ) − ◦ γ B + = ∆ B + so γ B + = θ B + . Altogether, we have proved that there are θ A − : x SA ∼ = S x A , ω S : x S ∼ = S y S , θ B + : x SB ∼ = + B x B , unique making the following diagram commutes: x SAθ A − | | ①①①①① ω SA (cid:15) (cid:15) x SBω SB (cid:15) (cid:15) θ B + / / x B Θ B (cid:15) (cid:15) z B Ψ B − o o x A x C y SAψ A − b b ❋❋❋❋ y SB y B Ψ B + E E ✡✡✡✡✡✡✡✡✡ y TC y TC ψ C + < < ①①①① The lemma follows by performing the exact same reasoning on the right hand side.
With Lemmas 21 and 22 we have done the hardest part of the job; but to collect the fruitsof that work we need to introduce some additional notation. We write S ( x B ) for the setof endosymmetries on x B . Let us fix a choice, for every x ∈ x B , of some κ x : x ∼ = B x B . Transporting through ( κ x ) x ∈ x B gives a bijection, for any two x, y ∈ x B , between the setof symmetries x ∼ = B y and the set S ( x B ). If θ ∈ S ( x B ) and x, y ∈ x B , let us write θ [ x, y ] : x ∼ = B y the transported symmetry obtained as κ − y ◦ θ ◦ κ x . Corollary 23.
There is a bijection
Υ : ∼ + - wit + σ ( x A , x B ) • ∼ + - wit + τ ( x B , x C ) → ∼ + - int + τ ⊛ σ ( x A , x B , x C ) × S ( x B ) such that for every pair of causally compatible witnesses ( w , w ) = (( θ A − , x S , θ B + ) , (Ω B − , x T , Ω C + )) ∈ ∼ + - wit + σ ( x A , x B ) • ∼ + - wit + τ ( x B , x C ) , writing (( ψ A − , y T ⊛ y S , ψ C + ) , ϕ ) = Υ( w , w ) , y B = y SB = y TB , Θ B = ϕ [ x B , y B ] , there are ω S : x S ∼ = S y T and ν T : x T ∼ = T y T such that the following diagrams commute: x SAθ A − y y ssssss ω SA (cid:15) (cid:15) x SB θ B + / / ω SB (cid:15) (cid:15) x B Θ B (cid:15) (cid:15) x TBν TB (cid:15) (cid:15) Ω B − o o x TC Ω C + % % ❑❑❑❑❑❑ ν TC (cid:15) (cid:15) x A x C y SAψ A − e e ❑❑❑❑❑❑ y SB y B y TB y TC ψ C + ssssss roof. Straightforward from Lemmas 21 and 22. Note that ω S and ν T are unique; therequirements of the diagrams constrain them entirely due to local injectivity of σ, τ .The commutation of this diagram is required for situations where one would exploit thisin the presence of valuations on configurations that are typed and transported coherentlythrough symmetry, such as for quantum valuations [7]. However, if one is merely interestedin counting the witnesses, then the take home message is: Corollary 24.
For any x A ∈ C ∼ = ( A ) , x B ∈ C ∼ = ( B ) and x C ∈ C ∼ = ( C ) , we have |∼ + - wit + σ ( x A , x B ) • ∼ + - wit + τ ( x B , x C ) | = |∼ + - int + τ ⊛ σ ( x A , x B , x C ) | × | S ( x B ) | . (4)If we know that the strategies to be composed do not deadlock, then this can besimplified further. Corollary 25.
Assume σ : A S → B and τ : B T → C do not deadlock. Then, |∼ + - wit + σ ( x A , x B ) | × |∼ + - wit + τ ( x B , x C ) | = |∼ + - int + τ ⊛ σ ( x A , x B , x C ) | × | S ( x B ) | , Proof.
By hypothesis, causal compatibility is always satisfied. Therefore, ∼ + - wit + σ ( x A , x B ) • ∼ + - wit + τ ( x B , x C ) = ∼ + - wit + σ ( x A , x B ) × ∼ + - wit + τ ( x B , x C )and the result follows from Corollary 24.This takes us close to Equation 2. One may wonder what is left to conclude; a hint isthe fact that for now, in this section, we have not used canonicity of representatives . The moral of Equation 4 seems clear: on the left hand side witnesses have the liberty topick any positive symmetry on respectively B and B ⊥ to interact, whereas on the righthand side they must match on the nose. Adding | S ( x B ) | on the right balances this out.Let us look deeper into this. From now on, we will rely heavily on canonicity ofrepresentatives. A first consequence of that is the following: Lemma 26. If B is representable, then for all x B ∈ C ∼ = ( B ) , we have | S ( x B ) | = | S − ( x B ) | × | S + ( x B ) | . Proof.
Obvious consequence of the definition of canonicity.Indeed this is almost the definition of canonicity, which states that every endosymmetryon x B factors uniquely as the composition of a positive and a negative endosymmetries of x B . Almost as obvious is the following fact: 23 emma 27. For any σ : A S → B, τ : B T → C , x A ∈ C ∼ = ( A ) , x B ∈ C ∼ = ( B ) , and x C ∈ C ∼ = ( C ) , |∼ + - wit + σ ( x A , x B ) | = | S − ( x A ) | × | wit + σ ( x A , x B ) | × | S + ( x B ) ||∼ + - int + τ ⊛ σ ( x A , x B , x C ) | = | S − ( x A ) | × | int + τ ⊛ σ ( x A , x B , x C ) | × | S + ( x C ) | Proof.
We only detail the first equality, the reasoning for the other is identical. Let uschoose, for every x ∈ x B such that x ∼ = + B x B , some positive symmetry κ Bx : x B ∼ = + B x .Likewise we choose, for each y ∈ x A such that y ∼ = − A x A , some κ Ay : x A ∼ = − A y .Now, we form the function: G : ∼ + - wit + σ ( x A , x B ) → S − ( x A ) × wit + σ ( x A , x B ) × S + ( x B )( θ A − , x S , θ B + ) ( θ A − ◦ κ Ax SA , x S , θ B + ◦ κ Bx SB )which is clearly a bijection as positive and negative symmetries are invertible. Finally, we are now in position to prove:
Theorem 28.
Consider σ : A S → B and τ : B T → C that do not deadlock, and assume that B is representable. Then, for all x A ∈ C ∼ = ( A ) , x C ∈ C ∼ = ( C ) , we have ( ∫ ( τ ⊙ σ )) x A , x C = X x B ∈ C ∼ = ( B ) ( ∫ σ ) x A , x B · ( ∫ τ ) x B , x C . Proof.
We calculate( ∫ ( τ ⊙ σ )) x A , x C = | wit + τ ⊙ σ ( x A , x C ) | (5)= | int + τ ⊛ σ ( x A , x C ) | (6)= X x B ∈ C ∼ = ( B ) | int + τ ⊛ σ ( x A , x B , x C ) | (7)= X x B ∈ C ∼ = ( B ) |∼ + - int + τ ⊛ σ ( x A , x B , x C ) || S − ( A ) | · | S + ( C ) | (8)= X x B ∈ C ∼ = ( B ) |∼ + - wit + σ ( x A , x B ) | · |∼ + - wit + τ ( x B , x C ) || S − ( A ) | · | S ( B ) | · | S + ( C ) | (9)= X x B ∈ C ∼ = ( B ) | S + ( x B ) | · | S − ( x B ) || S ( x B ) | · | wit + σ ( x A , x B ) | · | wit + τ ( x B , x C ) | (10)= X x B ∈ C ∼ = ( B ) | wit + σ ( x A , x B ) | · | wit + τ ( x B , x C ) | (11)= X x B ∈ C ∼ = ( B ) ( ∫ σ ) x A , x B × ( ∫ τ ) x B , x C (12)24here (5) is by definition, (6) is by Lemma 3, (7) is by Lemma 20, (8) is by Lemma 27,(9) is by Corollary 25, (10) is by Lemma 27 again, (11) is by Lemma 26 exploiting that B is representable; and (12) is by definition.This concludes the proof of (2). To conclude, we show that our original notion of witness based on symmetry classes ratherthan canonical representatives was, in fact, wrong. We give two counter-examples: the firstis geared towards simplicity, while the second aims to bring the counter-example as closeas possible to usual models of programming languages. The two examples are, however,powered by the same phenomenon.
Example . Consider the games A ⊥ , C = X formed of only one positive move. The game B = ⊖ ⊖ ⊕ has three moves, with ⊖ and ⊖ symmetric. We consider two strategies: σ : A S → B ⊖ ✵ t t| ♣♣♣♣♣♣ ❴ (cid:12) (cid:12)(cid:18) ⊖ ✯ q qx ❥❥❥❥❥❥❥❥❥❥❥ ❴ (cid:12) (cid:12)(cid:18) X ⊕ /o ⊕ τ : A T → C ⊕ ⊕ ⊖ ✑ $ $, ◗◗◗◗ X These are indeed valid strategies in the sense of [5]. Their composition is: τ ⊙ σ : A T ⊙ S → C X X /o X In particular, the non-deterministic choice on the right hand side originates from thechoice by σ : to which ⊖ i should it react? In particular, | wit τ ⊙ σ ( { X } , { X } ) | = 2 , as the two occurrences of X on the right are not symmetric (this boils down to the factthat ⊕ and ⊕ cannot be symmetric in τ , by thinness). On the other hand, the onlysymmetry class of B on which these two may interact is {⊖ , ⊖ , ⊕} . And we have: | wit σ ( { X } , {⊖ , ⊖ , ⊕} ) | = 1 | wit τ ( {⊕ , ⊕ , ⊖} , { X } ) | = 1In particular, the two configurations of σ responsible for the non-deterministic choice σ : A S → B ⊖ ✵ t t| ♣♣♣♣♣♣ ❴ (cid:12) (cid:12)(cid:18) ⊖ ✯ q qx ❥❥❥❥❥❥❥❥❥❥❥ X ⊕ σ : A S → B ⊖ ✵ t t| ♣♣♣♣♣♣ ⊖ ✯ q qx ❥❥❥❥❥❥❥❥❥❥❥ ❴ (cid:12) (cid:12)(cid:18) X ⊕ wit σ ( { X } , {⊖ , ⊖ , ⊕} ) – whereas they are two distinct elements of wit + σ ( { X } , {⊖ , ⊖ , ⊕} ).This also shows that it is not the case that configurations in wit + σ ( x ) are canonicalrepresentatives of symmetry classes – they are better than that, as they get it right wheresymmetry classes get it wrong.We now show essentially the same example in a more “programming language” style. Example . Consider a basic game o , with a unique move q − . Consider strategies: σ : ! o S → (! o ⊸ o ) ⊸ ! o ⊸ o q − ✮ q qx ✐✐✐✐✐✐✐✐✐✐ q + ✹ u u(cid:127) tttt q − i ✻ v v(cid:0) ✈✈✈✈ ✔ & &- ❚❚❚❚❚❚❚❚❚❚❚ q + i q + i τ : ((! o ⊸ o ) ⊸ ! o ⊸ o ) T → ! o ⊸ o q − ✮ q qx ✐✐✐✐✐✐✐✐✐✐✐✐ q + ✮ q qx ✐✐✐✐✐✐✐✐✐✐✐✐✷ u u} rrrrr q − ✰ r ry ❦❦❦❦❦❦❦❦❦✷ u u} rrrrr q − i ✕ & &- ❯❯❯❯❯❯❯❯❯❯❯❯ q +0 q +1 q + i Note that the moves on the left are only there to ensure the +-covered hypothesis.Their composition is: τ ⊙ σ : ! o T ⊙ S → ! o ⊸ o q − ✫ o ov ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢✬ p pw ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣✵ t t| ♣♣♣♣♣❇ { {(cid:5) q +0 q +1 q +0 q +1 Now, as for the previous example, we observe: | wit τ ⊙ σ ([ q , q ] , [ q ] ⊸ q ) | = 2using an intersection type like notation for symmetry classes on the game, which hopefullyis clear. The reader may check that there is a unique symmetry class on (! o ⊸ o ) ⊸ ! o ⊸ o on which the strategies may match to produce this via +-covered configurations, namely([ q , q ] ⊸ q ) ⊸ [ q ] ⊸ q in the same intersection-type like notation. And we have | wit σ ([ q , q ] , ([ q , q ] ⊸ q ) ⊸ [ q ] ⊸ q ) | = 1 | wit τ (([ q , q ] ⊸ q ) ⊸ [ q ] ⊸ q ) , ([ q ] , q )) | = 1for the same reason as in the previous example.These strategies are not quite terms but they are very well-behaved, in particular visibleand parallel innocent. This counter-example does not quite contradict the claims of [3]because there strategies are more constrained (in particular they are well-bracketed) andthe positions of interest (matching the points of the web) are complete. It is plausible thatthis makes this pathology disappear – in particular Example 30 exploits non-well bracketedbehaviour, but this is pure speculation. In any case concrete witnesses as developped hereare definitely better behaved, and are recommended in all situations.26 cknowledgments. This work is supported by ANR project DyVerSe (ANR-19-CE48-0010-01) and Labex MiLyon (ANR-10-LABX-0070) of Universit´e de Lyon, within the pro-gram “Investissements d’Avenir” (ANR-11-IDEX-0007), operated by the French NationalResearch Agency (ANR).
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