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The hypercoherence condition is strong enough to ensure that, on an additive resolution, the relation will define exactly one set of linkings. We make that clear in the following proposition. In the sequel,∆ denotes a sequent. Let us note that each element of a clique defines an additive resolution, and a linking. For instance, a list of A ⊕ B (respectively A & B) is either a list of A, or a list of B. Furthermore, as each element x comes from a nominal linear polarised relation, and that the list is nominal separated, one can associate to it a function relating its negative and positive literals, as done in Section 3.1.2.1. This one precisely checks the definition of a linking. Note that two elements on the same additive resolution defines the same linking if they are equivalent. This follows from the list being polarised separated.
Proposition 3.33. Let R : ~∆ a clique, and let us pick an element x of R . This one defines an additive resolution on∆, together with a set of linkings. Then every element y of R on the same additive resolution is equivalent to x.
For the proof we rely on what we proved in the above lemma: if R is a clique, then ˆR is a lax-clique.
Proof. We prove the property by contradiction, assuming that there is a y that implements a different linking on the same additive resolution.
We write λxfor the linking that x encodes, and∆ λx for the associated additive resolution
(and similarly for y). So let X⊥ be an occurrence of an atomic variable of ∆ λx, such that
λx(X⊥) , λy(X⊥).
Now, let a the name that appears for X⊥in x, and b the name that appears for X⊥in y. We apply a permutation (b, a) to y so that they share the same name a for this location. Furthermore, let cibe a name such that cihas same sort as ai. We apply substitutions [ai/ci] to x for all names
ai ∈ν(x) \ a, and similarly for y. Hence in all locations l of ∆ λxdifferent than X⊥and λx(X⊥),
x l = c. Similarly, in all locations l of ∆ λxdifferent than X⊥and λy(X⊥), we got y l = c.
Furthermore, x X⊥ = y X⊥ = a. So basically, we obtain two elements (that we keep on calling with their original names) x, y that are equal on all locations except λx(X⊥), λy(X⊥)
where they differ and are strictly incoherent.
We show that {x, y} < [Γ(~∆) by induction on the structure of ∆, seen as a unique formula (based on the equivalence between the interpretation of ` F1, .., Fn and ` F1M ... M Fn) . We
prove the following intermediate property: if two elements x, y ∈ ~F are strictly incoherent on one or several locations and equal everywhere else, then they are strictly incoherent.
We call X the location where they are strictly incoherent. The base case consists in F being this single location. Then they are incoherent in F by definition. So there are now two induction cases to tackle : F = F1⊗ F2or F = F1M F2. In the first case if X is in F1, then {x, y} F1is
strictly incoherent and therefore, by definition ofΓ(F1⊗ F2), {x, y} F is. The other cases are
dealt with on an equal footing.
Finally, we observe that the {x, y} from the main proof satisfies the required hypotheses, hence {x, y} is strictly incoherent. As R being a clique entails that ˆR is a lax-clique, this implies that R is not a clique. This is a contradiction, and the two elements y, x ∈ R were
equivalent.
Thus, the hypercoherence condition enables us to avoid bad relations, as presented in section 3.4.4. We now turn to study if they satisfy the condition (P10) ofMALL−proof structures 2.2.3.2, namely that for each &-resolution, the relation defines a unique ⊕-resolution on it. We say that an element x ∈ ~∆ is on a additive resolution of Ψ of ∆ if it is in the image of the natural embedding ~∆ Ψ → ~∆.
Proposition 3.34. Given∆ a sequent, R a clique of ~∆, Ψ a &-additive resolution of ∆, and x, y ∈ R such that x, y on Ψ. Then x, y are on the same additive-resolution.
Proof. The proof is done in a similar fashion as above. We take two lists x, y that are on the same &-resolution, but on a different ⊕-resolution. We then use substitutions to equalise all names in the list. Then there must some some sub-formulas F1 ⊕ F2 of F such that the x explores
F1 whereas y is on F2. Hence on these sub-formulas, the two lists are strictly lax-incoherent.
Thus, using the same reasoning as above, they are lax-incoherent. This is a contradiction, and therefore on each &-resolution, the relation can define only one additive-resolution.
However, nothing prevents the clique from being empty on a &-resolution. So in order for a clique R ⊆ ~∆ to form a valid proof-structure, we have to add the condition that for any &- resolutionΨ of ∆, there exists x ∈ R , such that x is on that Ψ. This condition will be explored in more details in the last chapter 7.3.2, where we shall notably prove that it composes.
Even with this additive property, the clique might still fail to form a proper proof structure as the property (P2) is not automatically verified. We recall that (P2) imposes that on each additive resolution, the linking defines aMLL-proof net. However, the hypercoherence condition fails at theMLL-level, allowing the mix-rule. For instance, the sequent ` A, A⊥, B, B⊥with its unique possible linking, has a valid encoding as a clique, despite being not provable.
As proven in [87] [83], hypercoherence is not strong enough for a completeness result for
MALL. However, the denotation of atomic formulas can be chosen such as a full completeness result holds forMLL−+mix. Furthermore, if the proof structure is (P2), then the hypercoherence model can be strong enough to enforce full completeness, although with a different modelling of atomic types than the one presented above. This will be further explored in the last chapter of this thesis.
Nominal Asynchronous Games
Nominal Structures for Asynchronous
Games
As nominal refinement of traditional static nominal models proved to be not fully complete for linear logic, we take another direction, using the aforementioned link between tensorial and linear logic as means to achieve full completeness. The goal of this section is to present nominal structures that extend the current ones already established for games for tensorial logic, by translating them within a nominal universe. Ultimately, this will allow us to project morphisms onto nominal relations, and hence, denotations of proofs of linear logic.
Dialogue games were introduced in [69], as the objects supporting the game semantics of tensorial logic, without proposional variables. We devise their nominal sibling, providing us with the appropriate arenas, that let the strategies play with names. Therefore, we can enforce them to capture the required linearity between negative and atomic type variables, and hence extend the already established semantics of tensorial logic to include atomic variables.
We take advantage of this shift to deepen the relation between syntax, that is, terms, and semantics. It was established in [22], that the linear head reductions of lambda-terms in normal form produce look-alike strategies, that correspond to the denotations of the terms as strategies in game semantics. This correspondence was later refined, establishing an analogy between the “tree of views” of an innocent strategy, and the Böhm tree of the lambda-term. As we rely on nominal constructions, we give a nominal structure to the tensorial lambda-calculus, and characterise precisely alpha-equivalence. This permits the definition of nominal Böhm trees. We relate those Böhm trees with the sub-graphs of our nominal arenas, that will correspond to strategies, that we define in the next chapter 5. Therefore, although it was not the primary goal of this work, our framework allows us to draw an almost perfect correspondence between our strategies, and the set of nominal paths that arise as traces from the α-equivalence classes of lambda terms. This correspondence fails primarily due to the greater symmetry our structures enjoy compared to the lambda calculus, where the intermediate resources generated by player are not taken into account.
To sum up, the material presented here extends previous work on asynchronous game se- mantics [69, 64, 65, 66] by enriching arenas with names, providing appropriate structures for a clean semantics of atomic variables, while strengthening its relation with syntax. We start by characterising graphs that form nominal trees 4.2. Equipping the tensorial lambda calculus with names, we redefine Böhm trees as nominal graphs 4.3. Next, we introduce nominal dialogue games, alongside exposing the denotation function from formulas of tensorial logic onto them 4.4. Dialogue games are the backbone behind arenas, however we still need to unravel some nominal structure between the two. We set to define nominal event structures together with their relation with nominal di-domains. We present the event structure associated with a dialogue game, and characterise its set of positions 4.5. At this stage, we make a pause, and expose how one can project maximal positions onto lists 4.5.6; projections that will later allow us to project strategies onto nominal relations. Finally, we expose how one can see the set of positions as a polarised nominal asynchronous graph 4.6. At last, the Böhm trees lead as well to asynchronous graphs, that form sub-graphs of their respective arenas 4.7.
In the sequel, we add a new infinite enumerable set Acellsof names, that will accommodate
the untyped cells. These will be a key element to describe our games, and notably the additive units. The set A now becomes A = (U
X∈TVarAX) ] Acells, and we write AT forUX∈TvarAX, the
set of typed names. Furthermore, we setPerm(A) =Perm(AT) ⊕Perm(Acells). We will refer to
the elements of Acellsas untyped names, or cell-names. Finally, we write νT(x) for ν(x) ∩ AT
and νcells(x) for ν(x) ∩ Acells. Similarly, we will sometimes use notations 'T, (respectively 'cells)
meaning that there are permutations of AT (respectively Acells) equalising the two elements.
4.1
Fraenkel-Mostowski sets
Nominal sets suffer that they do not allow us to consider elements with non-empty support as first-class objects. That is, given an element a of a nominal set, if a has non-empty support then the set {a} does not form a nominal set. To make up for this, we introduce Fraenkel- Mostowski set theory, that provides a model of set theory encompassing nominal sets. Notably, as it is closed under ∈ precedence, each element of a Fraenkel-Mostowski set forms a set of the Fraenkel-Mostowski model, abbreviated FM in the future.
The FM model of set theory is a model of set theory with atoms, that form the building blocks of the model. The atoms are primitive elements: no set can belong to an atom. More pre- cisely, the FM model of set theory VF M is built according to a cumulative hierarchy following
similar steps to those leading to the Von Neumann model of set theory. However, the starting point V0for FM consists of the set of names A instead of the empty-set ∅ for the Von Neumann
model. At each iteration n of the construction, the action of nominal permutations on Vn is
well defined by ∈-recursion. Notably, all related notions, such as support, are well-defined for elements of Vn. The inductive step consists in:
The resulting model VF Mleads to sets that have finite support, and such that each element
of the set has finite support. Working within this model allows us to consider functions VF M→
VF M, and reason about them. For instance, we will make use of the following proposition.
Proposition 4.1 ([32]). Let F an equivariant function. Then F(u)= v ⇒ ν(F(v)) ⊆ ν(F(u))
Looking at general constructions on sets (such as ∪, ∩, ...) as functions on VF M, this allows
us to deduce inequalities like ν(A ∪ B) ⊆ ν(A) ∪ ν(B) for instance.
As we will have some use of trees with non-empty support, we provide here some additional terminology. Given A ⊆fin A, we say that an object T of FM is A-nominal if its support belongs in A: ν(T ) ⊆ A. For instance, an A-nominal function between two A-nominal sets S → T is a function such that ν( f ) ⊆ A. Given an A-nominal object T , properties in T hold up to A-equivalence, that is, up to equivalence for permutations π such that π#A. For instance, an A-nominal function f satisfies: ∀π#A. f (π · x) = π · f (x). A nominal set is an ∅-nominal set. Finally, given an element x, or a subset S of an A-nominal set T , we write [x]A, [S ]A for their
A-orbits, [x]A= {π · x | π#A} ⊆ T and [S ]A = {π · x | x ∈ S, π#A} ⊆ T. Accordingly, we say that
two elements x, y are A-equivalent, written x 'Ay, if [x]A = [y]A.
Definition 4.2. A-nominal sets and A-nominal functions form a category, called the category of A-nominal sets, writtenNSetA.
We gather in the following proposition some relevant properties of A-nominal sets. Proposition 4.3. • Given A ⊆ B ⊆finA, there is a faithful functor F :NSetA →NSetB.
• Given any set S of VF Mwith finite support, S is aν(S )-nominal set.
• For all S ⊆ T, then [S ]ν(T )⊆ T .
The functor F is simply the functor that sends an object and a morphism to itself. The proof of the proposition is straightforward. This new framework allows us to produce the following definition.
Definition 4.4. An A-partially ordered set (S , ≤) is an A-nominal set together with a partial order ≤ such thatν(≤) ⊆ A. That is, ∀v1, v2∈ S, ∀π#A,
v1≤ v2 ⇔π · v1≤π · v2.
For more on this, we refer to [88].