Innocent strategies and weak sequentiality structures

Sequentiality structures have been introduced in [69], as a way to strengthen innocent strategies, and most of the properties-definition presented here are directly adapted from this paper. These consists of a set of total functions, relating negative and positive cells. We here introduce weak sequentiality structures, that corresponds to these “partial functions” that an innocent strategy naturally produce. Those will be pivotal in our proof that sequential and relational composition are equivalent.

We say that a cell α justifies a movem, written α `m, ifm= (α, v, S ). We say that a cell α

is accessible from a position x ∈Legal(A) in a dialogue game if:

∃m∈ x.m= (β, v, S ), α ∈ S , and ¬(∃m0∈ x.m0= (α, v0, S0)) .

In other terms, the cell α justifies no move in the position. Recalling the fact that the dialogue game is almost the syntactical tree of the formula, the accessible cells correspond to those sub- formulas that are yet to be explored by the position. We denote Axthe set of accessible cells of

the position x. We divide Ax into two subsets, A+x the subset of Ax of cells of positive polarity

and A−x those of negative polarity. Those of negative polarity are brought by a move of positive

polarity and justify moves of negative polarities, and inversely for those of negative polarities. Definition 5.17. Letσ be an innocent strategy, and x ∈ σ•. Letα, β ∈ A−_x two different opponent cells. Letσ |xα be defined as follows:

σ |x α = {s : x  y | ∃s0: ? x. s0.s ∈ σ ∧ ∀m∈ s.¬(∃α0 ∈ A−x \ {α}.α0`m)}

σ |x α is the part of the strategy above x that corresponds to a trigger by opponent of the cell

α. Given a movem, we say thatm∈ σ |x α, if there is a sequence s ∈ σ |x α such that mis a

move of s.

Lemma 5.18. The following property holds:

m∈σ |xα ⇒ ∀n∈σ |x β.mn.

This lemma is quite a strong property, also called the “separation of contexts”. It says that the strategy above two different cells explores two distinct sub-trees.

Proof. We consider a move $ of σ |x α (we use a Greek letter to distinguish it from the other

moves), and let s ∈ σ |x α such that $ ∈ s. Now, let t a sequence of σ |x β, with β , α. As

we could consider s0  s, t0  t such that s0 Cpost t, we assume without lost of generality that

s Cpost t. In that case $  n⇒ $ =n. We will prove that for all movesm ∈ s,n ∈ t.m ⇑n.

As the first movem1of s is justified by α, and the first onen1 of t by β, we already know that

m1 ⇑n1. By forward consistency ∀i, j ≤ 2.mi ⇑nj. So we proved the property for the paths of

length 2. The proof is done by induction on the lengths of the paths s and t.

We consider that the length of s is n+ 2, t is of length m, and that we proved the property hold for (n, m). We write s0 for the subpath of s that consists of its nth first moves. By using the forward consistent diagram of innocence and the inductive hypothesis, we can push the path t along the s0. We now need to prove thatmn+1 ⇑ nfor every move nof t. As mn+1 is an

opponent move, its departure cell is brought by a player move. By definition of s, its departure cell cannot be a cell available at x. Hence it is brought up by a player move in s0. As s0, t are independent (in the sense that ∀m∈ s,n∈ t0.m⇑n), the movemn+1is not related by the partial

order, not in conflict with the moves of t, and is post-compatible. Therefore,mm+1⇑ t (meaning

it is strongly compatible with every move of t). Hence, by repetitive applications of forward consistencynn+1⇑ t, finishing the inductive case.

If t is of length m+ 2, the induction is strongly similar, as the role being played by s, t are

interchangeable. 

There is a direct corollary to this lemma, that highlights why we speak about separation of contexts.

Corollary 5.19. Let x a position ofσ•, andα ∈ A−_x. Then there is a set, calleddominionx(α) ⊆

A+_x defined by:

dominionx(α)= {β ∈ A+x | ∃m∈σ |xα.β `m}

such thatσ |x α takes place above α tdominionx(α) and such that α , β ⇒ dominionx(α) ∩

This corollary enables the following definition.

Definition 5.20. Given an innocent strategyσ, there exists a family partial function {φx : A+x →

A−_x | x ∈σ•}, called weak sequentiality structure, such that: φx(α)= β ⇔ ∃m∈σ |x α, β `m.

A function φxis partial as it is undefined on cells without move above (that is, cells corre-

sponding to ¬0), or even on cells above whom the strategy does not explore. Given a cell α ∈ A−x,

we say thatdominion(α) is the context captured by α. Using the correspondence between un- typed cells and formulas of tensorial logic, this corollary is the game semantics counterpart of the following (wrong) logical rule:

Γ1` A Γ2 ` B

Γ1, Γ2, Γ3` A ⊗ B

.

Indeed, the formulas on the right hand side of the sequents correspond to negative cells, the ones on the right to positive ones. Now, the corollary tells us that the strategy splits in two independent parts, depending on the cell (formula) chosen by opponent. However, as one can clearly notice, the main problem is that the context is affine, there might be some parts of the context (that is some positive cells), that might not be explored. This will be targeted in Section 5.3.2.

We introduce an invariant of the sequentiality structure.

Lemma 5.21. Let s : ? x y ∈ σ, and α ∈ At −x ∩ Ay−. Thendominionx(α)=dominiony(α),

In other terms, let x, y ∈ σ• such that x ≤ y. Then if α ∈ A−x ∩ A−y, it holds that

dominionx(α)=dominiony(α).

Proof. We do the proof in the case where t = m.n. This proof generalises straight away in the general case.

Consider the set of plays s that belongs to σ |x α. As α is an accessible cell of both x and

y, it means that the play m.nbelongs to σ |x β for β , α. We proved above (in the proof of

5.18) that this entailsm.n⇑σ |x α. Consequently, the plays of σ |x α can be pushed abovem.n

(assuming they satisfy the necessary conditions of Cpost), and σ |y α = {s ∈ σ |x α | s Cpost y}.

Given a path s ∈ σ |x α, there exists a path s0 such that s0 ' s, s0 Cpost yand s0 ∈ σ |x α.

Consequently, if a cell of x justifies a move in σ |x α, it also justifies one in σ |yα. 

One of the key points of sequentiality structures is this simple property, that states that if φ(α) = β, then β has appeared after α in the sequence. This will turn to be a central point to later prove compositionality.

Definition 5.22. Letσ be an innocent strategy, s : ?  x ∈ σ, and let α ∈ Ax. We define kskα

as the length of the minimal prefix s0 : ? y of s such that α ∈ Ay.

Proposition 5.23. Let σ be an innocent strategy, s : ?  x ∈ σ and α, β ∈ Ax such that

α ∈dominion(β). Then kskα≤ kskβ.

Note that, one can equivalently write α ∈ dominionx(β) or φx(α)= β. Hence φx(α)= β ⇒

kskα ≤ kskβ.

Proof. Let s, α, β as before. Let s0: ?  y the smallest even prefix of s that reaches a position where β is available. Then, as φ−1y (β) = φ−1x (β), it entails that α ∈ A+y. Now, as β is a negative

cell, is has been brought by a player move. This one has to be the last move of s0. Similarly, as α is a positive cell, it has been brought by an opponent move, and therefore has to be already available before the last move of s0. That is, ks0k_α < ks0k_β. Finally, as for any given cell γ ∈ Ax∩ Ay, we naturally have kskγ = ks0kγ, we conclude. 

For instance, let us suppose that α appears at a position y in s. As α ∈ A+x, it is introduced

by an opponent movem, and we consider the positions σ• 3 x−→ ym −→ z ∈n σ•right below and above y in s. Suppose that φz(α)= β points to a cell available at x. This situation is pictured in

the figure below:

β λ v γ α w η m n φ

This entails there is a sequence in σ |x β that contains a movemabove α. Consequently, there

is a sequence in σ |xβ that contains a move above λ (as α is above λ), which is in contradiction

with the definition of σ |x β. Therefore, the cell that φ(α) points to appears after α; it is brought

up bynin z.

We end up this section with a final technical lemma.

Lemma 5.24. Letσ an innocent strategy, and x, y ∈ σ•such that x < y. Let Ax<y ⊆ Ax be the

subset of cellsα available at x such that ∃m.α ` mand x−_{→ y. Then φ}m x  Ax<y is a total,

Proof. We focus on totality. Let α ∈ A+_x<y. Then let s0 : x  y ∈ σ. This path has a movem

such that α ` m. Therefore, ∃ β ∈ A−_x<y such thatm ∈ σ | β. Then φx(α) = β and hence φ is

total.