2.2 A brief introduction to linear logic
2.2.3 Proof nets
In the next chapters, one will often rely on a geometric characterisation of invariants of linear proofs, called proof nets. The basic structures we rely on are proof-structures. We first consider theMLLcase, then theMALLone.
2.2.3.1 Proof structures forMLL
Proof structures form a common tool introduced in [33] to reason about invariants of linear logic proofs. Proofs structures are more general than proof invariants. That is, some proof structures do not correspond to proof invariants, but to all proof invariants corresponds one, or several, proof structures. Proof structures that indeed correspond to proof invariants are refereed to as proof-nets. Therefore, one seeks to find the right characterization, that discriminates proof- structures that are valid, that is, correspond to proofs. Proof structures can be composed, and form a category, of which the proof-nets form a sub-category. However, this topic exceeds the scope of this introduction.
Often, in the literature, one will encounter the definition restricted to MLL without units. The reason behind this choice is that, for this fragment, proof structures that are valid are in one-to-one correspondence to proof invariants of linear logic restricted to this fragment. However, when extended to units, we lose this property. That is, a single proof invariant can actually be encoded as various proof structures: proof structures are then too precise. One can bypass this issue by defining a quotient, using a method called Trimble’s empire rewriting [47, 48]. However, for brevity, we will not expose this method, and refer to the above references for its presentation.
⊥ M I ⊗ X M X⊥ , X ⊗ X⊥ , Y ⊗ ⊥ , Y⊥
M I
M M
⊗
⊗ ⊗ M
Figure 2.2: AMLL-proof structure
Definition 2.3. An MLL sequent is a multiset of formulas built out of the following grammar : F, F0 ::= X | X⊥
| I | ⊥ | F ⊗ F0 | F M F0,
where X ∈TVar. An MLL−sequent is a multiset of formulas built out of the following grammar:
F, F0
:= X | X⊥ | F ⊗ F0 | F M F0
where X ∈TVar. A sequent is said to be balanced if each atomic variable X appears the same number of times as its negation X⊥.
In the sequel, we will refer toMLL−andMLLto speak about the canonical fragment ofMALL
whose cut-free proofs are those with conclusions lying insideMLL−andMLLrespectively.
In the sequel, we identify formulas with their parse trees, and hence see them as tree-graphs. Therefore, a sequent is seen as a forest. The propositional variables X, Y, ... as well as the unit 1 are positive, whereas their negation X⊥, Y⊥, ... and ⊥ are negative. Given a balanced
MLL-sequent, we define a linking λ to be a function from its set of negative leaves to its set of positive ones, that is type preserving : λ(X⊥) = X, and such that it establishes a bijection between its set of occurrences of positive propositional variables and negative ones. This enforces that the sequent be balanced.
Definition 2.4. A proof structure is a graph, made out from a sequent ` Γ, seen as a forest, together with a linking functionλ on it, by adding edges between x and λ(x). When x, λ(x) are propositional variables, such an edge is called an axiom link. On the other hand, when x= ⊥, we call it a ⊥-link.
Such a proof-structure is presented in the figure 2.2. There is a famous criterion that characterizes precisely the proof structures that arise from the denotation of an actual proof of
MLL. It has been first presented by Danos and Regnier in [23] forMLL−. To implement it, we first need to define the notion of switching.
⊥ M I ⊗ A M A⊥ , A ⊗ A⊥ , B ⊗ ⊥ , B⊥
M I
M M
⊗
⊗ ⊗ M
Figure 2.3: AnMLLcorrection graph
in the parse tree, of the left or the right premise. That is, it is a function from the set of M- occurrences to the set {l,r}.
The correction graph consists in, given a proof structurePand a switching S, removing the edge between the premise not chosen by the switching and its conclusion. We say that a proof structure is acyclic (respectively connected) if the correction graphs are connected (respectively acyclic) for all the switchings S on it.
For instance, the correction graph in figure 2.3 is disconnected and acyclic.
Theorem 2.6. [23, 48]
• To every equivalence class of proofs ofMLL−, orMLL−+MIX, one can assign a unique
MLL−proof structure (that is, a unique linking). Furthermore, this assignment is faithful and functorial.
• AnMLL−proof structure is a denotation of anMLL−+MIXproof if and only if it is acyclic. • AnMLL−proof structure is a denotation of anMLL−proof it is acyclic and connected, that
is, a tree.
• To everyMLLproof structure satisfying the acyclicity criterion, one can canonically as- sign anMLL+MIXproof.
• To everyMLL proof-structure satisfying the acyclicity and the connectedness criterion, one can assign anMLLproof.
• To everyMLLproof, one can assign a unique equivalence class of MLLproof structures satisfying both criteria. This assignment is functorial, and an isomorphism of categories.
We will sometimes refer to these conditions (connectedness and acyclicity) as the Danos- Reigner criterion. We say that anMLL−proof structure is a proof-net if it satisfies the Danos- Reigner criterion. That is, a proof-net is a proof structure that can be assigned a proof.
2.2.3.2 Proof structures forMALL
In this section, we will restrict our attention toMALL−, the fragment of MALLwithout units, neither additive nor multiplicative. That is, we focus purely on the propositional part. The
grammar of the formulasMALL−is defined as follows :
F::= X | X⊥ | F ⊗ F0 | F M F0 | F ⊕ F0 | F & F0.
where X ∈ TVar. Historically, two notions of proof structures, and proof-nets, have been de- veloped forMALL−. The first one, presented by Girard in [36] is based on a notion of graph enriched with booleans, that tracks in what “branch” of a &-link we are. Although we will make use of them in the final chapter 7, they are hard to present, understand, and are mostly dealt with as a technical tool in the scope of this thesis. We will simply say here that their cor- rectness criterion is related to the one ofMLL; the notion of switching is much more elaborate, but the final correction graphs must also be acyclic and connected.
The second one, presented by Hughes in [46], satisfies a much simpler presentation, and its exposition might help the reader getting an idea of how invariants of proofs behave inMALL. The correctedness criterion, on the other hand, is a bit obscure, and we will restrict our attention to proof structures. We refer to the literature [46] for more details.
Once again, we see each sequent as a set of parse trees, hence a parse forest. A &-resolution onΓ is the result of erasing one argument sub-graph of each &-occurrence. An additive reso- lution is the result of deleting one argument sub-graph of each ⊕, &-occurrence. We say that an additive resolution is on a &-resolution, if it can be obtained through deleting argument sub-trees of the ⊕-occurrences of this &-resolution.
An axiom-link is an edge between complementary literal. A linking on a sequentΓ is a set λ of disjoint axiom links such that S λ partitions the set of leaves of an additive resolution of Γ, calledΓ λ. That is, each literal of the additive resolution is under an unique axiom-link.
AMALL-proof structure comprises aMALL-sequent, seen as a forest, together with a set of linkingsΛ = {λ1, ..., λn}, such that:
• (P1) For each &-resolution of Γ, there is a unique linking λ ∈ Λ such that Γ λ is on this &-resolution.
• (P2) Each graph Γ λ satisfies the Danos-Reigner criterion: for any switching (choice for each M of one of its subtree arguments), the correction graph is a tree.
We usually representMALLproof structure with all the axiom-links coming from the set of linkingsΛ at once, as presented in figure 2.4. As there is only once linking per &-resolution, one can recover the set of linkings by ranging through all &-resolutions.
For instance, there are only two possible &-resolutions on that proof structure, as there is only one &. The set of linkings defines two additive resolutions:
X⊥ ⊕ (Y ⊕ X) , (X & X) ⊗ (Z ⊕ Z) , (Z⊥ ⊗ Z) M Z⊥ ⊕ ⊕ & ⊗ ⊕ ⊗ M
Figure 2.4: An MALL proof structure
X⊥ , X ⊗ Z , (Z⊥⊗ Z) M Z⊥ ⊕ & ⊗ ⊕ ⊗ M
Figure 2.5: First additive resolution
is the first one, and the second is :
X , X ⊗ Z , (Z⊥⊗ Z) M Z⊥ ⊕ ⊕ & ⊗ ⊕ ⊗ M .
Figure 2.6: Second additive resolution
For a proof-structure of MALL− to be a proof-net, one also needs the toggling condition, that is too technical to be exposed here. Just as the switching condition enforces that sequen- tialisation is possible for the multiplicative part, the toggling condition is concerned with the additive fragment of the proof. For the next chapters, the relevant aspect of the definition of proof structure is the condition (P1), that can actually be divided into two sub-conditions :
• For each & resolution, the proof chooses exactly one ⊕-resolution on this &-resolution. • On this additive resolution, it defines a unique linking, that is, a unique set of axiom-links
that partitions the leaves of this resolution.
Therefore, the proofs structure can be seen as functions f from the domain D = ×&∈Γ{l&,r&},
where & ∈ Γ refers to the set of &-occurrences in Γ, to the set of linkings, such that given an element k ∈ D, Γ f (k) is a ⊕-resolution compatible with k. This path has notably been explored in [2].