Taylor expansion in linear logic is invertible

Each Multiplicative Exponential Linear Logic (MELL) proof-net can be expanded into a differential net, which is its Taylor expansion. We prove that two different MELL proof-nets have two different Taylor expansions. As a corollary, we prove a completeness result for MELL: We show that the relational model is injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the relational model is exactly axiomatized by cut-elimination.

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1. Syntax

1.1. Differential proof-structures

We introduce the syntactical objects we are interested in. As recalled in the introduction, correctness does not play any role, that is why we do not restrict our nets to be correct and we rather consider proof-structures (PS’s). Since it is convenient to represent formally our proof using differential nets possibly with boxes (differential PS’s), we define PS’s as differential PS’s satisfying some conditions (Definition 1.1). More generally, differential in-PS’s are defined by induction on the depth, which is the maximum level of box nesting: Definition 1.1, Definition 1.1 and Definition 1.1 concern what happens at depth , i.e. whenever there is no box; in particular, typed ground-structures allow to represent proofs of the multiplicative fragment (MLL).

We set .

A pre-net is a -tuple , where

• is a finite set; the elements of are the ports of ;

• is a function ; the element of is the label of in ;

• is a subset of ; the elements of are the wires of ;121212We identify a wire with its source port.

• is a partition of ; the elements of are the axioms of ;

• is a subset of such that ; the elements of are the cuts of ;

• is a function such that, for any , we have ; if , then is a premise of ; the arity of is the number of its premises;

• and is a subset of such that ; if such that , then is the left premise of ; if such that , then is the right premise of .

We set , , , , , and . The set is the set of conclusions of . For any , we set ; we set ; the set of exponential ports of is .

A pre-ground-structure is a pre-net such that .

Notice that, although we depict cuts as wires131313like wires between principal ports in the formalism of interaction nets [24] (but, in contrast with interaction nets, Definition 1.1 allows axiom-cuts) (see the content of the box of the PS - the third leftmost box at depth of Figure 11, p. 11 - for an example of a cut), cuts are not elements of the set . A wire goes from a port that has the same name to its target ; instead of using arrows in our figures to indicate the direction, we will use the following convention: Unless (but in this case there is no ambiguity since such a port can never be the target of any wire), whenever a wire goes from to some port , it will be depicted by an edge reaching underneath the vertice corresponding to .

Given a pre-net , we denote by the reflexive transitive closure of the binary relation on defined by .

A simple differential net (resp. a ground-structure) is a pre-net (resp. a pre-ground-structure) such that the relation is irreflexive and the relation is antisymmetric.

The ground-structure of the content of the box of the PS (the leftmost box of Figure 11) is defined by: , , , , , and .

Types will be used only in Subsection 4.2. We introduce them right now in order to help the reader to see how ground-structures can represent MLL proofs.

We are given a set of propositional variables. We set . We define the set of MELL types as follows: . We extend the operator from the set to the set by defining by induction on , for any , as follows: if ; ; ; ; ; ; .

A typed simple differential net (resp. a typed ground-structure) is a pair such that is a pre-net (resp. a pre-ground-structure) and is a function such that

• for any axiom of , there exists a propositional variable such that ;141414Our typed proof-structures are -expanded.

• for any cut of , there exists a MELL type such that ;

and, for any , the following properties hold:

• ;

• if , then , where (resp. ) is the left premise of (resp. the right premise of );

• if , then , where (resp. ) is the left premise of (resp. the right premise of );

• and, if is an exponential port of , then there exists a MELL type such that .

Notice that the ports labelled by “” are completely symmetric to the ports labelled by “”: They can have any number of premises and the typing rule systematically introduces the connector (see Figure 2, while in [17], there were three different kinds of cells: co-weakenings (of arity ) and co-derelictions (of arity ) that introduce the connector , and co-contractions (of arity ) that do not modify the type (see Figure 2).

Let be a typed ground-structure (resp. a typed simple differential net). Then is a ground-structure (resp. a simple differential net).

Proof.

It is enough to notice that, for any , the size of is greater than the size of . ∎

A ground-structure such that is essentially a PS of depth , so MLL proofs can be represented by typed ground-structures.

As we wrote in the introduction, the motivation for proof-nets was to have a canonical object to represent different sequent calculus proofs that should be identified. For instance, Figure 4, Figure 4 and Figure 5 are three different sequent calculus proofs of the same sequent,151515We underline some occurrences of propositional variables in order to distinguish between different occurrences of the same propositional variable instead of using explicitly the exchange rule. but the two first proofs are two different sequentializations of the same typed proof-net depicted in Figure 7, while the third proof is a sequentialization of the typed proof-net depicted in Figure 7. Let (resp. ) be the ground-structure that corresponds to the proof-net (resp. ).

We can define and as follows: ; ; with and ; and ; and , , , , , and .

We can now define our notion of PS: We recall that this notion formalizes Danos & Regnier’s new syntax, and not Girard’s original syntax. Figure 10, Figure 10 and Figure 10 illustrate some differences between the two syntaxes: Figure 10 and Figure 10 are two different objects in the original syntax, both of them are represented in the new syntax by the PS that is depicted in Figure 10. In particular, in the new syntax, auxiliary doors of boxes are always premises of contractions. Since between auxiliary doors and contractions several box boundaries might be crossed, we need the auxiliary notion of (differential) in-PS. Concerning differential PS’s, it is worth noticing that the content of each of their boxes is an in-PS, in particular every -port inside is always the main door of a box.

For any , we define, by induction on , the set of differential in-PS’s of depth (resp. the set of in-PS’s of depth ) and, for any differential in-PS of depth , the sets and . A differential -PS of depth (resp. an -PS of depth ) is a 4-tuple such that

• is a simple differential net (resp. a ground-structure); we set ;

• (resp. ) such that and, for any pair , we have and, if is a pair too, then ;161616We cannot simply disallow pairs in since in the definition of the differential in-PS (Definition 2) we will use pairs to denote copies of ports of the contents of the boxes that have been expanded. the elements of are the boxes of at depth ;171717We identify a box with its main door.

• is a function that associates with every an in-PS of depth that enjoys the following property: if , then there exists such that is an in-PS of depth ;181818The function maps boxes at depth to their contents. we set ; the elements of are the ports of ;

• is a partial function such that, for any , there is a unique , which we will denote by , such that ;191919The function maps to exponential ports at depth their premises that are doors of boxes. we set and ; the elements of (resp. of , resp. of ) are the (resp. shallow, resp. non-shallow) conclusions of .

We set (the elements of are the ports of at depth ) and, for any , we set . We set , , and . The function is defined by setting for any . The integer is defined by induction on :202020The supremum is taken in , hence, if is the empty PS, then .

 co-size(S)=sup({aS(p);p∈P0(S)}∪{co-size(B0(o));o∈B0(S)})

We set , and . For any , we set .212121Equivalently, . We denote by the function that associates with every the port of and with every , where and , the port of . We set for any . The set of boxes of is defined by induction on : . For any binary relation on , for any , we set and we define, by induction on , the set as follows: . We set and .

A differential PS (resp. a PS) is a differential in-PS (resp. an in-PS) such that .222222Equivalently, a differential PS (resp. a PS) is a differential in-PS (resp. an in-PS) such that .

The set of cocontractions of an in-PS is the set . Notice that an in-PS is a differential in-PS with no co-contraction.

It is worth noticing that the binary relation on the set defined by defines a tree with as the root.

If is the PS of depth depicted in Figure 11, then we have , , , , , and is the ground-structure of Example 1.1.

In the absence of axioms and cuts, our definition of PS through in-PS’s is equivalent to our definition of PS in Definition 4 of [10] through -PS’s. We removed -ports because we simplified the proof of Proposition 3.2 and after this simplification they would not play any role any more (actually we introduced a syntactic construction232323See Definition 1.3 that, roughly speaking, can be seen as a partial recovery of these -ports).

For any , for any , we denote by the PS of depth such that and .

For any , we define, by induction on , the set of typed differential in-PS’s of depth (resp. the set of typed in-PS’s of depth ): it is the set of pairs such that is a differential in-PS (resp. an in-PS) and is a function such that:

• is a typed simple differential net (resp. a typed ground-structure);

• for any , the pair is a typed simple differential net, where is the function defined by for any ;

• and, for any , we have .

A typed differential PS (resp. a typed PS) is a typed differential in-PS (resp. a typed in-PS) such that .

1.2. Isomorphisms

We want to consider PS’s up to the names of the ports, apart from the names of the shallow conclusions. We thus define the equivalence relation on PS’s; this relation is slightly finer than the equivalence relation , which ignores all the names of the ports.

For any simple differential nets and , an isomorphism from to is a bijection such that:

We write to denote that is an isomorphism from to ; we write if there exists such that .

Moreover, we write to denote that and ; we write if there exists such that .

For any differential in-PS of depth , for any differential in-PS , we define, by induction on , the set of isomorphims from to : an isomorphism from to is a function such that:

• and the function is an isomorphism ;

• ;

• and the function

 φo:P(BS(o))→P(BS′(φ(o)))p↦p′ such that φ(o,p)=(φ(o),p′)

is an isomorphism from to ;

• and, for any , we have .

We write to denote that is an isomorphism from to ; we write if there exists such that .

Moreover, we write to denote that and ; we write if there exists such that .

Now, if and are two sets of differential in-PS’s, we write if there exists a bijection such that, for any , we have .

Finally, if and are two typed differential in-PS’s, then we write if there exists such that .

Let and be two cut-free typed differential in-PS’s such that . If , then .

Another variant of the notion of isomorphism will be defined in the next subsection (Definition 1.3). A special case of isomorphism consists in renaming only ports at depth :

Let and be two differential in-PS’s. Let be a bijection , where . We say that is obtained from by renaming the ports via and we write if the following properties hold:

• and, for any , we have

• and