Programa Legal – Institucional

The rules of the sequent calculus of MALLare of two kinds: the synchronous ones, and the asynchronous ones. Asynchronous rules are those that can always be permuted with other rules, that is, pushed down the proof-tree. For instance, the introduction rule of ⊥ is asynchronous. They will later correspond to opponent moves in games semantics, and the synchronous ones correspond to player, or proponent moves. Seeing a proof as a strategy, if a proof is able to play a (player)-move at some point in the game, then it can still produce it after some more opponent moves. Therefore, the opponent-moves that happened after the player-move can be inverted with this move and played before-hand. On the other hand, in a proof, a player move cannot be inverted with the opponent moves that happened before, since it might depend on some of the informations produced by these opponent-moves.

By extension, the connectives of linear logic are given a polarity, in relation with their introduction rules. The connectives ⊗, ⊕ are positive, and their negations M, & are negative. Similarly, ⊥, > are negative, and I, 0 are positive. On the other hand, assigning a polarity to literals must be an arbitrary choice, since X, X⊥are introduced alongside by the same rule, but it is common to consider X as positive, and X⊥as negative [11], and corresponds to the idea that the axiom link is directed from X⊥ to X. According to the above remark, it follows that the formulas of linear logic can be split into two sets N, P of negative and positive formulas respectively.

P ::= X | F ⊗ F0 | F ⊕ F0 | 0 | 1 | N⊥ N ::= F M F0 | F & F0 | > | ⊥ | P⊥ where F, F0are formulas ofMALL.

If the polarity assigned to a rule is something rooted in the sequent calculus, its extension to formulas is a bit dubious. Therefore, the polarity of formulas should not be considered as a ground feature of linear logic, but more as a tool, that will later be used in order to devise properly the focussed sequent calculus.

These distinctions between asynchronous and synchronous rules lead to a second sequent calculus for linear logic, called focussed (or focalised in the literature, but here we will re- strain from using the term focalised since it will refer to another property) Before presenting it formally, we expose the general idea.

According to the definition of asynchronous rules, they can all be pushed backwards towards the root of the proof-tree, until they are blocked by a positive connective that happens before in the syntactic formula tree (for instance, one cannot push the & before the ⊕ when trying to prove A ⊕ (B & C)). Once this process is finished, we end up with a proof-stree such that starting from the root of the tree and going upward, we will encounter only negative rules until all formulas (except maybe the literals) in the sequent are positive. Now, a proof is focussed, if it chooses one of the positive decomposable formulas, and all the next rules are positive rules decomposing this formula until all the resulting sub-formulas are either negative or literals. Then, one can repeat this routine: a set of negative rules until all formulas in the sequent are positive, following by a choice of a focus, and the decomposition of the focussed formula. The stunning result established by Andreoli [11] is that every cut-free proof of linear logic is equivalent to a focussed one. This gives us a precious insight into the structure of cut-free proofs of linear logic.

Let P stands for a list of positive formulas, N be a list of negative formulas, X a list of negative atomic formulas, P, Q are positive formulas, N, M negative ones. The sequents are of two shapes: either ` P, N, X; or ` P, X; P. The focussed sequent calculus is defined as follows. Note that O is a formula of either positive or negative polarity.

Ax ` A⊥; A <sub>` P, X, P;</sub>` P, X; P foc ` P, N, A, B; M ` P, N, A M B; ` P, X; P ` P0, X0; Q ⊗ ` P, P0_{, X, X}0_{; P ⊗ Q} ` P, X; P ` P0, X0, M; ⊗ ` P, P0<sub>, X, X</sub>0<sub>; P ⊗ M</sub> ` P, X, M; ` P0, X0; P0 ⊗ ` P, P0_{, X, X}0 ; M ⊗ P0 ` P, X, M; ` P0, X0, N; ⊗ ` P, P0<sub>, X, X</sub>0 ; M ⊗ N ` P, N, X, A; ` P, N, X, B; & ` P, N, X, A & B; ` P, X; P ⊕<sub>1</sub> ` P, X; P ⊕ O ` P, X; P ⊕<sub>2</sub> ` P, X; O ⊕ P ` P, M, X; ⊕<sub>1</sub> ` P, X; M ⊕ O ` P, X, M; ⊕<sub>2</sub> ` P, X; O ⊕ M

1

`; 1 _{P, N, X, ⊥;}P, N, X; ⊥ P, N, X, >; >

A slightly weaker system is presented, called weakly focused. Just as in [59], where it was originally presented, we writeΠ for a sequent that is either empty or consists of a unique positive formula P. The sequents ofMALL_wfoc are of the form ` Γ; Π, where Γ is a multi-set of formulas. This system is more permissive thanMALLfoc as it allows rules to be applied to

negative formulas on the left-hand-side of the sequent even when the right-hand-side of the sequent is non-empty. Furthermore, the negative formulas are now pushed on the left-hand-side of the sequent thanks to a “unfoc”-rule, instead of a built-in machinery.

Ax ` A⊥; A <sub>`</sub>`<sub>Γ, P;</sub>Γ; P foc `<sub>`</sub>Γ, M;<sub>Γ; M</sub> unfoc `Γ, A, B; Π M `Γ, A M B; Π `Γ; A `∆; B ⊗ `Γ, ∆; A ⊗ B `Γ, A; Π `Γ, B; Π & `Γ, A & B; Π `Γ; A ⊕1 `Γ; A ⊕ B `Γ; B ⊕2 `Γ; A ⊕ B 1 `; 1 <sub>`</sub>`_{Γ, ⊥; Π}Γ; Π ⊥ `Γ, >; Π >

Theorem 2.1. [11] [59] Every proof of MALL is equivalent to a proof of MALL_wfoc (resp

MALLFoc), andMALLwfoc(respMALLFoc) can be seen as a subsystem ofMALL.

2.2.2.1 Global connectives

In this section, we strive to prove that proofs of linear logic can be represented, up to equivalence by two global connectives. We present in the paragraph an intermediate step, with four ones, that will be enough for our purposes. They consist of two positive ones, that encapsulate sequences of ⊗ and ⊕, and two negative ones, that encapsulate sequences of M and &. However, for such a system to work, one must consider formulas up to distributivity isomorphisms.

InMALL, there are four distributivity laws, coming from proofs of linear logic.

A ⊗(B ⊕ C) ' (A ⊗ B) ⊕ (A ⊗ C) (A ⊕ B) ⊗ C ' (A ⊗ C) ⊕ (B ⊗ C) A M (B & C) ' (A M C) & (A M C) (A & B) M C ' (A M C) & (B M C)

We write F1 ' F2 to indicate that there are two proofs π1 : F1 → F2, and π2 : F2 → F1

such that π1;cutπ2 ∼ idF1 and π2;cutπ1 ∼ idF1. We present these proofs for the first equation,

while the others can be designed in a similar way. ` A⊥, A ` B⊥, B ` A⊥, B⊥, A ⊗ B ` A⊥, B⊥, (A ⊗ B) ⊕ (A ⊗ C) ` A⊥, A ` C⊥, C ` A⊥, C⊥, A ⊗ C ` A⊥, C⊥, (A ⊗ B) ⊕ (A ⊗ C) ` A⊥, B⊥& C⊥, (A ⊗ B) ⊕ (A ⊗ C) ` A⊥M (B⊥& C⊥), (A ⊗ B) ⊕ (A ⊗ C) ` A⊥, A ` B⊥, B ` B⊥, B ⊕ C A⊥, B⊥, A ⊗ (B ⊕ C) A⊥M B⊥, A ⊕ B ⊕ C ` A⊥, A ` C⊥, C ` C⊥, B ⊕ C ` A⊥, C⊥, A ⊗ (B ⊕ C) ` A⊥M C⊥, A ⊗ (B ⊕ C) ` (A⊥M B⊥) & (A⊥M C⊥), A ⊗ (B ⊕ C)

As these proofs are isomorphisms, they only change the set of morphisms up to (set)- isomorphism. That is, given a formula A, such that, for instance, π1 can be applied to A,

π1 : A → A0, then C(A, B) ' C(A0, B). Therefore we consider the following rewriting sys-

tem: A ⊗(B ⊕ C) (A ⊗ B) ⊕ (A ⊗ C) (A ⊕ B) ⊗ C (A ⊗ C) ⊕ (B ⊗ C) A M (B & C) (A M C) & (A M C) (A & B) M C (A M C) & (B M C) A B ⇒ F[A] F[B]

This system is confluent and strongly normalizing. Therefore, for any given formula A, there exists a unique A0 such that A0 is the normal form of A. That is, A0 is obtained from A by a sequence of steps from the rewriting system, and one cannot rewrite A0 any further. Then given A, B any formula of linear logic, and A0, B0 their normal form, we have the following isomorphism:

C(A, B) ' C(A0, B0)

as each of the rewriting step corresponds to the application of an isomorphism. In other terms, when dealing with proof invariants, one can restrict to considering only formulas in normal form. This generalises straightforwardly to sequents, where a sequent is considered in normal form if each formula in it is.

A second important set of isomorphisms comes from the associativity of each of the con- nective of linear logic ⊗, M, &, ⊕. For instance:

This isomorphism allows us to forget the parentheses, and simply write it as A ⊗ B ⊗ C, without specifying the order in which the ⊗ connectives must be considered. This generalises straight- forwardly in the case where we consider n-tensored formulas, and we writeN_i=1...nAi in that

case. In terms of proofs, these isomorphisms tell us that the order in which the ⊗-rules are ap- plied does not matter. This reasoning applies to the other binary connectives ofMALLsimilarly.

By considering formulas up to these two sets of isomorphisms, we can restrict to formulas of two different forms. A negative formula in normal form, is, once considered up to associativity isomorphisms, shaped as follows:

_&

_i(

_M

_jAi, j), where each Ai, jis either a literal or a positive formula in normal form. On the other hand, a positive formula in normal form is of the shape: L

i(

N

jAi, j), where each Ai, j is either a literal or a negative formula in normal form. We will

say that these formulas are in distributive-associative normal form.

Now, let us consider a focused proof of linear logic, where the sequents are in normal form. Then the root of the proof is of the shape P, N, X. We see it as a unique formula (

_M

P) M (

_M

N ) M (

_M

X), and put it in distributive-associative normal form. That is, it is a formula of the shape

_&

_i(

_M

_jAi, j). If this ends up being the

_&

of 0 formula, that is of the shape

&

∅= >, then the proof proceeds with a >-rule. Otherwise, a proof of this sequent starts with a global

_&

-connective, that encapsulates a series of &-rules.

`

_M

_jA_{1, j}; ... `

_M

_jA_{i, j}; ... `

_M

_nA_{n, j};

&

`

_&

_i(

_M

_jAi, j);

Then, let us focus on one of the branch. The second step consists in decomposing the M: ` Ai,1, .., Ai, j, .., Am, j:

M

`

_M

_jAi, j;

And finally, if there is some ⊥ present among the Ai, j, then the proof removes them using a global ⊥-rule, whose shape is as follows:

` A1, .., An

⊥m

` A1, ..., An, ⊥1, ..., ⊥m

This is the end of the asynchronous phase. These three global connectives will be encapsulate into one negative move in the sequel, Chapter 5. At this point the synchronous phase begins. It starts with a choice of a focus, that is, a positive formula.

`Γ, P; Foc `Γ; P

Then, as we consider formulas in distributive-associative normal form, P = L

i(

N

jMi, j

N

kXi,k, Nl1i,l). Then the focused proof at this stage going to consists in three

distinct steps. The first one is going to consist in a series of ⊕ rule, that we can sum up into one majorL rule:

Γ ` Pi

⊕ Γ ` LiPi

Following that, the Pi is going to consists in tensored formulas Pi =

(N_jMi, jN_kXi,kN_l1i,l). The following sequence of ⊗-rules can be summed up into a single globalN-rule:

Γ1` B1 ... Γi` Bi ... Γn` Bn N Γ1, .., Γi, ..., Γn `

N

i=1..nBi

Finally, the third part of the synchronous phase is going to consist in dealing with all the branches just created. For instance, if the proof reaches a negative formula on the right-hand- side, then it pushes it on the left-hand-side, thanks to a unfoc-rule.

Γ, M; unfoc Γ; M

If, on the other hand, it reaches a sequent `; 1, then it applies the right 1 rule. Finally, if the sequent reached is of the formΓ; X, then we must have Γ = X, and the proof-leaf consists of an axiom-rule.

Therefore, we can consider the following proof-system, calledMALLfoc−glob. We decompose

the formulas into four sets, depending on what is their principal connective. N=

&

i Oi | > | l O=

M

i Pi

M

j ⊥j P= M i Ri | 0 | l R= O i Ni O j 1j l::= X | X⊥

where X ∈TVar. Note that > corresponds to

_&

∅, and 0 toL ∅. In the following, P denotes a list of P-formulas, N a list of N-formulas, and N, O, R, P formulas as above.

Ax ` X⊥; X <sub>` P, P;</sub>` P; P foc ` P, N;<sub>` P; N</sub> unfoc ` P, P1, ..., Pm, ⊥1, ..⊥n;

M

` P,

_M

_iPi

_M

j⊥j; ` P1; N1 ... ` Pi; Ni ... ` Pn; Nn N ` P₁, ..., Pn; N iNi ` P; Rk L k,n ` P;L_i=1..nRi

` P, O1; ... ` P, Oi; ... ` P, On;

&

` P,

_&

_i=1..nOi; 1 `; 1 <sub>` P, ⊥</sub>` P; ⊥m 1, ..., ⊥m; > P, >;

As explained above, the following theorem holds.

Theorem 2.2. Every proofπ :` Γ ofMALLis equivalent to a proofπ0: A ofMALL_foc−glob, where A is the distributive-associative normal form of

_M

Γ.

This system can be once again reformulated into a system with two global connectives: the positive one, that is a mix of L and N, and its negation, a global negative one. This is the idea underlying the creation of ludics [35], and that originates fully complete alternating game models, such as the one presented in the next chapter 5.

In document EVALUACIÓN DE LA POLÍTICA PÚBLICA PARA LA EXTRACCIÓN DEL MATERIAL DE ARCILLA EN EL MUNICIPIO DE NEMOCÓN - DEPARTAMENTO DE CUNDINAMARCA (página 49-55)