Cyclic Multiplicative Proof Nets of Linear Logic with an Application to Language Parsing

Abstract

This paper concerns a logical approach to natural language parsing based on proof nets (PNs), i.e. de-sequentialized proofs, of linear logic (LL). In particular, it presents a simple and intuitive syntax for PNs of the cyclic multiplicative fragment of linear logic (CyMLL). The proposed correctness criterion for CyMLL PNs can be considered as the non-commutative counterpart of the famous Danos-Regnier (DR) criterion for PNs of the pure multiplicative fragment (MLL) of LL. The main intuition relies on the fact that any DR-switching (i.e. any correction or test graph for a given PN) can be naturally viewed as a seaweed, i.e. a rootless planar tree inducing a cyclic order on the conclusions of the given PN. Dislike the most part of current syntaxes for non-commutative PNs, our syntax allows a sequentialization for the full class of CyMLL PNs, without requiring these latter must be cut-free. Moreover, we give a simple characterization of CyMLL PNs for Lambek Calculus and thus a geometrical (non inductive) way to parse phrases or sentences by means of Lambek PNs.

Technical Appendices

1.1 Proof of Theorem 1: Stability of PNs under Cut Reduction

Proof

Observe that condition 1 of Definition 4 follows as an almost immediate consequences of the next graph theoretical property (see pages 250-251 of [7]):

Property 1

(Euler-Poicaré invariance). Given a graph $\mathcal G$, then $(\sharp CC - \sharp Cy) = (\sharp V - \sharp E)$, where $\sharp CC$, $\sharp Cy$, $\sharp V$ and $\sharp E$ denotes the number of, respectively, connected components, cycles, vertices and edges of $\mathcal G$.

Condition 2 of Definition 4 follows by calculation. Assume $\pi$ reduces to $\pi '$ after the reduction of a cut between and $(Y^\perp \triangledown X^\perp )$ and assume, by absurdum, there exist a $\triangledown$-link labeled by a formula $A\triangledown B$ s.t. the triple (A, C, B) occurs in this wrong cyclic order in a seaweed $S(\pi ')$ restricted to A, B, i.e., $S(\pi ')\downarrow ^{(A,B)}$, for a pending leave C occurring in this restriction, i.e., $(A,C,B)\in S(\pi ')\downarrow ^{(A,B)}$. Then, two of the three paths , and must go through (i.e., they must contain) the two (sub)cut-links, $cut_1\frac{X\;\;\;X^\perp }{}$ and $cut_2\frac{Y\;\;\;Y^\perp }{}$, resulting from the cut reduction, otherwise $\pi$ would already be violating condition 2 of Definition 4; assume path (resp., ) goes through $cut_1$ link (resp., $cut_2$ link) as follows

This means there exist a seaweed $S'(\pi )$, a link $Y^\perp \triangledown X^\perp$ and a triple $Y^\perp ,C,X^\perp$ s.t. $(Y^\perp ,C,X^\perp )\in S'(\pi )\downarrow ^{(Y^\perp ,X^\perp )}$, violating condition 2 and so contradicting correctness of $\pi$ (see the right hand side picture above; since any switching of $\pi$ is acyclic, deleting the subgraph below $Y^\perp \triangledown X^\perp$ does not make disappear C).

The remaining case when path goes through $cut_1$ (resp., through $cut_2$) and either path or path goes through $cut_2$ (resp., through $cut_2$) is treated similarly and so omitted.

1.2 Proof of Lemma 1: Splitting

Proof

Assume $\pi$ is a CyMLL PN in splitting condition, then by the Splitting Lemma for standard commutative MLL PNs ([5]) $\pi$ must split either at a -link or a cut-link. We reason according these two cases.

• 1. Assume $\pi$ splits at in two components $\pi _A$ and $\pi _B$; we know that both components satisfy condition 1 (they eare MLL PNs); assume by absurdum $\pi _A$ is not a CyMLL PN, i.e., $\pi _A$ violates condition 2 of Definition 4. This means there exists a $\frac{X\;\;\;Y}{X\triangledown Y}$ and a restricted seaweed $S(\pi _A)\downarrow ^{(X,Y)}$ containing the triple X, A, Y in the wrong order, i.e., $(X,A,Y)\in S(\pi _A)\downarrow ^{(X,Y)}$ like Case 1 in picture below.

  

  This means there exists a restricted seaweed $S(\pi )\downarrow ^{(X,Y)}$ containing X, Y and C (where ) in the wrong cyclic order, i.e., $(X,C,Y)\in S(\pi )\downarrow ^{(X,Y)}$, contradicting the correctness of $\pi$.

• 2. Assume $\pi$ splits at the cut link $\frac{A\;\;\;A^\perp }{}$ in two components $\pi _A$ and $\pi _{A^\perp }$; assume by absurdum $\pi _A$ is not a CyMLL PN, hence $\pi _A$ must be violating condition 2 of Definition 4. Moreover, assume $\pi$ is such a minimal (w.r.t. the size, $\langle \sharp V,\sharp E\rangle$) PN in cut-splitting condition whose subproof $\pi _{A}$ is not a CyMLL PN. This means, as before, there exists a $\frac{X\;\;\;Y}{X\triangledown Y}$ and a restricted seaweed $S(\pi _A)\downarrow ^{(X,Y)}$ containing the triple X, A, Y in the wrong order, i.e., $(X,A,Y)\in S(\pi _A)\downarrow ^{(X,Y)}$ like Case 2 of the previous picture. Then, by correctness $\pi$, $\pi _{A^\perp }$ must have $A^\perp$ as its unique conclusion, otherwise there exists a restricted seaweed for $\pi$, $S(\pi )\downarrow ^{(X,Y)}$, containing a triple X, C, Y in the wrong order for a conclusion $C\ne A^\perp$. Moreover, $\pi _{A^\perp }$ cannot contain any cut, otherwise, by Theorem 1, we could replace in $\pi$ the redex $\pi _{A^\perp }$ by its reductum $\pi '_{A^\perp }$, contradicting the minimality of $\pi$. Now, observe this equality , relating the number of -nodes with the number of $\triangledown$-nodes, holds for any cut free proof net with an unique conclusion. Therefore, $\pi _{A^\perp }$ must contain at least a $\triangledown$-link, let us say $\frac{Z\;\;\;T}{Z\triangledown T}$. But then we can easily find a restricted seaweed for $\pi$, $S(\pi )\downarrow ^{(X,Y)}$, and a triple (X, Z, Y) occurring in $S(\pi )\downarrow ^{(X,Y)}$ with the wrong cyclic order, contradicting the correctness of $\pi$, like in Case 2.

1.3 Proof of Lemma 2: Cyclic Order Conclusions of a PN

Proof

By induction on the size $\langle \sharp V,\sharp E\rangle$ of $\pi$.

• 1. If $\pi$ is reduced to an axiom link, then obvious.

• 2. If $\pi$ contains at least a conclusion $A\triangledown B$, then $\varGamma =\varGamma ',A\triangledown B$; by hypothesis of induction the sub-proof net $\pi '$ with conclusion $\varGamma ',A,B$ has cyclic order $\sigma (\varGamma ',A,B)$, and so, by condition 2 of Definition 4 applied to $\pi$, we know that each restricted seaweed $S_i(\pi )\downarrow ^{(\varGamma ',A,B)}$ induces the same cyclic order $\sigma (\varGamma ',A,B)$; finally, by substituting $[A/A\triangledown B]$ (resp., $[B/A\triangledown B]$) in the restriction $S_i(\pi )\downarrow ^{(\varGamma ',A)}$ (resp., $S_i(\pi )\downarrow ^{(\varGamma ',B)}$), we get that each seaweed $S_i(\pi )\downarrow ^{(\varGamma ',A\triangledown B)}$ induces the same cyclic order $\sigma (\varGamma ',A\triangledown B)$.

• 3. Otherwise $\pi$ must contain a terminal splitting -link or cut-link. Assume $\pi$ contains a splitting -link, , and assume by absurdum that $\pi$ is such a minimal (w.r.t. the size) PN with at least two seaweeds $S_i(\pi )$ and $S_j(\pi )$ s.t. $(X,Y,Z)\in S_i(\pi )$ and $(X,Y,Z)\not \in S_j(\pi )$. We follow two sub-cases.

  • (a) It cannot be the case $X=B, Y=A$ and $Z=C$ otherwise, by definition of seaweeds, $S_i(\pi )$ and $S_j(\pi )$ will appear as follows:

    

    Now, by hypothesis of induction, all seaweeds on $\pi _A$ (resp., all seaweeds on $\pi _B$) induce the same order on $\varGamma _1,A$ (resp., $\varGamma _2,B$), then in particular,

    

    but this implies .

  • (b) Assume both X and Y belong to $\pi _A$ (resp., $\pi _B$) and Z belongs to $\pi _B$ (resp., $\pi _A$); moreover, assume for some i, j, and ; by Splitting Lemma 1, each seaweeds for $\pi$, $S_i(\pi )$ and $S_j(\pi )$, must appear as follows:

    

    so, by restriction, $(X,Y,A)\in S_i(\pi _A)\downarrow ^{\varGamma _1,A}$ and $(X,Y,A)\not \in S_j(\pi _A)\downarrow ^{\varGamma _1,A}$, contradicting the assumption (by minimality) that $\pi _A$ is a correct PN with a cyclic order on its conclusions $\varGamma '_1,X,Y,A=\varGamma _1,A$.

  The remaining case, $\pi$ contains a splitting cut, is similar and so omitted.

1.4 Proof of Theorem 5: Sequentialization of Lambek CyMLL PNs

Proof

Assume by absurdum there exists a pure Lambek CyMLL proof net $\pi$ that does not sequentialize into a Lambek CyMLL proof. We can chose $\pi$ minimal w.r.t. the size. Clearly, $\pi$ cannot be reduced to an axiom link; moreover $\pi$ contains neither a negative conclusion of type $A^\perp \triangledown B^\perp$ nor a positive conclusion of type $A^\perp \triangledown B$ (resp., $A\triangledown B^\perp$), otherwise, we could remove this terminal $\triangledown$-link and get a strictly smaller (than $\pi$) proof net $\pi '$ that is sequentializable, by minimality of $\pi$; this implies that also $\pi$ is sequentializable (last inference rule of the sequent proof will be an instance of $\triangledown$-rule) contradicting the assumption. For same reasons (minimality), the unique positive conclusion (e.g. ) of $\pi$ cannot be splitting. Therefore, since $\pi$ is not an axiom link $\frac{}{A^\perp \;\;\;A}$, by Lemmas 1 and 2, there must exist either a (negative) splitting -link (Case 1) or a splitting cut-link (Case 2).

Case 1. Assume a negative splitting conclusion (resp., ). By minimality, $\pi$ must split like in the next left hand side picture (we use $A^+$, resp. $A^-$, to denote positive, resp., negative, LF and $\varGamma ^-$ for sequence of negative LFs):

Now, let us reason on $\pi _1$ (reasoning on $\pi _2$ is symmetric): by minimality of $\pi$, $\pi _1$ cannot be reduced to an axiom link (otherwise $\varGamma ^-_1$ would not be negative); moreover, none of $\varGamma ^-_1$ is a (negative) splitting link, like e..g., , otherwise we could easily restrict to consider the sub-proof-net $\pi '$, obtained by erasing from $\pi$ the sub-proof-net $\pi ''_1$ (with conclusions $\varGamma ^{''-}_1,C$) together with the -link, like the graph enclosed in the dashed line above. Clearly, $\pi '$ would be a non sequentializable Lambek proof net strictly smaller than $\pi$. In addition, $\pi _1$ must be cut-free, otherwise by minimality, after a cut-step reduction we could easily build a non sequentializable reductum PN $\pi '$, strictly smaller than $\pi$, ($\pi '$ will have same conclusions of $\pi$). Therefore, there are only two sub-cases:

• 1. either $A^\perp =C^\perp \triangledown D^\perp$, then from the PN $\pi$ on the l.h.s. of the next figure, we can easily get the non sequentializable PN $\pi '$ (on the r.h.s.); $\pi '$ is strictly smaller than $\pi$, contradicting the minimality assumption:

  

• 2. or , then this -link must split by Lemma 1, since $\pi _1$ is a cut-free PN in splitting condition without other -splitting conclusion in $\varGamma ^-_1$; so from $\pi$ on the l.h.s., we can easily get the non sequentializable PN $\pi '$ on r.h.s.; $\pi '$ is strictly smaller of $\pi$, contradicting the minimality assumption:

  

Case 2. Assume $\pi$ contains a splitting cut link, like the leftmost hand side picture below, then we proceed like in Case 1. We reason on $\pi _1$ with two sub-cases:

• 1. either $A^\perp =C^\perp \triangledown D^\perp$, then we can easily get, starting from the PN $\pi$ on the middle side below, a non sequentializable PN $\pi '$, like the rightmost hand side picture; $\pi '$is strictly smaller than $\pi$, contradicting the minimality assumption:

  

• 2. or , then this $A^\perp$-link must be splitting by Lemma 1, since $\pi _1$ is a cut-free PN in splitting condition without any other -splitting conclusion in $\varGamma ^-_1$; so, we can easily get, starting from the PN $\pi$ on the l.h.s., a non sequentializable PN $\pi '$ that is strictly smaller than $\pi$ (on the r.h.s.), contradicting the minimality assumption.