Proof-Net as Graph, Taylor Expansion as Pullback

Abstract

We introduce a new graphical representation for multiplicative and exponential linear logic proof-structures, based only on standard labelled oriented graphs and standard notions of graph theory. The inductive structure of boxes is handled by means of a box-tree. Our proof-structures are canonical and allows for an elegant definition of their Taylor expansion by means of pullbacks.

Appendices

Technical Appendix

A Computing a Pullback in the Category of Graphs

The category of graphs has all pullbacks, a fact that we use extensively. We recall here all the definitions and facts that are packed in that affirmation.

Definition 11

(pullback). Let $\mathcal {C}$ be a category. Let $X$, $Y$, and $Z$ be three objects of $\mathcal {C}$ and $f:X\rightarrow Z$ and $g:Y\rightarrow Z$ be two arrows of $\mathcal {C}$.

The pullback of $X$ and $Y$ along $f$ and $g$ is the triple such that the diagram

commutes and, for every other $(B,h : B \rightarrow X,k : B \rightarrow Y)$ making the same diagram commute, there exists a unique arrow $p:B \rightarrow A$ such that:

It is unique (up to unique isomorphism), and it is customary to write $X \times _{Z} Y$ a pullback of $X$ and $Y$ over $Z$ (leaving $f$ and $g$ implicit) and a pullback diagram with a corner:

All pullbacks exist in the category of graphs. Explicitely, let $\tau = (F_{\tau }, V_{\tau }, \partial _{\tau }, j_{\tau })$, $\sigma = (F_{\sigma }, V_{\sigma }, \partial _{\sigma }, j_{\sigma })$ and $\rho = (F_{\rho }, V_{\rho }, \partial _{\rho }, j_{\rho })$ be three graphs and $g: \sigma \rightarrow \tau$, $h : \rho \rightarrow \tau$ be two graph morphisms. Consider the two sets

$$\begin{aligned} F&= \left\{ (f_1, f_2) \in F_{\sigma } \times F_{\rho } \mid g_F(f_1) = h_F(f_2) \right\} \\ V&= \left\{ (v_1, v_2) \in V_{\sigma } \times V_{\rho } \mid g_V(v_1) = h_V(v_2) \right\} \end{aligned}$$

They are both equiped with two projections, which we will write $\pi _{\sigma }^F, \pi _{\rho }^F, \pi _{\sigma }^V, \pi _{\rho }^V$. Let $f\in F$.

$$\begin{aligned} g_V \circ \partial _{\sigma }\circ \pi _{\sigma }^F (f)&= \partial _{\tau } \circ g_F \circ \pi _{\sigma }^F (f) \text {, because g is a graph morphism}\\&= \partial _{\tau } \circ h_F \circ \pi _{\rho }^F (f)\text {, by definition of F}\\&= h_V \circ \partial _{\rho } \circ \pi _{\rho }^F(f) \text {, because h is a graph morphism} \end{aligned}$$

Hence, we can define $\partial : F \rightarrow V$ by $\partial (f) = (\partial _{\sigma }\circ \pi _{\sigma }^F (f), \partial _{\rho } \circ \pi _{\rho }^F(f))$. In the same way, we define $j : F \rightarrow F$ by $j(f) = (j_{\sigma } \circ \pi _{\sigma }^F(f), j_{\rho } \circ \pi _{\rho }^F(f))$, and check that it is an involution.

Hence $\sigma \times _{\tau } \rho = (F,V,\partial ,j)$ is a graph and $\pi _{\sigma } = (\pi _{\sigma }^F, \pi _{\sigma }^V) : \sigma \times _{\tau } \rho \rightarrow \sigma$ and $\pi _{\rho } = (\pi _{\rho }^F, \pi _{\rho }^V) : \sigma \times _{\tau } \rho \rightarrow \rho$ are graph morphisms.

Consider now any $\mu = (F_{\mu },V_{\mu },\partial _{\mu },j_{\mu })$ such that the diagram

commutes. For $f\in F_{\mu }$, let $r_F(f) = (p_F(f),q_F(f))$ and for $v\in V_{\mu }$, let $r_V(v) = (p_V(v),q_V(v))$. We check that it defines a graph morphism $r:\mu \rightarrow \sigma \times _{\tau } \rho$ and it factorises $p$ and $q$.