## Graphic lambda calculus

This is a tutorial for graphic lambda calculus. Up to now the best exposition is arXiv:1305.5786   but there are some facts not covered there. See  also the related Chemical concrete machine  tutorial.

Sources:

The following list of papers (this list will be updated with new papers and with publication details):

• (arxiv) – Chemical concrete machine
• (arxiv) – Graphic lambda calculus, to appear in Complex Systems
• (arxiv) – On graphic lambda calculus and the dual of the graphic beta move
• (arxiv) – Graphic lambda calculus and knot diagrams
• (arxiv) – Local and global moves on locally planar trivalent graphs, lambda calculus and lambda-Scale
• (arxiv) – Lambda-Scale, a lambda calculus for spaces with dilations
• (arxiv) – Emergent algebras

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What is graphic lambda calculus?

Graphic lambda calculus is a formalism working with a set $GRAPH$ of oriented, locally planar, trivalent graphs, with decorated nodes (and also wires, loops and a termination gate).   The set $GRAPH$ is described in the Introduction to graphic lambda calculus.

There are  moves  acting on such graphs, which can be local or global moves.

Graphic lambda calculus contains  differential calculus in metric spaces, untyped (or simply typed)  lambda calculus and that part of knot theory which can be expressed by using knot diagrams.

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The set GRAPH.

This is the set of graphs which are subjected to the moves.  Any assembly of the following elementary graphs, called “gates” is a graph in $GRAPH$.

• The $\lambda$ gate, which corresponds to the lambda abstraction operation from lambda calculus, see lambda terms. It is this:

But wait! This gate looks like it has one input (the entry arrow) and two outputs (the left and right exit arrows respectively). This could not be a graph representing an operation, because an operation has two inputs and one output. For example, the lambda abstraction operation takes as inputs $x$ a variable name and $A$ a term and outputs the term $\lambda x.A$.

Remember that the graphic lambda calculus does not have variable names. There is a certain algorithm which transforms a lambda term into a graph in $GRAPH$, such that to any lambda abstraction which appears in the term corresponds a $\lambda$ gate. The algorithm starts with the representation of the lambda abstraction operation as a node with two inputs and one output, namely as an elementary gate which looks like the $\lambda$  gate, but the orientation of the left exit arrow is inverse than the one of the $\lambda$ gate. At some point in the algorithm the orientation is reversed and we get $\lambda$ gates as shown here. There is a reason for this, wait and see.

It is cool though that this $\lambda$ gate looks like it takes a term $A$ as input and it outputs at the left exit arrow the variable name $x$ and at the right exit arrow the term $\lambda x.A$. (It does not do this, properly, because there will be no variable names in the formalism, but it’s still cool.)

• The $\curlywedge$ graph, which corresponds to the application operation from lambda calculus, see lambda terms. It is this:

This looks like the graph of an operation, there are no clever tricks involved. The sign I use is like a curly join sign.

• The $\Upsilon$ graph, which will be used as a FAN-OUT gate, it is:

• The $\bar{\varepsilon}$ graph. For any element $\varepsilon \in \Gamma$ of an abelian group $\Gamma$ (think about $\Gamma$ as being $(\mathbb{Z}, +)$ or $((0,+\infty), \cdot)$ ) there is an “exploration gate”, or “dilation gate”, which looks like the graph of an operation:

(Therefore we have a family of operations, called “dilations”, indexed by the elements of an abelian group. This is a structure coming from emergent algebras.)

We use these elementary graphs for constructing the graphs in $GRAPH$. Any assembly of these gates, in any number, which respects the orientation of arrows, is in $GRAPH$.

Remark that we obtain trivalent graphs, with decorated nodes, each node having a cyclical order of his arrows (hence locally planar graphs).

There is a small thing to mention though: we may have arrows which input or output into nothing. Indeed, in particular the elementary graphs or gates are in $GRAPH$ and all the arrows of an elementary graph either input or output to nothing.

Technically, we may imagine that we complete a graph in $GRAPH$, if necessary, with univalent nodes, called “leaves” (they may be be decorated with “INPUT” or “OUTPUT”, depending on the orientation of the arrow where they sit onto).

• For this reason we admit into $GRAPH$ arrows without nodes which are elementary graphs, called wires

and loops (without nodes from the elementary graphs, nor leaves)

• Finally, we introduce an univalent gate, the termination gate:

The termination gate has an input leaf and no output.

and now, any graph which is a reunion of lines, loops and assemblies of the elementary graphs (termination graph included) is in $GRAPH$.

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The moves.

Still in beta version:

Yet more:

If there is no oriented path from “2″ to “1″ outside the left hand side picture then one may replace this picture by an edge. Conversely, if there is no oriented path connecting “2″ with “1″ then one may replace the edge with the graph from the left hand side of the following picture:

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Macros.

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Sectors.

A sector of the graphic lambda calculus is:

• a set of graphs, defined by a local or global condition,
• a set of moves from the list of all moves available.

The name “graphic lambda calculus” comes from the fact that there it has untyped lambda calculus as a sector. In fact, there are three four important sectors of graphic lambda calculus:

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Simply typed version of graphic lambda calculus:

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Where I use it  (work in progress) and other useful links:

In these two posts is used the tree formalism of the emergent algebras.

In order to understand how this works, see the following  posts, which aim to describe finite differential calculus as a graph rewriting system:

The chemical concrete machine project:

The neural networks project:

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