UPDATE (2020): If you search for the history of graphic lambda calculus and it’s relations with chemlambda then please use:

For more information Chemlambda page is the place to go, or arXiv:2003.14332 [cs.AI].

If you look only for graphic lambda calculus then see

(journal) (arxiv) – Graphic lambda calculus. Complex Systems 22, 4 (2013), 311-360

UPDATE (2020): The chemlambda collection of animations, salvaged from G+, is now available in an enhanced form. Instructive, with lots of links.

UPDATE (2019): There are three recent projects which continue after GLC. You may be most interested in the lambda calculus to chemlambda parser, mainly because you can play with it exclusively in javascript. I gradually add much more info than previously.

UPDATE (2019): You can explore graphs and reductions in chemlambda and in Interaction Combinators at this page. If you want to use this blog for learning more about graphic lambda calculus and chemlambda, then you should read the more recent explanatory posts, starting with this one.

UPDATE (2017): Graphic lambda calculus (GLC) and chemlambda are two different formalisms which were developed in parallel. In my opinion GLC is by far bested by chemlambda and also the main interest in chemlambda comes from molecular computers, not decentralized computing.

In GLC_computations_examples are collected some constructions with GLC which are earlier than chemlambda.

This is the logo of chemlambda:

UPDATE 06.11.2016: For the moment the best entry point to this universe is the README from the active branch of the chemlambda repository. As for some important ideas to take home, these are:

The whole algorithm, with it’s various parts, is the model, not only the rewrite system.

The model has chemlambda as a proof of concept.

Go as much as possible without global semantics and control

Everything is local and decentralized.

Lambda calculus is only a tiny part, serving only as inspiration for more interesting ideas, like for example understanding the predecessor and turning it into the first artificial life organism in chemlambda, later to give the notion of a chemlambda quine

Space is not at all understood in other approaches. Evidence of a thing is not the same as a thing.

Space is a plug-in, of the same nature as the rest of the model.

Nature is the fastest computer.

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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 chemlambda aka “chemical concrete machine” tutorial.

Visit the chemlambda demos page to SEE how this works.

Read the vision page to understand what is good for.

FAQ: chemlambda in real and virtual worlds

A decentralized computing model is described in the Distributed GLC tutorial.

Sources:

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What is graphic lambda calculus?

Graphic lambda calculus is a formalism working with a set of oriented, locally planar, trivalent graphs, with decorated nodes (and also wires, loops and a termination gate node). The set 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 .

The graph , 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 a variable name and a term and outputs the term .

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 , such that to any lambda abstraction which appears in the term corresponds a 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 gate, but the orientation of the left exit arrow is inverse than the one of the gate. At some point in the algorithm the orientation is reversed and we get gates as shown here. There is a reason for this, wait and see.

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

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

The graph. For any element of an abelian group (think about as being or ) 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 . Any assembly of these gates, in any number, which respects the orientation of arrows, is in .

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 and all the arrows of an elementary graph either input or output to nothing.

Technically, we may imagine that we complete a graph in , 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 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 .

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The moves.



Still in beta version:

Yet more:

The ext1 move, if you need extensionality.

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: