Towards an overall small graph rewrites system

Let’s see what we have.

1. Deviation from LIN measures curvature. LIN is equivalent with R3, which is an emergent rewrite, i.e. it can be deduced from R1, R2 and a passage to the limit.
2. Deviation from COLIN measures non-commutativity. Both these properties are described in arXiv:2110.08178, but the point 1 is much older, first time explained in sections 3-6 of arXiv:1103.6007.
3. R2 itself can be seen as a commutativity of numbers multiplication, like in this post, with the name of an rewrite. It is only a part of the rewrite NCOMM or CONCOMM, name not fixed
4. in Pure See we have a proof that the beta rewrite and the DIST rewrites are emergent, from SHUFFLE and a passage to the limit. Also LIN, COLIN, NCOMM are particular forms of SHUFFLE.
5. With the introduction of a star-triangle decomposition of a dilation node into inversions, we can reformulate any of the graph rewrite systems of interest (chemlambda, dirIC, ZSS) as a small graph rewrite system. Again a form of SHUFFLE appears as MIQ (related to miquelian geometry). We can now include projective geometry in our formalism.
6. So even if initially it seemed that graph rewriting formalisms coming from knot theory were very relevant, they are only emergent from other more simple formalisms, coming from SHUFFLE. Here is a list of posts which gradually build some small graph rewrite systems (to be merged): [1], [2], [3], [1-problems], [2-problems], [4], [5].
7. Knot theory appears also in a new alternative to Interaction Combinators. Indeed, as seen here, the IC appear as pairs (A,L) and (FI,FOE) of dirIC nodes, while in Zip-Slip-Smash we encounter a decomposition of crossings as pairs (FI.L) and (A,FOE), which shows that a graph rewriting formalism based on R1, R2, R3 and ZSS is equivalent with IC

All this seems to point to the existence of a small graph rewrite system which is universal in mathematics and logic, in the sense that it covers differential calculus and differential geometry, even non-commutative, projective and inversive geometry, linear algebra and multiplicative linear logic. It has moreover the property that it is further generalizable to new formalisms, differently from those (generalizations from too particular examples) which fashionable in applied category theory and logic.

Going even further down the hole, we remark that most of the computation effort is not even spent on the rewrites, but on (preudo-)random number generation, which itself is amenable to the same (class of) formalisms, which indicate a general shape of a model of the universe, so to say.

Although this model of the universe is very different from Wolfram physics project, there is one ingredient which is common, namely Wolfram’ Principle of Computational Equivalence, but I never seen it used as I do and you don’t know what I mean because I have not written here a word about this.

Definitely not an universe which is a giant graph where all rewrites are possible, among with God’s eye view of it or derived structures (branchial graph, ruliad, observers, etc).

This material has to be organized and structured in the near future. It is one my 3 projects I’m working on right now. I made this post more as a self-reminder about where some of the material is. As it took me so long to arrive here, I shall probably first do a switch to my other two projects, to have some fresh work. The last years I forgot how sane it is to always have three different things to work on 🙂

Inversive algebras and commutative numbers

The following list of posts is needed:

• [1] Permutation magic for inversion, which is the preliminary of the present post
• [2] How to select Heisenberg groups, formally, for the commutative numbers property
• [3] Star-triangle relations in emergent algebras, for the mention of inversive algebras
• [4] The wonderful group of emergent numbers, em-convex and COLIN, for the notion of a number

According to [3] an inversive algebra is obtained from the transformation of emergent algebra via star-triangle relations, i.e. in this case by the decomposition of dilations into pairs of inversions:

where all inversions are involutions

and such that of course the dilations satisfy the emergent algebras axioms.

Moreover, there is a supplementary axioms satisfied by an inversive algebra (previously qualified as miquelian, to be explained in a future post), namely the one described in the next picture which appears in [1]: let’s call this the MIQ relation

In [1] was given a proof of this relation based on manipulations of fractions, the question is what we really did there? Can we treat these fractions as if they are real fractions?

Here is basically the same proof, but when we add what should complete the graphical rewriting side of the formalism, namely a rewrite

:

and a rewrite baptized

:

but mind that this time is a number like explained in [4].

We don’t actually suppose in the proof the R2 axiom, nor SHUFFLE (of course). The is the definition of numbers multiplication.

In the following figure we also use the notation of pure see, used as well in [1].

The left hand side of MIQ reduces as explained in the next two pictures:

and continued

Observe that there is a number denoted by a circled star. It practically corresponds to a dilation with the complicated fraction from the LHS of last relation from the picture (taken from [1])

Now let’s reduce the RHS of the relation MIQ. There will be two steps

and

Remark the appearance of the second number denoted by a circled . This corresponds to the fraction from the RHS of the last relation in a previous picture.

These two numbers are equal if and only if we can reduce one number to the other. What does it mean exactly? Let’s see, in two steps:

and

So actually MIQ is equivalent with the commutativity of multiplication of numbers. In this picture I wrote “N-COMM”, but in the post about the Heisenberg group [2] I denoted this relation (or an equivalent one) by CONCOMM.

In conclusion inversive algebras imply CONCOMM, which implies we have to be in commutative (i.e. SHUFFLE) case or in the Heisenberg case!