Artificial Physics for Artificial Chemistries

(1) artificial chemistries
(2) chemlambda
(3) information content of the deviation from hamiltonian evolution
(4) chemlambda + physics
[Dittrich, P., Ziegler, J., Banzhaf, W. (2001)] - an artificial chemistry is a triple (S,R,A):
S - a class of molecule types
R - a class of chemical reactions
A - an algorithm for the application of the chemical reactions to a multiset of molecules
[K. Tomita, H. Kurokawa, S. Murata (2009)] - a graph-rewrite automaton is a triple (S,R,A):
S - a class of graphs for example trivalent graphs with colored nodes
R - a class of graph-rewrites
A - an algorithm for the application of the graph-rewrites to one graph

- an asynchronous graph-rewrite automaton is a triple (S,R,A):
S - a class of "molecules" (i.e. of graphs)
R - a class of "chemical reactions" (i.e. graph-rewrite rules)
A - a random and local algorithm for the application of graph-rewrites to one graph
λ calculus [A. Church (1936)] is a term rewrite system
Terms:
• variable: x, y, z, ...
• term:
• variable   |
• A B where A, B terms   |
• λx.A where x var, A term
Term rewrite rule:
• β-reduction: (λx.A)B → A[x=B]
λ calculus [A. Church (1936)] is a term rewrite system
Term rewrite rule:
• β-reduction: (λx.D)B → D[x=B]
(1971, C.P. Wadsworth, 1990, J. Lamping) graph rewrite system
[W. Fontana, L. W. Buss (1994)] - ALCHEMY = ALgorithmic Chemistry:
S - a class of λ- terms in normal form
R - a class of reductions

A + B → A + B + normal form of AB

A - algorithm of random application of a reaction to a random selection of terms
[G. Berry, G. Boudol (1992)] - CHAM = Chemical Abstract Machine: a programming language which uses molecules, solutions and transformation rules
[Y. Lafont (1997)] Interaction combinators
S - a class of trivalent fatgraphs
R - a class of graph rewrite rules, without conflicts
A - any algorithm will do for graphs which reduce in a finite number of steps
The Matrix: what does it mean?

"still he'd see the matrix in his sleep, bright lattices of logic unfolding across that colorless void" (William Gibson, Neuromancer)
versus

Turing machines and λ-terms
S - a class of trivalent fatgraphs

R - two main families of graph rewrite rules

which look like the IC rewrites

each IC node has only 1 active port, so no conflicts
however chemlambda nodes have 2 active ports, leads to conflicts

R - a COMB graph rewrite cycle to eliminate the Arrow elements and loops

R - and some TERM rewrites like the other IC rewrites

Dodecahedron multiplication
Tube pattern duplication
Tube pattern with Turing Machines, builts itself
The factorial of 5: (a,b) to (a+1, ab)
Properties:
Turing universal: we can represent λ-terms and rewrites, or TM.
[Ackermann(2,2) simulation]     [Demos]     [Molecular computers (2015)]     arXiv:1811.04960
gives many interesting graphs modified outside λ-calculus

artificial life: a chemlambda quine is a graph deterministically periodic, evolving under the random algorithm.
[Library of molecules]     [Collection of simulations]
The information content of the deviation from hamiltonian evolution, an extension of Hamiltonian mechanics. [Buliga (2009)]   [(2018)]   [(2019)]   [Buliga, de Saxce (2017)]
aka "Hamiltonian inclusions with convex dissipation" or "symplectic Brezis-Ekeland-Nayroles principle"
define the deviation from hamiltonian evolution vector η

impose that the evolution minimizes the information content

allows dissipation (a generalized Liouville theorem)

applies to: viscosity, plasticity, damage, fracture
also to unilateral contact (and friction)
[E. Fredkin, T. Toffoli (1982)] billiard ball computer

Let M be the space (billiard), then the information content function is

stochastic version

chemlambda+physics
A version of chemlambda is a billiard ball computer (with the physics as in the principle of minimal information content), with some of the state variables over a pair of permutohedra
chemlambda+physics
anatomy of nodes and edges:
chemlambda+physics
initial graph:
chemlambda+physics
a rewrite is a pair of permutations:
chemlambda+physics
we modify the hamiltonian of the force graph by using the state of the graph
we modify the information content and we use the stochastic version
... and it works (announcement).
CONCLUSION: is a game with a pair of permutohedron dices.

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