01      M. Buliga   (2019)   CC-BY-4-0


Artificial Physics for Artificial Chemistries

(1) artificial chemistries
(2) chemlambda
(3) information content of the deviation from hamiltonian evolution
(4) chemlambda + physics
  02      M. Buliga   (1) artificial chemistry    


[Dittrich, P., Ziegler, J., Banzhaf, W. (2001)] - an artificial chemistry is a triple (S,R,A):
S - a class of molecule types for example A, B
R - a class of chemical reactions for example

A - an algorithm for the application of the chemical reactions to a multiset of molecules, for example random application of a reaction to a random selection of molecules of the right type

  03      M. Buliga   (1) artificial chemistry    


[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 for example
A - an algorithm for the application of the graph-rewrites to one graph
  04      M. Buliga   (1) artificial chemistry    


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 - an algorithm for the application of graph-rewrites to one graph, which is random and local

  05      M. Buliga   (1) artificial chemistry    


λ calculus [A. Church (1936)] string rewrite system
  • variable: x, y, z, ...

  • term: variable   |   AB where A, B terms   |   λx.A where x var, A term

  • β-reduction: (λx.A)B → A[x=B]
[C.P. Wadsworth (1971)], [J. Lamping (1990)] graph rewrite system


  06      M. Buliga   (1) artificial chemistry    


[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
  07      M. Buliga   (1) artificial chemistry    


[Y. Lafont (1997)] Interaction combinators
S - a class of trivalent fatgraphs
R - a class of graph rewrite rules

A - any reasonable algorithm will do, because the graph-rewrites are strong confluent
  08      M. Buliga   (1) artificial chemistry    


contradictions

  09      M. Buliga   (1) artificial chemistry    


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
Blade Runner: is this alive?

  10      M. Buliga   (2) chemlambda    


Turing machines and λ-terms
  11      M. Buliga   (2) chemlambda    


[repository] [v2] Chemlambda:
S - a class of trivalent fatgraphs

  12      M. Buliga   (2) chemlambda    


Chemlambda:
R - two main families of graph rewrite rules

        beta rewrite

       a DIST rewrite

  13      M. Buliga   (2) chemlambda    


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

   COMB cycle

  14      M. Buliga   (2) chemlambda    


Chemlambda:
A - an algorithm for the application of graph-rewrites to one graph, which is random and local
  • start with a graph and pre-defined probabilities for each rewrite

  • randomly pick an edge

  • if the edge is part of a pattern for a graph-rewrite then flip a coin to perform the rewrite

  • do as many COMB rewrites as possible

  • (record the states and the transitions of the graph for further examination)

  • repeat until (no more rewrite possible OR too many nodes OR too many rewrites)

  • embed the record into a force graph simulation.
There is a deterministic algorithm which uses a list of priorities of rewrites (because they are not strongly confluent)
  15      M. Buliga   (2) chemlambda    


Dodecahedron multiplication
  16      M. Buliga   (2) chemlambda    


tube pattern duplication
  17      M. Buliga   (2) chemlambda    


tube pattern with Turing Machines, builts itself
  18      M. Buliga   (2) chemlambda    


the factorial of 5: (a,b) to (a+1, ab)
  19      M. Buliga   (2) chemlambda    


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]
  20      M. Buliga   (3) information content    


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

  21      M. Buliga   (3) information content    


allows dissipation (a generalized Liouville theorem)

  22      M. Buliga   (3) information content    


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

  23      M. Buliga   (3) information content    


stochastic version

  24      M. Buliga   (4) chemlambda+physics    


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
  25      M. Buliga   (4) chemlambda+physics    


chemlambda+physics
anatomy of nodes and edges:
  26      M. Buliga   (4) chemlambda+physics    


chemlambda+physics
initial graph:
  27      M. Buliga   (4) chemlambda+physics    


chemlambda+physics
a rewrite is a pair of permutations:
  28      M. Buliga   (4) chemlambda+physics    


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.
       M. Buliga   (2019)    


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