Function Repository Resource:

TotalCausalInvariantQ

Determine whether a given multiway system is causal invariant

Contributed by: Jonathan Gorard
 ResourceFunction["TotalCausalInvariantQ"][rules,n] returns True if the multiway system with the specified rules is total causal invariant after n steps, and False otherwise.

Details and Options

Rules can be specified in the following ways:
 {"lhs1"->"rhs1",…} string substitution system {{l11,l12,…}->{r11,r12,..},…} list substitution system WolframModel[rules] Wolfram Model system CellularAutomaton[rules] cellular automaton system "type"→rules system of the specified type
Supported rule types include:
 "StringSubstitutionSystem" rules given as replacements on strings "ListSubstitutionSystem" rules given as replacements on lists "CellularAutomaton" rules given as a list of CellularAutomaton rule specifications "WolframModel" rules given as replacements on hypergraphs
Causal invariance is a property of multiway systems in which all branches of evolution history yield causal networks that are isomorphic as acyclic graphs.
In the case of an abstract rewrite system it is a sufficient condition for confluence (or the Church-Rosser property), and in the case of a WolframModel it is equivalent to Lorentz covariance.

Examples

Basic Examples (2)

Determine total causal invariance for two string substitution systems:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=

TotalCausalInvariantQ can handle Wolfram Models and other system types:

 In[3]:=
 Out[3]=
 In[4]:=
 Out[4]=
 In[5]:=
 Out[5]=

Scope (3)

System Types (3)

TotalCausalInvariantQ supports both string and list substitution systems:

 In[6]:=
 Out[6]=
 In[7]:=
 Out[7]=

TotalCausalInvariantQ also supports multiway generalizations of cellular automata:

 In[8]:=
 Out[8]=

TotalCausalInvariantQ also supports multiway generalizations of Wolfram Models:

 In[9]:=
 Out[9]=
 In[10]:=
 Out[10]=

Applications (2)

Causal Invariant Rules (2)

Prove that the Xor and And functions are total causal invariant (due to associativity):

 In[11]:=
 Out[11]=
 In[12]:=
 Out[12]=

On the other hand, the Nand function is not associative, so it is not total causal invariant:

 In[13]:=
 Out[13]=

Version History

• 2.0.0 – 15 April 2020
• 1.0.0 – 20 March 2020