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AClass ofModelswith the
Potential to Represent
Fundamental Physics
Stephen Wolfram
A class of models intended to be as minimal and structureless as possible is introduced. Even in cases 
with simple rules, rich and complex behavior is found to emerge, and striking correspondences to some 
important core known features of fundamental physics are seen, suggesting the possibility that the 
models may provide a new approach to finding a fundamental theory of physics

1 | Introduction
Quantum mechanics and general relativity—both introduced more than a century ago—
have delivered many impressive successes in physics. But so far they have not allowed the 
formulation of a complete, fundamental theory of our universe, and at this point it seems 
worthwhile to try exploring other foundations from which space, time, general relativity, 
quantum mechanics and all the other known features of physics could emerge.

The purpose here is to introduce a class of models that could be relevant. The models are set 
up to be as minimal and structureless as possible, but despite the simplicity of their 
construction, they can nevertheless exhibit great complexity and structure in their behav‐
ior. Even independent of their possible relevance to fundamental physics, the models 
appear to be of significant interest in their own right, not least as sources of examples 
amenable to rich analysis by modern methods in mathematics and mathematical physics.

But what is potentially significant for physics is that with exceptionally little input, the models 
already seem able to reproduce some important and sophisticated features of known funda‐
mental physics—and give suggestive indications of being able to reproduce much more.

Our approach here is to carry out a fairly extensive empirical investigation of the models, 
then to use the results of this to make connections with known mathematical and other 
features of physics. We do not know a priori whether any model that we would recognize as 
simple can completely describe the operation of our universe—although the very existence of 
physical laws does seem to indicate some simplicity. But it is basically inevitable that if a 
simple model exists, then almost nothing about the universe as we normally perceive it—
including notions like space and time—will fit recognizably into the model.

1



And given this, the approach we take is to consider models that are as minimal and structure‐
less as possible, so that in effect there is the greatest opportunity for the phenomenon of
emergence to operate. The models introduced here have their origins in network‐based
models studied in the 1990s for [1], but the present models are more minimal and structure‐
less. They can be thought of as abstracted versions of a surprisingly wide range of types of
mathematical and computational systems, including combinatorial, functional, categorical,
algebraic and axiomatic ones.

In what follows, sections 2 through 7 describe features of our models, without specific
reference to physics. Section 8 discusses how the results of the preceding sections can
potentially be used to understand known fundamental features of physics.

An informal introduction to the ideas described here is given in [2].

2



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At the lowest level, the structures on which our models operate consist of collections of 
relations between identical (but labeled) discrete elements. One convenient way to repre-
sent such structures is as graphs (or, in general, hypergraphs). The elements are the nodes 
of the graph or hypergraph. The relations are the (directed) edges or hyperedges that 
connect these elements.

For example, the graph

3

4 1

2

corresponds to the collection of relations

{{1, 2}, {1, 3}, {2, 3}, {4, 1}}
The order in which these relations are stated is irrelevant, but the order in which elements 
appear within each relation is considered significant (and is reflected by the directions of 
the edges in the graph). The specific labels used for the elements (here 1, 2, 3, 4) are arbi-
trary; all that matters is that a particular label always refer to the same element.

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The core of our models are rules for rewriting collections of relations. A very simple exam-
ple of a rule is:

{{x, y}} → {{x, y}, {y, z}}
Here x, y and z  stand for any elements. (The elements they stand for need not be distinct; 
for example, x and y could both stand for the element 1.) The rule states that wherever a 

relation that matches {x,y} appears, it should be replaced by {{x ,y},{y,z}}, where z  is a new 

element. So given {{1, 2}} the rule will produce {{1,2},{2,}} where  is a new element. The 

label for the new element could be anything—so long as it is distinct from 1 and 2. Here we 

will use 3, so that the result of applying the rule to {{1,2}} becomes:

{{1, 2}, {2, 3}}
If one applies the rule again, it will now operate again on {1,2}, and also on {2,3}. On {1,2} it 
again gives {{1,2},{2,}}, but now the new node  cannot be labeled 3, because that label is 
already taken—so instead we will label it 4. When the rule operates on {2,3} it gives {{2,3},{3,}}, 
where again  is a new node, which can now be labeled 5. Combining these gives the final result:

{{1, 2}, {2, 4}, {2, 3}, {3, 5}}

3



(We have written this so that the results from {{1,2}} are followed by those from {{2,3}}—but 
there is no significance to the order in which the relations appear.)

In graphical terms, the rule we have used is:

x y x y z

and the sequence of steps is:

1

 2 1 , 1 2 3, 2 3 5

4

It is important to note that all that matters in these graphs is their connectivity. Where nodes 
are placed on the page in drawing the graph has no fundamental significance; it is usually 

just done to make the graphs as easy to read as possible.

Continuing to apply the same rule for three more steps gives:

 2 1 , 1 2 3,

1
6 11

1 6
16 14

2 3 5, 9 2 1
5 17 9 5 3 2

3 , 4 7 
4 10 13

7 8 12
8

4 15

Laying out nodes differently makes it easier to see some features of the graphs:

1 1 1 1

1
2

2
2

6 4 3

, 2 , , 4 3 , 10 7 8 5 
6

3 11 12 14
9

4 5
13 15 16

7 8
17

2 3 5 9

4



Continuing for a few more steps with the original layout gives the result:

Showing the last 3 steps with the other layout makes it a little clearer what is going on:

 , , 

The rule is generating a binomial tree, with 2n edges (relations) and 2n+1 nodes (distinct 
elements) at step n (and with Binomial[n, s –1] nodes at level s).

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Since order within each relation matters, the following is a different rule:

{{x, y}} → {{z, y}, {y, x}}
This rule can be represented graphically as:

x y x y z

5



Like the previous rule, running this rule also gives a tree, but now with a somewhat different 
structure:

 , , ,

, , ,

, , 

With the other rendering from above, the last 3 steps here are:

 , , 

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A relation can contain two identical elements, as in {0,0}, corresponding to a self-loop in a 

graph. Starting our first rule from a single self-loop, the self-loop effectively just stays 
marking the original node:

 , , ,

, , 

6



However, with for example the rule:

{{x, y}} → {{y, z}, {z, x}}

x y x z y

the self-loop effectively “takes over” the system, “inflating” to a 2n – gon:

 , , , , , 

The rule can also contain self-loops. An example is

{{x, x}} → {{y, y}, {y, y}, {x, y}}
represented graphically as:

x y x

Starting from a single self-loop, this rule produces a simple binary tree:

 , , ,

, , 

7



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Rules can involve several copies of the same relation, corresponding to multiedges in a 

graph. A simple example is the rule:

{{x, y}} → {{x, z}, {x, z}, {y, z}}

x y x z y

Running this rule produces a structure with 3n edges and 1 (3n + 3) nodes at step n:
6

 , , ,

, , 

Rules can both create and destroy multiedges. The rule

{{x, y}} → {{x, z}, {z, w}, {y, z}}

x y x y

z

w

8



generates a multiedge a�er one step, but then destroys it:

 , , ,

, , 

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The examples we have discussed so far all contain only relations involving two elements, 
which can readily be represented as ordinary directed graphs. But in the class of models we 

consider, it is also possible to have relations involving other numbers of elements, say three. 

As an example, consider:

{{1, 2, 3}, {3, 4, 5}}
which consists of two ternary relations. Such an object can be represented as a hypergraph 

consisting of two ternary hyperedges:
5 2

3

4 1

Because our relations are ordered, the hypergraph is directed, as indicated by the arrows 
around each hyperedge.

Note that hypergraphs can contain full or partial self-loops, as in the example of

{{1, 1, 1}, {1, 2, 3}, {3, 4, 4}}

9



which can be drawn as:

2

3 4

1

Rules can involve k -ary relations. Here is an example with ternary relations:

{{x, y, z}} → {{x, y, w}, {y, w, z}}
This rule can be represented as:

w

z x z x

y y

Starting from a single ternary self-loop, here are the first few steps obtained with this rule:

1

1 , 2 1
 , ,

3 4

2

6
9

1 23 2
8 22 6 1224 6

27
15 8 6

11 13 7
2 14

3 2 5
4 3

1 21 25 28

, 10, 29
1 20 4 

10
4 14 2 3 19 1 15

7 13 5 3216 8 30
5 6 11 189 31

7 12 17

10



Continuing with this rule gives the following result:

It is worth noting that in addition to having relations involving 3 or more elements, it is also 

possible to have relations with just one element. Here is an example of a rule involving 

unary relations:

{{x}} → {{x, y}, {y}, {y}}

x y x

Starting from a unary self-loop, this rule leads to a binary tree with double-unary self-loops 
as leaves:

 , , ,

, , 

11



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A crucial simplifying feature of the rules we have considered so far is that they depend only 

on one relation, so that in a collection of relations, the rule can be applied separately to each 

relation (cf. [1:p82]). Put another way, this means that all the rules we have considered 

always transform single edges or hyperedges independently.

But consider a rule like:

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
This can be represented graphically as:

y y

z z w

x x

Here is the result of running the rule for several steps:

1

1

 3 1 2, 2 4 , 5 4 2,

3

3

2
3

7
2

4 4
2 4 5 , , 13

9 16 1 6 
18 7 1

9
6 7 3

11 10 15 12
17 5

3 14 8 1
6

1 1 5 10
8

12



Here is the result for 10 steps:

Despite the simplicity of the underlying rule, the structure that is built (here a�er 15 steps, 
and involving 6974 elements and 13,944 relations) is complex:

13



In getting this result, we are, however, glossing over an important issue that will occupy us 
extensively in later sections, and that potentially seems intimately connected with founda-
tional features of physics. 

With a rule that just depends on a single relation, there is in a sense never any ambiguity in 

where the rule should be applied: it can always separately be used on any relation. But with 

a rule that depends on multiple relations, ambiguity is possible.

Consider the configuration:

{{1, 2}, {1, 3}, {1, 4}, {1, 4}}
4 2

1

3

The rule

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
can be applied here in two distinct, but overlapping, ways. First, one can take:

{x → 1, y → 2, z → 3}
4 y→2

x→1

z→3

giving the result:

{{1, 3}, {1, 5}, {2, 5}, {3, 5}, {1, 4}, {1, 4}}
4 2

1 5

3

14



But one can equally well take:

{x → 1, y → 3, z → 4}
z→4 2

x→1

y→3

giving the inequivalent result:

{{1, 2}, {1, 4}, {1, 5}, {3, 5}, {4, 5}, {1, 4}}
4

1 5

2 3

With a rule that just depends on a single relation, there is an obvious way to define a 

single complete step in the evolution of the system: just make it correspond to the result 
of applying the rule once to each relation. But when the rule involves multiple relations, 
we have seen that there can be ambiguity in how it is applied (cf. [1:p501]), and one 

consequence of this is that there is no longer an obvious unique way to define a single 

complete step of evolution. For our purposes at this point, however, we will take each step 

to be what is obtained by scanning the configuration of the system, and finding the largest 
number of non-overlapping updates that can be made (cf. [1:p487]). In other words, in a 

single step, we update as many edges (or hyperedges) as possible, while never updating 

any edge more than once.

For now, this will give us a good indication of what kind of typical behavior different rules 
can produce. Later, we will study the results of all possible updating orders. And while this 
will not affect our basic conclusions about typical behavior, it will have many important 
consequences for our understanding of the models presented here, and their potential 
relevance to fundamental physics.

15



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We have seen that there can be several ways to apply a particular rule to a configuration of 
one of our systems. It is also possible that there may be no way to apply a rule. This can 

happen trivially if the evolution of the system reduces the number of relations it contains, 
and at some point there are simply no relations le�. It can also happen if the rule involves, 
say, only k -ary relations, but there are no k -ary relations in the configuration of the system. 

In general, however, a rule can continue for any number of steps, but then get to a configura-
tion where it can no longer apply. The rule below, for example, takes 9 steps to go from 

{{0,0,0},{0,0}} to a configuration that contains only a single 3-edge, and no 2-edges that match 

the pattern for the rule:

{{x, y, z}, {u, x}} → {{x, u, v}, {z, y}, {z, u}}

 , , , ,

, , , , 

It can be arbitrarily difficult to predict if or when a particular rule will “halt”, and we will see 

later that this is to be expected on the basis of computational irreducibility [1:12.6]. 

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All the rules we have seen so far maintain connectedness. It is, however, straightforward to 

set up rules that do not. An obvious example is:

{{x, y}} → {{y, y}, {x, z}}

16



 , , ,

, , 

At step n, there are 2n +1 components altogether, with the largest component having n + 1 

relations.

Rules that are themselves connected can produce disconnected results:

{{x, y}} → {{x, x}, {z, x}}

 , , , 

Rules whose le�-hand sides are connected in a sense operate locally on hypergraphs. But 
rules with disconnected le�-hand sides (such as {{x},{y}}→{{x,y}}) can operate non-locally 

and in effect knit together elements from anywhere—though such a process is almost 
inevitably rife with ambiguity.

17






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Having introduced our class of models, we now begin to study the general distribution of 
behavior in them. Like with cellular automata [1:2] and other kinds of systems defined by 

what can be thought of as simple computational rules [1:3, 4, 5], we will find great 
diversity in behavior as well as unifying trends. 

Any one of our models is defined by a rule that specifies transformations between collec-
tions of relations. It is convenient to introduce the concept of the “signature” of a rule, 
defined as the number of relations of each arity that appear on the le� and right of each 

transformation. 

Thus, for example, the rule

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
has signature 22 → 42 (and involves a total of 4 distinct elements). Similarly, the rule

{{a, a, b}, {c, d}} → {{b, b, d}, {a, e, d}, {b, b}, {c, a}}
has signature 1312 → 2322 (and involves 5 distinct elements). 

So far, we have always used letters to indicate elements in a rule, to highlight the fact that 
these are merely placeholders for the particular elements that appear in the configuration to 

which the rule is applied. But in systematic studies it is o�en convenient just to use integers 
to represent elements in rules, even though these are still to be considered placeholders (or 
pattern variables), not specific elements. So as a result, the rule just mentioned can be 

written:

{{1, 1, 2}, {3, 4}} → {{2, 2, 4}, {1, 5, 4}, {2, 2}, {3, 1}}
It is important to note that there is a certain arbitrariness in the way rules are written. The 

names assigned to elements, and the order in which relations appear, can both be rear-
ranged without changing the meaning of the rule. In general, determining whether two 

presentations of a rule are equivalent is essentially a problem of hypergraph isomorphism. 
Here we will give rules in a particular canonical form obtained by permuting names of 
elements and orders of relations in all possible ways, numbering elements starting at 1, and 

using the lexicographically first form obtained. (This form has the property that DeleteDupli-
cates[Flatten[{lhs,rhs}]] is always a sequence of successive integers starting at 1.) 

Thus for example, both

{{1, 1}, {2, 4, 5}, {7, 5}} → {{3, 8}, {2, 7}, {5, 4, 1}, {4, 6}, {5, 1, 7}}
and

{{7, 3}, {4, 4}, {8, 5, 3}} → {{3, 4, 7}, {5, 6}, {8, 7}, {3, 5, 4}, {1, 2}}
would be given in the canonical form

19



{{1, 2, 3}, {4, 4}, {5, 3}} → {{3, 2, 4}, {3, 4, 5}, {1, 5}, {2, 6}, {7, 8}}
From the canonical form, it is possible to derive a single integer to represent the rule. The 

basic idea is to get the sequence Flatten[{lhs,rhs}] (in this case 

{1, 2, 3, 4, 4, 5, 3, 3, 2, 4, 3, 4, 5, 1, 5, 2, 6, 7, 8} ) and then find out (through a generalized 

pairing or “tupling” function [3]) where in a list of all possible tuples of this length this 
sequence occurs [4]. In this example, the result is 310528242279018009. 

But unlike for systems like cellular automata [5][1:p53][6] or Turing machines [1: p888][7] 
where it is straightforward to set up a dense sequence of rule numbers, only a small fraction 

of integers constructed like this represent inequivalent rules (most correspond to non-
canonical rule specifications).

In addition—for example for applications in physics—one is usually not even interested in all 
possible rules, but instead in a small number of somehow “notable” rules. And it is o�en 

convenient to refer to such notable rules by “short codes”. These can be obtained by hashing 

the canonical form of the rule, but since hashes can collide, it is necessary to maintain a 

central repository to ensure that short codes remain unique. In our Registry of Notable 

Universes [8], the rule just presented has short code wm8678.

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Given a particular signature, one may ask how many distinct possible canonical rules there 

are with that signature. As a first step, one can ask how many distinct elements can occur in 

the rule. If the rule signature has terms niki  on both le� and right, the maximum conceiv-
able number of distinct elements is ∑ni ki. (For example, a possible canonical 22 → 22 rule is 
{{1,2},{3,4}}→{{5,6},{7,8}}.) 

But for many purposes we will want to impose connectivity constraints on the rule. For 
example, we may want the hypergraph corresponding to the relations on the le�-hand side 

of the rule to be connected [9], and for elements in these relations to appear in some way on 

the right. Requiring this kind of “le� connectivity” reduces the maximum conceivable 

number of distinct elements to ∑i∈LHSni(ki-1)+∑i∈RHSni ki (or 6 for 22 → 22). (If the right-hand 

side is also required to be a connected hypergraph, the maximum number of distinct 
elements is 1+∑ni(ki-1), or 5 for 22 → 22.)

Given a maximum number of possible elements m, an immediate upper bound on the 

number of rules is m∑ni ki . But this is usually a dramatic overestimate, because most rules 
are not canonical. For example, it would imply 1,679,616 le�-connected 22→ 22 rules, but 
actually there are only 562 canonical such rules.

The following gives the number of le�-connected canonical rules for various rule signatures 
(for n1→anything there is always only one inequivalent le�-connected rule):

20



12 → 12 11 13 → 13 178 14 → 14 3915
12 → 22 73 13 → 23 9373 14 → 24 2 022 956
12 → 32 506 13 → 33 637 568 14 → 34 1.7×109

12 → 42 3740 13 → 43 53 644 781 14 → 44 2.1×1012

12 → 52 28 959 13 → 53 5.4×109 14 → 54 ≈ 4×1015

22 → 12 64 23 → 13 8413 24 → 14 1 891 285
22 → 22 562 23 → 23 772 696 24 → 24 2.3×109

22 → 32 4702 23 → 33 79 359 764 24 → 34 3.5×1012

22 → 42 40 405 23 → 43 9.2×109 24 → 44 ≈ 9×1015

22 → 52 353 462 23 → 53 1.2×1012 24 → 54 ≈ 3×1019

32 → 12 416 33 → 13 568 462 34 → 14 1.6×109

32 → 22 4688 33 → 23 8.4×107 34 → 24 3.8×1012

32 → 32 48 554 3 34 → 34 ≈ 1×10163 → 33 1.4×1010

42 → 12 3011 43 → 13 4.9×107 44 → 14 2.1×1012

42 → 22 42 955 4 44 → 24 ≈ 9×10153 → 23 1.1×1010

52 → 12 23 211 53 → 13 5.3×109 54 → 14 ≈ 4×1015

Although the exact computation of these numbers seems to be comparatively complex, it is 
possible to obtain fairly accurate lower-bound estimates in terms of Bell numbers [10]. If 
one ignores connectivity constraints, the number of canonical rules is bounded below by 

BellB[∑ni ki]/∏ni!. Here are some examples comparing the estimate with exact results both 

for the unconstrained and le�-connected cases:

estimate unconstrained left-connected
11 → 21 2.5 4 1
12 → 22 102 117 73
12 → 32 690 877 506
13 → 23 10 574 10 848 9373
22 → 22 1035 1252 562
22 → 32 9665 12 157 4702
22 → 42 87 783 117 121 40 405

Based on the estimates, we can say that the number of canonical rules typically increases 
faster than exponentially as either ni or ki increase. (For 5≤n≤10 874, one finds 2n < BellB[n] 
< 2n log n, and for larger n, 2n<BellB[n]<nn.)

Note that given an estimate for unconstrained rules, an estimate for the number of le�-connected 

rules can be found from the fraction of randomly sampled unconstrained rules that are le� 

connected. For signature 1p → 1q, the number of unconstrained canonical rules is BellB[p+q ], but 
given the constraint of le�-connectedness there is only ever one canonical rule in this case. When 

there are no connectivity constraints, the number of canonical rules for signature a → b  is the 

same as for signature b → a. With the constraint of le� connectivity, the number of 12 → 52 rules 
is slightly larger than 52 → 12 rules, because there are fewer constraints in the former case. 

21



For any given signature, we can ask how many distinct elements occur in different canoni-
cal rules. Here are histograms for a few cases:

12→32 12→42 12→52 22→42

1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 6 7

22→52 13→33 13→23 23→33

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 1011

There are a number of features of rules that are important when using them in our models. 
First, if there are relations of arity k  on the le�-hand side of the rule, there must be relations 
of the same arity on the right-hand side if those relations are not just going to be inert under 
that rule. Thus, for example, a rule with signature a2 → b3 will never (on its own) apply more 

than once, regardless of the values of a and b. 

In addition, if a rule is going to have a chance of leading to growth, the number of 
relations of some arity on the right-hand side must be greater than the number of that 
arity on the le�. 

These two constraints, however, do not always apply if a complete rule involves several 
individual rules. Thus, for example, a complete rule containing individual rules with 

signatures 22 → 32 21, 22 11 → 12 can show growth, and can involve all relations. Note that 
since canonicalization is independent between different individual rules, the total number 
of possible inequivalent complete rules is just the product of the number of possible individ-
ual inequivalent rules. 

When investigating cases where a large number of inequivalent rules are possible, it will 
o�en be convenient to do random sampling. If one picks a random rule first, and then 

canonicalizes it, some canonical rules will be significantly more common than others. But it 
is possible to pick with equal probability among canonical rules by choosing an integer 
between 1 and the total number of rules, then decoding this integer as discussed above to 

give the canonical rule.

22



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In addition to enumerating rules, we can also consider enumerating possible initial condi-
tions. Like each side of a rule, these k

 can be characterized by sequences n 1 n k
1 2 2... which 

give the number of relations ni of arity ki. 

The only possible inequivalent 12 initial conditions are {{1,1}}, corresponding a graph 

consisting of a single self-loop, and {{1,2}}, consisting of a single edge. The possible inequiva-
lent connected 22 initial conditions are: 

{{{1, 1}, {1, 1}}, {{1, 1}, {1, 2}}, {{1, 1}, {2, 1}},
 {{1, 2}, {1, 2}}, {{1, 2}, {2, 1}}, {{1, 2}, {1, 3}}, {{1, 2}, {2, 3}}, {{1, 2}, {3, 2}}}
These correspond to the graphs:

 , , , ,

, , , 

The possible inequivalent 13 initial conditions are: 

{{{1, 1, 1}}, {{1, 1, 2}}, {{1, 2, 1}}, {{1, 2, 2}}, {{1, 2, 3}}}
These correspond to the hypergraphs:

 , , , , 

There are 102 inequivalent connected 23 initial conditions. Ignoring ordering of relations, 
these correspond to hypergraphs with the following structures: 

 , , , , , ,

, , , , , 

Ignoring connectivity, the number of possible inequivalent n1 initial conditions is Partition-
sP[n], the number of 1n ones is BellB[n], while the number of n2 ones can be derived using 

cycle index polynomials (see also [11:A137975]). The number of inequivalent connected 

initial conditions for various small signatures is as follows (with essentially the same Bell 
number estimates applying as for rules) [12]: 

23



12 2 72 40211 33 3268 24 2032 35 2.3×108

22 8 82 293370 43 164391 34 678358 45 2.1×1012

32 32 92 2255406 5 44 4.2×1083 1.1×107 16 203
42 167 102 18201706 63 9.0×108 54 4.1×1011 26 2089513
52 928 1 36 1.1×10113 5 73 7.1×1010 15 52
62 5924 23 102 14 15 25 57109 17 877

A rule can only apply to a given initial condition if the initial condition contains at least 
enough relations to match all elements of the le�-hand side of the rule. In other words, for a 

rule with signature nk →… there must be at least n k -ary relations in the initial condition. 

One way to guarantee that a rule will be able to apply to an initial condition is to make the 

initial condition in effect be a copy of the le�-hand side of the rule, for example giving an 

initial condition {{1,2},{1,3}} for a rule with le�-hand side {{x,y},{x,z}}. But the initial condi-
tion that in effect has the most chance to match is what is in many ways the simplest 
possible initial condition: the “self-loop” one where all elements are identical, or in this case 

{{0,0},{0,0}}. In what follows we will usually use such self-loop initial conditions Table[0,n,k ]. 

�����������������������������������������������
The very simplest possible rules are ones that transform a single unary relation, for example 

the 11 → 21 rule: 

{{x}} → {{x}, {y}}

This rule generates a disconnected hypergraph, containing 2n disconnected unary hyper-
edges at step n:

, , , , , ,

24



To get less trivial behavior, one must introduce at least one binary relation. With the 

11 → 1211 rule

{{x}} → {{x, y}, {x}}

one just gets a figure with progressively more binary-edge “arms” being added to central 
unary hyperedge:

, , , , , ,

The rule

{{x}} → {{x, y}, {y}}

produces a growing linear structure, progressively “extruding” binary edges from the unary 

hyperedge:

, , , ,

With two unary relations and one binary relation (signature 11 → 1221) there are 16 possible 

rules; a�er 4 steps starting from a single unary relation, these give:

25



Many lead to disconnected hypergraphs; four lead to binary trees with structures we have 

already seen. ({{x}}→{{x,y},{x},{y}} is a 12 → 1221 rule that gives the same result as the very 

first 12 → 22 rule we saw.

26



Rules for a single unary relation can never give structures more complex than trees, though 

the morphology of the trees can become slightly more elaborate:

������������������������������������������������
There are 73 inequivalent le�-connected 12 → 22 rules, but none lead to structures more 

complex than trees. Starting each from a single self-loop, the results a�er 5 steps are (note 

that even a connected rule like {{x,y}}→{{x,z},{z,x}} can give a disconnected result): 

27



With an initial condition consisting of a square graph, the following very similar results are 

obtained:

There are 506 inequivalent le�-connected 12 → 32 rules. Running all these rules for 5 steps 
starting from a single self-loop, and keeping only distinct connected results, one gets (note 

that similar-looking results can differ in small-scale details):

28



Several distinct classes of behavior are visible. Beyond simple lines, loops, trees and radial 
“bursts”, there are nested (“cactus-like”) graphs such as 

 , , ,

, , 

obtained from the rule

{{x, y}} → {{z, z}, {x, z}, {y, z}}

The only slightly different rule

{{x, y}} → {{x, x}, {x, y}, {z, x}}

gives a rather different structure:

 , , , ,

, 

29



A layered rendering makes the behavior slightly clearer:

 , , , , , 

Another notable rule similar to one we saw in the previous section is:

{{x, y}} → {{x, z}, {x, z}, {z, y}}

From a single edge this gives:

 , , ,

, , 

Starting from a single self-loop gives a more complex topological structure (and copies of 
this structure appear when the initial condition is more complex): 

 , , ,

, , 

Another notable 12 → 22 rule is:

{{x, y}} → {{x, y}, {y, z}, {z, x}}

30



which produces an elaborately filled-in structure:

 , , ,

, , 

A�er 8 steps, the structure has the form:

31



A�er t  steps, there are 3t–1 nodes, and 1
 (3t + 1) edges. The graph diameter is 2 t – 1 if 
6

directions of edges are taken into account, and t – 1 if they are not. The maximum degree of 
any vertex is 2t—and all vertices have degrees of the form 2s, with the number of vertices of 
degree 2s being proportional to 3t–s. 

Starting from a single edge makes it slightly easier to understand what is going on: 

 , , ,

, , 

As the rule indicates, every edge of every triangle “sprouts” a new triangle at every step, in 

effect producing a sequence of “frills upon frills”. But even though this may seem compli-
cated, the whole structure basically corresponds just to a ternary tree in which each node is 
replaced by a triangle: 

Starting from a single self loop, all 12 → 32 rules give a�er n steps a number of relations that 
is either constant, or goes like 2 t – 1, 2t – 1 or 3t–1.

For 12 → 42, there are 3740 distinct le�-connected rules. As suggested by the random cases 
below, their behavior is typically similar to 12 → 32 rules, though the forms obtained can be 

somewhat more elaborate: 

32



For example, the rule

{{x, y}} → {{y, z}, {y, z}, {z, y}, {z, x}}

gives the following:

The rule

{{x, y}} → {{z, w}, {w, z}, {z, x}, {z, y}}

 gives a nested form:

33



The rule

{{x, y}} → {{y, z}, {y, w}, {z, w}, {z, x}}

gives

while the similar rule

{{x, y}} → {{x, z}, {x, w}, {z, w}, {z, y}}

34



gives

Successive steps in effect just fill in this shape, which seems somewhat irregular when 

rendered in 2D, but appears more regular if rendered in 3D. 

Another rule with a simple structure when rendered in 3D is

{{x, y}} → {{y, z}, {y, z}, {z, x}, {z, x}}

which yields:

35



The outputs from 12 → 42 rules all grow either linearly (for example, like 3 t – 2), or exponen-
tially, asymptotically like 2t , 3t  or 4t . The number of relations a�er t steps is always given by 

a linear recurrence relation; for the rule {{x,x}}→{{x,x},{x,x},{x,y},{x,y}} the recurrence is 
f[t]=3f[t–1]–2f[t–2] (with f[1]=1, f[2]=4), giving size 1 (3×2t – 4).

2

������������������������ �������������������
There are 9373 inequivalent le�-connected 13→ 23 rules. Here are typical examples of their 
behavior a�er 5 steps, starting from a single ternary self-loop:

Here are results from a few of these rules a�er 10 steps:

36



The number of relations in the evolution of 13→ 23 rules can grow in a slightly more compli-
cated way than for 12→ n2 rules. In addition to linear and 2t  growth, there is also, for 
example, quadratic growth: in the rule

{{x, x, y}} → {{y, y, z}, {x, y, x}}

each existing “arm” effectively grows by one element each step, and there is one new arm 

generated, yielding a total size of t k 1
 ∑k=2  = (t 2 + t – 2): 

2

{ , , , , , , }

The rule

{{x, x, y}}→ {{y, y, y}, {x, y, z}}

yields a Fibonacci tree, with size Fibonacci[t+2]–1 ~ ϕt :

{ , , , , , , }

13 → 23 rules can produce results that look fairly complex. But it is a consequence of their 
dependence only on a single relation that once such rules have established a large-scale 

structure, later updates (which are necessarily purely local) can in a sense only embellish it, 
not fundamentally change it:

37



There are 637,568 inequivalent le�-connected 13 → 33 rules; here are samples of their 
behavior:

The results can be more elaborate than for 13 → 23 rules—as the following examples illus-
trate—but remain qualitatively similar:

38



One notable 13 → 33 rule (that we will discuss below) in a sense directly implements the
recursive formation of a nested Sierpiński pattern:

{{x, y, z}}→ {{x, u, v}, {z, v, w}, {y, w, u}}

������������������������ ��������� �������������
����������� 2� 2 → 32 � ���
The smallest nontrivial signature that can lead to growth (and therefore unbounded evolu-
tion) is 22 → 32. There are 4702 distinct le�-connected rules with this signature. Here is a
random sample of the behavior they generate, starting from a double self-loop {{0,0},{0,0}}
and run for 8 steps:

Restricting to connected cases, there are 291 distinct outputs involving more than 10 rela-
tions a�er 8 steps:

39



The overall behavior we see here is very similar to what we saw with rules depending only 

on a single relation. But there is a new issue now to be addressed. With rules depending only 

on a single relation there is never any ambiguity about where the rule should be applied. But 
with rules that depend on more than one relation, there can be ambiguity, and the results 
one gets can potentially depend on the order in which updating is done. 

40



Consider the rule

{{x, y}, {x, z}} → {{x, w}, {y, w}, {z, w}}

With our standard updating order, the result of running this rule for 30 steps is:

But with 6 different choices of random updating orders one gets instead:

None of these graphs are isomorphic, but all of them are qualitatively similar. Later on, we 

will discuss in detail the consequences of different updating orders, and their potentially 

important implications for physics. But for now, suffice it to say that at a qualitative level 
different updating orders typically lead to similar behavior.

As an example of something of an exception, consider the 22 → 32 rule shown in the array 

above:

{{x, y}, {y, z}} → {{x, w}, {w, z}, {z, x}}

41



With our standard updating order, this rule behaves as follows, yielding complicated-
looking results with about 1.5n relations a�er n steps:

But with random updating order, the behavior is typically quite different. Here are six 

examples of results obtained a�er 10 steps—and all of them are disconnected:

 , , , , , 

����������� ��������������22→ 42
For 22 → 42, there are 40,405 inequivalent le�-connected rules. Of these, about 36% stay 

connected when they evolve. Starting from two self-loops {{0,0},{0,0}}, and running for 8 

steps, here is a sample of the 4000 or so distinct behaviors that are produced: 

42



43



Most of these rules show the same kinds of behaviors we have seen before. But there is one 

major new kind of behavior that is observed: in a little less than 1% of all cases, the rules 
produce globular structures that in effect continually add various forms of cross-connec-
tions. Here are a few examples (notably, even though 22 → 42 rules can involve up to 7 

distinct elements, these rules all involve just 4):

44



We will study these kinds of structures in more detail later. Note that the specific forms 
shown here depend on the underlying updating order used—though for example random 

orders typically seem to give similar results. It is also the case that the detailed visual layout 
of graphs can affect the impression of these structures; we will address this in the next 
section when we discuss various forms of quantitative analysis. 

It is remarkable how complex the structures are that can be created even from very simple 

rules. Here are three examples (with short codes wm5583, wm4519, wm2469) shown in more 

detail:

{{x, y}, {x, z}} → {{y, z}, {y, w}, {z, w}, {w, x}}

45



{{x, y}, {y, z}} → {{x, y}, {y, x}, {w, x}, {w, z}}

{{x, y}, {y, z}} → {{w, y}, {y, z}, {z, w}, {x, w}}

46



Much as we have seen in other systems such as cellular automata [1], there seems to be no 

simple way to deduce from the rules from our systems here what their behavior will be. And 

indeed even seemingly very similar rules can give dramatically different behavior, some-
times simple, and sometimes complex. 

������������������ ����������������������22 → 42
Going from signature 22 → 32 to signature 22 → 42 brought us the phenomenon of globular 
structures. Going to signature 22 → 52 and beyond does not seem to bring us any similarly 

widespread significant new form of behavior. The fraction of rules that yield connected 

results decreases, but among connected results, similar fractions of globular structures are 

seen, with examples from 22 → 52 including:

47



The last rule shown here has a feature that is seen in a few 22 → 42 rules, but is more promi-
nent in 22 → 52 rules: the presence of many “dangling ends” that at least visually obscure the 

structure. To see the structure better, one can take the evolution of this rule

and effectively just “edit” the graphs obtained at each step, removing all dangling ends:

48



In addition to increasing the number of relations on the right-hand side of the rule, one can
also increase the number on the le�. For example, one can consider 32 → 42 rules. These
much more o�en lead to termination than 22 →… rules, and appear to produce results
generally similar to 22 → 32 rules.

32 → 52 rules also produce globular structures, though more rarely than 22 → 42 rules, and
with slower growth. A few examples are:

������������������������������������������������
����������� �� 2� 3 → 33 � ���
There are 79,359,764 inequivalent le�-connected 23 → 33 rules. The fraction of these rules
showing continued growth is considerably smaller than for 22 →… rules. But here is a
typical sample of growth rules (note that different rules are run for different numbers of
steps to achieve a roughly balanced level of detail):

49



And even though there are only 3 relations on the right-hand side (rather than the 4 in 

22 → 42) these rules can produce globular structures. Some examples are:

50



A new phenomenon exhibited by 23 → 33 rules is the formation of globular structures by 

what amounts to slow grow. This is exemplified by a rule like:

{{x, y, z}, {x, u, v}} → {{x, w, u}, {v, w, y}, {w, y, z}}

This rule progressively builds up a structure by growing only in one place at a time (the 

position of the surviving self-loop):

 , , , , , ,

, , , , ,

, , , , 

A�er 1000 steps the rule has produced this structure containing 1000 ternary relations (plus 
the 2 already present in the initial condition):

51



Another example of slow growth occurs in the rule

{{x, x, y}, {z, u, x}} → {{u, u, z}, {v, u, v}, {v, y, x}}

which a�er 1000 steps generates:

Note the presence here of regions of square grids. These occur even more prominently in 

the rule

{{x, y, z}, {u, y, v}} → {{w, z, x}, {z, w, u}, {x, y, w}}

which a�er 500 steps produces:

52



As we will discuss in the next section, the grid here becomes quite explicit when the hyper-
graph is rendered in 3D. Notice that the grid is not evident even a�er 20 steps in the evolu-
tion of the rule; it takes longer to emerge:

 , , , , , , , , , , ,

, , , , , , , , , 

Once again, though, the rule adds just a single relation at each generation; in effect the grid 

is being “knitted” one node at a time.

The emergence of a grid is still easier to see in the rule:

{{x, y, z}, {x, u, v}} → {{z, z, w}, {w, w, v}, {u, v, w}}

which a�er 200 steps yields:

53



Once again, the “knitting” of this form is far from obvious in the first 20 steps of evolution:

Just sometimes, however, the behavior is quite easy to trace, as in this particularly direct 
example of “knitting”:

{{x, y, y}, {z, x, u}} → {{y, v, y}, {y, z, v}, {u, v, v}}

which a�er 200 steps yields:

As a different example of slow growth, consider the rule

{{x, y, y}, {y, z, u}} → {{u, z, z}, {u, x, v}, {y, u, v}}

54



A�er 200 steps this rule gives

while a�er 500 steps it gives:

Looking at all 79 million or so 23 → 33 rules in canonical order, one finds that rules with slow 

growth are quite rare and are strongly localized to about 10 broad regions in the space of 
possible rules. Of rules with slow growth, only a few percent form nontrivial globular 
structures. And of these, perhaps 10% exhibit obvious lattice-like patterns. 

The pictures below show additional examples. Note that—as we will discuss later—many of 
the patterns here are best visualized in 3D.

55



���������������������� ���������������������
There are about 9 billion inequivalent le�-connected 23 → 43 rules. About 20% lead to 

connected results, and of these about half show continued growth. Here is a random 

sampling of the behavior of such rules:

The fraction of complex behavior appears to be no higher than for 23 → 33 rules, and no 

obvious major new phenomena are seen. Much like in systems such as cellular automata 

(and as suggested by the Principle of Computational Equivalence [1:12]), above some low 

threshold, adding complexity to the rules does not appear to add complexity to the typical 
behavior produced.

The trend continues with 33 → 43 rules, with one notable feature here being an increased 

propensity for rules to yield results that become disconnected, though only a�er many 

steps. The general difficulty of predicting long-term behavior is illustrated for example by 

the evolution of this 33 → 53 rule, sampled every 10 steps:

56



������������ ���� ����������
So far essentially all the rules we have considered have “pure signatures” of the form 

mk → nk  for some arity k . Continued growth is never possible unless the right-hand side of a 

rule contains some relations with the same arity as appear on the le�. But, for example, it is 
perfectly possible to have growth in rules with signatures like 12 → 2221. Such rules produce 

unary relations, which can serve as “markers” for the application of the rule, but cannot 
themselves affect how or where the rule is used:

57



The 634 rules with signature 12 → 1312 all show very simple behavior (as do the 2212 rules 
with signature 12 → 131211), with not even trees being possible. But among the 7652 12→ 1322 
rules there are not only many trees, but also closed structures such as:

Previously we had only seen structures like the first one above in rules that depend on more 

than one relation. But as this illustrates, such structures can be produced even with just a 

single relation on the le�:

{{x, y}} → {{x, x, y}, {x, z}, {z, y}}

 , , , , ,

, , , 

The 44,686 rules with signature 12 → 2312 cannot even produce trees. Rules with signature 

13 → 2312 can produce trees, as well as closed structures similar to those seen in 12 → 1322 rules.

A minimal way to add mixed arity to the le�-hand sides of rule is to introduce unary rela-
tions—but the presence of these seems to inhibit the production of any more complex forms 
of behavior.

Looking at mixed binary and ternary le�-hand sides, none of the 1,141,692 rules with 

signature 1312→ 1322 seem to produce even trees. But rules with signature 1312 → 2322 
readily produce structures such as:

58



One can go on and look at rules with higher signatures, and probably the most notable 

finding is that—in keeping with the Principle of Computational Equivalence [1:12]—the 

overall behavior seen does not appear to change at all. Here are nevertheless a few exam-
ples of slightly unusual behavior found in 2312 → 3322 and 2312→ 4342 rules:

������ ����������������������������
So far we have always considered having just a single possible transformation rule which 

can be used wherever it applies. It is also possible to have multiple transformation rules 
which are used wherever they apply. A single transformation rule can either increase or 
decrease the number of relations, but must do the same every time it is used. With multiple 

transformations, some can increase the number of relations while others decrease it. 

As a minimal example, consider the rule

{{{x, x}} → {{y, x}, {x, z}}, {{x, y}, {y, z}} → {{x, x}}}

59



On successive steps, this rule simply alternates between two cases:

 , , , , , , , 

As another example, consider the rule

{{{x, x}} → {{y, x}, {y, x}, {z, x}}, {{x, y}, {z, y}} → {{y, y}}}

This rule produces results that alternately grow and shrink on successive steps:

 , , , , , , ,

, , , , , , ,

, , , , , , 

It is fairly common with multiple transformation rules to find that one transformation is 
occasionally applied. But at least with our standard updating order, it is difficult to find rules 
in which, for example, the total size of the results varies in anything but a fairly regular way 

from step to step. 

�����������������������������������������
We have mostly restricted ourselves so far to cases where the results generated by a rule 

remain connected. But in fact if one looks at all possible rules the majority generate discon-
nected pieces, or at least can do so for certain initial conditions. Among the 73 rules with 

signature 12 → 22, only 33 generate connected results starting from initial condition {{0,0}} 
(and a further 10 terminate from this initial condition). 

(Note that we are ignoring order in determining connectivity, so that, for example, the 

relation {1,2} is considered connected not only to {2,3} but also to {1,3}. Translating binary 

relations like these into directed edges in a graph, this means we are considering weak 

connectivity, or, equivalently, we are looking only at the undirected version of the graph.)

60



Most 12 → 22 rules that yield disconnected results essentially just produce exponentially 

more copies of the same structure:

{{x, y}} → {{y, z}, {y, z}}

 , , , , , 

(Note that this rule is an example of one that yields disconnected results even though the 

rule itself is not disconnected.)

A few rules show slightly more complicated behavior. Examples are (wm575, wm879):

{{x, y}} → {{y, y}, {x, z}}

 , , , , , 

{{x, y}} → {{x, x}, {y, z}}

 , , , , , 

Both these rules still show exponentially increasing numbers of connected components. In 

the first case, at step t  there are components of all sizes 1 through t , in exponentially 

decreasing numbers. In the second case, the size of the largest component is

1, 2, 2, 3, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, ...

or asymptotically ~ϕn (it follows the recurrence f[n]=2f[n–1]–f[n–3]).

61



Note that if one tracks only the largest component, one gets a sequence of results that could 

only be generated by a rule involving several separate transformations (in this case 

{{x,x}}→{{x,y},{x,x}} and {{x,y}}→{{x,x}}). In general, with a single transformation, the total 
number of relations must either always increase or always decrease. But if there are discon-
nected pieces, and one tracks, say, only the largest component, one can get a sequence of 
results that can both increase and decrease in size. 

As an example, consider the rule: 

{{x, y}, {x, z}} → {{y, z}, {z, y}, {x, w}}

Evolving this rule with our standard updating order gives:

 , , , , ,

, , , ,

, , , 

The total number of relations increases roughly exponentially. But tracing only the largest 
component, we see that it oscillates in size, eventually settling into the cycle 5,8,9,8:

 , , , , , , , ,

, , , , , , , , ,

, , , , , , , , 

62



Note that this result is quite specific to the use of our standard updating order. A random 

updating order, for example, will typically give larger results for the largest component, and 

no cycle will normally be seen.

It is quite common to see rules that sometimes yield connected results, and sometimes do 

not. (In fact, proving that a given rule in a given case can never generate disconnected 

components can be arbitrarily difficult.) Sometimes there can be a large component with a 

complex structure, with small disconnected pieces occasionally getting “thrown off”. 
Consider for example the rule: 

{{x, y}, {y, z}} → {{x, w}, {w, x}, {z, x}}

With the standard updating order, it remains connected for 10 steps, then suddenly starts 
throwing off small disconnected pieces. 

As a more elaborate example, consider the rule: 

{{x, y, z}, {u, v, z}} → {{y, w, u}, {w, x, y}, {u, y, x}}

63



This remains connected for 16 steps, then starts throwing off disconnected pieces:

 , , , , , ,

, , , , ,

, , , , ,

, , , , 

With a rule like this, once components become disconnected, they can in a sense never 
interact again; their evolutions become completely separate. The only way for disconnected 

components to interact is to have a rule which itself has a disconnected le�-hand side. 

For example, a rule like

{{x}, {y}} → {{x, y}}

will collect even completely disconnected unary relations, and connect pairs of them into 

binary relations:

 , 

Connected unary relations (i.e. such as {1}, {1}, ...) can end up in the same component, but 
the result depends critically on the order in which updates are done: 

 , 

For now, we will not consider any further rules with disconnected le�-hand sides—and the 

extreme nonlocality they represent.

64



�����������������
Not all rules continue to evolve forever from a given initial state. Instead they can reach a 

fixed point where the rule no longer applies. If the rule depends only on a single relation, 
this can only happen at the very first step. But if the rule depends on multiple relations, it 
can happen a�er multiple steps. Among the 4702 rules with signature 22 → 32, 1788 rules 
eventually reach a fixed point starting from a self-loop initial condition, at least using our 
standard updating order. Their “halting time” decreases roughly exponentially, with the 

maximum being 7 steps, achieved by the rule:

{{x, y}, {z, y}} → {{y, u}, {u, x}, {v, z}}

 , , , , , , , 

The longest halting time for which connectedness is maintained is 3 steps, achieved for 
example by:

{{x, y}, {y, z}} → {{y, z}, {y, u}, {v, z}}

 , , , 

Among the 40,405 22 → 42 rules, 10,480 evolve to fixed points starting from self-loops. The 

maximum halting time is 13 steps; the maximum maintaining connectedness is 6 steps, 
achieved by:

{{x, x}, {y, x}} → {{y, y}, {y, z}, {z, x}, {w, z}}

 , , , , , , 

Among the 353,462 22→ 52 rules, 67,817 (or about 19%) evolve to fixed points. The maximum 

halting time is 24 steps; the maximum maintaining connectedness is 10 steps, achieved for 
example by

{{x, x}, {y, x}} → {{y, y}, {y, z}, {y, z}, {z, x}, {w, z}}

65



 , , , , ,

, , , , , 

Among 23 → 33 rules

{{x, y, z}, {x, u, v}} → {{y, x, w}, {w, u, s}, {v, z, u}}
has halting time 20:

 , , , , , , ,

, , , , , , ,

, , , , , , 

��������������������������������������
For rules that depend on only a single relation, adding relations to initial conditions always 
just leads to replication of identical structures, as in these examples for the rule

{{x, y}} → {{y, z}, {z, x}}

 , , , , , 

{ , , , , , }

66



 , , , , , 

{ , , , , , }

Sometimes, however, the layout of hypergraphs for visualization can make the replication of 
structures a little less obvious, as in this example for the rule

{ , , , , , }

For rules depending on more than one relation, initial conditions can have more important 
effects. Starting the rule

{{x, y}, {y, z}} → {{w, z}, {w, z}, {x, w}, {y, z}}
from all 8 inequivalent 2-relation and all 32 inequivalent 3-relation initial conditions, one 

sees quite a range of behavior:

But in other rules—particularly many of those such as

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}

67



that yield globular structures—different initial conditions (so long as they lead to growth at 
all) produce behavior that is different in detail but similar in overall features, a bit like what 
happens in class 3 cellular automata such as rule 30 [1:p251]):

For the evolution of a rule to not just immediately terminate, the le�-hand side of the rule 

must be able to match the initial conditions given (and so must be a sub-hypergraph of the 

initial conditions). This is guaranteed if the initial conditions are in effect just a copy of the 

le�-hand side. But the most “fertile” initial conditions, with the most possibility for different 
matches, are always self-loops: in particular, n k -ary self-loops for a rule with signature 

nk →…. And in what follows, this is the form of initial conditions that we will most o�en use. 

One practical issue with self-loop initial conditions, however, is that they can make it 
visually more difficult to tell what is going on. Sometimes, for example, initial conditions 
that lead to slightly less activity, or enforce some particular symmetry, can help. Note, 
however, that in the evolution of rules that depend on more than one relation, there may be 

no way to preserve symmetry, at least with any specific updating order (see section 6). Thus, 
for example, the rule

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
with our standard updating order gives:

68



Another feature of initial conditions is that they can affect the connectivity of the results 
from a rule. Thus, for example, even in the case of the rule above that generates a grid, 
initial conditions consisting of different numbers of 3-ary self-loops lead to differently 

connected results:

 , , , ,

4
2

3 5

, , , , 

6 7 8 9 10

��������������� ������� �����
Essentially all the rules we have considered so far have been set up to add relations. But with 

extended initial conditions, it makes sense also to consider rules that maintain a fixed 

number of relations. 

Rules with signatures like 12 → 12 that depend only on one relation cannot give rise to 

nontrivial behavior. But rules with signature 22 → 22 (of which a total of 562 are inequiva-
lent) already can. 

Consider the rule

{{x, y}, {y, z}} → {{y, x}, {z, x}}

Starting from a chain of 5 binary relations, this is the behavior of the rule:

 , , , , ,

, , , , , 

69



Given that only a finite number of elements and relations are involved, the total number of 
possible states of the system is finite, so it is inevitable that the evolution of the system must 
eventually repeat. In the particular case shown here, the repetition period is only 3 steps. (Note 

that the detailed behavior—and the repetition period—can depend on the updating order used.)

In general, the total number of possible states of the system is given by the number of 
distinct hypergraphs of a certain size. One can then construct a state transition graph for 
these states under a rule. Here is the result for the rule above with the 32 distinct connected 

32 hypergraphs (note that with this rule, the hypergraphs always remain connected): 

The result for all 928 52 hypergraphs is:

70



This graph contains trees corresponding to transients, leading into cycles. The maximum 

cycle length in this case is 5. But when the size of the system increases, the lengths of cycles 
can increase rapidly (cf. [1:6.4]). The length is bounded by the number of distinct nk  hyper-
graphs, which grows faster than exponentially with n. The plot below shows the lengths of 
cycles and transients in the rule above for initial conditions consisting of progressively 

longer chains of relations: 

500

100
50 cycles
10
5 transients
1
0 5 10 15 20 25 30

����������������������� ��������������������������������
Here are samples of random rules with various signatures (only connected results are 

included):

22→32

22→42

22→52

22→62

22→72

22→82

22→92

22→102

71



32→42

32→52

32→62

32→72

32→82

32→92

32→102

32→42

32→52

32→62

32→72

32→82

32→92

32→102

As expected from the Principle of Computational Equivalence [1:12], above a low threshold 

more complex rules do not generally lead to more complex behavior, although the frequen-
cies of different kinds of behavior do change somewhat. 

At a basic visual level, one can identify several general types of behavior:
• Line-like: elements are connected primarily in sequences (lines, circles, etc.)
• Radial: most elements are connected to just a few core elements
• Tree-like: elements repeatedly form independent branches
• Globular: more complex, closed structures

Inevitably, these types of behavior are neither mutually exclusive, nor precisely defined. There are 

certainly specific graph-theoretic and other methods that could be used to discriminate different types, 
but there will always be ambiguous cases (and sometimes it will even be formally undecidable what 
category something is in). But just like for cellular automata—or for many systems in the empirical 
sciences—classifications can still be useful in practice even if their definitions are not unique or precise.

72



As an alternative to categorical classification, one can also consider systematically arranging 

behaviors in a continuous feature space (e.g. [13]). The results inevitably depend on how 

features are extracted. Here is what happens if one takes images like the ones above, and 

directly applies a feature extractor trained on images of picturable nouns in human lan-
guage [14]:

Here is the fairly similar result based on feature extraction of underlying adjacency matrices:

73



In addition to characterizing the behavior of individual rules, one can also ask to what extent 
behavior is clustered in rule space. Here are samples of what happens if one starts from 

particular 22 → 72 rules, then looks at a collection of “nearby” rules that differ by one 

element in one relation:

 , {,

 , {,

 , {,

 , {

And what we see is that even though there are only 68 million or so 22 → 72 rules, changing 

one element (out of 14) still usually gives a rule whose overall behavior is similar.

74



� |�� ��� �������� ���������������
��������

������������������ �������
Particularly for potential applications to fundamental physics, it will be of great importance
to understand what happens if we run our models for many steps—and to find ways to
characterize overall behavior that emerges. Sometimes the characterization is easy. One gets
a loop with progressively more links:

{{x, y}}→ {{y, z}, {z, x}}

 , , , , , 

Or one gets a tree with progressively more levels of branching:

{{x}} → {{x, y}, {y}, {y}}

 , , , , , , , 

But what about a case like the following? Is there any way to characterize the limiting
behavior here?

{{x, y}, {x, z}}→ {{x, y}, {x, w}, {y, w}, {z, w}}

 , , , , ,

, , , , 

75



It turns out that a rare phenomenon that we saw in the previous section gives a critical clue.
Consider the rule:

{{x, y, y}, {z, x, u}}→ {{y, v, y}, {y, z, v}, {u, v, v}}

Looking at the first 10 steps it is not clear what it will do:

 , , , , ,

, , , , , 

But showing the results every 10 steps therea�er it starts to become clearer:

 , , , , , , , , , 

And a�er 1000 steps it is very clear: the rule has basically produced a simple grid:

Geometrical though this looks, it is important to understand that at a fundamental level
there is no geometry in our model: it just involves abstract collections of relations. Our
visualization methods make it evident, however, that the pattern of these relations corre-
sponds to the pattern of connections in a grid.

76



In other words, from the purely combinatorial structure of the model, what we can interpret 
as geometrical structure has emerged. And if we continue running the model, the grid in our 
picture will get finer and finer, until eventually it approximates a triangular piece of continu-
ous two-dimensional space.

Consider now the rule 

{{x, x, y}, {x, z, u}} → {{u, u, z}, {y, v, z}, {y, v, z}}

Looking at the first 10 steps of evolution it is again not clear what will happen:

 , , , , ,

, , , , , 

But a�er 1000 steps a definite geometric structure has emerged:

77



There is evidence of a grid, but now it is no longer flat. Visualizing in 3D makes it clearer 
what is going on: the grid is effectively defining a 2D surface in 3D:

To make its form clearer, we can go for 2000 steps, and include an approximate surface 

reconstruction [15]:

The result is that we can identify that in the limit this rule can be characterized as creating 

what is essentially a cone. 

Other rules produce other shapes. For example, the rule

{{x, y, z}, {u, y, v}} → {{w, z, x}, {z, w, u}, {x, y, w}}

78



gives a�er 1000 steps:

The structure is clearer when visualized in 3D:

Despite its smooth form, there does not seem to be a simple mathematical characterization 

of this surface. (Its three-lobed structure means it cannot be an ordinary algebraic surface 

[16]; it is similar but not the same as the surface r = sin(ϕ) in spherical coordinates.)

79



Changing the initial condition from to  yields the rather different surface:

This rule gives a closer approximation to a sphere, though there is a definite threefold 

structure to be seen:

{{x, y, y}, {x, z, u}} → {{u, v, v}, {v, z, y}, {x, y, v}}

80



Simpler forms, such as cylindrical tubes, also appear:

{{x, x, y}, {z, u, y}} → {{u, u, y}, {x, v, y}, {x, v, z}}

It is worth pausing for a moment to consider in what sense the limiting object here “is” a 

tube. What we ultimately have is a collection of relations which define a hypergraph. But 
there is an obvious measure of distance on the hypergraph: how many relations you have to 

follow to go from one element to another. So now one can ask whether there is a way to 

make this hypergraph distance between elements correspond to an ordinary geometrical 
distance. Can one assign positions in space to the elements so that the spatial distances 
between them agree with the hypergraph distances? 

The answer in this case is that one can—by placing the elements at a lattice of positions on 

the surface of a cylinder in three-dimensional space. (And, conveniently, it so happens that 
our visualization method for hypergraphs basically automatically does this.) But it is impor-
tant to realize that such a direct correspondence with an easy-to-describe surface is a rare 

and special feature of the particular rule used here.

Consider the rule:

{{x, x, y}, {x, z, u}} → {{u, u, v}, {v, u, y}, {z, y, v}}

81



A�er 1000 steps, this rule produces:

In 3D, this can be visualized as:

There are many subtle issues here. First, at every step the rule adds more elements, and in 

principle this could change the emergent geometry. But it appears that a�er enough steps, 
there is a definite limiting shape. Unlike in the case of a cylinder, however, it is much less 
clear how to assign spatial coordinates to different elements. It does not help that the 

limiting shape does not appear to have a completely smooth surface; instead there are 

places at which it appears to form cusps (reminiscent of an orbifold [17]).

82



There are rules that give more obvious “singularities”; an example is:

{{x, x, y}, {y, z, u}} → {{v, v, u}, {v, u, x}, {z, y, v}}

Some rules produce surfaces with complex folds: 

{{x, x, y}, {z, u, x}} → {{z, z, v}, {y, v, x}, {y, w, v}}

83



It is also perfectly possible for the emergent geometry to have nontrivial topology. This rule 

produces a (strangely twisted) torus:

{{x, x, y}, {z, u, x}} → {{x, x, z}, {u, v, x}, {y, v, z}}

All the emergent geometries we have seen so far in effect involve a regular mesh. But this 
rule, instead uses a mixture of triangles, quadrilaterals and pentagons to cover a region:

{{x, y, x}, {x, z, u}} → {{u, v, u}, {v, u, z}, {x, y, v}}

84



��������������������
Among all possible rules, the formation of geometrical shapes of the kind we have just been 

discussing is very rare. Slightly more common is the type of behavior that we see in a rule 

like:

{{x, y}, {y, z}} → {{w, x}, {w, y}, {x, y}, {y, z}}

 , , , , ,

, , , , 

Essentially the same behavior also occurs in a mixed-arity rule with a single relation on the 

le�-hand side:

{{x, y}} → {{z, y, x}, {y, z}, {z, x}}

 , , , , ,

, , , , 

85



We can think of the structure that is produced as being like a binary tree of triangles:

 , , , , , 

The same structure can be produced from an Apollonian circle packing (e.g. [18][1: p985]):

 , , , 

If each triangle is required to have the same area, the structure can be rendered in 2D as:

 , , , , , 

If we tried to render this with every triangle roughly the same size, then even in 3D the best 
we could do would be to have something that crinkles up more and more at the edge, like an 

idealized lettuce leaf:

86



But just as we can think of the grids we discussed before as being regularly laid out in 

ordinary 2D or 3D space, so now we can think of the object we have here as being regularly 

laid out in a hyperbolic space [19][20] of constant negative curvature. 

In particular, the object corresponds to an infinite-order triangular tiling of the hyperbolic 

plane (with Schläfli symbol {3,∞}). There are a variety of ways to visualize the hyperbolic 

plane. One example is the Poincaré disk model in which hyperbolic-space straight lines are 

rendered as arcs of circles orthogonal to the boundary:

 , , , , 

(The particular graph here happens to be the Farey graph [21].)

����� ������������������������
The grids and surfaces that we saw above were all produced by rules that end up executing a 

laborious “knitting” process in which they add just a single relation at each step. But it is also 

possible to generate recognizable geometric forms more quickly—in effect by a process of 
repeated subdivision. 

Consider the 2312 → 4342 rule:

At each step, this rule doubles the number of relations—and quickly produces a structure
with a definite emergent geometrical form:

87



A�er 10 steps the rule has generated 2560 relations, in the following structure:

Visualized in 3D, this becomes:

Once again, this corresponds to a smooth surface, but with 3 cusps. The surface is defined 

not by a simple triangular grid, but instead by an octagon-square (“truncated square”) tiling—
that in this case becomes twice as fine at every step.

Changing the initial conditions can give a somewhat different structure: 

88



Visualized in 3D a�er 10 steps (and reconstructing less of the surface), this becomes:

��������������������
Consider the 13→ 33 rule:

Starting from a single ternary relation with three distinct elements {{1,2,3}}, this gives a 

classic Sierpiński triangle structure:

89



Starting instead from a ternary self loop {{0,0,0}} one gets what amounts to a tetrahedron of
Sierpiński triangles:

This is exactly the same as one would get by starting with a tetrahedron graph, and repeat-
edly replacing every trivalent vertex with a triangle of vertices [1:p509]:

 , , , , 

In an ordinary Sierpiński triangle, the points on the edges have different neighborhoods
from those in the interior. But in the structure shown here, all points have the same neigh-
borhoods (so there is an isometry).

Many of the rules we have used have completely different behavior if the order of elements
in their relations are changed. But in this case the limiting shape is always the same, regard-
less of ordering, as in these examples:

 , , , , , 

The rule we have discussed so far in this section in a sense directly implements the recursive
construction of nested patterns [1:5.4]. But the formation of nested patterns is also a com-
mon feature of the limiting behavior of many rules that do not exhibit any such obvious
construction.

90



As an example, consider the 13 → 23 rule

{{x, y, z}} → {{z, w, w}, {y, w, x}}

This rule effectively constructs a nested sequence of self-similar “segments”:

Similar behavior is seen in rules with binary relations, such as the 12 → 42 rule:

{{x, y}} → {{z, w}, {z, x}, {w, x}, {y, w}}

A clear “naturally occurring” Sierpiński pattern appears in the limiting behavior of the 

22 → 42 rule

{{x, y}, {z, y}} → {{y, w}, {y, w}, {w, x}, {z, w}}

91



A�er 15 steps, the rule yields:

��������������������� ������
In traditional geometry, a basic feature of any continuous space is its dimension. And we 

have seen that at least in certain cases we can characterize the limiting behavior of our 
models in terms of the emergence of recognizable geometry—with definite dimension. So 

this suggests that perhaps we might be able to use a notion of dimension to characterize the 

limiting behavior of our models even when we do not readily recognize traditional geometri-
cal structure in them.

For standard continuous spaces it is straightforward to define dimension, normally in terms 
of the number of coordinates needed to specify a position. If we make a discrete approxima-
tion to a continuous space, say with a progressively finer grid, we can still identify dimen-
sion in terms of the number of coordinates on the grid. But now imagine we only have a 

connectivity graph for a grid. Can we deduce what dimension it corresponds to?

92



Wemight choose to draw the grids so they lay out according to coordinates, here in 1-, 2-
and 3-dimensional Euclidean space:

But these are all the same graph, with the same connectivity information:

 , , , 

So just from intrinsic information about a graph—or, more accurately, from information
about a sequence of larger and larger graphs—can we deduce what dimension of space it
might correspond to?

The procedure we will follow is straightforward (cf. [1:p479][22]). For any point X in the
graph define Vr(X ) to be the number of points in the graph that can be reached by going at
most graph distance r. This can be thought of as the volume of a ball of radius r in the graph
centered at X .

For a square grid, the region that defines Vr(X ) for successive r starting at a point in the
center is:

 , , , , , , , 

93



For an infinite grid we then have:

Vr = 2 r2 + 2 r + 1

For a 1D grid the corresponding result is:

{ , , , , , , , }

Vr = 2 r + 1

And for a 3D grid it is:

 , , , , , 

4 r 3 8 r
Vr = + 2 r 2 + + 1

3 3
In general, for a d-dimensional cubic grid (cf. [1:p1031]) the result is a terminating hypergeo-
metric series (and the coefficient of zd in the expansion of (z+1)r/(z-1)r+1):

r 2d 2d–1 2d–2 (d + 1)
2F1(–d, r + 1; –d + r + 1; –1) ( –1

d) = rd+ rd + rd–2 +…
d! (d – 1)! 3 (d – 2)!

But the important feature for us is that the leading term—which is computable purely from 

connectivity information about the graph—is proportional to rd.  

What will happen for a graph that is less regular than a grid? Here is a graph made by 

random triangulation of a 2D region:

And once again, the number of points reached at graph distance r grows like r2:

 , , , , , 

94



In ordinary d-dimensional continuous Euclidean space, the volume of a ball is exactly

πd/2
rd

(d/2)!

And we should expect that if in some sense our graphs limit to d-dimensional space, then in 

correspondence with this, Vr should always show rd growth. 

There are, however, many subtle issues. The first—immediately evident in practice—is that 
if our graph is finite (like the grids above) then there are edge effects that prevent rd growth 

in Vr when the radius of the ball becomes comparable to the radius of the graph. The 

pictures below show what happens for a grid with side length 11, compared to an infinite 

grid, and the rd term on its own:

12 120
10 d = 1 d = 2 1200 d = 3

100 1000 rd
8 80 800

 6 , 60 , 600 infinite grid 
4 40 400
2 20 200 finite grid
0 0 0
0 1 2 3 4 5 6 0 2 4 6 8 10 0 5 10 15

One might imagine that edge effects would be avoided if one had a toroidal grid graph such 

as: 

But actually the results for Vr(X ) for any point on a toroidal graph are exactly the same as 
those for the center point in an ordinary grid; it is just that now finite-size effects come from 

paths in the graph that wrap around the torus.

Still, so long as r is small compared to the radius of the graph—but large enough that we can 

see overall rd growth—we can potentially deduce an effective dimension from measure-
ments of Vr.

In practice, a convenient way to assess the form of Vr, and to make estimates of dimension, 
is to compute log differences as a function of r:

log(Vr+1) – log(Vr)
Δ(r) =

log(r + 1) – log(r)

95



Here are results for the center points of grid graphs (or for any point in the analogous 
toroidal graphs):

1.4 3.0
1.2 2.0

2.5
1.0 1.5 2.0
0.8
0.6 1.0 1.5

0.4 1.0
0.5

0.2 0.5
0.0 0.0 0.0

0 5 10 15 20 25 0 10 20 30 40 50 0 5 10 15 20 25 30

The results are far from perfect. For small r one is sensitive to the detailed structure of the 

grid, and for large r to the finite overall size of the graph. But, for example, for a 2D grid 

graph, as the size of the graph is progressively increased, we see that there is an expanding 

region of values of r at which our estimate of dimension is accurate: 

2.0

1.5

1.0

0.5

0.0
0 50 100 150 200

A notable feature of measuring dimension from the growth rate of Vr(X ) is that the measure-
ment is in some sense local: it starts from a particular position X . Of course, in looking at 
successively larger balls, Vr(X ) will be sensitive to parts of the graph progressively further 
away from X . But still, the results can depend on the choice of X . And unless the graph is 
homogeneous (like our toroidal grids above), one will o�en want to average over at least a 

range of possible positions X . Here is an example of doing such averaging for a collection of 
starting points in the center of the random 2D graph above. The error bars indicate 1σ 

ranges in the distribution of values obtained from different points X .

2.5

2.0

1.5

1.0

0.5

0.0
0 2 4 6 8 10 12 14

96



So far we have looked at graphs that approximate standard integer-dimensional spaces. But 
what about fractal spaces [23]? Let us consider a Sierpiński graph, and look at the growth of 
a ball in the graph:

 , , , 

r = 5 r = 10 r = 15 r = 20

Estimating dimension from Vr(X ) averaged over all points we get (for graphs made from 6 

and 7 recursive subdivisions):

2.0 2.0

1.5 1.5

1.0 1.0

0.5 0.5

0.0 0.0
0 10 20 30 40 0 20 40 60 80

The dotted line indicates the standard Hausdorff dimension log2(3)≈1.58 for a Sierpiński 
triangle [23]. And what the pictures suggest is that the growth rate of Vr approximates this 
value. But to get the exact value we see that in addition to everything else, we will need 

average estimates of dimension over different values of r.

In the end, therefore, we have quite a collection of limits to take. First, we need the overall 
size of our graph to be large. Second, we need the range of values of r for measuring Vr to be 

small compared to the size of the graph. Third, we need these values to be large relative to 

individual nodes in the graph, and to be large enough that we can readily measure the 

leading order growth of Vr—and that this will be of the form rd. In addition, if the graph is 
not homogeneous we need to be averaging over a region X  that is large compared to the size 

of inhomogeneities in the graph, but small compared to the values of r we will use in 

estimating the growth of Vr. And finally, as we have just seen, we may need to average over 
different ranges of r in estimating overall dimension.

97



If we have something like a grid graph, all of this will work out fine. But there are certainly 

cases where we can immediately tell that it will not work. Consider, for example, first the 

case of a complete graph, and second of a tree:

 , 

For a complete graph there is no way to have a range of r values “smaller than the radius of 
graph” from which to estimate a growth rate for Vr. For a tree, Vr grows exponentially 

rather than as a power of r, so our estimate of dimension Δ(r) will just continually increase 

with r:

4

3

2

1

0
0 2 4 6 8 10

But notwithstanding these issues, we can try applying our approach to the objects generated 

by our models. As constructed, these objects correspond to directed graphs or hypergraphs. 
But for our current purposes, we will ignore directedness in determining distance, effec-
tively taking all elements in a particular k -ary relation—regardless of their ordering—to be at 
unit distance from each other. 

98



As a first example, consider the 23 → 33 rule we discussed above that “knits” a simple grid:

As we run the rule, the structure it produces gets larger, so it becomes easier to estimate the 

growth rate of Vr. The picture below shows Δ(r) (starting at the center point) computed a�er 

successively more steps. And we see that, as expected, the dimension estimate appears to 

converge to value 2:

2.5

2.0

1.5

1.0

0.5

0.0
0 10 20 30 40

It is worth mentioning that if we did not compute Vr(X ) by starting at the center point, but 
instead averaged over all points, we would get a less useful result, dominated by edge 

effects: 

2.5

2.0

1.5

1.0

0.5

0.0
0 10 20 30 40

99



As a second example, consider the 23→ 33 rule that slowly generates a somewhat complex 

kind of surface:

As we run this longer, we see what appears to be increasingly close approximation to 

dimension 2, reflecting the fact that even though we can best draw this object embedded in 

3D space, its intrinsic surface is two-dimensional (though, as we will discuss later, it also 

shows the effects of curvature):

2.5

2.0

1.5

1.0

0.5

0.0
0 5 10 15 20 25

The successive dimension estimates shown above are spaced by 500 steps in the evolution of 
the rule. As another example, consider the 2312 → 4342 rule, in which geometry emerges 
rapidly through a process of subdivision:

100



These are dimension estimates for all of the first 10 steps in the evolution of this rule:

2.0

1.5

1.0

0.5

0.0
0 20 40 60 80

We can also validate our approach by looking at rules that generate obviously nested
structures. An example is the 22 → 42 rule that produces:

The results for each of the first 15 steps show good correspondence to dimension
log2(3)≈1.58:

2.0

1.5

1.0

0.5

0.0
0 20 40 60 80 100 120

��� �� �������������� �����������������
Having seen how our notion of dimension works in cases where we can readily recognize
emergent geometry, we now turn to using it to study the more general limiting behavior of
our models.

As a first example, consider the 22 → 42 rule

{{x, y}, {x, z}}→ {{x, y}, {x, w}, {y, w}, {z, w}}

101



which generates results such as (with about 1.84t relations at step t):

If we attempt to reconstruct a surface from successive steps in the evolution of this rule, no
clearly recognizable geometry emerges:


 , , ,

, ,

, 

102



But instead we can try to characterize the results using Vr(X ) and our notion of dimension.
We compute Vr(X ) as we do elsewhere: by starting at a point in the structure and construct-
ing successively larger balls:

 , , ,

, , 

Computing the Δ(r) for all points over the first 16 steps of evolution gives:
3.0

2.5

2.0

1.5

1.0

0.5

0.0
0 5 10 15 20 25 30 35

The most important feature of this plot is that it suggests Δ(r)might approach a definite
limit as the number of steps increases. And from the increasing region of flatness there is
some evidence that perhaps Vr might approach a stable rd form, with d ≈ 2.7, suggesting
that in the limit this rule might produce some kind of emergent geometry with dimension
around 2.7.

103



What about other rules? Here are some examples for rules we have discussed above:

2.0 2.5
2.0 2.0

1.5 2.0
1.5 1.5 1.5

1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5
0.0 0.0 0.0 0.0

0 5 10 15 0 10 20 30 40 50 0 5 10 15 20 25 0 5 10 15 20 25

2.5 3.0
2.0 2.5

2.0 2.5
2.0

1.5 1.5 2.0
1.0 1.5 1.5

1.0 1.0 1.0
0.5 0.5 0.5 0.5
0.0 0.0 0.0 0.0

0 5 10 15 20 0 5 10 15 20 0 2 4 6 8 10 12 0 5 10 15 20

3.0 3.5 3.0
2.5 3.0 3.0

2.5
2.0 2.5 2.5

2.0 2.0 2.0
1.5 1.5 1.5 1.5
1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5
0.0 0.0 0.0 0.0

0 2 4 6 8 10 12 0 2 4 6 8 10 0 2 4 6 8 0 5 10 15 20

Some rules do not show convergence, at least over the number of steps sampled here. Other 
rules show quite stable limiting forms, o�en with a flat region which suggests a structure 

with definite dimension. Sometimes this dimension is an integer, like 1 or 2; o�en it is not. 
Still other rules seem to show linear increase in log differences of Vr, implying an exponen-
tial form for Vr itself, characteristic of tree-like behavior.

104



��������������
In ordinary plane geometry, the area of a circle is πr2. But if the circle is drawn on the 

surface of a sphere of radius a, the area of the spherical region enclosed by the circle is 
instead:

r r2 r4
2πa2 (1 – cos ( )) = πr2 (1 – + – ...)

a 12a2 360a4

In other words, curvature in the underlying space introduces a correction to the growth rate 

for the area of the circle as a function of radius. And in general there is a similar correction 

for the volume of a d-dimensional ball in a curved space (e.g. [24][1:p1050]):

πd/2 r2
rd(1 – R + O(r4) )

(d /2)! 6 (d + 2)
where here R  is the Ricci scalar curvature of the space [25][26][27]. (For example, for the d-
dimensional surface of a (d+1)-dimensional sphere of radius a, R= (d–1) d

a2
.) 

Now consider the sequence of “sphere” graphs:

 , , , , , 

We can compute Vr for each of these graphs. Here are the log differences Δ(r) (the error 
bars come from the different neighborhoods associated with hexagonal and pentagonal 
“faces” in the graph):

2.0

1.5

1.0

0.5

0.0
0 10 20 30 40

105



We immediately see the effect of curvature: even though in the limit the graphs effectively 

define 2D surfaces, the presence of curvature introduces a negative correction to pure r2 
growth in Vr. (Somewhat confusingly, there is only one scale defined for the kind of “pure 

sphere” graphs shown here, so they all have the same curvature, independent of size.)

A torus, unlike a sphere, has no intrinsic surface curvature. So torus graphs of the form

give flat log differences for Vr:

2.0

1.5

1.0

0.5

0.0
0 20 40 60 80 100

106



A graph based on a tiling in hyperbolic space

has negative curvature, so leads to a positive correction to Vr:

4

3

2

1

0
0 1 2 3 4 5 6

(One can imagine getting other examples by taking 3D objects and putting meshes on their 
surfaces. And indeed if the meshes are sufficiently faithful to the intrinsic geometry of the 

surfaces—say based on their geodesics—then the Vr(X ) for the connectivity graphs of these 

meshes [28] will reflect the intrinsic curvatures of the surfaces. In practical computational 
geometry, though, meshes tend to be based on things like coordinate parametrizations, and 

so do not reflect intrinsic geometry.)

107



Many structures produced by our models exhibit curvature. There are cases of negative 

curvature:

6

5

4

3

2

1

0
0 1 2 3 4 5 6

As well as positive curvature:

3.0

2.5

2.0

1.5

1.0

0.5

0.0
0 5 10 15 20

The most obvious examples of nested structures have fractional dimension, but no 

curvature:

2.0

1.5

1.0

0.5

0.0
0 10 20 30 40

But even though it is not well characterized using ideas from traditional calculus, there is 
every reason to expect that the limits of our models can exhibit a combination of fractional 
dimension and curvature.

108



In general, though, there is no obvious constraint on the possible limiting form of Vr. 
Curvature can be thought of as associated with the O(r2) term in a Taylor expansion of Vr 
about r = 0, a�er factoring out rd. But there is nothing to say that the leading behavior of Vr 
should match a form like rd. In addition to exponentials like λr it could show an infinite 

collection of intermediate asymptotic scales, like 2r log(r) or r r [29][30]. 

��������������������������� ������������������
In studying Vr we are looking at the total size of the neighborhood up to distance r around a 

point in a graph. But what about the actual local structure of the neighborhood? 

In general, it can be different for every point on the graph. Thus, for example, in 

obtained from 10 steps of the rule {{x,y},{x,z}}→{{x,z},{x,w },{y,w },{z,w }} the collection of 
distinct range-1 neighborhoods (with their counts) is:

 → 139, → 29, → 25, → 19, → 19, → 14, → 10, → 10, → 7,

→ 7, → 7, → 7, → 5, → 4, → 3, → 2, → 2, → 2, → 2,

→ 1, → 1, → 1, → 1, → 1, → 1, → 1, → 1, → 1, → 1

The corresponding result a�er 12 steps is:

 → 478, → 94, → 79, → 67, → 62, → 55, → 36,

→ 26, → 24, → 23, → 20, → 18, → 13, → 13, → 10,

→ 9, → 7, → 7, → 6, → 4, → 3, → 3, → 2, → 2,

→ 2, → 2, → 2, → 2, → 1, → 1, → 1, → 1, → 1,

→ 1, → 1, → 1, → 1, → 1, → 1, → 1, → 1

And it seems that for this rule the distribution of different forms for a given range of neigh-
borhood generally stabilizes as the number of steps increases. (It may be possible to charac-
terize it as limiting to an invariant measure in the space of possible hypergraphs, perhaps 
with some related entropy (cf. [1:p958][31]).)

109



One sees the same kind of stabilization for most rules, though, for example, in a case like

from the rule {{x,y}}→{{x,y},{y,z},{z,x}} one always gets some neighborhoods with new 

forms at each step: 

 → 6, → 2, → 2, → 1, → 1, → 1, → 1
 → 14, → 12, → 6, → 2, → 2, → 1, → 1, → 1, → 1, → 1

 → 50, → 30, → 14, → 12, → 6,

→ 2, → 2, → 1, → 1, → 1, → 1, → 1, → 1

 → 180, → 62, → 50, → 30, → 14, → 12, → 6,

→ 2, → 2, → 1, → 1, → 1, → 1, → 1, → 1, → 1

In general, the presence of many identical neighborhoods reflects a certain kind of approxi-
mate symmetry or isometry of the emergent geometry of the system. 

In a torus graph, for example, the symmetry is exact, and all local neighborhoods of a given 

range are the same: 

 ,  ,  ,  ,  

The same is true for a 3D torus graph:

 ,  ,  ,  ,  

For a sphere graph not every point has the exact same local neighborhood, but there are a 

limited number of neighborhoods of a given range:

 → 500,  → 440, → 60,

 → 380, → 60, → 60,  → 260, → 120, → 60, → 60

110



And from the dual graph it becomes clear that these are associated with hexagonal and 

pentagonal “faces”:

 → 240, → 12,  → 180, → 60, → 12,

 → 60, → 60, → 60, → 60, → 12

For a (spherical) Sierpiński graph, there are also a limited number of neighborhoods of a 

given range:

 → 108,  → 108,  → 72, → 36,  → 72, → 36

Whenever every local neighborhood is essentially identical, Vr(X ) will have the same form 

for every point X  in a graph or hypergraph. But in general Vr(X ) (and the log differences 
Δr(X )) will depend on X . The picture below shows the relative values of Δr(X ) at each point 
in the structure we showed above:

 , , ,

r = 1 r = 2 r = 3

, , , 

r = 4 r = 5 r = 6 r = 7

We can also compute the distribution of values for Δr(X ) across the structure, as a function 

of r:

r = 1 r = 2 r = 3 r = 4

 , , , ,

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

r = 5 r = 6 r = 7 r = 8 r = 9

, , , , 

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

111



Both these pictures indicate a certain statistical uniformity in Vr(X ). This is also seen if we 

look at the evolution of the distribution of Δr(X ), here shown for the specific value r = 6, for 
steps 8 through 16:

 , , , ,

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

, , , , 

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

�������������� �����������������������������
We have made explicit visualizations of the connectivity structures of the graphs (and 

hypergraphs) generated by our models. But an alternative approach is to look at adjacency 

matrices (or tensors). In our models, there is a natural way to index the nodes in the graph: 
the order in which they were created. Here are the adjacency matrices for the first 14 steps 
in the evolution of the rule {{x,y},{x,z}}→{{x,z},{x,w },{y,w },{z,w }} discussed above:

It is notable that even though these adjacency matrices grow by roughly a factor of 1.84 at 
each step, they maintain many consistent features—and something similar is seen in many 

other rules.

112



Our models evolve by continually adding new relations, and for example in the rule we are 

currently considering, there are roughly exponentially more relations at each step. The 

result, as shown below for step 14, is that at a given step the relations that exist will almost 
all be from the most recent step (shown in red):

Other rules can show quite different age distributions. Here are age distributions for a few 

rules that “knit” their structures one relation at a time:

 , ,

, 

113



������ ����� ���������������
There are many graph and hypergraph properties that can be studied for the output of our 
models. Here we primarily give examples for the rule {{x,y},{x,z}}→{{x,z},{x,w },{y,w },{z,w }} 
discussed above.

A basic question is how the numbers of vertices and edges (elements and relations) grow 

with successive steps. Plotting on a logarithmic scale suggests eventually roughly exponen-
tial growth in this case:

104 edges

vertices
1000

100

10

1
0 5 10 15

We can also compute the growth of the graph diameter (greatest distance between vertices) 
and graph radius:

50 diameter

radius

10

5

1

0 5 10 15

If one assumes that the total vertex count V  is related to diameter D by V = Dd, then plotting 

d  gives (to be compared to dimension approaching ≈2.68 computed from the growth of Vr): 

2.2

2.0

1.8

1.6

0 5 10 15

114



There are many measures of graph structure which basically support the expectation that 
a�er many steps, the outputs from the model somehow converge to a kind of statistically 

invariant “equilibrium” state:

0.6 0.6 0.0
0.5 0.5 -0.2
0.4 0.4

-0.4
 0.3 , 0.3 , -0.6 
0.2 0.2
0.1 0.1 -0.8

0.0 0.0 -1.0
0 5 10 15 0 5 10 15 0 5 10 15

mean clustering coefficient global clustering coefficient assortativity

Some centrality measures [32][33] start (here at step 10) somewhat concentrated, but rapidly 

diffuse to be much more broadly distributed:

 , , 

degree centrality betweenness centrality eigenvector centrality

There are local features of the graph that are closely related to V1(X ) and V2(X ):

 , 

vertex degree local clustering coefficient

Another feature of our graphs to study is their cycle structure. At the outset, our graphs give 

us only connectivity information. But one way to imagine identifying “faces” that could be 

used to infer emergent topology is to look at the fundamental cycles in the graph: 

115



 , , ,

cycle length 3 cycle length 4 cycle length 5

, , 

cycle length 6 cycle length 7 cycle length 8

In this particular graph, there are altogether 320 fundamental cycles, with the longest one 

being of length 24. The distribution of cycle lengths on successive steps once again seems to 

approach an “equilibrium” form:

t = 10 t = 11 t = 12

 , , ,

5 10 15 20 25 5 10 15 20 25 30 5 10 15 20 25 30 35 40

t = 13 t = 14 t = 15

, , 

10 20 30 40 10 20 30 40 10 20 30 40

116



One way to probe overall properties of a graph is to consider the evolution of some dynami-
cal process on the graph. For example, one could run a totalistic cellular automaton with 

values at nodes of the graph. Another possibility is to solve a discretized PDE. For example, 
having computed a graph Laplacian [34] (or its higher order analogs) one can determine the 

distribution of eigenvalues, or the eigenmodes, for a particular graph [35]. The density of 
eigenvalues is then closely related to Vr and our estimates of dimension and curvature.  

������ ����������������������������������
Many rules (at least when they exhibit complex behavior) seem to lead to statistically similar 
behavior, independent of their initial conditions. But there could still be disjoint families of 
states that can be reached from different initial conditions, perhaps characterized by 

different graph or hypergraph invariants. 

As one example, we can ask whether there are rules that preserve the planarity of graphs. 
All rules with signature 12→ 22 inevitably do this. A rule like

{{x, y}} → {{x, y}, {y, z}, {z, x}}

might not at first appear to:

 , , , 

But a different graph layout shows that actually all these graphs are planar [36]:

 , , , 

117



Among larger rules, many still preserve planarity. But for example, 
{{x,y},{x,z}}→{{x,z},{x,w },{y,w },{z,w }} does not, since it transforms the planar graph

to the nonplanar one:

In general, a graph is planar so long as it does not contain as a subgraph either of [37]

 , 

so a rule preserves planarity if (and only if) it never generates either of these subgraphs.

Planarity is one of a class of properties of graphs that are preserved under deletion of 
vertices and edges, and contraction of edges. Another such property is whether a graph can 

be drawn without crossings on a 2D surface of any specific genus g [38]. It turns out [38] that 
for any such property it is known that there are in principle only a finite number of sub-
graphs that can “obstruct” the property—so if a rule never generates any of these, it must
preserve the property. 

������������������������������ �����������
The phenomenon of intrinsic randomness generation is an important and ubiquitous 
feature of computational systems [1:7.5][39]—the rule 30 cellular automaton [1:2.1] being a 

quintessential example. In our models the phenomenon definitely o�en occurs, but two 

issues make it slightly more difficult to identify. 

First, there is considerable arbitrariness in the way we choose to present or visualize graphs 
or hypergraphs—so it is more difficult to tell whether apparent randomness we see is a 

genuine feature of our system, or just a reflection of some aspect of our presentation or 
visualization method. 

118



And second, there may be many possible choices of updating orders, and the specific results 
we get may depend on the order we choose. Later we will discuss the phenomenon of causal 
invariance, and we will see that there are causal graphs that can be independent of updating 

order. But for now, we can consider our updating process to just be another deterministic 

procedure added to the rules of our system, and we can ask about apparent randomness for 
this combined system.

And to avoid the arbitrariness of different graph or hypergraph presentations, we can look 

at graph or hypergraph invariants, which are the same for all isomorphic graphs or hyper-
graphs, independent of their presentation or visualization.

The most obvious invariants to start with are the total numbers of elements and relations 
(nodes and edges) in the system. For rules that involve only a single relation on the le�-hand 

side, it is inevitable that these numbers must be determined by a linear recurrence (cf. 
[1:p890]). (For 1k → nk  rules, up to a k-term linear recurrence may be involved.)

For rules that involve more than one relation more complicated behavior is common. 
Consider for example the rule: 

{{x, y}, {x, z}} → {{y, w}, {w, x}, {z, w}}

 , , , , , ,

, , , , , , , ,

, , , , , , 

The number of relations generated over the first 50 steps (using our standard updating 

order) is:

{2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 18, 22, 26, 30, 35, 42, 50, 58, 67, 79, 94, 110, 127, 148, 175, 206, 239, 277, 325, 383, 447,
518, 604, 710, 832, 967, 1124, 1316, 1544, 1801, 2093, 2442, 2862, 3347, 3896, 4537, 5306, 6211, 7245, 8435, 9845}

Taking third differences yields: 

{0, 0, 0, 1, -1, 0, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 2, 0, -2, 0, 3, 2, -2,
-2, 3, 5, 0, -4, 1, 8, 5, -4, -3, 9, 13, 1, -7, 6, 22, 14, -6, -1, 28, 36, 8, -7, 27, 64}

20
15
10
5
0
-5

0 10 20 30 40

119



One can consider many other invariants, including counts of in and out degrees of elements, 
and counts of cycles. In general, one can construct invariant “fingerprints” of hypergraphs—
and the typical observation is that for rules whose behavior seems complex, almost all of 
these will exhibit extensive apparent randomness.

������������������ ��������
Features like dimension and curvature can be used to probe the consistent large-scale 

structure of the limiting behavior of our models. But particularly insofar as our models 
generate apparent randomness, it also makes sense to study their statistical features. We 

discussed above the overall distribution of values of Vr(X ) and Δr(X ). But we can also 

consider fluctuations and correlations. 

For example, we can look at a 2-point correlation function 

Sr(s)=(〈Vr(X)Vr(Y )〉 – 〈Vr(X )〉2)/〈Vr(X)〉2 for points X  and Y  separated by graph distance s. 
For a uniform graph such as the torus graph, Sr(s) always vanishes. For the buckyball 
approximation to the sphere that we used above, Sr(s) shows peaks at the distances between 

“pentagons” in the graph. 

For the rule {{x,y},{x,z}}→{{x,z},{x,w },{y,w },{z,w }}, Sr(s) steadily expands the region of s 
over which it shows positive correlations, and perhaps (at least for larger r, indicated by 

redder curves) approaches a limiting form: 

 , , , 

0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40

It is conceivable that for this or other rules there might be systematic rescalings of distance 

and number of steps that would lead to fixed limiting forms.

In statistical mechanics, it is common to think about the ensemble of all possible states of a 

system—and for example to discuss evolution from all possible initial conditions. But typical 
systems in statistical mechanics can basically be discussed in terms of a fixed number of 
degrees of freedom (either coordinates or values). 

For our models, there is no obvious way to apply the rules but, for example, to limit the total 
number of relations—making it difficult to do analysis in terms of ensembles of states.

One can certainly imagine the set of all possible hypergraphs (and even have Ramsey-theory-
style results about it), but this set does not appear to have the kind of geometry or structure 

that has typically been necessary for results in statistical mechanics or dynamical systems 
theory. (One could however potentially think in terms of a distribution of adjacency matri-
ces, limiting to graphon-like functions [40] for infinite graphs.)

120



���������������������������������
Imagine that at some step in the evolution of a rule one reverses a single relation. What 
effect will it have? Here is an example for the rule {{x,y},{x,z}}→{{x,z},{x,w },{y,w },{z,w }}. 
The first row is the original evolution; the second is the evolution a�er reversing the relation:

 , , , , 

 , , , , 

We can illustrate the effect by coloring edges in the first row of graphs that are different in 

the second one (taking account of graph isomorphism) [41]:

 , , , , 

Visualizing the second and third graphs in 3D makes it more obvious that the changed edges 
are mostly connected:

 , 

It takes only a few steps before the effect of the change has spread to essentially all parts of 
the system. (In this particular case, with the updating order used, about 20% of edges are 

still unaffected a�er 5 steps, with the fraction slowly decreasing, even as the number of new 

edges increases.)

121



In rules with fairly simple behavior, it is common for changes to remain localized:

 , , , 

However, when complex behavior occurs, changes tend to spread. This is analogous to what 
is seen, for example, in the much simpler case of class 2 versus class 3 cellular automata 

[31][1:6.3]:

 , , , 

Cellular automata are also known [31] to exhibit the important phenomenon of class 4 

behavior—in which there is a discrete set of localized “particle-like” structures through 

which changes typically propagate: 

 , 

In cellular automata, there is a fixed lattice on which local rules operate, making it straight-
forward [1:6.3] to identify the region that can in principle be affected by a change in initial 
conditions. In the models here, however, everything is dynamic, and so even the question of 
what parts can in principle be affected by a change in initial conditions is nontrivial. 

As we will discuss at length later, however, it is always possible to trace which updating 

events in a particular evolution depend on which others, and which relations are associated 

with these. The result will always be a superset of the actual effect of a change in the initial 
condition:

 , , , , 

122



We discussed above the quantity Vr(X ) obtained by “statically” looking at the number of 
nodes in a hypergraph reached by going graph distance r—in effect computing the volume 

of a ball of radius r in the hypergraph. By looking at the dependence of updating events in t  
successive steps of evolution, we can define another quantity Ct (X ) which in effect mea-
sures the volume of a cone of dependencies in the evolution of the system. 

Vr(X ) is in a sense a quantity that is “applied” to the system from outside; Ct (X ) is in a sense 

intrinsic. But as we will discuss later, Vr(X ) is in some sense an approximation to Ct (X )—and 

particularly when we can reasonably consider the evolution of a model to have reached 

some kind of “equilibrium”, Vr(X ) will provide a useful characterization of the “state” of a 

model.

������ ��������
Given any two points in a graph or hypergraph one can find a (not necessarily unique) 
shortest path (or “geodesic”) between them, as measured by the number of edges or hyper-
edges traversed to go from one point to the other. Here are a few examples of such 

geodesics: 

 , , 

 , 

A geodesic in effect defines the analog of a straight line in a graph or hypergraph, and by 

analogy with the way geodesics work in continuous spaces, we can use them to probe 

emergent geometry. 

123



For example, in the case of positive curvature, we can expect that nearby geodesics diverge, 
while in the case of negative curvature they converge:

 , , 

One can see the same effect in sufficiently large graphs (although it can be obscured by 

regularities in graphs which lead to large numbers of “degenerate” geodesics, all of the same 

length):

We saw before that the growth rate of the volume Vr(X ) of a ball centered at some point X  in 

a graph could be identified as giving a measure of the Ricci scalar curvature R  at X . But we 

can now consider tubes formed from balls centered at each point on a geodesic. And from 

the growth rates of volumes of these tubes we will be able to measure what can be identified 

as different components of curvature associated with the Ricci tensor (cf. [1:p1048][27][42]).

In a continuous space (or, more precisely, on a Riemannian manifold) the infinitesimal 
volume element at a point X  is given in terms of the metric tensor g by det g(X ) . If we 

look at a nearby point X + δx we can expand in a power series in δx (e.g. [43]):

1
det g(X + δx) = det g(X ) (1 – R 3

ij(X ) δxi δxj + O(δx ) + ....)
6

where Rij is the Ricci tensor and the δi (contravariant vectors) are orthogonal components of 
δx (say along axes defined by some coordinate system). 

124



If we integrate over a ball of radius r in d  dimensions, we recover our previous formula for 
the volume of a ball 

πd/2 r2
Vr(X ) =  det g(X + δx) ddδx = rd(1 – R i

i + O(r4) )
(d /2)! 6 (d+2)

where R=R i
i  is the Ricci scalar curvature. 

But now let us consider integrating over a tube of radius r that goes a distance δ along a 

geodesic starting at X . Then we get a formula for the volume of the tube (cf. [44])
d–1

_ π 2 d – 1
Vr,δx (X )= rd–1 δx (1 – ( ) (R j

–R ⋀ i 2
ij δx

⋀
δx ) r + O(r 3 + r 2δx ) + ...)

( d–1 )! d + 1
2
⋀

where the δxi are components of unit vectors along the geodesic.

There is now a direct analog in our hypergraphs: just as we measured the growth rates of 
geodesic balls to find Ricci scalar curvature, we can now measure growth rates of geodesic 

tubes to probe full Ricci curvature. 

To construct an example, consider a graph formed from a mesh on the surface of an ellip-
soid. (It is important that this mesh is intrinsic to the surface, with each mesh element 
corresponding to about the same surface area—and that the mesh does not just come from, 
say, a standard θ, ϕ coordinate grid.)

As a first step, consider balls of progressively larger radii at different points on the ellipsoid 

mesh graph:

 , , 

125



In the region of higher curvature near the tip, the area of the ball for a given radius is 
smaller, reflecting higher values of the Ricci scalar curvature R  there: 

4000

3000

2000

1000

0
0 10 20 30 40 50 60

But now consider tubes around geodesics on the ellipsoid mesh graph. Instead of measuring 

the scalar curvature R , these instead in effect measure components of the Ricci tensor along 

these geodesics.

To measure all the components of the Ricci tensor, we could consider not just a tube but a 

bundle of geodesics, and we could look at the sectional curvature associated with deforma-
tions of the shape of this bundle. Or, as an alternative, we could consider tubes along not 
just one, but two geodesics through a given point. But in both cases, the analogy with the 

continuous case is easiest if we can identify something that we can consider an orthogonal 
direction.

One way to do this on a graph is to start from a particular geodesic through a given point, 
then to look at all other geodesics through that point, and work out which ones are the 

largest graph distance away. These show sequences of progressively more distant geodesics 
(as measured by the graph distance to the original geodesic of their endpoints):

 , , , , , , , , 

 , , , , , , 

 , , , , , , , , 

In general there may be many choices of these geodesics—and in a sense these correspond 

to different local choices of coordinates. But given particular choices of geodesics we can 

imagine using them to form a grid.
Looking at growth rates of volumes on this grid then gives us results not just about the Ricci 
tensor, but also about the Riemann tensor, about parallel transport and about covariant 
derivatives (cf. [27]).

126



The examples we have shown so far all involve graphs that have a straightforward correspon-
dence with familiar geometry. But exactly the same methods can be used on the kinds of 
graphs and hypergraphs that arise from our models. This shows tubes of successively larger 
radii along two different geodesics:

 , , 

 , , 

In the limit of a large number of steps, we can measure the volumes of tubes like these to 

compute approximations to projections of the Ricci tensor—and for example determine the 

level of isotropy of the emergent geometry of our models.

������������������� �����
Traditional Riemannian manifolds are full of structure that our hypergraphs do not have. 
Nevertheless, we are beginning to see that there are analogs of many ideas from geometry 

and calculus on manifolds that can be applied to our hypergraphs—at least in some appropri-
ate limit as they become sufficiently large.

To continue the analogy, consider trying to define a function on a hypergraph. For a scalar 
function, we might just assign a value to each node of the hypergraph. And if we want the 

function to be somehow smooth, we should make sure that nearby nodes are assigned 

similar values. 

But what about a vector function? An obvious approach is just to assign values to each 

directed edge of the hypergraph. And given this, we can find the component in a direction 

corresponding to a particular geodesic just by averaging over all edges of the hypergraph 

along that geodesic. (To recover results for continuous spaces, we must take all sorts of 
potentially intricate limits.)

(At a slightly more formal mathematical level, to define vectors in our system, we need some 

analog of a tangent space. On manifolds, the tangent space at a point can be defined in 

terms of the equivalence class of geodesics passing through that point. In our systems, the 

obvious analog is to look at the edges around a point, which are exactly what any geodesic 

through that point must traverse.)

127



For a rank-p  tensor function, we can assign values to p  edges associated either with a single 

node, or with a neighborhood of nearby nodes. And, once again, we can compute “proje-
ctions” of the tensor in particular “directions” by averaging values along p  geodesics.

The gradient of a scalar function ∇f  at a particular point X  can be defined by starting at X  

and seeing along what geodesic the (suitably averaged) values decrease fastest, and at what 
rate. The results of this can then be assigned to the edges along the geodesic so as to specify 

a vector function.

The divergence of a vector function ∇.f  can be defined by looking at a ball in the hyper-
graph, and asking for the total of the values of the function on all hyperedges in the ball. The 

analog of Gauss’s theorem then becomes a fairly straightforward “continuity equation” 
statement about sums of values on edges inside and at the surface of part of a hypergraph. 

������ ������������� �����������
We saw above a rule which generates a sequence of hypergraphs like this:

We can think of this a�er an infinite number of steps as giving an infinitely fine mesh—or in 

effect a structure which limits to a two-dimensional manifold. In standard mathematics, the 

defining feature of a manifold is that it is locally like Euclidean space (in some number of 
dimensions d) [45]. By using Euclidean space as a model many things can be defined and 

computed about manifolds (e.g. [26]).

Some of our models here yield emergent geometry whose limit is an ordinary manifold. But 
the question arises of what mathematical structures might be appropriate for describing the 

limiting behavior of other cases. Is there perhaps some other kind of model space whose 

properties can be transferred? 

It is tempting to try to start from Euclidean space (or n), and define some subset such as a 

Cantor set. But it seems more likely to be fruitful to start from convenient discrete struc-
tures, and see how their limits might correspond to what we have. One important feature of 
Euclidean space is its uniformity: every point is in a sense like every other, even if different 
points can be labeled by different coordinates.

128



So this suggests that by analogy we could consider graphs (and hypergraphs) whose vertices 
all have the same graph neighborhood (vertex transitive graphs). Several obvious infinite 

examples are the limits of: 

 , , , , 

Any uniform tessellation or regular tree provides an example. One might think that another 
example would be graphs formed by uniform application of a vertex substitution rule—such 

as “spherical Sierpiński graphs” starting from a tetrahedron, dodecahedron or buckyball 
graph:

 , , 

But (as mentioned in 4.8) in these graphs not every vertex has exactly the same neighbor-
hood, at least if one goes beyond geodesic distance 2. The number of distinct neighborhoods 
does, however, grow fairly slowly, suggesting that it may be possible to consider such graphs 
“quasi vertex transitive” (in rough analogy to quasiconformal).

But one important class of graphs that are precisely vertex transitive are Cayley graphs of 
groups—and indeed the infinite tessellation and tree graphs above are all examples of these. 
(Note that not all vertex-transitive graphs are Cayley graphs; the Petersen graph is an 

example [46]. It is also known that there are infinite vertex-transitive graphs that are not 
Cayley graphs [47], and are not even “close” to any such graphs [48].) 

In a Cayley graph for a group, each node represents an element of the group, and each edge 

is labeled with a generator of the group. (Different presentations of the group—with differ-
ent choices of generators and relations—can have slightly different Cayley graphs, but their 
infinite limits can be considered the same.) Each point in the Cayley graph can then be 

labeled (typically not uniquely) by a word in the group, specified as a product of generators 
of the group.

129



One can imagine progressively building up a Cayley graph by looking at longer and longer 
words. In the case of a free group with two generators A and B, this yields:

_
AA BB _

AB BA
_ _

A B AA BB

_

, AB A B
 BA

_ _ _ , 
AA BB

_ _
_ A B

B A _ _ _
AB BA

_ _ _
AA _ _ _ BB

AB BA

If one adds the relation AB = BA, defining an Abelian group, the Cayley graph is instead a 

grid, with “coordinates” given by the numbers of As and Bs (or their inverses ):
_ _ _ _
AA _ _ BB

AB AAA _ _ _ _
AAB _ _

ABB BBB
_

_ AB
_

B B A
_ AA _ _

_ BB
B B

AAB A _ _ _
ABB

 , _ _
AB AB , AB _ _

AB 

ABB _
A _ _ _

AAB
B

_ B
A A

A _
BB AB _ _

_ AA
BB ABB _ _

AB B AAB _ _ _
AAA

BB AA

Here are the Cayley graphs for the first few symmetric and alternating (finite) groups:

S2 S3 S4 S5

A3 A4 A5 A6

130



For a given infinite Cayley graph, one can compute a limiting Vr just as we have for hyper-
graphs. If one picks a finite number of generators and relations at random, one will usually 

get a Cayley graph that has a basically tree-like structure, with a Vr that grows exponentially. 
For nilpotent groups, however, Vr always has polynomial growth—an example being the 

Heisenberg group H3 whose Cayley graph is the limit of [49]:

There are also groups known that yield growth intermediate between polynomial and 

exponential [50]. There do not, however, appear to be groups that yield fractional-power 
growth, corresponding to finite but fractional dimension. 

It is possible that one could view the evolution of one of our models as being directly 

analogous to the growth of the Cayley graph for a group—or at least somehow approximated 

by it. As we discussed above, the hypergraphs generated by most of our systems are not, 
however, uniform, in the sense that the structures of the neighborhoods around different 
points in the hypergraph can be different. But this does not mean that a Cayley graph could 

not provide a good (at least approximate) local model for a part of the hypergraph. And if 
this connection could be made, there might be useful results from modern geometric group 

theory that could be applied, for example in classifying different kinds of limiting behaviors 
of our systems.

On a sufficiently small scale, any manifold is defined to be like Euclidean space. But if one 

goes to a slightly larger scale, one needs to represent deviations from Euclidean space. And a 

convenient way to do this is again to consider model spaces. The most obvious is a sphere 

(or in general a hyperellipsoid)—and this is what gives the notion of curvature. Quite what 
the appropriate analog even of this is in fractional dimensional space is not clear, but it 
would potentially be useful in studying our systems. And when there is a possibility for 
change in dimension as well as change in curvature, the situation is even less clear.

131



132



� | ���� ����� ������������������ ��

���������� ���������

�������������������������������
The basic concept of our models is to define rules for updating collections of relations. But 
for a particular collection of relations, there are o�en multiple ways in which a given rule 

can be applied, and there is considerable subtlety in the question of what effects different 
choices can have.

To begin exploring this, we will first consider in this section the somewhat simpler case of 
string substitution systems (e.g. [1:3.5][51]). String substitution systems have arisen in many 

different settings under many different names [1:p893], but in all cases they involve strings 

whose elements are repeatedly replaced according to fixed substitution rules. 

As a simple example, consider the string substitution system with rules {A→AB,B→BA}. 
Starting with A and repeatedly applying these rules wherever possible gives a sequence of 
results beginning with:
{A, AB, ABBA, ABBABAAB, ABBABAABBAABABBA, 
ABBABAABBAABABBABAABABBAABBABAAB}
The application of these particular rules is simple and unambiguous. At each step, every 

occurrence of A or B is independently replaced, in a way that does not depend on its neigh-
bors. One can visualize the process as a tree: 

A

A B

A B B A

A B B A B A A B

A B B A B A A B B A A B A B B A

At step n, there are 2n elements, and the kth
 element is determined by whether the number 

of 1s in the base-2 decomposition of k  is even or odd. (This particular case corresponds to 

the Thue–Morse sequence.)

The evolution of the string substitution system can still be represented by a tree even if the 

replacements are not the same length. The rules {A→B. B→AB} yield the “Fibonacci tree”: 

133



A

B

A B

B A B

A B B A B

B A B A B B A B

A B B A B B A B A B B A B

In this case, the number of elements on the t th step is the t th Fibonacci number. For any 

neighbor-independent substitution system, the number of elements on the nth
 step is 

determined by a linear recurrence, and usually, but not always, grows exponentially. 
(Equivalently, the number of elements of each type on the nth

 step can be determined from 

the nth
 power of the transition matrix.)

But consider now a substitution system with rules {A→BBB, BB→A}. If one starts with A, the 

first step is unambiguous: just replace A by BBB. But now there are two possible replace-
ments for BBB: either replace the first BB to get AB, or replace the second one to get BA.

We can represent all the different possibilities as a multiway system [1:5.6] in which there 

can be multiple outcomes at each step, and there are multiple possible paths of evolution 

(here shown over the course of 5 steps):

A

B B B

A B B A

B B B B

A B B B A B B B A

A A B B B B B

134



We can represent the possible paths of 5 steps of evolution between states of the system by a 

graph: 

A

BBB

AB BA

BBBB

ABB BBA BAB

AA BBBBB

In effect each path through this graph represents a different possible history for the system, 
based on applying a different sequence of possible updates.

By adding something extra to our model, we can of course force a particular history. For 

example, we could consider sequential substitution systems (analogous to search-and-
replace in a text editor) in which we always do only the first possible replacement in a le�-to-
right scan of the state reached at each step. With this setup we get the following history for 

the system shown above:
{A, BBB, AB, BBBB, ABB, BBBBB, ABBB, BBBBBB, ABBBB, BBBBBBB, ABBBBB}
An alternative strategy (analogous, for example, to the operation of StringReplace in the 

Wolfram Language) is to scan from le� to right, but rather than just doing the first possible 

replacement at each step, instead keep scanning a�er the first replacement, and also carry 

out every subsequent replacement that can independently be done. With this “maximum 

scan” strategy, the sequence of states reached in the example above becomes: 
{A, BBB, AB, BBBB, AA, BBBBBB, AAA, BBBBBBBBB, AAAAB, BBBBBBBBBBBBB, AAAAAAB}
(The first deviation occurs at BBBB. A�er replacing the first BB, the maximum scan strategy 

can continue and replace the second BB as well, thereby in effect “skipping a step” in the 

multiway evolution graph shown above.)

Note that in both the strategies just described, the evolution obtained can depend on the 

order in which different replacements are stated in the rule. With the rule {BB→A, A→BBB} 
instead of {A→BBB, BB→A}, the sequential substitution system updating scheme yields:
{A, BBB, AB, BBBB, ABB, AA, BBBA, ABA, BBBBA, ABBA, AAA}
instead of:

{A, BBB, AB, BBBB, ABB, BBBBB, ABBB, BBBBBB, ABBBB, BBBBBBB, ABBBBB}
Given a particular multiway system, one can ask whether it can ever generate a given string. 
In other words, does there exist any sequence of replacements that leads to a given string? 

135



 any
In the case of {A→BBB, BB→A}, starting from A, the strings B and BB cannot be generated, 
though with this particular rule, all other strings eventually can be generated (it takes 

5 (k – 1) steps to get all 2k  strings of length k). 

One of the applications of multiway systems is as an idealization of derivations in equational 
logic [1:p777], in which the rules of the multiway system correspond to axioms that define 

transformations between equivalent expressions in the logical system. Starting from a state 

corresponding to a particular expression, the states generated by the multiway system are 

expressions that are ultimately equivalent to the original expression. The paths in the 

multiway system are then chains of transformations that represent proofs of equivalences 

between expressions—and the problem of whether a particular equivalence between 

expression holds is reduced to the (still potentially very difficult) problem of determining 

whether there is a path in the multiway system that connects the states corresponding to 

these expressions.

Thus, for example, with the transformations {A→BBB,BB→A}, it is possible to see that A can 

be transformed to AAA, but the path required is 10 steps long:

A

BBB

AB BA

BBBB

ABB BBA BAB

AA BBBBB

ABBB BBBA BBAB BABB

ABA AAB BBBBBB BAA

ABBBB BBBAB BBABB BBBBA BABBB

ABAB AABB ABBA BBBBBBB BAAB BBAA BABA

ABBBBB BBBABB BBBBAB AAA BBABBB BBBBBA BABBBB

136



In general, there is no upper bound on how long a path may be required to reach a particu-
lar string in a multiway system, and the question of whether a given string can ever be 

reached is in general undecidable [52][53][1:p778].

���������������������������������������
Consider the rule {BA→AB}, starting from BABABA:

BABABA

ABBABA BAABBA BABAAB

ABABBA ABBAAB BAABAB

AABBBA ABABAB BAAABB

AABBAB ABAABB

AABABB

AAABBB

As before, there are different possible paths through this graph, corresponding to different 
possible histories for the system. But now all these paths converge to a single final state. And 

in this particular case, there is a simple interpretation: the rule is effectively sorting As in 

front of Bs by repeatedly doing the transposition BA→AB. And while there are multiple 

different possible sequences of transpositions that can be used, all of them eventually lead 

to the same answer: the sorted state AAABBB.

There are many practical examples of systems that behave in this kind of way, allowing opera-
tions to be carried out in different orders, generating different intermediate states, while always 
leading to the same final answer. Evaluation or simplification of (parenthesized) arithmetic (e.g. 
[54]), algebraic or Boolean expressions are examples, as is lambda function evaluation [55]. 

Many substitution systems, however, do not have this property. For example, consider the 

rule {AB→AA,AB→BA}, again starting from BABABA. Like the sorting rule, a�er a limited 

number of steps this rule gets into a final state that no longer changes. But unlike the sorting 

rule, it does not have a unique final state. Depending on what path is taken, it goes in this 

case to one of three possible final states:

137



BABABA

BAAABA BABBAA

BAABAA BBAABA

BABAAA BBABAA

BAAAAA BBAAAA BBBAAA

But what about systems that do not “terminate” at particular final states? Is there some way
to define a notion of “path independence” [55]—or what we will call “causal invariance”
[1:9.10]—for these?

Consider again the rule {A→BBB,BB→A} that we discussed above:

A

BBB

AB BA

BBBB

ABB BBA BAB

AA BBBBB

ABBB BBBA BBAB BABB

ABA AAB BBBBBB BAA

ABBBB BBBAB BBBBA BBABB BABBB

The state BBB at step 2 has two possible successors: AB and BA. But a�er another step, AB
and BA converge again to the state BBBB. And in fact the same kind of thing happens
throughout the graph: every time two paths diverge, they always reconverge a�er just one
more step. This means that the graph in effect consists of a collection of “diamonds” of
edges [56]:

138



There is no need, however, for reconvergence to happen in just one step. Consider for 

example the rule {A→AA, AA→AB}:

AAA

AAAA ABA AAB

AAAAA AABA ABAA AAAB ABB

AAAAAA AAABA ABAAA AABAA AAAAB ABBA ABAB AABB

As the picture indicates, there are paths from AAA leading to both ABA and AAB—but these 

only reconverge again (to ABAB) a�er two more steps. (In general, it can take an arbitrary 

number of steps for reconvergence to occur.)

Whether a system is causal invariant may depend on its initial conditions. Consider, for 

example, the rule AA→AAB. With initial condition AABAA the rule is causal invariant, but 
with initial condition AAA it is not

139



AABAA AAA

AABAAB AABBAA AAAB AABA

 , 

AABAABB AABBAAB AABBBAA AAABB AABAB AABBA

AABAABBB AABBAABB AABBBAAB AABBBBAA AAABBB AABABB AABBAB AABBBA

When a system is causal invariant for all possible initial conditions, we will say that it is 

totally causally invariant. (This is essentially the confluence property discussed in the theory 

of term-rewriting systems.) Later, we will discuss how to systematically test for causal 
invariance—and we will see that it is o�en easier to test for total causal invariance than for 

causal invariance for specific initial conditions.

Causal invariance may at first seem like a rather obscure property. But in the context of our 

models, we will see in what follows that it may in fact be the key to a remarkable range of 
fundamental features of physics, including relativistic invariance, general covariance, and 

local gauge invariance, as well as the possibility of objective reality in quantum mechanics.

����������� �����
If there are multiple successors to a particular state in a substitution system one thing to do 

would be just to assume that each of these successors is a new, unique state. The result of 
this will always be to produce a tree of states, here shown for the rule {A→BBB,BB→A}: 

A

BBB

AB BA

BBBB BBBB

BBA ABB BAB BBA ABB BAB

BBBBB AA AA BBBBB BBBBB BBBBB AA AA BBBBB BBBBB

But in our construction of multiway systems, we assume that actually the states produced 

are not all unique, and instead that states at a given step consisting of the same string can be 

merged, thereby reducing the tree above to the directed graph:

140



A

BBB

AB BA

BBBB

ABB BBA BAB

AA BBBBB

But if we are going to merge identical states, why do so only at a given step? Why not merge 

identical states whenever they occur in the evolution of the system? A�er all, given the 

setup, a particular state—wherever it occurs—will always evolve in the same way, so in some 

sense it is redundant to show it multiple times.

The particular rule {A→BBB,BB→A} that we have just used as an example has the special 
feature that it always “makes progress” and never repeats itself—with the result that a given 

string only ever appears once in its evolution. Most rules, however, do not have this 

property.

Consider for example the rule {AB→BAB, BA→A}. Starting from ABA, here is our normal 
“evolution graph”:

ABA

AA BABA

BAA ABA BBABA

AA BABA BBAA BBBABA

ABA BAA BBABA BBBAA BBBBABA

141



Notice that in this evolution even the original state ABA appears again, both at step 3 and at 
step 5—and each time it appears, it necessarily makes a complete repeat of the same evolu-
tion graph. To remove this redundancy, we can make a graph in which we effectively merge 

all instances of a given state, so that we show each state only once, connecting it to whatever 

states it evolves to under the rule:

BBBABA ABA

BBBAA BBBBABA BBABA

BBAA BABA

BAA

AA

But in this graph there are, for example, two connections from ABA to BABA, because this 

transformation happens twice in the 4 steps of evolution that we are considering. But in a 

sense this multiplicity again always gives redundant information, so in what we will call our 

“states graph” [1:p209], we only ever keep one connection between any given pair of states, 
so that in this case we get:

BBBABA ABA

BBBAA BBBBABA BBABA

BBAA BABA

BAA

AA

There is one further subtlety in the construction of a states graph. The graph only records 

which state can be transformed into which other: it does record how many different replace-
ments could be applied to achieve this. In the rule we just showed, it is never possible to 

have different replacements on a single string yield the same result. 

142



Consider the rule {A→AA, A→B} starting with AA. There are, for example, two different ways 

that this rule can be applied to AA to get AAA: 

AA

AAA AB BA

AAAA ABA AAB BAA BB

In our standard states graph, however, we show only that AA is transformed to AAA, and we 

do not record how many different possible replacements can achieve this:

AA

AAA AB BA

AAAA ABA AAB BAA BB

The degree of compression achieved in going from evolution graphs to states graphs can be 

quite dramatic. For example, for the rule {BA→AB, AB→BA} the evolution graph is:

BABA

ABBA BAAB BBAA

ABAB BABA

AABB ABBA BAAB BBAA

ABAB BABA

AABB ABBA BAAB BBAA

ABAB BABA

143



and the states graph is:

BABA AABB

BBAA ABAB

ABBA BAAB

or in our standard rendering:

BAAB

BBAA BABA ABAB AABB

ABBA

Note that causal invariance works the same in states graphs as it does in evolution graphs: if 
a rule is causal invariant, any two paths that diverge must eventually reconverge. 

������������ ���� ��� ���������������
Before considering further features of the updating process, it is helpful to discuss the 

typical kinds of multiway graphs that are generated from string substitution systems. Much 

as for our primary models based on general relations (hypergraphs), we can assign signa-
tures to string substitution system rules based on the lengths of the strings in each transfor-
mation. For example, we will say that the rule {A→BBB,BB→A} has signature 2: 1→3, 2→1, 
where the initial 2 indicates the number of distinct possible elements (here A and B) that 
occur in the rule.

With a signature of the form k: n1→n2, n3→n4, … there are nominally k∑ni possible rules. 
However, many of these rules are equivalent under renaming of elements or reversal of 
strings. Taking this into account, the number of inequivalent possible rules for various cases is:

144



k=2 k=3 k=4 k=2 k=3 k=4 k=2 k=3 k=4
1 → 1 2 2 2 1 → 6 36 196 379 1 → 2, 1 → 4 72 574 1451
1 → 2 3 4 4 2 → 5 36 196 379 1 → 2, 4 → 1 72 574 1451
1 → 3 6 10 11 3 → 4 36 196 379 1 → 4, 2 → 1 72 574 1451
2 → 2 6 10 11 1 → 1, 1 → 4 40 205 395 1 → 1, 3 → 3 80 610 1515

1 → 1, 1 → 1 8 14 15 1 → 1, 2 → 3 40 205 395 1 → 3, 1 → 3 80 610 1515
1 → 4 10 25 31 1 → 2, 1 → 3 40 205 395 1 → 3, 3 → 1 80 610 1515
2 → 3 10 25 31 1 → 2, 3 → 1 40 205 395 1 → 2, 2 → 3 72 574 1451

1 → 1, 1 → 2 12 28 35 1 → 3, 2 → 1 40 205 395 1 → 2, 3 → 2 72 574 1451
1 → 5 20 70 107 1 → 2, 2 → 2 36 196 379 1 → 3, 2 → 2 72 574 1451
2 → 4 20 70 107 1 → 1, 1 → 1, 1 → 2 48 244 459 2 → 1, 2 → 3 72 574 1451
3 → 3 20 70 107 1 → 7 72 574 1451 1 → 1, 1 → 1, 1 → 3 96 730 1771

1 → 1, 1 → 3 24 82 123 2 → 6 72 574 1451 2 → 2, 2 → 2 72 574 1451
1 → 1, 2 → 2 20 70 107 3 → 5 72 574 1451 1 → 1, 1 → 1, 2 → 2 80 610 1515
1 → 2, 1 → 2 20 70 107 1 → 1, 1 → 5 80 610 1515 1 → 1, 1 → 2, 1 → 2 80 610 1515
1 → 2, 2 → 1 20 70 107 4 → 4 72 574 1451 1 → 1, 1 → 2, 2 → 1 80 610 1515

1 → 1, 1 → 1, 1 → 1 32 122 187 1 → 1, 2 → 4 72 574 1451 1 → 1, 1 → 1, 1 → 1, 1 → 1 128 1094 2795

Rules with signature k: 1→n must always be totally causal invariant. If they are started from
strings of length 1, their states graphs can never branch. However, with strings of length
more than 1, branching can (but may not) occur, as in this example of A→AB started from
three strings of length 2:

AA AB BA

AAB ABA ABB BAB

AABB ABAB ABBA ABBB BABB

AABBB ABABB ABBAB ABBBA ABBBB BABBB

AABBBB ABABBB ABBABB ABBBAB ABBBBA ABBBBB BABBBB

With initial conditions AAA and AAAA, this rule produces the following states graphs:

 , {  , {

Even the rule A→AA can produce states graphs like these if it has initial conditions that
contain Bs as “separators”:

145



ABA

AABA ABAA

AAABA AABAA ABAAA

AAAABA AAABAA AABAAA ABAAAA

AAAAABA AAAABAA AAABAAA AABAAAA ABAAAAA

And as a seemingly even more trivial example, the rule A→B with an initial condition contain-
ing n As gives a states graph corresponding to an n-dimensional cube (with a total of 2n nodes):

AAAA
AA AAA

AAAB AABA ABAA BAAA

AAB ABA BAA

 AB BA , , AABB ABAB ABBA BAAB BABA BBAA 

ABB BAB BBA

ABBB BABB BBAB BBBA

BB BBB
BBBB

With multiple rules, one can get tree-like structures, with exponentially increasing numbers 

of states. The simplest case is the 2: 0→1, 0→1 rule {""→A, ""→B} starting with the null string "":

A B

AA AB BA BB

AAA AAB ABA BAA ABB BAB BBA BBB

With multiple rules, even with single-symbol le�-hand sides, causal invariance is no longer 

guaranteed. Of the 8 inequivalent 2: 1→1, 1→1 rules, all are totally causal invariant, although 

the rule {A→B, B→A} achieves this through cyclic behavior:

146



BA BB

AA AB

Among the 14 inequivalent 3: 1→1, 1→1 rules, all but two are totally causal invariant. The 

exceptions are the cyclic rule, and also the rule {A→B, A→C}, which in effect terminates 

before its B and C branches can reconverge:

AA

BA AB AC CA

BB BC CB CC

The 12 inequivalent 2: 1→2, 1→1 rules yield the following states graphs when run for 4 steps 

starting from all possible length-3 strings of As and Bs:

 , , , , , ,

, , , , , 

147



All but three of these rules are totally causal invariant. Among causal invariant ones are:

A

A

BB
AB

 AA ABB , AB BA 

AAB ABA ABBB
AA BBB

AAA AABB ABAB ABBA ABBBB

{A → AB, B → A} ABB BBA BAB

{A → BB, B → A}

A�er more steps these yield: 

 , 

Examples of non-causal invariant 2: 1→2, 1→1 rules are:

A
A

B AA B AB

 AB BA AAA , BB ABB 

BB AAB BAA AAAA ABA
BBB ABBB

BAB AAAB ABB BAAA AAAAA BBA AABA ABAA
ABBBB BBBB

{A → AA, A → B}
{A → AB, A → B}

148



A�er more steps these yield:

 , 

When we look at rules with larger signatures, the vast majority at least superficially show 

the same kinds of behavior that we have already seen. 

Like for the hypergraphs from our models that we considered in previous sections, we can 

study the limiting structure of states graphs generated in multiway systems, and see what 
emergent geometry they may have. And in analogy to Vr(X) for hypergraphs, we can define 

a quantity Mt(S) which specifies the total number of distinct states reached in the multiway 

system a�er t  steps of evolution starting from a state S.

For the rule {A→AB} mentioned above, the geometry of the multiway graph obtained by 

starting from n As is effectively a regular n-dimensional grid:

A AB ABB ABBB ABBBB ABBBBB

AA

AAB ABA

AABB ABAB ABBA

AABBB
ABABB ABBBA

ABBAB
AABBBBB AABBBB ABBBBA ABBBBBA

ABABBB
ABBABB ABBBAB

ABABBBB ABBBBAB
ABBABBB ABBBABB

149



 Mt(An) in this case is then an n-fold nested sum of t  1s:
n–1 (t – k) tn

Mt (An) = ~ (1 – (n
n

k= n! n 2) + ...)
0 ! t

Some rules create states graphs that are trees, with Mt~mt . Other rules do not explicitly give 

trees, but still give exponentially increasing Mt. An example is {A→AB,B→A}, for which Mt  is 

a sum of Fibonacci numbers, and successive states graphs can be rendered as:

 , , , , , , 

Note that rules with both polynomial and exponential growth in Mt  can exhibit causal 
invariance.

It is not uncommon to find rules with fairly long transients. A simple example is the totally 

causal invariant rule {A→BBB,BBBB→A}. When started from n As this stabilizes a�er 7n 

steps, with ~2.8nstates, with a states graph like:

Rules typically seem to have either asymptotically exponential or asymptotically polynomial 
Mt. (This may have some analogy with what is seen in the growth of groups [57][58][22].) 
Among rules with polynomial growth, it is typical to see fairly regular grid-like structures. 
An example is the rule
{AB → A, ABA → BBAABB}
which from initial condition ABAA gives states graph:

150



With the slightly different initial condition ABAAA, the states graph has a more elaborate 

structure: 

and the form of Mt  is more complicated (though ~t2 and ultimately quasiperiodic); the 

second differences of Mt  in this case are:
20

10

0

-10

0 20 40 60 80 100

Another example of a somewhat complex structure occurs in the rule

{ABA → BBAA, BAA → AAB}
that was discussed in [1:p 205]. Starting from BABBAAB it gives the states graph:

151



���������������������������������
Causal invariance implies that regardless of the order in which updates are made, it is
always possible to get to the same outcome. One way this can happen is if different updates
can never interfere with each other. Consider the rule {A→AA, B→BB}. Every time one sees
an A, it can be “doubled”, and similarly with a B. But these doublings can happen indepen-
dently, in any order. Looking at an evolution graph for this rule we see that at each state
there can be an “A replacement” applied, or a “B replacement”. There are two branches of
evolution depending on which replacement is chosen, but one can always eventually reach
the same outcome, regardless of which choice was made—and as a result the rule is causal
invariant:

AB

A B

AAB ABB

A B A B

AAAB AABB ABBB

A B A B A B

AAAAB AAABB AABBB ABBBB

A B A B A B A B

AAAAAB AAAABB AAABBB AABBBB ABBBBB

The replacements A→AA and B→BB trivially cannot interfere because their le�-hand sides
involve different elements. But a more general way to guarantee that interference cannot
occur is for the le�-hand sides of replacements not to be able to overlap with each other—or
with themselves.
In general, the set of strings of As and Bs up to length 5 that do not overlap themselves are 

 

(up to A,B interchange and reversal): [1:p1033][59]

{A, AB, AAB, AAAB, AABB, AAAAB, AAABB, AABAB}
These can be formed into pairs in the following ways:

{{A, B}, {AABAB, AABB}, {AABB, ABABB}, {AAABB, AABAB}, {AAABB, ABABB}}
The first triples with no overlaps are of length 6. An example is {AAABB, ABAABB, ABABB}.
And whenever there is a set of strings with no overlaps being used as the le�-hand side of
replacements, one is guaranteed to have a system that is totally causal invariant.

152



It is also perfectly possible for the right-hand sides of rules to be such that the system is 

totally causal invariant even though their le�-hand sides overlap. An example is the simple 

rule {A→AA, AA→A}: 

A

AA

A AAA

AA AAAA

A AAA AAAAA

AA AAAA AAAAAA

At every branch point in a multiway system, one can identify all pairs of strings that can be 

generated. We will call such pairs of strings branch pairs (they are o�en called “critical 
pairs” [60]). And the question of whether a system is causal invariant is then equivalent to 

the question of whether all branch pairs that can be generated will eventually resolve, in the 

sense that there is a common successor for both members of the pair [61][62][63][64].

Consider for example the rule {A→AB, B→A}:

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

A�er two steps, a branch pair appears: {AA, ABB}. But a�er just one more step it resolves. 
However, two more branch pairs are generated: {AAB, ABA} and {ABA, ABBB}. But a�er 

another step, these also resolve. And in fact in this system all branch pairs that are ever 

generated resolve (actually always in just one step), and so the system is causal invariant.

In general, it can, however, take more than one step for a branch pair to resolve. The 

simplest case involving resolution a�er two steps involves the rule:
{A → B, AB → AA}

153



The state AB generates the branch pair {BB, AA}: 

AB

BB AA

AB BA

AA BB

BA AB

In the evolution graph, we do not see a resolution of this branch pair. But looking at the 

states graph, we see that the branch pair does indeed resolve in two steps, though with BB 

being a terminating state:

AA

AB BA

BB

The same kind of thing can also happen without a terminating state. Consider for example

{A → AA, AB → BA}
AB

AAB

AAAB ABA BA

AAAAB AABA ABAA BAA

BAAA

where the branch pair {AAB, BA} takes 2 steps to resolve.

In the rule

{A → AA, AAB → BA}

154



it takes 3 steps for the branch pair {AAAB, BA} to resolve: 

AAB

AAAB

AAAAB ABA

BA AAAAAB AABA

AAAAAAB BAA AAABA

AAAAAAAB AAAABA ABAA

AABAA ABAAA

BAAA AAABAA AABAAA ABAAAA

BAAAA

BAAAAA

Things can get quite complicated even with simple rules. For example, in the rule

{A → AA, AA → BAB}
the branch pair {AAA, BAB} resolves a�er 4 steps. The following is essentially a proof of this:

AAA BAB

AAAA BAAB

AAAAA BAAAB

AAABAB BAAAAB

BABABAB

But finding this in the 67-node 4-step states graph is quite complicated: 

155



In the rule 

{A → AAB, ABBA → A}
it takes 7 steps for the branch pair

{A, AABBBA}
to resolve to the common successor AAAABBBAAB. The 7-step states graph involved has 

5869 nodes, and the proof that the branch pair resolves is: 

A AABBBA

AAB AABABBBA

AABAB AAABBABBBA

AAABBAB AAABABBABBBA

AAAB AAAABBABBABBBA

AABAAB AAAABBABBABBBAAB

AAABBAAB AAAABBABBBAAB

AAAABBBAAB

In general, there is no upper bound on how long it may take a branch pair to resolve, or for 

example how long its common successor—or intermediate strings involved in reaching it—
may be. Here are the simplest rules with two distinct elements that take successively longer 

to resolve (the last column gives the size of the states graph when resolution is found):

1 {A → A} A → {A, A} → A 2
2 {A → B, AB → AA} AB → {BB, AA} → BB 5
3 {A → AA, A → BAB} A → {AA, BAB} → BABABAB 22
4 {A → AA, A → BAAB} A → {AA, BAAB} → BAABAABAAB 121
5 {A → AA, A → BAABB} A → {AA, BAABB} → BAABBAABBAABBABB 515
6 {A → AAB, ABAA → A} ABAA → {AABBAA, A} → AABBAABB 1664
7 {A → AAB, ABBA → A} ABBA → {AABBBA, A} → AAAABBBAAB 2401
8 {A → AA, AA → BABBB} AA → {AAA, BABBB} → BABBBABBBABBBBBBBBB 759
9 {A → B, BB → A, AAAA → B} AAAA → {BAAA, B} → B 30
10 {A → AA, AA → BABBBB} AA → {AAA, BABBBB} → BABBBBABBBBABBBBBBBBBBBBBBBB 4020
11 {A → AA, AAA → BABBB} AAA → {AAAA, BABBB} → BABBBABBBABBBBBBBBB 2294
12 {A → AA, AA → B, BBBB → A} AA → {AAA, B} → B 405
13 {A → B, BB → A, AAAAAB → A} AAAAAB → {BAAAAB, A} → A 94
14 {A → AB, BAA → A, BBB → A} BBBAA → {AAA, BBA} → AA 2698
15 {A → AA, BBB → A, AAAA → B} AAAA → {AAAAA, B} → B 430
16 {A → AA, AAA → B, BBBB → A} AAA → {AAAA, B} → B 906

Note that—like the first case of a 2-step resolution that we showed above—quite a few of 
these longest-to-resolve rules actually terminate, with a member of the branch pair being 

156



 actually
their final output. Thus, for example, the last case listed is really just a reflection of the fact 
that with this rule, AAAA takes 16 steps to reach the termination state B:
{AAAA, AAAAA, AAAAAA, AAAAAAA, AAAAAAAA, AAAAAAAAA, AAAAAAAAAA, 
AAAAAAAAAAA, AAAAAAAAAAAA, AAAAAAAAAAAAA, AAAAAAAAAAAAAA, 
AAAAAAAAAAAB, AAAAAAAABB, AAAAABBB, AABBBB, AAA, B}
(With 3 distinct elements similar results are seen; the first time a shorter example is seen is 

{A→BAC,A→AABA} at resolution-length 6.)

Despite the existence of long-to-resolve cases, most branch pairs in most rules in practice 

resolve quickly: the fraction that take τ steps seems to decrease roughly like 2–τ. But there is 

still in a sense an arbitrarily long tail—and the general problem of determining whether a 

branch pair will resolve is known to be formally undecidable (e.g. [65]). 

One interesting feature of causal invariance testing is that (while still in principle undecid-
able) it is in some ways easier to test for total causal invariance than to test for partial causal 
invariance for specific initial conditions. The reason is that if a rule is going to be totally 

causal invariant then there is a certain core set of branch pairs that must resolve, and if all 
these resolve then the rule is guaranteed to be totally causal invariant. This core set of 
branch pairs is derived from the possible overlaps between le�-hand sides of replacements, 
and are the successors of the minimal “unifications” of these le�-hand sides, formed by 

minimally overlapping the strings. 

Consider the rule:

{AA → A, AB → BAA}
The only possible overlap between the le�-hand sides is A. This leads to the minimal 
unification AAB, which yields the branch pair {ABAA,AB}:

AAB

ABAA AB

157



From this branch pair we construct a states graph:

ABAA

ABA BAAAA

AB BAAA

BAA

BA

And from this we see that the branch pair resolves in 3 steps. And because this is the only 

branch pair that can arise through overlaps between the le�-hand sides, this resolution now
establishes the total causal invariance of this rule. 

In the rule

{AB → BA, BAA → A}
there are two possible overlaps between the le�-hand sides: A and B. These lead to two 

different minimal unifications: BAAB and ABAA. And these two unifications yield two 

branch pairs, {BABA, AB} and {BAAA, AA}. But now we can establish that both of these 

resolve, thus showing the total causal invariance of the original rule. 

In general, one might in principle have to continue for arbitrarily many steps to determine if 
a given branch pair resolves. But the crucial point here is that because the number of 
possible overlaps (and therefore unifications) is fine, there are only a finite number of 
branch pairs one needs to consider in order to determine if a rule is totally causal invariant. 
There is no need to look at branch pairs that arise from larger strings than the unifications; 
any additional elements are basically just “padding” that cannot affect the basic interference 

between replacements that leads to a breakdown of causal invariance.

Looking at branch pairs from all possible unifications is a way to determine total causal 
invariance—and thus to determine whether a rule will be causal invariant from all possible 

initial conditions. But even if a rule is not totally causal invariant, it may still be causal 
invariant for particular initial conditions—effectively because whatever branch pairs might 
break causal invariance simply never occur in evolution from those initial conditions. 

In practice, it is fairly common to have rules that are causal invariant for some initial 
conditions, but not others. In effect, there is some “conservation law” that keeps the rule 

away from branch pairs that would break causal invariance—and that keeps the rule operat-
ing with some subset of its possible states that happen to yield causal invariance.

158



��������������������������������������
The plots below show the fractions of rules found to be totally causal invariant [66], as a 

function of the total number of elements they involve, for the cases of k = 2 (A, B) and k = 3 

(A, B, C):

1.0 1.0
{A,B} {A,B,C}

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10

The darker colors indicate larger numbers of steps to resolve branch pairs. (There is some 

uncertainty in these plots—conceivably as much as 9% for 10 total elements with k = 3—
since in some cases the states graph became too big to compute before it could be deter-
mined whether all branch pairs resolved.) 

The dotted lines indicate rules are in a sense inevitably causal invariant because their le�-
hand sides involve strings that do not overlap themselves or each other, thereby guarantee-
ing total causal invariance. Rules such as AA→AAA are causal invariant despite having 

overlapping le�-hand sides because their right-hand sides in a sense give the same result 
whatever overlap occurs.

Ignoring the structure of rules, one can just ask what fraction of strings are non-overlapping 

[67]. Out of the total of kn
 possible strings with length n containing k  distinct elements the 

number that do not overlap themselves is given by [1:p1033]:

a[0] = 1; a[n_] := k a[n-1]- If[EvenQ[n], a[n /2], 0]

This yields the following fractions (for the limit see e.g. [68][@Finch: 5.17]): 

n k = 2 k = 3 k = 4 k = 5
2 0.500 0.667 0.750 0.800
3 0.500 0.667 0.750 0.800
4 0.375 0.593 0.703 0.768
5 0.375 0.593 0.703 0.768
6 0.313 0.568 0.691 0.762
7 0.313 0.568 0.691 0.762
8 0.289 0.561 0.689 0.760
9 0.289 0.561 0.689 0.760
10 0.277 0.558 0.688 0.760
∞ 0.268 0.557 0.688 0.760

159



One can also look at how many possible sets of s strings of length up to n allow no overlaps 

with themselves or each other [1:p1033]. The numbers and fractions for k = 2 are as follows:

n s = 1 s = 2 s = 3 s = 4 s = 5
2 2 (0.5) 2 (0.2) 2 (0.1) 2 (0.057) 2 (0.036)

3 4 (0.5) 4 (0.11) 4 (0.033) 4 (0.012) 4 (0.0051)

4 6 (0.38) 6 (0.044) 6 (0.0074) 6 (0.0015) 6 (0.00039)

5 12 (0.38) 20 (0.038) 28 (0.0047) 36 (0.00069) 44 (0.00012)

6 20 (0.31) 54 (0.026) 104 (0.0023) 170 (0.00022) 252 (0.000024)

7 40 (0.31) 220 (0.027) 728 (0.002) 1788 (0.00015) 3672 (0.000012)

8 74 (0.29) 798 (0.024) 4806 (0.0017) 19708 (0.00011) 62668 (6.6× 10-6)

To get a sense of the distribution of non-overlapping strings, one can make an array that 
shows which pairs of strings (ordered lexicographically) do not allow overlaps. Here are the 

results for k = 2 and k = 3 for strings respectively up to length 6 and length 4, showing clear 

structure in the space of possible strings [51]:

,

�����������������������������������������
So far the nodes in our graphs have always been states generated by substitution systems. 
But we can also introduce nodes to represent the “updating events” associated with replace-
ments performed on strings. Here is the result for evolution according to the rule {AB→
BAB,BA→A} starting from ABA—with each event node indicating the string replacement to 

which it corresponds:

160



ABA

ABA AB
A BABA

AA BABA

BABA BA
A A BA B AB

BABA

BAA ABA BBABA

BAA ABA AB AB
A A BABA BBBABA BBAA BA BBABAA

AA BBBABA BABA BBAA

BBB AB
BABA BBBABA BBBABA B AB

A BABA
BA

A A BA BABAA BBAA A

BBBBABA BBABA BBBAA ABA BAA

We can also show this as a states graph, where we have merged instances of the same state 

that occur at different steps: 

ABA

AB
BABA BBAA BA

BABA

BBABAA BBAA A BBBAA BA BA
A BA B AB

BABA BABAA

BBAA BBABA BAA

BBB AB
BABA BBBABA BB AB

BABA
BA
A A ABAA A

BBBBABA BBBAA BBBABA AA

161



States are connected through events. But how are events connected? Given two events the 

key question to ask is whether they are causally related. Does one event depend on the 

other—in the sense that all or part of its input comes from the output of the other event?

Looking at the graph above, for example, the event AB
 BABA depends on BA

 A BA because AB
 BABA 

uses as input the A that arises as output from BA
 A BA. On the other hand, ABAA  does not 

depend BA
 on A BA because the BA it consumes was not generated by BA

 A BA.

We can add this dependency information to the evolution graph by putting dotted lines 

between events that are causally related:

ABA

ABA AB
A BABA

AA BABA

BABA BA
A A BA B AB

BABA

BAA ABA BBABA

BA BA AB
A A A A BABA BBAA BA BBABAA BB AB

BABA

AA BABA BBAA BBBABA

B AB BA
B A BA BABA BBBA BA AB
ABA A A BA B A A BBBBABA BBBABAA

BBABA ABA BAA BBBBABA BBBAA

We can also do this in the states graph, in which we have merged instances of states from 

different steps:

ABA

BBA AB
A BA BABA

BBABAA BABA ABAA

BBBA BBAA BA
A BA A BA B AB

BABA

BBB AB
BABA BBABA BBBABAA BBAA A BABAA

BBBBABA BB AB
BABA BBBAA BAA

BBBABA BA
A A

AA

162



We can redraw this graph without the layering that puts the initial state at the top. We have 

added an “initialization event” ABA " to indicate the creation of the initial condition:

ABA

BBBBABA

ABA
BBB AB

BABA BA
A BA

BBAA BA
ABAA

BBBABA BB AB BABA AB
BABA BABA

BBBABA BBABA
A BBBAA BA B AB

BABA AA
BBBAA BABAA

BA
A A

BBABAA
BAA

BBAA A

BBAA

If we want to focus on causal relationships, we can now drop the state nodes altogether, and 

get a multiway causal graph that represents possible causal relationships between events:

BBAA A

BBABAA

BABAA

BBBABA B AB BA
BABA A BA

A
BBAA BA

BB AB
BABA AB

BABA

BAA
BBBAA BA A

BBB AB
BABA ABAA

This causal graph is dual to our original evolution graph in the sense that edges in the 

original evolution graph correspond to events—which now become nodes in the causal 
graph. Similarly, each edge in the causal graph is associated with some state which appears 

as a node in the evolution graph.

163



We can get a sense of “possible causal histories” of our system by arranging the multiway 

causal graph in layers:

AB
BABA

BA B
A A A A

A B AB
BABA

BABA BBAA A BBABA BB AB
A BABA BBAA A BA

BBB AB
BABA BBBA BA

A BA BBBABAA A BA

Just like every possible path through the multiway evolution graph gives a possible sequence 

of states that can occur in the evolution of a system, so also every possible path through the 

multiway causal graph gives a possible sequence of events that can occur in the evolution of 
the system.

A�er 10 steps, the graph in our example has become quite complicated:

164



����������� ���������������������������������������
The multiway causal graph that we have just constructed shows the causal relationships for 

all possible paths of evolution in a multiway system. But what if we just pick a single path of 
evolution? Instead of looking at the results for every possible updating order, let us pick a 

particular updating order.

For example, for the rule above we could pick “sequential updating”, in which at each step, 
first for AB→BAB and then for BA→A, we scan the string from le� to right, doing the first 
replacement we can [1:3.6]. This leads to a specific single path of evolution for the system 

(now drawn across the page): 

ABA AB
BABA BABA B AB

BABA BBABA BB AB
BABA BBBABA BBB AB

BABA BBBBABA

We can show the causal relationships between the events in this evolution: 

ABA

AB
BABA

BABA

B AB
BABA

BBABA

BB AB
BABA

BBBABA

BBB AB
BABA

BBBBABA

And we can generate a causal graph—which is very simple:

AB
BABA B AB

BABA BB AB
BABA BBB AB

BABA

But now let us consider a different updating scheme, in which now as we scan the string, we 

try at each position both AB→BAB and BA→A, then do the first replacement we can. This 

procedure again leads to a specific single path of evolution, but it is a different one from 

before:

ABA AB
BABA BABA BA

A BA ABA AB
BABA BABA BA

A BA ABA

This particular path just involves an alternation between the states ABA and BABA, so the 

states graph is a cycle: 

165



BA
A BA

ABA BABA

AB
BABA

Including causal relationships here we get:

BA
A BA

BABA ABA

AB
BABA

The resulting causal graph is then: 

BA
A BA AB

BABA

This causal graph is quite different from the one we got for the previous updating scheme. 
But both individual causal graphs necessarily occur in the whole multiway causal graph:

AB B
BABA

A
BABA

BA B AB BA BA AB
 A A A A

A BBABA , A A A A BBABA 

BABA BBAA A BBABA B AB
BABA BBAA B A BA BABAA BBAA A BBABAA BB AB

A BABA BBAA BA

BBB AB
BABA BBBAA BA BBBABA BABA BBB AB BA

A A BABA BBBAA BA BBBABAA A BA

166



Given the multiway causal graph, we can explicitly find all possible individual causal graphs 

(not showing individual loop configurations separately):
AB A AB AB
BAB BABA

AB AB AB
BABA BABA BABA BABA

BA B
A BA B A

BABA B AB
BABA B AB

BABA

 , , , , , ,
AB BA
BABA B A BA BB AB

BABA BB AB
BABA

BABA BA BA BA AB B BA
A A A A BA A A BBABA B A

BABA BB A BA BBB AB
BABA

AB AB AB AB AB AB
BABA BABA BABA BABA BABA BABA

BA
A BA B AB BA

BABA A BA B AB
BABA

, , , B AB
BABA , B AB

BABA , 
AB
BABA BBA AB

A BA BABA BB AB
BABA

BA BA BA
A BA A BA BABAA BBABAA B A A BBAA BA BABAA BBBABAA

And what this shows is that at least with the particular rule we are looking at, there are many 

different classes of evolution paths, that can lead to distinctly different individual causal 
graphs—and therefore causal histories.

����������������������������������������
One might think that all rules would work like the one we just studied, and would give 

different causal graphs depending on what specific path of evolution one chose, or what 
updating scheme one used. But a crucial fact that will be central to the potential application 

of our models to physics is this is not the case.

Instead, whenever a rule is causal invariant, it does not produce many different causal 
graphs. Instead, whatever specific path of evolution one choses, the rule always yields an 

exactly equivalent causal graph.

The rule we just studied is not causal invariant. But consider instead the simple causal 
invariant rule {BA→AB} evolving from the state BBBAA:

167



BBBAA

BBBAABA

BBABA

BBABAAB BBAABBA

BBAAB BABBA

BBA BA
ABAB BABBAAB ABBBA

BABAB ABBBA

BABAABB
BA
ABBAB ABBBAAB

BAABB ABBAB

BA
ABABB ABBAABB

ABABB

Adding in causal relationships this becomes:

BBBAA

BBBAABA

BBABA

BBAABBA BBABAAB

BABBA BBAAB

BA
ABBBA BABBAAB BBAABAB

ABBBA BABAB

ABBBA BA
ABBAB BABAAB ABB

ABBAB BAABB

ABBA BA
ABB ABABB

ABABB

168



The corresponding multiway causal graph is:

BBBAABA

ABBBA ABBA B A BBAAB B AB BBA A
AB BBABAB ABBA

ABBAABB BABAB BA
AB ABABB

BA
ABBAB

BA
ABBBA

But now consider the individual causal graphs, corresponding to different possible paths of 
evolution. From the multiway causal graph we can extract all of these, and the result is: 

BBBAABA BBBAABA BBBAABA BBBAABA BBBAABA

 BBABA BA
AB BABAB , BBABAB BBABAA AB , BABBAAB BBAABBA , BBAABBA BABBAAB , BBAABBA ABBBAAB 

BABA A BA
ABB

B
ABABB ABBAB ABBA A BA BA

ABB BABAABB
B
ABABB ABBAB ABBAABB ABBBA ABBAABB

There are five different cases. But the remarkable fact is that they all correspond to isomor-
phic graphs. In other words. even though the specific sequence of states that the system 

visits is different in each case, the network of causal relationships is always exactly the 

same, regardless of what path is followed.

And this is a general feature of any causal invariant system. The underlying evolution rules 

for the system may allow many different paths of evolution—and many different sequences 

of states. But when the rules for the system are causal invariant, it means that the network 

of relationships between updating events is always the same. Depending on the particular
order in which updates are done, one can see different paths of evolution; but if one looks 

only at event updates and their relationships, there is just one thing the system does. 

Here is a slightly larger example for the rule {BA→AB} starting from BBBBAAAA. In each 

case a different random updating order is used, leading to a different sequence of states. But 
the final causal graphs representing the causal relationships between events are exactly the 

same: 

169



BBBBAABAAA BBBBAABAAA BBBBAABAAA

BBBAABBAAA BBABBAABAA BBBAABAAAB BBBABAABAA BBBAABBAAA BABBBAABAA

BBAABAABBA BBABAABBAA BBAABBAABA BBA ABBBAABBAAAB ABAAB BBBAABAABA BBA A A
ABBBAAA BABBABABA BABBABABA

 BA
ABBAAABB ABBAABAABB BBAABAABBA BABAABBA , BA

AB ABBBAAAB ABBAABBAAB ABABBAABAB BBBAAABAAB , BA
ABBABABA ABBAABBABA ABABBAABBA ABABABBAAB 

ABAABBAABB AABBAABABB BABAABAABB ABAABAABBB ABABAABBAB ABAABBAABB ABAABBAABB AABBAABABB ABABABAABB

AABAABABBB AABABAABBB AABAABABBB ABAABAABBB AABAABBABB AAABBAABBB

AAABAABBBB AAABAABBBB AAABAABBBB

���������������������������������������
We have discussed causal invariance in terms of path independence in multiway systems. 
But we can also explore it in terms of specific evolution histories for the underlying substitu-
tion system. Consider the rule {BA→AB}. Here is a representation of one way it can act on a 

particular initial string: 

B B A A A A B A A B B A B B B B B A A A

B A B A A A A B A B A B B B B B A B A A

B A A B A A A B A A B B B B B A B A B A

A B A B A A A A B A B B B B A B B A A B

A A B A B A A A B A B B B A B B A B A B

A A B A A B A A A B B B A B B B A A B B

A A A B A A B A A B B A B B B A B A B B

A A A A B A A B A B B A B B A B A B B B

A A A A A B A A B B A B B A B B A B B B

A A A A A A B A B A B B A B B A B B B B

A A A A A A A B A B B A B B A B B B B B

A A A A A A A A B B A B B A B B B B B B

A A A A A A A A B A B B A B B B B B B B

A A A A A A A A A B B A B B B B B B B B

A A A A A A A A A B A B B B B B B B B B

A A A A A A A A A A B B B B B B B B B B

In a multiway system, each path represents a particular sequence of updates that occur one 

a�er another. But here in showing the action of {BA→AB} we are choosing to draw several 
updates on the same row. If we want, we can think of these updates as being done in 

sequence, but since they are all independent, it is consistent to show them as we do.

170



If we annotate the picture by showing causal connections between updates, we see the 

causal graph for the evolution—and we see that the updates we have drawn on the same row 

are indeed independent: they are not connected in the causal graph:

The picture above in effect uses a particular updating order. The pictures below show three
possible random choices of updating orders. In each case, the final result of the evolution is 

the same. The intermediate steps, however, are different. But because our rule is causal 
invariant, the causal graph of causal relationships always has exactly the same form:

In picking updating orders there is always one constraint: no update can happen until the 

input for it is ready. In other words, if the input for update V  comes from the output of 
update U, then U must already have happened before V  can be done. But so long as this 

constraint is satisfied, we can pick whatever order of updates we want (cf. [1:9.10]).

171



It is sometimes convenient, however, to think in terms of “complete steps of evolution” in 

which all updates that could yet be done have been done. And for the particular rule we are 

currently discussing, we can readily do this, separating each “complete step” in the pictures 

below with a red line: 

Where the dotted lines go depends on the update order we choose. But because of causal 
invariance it is always possible to draw them to delineate steps in which a collection of 
independent updates occur. 

Each choice of how to assign updates to steps in effect defines a foliation of the evolution. 
We will call foliations in which the updates at each step are causally independent “causal 
foliations”. Such causal foliations are in effect orthogonal to the connections defined by the 

causal graph. (In physics, the analogy is that the causal foliations are like foliations of 
spacetime defined by a sequence of spacelike hypersurfaces, with connections in the causal 
graph being timelike.)

The fact that our underlying substitutions (in this case just BA→AB) involve neighboring 

elements implies a certain locality to the process of evolution. The consequence of this is 

that we can meaningfully discuss the “spatial” spreading of causal effects. For example, 
consider tracing the causal connections from one event in our system: 

172



In effect there is a cone (here just two-dimensional) of elements in the system that can be 

affected. We will call this the causal cone for the evolution. (In physics, the analogy is a light 
cone.) If we pick a different updating order, the causal cone is distorted. But viewed in terms 

of the causal foliations, it is exactly the same:

173



This is the result purely in terms of the causal graph:

Now let us turn things around. Imagine we have a causal graph. Then we can ask how it 
relates to an actual sequence of states generated by a particular path of evolution. The 

pictures below show how we can arrange a causal graph so that its nodes—corresponding to
events—appear at positions down the page that correspond to a particular causal foliation 

and thus a particular path of evolution: 

It is worth noticing that at least for the rule we are using here the intrinsic structure of the 

causal graph defines a convenient foliation in which successive events are simply arranged 

in layers:

174



The rule BA→AB that we are using has many simplifying features. But the concepts of causal 
foliations and causal cones are general, and will be important in much of what follows.

As it happens, we have already implicitly seen both ideas. The “standard updating order” for 

our main models defines a foliation (similar, in fact, to the last one we showed here, in 

which in some sense “as much gets done as possible” at each step)—though the foliation is 

only a causal one if the rule used is causal invariant.

In addition, in 4.14 we discussed how the effect of a small change in the state in one of our 

models can spread on subsequent steps, and this is just like the causal cone we are dis-
cussing here.

������������ ����������������������������
One of the simplifying features of the rule BA→AB discussed in the previous subsection is 

that for any finite initial condition, it always evolves to a definite final state a�er a finite 

number of steps—so it is possible to construct a complete multiway causal graph for it, and 

for example to verify that all the causal graphs for specific paths of evolution are identical. 

175



But consider the rule {A→BB,B→A}:

A

BB

AB BA

AA BBB

ABB BBA BAB

ABA AAB BBBB BAA

This rule is causal invariant, but never evolves to a definite state, and instead keeps growing 

forever. Including events in the evolution we get:

A

A
BB

BB

B
AB BBA

AB BA

AB A B
A BBB AA B A

BB

AA BBB

A A A B B
BB BBA ABB BBBA BAB

ABB BBA BAB

ABB A ABB B B B
A BBBB A ABA BB A B A B

BB BAA BBB AAB BAA

ABA BBBB AAB BAA

176



The corresponding multiway causal graph is:

A
BB

B B
AA AB AB B

A BA

A
BBA

A
BBB A A

BB B A
BB

B B B
ABA AAB BB B

ABB A ABAA BBB AB AB BABA BBBA A

A
BBBB B A

BBB BB A
BB

And now if we extract possible individual causal graphs from this, we get:

 , , , , , , , , ,

, , , , , , , , , , ,

, , , , , , , , , 

These look somewhat similar, but they are not directly equivalent. And the reason for this 

has to do with how we are “counting steps” in the evolution of our system. If we evolve for 

longer, the effect becomes progressively less important. Here are causal graphs generated 

by a few different randomly chosen specific sequences of updates (each corresponding to a 

specific path through the multiway system):

A A A A
BB BB BB BB

BBA B
AB

B
ABABA BB

B A
AA BBA

B A
BB A

BBB B A
BB

A
BBBA A A

BB BB B
AB BAA

 , , B
AAB BBAB , BBABA BBBA 

B A
B
ABAA ABB B

A BBAA
BBA

A
BBAAB A A

BBB B A
BBBA BB A

BB
BBABA BBBBAA

A
BBBBBA AB A

BBA B
ABAAB ABAAB ABBAB ABAB

B A BBBABA AA BBBBA
BBBA BAB A

BBA

BBBBABA BBBABAA BBABAA BABBABA
A
BBBABBB ABA A

BBB AB A
BBAA

177



Here are the corresponding results a�er a few more updates:

 , , , 

This is a different rendering:

 , , , 

And this is what happens a�er many more updates (with a somewhat more systematic 

ordering):

If we continued for an infinite number of updates, all these would give the same result—and 

the same infinite causal graph, just as we expect from causal invariance. But in the particu-
lar cases we are showing, they are being cut off in different ways. And this is directly related 

to the causal foliations we discussed in the previous subsection. 

178



Here are examples of evolution with specific choices of updating orders:

A A A

B B B B B B

A A A A A A

B B B B B B B B B B B B

A A A A A A A A A A A A

B B B B B B B B B B B B B B B B B B B B B B B B

B A A B A B A B A B B B A A A B B A A B A B A B

B B B A B A A B B A A B A B B B B B B B B B A B B A A A B B B

B B B A B B B B B A A A A A A B A B A B A B B B A A B B B A B B A B B

B B A A B A A B B A A B B A A B B B A B A B B B A A B A A B A A A B B B B A B

B A A B B B A A B B A B B A B B B A B B B A A A B A B B B A B A A B A B B A B B A A A A

Adding causal graphs we get:

179



Here is what happens if we continue these for longer:

And here are the causal graphs that correspond to these evolutions:

�������������������� �����
Any causal invariant system always ultimately has a unique causal graph. The graph can be
found by analyzing any possible evolution for the system, with any updating scheme—
though for visualization purposes, it is usually useful to use an updating scheme where as
much happens as possible at each step.

180



The trivial causal invariant rule A→A starting from A has causal graph: 

Starting from a string of 10 As it has causal graph:

A→AA has a causal graph starting from A that is a binary tree 

which can also be rendered:

The rule

{A → A, A → AA}

181



t
starting from A gives a “two-step binary tree” with 22

 nodes at level t :

One does not have to go beyond rules involving just a single element (all of which are causal 
invariant) to find a range of causal graph structures. For example, here are all the forms 

obtained by rules allowing up to 6 instance of a single element A, with initial condition AA:

 , , , ,
{A → A} {AA → A}

{A → AAA}
{A → AA}

, , , , ,
{AA → AA} {A → A, A → A}

{A → AAAA} {AA → AAA} {A → A, A → AA}

, , , , ,
{A → A, AA → A} {A → A, AAA → A}

{A → AAAAA} {AA → AAAA} {A → A, A → AAA}

, , , , 
{A → A, AA → AA} {A → AA, AA → A} {AA → A, AA → A} {A → A, A → A, A → A}

{A → AA, A → AA}

A notable case is the rule:

{AA → AAA}

182



Shown in layered form, the first few steps give:

 , , , , , , , 

A�er a few more steps, this can be rendered as:

Running the underlying substitution system AA→AAA updating as much as possible at each 

step (the StringReplace scheme), one gets strings with successive lengths

{2, 3, 4, 6, 9, 13, 19, 28, 42, 63, 94, 141, 211, 316, 474, 711,
1066, 1599, 2398, 3597, 5395, 8092, 12138, 18207, 27310, 40965}

which follow the recurrence:

a[1] = 2; a[n_] := If[EvenQ[n], 3 n /2, (3 n-1) /2]

183



Other rules of the form Ap→Aq
 for non-commensurate p  and q  give similar results, analo-

gous to tessellations in hyperbolic space:

 , , , ,

{2, 3} {2, 5} {3, 4}
{2, 7}

, , , ,

{3, 5} {3, 7} {4, 5} {4, 7}

, , , 

{5, 6} {6, 7}
{4, 9} {5, 7}

Rules that involve multiple replacements can give similar behavior even starting from a 

single A:

 , , 

{A → A, A → AAA, AA → A} {A → AA, A → AA, AAA → A} {A → AAAAA, AAA → A}

Rules just containing only As cannot progressively grow to produce ordinary tilings. One can 

get these with the “sorting rule”
{BA → AB}

184



which when started with 20 BAs yields:

There are also rules which “grow” grid-like tilings. For example, the rule

{A → AB, BB →  BB}
starting from a single A produces 

which is equivalent to a square grid:

185



There is also a simple rule that generates essentially a hexagonal grid:

{A → B, B →  AB, BA → A}

Other forms of causal graphs produced by simple causal invariant substitution systems 

include (starting from A, AB or ABA):

{A→BB,BB→AB} {A→AB,BB→BA} {AB→AABAB} {A→A A→A B→BB} {A→A,B→AAAB}

{AB→BAABBA} {ABA→ABABA} {A→AAB,BAA→A} {A→BB,BB→ABA} {A→AA,AAA→AA}

When rules terminate they yield finite causal graphs. But these can o�en be quite compli-
cated. For example, the rule
{A → BBB, BBBB →  A}

186



started from strings consisting of from 1 to 6 As yields the following finite causal graphs:

 , , ,

, , 

With a string of 50 As, the rule gives the finite causal graph:

Compared to the hypergraphs we studied in previous sections, or even the multiway graphs 

from earlier in this section, the causal graphs here may seem to have rather simple struc-
tures. But there is a good reason for this. While there can be many updating events in the 

evolution of a string substitution system, all of them are in a sense arranged on the same 

one-dimensional structure that is the underlying string. And since the updating rules we 

consider involve strings of limited length, there is inevitably a linear ordering to the events 

along the string. This greatly simplifies the possible forms of causal graphs that can occur, 
for example requiring them always to remain planar. In the next section, we will see that for 

our hypergraph-based models—which have no simplifying underlying structure—causal 
graphs can be considerably more complex.

187



������� �������������� �����
Much as we did for hypergraphs in section 4, we can consider the limiting structures of
causal graphs a�er a large number of steps. And much as for hypergraphs, we can poten-
tially describe these limiting structures in terms of emergent geometry. But one difference 

from what we did for hypergraphs is that for causal graphs, it is essential to take account of 
the directedness of their edges. It is still perfectly possible to have a limit that is like a 

manifold, but now to measure its properties we must generate the analog of cones, rather 

than balls.

Consider for example the simple directed grid graph:

Now consider starting from a particular node, and constructing progressively larger “cones”:

 , , , , 

We can call the number of nodes in this cone a�er t  steps Ct . In this case the result (for t  

below the diameter of the graph) is:

1
Ct = t (1 + t)

2
And in the limit of large graphs, we will have:

Ct~ t2

188



We can also set up a 3D directed grid graph

and generate a similar cone

 , , , , 

for which now Ct~ t3. 

In general, we can think about the limits of these grid graphs as generating a d-dimensional
“directed space”. There is also nothing to prevent having cyclic versions, such as

and in general a family of graphs that are going to behave like d-dimensional directed space 

in the limit will have Ct~ td.

In direct analogy to what we did with hypergraphs in section 4, we can compute Ct  for 

causal graphs, and then estimate effective dimension by looking at its growth rate. (There 

are some additional subtleties, though, because whereas at any given step in the evolution of 
the system, Vr can be computed for any r for any point in a hypergraph, Ct  can be computed 

only until t  “reaches the edge of the causal graph” from that starting point—and later we will 
see that the cutoff can also depend on the foliation one uses.)

189



Consider the substitution system:

{A → AB, BB →  BB}
This generates the causal graph: 

The log differences of Ct  averaged over all points for causal graphs obtained from 10 

through 100 steps of evolution have the form (the larger error bars for larger t  in each case 

are the result of fewer starting points being able to contribute):

2.0

1.5

1.0

0.5

0.0
0 20 40 60 80 100

As expected, for t  small compared to the number of steps, the limiting estimated dimension
is 2. 

190



Most of the other causal graphs shown in the previous subsection do not have finite dimen-
sion, however. For example, for the rule AA→AAA the causal graph has the form

which increases exponentially, with Ct~ ( 3 )t .2

The limits of the grid graphs we showed above essentially correspond to flat d-dimensional 
directed space. But we can also consider d-dimensional directed space with curvature. 
Although we cannot construct a complete sphere graph that is consistently directed, we can 

construct a partial sphere graph:

191



In a layered rendering, this is:

Once again, we can compute the log differences of Ct  (the similarity of sphere graphs means 

that using larger versions does not change the result):

2.0

1.5

1.0

0.5

0.0
0 5 10 15 20 25

The systematic deviation from the d = 2 result is—like in the hypergraph case—a reflection 

of curvature.

������������������������������������������ �����
One way to describe a causal graph is to say that it defines the partial ordering of events in a 

system—or, in other words, it is a representation of a poset. (The actual graph is essentially 

the Hasse diagram of the poset (e.g. [69]).) Any particular sequence of updating events can 

then be thought of as a particular total ordering of the events.

192



As a simple example, consider a grid causal graph (as generated, for example, by the rule 

BA→AB):
1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

One total ordering of events consistent with all the causal relations in the graph is a 

“breadth-first scan” [70]:
1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

But another possible ordering is a “depth-first scan” [71]:
1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

193



Another conceivable ordering would be:
1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

And in general there are many orderings consistent with the relations defined by the causal 
graph. For the particular graph shown here, out of the n! conceivable orderings of nodes 1 

through n, the orderings consistent with the causal relations correspond to possible Young 

tableaux, and the number of them is equal to the number of involutions (self-inverse 

permutations) of n elements (e.g. [11:A000085]) which is asymptotically a little larger than 

( nⅇ )
- n
2
 times n!.

Here are the possible causal orderings that visit nodes 1 through 6 above (out of all 6! = 720 

orderings): 

1 4 6 1 3 6 1 2 6 1 3 6 1 2 6 1 4 5 1 3 5 1 2 5
2 5 2 5 3 5 2 4 3 4 2 6 2 6 3 6
3 4 4 5 5 3 4 4

1 3 4 1 2 4 1 2 3 1 3 5 1 2 5 1 3 4 1 2 4 1 2 3
2 6 3 6 4 6 2 4 3 4 2 5 3 5 4 5
5 5 5 6 6 6 6 6

And here are all 76 possible causal orderings that visit any six nodes starting with node 1:

194



But while a great many total orderings are in principle possible, if one wants, for example, 
to think about large-scale limits, one usually wants to restrict oneself to orderings that can 

specified by (“reasonable”) foliations.

The idea of a foliation is to define a sequence of slices with the property that events on 

successive slices must occur in the order of the slices, but that events within a slice can 

occur in any order. So, for example, an immediate possible foliation of the causal graph 

above is just:

1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54 55

This foliation specifies that event 1 must happen first, but then events 2 and 3 can happen in 

any order, followed by events 4, 5 and 6 in order, and so on.

But another consistent foliation takes diagonal slices (with actual locations of events in the 

diagram being thought of as the centers of the boxes):

1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54 55

195



And so long as the diagonals are not steeper than the connections in the causal graph, this 

foliation will again lead to orderings that are consistent with the partial order defined by the 

causal graph. (For example, here event 1 must occur first, followed by event 2, followed by 

events 3 and 4 in any order, and so on.)

Particularly if the diagonals are steeper, multiple events will o�en happen in a single slice 

(as we see with events 4 and 7 here):

1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54 55

Now let us consider how this relates to taking the limit of a large number of events. If it were 

not for the directedness of the graph, we could do as we did in 4.17, and just imagine a 

process of refinement that leads to a manifold. But the Euclidean space that is the model for 

the manifold does not immediately have a way to capture the directedness of the graph, and 

so we need to do a little more.

But this is a place where foliations help. Because within a slice of a foliation we have events 

that can happen in any order. And at least for our string substitution system, the events can 

be thought of in the limit as being on a one-dimensional manifold, with a coordinate related 

to position on the string. And then there is just a second coordinate that is the index of the 

slices in the foliation. 

But if the limit of our causal graph is a continuous space, we should be able to have a 

consistent notion of “distance between events”. For events that are “out of order”, the 

distance should be undefined (or perhaps infinite). But for other events, we should be able 

to compute the distance in terms of the coordinate (say t ) that indexes the slices in the 

foliation and the coordinate (say x) within the slices. 

Given the particular setup of diagonal slices on a causal graph that is a grid, there is a unique 

distance function that is independent of the angle of the slices, which can be expressed in 

terms of the coordinate differences Δt  and Δx:

Δt 2 – Δx2

196



This function is exactly the standard Minkowski metric for a Lorentzian manifold [72][73], 
and we will encounter it again in section 8 when we discuss potential connections to 

physics. But here the metric is simply an abstract way to express distance in the limit of our 

causal graphs for string substitution systems. 

What happens if we use a different foliation? For example, a foliation like the following also 

leads to orderings of events that are consistent with the partial ordering required by the 

causal graph:

1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

The process of limiting to a manifold is more complicated here. We can start by defining a 

“lapse function” α (t ,x) (in analogy with the ADM formalism of general relativity [74][75]) 
which effectively says “how thick” each slice of the foliation is at each position. (If we also 

wanted to skew our foliations, we could include a “shi� vector” as well.) And in the limit we 

can potentially define a distance by integrating α(t ,x)2 δt 2 – δx2 along the shortest path 

from one point to another. 

In a sense, however, even by imagining that there is a reasonable function α (t ,x) that
depends on real variables t  and x we are implicitly assuming that our foliation has a certain 

simplicity and structure—and is not trying to reproduce some of the more circuitous total 
orderings at the beginning of this subsection. 

But in a sense the question of what type of foliation we need to consider depends on what we 

want to use it for. And in making potential connections with physics, foliations will in effect 
be how we parametrize observers. And as soon as we assume that observers are limited in 

their computational capabilities, this puts constraints on the types of foliations we need to 

consider. 

197



������������������������������ �����
Causal graphs provide one kind of summary of the evolution of a system, based on capturing 

the causal relationships between events. What we call branchial graphs provide another 

kind of summary, based on capturing relationships between states on different branches of 
a multiway system. And whereas causal graphs capture relationships between events at 
different steps in the evolution of a system, branchial graphs capture relationships between 

states on different branches at a given step. And in a sense they define a map for exploring 

branchial space in a multiway system.

One might perhaps have imagined that states on different branches of a multiway system 

would be completely independent. But when causal invariance is present they are definitely 

not—because for example whenever they split (to form a branch pair), they will always 

merge again. 

Consider the multiway evolution graph (for the rule {A→AB,B→A}):

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

Look at the second-to-last step shown. This contains the following unresolved branch pairs:

{{AAA, AABB}, {AAA, ABAB}, {AAA, ABBA}, {AABB, ABAB}, {AABB, ABBA}, {AABB, ABBBB}, 
{ABAB, ABBA}, {ABAB, ABBBB}, {ABBA, ABBBB}}
The two states in each of these branch pairs are related, in the sense that they diverged from 

a common ancestor—and will converge to a common successor. We form the branchial 
graph by connecting the states that appear in newly unresolved branch pairs at a given step. 
For the steps shown in the evolution graph above, the successive branchial graphs are:

198



ABA

 ABB AA , ,

AAB ABBB

ABBA AABBB AAAB

ABABB

AAA ABAB ABBBB , ABBBBB AABA 
ABBAB

AABB ABBBA ABAA

For the next few steps, the states branchial graphs are:

 , , 

For the rule shown here, the number of nodes at the t th step is the t th Fibonacci number, or 

~ϕt
 for large t . The graphs are highly connected, but far from completely so. The number of 

edges ~ 2t , while the diameter is  t . The degree of transitivity (i.e. whether X  being con-
2

nected to Y  and Y  being connected Z  implies X  being connected to Z) [76] gradually 

decreases with t . As a measure of uniformity, one can look at the local clustering coefficient 
[77] (which measures to what extent there are local complete graphs):

 , , 

199



The graphs have somewhat complex vertex degree distributions (here shown a�er 10 and 15 

steps):

80
15

60
10 40
5 20

0 0
10 15 20 25 30 35 20 30 40 50 60 70 80

For the rule we are showing here, the branchial graph turns out to have a particularly 

simple interpretation as a map of the level of common ancestry of states. By definition, if 
two states are directly connected on the branchial graph, it means they had an immediate 

common ancestor one step before. But what does it mean if two states are graph distance 2 

apart?

The particular rule shown here has the property that it is causal invariant, but also that all 
branch pairs resolve in just one step. And from this it follows that states that are distance 2 

apart on the branchial graph must have a common ancestor 2 steps back. And in general the 

distance on the branchial graph is equal to the number of steps one must go back before one 

gets to a common ancestor.

The following histograms show the distribution of graph distances between all pairs of 
states at steps 10 and 15—or alternatively, the distribution of how many steps it has been 

since the states had a common ancestor (for this rule the mean increases roughly like t
 ):
6

0 1 2 3 4 5 0 2 4 6

We are defining the branchial graph to be based on looking at branch pairs that are unre-
solved at a particular step, and are new at that step. But we could generalize to consider a 

“thickened” branchial graph that includes all branch pairs that have been new within the 

past m  steps. By doing this we can capture common ancestry of states even when they are 

associated with branch pairs that take up to m  steps to resolve—but when we do this it is at 
the cost of having many additions to the graph associated with branch pairs that have “come 

and gone” within our thickening depth. 

It should be noted that any possible interpretation of branchial graphs in terms of common 

ancestors depends on having causal invariance. Absent causal invariance, there is no 

guarantee that states with common ancestors will even be connected in the branchial graph. 
As an extreme example, consider the rule: 
{A → AB, A → AC}

200



The multiway graph for this rule is a tree

A

AB AC

ABB ACB ABC ACC

ABBB ACBB ABCB ACCB ABBC ACBC ABCC ACCC

and its branchial graph on successive steps just consists of a collection of disconnected 

pieces:

ACBB ABBB

ACB ABB ACBC ABBC
 AC AB , , 

ACC ABC ACCB ABCB

ACCC ABCC

By thickening the branchial graph by m  steps, one could capture m  steps of common 

ancestry. And in general one could imagine infinitely thickening the graph, so that one looks 

all the way back to the initial conditions. But the branchial graph one would get in this way 

would essentially just be a copy of the whole multiway graph.

When a rule has branch pairs that take several steps to resolve, it is possible for the 

branchial graph to be disconnected, even when the rule is causal invariant. Consider for 

example the rule
{A → AA, A → BAB}

201



in which the branch pair {BAAB,BBABB} takes 3 steps to resolve. The first few branchial 
graphs for this rule are:

BABA ABAB

 BAB AA , , ,

AAA

BBABB BAAB

, , 

It is fairly common to get branchial graphs in which a few disconnected pieces are “thrown 

off” in the first few steps, and never recombine. Sometimes, however, disconnected pieces 

can recombine. The rule
{A → BB, BB → AA}
starting from initial condition A yields the sequence of branchial graphs:

AABB

 BBA ABB , BBBB AAA , BAAB ABBA ,
BBAA

BBBBA BBABB

, , , 

ABBBB AAAA

BBBAB BABBB

202



Sometimes the branchial graph can break into many components. This happens for the rule

{AA → AABB}
starting from initial condition AA:

 , , ,

, , , 

The causal graph for this rule reveals that there is also causal disconnection in this case:

�������������������������������� �����
As a first example, consider the (causal invariant) rule which effectively just creates either A 

or B from nothing:
{ → A, → B}

203



At step t , this rule produces all 2t  possible strings. Its multiway way graph is:

A B

AA AB BA BB

AAA AAB ABA BAA ABB BAB BBA BBB

The succession of branchial graphs is then:

AB

 B A , BB AA ,

BA

AABB

ABBB AAAB
ABAB

BBA BAA BABB AABA

BBB BAB ABA AAA , BBBB ABBA BAAB AAAA 

BBAB ABAA
ABB AAB

BABA
BBBA BAAA

BBAA

The graph on step t  has 2t  nodes and 2t–2 (t 2 – t + 4) – 1 edges. The distance on the graph 

between two states is precisely the difference in the total number of As (or Bs) between 

them, plus 1—so combining states which differ only through ordering of A and B the last 
graph becomes:

AAAA AAAB AABB ABBB BBBB

204



With the rule

{ → A, → B, → C}
the sequence of branchial graphs is

AAA
B BB BA AA

AB
BAA CAA

AAB AAC
AC ABA ACA

 , CB , 
BC

CA BCA
BAB CBA

BBA BAC CCA
ACB CAC

CAB
ABB ABC ACC

A C CC BCB CBC BCC
BBB CBB BBC CCC

CCB

and in the last case combining states which differ only in the order of elements, one gets: 

AAA C
AAC ACC CC

AAB ABC BCC

ABB BBC

BBB

Note that no rule involving only As can have a nontrivial branchial graph, since all branch 

pairs immediately resolve.

Consider now the rule:

{A → AB}
As mentioned in 5.4, with initial condition AA this rule gives a multiway graph that corre-
sponds to a 2D grid: 

AA

AAB ABA

AABB ABAB ABBA

AABBB ABABB ABBAB ABBBA

205



The corresponding branchial graphs are 1D:

 ABA AAB , AABB ABAB ABBA , AABBB ABABB ABBAB ABBBA 

With initial condition AAA, the multiway graph is effectively a 3D grid, and the branchial 
graph is a 2D grid:

AABBBA AABBAB AABABB AAABBB

AABA AABBA

ABABBA ABABAB ABAABB

 , ABABA AABAB , 

ABBABA ABBAAB

AAAB ABAA
ABBAA ABAAB AAABB

ABBBAA

Some rules produce only finite sequences of branchial graphs. For example, the rule

{A → B}
with initial condition AAAA yields what are effectively sections through a cube oriented on 

its corner:

AABA BABB
BABA ABBA

 AAAB ABAA , AABB BBAA , ABBB BBAB 

BAAB ABAB
BAAA BBBA

As another example producing a finite sequence of branchial graphs, consider the rule:

{BA → AB}

206



Starting from BBBAAA it gives:

BAABBA

 BBAABA BABBAA , ABBBAA BABABA BBAAAB , ,

ABBABA BABAAB

ABBAAB

, AABBBA ABABAB BAAABB , ABAABB AABBAB 

ABABBA BAABAB

Starting from BBBBBAAAAA it gives:

 , , , , , , , , , ,

, , , , , , ,

, , , , , , 

One can think of this as showing the “shapes” of successive slices through the multiway
system for the evolution of this rule.

As another example of a rule yielding an infinite sequence of branchial graphs, consider:

{A → AAB}

207



This yields the following branchial graphs:

AAABBAB AAAABBB

 AABAB AAABB , AABAABB ,
AABABAB AAABABB

, , 

In a 3D rendering, the graph on the next step is:

208



The following are the distinct forms of branchial graphs obtained from rules involving a 

total of up to 6 As and Bs (starting from a single A):

{A→AAB} {A→AABA}

{A→A {A→AAAB} {A→B,B→AA}
}

{A→B,B→AB} {A→AAABA} {A→AABAA}
{A→B,B→AAA}

{A→AAAAB}

{A→B,B→ABA} {A→B,B→BAB}
{A→B,B→AAB} {A→AA,A→AB}

{A→B,B→ABB}

{A→AB,A→BA} {A→AA,AA→A}

{A→AB,BB→A}

{A→BB,B→AA}

We have seen that branchial graphs can form regular grids. But many branchial graphs have 

much higher levels of connectivity. No branchial graph can continue to be a complete graph 

(with all neighbors having distance 1) for more than a limited number of steps. However, 
the diameters of branchial graphs do tend to grow slowly, and on step t  they can be no 

larger than t . Some branchial graphs show linear or polynomial growth with the number of 
steps in vertex and edge count, but many show exponential growth. 

In analogy to what we did for hypergraphs and causal graphs, we can define a quantity 

Bb which measures the number of nodes in the branchial graph reached by going out to 

graph distance b  from a given node. 

209



Consider for example the rule:

{ → A, → B}
The sequence of forms for Bb as a function of b  on successive steps is:

4000

3000

2000

1000

0
0 2 4 6 8 10 12

At step t , the diameter of the graph is just t , and Bb=t = 2t . For smaller b, the ratios of the Bb 

for given b  at successive steps t  steadily decrease, perhaps suggesting ultimately less-than-
exponential growth:

2.0 9

8

1.8 7

6

1.6 5

4
1.4 3

2
1.2

1

0 2 4 6 8

One can ask what the limit of a branchial graph a�er a large number of steps may be. As an 

initial possible model, consider graphs representing n-cubes in progressively higher 

dimensions:

 , , , , , 

210



The graph distances between nodes in these graphs are exactly the same as the Euclidean 

distances between the 2n possible tuples of 0s and 1s (here shown in distance matrices 

arranged in lexicographic order):

 , , , , , 

The values of Bb in this case can be found from [11:A008949]:

Accumulate[Table[Binomial[n, k], {k, 0, n}]
1000

800

600

400

200

0
0 2 4 6 8 10

This shows the ratios of Bb for given b  for successive n:
2.0

1.8

1.6

1.4

1.2

1.0
0 5 10 15 20

Much as one can consider progressively larger grid graphs as limiting to a manifold, so 

perhaps one may consider higher and higher “dimensional” cube graphs as limiting to a 

Hilbert space.

It is also conceivable that limits of branchial graphs may be related to projective spaces [78]. 
As one potential connection, one can look at discrete models of incidence geometry [79]. For 

example, with integers representing points and triples representing lines, the Fano plane
{{1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 4, 6}, {2, 5, 7}, {3, 4, 7}, {3, 5, 6}}

211



is a discrete model of the projective plane. One can consider the sequence of such objects as 

hypergraphs

 , , 

and representing both points and lines here as nodes, these correspond to the graphs: 

 , , 

But for such graphs one finds that Bb has a very different form from typical branchial 
graphs:
500

400

300

200

100

0
0 1 2 3 4 5

An alternative approach to connecting with discrete models of projective space is to think in 

terms of lattice theory [69][80][81]. A multiway graph can be interpreted as a lattice (in the 

algebraic sense), with its evolution defining the partial order in the lattice. The states in the 

multiway system are elements in the lattice, and the meet and join operations in the lattice 

correspond to finding the common ancestors and common successors of states. 

The analogy with projective geometry is based on thinking of states in the multiway system 

(which correspond to elements in the lattice) as points, connected by lines that correspond 

to their evolution in the multiway system. Points are considered collinear if they have the 

same common successor. But (assuming the multiway system starts from a single state), 
causal invariance is exactly the condition that any set of points will eventually have a 

common successor—or in other words, that all lines will eventually intersect, suggesting that 
the multiway graph is indeed in some sense a discrete model of projective space—so that 
branchial graphs may also be models of projective Hilbert spaces.

212



����������������������� ���� ��� ��������������������������
���������������
Just as states that occur at successive steps in the evolution of our underlying systems can be 

thought of as associated with successive slices in a foliation of the causal graph, so also 

branchial graphs can be thought of as being associated with successive slices in a foliation of 
the multiway graph. 

As we discussed above, different foliations of the causal graph define different relative 

orderings of updating events within our underlying system. But we can now think about this 

at a higher level and consider foliations of the multiway graph, that in effect define different 
relative orders of updating events on different branches of the multiway system. A foliation 

of the causal graph in effect defines how we should line up our notion of “time” for events in 

different parts of our underlying system; a foliation of the multiway graph now also defines 

how we should line up our notion of “time” for events on different branches of the multiway 

system.

For example, with the rule 

{ A → AB}
starting from AA, we can define the following foliation of the multiway graph: 

AA

ABA AAB

ABBA ABAB AABB

ABBBA ABBAB ABABB AABBB

ABBBBA ABBBAB ABBABB ABABBB AABBBB

ABBBBBA ABBBBAB ABBBABB ABBABBB ABABBBB AABBBBB

ABBBBBBA ABBBBBAB ABBBBABB ABBBABBB ABBABBBB ABABBBBB AABBBBBB

ABBBBBBBA ABBBBBBAB ABBBBBABB ABBBBABBB ABBBABBBB ABBABBBBB ABABBBBBB AABBBBBBB

213



This yields the branchial graphs:

ABA AAB

AABB ABAB ABBA

AABBB ABABB ABBAB ABBBA

AABBBB ABABBB ABBABB ABBBAB ABBBBA

AABBBBB ABABBBB ABBABBB ABBBABB ABBBBAB ABBBBBA

AABBBBBB ABABBBBB ABBABBBB ABBBABBB ABBBBABB ABBBBBAB ABBBBBBA

AABBBBBBB ABABBBBBB ABBABBBBB ABBBABBBB ABBBBABBB ABBBBBABB ABBBBBBAB ABBBBBBBA

Just as the causal graph defines a partial order for events, so now the multiway graph 

defines a partial order for states. And so long as it is consistent with this partial order, we 

can pick any total order for the states. And we can parametrize some of these total orders as 

foliations. 

For the particularly simple case shown here, an alternative foliation consistent with the 

partial order defined by the multiway system is: 

AA

ABA AAB

ABBA ABAB AABB

ABBBA ABBAB ABABB AABBB

ABBBBA ABBBAB ABBABB ABABBB AABBBB

ABBBBBA ABBBBAB ABBBABB ABBABBB ABABBBB AABBBBB

ABBBBBBA ABBBBBAB ABBBBABB ABBBABBB ABBABBBB ABABBBBB AABBBBBB

ABBBBBBBA ABBBBBBAB ABBBBBABB ABBBBABBB ABBBABBBB ABBABBBBB ABABBBBBB AABBBBBBB

214



And if we use this foliation, we get a different sequence of branchial graphs, now no longer 

connected:

ABABB ABBAB
AAB ABA

, AABB ABAB
 , ABBBAB ABBBBA ,

ABBA ABBBA
ABBBBBA

AABBBB ABABBB ABABBBB ABBBBBABB

ABBBBBAB ABBBBBBA , ABBABBB ABBBABB
AABBB ABBABB ABBBBAB , ,

ABBBBBBAB ABBBBBBBA

ABBBBABB ABBBABBB

AABBBBBB
ABABBBBB ABBABBBB

ABBBABBBB ABBBBABBB , , AABBBBBBB ABABBBBBB 

AABBBBB
ABBABBBBB

This example is particularly obvious in its analogy with the causal graphs we discussed 

above. But what makes this work is a special feature of the branchial graphs in this case: the 

fact that the states that appear in them can in effect be assigned simple 1D “coordinates”. In 

the original foliation with unslanted (“one event per slice”) slices, the “coordinate” is 

effectively just the position of the second A in the string. And with respect to the “space” 
defined by this “coordinate”, the branchial graphs can readily be laid out in one dimension, 
and we can readily set up “slanted” foliations, just like we did for causal graphs. 

With the underlying systems we are discussing in this section being based on strings of 
elements, it is inevitable that there will be 1D coordinates that can be assigned to the events 

that occur in the causal graph. But nothing like this need be true of the branchial graph, and 

indeed most branchial graphs have a significantly more complex structure.

215



Consider the same rule as above, but now started from AAA. The multiway graph in this 

case—with a foliation indicated—is then:

The branchial graphs now have a two-dimensional structure—with the positions of the 

second and third As in each string providing potential “coordinates”:

AABBBA AABBAB AABABB AAABBB

AABBA

ABABBA ABABAB ABAABB

 ABABA AABAB , ,

ABBABA ABBAAB

ABBAA ABAAB AAABB

ABBBAA

, , , 

But consider now the rule

{BA → AB}

216



started from BABABABA. The multiway graph in this case is:

And the sequence of branchial graphs based on the foliation above is now:

 , , ,

, , , 

But here it is already much less clear how to assign “coordinates” in “branchial space”, or 

how to create a meaningful family of foliations of the multiway graph.

In thinking about multiway graphs and their foliations there is another complication that 
can arise for some rules. Consider two versions of the multiway graph for the rule
{AB → BAB, BA → A}

217



ABA ABA

BABA AA AA BABA

 BBABA BAA , BAA ABA BBABA 

BBBABA BBAA AA BABA BBAA BBBABA

BBBAA BBBBABA ABA BAA BBABA BBBAA BBBBABA

In the first version, every distinct state is shown only once. But in the second case, the 

evolution is “partially unrolled” to show separately different instances of the same state, 
produced a�er different numbers of updating events. With a foliation whose slices corre-
spond to the layers in the renderings above, the first version of the multiway system yields 

the branchial graphs: 

BAA BBAA BBBAA

 BABA AA , , , 

ABA BBABA BABA BBBABA BBABA BBBBABA

The second version, however, yields different branchial graphs:

BAA BBAA BBBAA ABA

 BABA AA , , AA BABA , BBABA 

ABA BBABA BBBABA BBBBABA BAA

To some extent this difference is just like taking a different foliation. But there is more to it, 
because the second version of the multiway graph actually defines different ordering 

constraints than the first one. In the second version, there is a true partial ordering, defined 

by the directed edges in the multiway graph. But in the first version, there can be loops, and 

so no strict partial order is defined. (We will discuss this phenomenon more in 6.9.)

218



������������������������������ ��������������� ���� ���
������� �����

In the course of this section, we have seen various ways of describing and relating the 

possible behaviors of systems. In many ways the general is the combined multiway evolu-
tion and multiway causal graph.

For the rule

{A → AB}
starting from AA this graph has the form:

Continuing for another step, we have:

219



There are several different kinds of descriptions that we can derive from this graph. The 

standard multiway graph gives the evolution relationship between states:

AA

AAB ABA

AABB ABAB ABBA

AABBB ABABB ABBAB ABBBA

AABBBB ABABBB ABBABB ABBBAB ABBBBA

Each possible path through this graph corresponds to a possible evolution history for the 

system. 

The multiway causal graph gives the causal relationships between all events that can happen 

on any branch. The full multiway causal graph for the rule shown here is infinite. But 
truncating to show only the part contained in the graph above, one gets:

A A
ABAB

A
ABABB ABA ABB A AB A

AB AB A A
AB

A A A
ABBAB ABBABB ABBA ABB A

ABB AB A A A
ABB ABB

A B A A
AB B B ABBBA AB A

ABBB A A
ABBB

A
ABBBBA A A

ABBBB

A
ABABBB ABBB A

AB

Continuing for more steps one gets:

From the multiway causal graph, one can project out the specific causal graph for each 

possible evolution history, corresponding to each possible branch in the multiway system. 

220



 history,  multiway system.
But for rules like the one shown here that have the property of causal invariance, every one 

of these specific causal graphs (at least if extended far enough) must have exactly the same 

structure. For the particular rule shown here, this structure is extremely simple:

(In effect, the nodes here are “generic events” in the system, and could be labeled just by 

copies of the underlying local rule.)

The multiway graph and multiway causal graph effectively give two different “vertical 
views” of the original graph—using respectively states as nodes and and events as nodes. But 
an alternative is to view the graph in terms of “horizontal slices”. To get such slices we have 

to do foliations.

But now if we look at horizontal slices associated with states, we get the branchial graphs, 
which for this rule with this initial condition are rather trivial:

ABA AAB

AABB ABAB ABBA

AABBB ABABB ABBAB ABBBA

AABBBB ABABBB ABBABB ABBBAB ABBBBA

In principle we could also ask about horizontal slices associated with events. But by construc-
tion the events analog of the branchial graph must just consist of a collection of complete 

graphs.

However, a particular sequence of slices through any particular causal graph defines an 

actual sequence of states for the underlying system, and thus a possible evolution history, 
such as:
{AA, ABAB, ABBABB, ABBBABBB, ABBBBABBBB}
As a slightly cleaner example with similar behavior, consider the rule:

{A → AB, A → BA}

221



The combined multiway evolution and multiway causal graph in this case is

A

A A
AB BA

AB BA

A A
ABB BAB B A A

AB BBA

ABB BAB BBA

A
ABBB

A
BABB B A

ABB B A
BAB BB A

AB BB A
BA

ABBB BABB BBAB BBBA

A
ABBBB

A A
BABBB BABBB B A A

BABB BB A
ABB BBBAB BBB A

AB BBB A
BA

ABBBB BABBB BBABB BBBAB BBBBA

and the individual multiway evolution and causal graphs are both regular 2D grids:

 , 

The rules we have used as an example so far have behavior that is in a sense fairly trivial. 
But consider now the related rule with slightly less trivial behavior:
{A → AB, B → A}

222



For this rule, the combined multiway evolution and causal graph has the form:

On their own, the state evolution graph and the event causal graph have the forms:

 , 

The sequence of branchial graphs is:

 , , , , 

This rule is causal invariant, and so the multiway causal graph decomposes into many 

identical copies of causal graphs for all individual possible paths of evolution. In this case, 
these graphs all have the form:

But even though the multiway causal graph can be decomposed into identical pieces, it still 
contains more information than any of them. Because in effect it describes not only “spatial” 

223



 any  only
causal relationships between events happening in different places in the underlying string, 
but also “branchial” causal relationships between events happening on different branches of 
the multiway system.

And just like for other graphs, we can study the large-scale structure of multiway causal 
graphs. We can M

 define a quantity Ct  which is the multiway analog of the cone volume Ct  for 

individual causal graphs. For the rule shown here, the various graph growth rates (as 

computed with our standard foliation) have the forms:
600
500 Mt 200 Bb

400 150
300 100
200
100 50

0 0
0 2 4 6 8 10 12 0 1 2 3 4 5 6

500
Ct 800 C M

t
400

600
300

400
200

100 200

0 0
0 2 4 6 8 10 12 14 0 2 4 6 8

As another example, consider the “sorting” rule

{BA → AB}
starting from BABABABA. The combined multiway evolution and causal graph has the 

(terminating) form:

224



The multiway evolution and causal graphs on their own are:

 , 

The branchial graphs are

 , , , , , , , 

and the causal graph for a single (finite) path of evolution is:

Earlier in this section we looked at the multiway evolution graphs generated by all 12 

inequivalent 2: 1→2, 1→1 rules. The pictures below now compare these with the multiway 

causal graphs for the same rules (starting from all possible length-3 strings of As and Bs, and 

run for 4 steps of our standard multiway foliation): 

 , , ,

, , ,

, , ,

225



, , 

The multiway causal graph is in some ways a kind of dual to the multiway evolution graph—
resulting in many similarities among the graphs in the pictures above. Like Mt  for multiway 

evolution graphs, CM
t  for multiway causal graphs typically seems to grow either polynomi-

ally or exponentially.

But even in a case like the rule

{AA → AAA}
where the causal graph for a single evolution has the fairly regular form

the full multiway causal graph is quite complex. This shows how it builds up over the first 
few steps (in our standard multiway foliation):

226



AA AA
AAA

A AA
AAA

 AA
AAA , A AA AA AA

AAA AAA AAAA , A AA
AAAA

AA
AAA ,

AA
AAAA

AA
AAAAA

AA
AAA

AA
AAAAAA AA

AAAAA
AA
AAAAAAA

AAAA AA
AAA

A AA
AAA AA

A AA
A AA AAAAA AA AAAA

AAAAAAAA AAA
AAA AA

AAA
, AAA

AA 
AAAAA

AA
AAAA A AA

AAAA
AA AA

A AA
AAAA

AAA
AA AA AA AA

AAAA A AA AAAA
AAAAA

AAA AA A AA
AAAAAA AAA AA

AAA AA AA AAAA
AAA

AA AA
AAAAA

And here are 3D renderings a�er 8 and 9 steps:

 , 

����� �������� ���� ��� �����
In a multiway system, there are in general multiple paths that can produce the same state. 
But in our usual construction of the multiway graph, we record only what states are pro-
duced, not how many paths can do it. 

Consider the rule

{A → AA, A → A}

227



The full form of its multiway graph—including an edge for every possible event—is:

A

A AA

A AA AAA

A AA AAA AAAA

A AA AAA AAAA AAAAA

Here is the same graph, with a count included at each node for the number of distinct paths 

from the root that reach it: 
1
A

1 1
A AA

1 3 2
A AA AAA

1 7 12 6
A AA AAA AAAA

A AA AAA AAAA AAAAA
1 15 50 60 24

An alternative weighting scheme might be to start with weight 1 for the initial state, then at 
each state we reach, to distribute the weight to its successors, dividing it equally among 

possible events: 
1
A

1 1

2 2
A AA

1 1 1

6 2 3
A AA AAA

1 7 6 3

26 26 13 13
A AA AAA AAAA

A AA AAA AAAA AAAAA
1 1 1 2 4

150 10 3 5 25

This approach has the feature that it gives normalized weights (summing to 1) at each 

successive layer in a graph like this. But in general the approach is not robust, and if we 

228



 layer
even took a different foliation through the graph above, the weights on each slice would no 

longer be normalized. In addition, if we were to combine identical states from different 
steps, we would not know what weights to assign. Pure counting of paths, however, still 
works even in this case, although any normalization has to be done only a�er all the counts 

are known:
5 26 64 66 24

A AA AAA AAAA AAAAA

Note that even the counting of paths becomes difficult to define if there is a loop in the 

multiway graph—though one can adopt the convention that one counts paths only to the first 
encounter with any given state:

4
BBABA

3 1 1
BABA BBAA BBBABA

4 4 1 1
ABA BAA BBBAA BBBBABA

3
AA

Weights on the multiway graph can also be inherited by branchial graphs. Consider for 

example the rule 

{A → AB, B → BA}
The multiway graph for this rule, weighted with path counts, is:

1
AB

1 1
AAB BB

1 2 2
AAAB ABB BAB

1 3 5 3 4
AAAAB AABB ABAB BAAB BBB

1 4 9 4 9 12 10 12
AAAAAB AAABB AABAB BAAAB ABAAB ABBB BABB BBAB

229



The corresponding weighted branchial graphs are:

2 3 12
4

9

1
 , 1 5 4 0

, 1,
9

2 3 12
1 4

25
5

14 18
35 34

1 19 35 , , 
5 42

14
25

The weights in effect define a measure on the branchial graph. A case with a particular 

straightforward limiting measure is the rule:
{A → AB}
With initial condition A this gives weights that reproduce Pascal’s triangle, and yield a 

limiting Gaussian:
1
AA

1 1
AAB ABA

1 2 1
AABB ABAB ABBA

1 3 3 1
AABBB ABABB ABBAB ABBBA

AABBBB ABABBB ABBABB ABBBAB ABBBBA
1 4 6 4 1

230



With initial condition AAA, the weights in the branchial graph limit to a 2D Gaussian:

1 1
9 126 12 84 36 9

36 84 6
9 63 504 252 72 9

72 252 504 0

36 252 756 1260 1260 756 252 36

84 504 1260 1680 1260 504 84

126 630 1260 1260 630 126

126 504 756 504 126

84 252 252 84

36 72 36

9 9

1

In general, a�er sufficiently many steps one can expect that the weights will define an 

invariant measure, although a complexity is that the branchial graph will typically continue 

to grow. As one indication of the limiting measure, one can compute the distribution of 
values of the weights. 

The results for the rule {A→B,B→AB} above illustrate slow convergence to a limiting form:

t = 10 t = 11 t = 12

1 10 100 1000 104 105 1 10 100 1000 104 105 1 10 100 1000 104 105 106

We discussed in a previous subsection probing the structure of the branchial graph by 

computing the number of nodes Bb at most graph distance b  from a given point. We can 

now generalize this to computing a path-weighted quantity Bμ
b (cf. [1:p959]). At least for 

simple multiway graphs, this may be related in the limit to the results of solving a PDE on 

the multiway graph.

��������������������������������
In any causal invariant rule, all branch pairs that are generated must eventually resolve. But 
by looking at how many branch pairs are resolved—and unresolved—at each step, we can get 
a sense of “how far” a rule is from causal invariance. 

Consider the causal invariant rule:

{A → AB, B → A}

231



With this rule, all branch pairs resolve in one step

AA

AAB ABA

AABB AAA ABAB ABBA

AABBB AAAB ABABB AABA ABAA ABBAB ABBBA

and the total number of branch pairs that have already resolved on successive steps is 

(roughly 2t ):

{0, 6, 22, 66, 174, 420, 951, 2053, 4273, 8643}

But now consider the slightly different—and non-causal-invariant—rule:

{A→ AB, BA→ BB}

AA

AAB ABA

AABB ABAB ABB ABBA

AABBB ABABB ABBAB ABBB ABBBA

AABBBB ABABBB ABBABB ABBBB ABBBAB ABBBBA

The number of resolved branch pairs goes up on successive steps—in this case quadratically:

{1, 5, 11, 19, 29, 41, 55, 71, 89, 109}

But now there is a “residue” of new unresolved branch pairs at each step that reflect the lack 

of causal invariance

{4, 6, 8, 10, 12, 14, 16, 18, 20, 22}

There are other rules in which the deviation from causal invariance is in a sense larger. 
Consider the rule:

{AA→ AAB, AA→ B}

232



The multiway graph for this rule shows various “dead ends”

AA

BA AB AAB ABA

BB BAB AABB ABB ABAB BBA ABBA

BABB AABBB ABABB BBB ABBB BBAB ABBAB BBBA ABBBA

BABBB AABBBB ABABBB BBABB ABBABB BBBB ABBBB BBBAB ABBBAB ABBBBA BBBBA

and now the number of resolved branch pairs is

{5, 15, 30, 50, 75, 105, 140, 180, 225, 275}

while the number of unresolved ones grows at a similar rate:

{14, 23, 33, 44, 56, 69, 83, 98, 114, 131}

When a system is not causal invariant, what it means is that in a sense the system can reach 

states from which it cannot ever “get back” to other states. But this suggests that by extend-
ing the rules for the system, one might be able to make it causal invariant. 

Consider the rule

{A → AA, A → B}
with multiway graph:

A

B AA

AB BA AAA

BB AAB BAA AAAA ABA

BAB AAAB ABB BAAA AAAAA BBA AABA ABAA

With this rule, the branch pair {B, AA} never resolves. But now let us just extend the rule by 

adding B→AA:
{A → AA, A → B, B → AA}

233



The multiway graph becomes

A

B

AA

AB BA

BB AAA

AAB ABA BAA

ABB BAB BBA AAAA

AAAB AABA ABAA AAAAA BAAA

and the rule is now causal invariant. 

In general, given a rule that is not causal invariant, we can consider extending it by includ-
ing transformations that relate strings in branch pairs that do not resolve. For the rule
{AA → AAB, AA → B}
discussed above it turns out that the minimal additions to achieve causal invariance are:

{AB → AAAB, BA → AB}
Having added these transformations, the multiway graph now begins:

AAA

BA

AABA AB

AAAABA AABBA BBA AAAB

AAAAAB AAABB AABAB ABB BAB

234



A�er more steps, the multiway graph is:

This kind of “completion” procedure of adding “relations” in order to achieve what amounts 

to causal invariance is familiar in automated theorem proving [82] and related areas [60]. 
For our purposes we can think of it as a kind of “coarse graining” of our systems, in which 

the additional rules in effect define equivalences between states that would otherwise be 

distinct. 

If a particular multiway graph terminates a�er a finite number of steps, then it is always 

possible to add enough completion rules to the system to ensure causal invariance [63][64]. 
But if the multiway graph grows without bound, this may not be possible. Sometimes one 

may succeed in adding enough completions to achieve causal invariance for a certain 

number of steps, only to have it fail a�er more steps. And in general, like determining 

whether branch pairs will resolve, there is ultimately no upper bound on how long one may 

have to wait, making the problem of completion ultimately formally undecidable [83]. 

But if (as is o�en the case) one only has to add a small number of completion rules to make a 

system causal invariant, then one can take this to mean that the system is not far from 

causal invariance, so that it is likely that many of the large-scale properties of the comple-
tion of the system will be shared by the system itself.

����� ���������������������
Systems like cellular automata always update every element at every step in their evolution. But 
in string substitution systems (as well as in our hypergraph-based models), the presence of 
overlaps between possible updating events typically means that there is no single, consistent way 

to do this kind of parallel updating. Nevertheless, in studying our models in earlier sections, we 

o�en used “steps” of evolution in which we updated as many elements as we consistently could. 
And we can also apply this kind of “generation-based” updating to string substitution systems. 

235



Consider the rule:

{A → AB, B → A}
We can construct the multiway graph for this rule by considering how one state is produced 

from another by a single updating event, corresponding to a single application of one of the 

transformations in the rule:

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

AAAA AAABB AABAB ABAAB ABABA AABBA ABBAA ABABBB ABBABB AABBBB ABBBAB ABBBBA ABBBBBB

But we can also consider producing a “generational multiway graph” in which we do as 

many non-overlapping updates as possible on any given string. For this particular rule, 
doing this is straightforward, since every A and every B in the string can be transformed 

separately. 

But the result is now a radically simplified multiway graph, in which there is just a single 

path of evolution:

A AB ABA ABAAB ABAABABA ABAABABAABAAB

The “generational steps” here involve an increasing number of update events, as we can see 

from this rendering of the evolution:

A

A B

A B A

A B A A B

A B A A B A B A

A B A A B A B A A B A A B

A B A A B A B A A B A A B A B A A B A B A

236



All the states obtained at generational steps do appear somewhere in the full multiway 

graph, but the full graph also contains many additional states—that, among other things, can 

be thought of as representing all possible intermediate stages of generational steps: 

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

AAAA AAABB AABAB ABAAB ABABA AABBA ABBAA ABABBB ABBABB AABBBB ABBBAB ABBBBA ABBBBBB

For a rule like

{A → AB}
the generational steps

AA ABAB ABBABB ABBBABBB ABBBBABBBB

correspond to a particularly simple trajectory through the full multiway graph:

AA

AAB ABA

AABB ABAB ABBA

AABBB ABABB ABBAB ABBBA

AABBBB ABABBB ABBABB ABBBAB ABBBBA

AABBBBB ABABBBB ABBABBB ABBBABB ABBBBAB ABBBBBA

AABBBBBB ABABBBBB ABBABBBB ABBBABBB ABBBBABB ABBBBBAB ABBBBBBA

237



It is not always the case that the generational multiway graph involves just a single path. 
Consider the rule:
{A → AB, A → B}
The ordinary multiway graph for this rule starting from AA is:

AA

BA AB AAB ABA

BB BAB AABB ABB ABAB BBA ABBA

BABB AABBB ABABB BBB ABBB BBAB ABBAB BBBA ABBBA

BABBB AABBBB ABABBB BBABB ABBABB BBBB ABBBB BBBAB ABBBAB ABBBBA BBBBA

And the generational multiway graph is now:

AA

BB BAB ABB

BABB BBB ABBB ABAB

BABBB BBBB ABBBB ABBABB BBABB

ABBBBB ABBBABBB ABBBBBB BBBABBB BBBBB BBBBBB BBABBB

The generational multiway graph is always in a sense a compression of the full multiway 

graph. And one way to think of it is as being derived from the full multiway graph by 

combining sequences of edges when they correspond to updating events that do not overlap 

on the string.

But there is also another view of generational evolution. Consider a branchial graph from 

the full multiway graph above (this branchial graph is derived from the layered foliation 

shown):

238



AABB

ABAB

BAB
ABBA

ABB

BBA
BB

The branch pairs in this branchial graph (shown as adjacent nodes) can be thought of as 

being of two kinds. The first are produced by applying different rules to a single part of a 

string (e.g. ABA→{ABBA,BBA}). And the second (highlighted in the graph above) by applying 

rules to different parts of a string (e.g. ABA→{ABBA,ABAB}).

In the full multiway graph, no distinction is made between these two kinds of branch pairs, 
and the graph includes both of them. But in a generational multiway system, strings in the 

second kind of branch pairs can be combined.

And indeed this provides another way to construct a generational multiway system: look at 
branchial graphs and take pairs of strings corresponding to “spatially” disjoint updating 

events, and then knit these together to form generational steps. And if there is only one way 

to do this for each branchial graph, one will get a single path of generational evolution. But 
if there are multiple ways, then the generational multiway graph will be more complicated. 

For the rule

{A → AB, B → A}
that we discussed above, the sequence of branchial graphs is

ABA

 ABB AA , ,

AAB ABBB

AABBB AAAB
ABBA

ABABB

AAA ABAB ABBBB , ABBBBB AABA 

ABBAB

AABB
ABBBA ABAA

239



and it is readily possible to assemble “string fragments” (such as those highlighted) to 

produce the states at successive generational steps:
{{AA}, {ABAB}, {ABBABB}, {ABBBABBB}}
The branchial graphs are determined by the foliation of the multiway graph that one uses. 
But given a foliation, one can then assemble strings corresponding to generational steps 

using the procedure above. 

There is always an alternative, however: instead of combining strings from a branchial 
graph to produce the state for a generational step, one can always in a sense just wait, and 

eventually the full multiway system will have done the necessary sequence of updates to 

produce the complete state for the generational step.

Whenever there are no possible overlaps in the application of rules, the generational 
multiway graph must always yield a single path of history. But there is also another feature 

of such rules: they are guaranteed to be causal invariant. There are, however, plenty of rules 

that are causal invariant even though they allow overlaps—in a sense because their right-
hand sides also appropriately agree. And for such rules, the generational multiway graph 

may have multiple paths of history.

A simple example is the rule:

{A → AB, A → BA}
This rule is causal invariant, and starting from A yields the full multiway graph:

A

AB BA

ABB BAB BBA

ABBB BABB BBAB BBBA

ABBBB BABBB BBABB BBBAB BBBBA

240



Its generational multiway graph is actually identical in this case—because all the rule ever 

does is to apply one transformation or the other to the single A that appears:

A

AB BA

ABB BAB BBA

ABBB BABB BBAB BBBA

ABBBB BABBB BBABB BBBAB BBBBA

Starting from AA, however, the rule yields the full multiway graph

AA

AAB ABA BAA

AABB ABAB ABBA BAAB BABA BBAA

AABBB ABABB ABBAB BAABB ABBBA BABAB BABBA BBAAB BBABA BBBAA

and the generational multiway graph

AA

ABAB ABBA BAAB BABA

ABBABB ABBBAB ABBBBA BABABB BABBAB BABBBA BBAABB BBABAB BBABBA

In an alternative rendering, these graphs a�er a few more steps become, respectively:

241



 , 

Note that in both cases, the number of states reached a�er t  steps grows like t 2 (in the first 
case it is 1

 t (t + 1); in the second case exactly t 2).
2

In the case of a rule like

{A → AB, A → BB}
the presence of many potential overlaps in where updates can be applied makes many of the 

possible states in the full multiway graph also appear in the generational multiway graph (in 

the limit about 64% of all possible states are generational results):

Generational multiway graphs share many features with full multiway graphs. For example, 
generational multiway graphs can also show causal invariance—and indeed unless strings 

grow too fast, any deviation from causal invariance must also appear in the generational 
multiway graph.

The basic construction of ordinary multiway graphs ensures that the number of states Mt  
a�er t  steps can grow at most exponentially with t . In a generational multiway graph, there 

can be faster growth. 

242



Consider for example the rule (whose full multiway graph grows in a Fibonacci sequence 

≈ϕt ):
{A → AA, A → B}
Its full multiway graph grows in a Fibonacci sequence ≈ϕt :

A

B AA

AB BA AAA

BB AAB BAA ABA AAAA

BAB ABB BBA AAAB BAAA AABA ABAA AAAAA

BBB BAAB AABB BBAA ABAB BABA AAAAB BAAAA ABBA AAABA AABAA ABAAA AAAAAA

But its generational multiway graph grows much faster. A�er 3 generational steps it has the form

while a�er 4 steps (in a different rendering) it is:

243



The number of states reached in successive steps is:

{1, 2, 5, 24, 455, 128702}

Although there is a distribution of lengths for the strings, say, at steps 4 and 5

 , 

5 10 15 5 10 15 20 25 30

the fact that the maximum string length at generational step t  is 2t–1—combined with the 

lack of causal invariance for this rule—-allows for double exponential growth in the number 

of possible states with generational steps. In fact, with this particular rule, by step t  almost 
all sequences of up to 2t–1 Bs and AAs have appeared (the missing fractions on steps 3, 4, 5 

are 0.13, 0.078, 0.017) so at step t  the total number of states approaches 

22t–1

244



6 |The Updating Process in Our
Models

6.1 Updating Events and Causal Dependence
Consider the rule:

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}

When we discussed this rule previously, we showed the first few steps in its evolution as:

 , , , , 

But to understand the updating process in our models in more detail, it is helpful to “look
inside” these steps, and see the individual updating events of which they are comprised:

2 1
0

 0 , 1 0
 ,  , 2 3,

1
0

5

0 4 6 2
0 4

 3 , 3 2, 1 ,
1 1

2
3 0 5

4

7 7 7
6 6 2 5 10 3 6 7

0 2 3 2 5 7 0
9 8

 4 2 3, 3 1 2
1 0

5 0 , , 1
6

1 1 , 1
4 0 

5 1 8 3 4 5
6 4 9

8 4
9 8 10

The 22 → 42 signature of the rule means that in each updating event, two relations are
destroyed, and four new ones are created. In the pictures above, new relations from each
event are shown in red; the ones that will disappear in the next event are shown dotted. The
elements are numbers in the sequence they are created.

2435



There are in general many possible sequences of updating events that are consistent with 
the rule. But in making the pictures above (and in much of our discussion in previous 
sections), we have used our “standard updating order”, in which each step in the overall 
evolution in effect includes as many non‐overlapping updates as will “fit”. (In more detail, 
what is done is that in each overall step, relations are scanned from oldest to newest, in each 
case using them in an update event so long as this can be done without using any relation 
that has already been updated in this overall step.)

Our models—and the hypergraphs on which they operate—are in many ways more difficult 
to handle than the string‐based systems we discussed in the previous section. But one way in 
which they are simpler is that they more directly expose causal relationships between events. 
To see if an event B depends on an event A, all we need do is to see whether ele‐ments that 
were involved in A are also involved in B.

Looking at the sequence of updates above, therefore, we can immediately construct a causal graph:

0 1 0

3
1 0 1

0 2 1 1 0
0

2 0 0 4 2 0 5 0 1 6 2
2 1 1 3 2 3

2 1

Another feature of our models is that every element can be thought of as having a unique 
“lineage”, in that it was created by a particular updating event, which in turn was the result 
of some other updating event, and so on. When we introduced our models in section 2, we 
just said that any element created by applying a rule should be new and distinct from all 
others. If we were implementing the model, this might then make us imagine that the 
element would have a name based on some global counter, or a UUID.

But there is another, more deterministic (as well as more local and distributed) alternative: 
think of each new element as being a kind of encapsulation of its lineage (analogous to a 
chain of pointers, or to a hash like in blockchains [84] or Git). In the evolution above, for 
example, we could describe element 10 by just saying it was created as part of the relation 
{2,10} from the relations {{2,4},{2,5}} (as the second part of the output of an update that uses 
them)—but then we could say that these relations were in turn created from earlier rela‐
tions, and so on recursively, all the way back to the initial state of the system:

246



{{2, 10}}
{{{2, 4}, {2, 5}} ⊳ 2}
{{{{0, 1}, {0, 2}} ⊳ 4, {{0, 2}, {0, 1}} ⊳ 3} ⊳ 2}
{{{{{0, 0}, {0, 1}} ⊳ 1, {{0, 0}, {0, 1}} ⊳ 2} ⊳ 4, {{{0, 0}, {0, 1}} ⊳ 3, {{0, 1}, {0, 1}} ⊳ 1} ⊳ 3} ⊳ 2}
{{{{{{0, 0}, {0, 0}} ⊳ 1, {{0, 0}, {0, 0}} ⊳ 2} ⊳ 1, {{{0, 0}, {0, 0}} ⊳ 1, {{0, 0}, {0, 0}} ⊳ 2} ⊳ 2} ⊳ 4,

{{{{0, 0}, {0, 0}} ⊳ 1, {{0, 0}, {0, 0}} ⊳ 2} ⊳ 3, {{{0, 0}, {0, 0}} ⊳ 3, {{0, 0}, {0, 0}} ⊳ 4} ⊳ 1} ⊳ 3} ⊳ 2}

The final expression here can also be written as:

•• •• 1 •• •• 2 •• •• 1 •• •• 2 •• •• 1 •• •• 2 •• •• 3 •• •• 4
1 2 4 3 1 3 2

Roughly what this is doing is specifying an element (in this case the one we originally
labeled simply as 10) by giving a symbolic representation of the path in the causal graph that
led to its creation. And we can then use this to create a unique symbolic name for the
element. But while this may be structurally interesting, when it comes to actually using an
element as a node in a hypergraph, the name we choose to use for the element is irrelevant;
all that matters is what elements are the same, and what are different.

����� ���� ��������������� ��� �����
Just like for the string substitution systems of section 5, we can construct multiway systems
[1:5.6] for our models, in which we include a separate path for every possible updating
event that can occur:

247



For string systems, it is straightforward to determine when states in the system should be
merged: one just has see whether the strings corresponding to them are identical. For our
systems, it is more complicated: we have to determine whether the hypergraphs associated
with states are isomorphic [85], in the sense that they are structurally the same, independent
of how their nodes might be labeled.

Continuing one more step with our rule, we see some cases of merging:

Here is an alternative rendering, now also showing the particular path obtained by following
our “standard updating order”:

248



In general, each path in the multiway system corresponds to a possible sequence of updat‐
ing events—here shown along with the causal relationships that exist between them:

6.3 Causal Invariance
Like string substitution systems, our models can have the important feature of causal
invariance [1:9.9]. In analogy with neighbor‐independent string substitution systems, causal
invariance is guaranteed if there is just a single relation on the le�‐hand side of a rule.

Consider for example the rule:

{{x, y}} → {{x, y}, {y, z}}

249



Starting from a single self‐loop, this gives the multiway system:

As implied by causal invariance, every pair of paths that diverge must reconverge. And
looking at a few more steps, we can see that in fact with this particular rule, branches always
recombine a�er just one step:

The different paths here lead to hypergraphs that look fairly different. But causal invariance 
implies that every time there is divergence, there must always eventually be reconvergence.

And for some rules, different paths give hypergraphs that do look very similar. An example is 
the rule

{{x, y}} → {{y, z}, {z, x}}

250



where the hypergraphs produced on different paths differ only by the directions of their hyperedges:

For rules that depend on more than one relation, causal invariance is not guaranteed, and in 
fact is fairly rare. Of the 4702 inequivalent 22 → 32 rules, perhaps 5% are causal invariant.

In some cases, the causal invariance is rather trivial. For example, the rule
{{x, y}, {x, y}} → {{z, z}, {z, z}, {y, z}}
leads to a multiway graph that only allows one path of evolution:

A less trivial example is the rule

{{x, y}, {z, y}} → {{x, w}, {y, w}, {z, w}}

251



which yields the multiway system:

With our standard updating order, this rule eventually produces forms like

but the multiway system shows that other structures are also possible.

As another example, consider the rule

{{x, y}, {z, y}} → {{x, z}, {y, z}, {w, z}}
which with our standard updating order gives:

 , , , , , ,

, , , , 

252



The multiway system for this rule branches rapidly

but every pair of branches still reconverges in one step.   

The rule

{{x, y}, {x, z}} → {{y, w}, {y, z}, {w, x}}
provides an example of causal invariance in which branches can take 3 steps to converge:

Among all possible rules, causal invariance is much more common for rules that generate 

disconnected hypergraphs.  It is also perhaps slightly less common for rules with ternary 

relations instead of binary ones.  

253



6.4 Testing for Causal Invariance
Testing for causal invariance in our models is similar in principle to the case of strings. 
Failure of causal invariance is again the result of branch pairs that do not resolve. And just 
like for strings, it is possible to test for total causal invariance by determining whether a 
certain finite set of core branch pairs resolve. (Again in analogy with strings, the finiteness 
of this set is a consequence of the finiteness of the hypergraphs involved in our rules.)

The core branch pairs that we need to test represent the minimal cases of overlap between 
le�‐hand sides of rules—or, in a sense, the minimal unifications of the hypergraphs that 
appear. For the two hypergraphs

w

b a x
 , y

z

there are two possible unifications (where the purple edge shows the overlap):

w z

 y,b x,a , b x,y,a 

z w

For a single rule with le�‐hand side

{{x, y}, {x, z}}
the core branch pairs arise from unifications associated with the possible self‐overlaps of
this small hypergraph. Representing two copies of the hypergraph as

 z x y, c a b

the possible unifications are:

z b,z

 , a,x , ,
a,x c,z b,y a,x

b,y c y c

254



z c,z

a,x , , 
c,y b,z a,x a,x

c,y b y b

In the case of strings, all that matters is what symbols appear within the unification. In the 
case of hypergraphs, one also has to know how the unification in effect “attaches”, and so 
one has to distinguish different labelings of the nodes.

Starting from the unifications, one applies the rule to find what branch pairs can be pro‐
duced. These branch pairs form the core set of branch pairs for the rule—and determining 
whether the rule is causal invariant then becomes a matter of finding out whether these 
branch pairs resolve.

In the case of

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
the application of the rule to the unifications above yields the following 58 core branch pairs:

{ , }, { , },  , , { , },  , , { , }, { , }, { , },

 , ,  , ,  , ,  , ,  , ,  , ,  , ,

 , , { , },  , ,  , , { , }, { , }, { , },

 , ,  , ,  , ,  , ,  , ,  , ,  , ,

 , , { , },  , , { , },  , , { , }, { , },  , ,

{ , },  , , { , },  , , { , }, { , },  , ,

{ , },  , , { , },  , , { , }, { , },  , ,

 , ,  , ,  , ,  , ,  , ,  , ,  , 

Running the rule for one step yields resolutions for 6 of these branch pairs:

{ , }→ , { , }→ ,  ,  → ,

 ,  → ,  ,  → ,  ,  →

Running for another step resolves no additional branch pairs.

Will the rule turn out to be causal invariant in the end? As a comparison, consider the rule

{{x, y}, {x, z}} → {{y, w}, {y, z}, {w, x}}

255



discussed in the previous subsection.  This rule starts with 14 core branch pairs:

{ , },  , ,  , ,  , ,

 , ,  , , { , }, { , },  , ,

 , ,  , ,  , , { , },  , 

A�er one step, 6 of them resolve:

 ,  → ,  ,  → ,  ,  → ,

 ,  → ,  ,  → ,  ,  → 

Then a�er another step the 8 remaining ones resolve, establishing that the rule is indeed 

causal invariant:

{ , }→ ,  ,  → ,

 ,  → ,  ,  → , { , }→ ,

{ , }→ ,  ,  → , { , }→ 

But in general there is no upper bound on the number of steps it can take for core branch 

pairs to resolve.  Perhaps the fact that so many additional branch pairs are generated at each 

step in the rule {{x,y},{x,z}}→{{x,z},{x,w},{y,w},{z,w}} makes it seem unlikely that they will all 
resolve, but ultimately this is not clear. 

And even if the rule does not show total causal invariance, it is still perfectly possible that it 
will be causal invariant for the particular set of states generated from a certain initial 
condition.  However, determining this kind of partial causal invariance seems even more 

difficult  than determining total causal invariance. 

Note that if one looks at all 4702 22 → 32 rules, the largest number of core branch pairs for 

any rule is 554; the  largest number that resolve in 1 step is 132, and the largest number that 
remain unresolved is 430.  

6.5  Causal Graphs for Causal Invariant Rules
An important consequence of causal invariance is that it establishes that a rule produces the 

same causal graph independent of the particular order in which update events occurred. 
And so this means, for example, that we can generate causal graphs just by looking at 
evolution with our standard updating order. 

256



For rules that depend on only one relation, the causal graph is always just a tree

regardless of whether the structure generated is also a tree

or has a more compact form:

But as soon as a rule depends on more than one relation, the causal graph can immediately 

be more complicated.  For example, consider even the rule:

{{x}, {x}} → {{x}, {x}, {x}}
The multiway system for this rule shows that only one path is possible (immediately demon‐
strating causal invariance):

257



But the causal relationships between steps are not so straightforward

and a�er 15 steps the causal graph has the form

258



or in an alternative rendering:

The fact that the multiway system is nontrivial does not mean that the causal graph for a 

particular rule evolution will be nontrivial.  Consider for example a causal invariant rule that 
we discussed above: 

{{x, y}, {z, y}} → {{x, w}, {y, w}, {z, w}}
The multiway system for this rule, with causal connections shown, is:

259



This yields the multiway causal graph:

But the causal graph for any individual evolution is just:

For the causal invariant rule (also discussed above)

{{x, y}, {z, y}} → {{x, z}, {y, z}, {w, z}}
the multiway system a�er 5 steps has the form:

260



A�er 20 steps of evolution with our standard updating order gives:

The causal graph for this rule a�er 10 steps is

261



and a�er 20 steps, in a different rendering, it becomes:

As another example, consider the rule (also discussed above):

{{x, y}, {x, z}} → {{y, w}, {y, z}, {w, x}}
The multiway system for this rule (with events included) has the form:

262



A�er 20 steps, the causal graph is:

A�er 100 steps it is:

263



A�er 500 steps, in an alternative rendering, a grid‐like structure emerges (the directed edges 

point outward from the center):

A�er 5000 steps, rendering the graph in 3D with surface reconstruction reveals an elaborate 

effective geometry:

264



6.6 The Role of Causal Graphs
Even if we do not know that a rule is causal invariant, we can still construct a causal graph 
for it based on a particular updating order—and o�en different updating orders will give at 
least similar causal graphs.

Thus, for example, for the rule
{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
applying our standard updating order for 5 steps gives the causal graph

Continuing for 10 steps, we get:

265



This can also be rendered as:

A�er 15 steps, there are 10,346 nodes: 

266



In effect the successive steps in the evolution of the system correspond to successive slices
through this causal graph. In the case of causal invariant rules, any possible updating order
must correspond to a possible causal foliation of the graph. But here we can at least say that
the foliation obtained by looking at successive layers starting from the root corresponds to
successive steps of evolution with our standard updating order.

For any system whose evolution continues infinitely, the causal graph will ultimately be
infinite. But by slicing the graph as we have above, we are effectively showing the events
that contribute to forming the state of the system a�er 15 steps of evolution (in this case,
with our standard updating order):

(Note that with this particular rule, the vast majority of relations that appear at step 15 were
added specifically at that step, so in a sense most of the state at step 15 is associated just with
the slice of the causal graph at layer 15.)

6.7 Typical Causal Graphs
As we discussed in 5.12, causal graphs for string substitution systems tend to have fairly
simple structures. Causal graphs for our models tend to be considerably more complicated,
and even among 22 → 32 rules considerable diversity is observed. A typical random sample
of different forms is:

267



Even rules whose states seem quite simple can produce quite complex causal graphs:

 , , , {,

 , , , {,

 , , , {,

 , , , {

268



As a first example, consider the rule: 

{{x, x}, {x, y}} → {{y, y}, {z, y}, {x, z}}
At each step, the self‐loop just adds a relation, and effectively moves around the growing loop: 

 , , , , , , , ,

, , , , , , , 

The causal graph captures the causal connections created by the self‐loop encountering the 

same relations again a�er it goes all the way around:

The structure gets progressively more complicated:

269



Re‐rendering this gives

or a�er 500 steps:

As another example, consider the rule: 

{{x, y}, {x, z}} → {{y, w}, {y, x}, {w, x}}
Here are the first 25 steps in its evolution (using our standard updating order):

 , , , , , , , ,

, , , , , , , , ,

, , , , , , , , 

270



A�er a few steps all that happens is that there is a small structure that successively moves 

around the loop creating new “hairs”.  The causal graph (here shown a�er 25 steps) captures 

this process:

An alternative rendering shows that a grid structure emerges:

Here are the corresponding results a�er 100 steps:

 , 

271



As a somewhat different example, consider the rule:

{{x, y}, {y, z}} → {{x, w}, {x, y}, {w, z}}

 , , , , , , , ,

, , , , , , , 

A�er the same number of steps, one can effectively see the separate trees in the causal graph:

Re‐rendering the causal graph, it has a structure that is quite similar to the actual state of 
the system:

272



Continuing for a few more steps, a definite tree structure emerges:

It is not uncommon for a causal graph to “look like” the actual hypergraph generated by one 

of our models.  For example, rules that produce globular structures tend to produce similar 

“globular” causal graphs (here shown for three 22 → 42 rules from section 3):

 , , {,

 , , {,

 , , {

273



Rules that exhibit slow growth o�en yield either grid‐like or “hyperbolic” causal graphs 

(here shown for some 23 → 33 rules from section 3):

 , , {,

 , , {,

 , , {,

 , , {,

 , , {,

274



 , , {,

 , , {,

 , , {,

 , , {,

 , , {,

 , , {

275



A typical source of grid‐like causal graphs [1:p489] is rules where in a sense only one thing 

ever happens at a time, or, in effect, the rules operate like a mobile automaton [1:3.3] or a 

Turing machine, with a single active element.  As an example, consider the rule (see 3.10):

{{x, y, y}, {z, x, u}} → {{y, v, y}, {y, z, v}, {u, v, v}}

 , , , , , , ,

, , , , , , , ,

, , , , , , , ,

, , , , , , , 

Updates can only occur at the position of the self‐loop, which progressively “moves around”, 
“knitting” a grid pattern.  The causal graph captures the fact that “only one thing happens at 
a time”: 

1
2

3
4

5
6

7
8

9
10

11
12

13
14

15
16

17
18

19
20
21

22
23

24
25

26
27

28
29
30

276



But what is notable is that if we ask about the overall causal relationships between events, 
we realize that even events that happened many steps apart in the evolution as shown here 

are actually directly causally connected, because in a sense “nothing else happened in 

between”.  Re‐rendering the causal graph illustrates this, and shows how a grid is built up:
1

2

4 5
3

6 13
7 14

12
8 26 27

15
11 28

9 10 16 25
29

17
24

19 18 30
23

22

20 21

Sometimes the actual growth process can be more complicated, as in the case of the rule 

{{x, y, y}, {z, y, u}} → {{v, z, v}, {v, u, u}, {u, v, x}}

 , , , , , , ,

, , , , , , , ,

, , , , , , , ,

, , , , , , , 

A�er 200 steps this yields:

277



And a�er 1000 steps it gives:

But despite this elaborate structure, the causal graph is very simple:

A�er 200 steps, the grid structure is clear:

Sometimes the causal graph can locally be like a grid, while having a more complicated 

overall topological structure.  Consider for example the rule:

{{x, y, y}, {z, x, u}} → {{y, z, y}, {z, u, u}, {y, u, v}}

278



A�er 200 steps this gives:

The corresponding causal graph is:

A�er 1000 steps with surface reconstruction this gives:

279



Rules (such as those with signature 22 → 22) that cannot exhibit growth inevitably terminate 
or repeat, thus leading to causal graphs that are either finite or repetitive—but may still have 
fairly complex structure. Consider for example the rule (compare 3.15):

{{x, y}, {y, z}} → {{z, x}, {z, y}}
Evolution from a chain of 9 relations leads to a 31‐step transient, then a 9‐step cycle:

 , , , , , , ,

, , , , , , , ,

, , , , , , , ,

, , , , , , , 

The first 30 layers in the causal graph are:

In an alternative rendering, the graph is:

280



A�er 50 more steps, the repetitive structure becomes clear:

Sometimes the structure of the causal graph may be very much a reflection of the updating
order used. Consider for example the rather trivial “identity” rule:

{{x, y}, {y, z}} → {{x, y}, {y, z}}
Starting with a chain of 3 relations, this shows update events according to our standard
updating order (note that the same relation can be both created and destroyed at a particular
step):

{ , , ,
, , ,
, , }

The corresponding causal graph is:

For a chain of length of 21 the causal graph consists largely of independent regions—except 
for the connection created by updates fitting differently at different steps:

281



Re‐rendering this gives a seemingly elaborate structure:

A�er 100 steps, though, its repetitive character becomes clear:

Note that if the initial condition is a ring rather than a chain, one gets

together with the tube‐like structure:

282



6.8 Large-Scale Structure of Causal Graphs
In section 5 we used the cone volume Ct to probe the large‐scale structure of causal graphs
generated by string substitution systems. Now we use Ct to probe the large‐scale structure of
causal graphs generated by our models.

Consider for example the rule

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
We found in section 4 that a�er a few steps, the volumes Vr of balls in the hypergraphs
generated by this rule grow roughly like r2.6, suggesting that in the limit the hypergraphs
behave like a finite‐dimensional space, with dimension ≈2.6.

The pictures below show the log differences in Vr and Ct for this rule a�er 15 steps of evolution:

3.0 8

2.5
6

2.0

1.5 , 4 
1.0

2
0.5

0.0 0
0 10 20 30 40 50 0 2 4 6 8 10 12 14

The linear increase in this plot implies exponential growth in Ct and indeed we find that for 
this rule:

Ct~ 2.2t

This exponential growth—compared with the polynomial growth of Vr—implies that expan‐
sion according to this rule is in a sense sufficiently rapid that there is increasing causal 
disconnection between different parts of the system.

The other three 22 → 42 globular‐hypergraph‐generating rules shown in the previous 
subsection show similar exponential growth in Ct, at least over the number of steps of 
evolution tested.

A rule such as

{{x, y, y}, {x, z, u}} → {{u, v, v}, {v, z, y}, {x, y, v}}

283



whose hypergraph and causal graph (a�er 500 steps) are respectively

 , 

gives the following for the log differences of Vr and Ct a�er 10,000 steps: 

2.5
2.5

2.0 2.0

1.5 1.5
 , 
1.0 1.0

0.5 0.5

0.0
0 10 20 30 40 50 60 70 0.0

0 20 40 60 80 100 120

This implies that for this rule the hypergraphs it generates and its causal graph both effec‐
tively limit to finite‐dimensional spaces, with the hypergraphs having dimension perhaps 

slightly over 2, and the causal graph having dimension 2.

Consider now the rule:

{{x, y, x}, {x, z, u}} → {{u, v, u}, {v, u, z}, {x, y, v}}
The hypergraph and causal graph (a�er 1500 steps) for this rule are respectively:

 , 

284



The log differences of Vr and Ct a�er 10,000 steps are then:

3.5 3.5

3.0 3.0

2.5 2.5

2.0 2.0
 , 
1.5 1.5

1.0 1.0

0.5 0.5

0.0 0.0
0 10 20 30 40 50 0 10 20 30 40 50 60

Both suggest limiting spaces with dimension 2, but with a certain amount of (negative) 
curvature.

6.9 Foliations of Causal Graphs
At least in a causal invariant system, the structure of the causal graph is always the same, 
and it defines the causal relationships that exist between updating events. But in relating 
the causal graph to actual underlying evolution histories for the system, we need to specify 
how we want to foliate the causal graph—or, in effect, how we want to define “steps” in the 
evolution of the system.

As an example, consider the rule:

{{x, y}, {z, y}} → {{x, z}, {y, z}, {w, z}}
(This rule is probably not causal invariant, but this fact will not affect our discussion here.) The 
most obvious foliation for the causal graph basically follows our standard updating order:

285



But this is not the only foliation we can use.  In fact, we can divide the graph into slices in 

any way, so long as the slices respect the causal relationships defined by the graph, in the 

sense that within a slice the causal relationships allow the events to occur in any order, and 

between successive slices events must occur in the order of the slices.  And with these 

criteria, for example, another possible foliation is:

With the first foliation shown above, the hypergraphs from what we consider to be the first 
“few steps” in the evolution of the underlying rule are: 

 , , , ,

, , , 

But the second foliation in effect has a different (and coarser) definition of “steps”, and with 

this foliation the first few steps would be: 

 , , , , , 

When we discussed foliations in the context of string substitution systems, there were a 

number of simplifying features in our discussion.  First, the underlying system fundamen‐
tally involved a linear string of elements.  And second, the main causal graph we actually 

considered was a simple grid.  

286



With a rule like

{{x, y, y}, {y, z}} → {{x, y}, {y, z, z}}

we can also get a simple grid causal graph (and this rule happens to be causal invariant).  
With the obvious foliation

the steps in the evolution of the underlying system from a particular initial condition are: 

But given the grid structure of the causal graph, we can use the same diagonal slice method 

for generating foliations that we did in 5.14.  And for example with the foliation

287



there are more steps involved in the evolution of the system:

But when the causal graph does not have such a simple structure, the definition of foliations can 

be much more complicated.  When the causal graph at least in some statistical sense limits to a 

sufficiently uniform structure, it should be possible to set up foliations that are analogous to the 

diagonal slices.  And even in other cases, it will o�en be possible to set up foliations that can be 

described, for example, by the kind of lapse functions we discussed in 5.14.

But there is one issue that can make it impossible to set up any reasonable “progressive” 
foliation of a causal graph at all, and that is the issue of loops.  This issue is actually already 

present even in the case of string substitution systems (and even causal invariant ones).  
Consider for example the rule:

{AA → A, A → AA}
Starting from AA the multiway causal graph for this rule is:

A
AA

A A
AA

AA
A A

A
AAA

A
AAAAA A

AAA AA A AAAAA A

A A A A
AAAA

AA
AAAA A A A AA

A AA A AAA A
A AAAAA A AAA A

A AA A

288



But note here the presence of several loops. And looking at the states graph in this case

A AA AAA AAAA AAAAA

one can see where these loops come from: they are reflections of the fact that in the evolu‐
tion of the system, there are states that can repeat—and where in a sense a state can return 
to its past.

Whenever this happens, there is no way to make a progressive foliation in which events in 
future slices systematically depend only on events in earlier slices. (In the continuum limit, 
the analog is failure of strong hyperbolicity [86]; loops are the analog of closed timelike 
curves (e.g. [75])) (Self‐loops also cause trouble for progressive foliations by forcing events 
to happen repeatedly within a slice, rather than only affecting later slices.)

The phenomenon of loops is quite common in string substitution systems, and already 
happens with the trivial rule A→A. It also happens for example with a rule like:

{AB → BAB, BA → A}
Starting with ABA, this gives the causal graph

and has a states graph:

BBBABA ABA

BBBAA BBBBABA BBABA

BBAA BABA

BAA

AA

289



Loops can also happen in our models. Consider for example the very simple rule:

{{{x}, {x}} → {{x}}, {{x}} → {{x}, {x}}}

The multiway graph for this rule is:

This contains loops, as does the corresponding causal graph:

(Note that the issue discussed in 6.1 of when we consider states “identical” as opposed to
“equivalent” can again arise here. There are similar issues when we consider finite‐size 
systems where the whole state inevitably repeats—and where in principle we can define a 
cyclic analog of our foliations.

6.10 Causal Disconnection
In 2.9 we discussed the fact that some rules—even though the rules themselves are connect‐
ed—can lead to hypergraphs that are disconnected. And—unless one is dealing with rules 
with disconnected le�‐hand sides—any hypergraphs that are disconnected must also be 
causally disconnected.

As a simple example of what can happen, consider the rule:

{{x, y}} → {{y, z}, {y, z}}

290



The evolution of this rule quickly leads to disconnected hypergraphs:

 , , , , 

The corresponding causal graph is a tree:

In this particular case, the different branches happen to correspond to isomorphic hyper‐
graphs, so that in our usual way of creating a multiway graph, this rule leads to a connected 

multiway graph, which even shows causal invariance:

(Note that causal invariance is in a sense easier to achieve with disconnected hypergraphs, 
because there is no possibility of overlap, or of ambiguity in updates.)

In the case of an extremely simple rule like

{{x}} → {{y}, {z}}

291



the evolution is immediately disconnected

 , , , , 

the causal graph is a tree

but the multiway graph consists of a simple “counting” sequence of states:

In other rules, the disconnected pieces are not isomorphic, and the multiway graph can 

split.  An example where this occurs is the rule:

{{x, y}, {x, z}} → {{x, x}, {y, u}, {u, v}}

 , , , , , , 

The multiway graph in this case is a tree:

292



The multiway causal graph, however, does not have an exponential tree structure, but 
instead effectively just has one branch for each disconnected component in the hypergraph:

As a result, for this rule the ordinary causal graph has a simple, sequential form:

As a related example, consider the rule: 

{{x, y}, {y, z}} → {{u, v}, {v, x}, {x, y}}

 , , , , , , 

In this case, the multiway graph has the two‐branch form

293



and the multiway causal graph has the similarly two‐branch form

though the ordinary causal graph is still just:

Sometimes there can be multiple branches in both the multiway graph and the ordinary 

causal graph.  An example occurs in the rule

{{{x, y}, {x, z}} → {{y, y}, {z, u}, {z, v}}}

 , , , , , , 

where the multiway graph is

294



the multiway causal graph is

and the ordinary causal graph is:

Is it possible to have both an infinitely branching multiway graph, and an infinitely branch‐
ing ordinary causal graph?  One of the issues is that in general it can be undecidable whether 

this is ultimately infinite branching.  Consider for example the rule:

{{{x, x}, {x, y}} → {{y, y}, {y, y}, {x, z}}}

 , , , , , , 

295



The ordinary causal graph for this rule has the form

or in a different rendering a�er more steps:

The multiway causal graph in this case is:

296



But now the multiway graph is:

Continuing for more steps yields:

But while it is fairly clear that this multiway graph does not show causal invariance, it is not 
clear whether it will branch forever or not. 

As a similar example, consider the rule:

{{x, y}, {y, z}} → {{x, x}, {x, y}, {w, y}}

 , , , , , , 

This yields the same ordinary causal graph as the previous rule

297



but now its multiway graph has a form that appears slightly more likely to branch forever:

All the examples we have seen so far involve explicit disconnection of hypergraphs.  How‐
ever, it is also possible to have causal disconnection even without explicit disconnection of 
hypergraphs.  As a very simple example, consider the rule:

{{x, x}} → {{x, x}, {x, x}}

 , , , , , 

The causal graph in this case is

although the multiway graph is just:

For the rule

{{x, y}} → {{x, x}, {x, z}}

 , , , , , 

the causal graph is again a tree

298



but now the multiway graph is:

The rule

{{x, y}} → {{x, y}, {y, z}}
gives exactly the same causal graph, but now its hypergraph is a tree:

 , , , , , 

Like some of the rules shown above, its multiway graph is somewhat complex:

It is actually fairly common to have causal graphs that look like the corresponding hyper‐
graphs in the case of rules where effectively only one update happens at a time.  An example 

occurs in the case of the rule (see 3.10):

299



 , 

So far, we have only considered fairly minimal conditions.  But as soon as it is possible to 

have multiple independent events occur in the initial conditions, it is also possible to get 
completely disconnected causal graphs.  (Note that if an “initial creation event” to create the 

initial conditions was added, then the causal graphs would again be connected.)  As an 

example of disconnected causal graphs, consider the rule

{{x}} → {{x}, {x}}
with an initial condition consisting of connected unary relations:

 , , , , 

This rule yields a disconnected causal graph:

The multiway graph in this case is connected, and shows causal invariance:

300



Sometimes the relationship between disconnection in the hypergraph and the existence of
disconnected causal graphs can be somewhat complex. This shows results for the rule
above with initial conditions consisting of increasing numbers of self‐loops:

Even with rather simple rules, the forms of branching in causal graphs can be quite com‐
plex—even when the actual hypergraphs remain simple. Here are a few examples:

301



6.11  Global Symmetries and Conservation Laws
Given the rule (stated here using numbers rather than our usual letters)

{{1, 2, 3}, {3, 4, 5}} → {{6, 7, 1}, {6, 3, 8}, {5, 7, 8}}
if we reverse the elements in each relation we get:

{{3, 2, 1}, {5, 4, 3}} → {{1, 7, 6}, {8, 3, 6}, {8, 7, 5}}
But the canonical version of this rule is:

{{1, 2, 3}, {3, 4, 5}} → {{6, 7, 1}, {6, 3, 8}, {5, 7, 8}}
In graphical form, the rule and its transform are: 

5 5 2
1 3 1 3

4 6
7 8

 , 7 8 
4

1 3 1 6 3

2 5 5

Most rules would not be le� invariant under a reversal of each relation.  For example, the rule 

{{1, 2, 3}, {2, 4, 5}} → {{5, 6, 4}, {6, 5, 3}, {7, 8, 5}}
yields a�er reversal of each relation

{{3, 2, 1}, {5, 4, 2}} → {{4, 6, 5}, {3, 5, 6}, {5, 8, 7}}
but the canonical form of this is

{{1, 2, 3}, {4, 3, 5}} → {{4, 1, 6}, {2, 6, 1}, {1, 7, 8}}
which is not the same as the original rule.

If a rule is invariant under a symmetry operation such as reversing each relation, it implies 

that the rule commutes with the symmetry operation.  So given a rule R and a symmetry 

operation Θ, this means that for any state S, R (Θ S) must be the same as Θ (R S). 

With the symmetric rule above, evolving from a particular initial state gives:

 , , , 

302



But now reversing the relations in the initial state gives essentially the same evolution, but 
with states whose relations have been reversed:

 , , , 

For the nonsymmetric rule above, evolution from a particular initial state gives:

 , , , 

But if one now reverses the relations in the initial state, the evolution is completely different:

 , , , 

For rules with binary relations, the only symmetry operation that can operate on relations is 

reversal, corresponding to the permutation {2,1}.  Of the 73 distinct 12 → 22 rules, 11 have 

this symmetry.  Of the 4702 22 → 32 rules, 92 have the symmetry.  Of the 40,405 22 → 42 rules, 
363 have the symmetry.  Those with the most complex behavior are:

{{1, 2}, {2, 3}} → {{1, 2}, {1, 4}, {2, 3}, {4, 3}}

 ,  , , , , , , , 

{{1, 2}, {2, 3}} → {{1, 4}, {1, 3}, {4, 5}, {5, 3}}

 ,  , , , , , , , 

For rules with ternary relations, there are six distinct symmetry classes corresponding to the 

six subgroups of the symmetric group S3: no invariance, invariance under transposition of 
two elements (3 cases of S2) ({1,3,2}, {3,2,1} or {2,1,3} only), invariance under cyclic rotation 

(A3) ({2,3,1} and {3,1,2}), or invariance under any permutation (full S3).  Here are the num‐
bers of rules of various signatures with these different symmetries:

303



13 → 13 13 → 23 13 → 33 23 → 13 23 → 23 23 → 33 33 → 13
none 114 8520 627 072 7662 759444 79170508 559602

S2 (each of 3) 20 282 3475 248 4413 63028 2933
A3 2 4 46 4 8 131 40
S3 2 3 25 3 5 41 21

Examples of rules with full S3 symmetry include (compare 7.2):

{{1, 2, 3}, {2, 3, 1}} → {{2, 2, 4}, {4, 3, 3}, {1, 4, 1}}

 ,  , , , , , , , 

{{1, 2, 3}, {2, 3, 1}} → {{4, 4, 2}, {4, 1, 4}, {3, 4, 4}}

 ,  , , , , , , , 

An example of a rule with only cyclic (A3) symmetry is:

{{1, 2, 3}, {2, 3, 1}} → {{1, 1, 4}, {4, 2, 2}, {3, 4, 3}}

 ,  , , , , , , , 

The existence of symmetry in a rule has implications for its multiway graph, effectively 
breaking its state transition graph into pieces corresponding to different cosets (compare 
[1:p963]). For example, starting from all 102 distinct 2‐element ternary hypergraphs, the 
first completely symmetric rule above gives multiway system:

304



A somewhat simpler example of a completely symmetric rule is:

{{1, 2, 3}, {2, 3, 1}} → {{1, 2, 3}, {2, 3, 1}, {3, 1, 2}}

This rule has a simple conservation law: it generates new relations but not new elements.  
And as a result its multiway graph breaks into multiple separate components. 

In general one can imagine many different kinds of conservation laws, some associated with 

identifiable symmetries, and some not.  To get a sense of what can happen, let us consider 

the simpler case of string substitution systems.

305



The rule (which has reversal symmetry)

{BA → AB, AB → BA}
gives a multiway graph which consists of separate components distinguished by their total 
numbers of As and Bs:

AABBA ABBAA
ABABA

AAABB AABAB BABAA BBAAA

ABAAB BAABA

BAAAB

ABBBA

BABBA ABBAB

BBBAA BBABA ABABB AABBB
BABAB

BBAAB BAABB

AAAAB AAABA AABAA ABAAA BAAAA

ABBBB BABBB BBABB BBBAB BBBBA

The rule 

{AA → BB, BB → AA}

BAABB BBAAB
BBBBB

BABBA BAAAA AAAAB ABBAB

BBBAA AABBB

AABAA

AABBA ABBAA
AAAAA

BAABA BBBBA ABBBB ABAAB

BBAAA AAABB

BBABB

ABABB ABAAA ABBBA AAABA BBABA

AABAB BBBAB BAAAB BABBB BABAA

306



gives the same basic structure, but now what distinguishes the components is the difference 

in the number of ABs vs. BAs that occur in each string.  In both these examples, the number 

of distinct components increases linearly with the length of the strings.  

The rule

{AA → BB, AB → BA}
already gives exactly two components, one with an even number of Bs, and one with an 

odd number:

AAAAA AAAAB

AAABB AAABA

AABAB AABAA

ABAAB AABBA ABAAA AABBB

BAAAB ABABA BAAAA ABABB

BAABABBAA BAABBABBAB

ABBBB BABAA ABBBA BABAB

BABBB BBAAA BABBA BBAAB

BBABB BBABA

BBBAB BBBAA

BBBBA BBBBB

The rule

{AB → AA, BB → BA}
also gives two components, but now these just correspond to strings that start with A or 

start with B.

BBBBB ABBBB

BABBB BBABB BBBAB BBBBA AABBB ABABB ABBAB ABBBA

BAABB BABAB BBAAB BABBA BBABA BBBAA AAABB AABAB ABAAB AABBA ABABA ABBAA

BAAAB BAABA BABAA BBAAA AAAAB AAABA AABAA ABAAA

BAAAA AAAAA

307



6.12  Local Symmetries
In the previous subsection, we considered symmetries associated with global transforma‐
tions made on all relations in a system.  Here we will consider symmetries associated with 

local transformations on relations involved in particular rule applications.

Every time one does an update with a given rule, say

{{x, y}, {z, y}} → {{x, x}, {y, y}, {z, w}}
one needs to match the “variables” that appear on the le�‐hand side with actual elements in 

the hypergraph.  But in general there may be multiple ways to do this.  For example, with 

the hypergraph

{{1, 2}, {3, 2}}
one could either match

{x → 1, y → 2, z → 3}
or:

{z → 1, y → 2, x → 3}
The possible permutations of matches correspond to the automorphism group of the 

hypergraph that represents the le�‐hand side of the rule.

For 22 hypergraphs of which {{x,y},{y,z}} is an example, there are only two possible automor‐
phism groups: the trivial group (i.e. no invariances), and the group S2 (i.e. permutations 

{2,3}, {1,3} or {1,2}).

Here are automorphism groups for binary and ternary hypergraphs with various signatures.  
In each case the group order is included, as are a couple of sample hypergraphs:

E 1 120

E 1 25 ℤ2 2 38
ℤ2×ℤ2 4 2

E 1 5 ℤ2 2 4
 , , 

ℤ3 3 1 ℤ4 4 1
ℤ2 2 3

S3 6 2 S3 6 4
22

32 S4 24 2

42

308



E 1 155542
ℤ2 2 8427
ℤ3 3 30

E 1 3042

E 1 81 ℤ2 2 204 ℤ2×ℤ2 4 179
 , , 

ℤ2 2 21 ℤ3 3 10 ℤ4 4 15
23 S3 6 12

S3 6 174
33 D4 8 12

S4 24 12

43

If the right‐hand side of a rule has at least as high a symmetry as the le�‐hand side, then any 
possible permutation of matches of elements will lead to the same result—which means that 
the same update will occur, and the only consequence will be a potential change in path 
weightings in the multiway graph.

But if the right‐hand side of the rule has lower symmetry than the le�‐hand side (i.e. its 
automorphism group is a proper subgroup), then different permutations of matches can 
lead to different outcomes, on different branches of the multiway system. It may still 
nevertheless be the case that some permutations will lead to identical outcomes—and this 
will happen whenever the canonical form of the rule is the same a�er a permutation of the 
elements on the le�‐hand side (cf. [87]).

Thus for example the rule

{{x, y}, {z, y}} → {{x, x}, {y, y}, {z, w}}
is invariant under any of the permutations

{{1, 2, 3}, {2, 1, 3}, {3, 1, 2}, {3, 2, 1}}
of the elements corresponding to {x, y, z}. Note that the permutations that appear here do 
not form a group. To compose multiple such transformations one must take account of 
relabeling on the right‐hand side as well as the le�‐hand side.

For the 3138 22 → 32 rules that involve 3 elements on the le�, the following lists the 10 of 64 
subsets of the 6 possible permutations that occur:

309



1164

808

113

808

2

16

94

97

19

17

In a sense, we can characterize the local symmetry of a rule by determining what permuta‐
tions of inputs it leaves invariant. But we can do this not only for a single update, but for a 
sequence of multiple updates. In effect, all we have to do is to form a power of the rule, and 
then apply the same procedure as above.

There are several ways to put a notion of powers (or in general, products) of rules. As one 
example, we can consider situations in which a rule is applied repeatedly to an overlapping 
set of elements—so that in effect the successive rule applications are causally connected.

In this case—much as we did for testing total causal invariance—we need to work out the 
unifications of the possible initial conditions. Then we effectively just need to trace the 
multiway evolution from each of these unified initial conditions.

Consider for example the rule:

{{x, y}} → {{x, y}, {y, z}}

310



The “square” of this rule is:

 , 

And its cube is:

 , , , 

The multiway graph for the original rule a�er 4 updates is:

The “square” of the rule generates the same states in only 2 updates:

We can use the same approach to find the “square” of a rule like: 

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}

311



The result is:

 , , , ,

, , , 

Using this for actual evolution gives the result:

 , , , , , ,

And now that we can compute the power of a rule, we have a way to compute the effective 
symmetry for multiple updates according to a rule. In general, a�er t updates we will end up 
with a collection of permutations of variables that leave the effective “power rule” invariant.

But if we now consider increasingly large values of t, we can ask whether the collections of 
permutations we get somehow converge to a definite limit. In direct analogy to the way that 
our hypergraphs can limit to manifolds, we may wonder whether these collections of 
permutations could limit to a Lie group (cf. [88]).

As a simple example, say that the permutations are of length n, but of the n! possibilities, we 
only have the n cyclic permutations, say for n = 4:

{{1, 2, 3, 4}, {2, 3, 4, 1}, {3, 4, 1, 2}, {4, 1, 2, 3}}
As n → ∞ we can consider this to limit to the Lie group U(1), corresponding to rotations by 
any angle θ on a circle.

It is not so clear [89][90] how to deal in more generality with collections of permutations, 
although one could imagine an analog of a manifold reconstruction procedure. To get an 
idea of how this might work, consider the inverse problem of approximating a Lie group by 
permutations. (Note that things would be much more straightforward if we could build up 
matrix representations, but this is not the setup we have.)

In some cases, there are definite known finite subgroups of Lie groups—such as the icosahe‐
dral group A5 as a subset of the 3D rotation group SO(3). In such cases one can then explic‐
itly consider the permutation representation of the finite group. It is also possible to imagine 
just taking a lattice (or perhaps some more general structure of the kind that might be used 
in symbolic dynamics [91][92]) and applying random elements of a particular Lie

312



group to it, then in each case recording the transformation of lattice points that this yields. 
Typically these transformations will not be permutations, but it may be possible to approxi‐
mate them as such. By inverting this kind of procedure, one can imagine potentially being 
able to go from a collection of permutations to an approximating Lie group.

6.13 Branchial Graphs and Multiway Causal Graphs
Consider the rule:

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
If we pick a foliation for the first few steps in the multiway graph for this rule

then just as in 5.15 for string substitution systems, we can generate branchial graphs that repre‐
sent the connections defined by branch pairs between the states at each slice in the foliation:

313



Branchial graphs provide one form of summary of the multiway evolution.  Another sum‐
mary is provided by the multiway causal graph, which includes causal connections between 

parts of hypergraphs both within a branch of the multiway system, and across different 
branches:

The multiway causal graph is in many respects the richest summary of the behavior of our 

models, and it will be important in our discussion of possible connections to physics.

In a case like the rule shown, the structure of branchial and multiway causal graphs is quite 

complex.  As a simpler example, consider the causal invariant rule:

{{x, y}} → {{x, y}, {y, z}}

 , , , , , , 

314



With this rule, the multiway graph has the form:

A�er more steps, and with a different rendering, the multiway graph is:

(In this case, the size of the multiway graph as measured by Σt increases slightly faster than 2t.)

The branchial graphs with the standard layered foliation in this case are:

 , , , , 

315



The volumes Bt in the branchial graph grow on successive steps like: 

0 1 2 3 4 5 6 7

The multiway causal graph in this case is

or with more steps and a different layout:

316



Note that in this case the ordinary multiway graph is simply:

As a slightly more complicated example, consider the causal invariant 22 → 32 rule:

{{x, y}, {x, z}} → {{y, w}, {y, z}, {w, x}}

 , , , , , , , ,

, , , , , , , 

The multiway system in this case has the form:

317



The sequence of branchial graphs in this case are:

 , , ,

, , 

The causal graph for this rule is:

The multiway causal graph has many repeated edges:

318



Here it is in a different rendering:

Note that in our models, even when the hypergraphs are disconnected, the branchial graphs
can still be connected, as in the case of the rule:

{{x, y}} → {{y, z}, {z, w}}

 , , , , , , 

 , , , , , 

319






7 | Equivalence and Computation in 

Our Models
7.1  Correspondence with Other Systems
Our goal with the models introduced here is to have systems that are intrinsically as structure‐
less as possible, and are therefore in a sense as flexible and general as possible. And one way 

to see how successful we have been is to look at what is involved in reproducing other systems 

using our models.

As a first example, consider the case of string substitution systems (e.g [1:3.5]). An obvious 

way to represent a string is to use a sequence of relations to set up what amounts to a linked 

list. The “payload” at each node of the linked list is an element of the string. If one has two 

kinds of string elements A and B, these can for example be represented respectively by 3‐ary 

and 4‐ary relations. Thus, for example, the string ABBAB could be (where the Bs can be 

identified from the “inner self‐loops” in their 4‐ary relations): 

{{0, A1, 1}, {1, B2, B2, 2}, {2, B3, B3, 3}, {3, A4, 4}, {4, B5, B5, 5}}

B2 A4
B3

0 2 3 B5
1 4

A1 5

Note that the labels of the elements and the order of relations are not significant, so 

equivalent forms are

{{1, 7, 2}, {2, 9, 9, 3}, {3, 10, 10, 4}, {4, 8, 5}, {5, 11, 11, 6}}

or, with our standard canonicalization:

{{1, 2, 2, 3}, {3, 4, 4, 5}, {6, 7, 7, 8}, {5, 9, 6}, {10, 11, 1}}

A rule like

{A → BA, B → A}
can then be translated to

{{{0, A1, 1}}→ {{0, B1, B1, -2}, {-2, A2, 1}}, {{0, B1, B1, 1}}→ {{0, A1, 1}}}

or:

{{{x, y, z}} → {{x, u, u, v}, {v, w, z}}, {{x, y, y, z}} → {{x, u, z}}}

321



Starting with A, the original rule gives: 

{A, BA, ABA, BAABA, ABABAABA, BAABAABABAABA, ABABAABABAABAABABAABA}
In terms of our translation, this is now:

 , , , ,

, , 

The causal graph for the rule in effect shows the dependence of the string elements

corresponding to the evolution graph (see 5.1):

A

B

A B

B A B

A B B A B

B A B A B B A B

A B B A B B A B A B B A B

322



We can also take our translation of the string substitution system, and use it in a multiway system:

The result is a direct translation of what we could get with the underlying string system:

A

BA

AA BBA

ABA BAA BBBA

AAA ABBA BABA BBAA BBBBA

AABA ABAA BAAA BABBA ABBBA BBABA BBBAA BBBBBA

Having seen how our models can reproduce string substitution systems, we consider next 
the slightly more complex case of reproducing Turing machines [93].

As an example, consider the simplest universal Turing machine [1:p709][94][95] [96], which 

has the 2‐state 3‐color rule:

Given a tape with the Turing machine head at a certain position

323



a possible encoding uses different‐arity hyperedges to represent different values on the tape, 
and different states for the head, then attaches the head to a certain position on the tape, and 

uses special (in this case 6‐ary) hyperedges to provide “extensible end caps” to the tape: 

With this setup, the rule can be encoded as:

 , , , ,

, , , 

Starting from a representation of a blank tape, the first few steps of evolution are (note that 
the tape is extended as needed)

 , , ,

, , ,

, , 

324



which corresponds to the first few steps of the Turing machine evolution:

The causal graph for our model directly reflects the motion of the Turing machine head, as 

well as the causal connections generated by symbols “remembered” on the tape between 

head traversals:

325



Re‐rendering this causal graph, we see that it begins to form a grid:

It is notable that even though in the underlying Turing machine only one action happens at 
each step, the causal graph still connects many events in parallel (cf. [1:p489]). A�er 1000 

steps the graph has become a closer approximation to a flat 2D manifold, with the specific 

Turing machine evolution reflected in the detailed “knitting” of connections on its surface:

The rule we have set up allows only one thread of history, so the multiway system is trivial: 

But with our model the underlying setup is general enough that it can handle not only 

ordinary deterministic Turing machines in which each possible case leads to a specific 

outcome, but also non‐deterministic ones (as used in formulating NP problems) (e.g. [97]), 
in which there are multiple outcomes for some cases: 

326



For a non‐deterministic Turing machine, there can be multiple paths in the multiway system:

Continuing this, we see that the non‐deterministic Turing machine shows a fairly complex 

pattern of branching and merging in the multiway system (this particular example is not 
causal invariant):

327



A�er a few more steps, and using a different rendering, the multiway system has the form:

(Note that in actually using a non‐deterministic Turing machine, say to solve an NP‐
complete problem, one needs to check the results on each branch of the multiway system—
with different search strategies corresponding to using different foliations in exploring the 

multiway system.)

As a final example, consider using our models to reproduce cellular automata. Our models 

are in a sense intended to be as flexible as possible, while cellular automata have a simple 

but rigid structure. In particular, a cellular automaton consists of a rigid array of cells, with 

specific, discrete values that are updated in parallel at each step. In our models, on the other 

hand, there is no intrinsic geometry, no built‐in notion of “values”, and different updating 

events are treated as independent and “asynchronous”, subject only to the partial ordering 

imposed by causal relations.

In reproducing a Turing machine using our models, we already needed a definite tape that 
encodes values, but we only had to deal with one action happening at a time, so there was no 

issue of synchronization. For a cellular automaton, however, we have to arrange for synchro‐
nization of updates across all cells. But as we will see, even though our models ultimately 

work quite differently, there is no fundamental problem in doing this with the models.

328



For example, given the rule 30 cellular automaton

we encode a state like 

in the form

where the 6‐ary self‐loops represent black cells. Note that there is a quite complex structure 

that in effect maintains the cellular automaton array, complete with “extensible end caps” 
that allow it to grow.

Given this structure, the rule corresponding to the rule 30 cellular automaton becomes

 , , ,

, , ,

, , , 

329



where the first two transformations relate to the end caps, and the remaining 8 actually 

implement the various cases of the cellular automata rule. Applying the rule for our model 
for a few steps to an initial condition consisting of a single black cell, we get:

 , , ,

, , ,



Each of these steps has information on certain cells in the cellular automaton at a certain 

step in the cellular automaton evolution. “Decoding” each of the steps in our model shown 

above, we get the following, in which the “front” of cellular automaton cells whose values 

are present at that step in our model are highlighted:

 , , , ,

, 

The particular foliation we have used to determine the steps in the evolution of our model 
corresponds to a particular foliation of the evolution of the cellular automaton:

330



The final “spacetime” cellular automaton pattern is the same, but the foliation defines a 

specific order for building it up. We can visualize the way the data flows in the computation 

by looking at the causal graph (with events forming cells with different colors indicated): 

Here is the foliation of the causal graph that corresponds to each step in a traditional 
synchronized parallel cellular automaton updating:

7.2  Alternative Formulations
We have formulated our models in terms of the rewriting of collections of relations between 

elements. And in this formulation, we might represent a state in one of our models as a list 
of (here 3‐ary) relations

{{1, 2, 2}, {3, 1, 4}, {3, 5, 1}, {6, 5, 4}, {2, 7, 6}, {8, 7, 4}}

and the rule for the model as:

{{x, y, z}, {z, u, v}} → {{w, z, v}, {z, x, w}, {w, y, u}}
where x, y, ... are taken to be pattern or quantified variables, suggesting notations like [98]

{{x_, y_, z_}, {z_, u_, v_}}→ {{w, z, v}, {z, x, w}, {w, y, u}}

331



or [99]

∀{x,y,z,u,v} ({{x, y, z}, {z, u, v}}→ {{w, z, v}, {z, x, w}, {w, y, u}})

An alternative to these kinds of symbolic representations is to think—as we have o�en done 

here—in terms of transformations of directed hypergraphs. The state of one of our models 

might then be represented by a directed hypergraph such as

while the rule would be:

But in an effort to understand the generality of our models—as well as to see how best to 

enumerate instances of them—it is worthwhile to consider alternative formulations.

One possibility to consider is ordinary graphs. If we are dealing only with binary relations, 
then our models are immediately equivalent to transformations of directed graphs.

But if we have general k‐ary relations in our models, there is no immediate equivalence to 

ordinary graphs. In principle we can represent a k‐ary hyperedge (at least for k > 0) by a 

sequence of ordinary graph edges:

1
 11 1 , 2 1 2 12 11 1 ,

1 3 2 4 3 4 1

14 11
12 , 

1 13 1
1 1 2

3
2 1 3 1 2 3 2

332



For the hypergraph above, this then yields:

The rule above can be stated in terms of ordinary directed graphs as:

In terms of hypergraphs, the result of 5 and 10 steps of evolution according to this rule is

 , 

and the corresponding result in terms of ordinary directed graphs is:

 , 

In thinking about ordinary graphs, it is natural also to consider the undirected case. And 

indeed—as was done extensively in [1:𝕔9]—it is possible to study many of the same things we 

do here with our models also in the context of undirected graphs. However, transformations 

of undirected graphs lack some of the flexibility and generality that exist in our models 

based on directed hypergraphs.

333



It is straightforward to convert from a system described in terms of undirected graphs to one 

described using our models: just represent each edge in the undirected graph as a pair of 
directed binary hyperedges, as in:

 , 

Transformations of undirected graphs work the same—though with paired edges. So, for 

example, the rule

which yields

becomes

334



which yields:

In dealing with undirected graphs—as in [1:𝕔9]—it is natural to make the further simplifica‐
tion that all graphs are trivalent (or “cubic”). In the context of ordinary graphs, nothing is 

lost by this assumption: any higher‐valence node can always be represented directly as a 

combination of trivalent nodes. But the point about restricting to trivalent graphs is that it 
makes the set of possible rules better defined—because without this restriction, one can 

easily end up having to specify an infinite family of rules to cover graphs of arbitrary 

valence that are generated. (In our models based on transformations for arbitrary relations, 
no analogous issue comes up.) 

It is particularly easy to get intricate nested structures from rules based on undirected 

trivalent graphs; it is considerably more difficult  to get more complex behavior:

 , {,

 , {

335



Another issue in models based on undirected graphs has to do with the fact that the objects 

that appear in their transformation rules do not have exactly the same character as the 

objects on which they act. In our hypergraph‐based models, both sides of a transformation 

are collections of relations (that can be represented by hypergraphs)—just like what appears 

in the states on which these transformations act. But in models based on undirected graphs, 
what appears in a transformation is not an ordinary graph: instead it is a subgraph with 

“dangling connections” (or “half‐edges”) that must be matched up with part of the graph on 

which the transformation acts. 

Given this setup, it is then unclear, for example, whether or not the rule above—stated in 

terms of undirected graphs—should be considered to match the graph:

(In a sense, the issue is that while our models are based on applying rules to collections of 
complete hyperedges, models based on undirected graphs effectively apply rules to collec‐
tions of nodes, requiring “dangling connections” to be treated separately.) 

Another apparent problem with undirected trivalent graphs is that if the right‐hand side of a 

transformation has lower symmetry than the le�‐hand side, as in 

then it can seem “undefined” how the right‐hand side should be inserted into the final 
graph. Having seen our models here, however, it is now clear that this is just one of many 

examples where multiple different updates can be applied, as represented by multiway 

systems.

A further issue with systems based on undirected trivalent graphs has to do with the enumer‐
ation of possible states and possible rules. If a graph is represented by pairs of vertices
corresponding to edges, as in

{{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}

the fact that the graph is trivalent in a sense corresponds to a global constraint that each 

vertex must appear exactly three times. The alternate “vertex‐based” representation

{1→ {2, 3, 4}, 2→ {1, 3, 4}, 3→ {1, 2, 4}, 4→ {1, 2, 3}}

336



does not overcome this issue. In our models based on collections of relations, however, 
there are no such global constraints, and enumeration of possible states—and rules—is 

straightforward. (In our models, as in trivalent undirected graphs, there is, however, still the 

issue of canonicalization.) 

In the end, though, it is still perfectly possible to enumerate distinct trivalent undirected 

graphs (here dropping cases with self‐loops and multiple edges)

 , , , , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , , , , 

as well as rules for transforming them, and indeed to build up a rich analysis of their 

behavior [1:9.12]. Notions such as causal invariance are also immediately applicable, and for 

example one finds that the simplest subgraphs that do not overlap themselves, and so 

guarantee causal invariance, are [1:p515][87]:

 , , , , , , , , ,

, , , , , , , , ,

, , , , , , , , 

Directed graphs define an ordering for every edge. But it is also possible to have ordered 

graphs in which the individual edges are undirected, but an order is defined for the edges at 
any given vertex [87]. Trivalent such ordered graphs can be represented by collections of 
ordered triples, where each triple corresponds to a vertex, and each number in each triple 

specifies the destination in the whole list of a particular edge:

{{2, 1, 6}, {5, 4, 3}}

For visualization purposes one can “name” each element of each triple by a color

{ → 2, → 1, → 6, → 5, → 4, → 3}

337



and then the ordered graph can be rendered as:

In the context of our models, an ordered trivalent graph can immediately be represented as 

a hypergraph with ternary hyperedges corresponding to the trivalent nodes, and binary 

hyperedges corresponding to the edges that connect these nodes:

 , 

To give rules for ordered trivalent graphs, one must specify how to transform subgraphs 

with “dangling connections”. Given the rule (where letters represent dangling connections)

{{4, a, b}, {1, c, d}} → {{4, 8, a}, {1, 11, b}, {10, 2, c}, {7, 5, d}}

the evolution of the system is:

 , , , ,

, , , 

338



The corresponding rule for hypergraphs would be

and the corresponding evolution is:

 , , ,

, , , 

The rule just shown is example of a rule with 2 → 4 internal nodes and 4 dangling connec‐
tions—which is the smallest class that supports growth from minimal initial conditions. 
There are altogether 264 rules of this type, with rules of the following forms (up to vertex 

orderings) [87]:

 , , , 

These rules produce the following distinct outcomes:

339



Even though there is a direct translation between ordered trivalent graphs and our models, 
what is considered a simple rule (for example for purposes of enumeration) is different in 

the two cases. And while it is more difficult  to find valid rules with ordered trivalent graphs, 
it is notable that even some of the very simplest such rules generate structures with limiting 

manifold features that we see only a�er exploring thousands of rules in our models: 

340



Our models are based on directed (or ordered) hypergraphs. And although the notion is not as 

natural as for ordinary graphs, one can also consider undirected (or unordered) hypergraphs, 
in which all elements in a hyperedge are in effect unordered and equivalent. (In general one 

can also imagine considering any specific set of permutations of elements to be equivalent.) 

For unordered hypergraphs one can still use a representation like

{{1, 2, 3}, {1, 2, 4}, {3, 4, 5}}

but now there are no arrows needed within each hyperedge:
1 3

5

2 4

341



There are considerably fewer unordered hypergraphs with a given signature than ordered ones:

ordered unordered ordered unordered ordered unordered
12 2 2 82 293 370 4779 14 15 5
22 8 4 92 2 255 406 19 902 24 2032 51
32 32 11 102 18 201 706 86 682 34 678 358 1048
42 167 30 13 5 3 15 52 7
52 928 95 23 102 15 25 57 109 164
62 5924 328 33 3268 107 16 203 11
72 40 211 1211 43 164 391 1098 26 2 089 513 499

There is a translation between unordered hypergraphs and ordered ones, or specifically 

between unordered hypergraphs and directed graphs. Essentially one creates an incidence 

graph in which each node and each hyperedge in the unordered hypergraph becomes a 

node in the directed graph—so that the unordered hypergraph above becomes:
1 3

e3
e1
e 5
2

2 4

But despite this equivalence, just as in the case of ordered graphs, the sequence of rules will 
be different in an enumeration based on unordered hypergraphs from one based on ordered 

hypergraphs. 

There are many fewer rules with a given signature for unordered hypergraphs than for 

ordered ones:

unordered ordered unordered ordered
12 → 12 5 11 32 → 12 59 416
12 → 22 19 73 32 → 22 347 4688
12 → 32 71 506 32 → 32 1900 48 554
12 → 42 296 3740 42 → 12 235 3011
12 → 52 1266 28 959 42 → 22 1697 42 955
22 → 12 16 64 52 → 12 998 23 211
22 → 22 76 562 13 → 13 22 178
22 → 32 348 4702 13 → 23 257 9373
22 → 42 1657 40 405 23 → 13 223 8413
22 → 52 7992 353 462 14 → 14 84 3915

342



Here is an example of a 23 → 33 rule for unordered hypergraphs:

{{x, y, z}, {u, v, z}} → {{x, x, w}, {u, v, x}, {y, z, y}}

Starting from an unordered double ternary self‐loop, this evolves as:

 , , , , , ,

, , , , 

In general the behavior seen for unordered rules with a given signature is considerably
simpler than for ordered rules with the same signature. For example, here is typical behav‐
ior seen with a random set of unordered 23 → 33 rules:

In ordered 23 → 33 rules, globular structures are quite common; in the unordered case they
are not. Once one reaches 23 → 43 rules, however, globular structures become common even
for unordered hypergraph rules:

{{x, y, z}, {u, y, v}} → {{x, w, w}, {x, s, z}, {z, s, u}, {y, v, w}}

343



 , , , , , 

It is worth noting that the concept of unordered hypergraphs can also be applied for binary 

hyperedges, in which case it corresponds to undirected ordinary graphs. We discussed 

above the specific case of trivalent undirected graphs, but one can also consider enumerat‐
ing rules that allow any valence. 

An example is

{{x, y}, {x, z}} → {{x, w}, {y, z}, {y, w}, {z, w}}

344



which evolves from an undirected double self‐loop according to:

 , , , , ,

, , , , 

This rule is similar, but not identical, to a rule we have o�en used as an example:

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
Interpreting this rule as referring to undirected graphs, it evolves according to:

 , , , ,

, , , , 

In general, rules for undirected graphs of a given signature yield significantly simpler 
behavior than rules of the same signature for directed graphs. And, for example, even among 

all the 7992 distinct 22 → 52 rules for undirected graphs, no globular structures are seen.

Hypergraphs provide a convenient approach to representing our models. But there are other 

approaches that focus more on the symbolic structure of the models. For example, we can 

think of a rule such as

{{x, y, z}, {z, u, v}} → {{w, z, v}, {z, x, w}, {w, y, u}}
as defining a transformation for expressions involving a ternary operator f together with a 

commutative and associative (n‐ary) operator ∘:

f[x, y, z]∘ f[z, u, v]→ f[w, z, v]∘ f[z, x, w]∘ f[w, y, u]

In this formulation, the ∘ operator can effectively be arbitrarily nested. But in the usual 
setup of our models, f cannot be nested. One could certainly imagine a generalization in 

which one considers (much as in [98]) transformations on symbolic expressions with 

arbitrary structures, represented by pattern rules like

f[g[x_, y_], z_]∘ f[h[g[z_, x_], x_]]→…

345



or even:

f[g[x_, y_], z_]∘ f[h_[g[z_, x_], h_[x_]]]→…

And much as in the previous subsection, it is always possible to represent such transformations 
in our models, for example by having fixed subhypergraphs that act as “markers” to distinguish 

different functional heads or different “types”. (Similar methods can be used to have literals in 

addition to pattern variables in the transformations, as well as “named slots” [100].)

Our models can be thought of as abstract rewriting (or reduction) systems that operate on 

hypergraphs, or general collections of relations. Frameworks such as lambda calculus 

[101][102] and combinatory logic [103][104] have some similarities, but focus on defining 

reductions for tree structures, rather than general graphs or hypergraphs. 

One can ask how our models relate to traditional mathematical systems, for example from 

universal algebra [105][106]. One major difference is that our models focus on transforma‐
tions, whereas traditional axiomatic systems tend to focus on equalities. However, it is 

always possible to define two‐way rules or pairs of rules X→Y , Y→X which in effect repre‐
sent equalities, and on which a variety of methods from logic and mathematics can be used.

The general case of our models seems to be somewhat out of the scope of traditional mathe‐
matical systems. However, particularly if one considers the simpler case of string substitu‐
tion systems, it is possible to see a variety of connections [1:p938]. For example, two‐way 

string rewrites can be thought of as defining the relations for a semigroup (or, more specifi‐
cally, a monoid). If one adds inverse elements, then one has a group.

One thinks of the strings as corresponding to words in the group. Then the multiway 

evolution of the system corresponds to starting with particular words and repeatedly 

applying relations to them—to produce other words which for the purposes of the group are 

considered equivalent. 

This is in a sense a dual operation to what happens in constructing the Cayley graph of a 

group, where one repeatedly adds generators to words, always reducing by using the 

relations in the group (see 4.17). 

For example, consider the multiway system defined by the rule:

{AB → BA, BA → AB}

346



The first part of the multiway (states) graph associated with this rule is:

BAAB

AABB ABAB BABA BBAA

ABBA

BAAA ABAA AABA AAAB

BBBA BBAB BABB ABBB

BAA ABA AAB

BBA BAB ABB AB BA

Ignoring inverse elements (which in this case just make double edges) the first part of the 

infinite Cayley graph for the group with relations ABBA has the form:

B A

BB AB AA

BBBB BBB AAA AAAA
ABB AAB

ABBB AAAB
AABB

One can think of the Cayley graph as being created by starting with a tree, corresponding to 

the Cayley graph for a free group, then identifying nodes that are related by relations. The 

edges in the multiway graph (which correspond to updating events) thus have a correspon-
dence to cycles in the Cayley graph.

As one further example, consider the (finite) group S3  which can be thought of as being 

specified by the relations:

{ AA, AA BB, BB ABABAB}

The Cayley graph in this case is simply: 



The multiway graph in this case begins:

AAAAAA ABABAB
AABB

AAAA BBAA BB

ABBA BAAB

AA

Continuing for a few more steps gives:

On successive steps, the volumes Σt in these multiway graphs grow like:

2000

1500

1000

500

0
0 2 4 6 8 10 12 14

There does not appear to be any direct correspondence to quantities such as growth rates of 
Cayley graphs (cf. [22]).

348



7.3  Computational Capabilities of Our Models
An important way to characterize our models is in terms of their computational capabilities. 
We can always think of the evolution of one of our models as corresponding to a computa‐
tion: the system starts from an initial state, then follows its rules, in effect carrying out a 

computation to generate a sequence of results. 

The Principle of Computational Equivalence [1:𝕔12] suggests that when the behavior of our 

models is not obviously simple it will typically correspond to a computation of effectively 

maximal sophistication. And an important piece of evidence for this is that our models are 

capable of universal computation.

We saw above that our models can emulate a variety of other kinds of systems. Among these 

are Turing machines and cellular automata. And in fact we already saw above how our models 

can emulate what is known to be the simplest universal Turing machine [1:p709][94][95] [96]. 
We also showed how our models can emulate the rule 30 cellular automaton, and we can use 

the same construction to emulate the rule 110 cellular automaton, which is known to be 

computation universal [1:11.8].

So what this means is that we can set up one of our models and then “program” it, by giving 

appropriate initial conditions, to make it do any computation, or emulate any other computa‐
tional system. We have seen that our models can produce all sorts of behavior; what this 

shows is that at least in principle our models can produce any behavior that any computa‐
tional system can produce.

But showing that we can set up one of our models to emulate a universal Turing machine is 

one thing; it is something different to ask what computations a random one of our models 

typically performs. To establish this for certain is difficult,  but experience with the Principle 

of Computational Equivalence [1:𝕔12] in a wide range of other kinds of systems with simple 

underlying rules strongly suggests that not only is sophisticated computation possible to 

achieve in our models, it is also ubiquitous, and will occur basically whenever the behavior 

we see is not obviously simple.

This notion has many consequences, but a particularly important one is computational 
irreducibility [1:12.6]. Given the simplicity of the underlying rules for our models, we 

might imagine that it would always be possible—by using some appropriately sophisti‐
cated mathematical or computational technique—to predict what the model would do 

a�er any number of steps. But in fact what the Principle of Computational Equivalence 

implies is that more or less whenever it is not obviously straightforward to do, making this 

prediction will actually take an irreducible amount of computational work—and that in 

effect we will not be able to compute what the system will do much more efficiently  than 

by just following the steps of the actual evolution of the system itself.

349



Much of what we have done in studying our models here has been based on just explicitly
running the models and seeing what they do. Computational irreducibility implies that this
is not just something that is convenient in practice; instead it is something that cannot
theoretically be avoided, at least in general.

Having said this, however, it is an inevitable feature of computational irreducibility that
there is always an endless sequence of “pockets” of computational reducibility: specific
features or questions that are amenable to computation or prediction without irreducible
amounts of computational work.

But another consequence of computational irreducibility is the appearance of undecidability
[107][108]. If we want to know what will happen in one of our models a�er a certain number
of steps, then in the worst case we can just run the model for that many steps and see what it
does. But if we want to know if the model will ever do some particular thing—even a�er an
arbitrarily long time—then there can be no way to determine that with any guaranteed finite
amount of effort, and therefore we must consider the question formally undecidable.

Will a particular rule ever terminate when running from a particular initial state? Will the
hypergraphs it generates ever become disconnected? Will some branch pair generated in a
multiway system ever resolve?

These are all questions that are in general undecidable in our models. And what the Princi‐
ple of Computational Equivalence implies is that not only is this the case in principle; it is
something ubiquitous, that can be expected to be encountered in studying any of our models
that do not show obviously simple behavior.

It is worth pointing out that undecidability and computational irreducibility apply both to
specific paths of evolution in our models, and to multiway systems. Multiway systems
correspond to what are traditionally called non‐deterministic computations [109]. And just
as a single path of evolution in one of our models can reproduce the behavior of any ordi‐
nary deterministic Turing machine, so also the multiway evolution of our models can
reproduce any non‐deterministic Turing machine.

The fact that our models show computation universality means that if some system—like our
universe—can be represented using computation of the kind done, for example, by a Turing
machine, then it is inevitable that in principle our models will be able to reproduce it. But
the important issue is not whether some behavior can in principle be programmed, but
whether we can find a model that faithfully and efficiently reflects what the system we are
modeling does. Put another way: we do not want to have to set up some elaborate program
in the initial conditions for the model we use; we want there to be a direct way to get the
initial conditions for the model from the system we are modeling.

350



There is another important point, particularly relevant, for example, in the effort to use our
models in a search for a fundamental theory of physics. The presence of computation
universality implies that any given model can in principle encode any other. But in practice
this encoding can be arbitrarily complicated, and if one is going to make an enumeration of
possible models, different choices of encoding can in effect produce arbitrarily large
changes in the enumeration.

One can think of different classes of models as corresponding to different languages for
describing systems. It is always in principle possible to translate between them, but the
translation may be arbitrarily difficult, and if one wants a description that is going to be
useful in practice, one needs to have a suitable language for it.

351






8 | Potential Relation to Physics
8.1  Introduction
Having explored our models and some of their behavior, we are now in a position to discuss 

their potential for application to physics. We shall see that the models generically show 

remarkable correspondence with a surprisingly wide range of known features of physics, 
inspiring the hope that perhaps a specific model can be found that precisely reproduces all 
details of physics. It should be emphasized at the outset that there is much le� to explore in 

the potential correspondence between our models and physics, and what will be said here is 

merely an indication—and sometimes a speculative one—of how this might turn out.

(See also Notes & Further References.)

8.2  Basic Concepts
The basic concept of applying our models to physics is to imagine that the complete struc‐
ture and content of the universe is represented by an evolving hypergraph. There is no 

intrinsic notion of space; space and its apparent continuum character are merely an emer‐
gent large‐scale feature of the hypergraph. There is also no intrinsic notion of matter: 
everything in the universe just corresponds to features of the hypergraph.

There is also no intrinsic notion of time. The rule specifies possible updates in the hyper‐
graph, and the passage of time essentially corresponds to these update events occurring. 
There are, however, many choices for the sequences in which the events can occur, and the 

idea is that all possible branches in some sense do occur. 

But the concept is then that there is a crucial simplifying feature: the phenomenon of causal 
invariance. Causal invariance is a property (or perhaps effective property) of certain underly‐
ing rules that implies that when it comes to causal relationships between events, all possible 

branches give the same ultimate results.

As we will discuss, this equivalence seems to yield several core known features of physics, 
notably Lorentz invariance in special relativity, general covariance in general relativity, as 

well as local gauge invariance, and the perception of objective reality in quantum mechanics. 

Our models ultimately just consist of rules about elements and relations. But we have seen 

that even with very simple such rules, highly complex structures can be produced. In 

particular, it is possible for the models to generate hypergraphs that can be considered to 

approximate flat or curved d‐dimensional space. The dimension is not intrinsic to the 

model; it must emerge from the behavior of the model, and can be variable. 

353



The evolving hypergraphs in our models must represent not just space, but also everything 

in it. At a bulk level, energy and momentum potentially correspond to certain specific 

measures of the local density of evolution in the hypergraph. Particles potentially corre‐
spond to evolution‐stable local features of the hypergraph. 

The multiway branching of possible updating events is potentially closely related to quan‐
tum mechanics, and much as large‐scale limits of our hypergraphs may correspond to 

physical space, so large‐scale limits of relations between branches may correspond to 

Hilbert spaces of states in quantum mechanics. 

In the case of physical space, one can view different choices of updating orders as corre‐
sponding to different reference frames—with causal invariance implying equivalence 

between them. In multiway space, one can view different updating orders as different 
sequences of applications of quantum operators—with causal invariance implying equiva‐
lence between them that lead different observers to experience the same reality.

In attempting to apply our models to fundamental physics, it is notable how many features 

that are effectively implicitly assumed in the traditional formalism of physics can now 

potentially be explicitly derived. 

It is inevitable that our models will show computational irreducibility, in the sense that 
irreducible amounts of computational work will in general be needed to determine the 

outcome of their behavior. But a surprising discovery is that many important features of 
physics seem to emerge quite generically in our models, and can be analyzed without 
explicitly running particular models.

It is to be expected, however, that specific aspects of our universe—such as the dimensional‐
ity of space and the masses and charges of particles—will require tracing the detailed 

behavior of models with particular rules.

It is already clear that modern mathematical methods can provide significant insight into 

certain aspects of the behavior of our models. One complication in the application of these 

methods is that in attempting to make correspondence between our models and physics, 
many levels of limits effectively have to be taken, and the mathematical definitions of these 

limits are likely to be subtle and complex.

In traditional approaches to physics, it is common to study some aspect of the physical 
world, but ignore or idealize away other parts. In our models, there are inevitably close 

connections between essentially all aspects of physics, making this kind of factored 

approach—as well as idealized partial models—much more difficult.  
Even if the general structure of our models provides an effective framework for representing 

our physical universe at the lowest level, there does not seem to be any way to know within a 

wide margin just how simple or complex the specific rule—or class of equivalent rules—for 
our particular universe might be. But assuming a certain degree of simplicity, it is likely that 
fitting even a modest number of details of our universe will completely determine the rule.

354



The result of this would almost certainly be a large number of specific predictions about the 

universe that could be made even without irreducibly large amounts of computation. But 
even absent the determination of a specific rule, it seems increasingly likely that experimen‐
tally accessible predictions will be possible just from general features of our models. 

8.3  Potential Basic Translations
As a guide to the potential application of our models to physics, we list here some current 
expectations about possible translations between features of physics and features of our 
models. This should be considered a rough summary, with every item requiring significant 
explanation and qualification. In addition, it should be noted that in an effort to clarify presenta‐
tion, many highly abstract concepts have been indicated here by more mechanistic analogies. 

Basic Physics Concepts
space: general limiting structure of basic hypergraph

time: index of causal foliations of hypergraph rewriting

matter (in bulk): local fluctuations of features of basic hypergraph

energy: flux of edges in the multiway causal graph through spacelike (or branchlike) hypersurfaces

momentum: flux of edges in the multiway causal graph through timelike hypersurfaces

(rest) mass: numbers of nodes in the hypergraph being reused in updating events

motion: possible because of causal invariance; associated with change of causal foliations

particles: locally stable configurations in the hypergraph

charge, spin, etc.: associated with local configurations of hyperedges

quantum indeterminacy: different  foliations (of branchlike hypersurfaces) in the 

multiway graph

quantum effects: associated with locally unresolved branching in the multiway graph

quantum states: (instantaneously) nodes in the branchial graph

quantum entanglement: shared ancestry in the multiway graph / distance in 

branchial graph

quantum amplitudes: path counting and branchial directions in the multiway graph

quantum action density (Lagrangian): total flux (divergence) of multiway causal 
graph edges

355



Physical Theories & Principles
special relativity:  global consequence of causal invariance in hypergraph rewriting

general relativity / general covariance: effect of causal invariance in the causal graph

locality / causality: consequence of locality of hypergraph rewriting and causal invariance

rotational invariance: limiting homogeneity of the hypergraph

Lorentz invariance: consequence of causal invariance in the causal graph

time dilation: effect of different foliations of the causal graph

relativistic mass increase: effect of different foliations of the causal graph

local gauge invariance: consequence of causal invariance in the multiway graph

lack of quantum cosmological constant: space is effectively created by quantum fluctuations

cosmological homogeneity: early universe can have higher effective spatial dimension

expansion of universe: growth of hypergraph

conservation of energy:  equilibrium in the causal graph

conservation of momentum: balance of different hyperedges during rewritings

principle of equivalence: gravitational and inertial mass both arise from features 

of the hypergraph

discrete conservation laws: features of the ways local hypergraph structures can combine

microscopic reversibility: limiting equilibrium of hypergraph rewriting processes

quantum mechanics: consequence of branching in the multiway system

observer in quantum mechanics: branchlike hypersurface foliation

quantum objective reality: equivalence of quantum observation frames in the multiway graph

quantum measurements: updating events with choice of outcomes, that can be frozen 

by a foliation

quantum eigenstates: branches in multiway system

quantum linear superposition: additivity of path counts in the multiway graph

uncertainty principle: non‐commutation of update events in the multiway graph

wave‐particle duality: relation between spacelike and branchlike projections of the 

multiway causal graph

356



operator‐state correspondence: states in the multiway graph are generated by 

events (operators)

path integral: turning of paths in the multiway graph is proportional to causal edge density

violation of Bellʼs inequalities, etc.: existence of causal connections in the multiway graph

quantum numbers: associated with discrete local properties of the hypergraph

quantization of charge, etc.: consequence of the discrete hypergraph structure

black holes / singularities: causal disconnection in the causal graph

dark matter: (possibly) relic oligons / dimension changes in of space

virtual particles: local structures continually generated in the spatial and multiway graphs

black hole radiation / information: causal disconnection of branch pairs

holographic principle: correspondence between spatial and branchial structure

Physical Quantities & Constructs
dimension of space: growth rate exponent in hypergraph / causal cones

curvature of space: polynomial part of growth rate in hypergraph / causal cones

local gauge group: limiting automorphisms of local hypergraph configurations

speed of light (c): measure of edges in spatial graph vs. causal graph

light cones: causal cones in the causal graph

unit of energy: count of edges in the causal graph

momentum space: limiting structure of causal graph in terms of edges

gravitational constant: proportionality between node counts and spatial volume

quantum parameter (ℏ): measure of edges in the branchial graph (maximum speed 

of measurement)

elementary unit of entanglement: branching of single branch pair

electric/gauge charges: counts of local hyperedge configurations

spectrum of particles: spectrum of locally stable configurations in the hypergraph

Idealizations, etc. Used in Physics
inertial frame: parallel foliation of causal graph

rest frame of universe: geodesically layered foliation of causal graph

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flat space: uniform hypergraph (typically not maintained by rules)

Minkowski space: effectively uniform causal graph

cosmological constant: uniform curvature in the hypergraph

de Sitter space: cyclically connected hypergraph

closed timelike curves: loops in the causal graph (only possible in some rules)

point particle: a persistent structure in the hypergraph involving comparatively few nodes 

purely empty space: not possible in our models (space is maintained by rule evolution)

vacuum: statistically uniform regions of the spatial hypergraph

vacuum energy: causal connections attributed purely to establishing the structure of space

isolated quantum system: disconnected part of the branchial/multiway graph

collapse of the wave function: degenerate foliation that infinitely retards 

branchlike entanglement

non‐interacting observer in quantum mechanics: “parallel” foliation of multiway graph

free field theory: e.g. pure branching in the multiway system

quantum computation: following multiple branches in multiway system (limited by 

causal invariance)

string field  theory: (potentially) continuous analog of the multiway causal graph for 

string substitutions

8.4  The Structure of Space
In our models, the structure of spacetime is defined by the structure of the evolving hyper‐
graph. Causal foliations of the evolution can be used to define spacelike hypersurfaces. The 

instantaneous structure of space (on a particular spacelike hypersurface) corresponds to a 

particular state of the hypergraph.

A position in space is defined by a node in the hypergraph. A geometrical distance between 

positions can be defined as the number of hyperedges on the shortest path in the hyper‐
graph between them. Although the underlying rules for hypergraph rewriting in our models 

depend on the ordering of elements in hyperedges, this is ignored in computing geometrical 
distance. (The geometrical distance discussed here is basically just a proxy for a true physi‐
cal distance measured from dynamic information transmission between positions.) A 

shortest path on the hypergraph between two positions defines a geodesic between them, 
and can be considered to define a straight line.

The only information available to define the structure of space is the connectivity of the 

hypergraph; there is no predefined embedding or topological information. The continuum 

358



hypergraph;
character of space assumed in traditional physics must emerge as a large‐scale limit of the 

hypergraph (somewhat analogously to the way the continuum character of fluids emerges as 

a large‐scale limit of discrete molecular dynamics (e.g. [1:p378][110] ). Although our models 

follow definite rules, they can intrinsically generate effective randomness (much like the rule 

30 cellular automaton, or the computation of the digits of π). This effective randomness 

makes large‐scale behavior typically approximate statistical averages of small‐scale dynamics.

In our models, space has no intrinsic dimension defined; its effective dimension must 
emerge from the large‐scale structure of the hypergraph. Around every node at position X 

consider a geodesic ball consisting of all nodes that are a hypergraph distance not more than 

r away. Let Vr(X) be the total number of nodes in this ball. Then the hypergraph can be 

considered to approximate d‐dimensional space if

Vr(X) ~ rd

for a suitable range of values of r. Here we encounter the first of many limits that must be 

taken. We want to consider the limit of a large hypergraph (say as generated by a large 

number of steps of evolution), and we want r to be large compared to 1, but small compared 

to the overall diameter of the hypergraph. 

As a simple example, consider the hypergraph created by the rule

{{x, y}, {x, z}} → {{x, y}, {x, w}, {y, w}, {z, w}}
y y

z z w

x x

Starting from a minimal initial condition of two self‐loops, the first few steps of evolution 

with our standard updating order are: 

 , , , , , ,

, , , , 

359



The hypergraph obtained a�er 12 steps has 1651 nodes and can be rendered as:

This plots the effective “dimension exponent” of r in Vr as a function of r, averaged over all 
nodes in the hypergraph, for a succession of steps in the evolution:
3.0

2.5

2.0

1.5

1.0

0.5

0.0
0 5 10 15 20 25

A constant limiting value d indicates approximation to a “flat” d‐dimensional space. For 

integer d, this corresponds to ordinary d‐dimensional Euclidean space, but in our models d 

o�en does not end up being integer valued, nor does it need to be constant at different 
positions, or through the course of evolution. It is also important to note that only some 

rules give Vr~ rd; exponential or more complex behavior is common.

Even when to leading order Vr~ rd, there are corrections. For small r (measured, say, relative 

to the diameter of the hypergraph) one can consider a power series expansion in r. By 

comparison to ordinary manifolds one can then write (e.g. [24][1:p1050])

r2
Vr~ rd(1 – R + O(r4) )

6 (d + 2)

360



where R can be identified as the (Ricci) scalar curvature [25][26] of the limiting space. The
value of this curvature is again purely determined by the (limiting) structure of the hyper‐
graph. (Note that particularly if one goes beyond a pure power series, there is the potential for 
subtle interplay between change in dimension and what one might attribute to curvature.)

It is also possible to identify other limiting features of the hypergraph. For example, con‐
sider a small stretch of geodesic (where by “small” we mean still large compared to individ‐
ual connections in the hypergraph, but small compared to the scale on which statistical 
features of the hypergraph change). Now create a tube of radius r by including every node 

with distance up to r from any node on the geodesic. The growth rate of the number of 
nodes in this tube can then be approximated as [44]

~ r2
Vr ~ rd(1 + R i

ij δx δxj + O(r4) )
6

where now Rij δxi δxj
 is the projection of the Ricci tensor along the direction of the 

geodesic. (The Ricci tensor measures the change in cross‐sectional area for a bundle of 
geodesics, associated with their respective convergence and divergence for positive and 

negative curvature.)

In a suitable limit, the nodes in the hypergraph correspond to points in a space. A tangent 
bundle at each point can be defined in terms of the equivalence class of geodesics through 

that point, or in our case the equivalence class of sequences of hyperedges that pass through 

the corresponding node in the hypergraph.

One can set up what in the limit can be viewed as a rank‐p tensor field on the hypergraph by 

associating values with p hyperedges at each node. When these values correspond to 

intrinsic features of the hypergraph (such as Vr), their limits give intrinsic properties of the 

space associated with the hypergraph. And for example the Riemann tensor can be seen as 

emerging from essentially measuring areas of “rectangles” defined by loops in the hyper‐
graph, though in this case multiple limits need to be taken. 

8.5  Time and Spacetime
In our models, the passage of time basically corresponds to the progressive updating of the 

hypergraph. Time is therefore fundamentally computational: its passage reflects the 

performance of a computation—and typically one that is computationally irreducible. It is 

notable that in a sense the progression of time is necessary even to maintain the structure of 
space. And this effectively forces the entropic arrow of time (reflected in the effective 

randomization associated with irreducible computation) to be aligned with the cosmological 
arrow of time (defined by the overall evolution of the structure of space).

At the outset, time in our models has a very different character from space. The phe‐
nomenon of causal invariance, however, implies a link which leads to relativistic invariance. 

361



To see this, we can begin much as in the traditional development of special relativity [111] by 

considering what constitutes a physically realizable observer. In our model, everything is 

represented by the evolving hypergraph, including all of the internal state of any observer. 
One consequence of this is that the only way an observer can “sense” anything about the 

universe is by some updating event happening within the observer.

And indeed in the end all that any observer can ultimately be sensitive to is the causal 
relationships between different updating events that occur. From a particular evolution 

history of a hypergraph, we can construct a causal graph whose nodes correspond to 

updating events, and whose directed edges represent the causal relations between these 

events—in the sense that there is an edge between events A and B if the input to B involves 

output from A. For the evolution shown above, the beginning of the causal graph is:

We can think of this causal graph as representing the evolution of our system in spacetime. 
The analog of a light cone is then the set of nodes that can be reached from a given node in 

the graph. Every edge in the graph represents a timelike relationship between events, and 

can be thought of as corresponding to a timelike direction in spacetime. Nodes that cannot 
be reached from each other by following edges of the graph can be thought of as spacelike 

separated. Just as for space with the “spatial hypergraphs” we discussed above, there is 

nothing in the abstract that defines the geometry of spacetime associated with the causal 
graph; everything must emerge from the pattern of connections in the graph, which in turn 

are generated by the operation of the underlying rules for our models.

In its original construction, a causal graph is in a sense a causal summary of a particular 

evolution history for a given rule, with a particular sequence of updating events. But when 

the underlying rule has the property of causal invariance, this has the important conse‐
quence that in the appropriate limit the causal graph obtained always has the same form, 
independent of the particular sequence of updating events. In other words, when there is 

causal invariance, the system in a sense always has a unique causal history.

362



The interpretation of this causal history in terms of a spacetime history, however, depends 

on what amount to definitions made by an observer. In particular, to define what can be 

interpreted as a time coordinate, one must set up a foliation of the causal graph, with 

successive slices corresponding to successive steps in time. 

There are many such foliations that can be set up. The only fundamental constraint is that 
events in a given slice cannot be directly connected by an edge in the causal graph—or, in 

other words, they must be spacelike separated. The possible foliations thus correspond to 

possible sequences of spacelike hypersurfaces, analogous to those in standard discussions of 
spacetime.

(Note that the causal graph ultimately just defines a partial order on the set of events, and 

one could in principle imagine having arbitrarily complex foliations set up to imply any 

given total order of events. But such foliations are not realistic for macroscopic observers 

with bounded computational resources, and in our analysis of observable continuum limits 

we can ignore them.)

When one reaches a particular spacelike hypersurface, it represents a particular set of 
events having occurred, and thus a particular state of the underlying system having been 

reached, represented by a particular hypergraph. Different sequences of spacelike hypersur‐
faces thus correspond to different sequences of “instantaneous states” having been 

reached—corresponding to different evolution histories. But the crucial point is that causal 
invariance implies that even though the sequences of instantaneous states are different, the 

causal graphs representing the causal relationships between events that occur in them are 

always the same. And this is the essence of how the phenomena of relativistic invariance—
and general covariance—are achieved. 

8.6  Motion and Special Relativity
In the traditional formalism of physics, the principles of special relativity are in a sense 

introduced as axioms, and then their consequences are derived. In our models, what 
amount to these principles can in effect emerge directly from the models themselves, 
without having to be introduced from outside. 

To see how this works, consider the phenomenon of motion. In standard physics, one thinks 

of different states of uniform motion as corresponding to different inertial reference frames 

(e.g. [111][112]). These different reference frames in turn correspond to different choices of 
sequences of spacelike hypersurfaces, or, in our setup, different foliations of the causal graph.

363



As a simple example, consider the string substitution system BA→AB, starting from 

...BABABA... The causal graph for the evolution of this system can be drawn as a grid:

A simple foliation is just to form successive layers:

With this foliation, the sequence of states in the underlying string substitution system is:

B A B A B A B A B A B A B A B A B A B A

A B A B A B A B A B A B A B A B A B A B

A A B A B A B A B A B A B A B A B A B B

A A A B A B A B A B A B A B A B A B B B

A A A A B A B A B A B A B A B A B B B B

A A A A A B A B A B A B A B A B B B B B

A A A A A A B A B A B A B A B B B B B B

A A A A A A A B A B A B A B B B B B B B

A A A A A A A A B A B A B B B B B B B B

A A A A A A A A A B A B B B B B B B B B

A A A A A A A A A A B B B B B B B B B B

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In drawing our foliation of the causal graph, we can think of time as being vertical, and 

space horizontal. Now imagine we want to represent uniform motion. We can do this by 

making our foliation use slices with a slope proportional to velocity: 

But imagine we want to show time vertically, while not destroying the partial order in our 

causal network. The unique way to do it (if we want to preserve straight lines) is to transform 

a point {t, x} to {t – βx, x – βt}/ 1 – β2 :

But this is precisely the usual Lorentz transformation of special relativity. And time dilation 

is then, for example, associated with the fact that to reach what corresponds to an event at 
slice t in the original foliation, one now has to go through a sequence of events that is longer 

by a factor of γ = 1/ 1 – β2 . 

Normally one would argue for these results on the basis of principles supplied by special 
relativity. But the crucial point here is that in our models the results can be derived purely 

from the behavior of the models, without introducing additional principles. 

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Imagine simply using the transformed causal graph to determine the order of updating 

events in the underlying substitution system:

B A B A B A B A B A B A B A B A B A B A

A B B A B A B A B A B A B A B A B A B A

A B A B A B B A B A B A B A B A B A B A

A A B B A B A B B A B A B A B A B A B A

A A B A B B A B A B A B B A B A B A B A

A A B A B A B A B B A B A B A B B A B A

A A A B A B B A B A B B A B A B A B B A

A A A B A B A B B A B A B A B B A B A B

A A A A B B A B A B A B B A B A B A B B

A A A A B A B A B B A B A B B A B A B B

A A A A B A B A B A B B A B A B A B B B

A A A A A B A B B A B A B A B B A B B B

A A A A A B A B A B B A B A B A B B B B

A A A A A A B B A B A B A B B A B B B B

A A A A A A B A B A B B A B A B B B B B

A A A A A A A B B A B A B B A B B B B B

A A A A A A A B A B B A B A B B B B B B

A A A A A A A B A B A B A B B B B B B B

A A A A A A A A B B A B A B B B B B B B

A A A A A A A A B A B A B B B B B B B B

A A A A A A A A A B B A B B B B B B B B

A A A A A A A A A B A B B B B B B B B B

A A A A A A A A A A B B B B B B B B B B

If we look vertically down the picture we see a different sequence of states of the system. But 
the crucial point is that the final outcome of the evolution is exactly the same as it was with 

the original foliation. In some sense the “physics” is the same, independent of the reference 

frame. And this is the essence of relativistic invariance (and here we immediately see some 

of its consequences, like time dilation). 

But in the context of the string substitution system, we can now see its origin of the invari‐
ance. It is the fact that the underlying rule we have used is causal invariant, so that regard‐
less of the specific order in which updating events occur, the same causal graph is obtained, 
with the same final output.

In our actual models based on infinitely evolving hypergraphs, the details are considerably more 

complicated. But the principles are exactly the same: if the underlying rule has causal invariance, 
its limiting behavior will show relativistic invariance, and (so long as it has limiting geometry 

corresponding to flat d‐dimensional space) all the usual phenomena of special relativity. 

(Note that the concept of a finite speed of light, leading effectively to locality in the causal 
graph, is related to the fact that the underlying rules involve rewriting hypergraphs only 

of bounded size.)

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8.7  The Vacuum Einstein Equations
In discussing the structure of space, we considered how the volumes of geodesic balls grow 

with radius. In discussing spacetime, we want to consider the analogous question of how the 

volumes of light cones [1:p1052]) grow with time. But to do this, we have to say what we 

mean by time, since—as we saw in the previous subsection—different foliations can lead to 

different identifications. 

Any particular foliation—with its sequence of spacelike hypersurfaces—provides at every 

point a timelike vector that defines a time direction in spacetime. So if we start at any point 
in the causal graph, we can look at the forward light cone from this point, and follow the 

connections in the causal graph until we have gone a proper time t in the time direction we 

have defined. Then we can ask how how many nodes we have reached in the causal graph.

The result will depend on the underlying rule for the system. But if in the limit it is going to 

correspond to flat (d + 1)‐dimensional spacetime, at any spacetime position X it must grow like:

Ct(X) ~ td+1

If we include the possibility of curvature, we get to first order

1
Ct(X) ~ td+1(1 – δtμ δtν Rμν (X) + ...)

6
where Rμν  is the spacetime Ricci tensor, and δtμ δtν Rμν is effectively its projection along the 

infinitesimal timelike vector δtμ. 

For any particular underlying rule, Ct(X) will take on a definite form. But in making connec‐
tions with traditional continuum spacetime, we are interested in its limiting behavior. 

Assume, to begin, that we have scaled t to be measured relative to the size of the whole 

causal graph. Then for small t we can expand Ct(X) to get the expression involving curvature 

above. But now imagine scaling up t. Eventually it is inevitable that the curvature term has 

the potential to affect the overall t dependence, and potentially change the effective expo‐
nent of t. But if the overall continuum limit is going to correspond to a (d + 1)‐dimensional 
spacetime, this cannot happen. And what this means is that at least a suitably averaged 

version of the curvature term must not in fact grow [1:9.15].

The details are slightly complicated [113], but suffice  it to say here that the constraint on Rμν 

is obtained by averaging over directions, then averaging over positions with a weighting 

determined by the volume element g  associated with the metric gμν defined by our choice 

of hypersurfaces. The requirement that this average not grow when t is scaled up can then 

be expressed as the vanishing of the variation of ∫ R g , which is precisely the usual 

367



Einstein–Hilbert action—thereby leading to the conclusion that Rμν must satisfy exactly the 

usual vacuum Einstein equations [114][115][75][116]: 

1
Rμν – R gμν = 0

2
A full derivation of this is given in [113]. Causal invariance plays a crucial role, ensuring for 
example that timelike directions ti associated with different foliations give invariant results. 
Much like in the derivation of continuum fluid behavior from microscopic molecular dynamics 
(e.g. [110]), one also needs to take a variety of fairly subtle limits, and one needs sufficient 
intrinsic generation of effective randomness [1:7.5] to justify the use of certain statistical averages.

But there is a fairly simple interpretation of the result above. Imagine all the geodesics that 
start at a particular point in the causal graph. The further we go, the more possible geodesic 

paths there will be in the graph. To achieve a power law corresponding to a definite dimen‐
sion, the geodesics must in a sense just “stream outwards”, evenly distributed in direction.

But the Ricci tensor specifically measures the rate at which bundles of geodesics change 

their cross‐sectional area. And as soon as this change is nonzero, it will inevitably change 

the local density of geodesics and eventually grow to disrupt the power law. And so the only 

way a fixed limiting dimension can be achieved is for the Ricci curvature to vanish, just as it 
does according to the vacuum Einstein equations. (Note that higher‐order terms, involving 

for example the Weyl tensor and other components of the Riemann tensor, yield changes in 

the shape of bundles of geodesics, but not in their cross‐sectional area, and are therefore not 
constrained by the requirement of fixed limiting dimension.)

8.8  Matter, Energy and Gravitation
In our models, not only space, but also everything “in space”, must be represented by 

features of our evolving hypergraphs. There is no notion of “empty space”, with “matter” in 

it. Instead, space itself is a dynamic construct created and maintained by ongoing updating 

events in the hypergraph. And what we call “matter”—as well as things like energy—must 
just correspond to features of the evolving hypergraph that somehow deviate from the 

background activity that we call “space”.

Anything we directly observe must ultimately have a signature in the causal graph. And a 

potential hypothesis about energy and momentum is that they may simply correspond to 

excess “fluxes” of causal edges in time and space. Consider a simple causal graph in which 

we have marked spacelike and timelike hypersurfaces:

368



 , 

The basic idea is that the number of causal edges that cross spacelike hypersurfaces would 

correspond to energy, and the number that cross timelike hypersurfaces would correspond 

to momentum (in the spatial direction defined by a given hypersurface). Inevitably the 

results one gets would depend on the hypersurfaces one chooses, and so would differ from 

one observer to another.

And one important feature of this identification of energy and momentum is that it would 

explain why they follow the same relativistic transformations as time and space. In effect 
space and time are probing distances between nodes in the causal graph (as measured 

relative to a particular foliation), while momentum and energy are probing a directly dual 
property: the density of edges.

There is additional subtlety here, though, because causal edges are needed just to maintain 

the structure of spacetime—and whatever we measure as energy and momentum must just 
be some excess in the density of causal edges over the “background” corresponding to 

space. But even to know what we mean by density we have to have some notion of volume, 
but this is also itself defined in terms of edges in the causal graph.

But as a rough idealized picture, we might imagine that we have a causal graph that main‐
tains the same overall structure, but adds some extra connections:

 , 

In our actual models, the causal graphs one gets are considerably more complicated. But 
one can still identify some features from the simple idealization. The basic concept is that 
energy and momentum add “extra causal connections” that are not “necessary” to define the 

basic structure of spacetime. In a sense the core thing that defines the structure of space‐
time is the way that “elementary light cones” are knitted together. 

369



Consider a causal graph like:
1

3 2

5 6 4

11 7 9 10 12 8

21 13 14 18 20 15 19 17 22 16

39 24 23 36 27 25 34 37 28 38 26 32 35 29 33 40 30 31

One can think of a set of edges like the ones indicated as in effect “outlining” the causal 
graph. But then there are other edges that add “extra connections”. The edges that “outline 

the graph” in effect maximally connect spatially separated regions—or in a sense transmit 
causal information at a maximum speed. The other edges one can think of as having slower 

speeds—so they are typically drawn closer to vertical in a rendering like the one above.

But now let us return to our simple grid idealization of the causal graph—with additional 
vertical edges added. Now do foliations like the ones we used above to represent inertial 
frames, parametrized by a velocity ratio β relative to the maximum speed (taken to be 1). 
Define E(β) to be the density of causal edge crossing of the spacelike hypersurfaces, and p(β) 
the corresponding quantity for timelike hypersurfaces. Then for speed 1 edges, we have (up 

to an overall multiplier) (cf. [111][112]):

1 + β
E(β) = p(β) =

1 - β2

But in general for edges with speed α we have

1 – α β α – β
E(β) = , p(β) =

1 – β2 1 – β2

which means that for any β

E (β)2 – p (β)2 = 1 – α2

thus showing that our crossing densities transform like energy and momentum for a particle 

with mass 1 – α2 . In other words, we can potentially identify edges that are not maximum 

speed in the causal graph as corresponding to “matter” with nonzero rest mass. Perhaps not 
surprisingly, this whole setup is quite analogous to thinking about world lines of massive 

particles in elementary treatments of relativity.

370



But in our context, all of this must emerge from underlying features of the evolving hyper‐
graph. Causal connections that transfer information at maximum speed can be thought of as 

arising from updating events that involve maximally separate nodes, and that are somehow 

always entraining “fresh” nodes. But causal connections that transfer information more slowly 

are associated with sequences of updating events that in effect reuse nodes. So in other words, 
rest mass can be thought of as being associated with local collections of nodes in the hyper‐
graph that allow repeated updating events to occur without the involvement of other nodes.

Given this setup, it is possible to derive other features of energy, momentum and mass by 

methods similar to those used in typical discussions of relativity. It is first helpful to include units 
in the quantities we have introduced. If an elementary light cone has timeline extent T then we 

can consider its spacelike extent to be c T, where c is the speed of light. Within the light cone let 
us say that there effectively μ causal edges oriented in the timelike direction. With the inertial 
frame foliations used above, the contribution of these causal edges to energy and momentum will 
be (the factor c in the energy case comes from the spacelike extent of the light cone):

μ β2
E(β) = c = c μ (1 + + O(β4))

1 – β2 2
β

p(β) = μ = μ β ( 1 + O(β2) )
1 – β2

But if we define the mass m as μ
 and substitute β = v

 , we get the standard formulas of 
c c

special relativity [111][112], or to first order

1
E = m c2 + m v2

2
p = m v

establishing in our model the relation E = m c2 between energy and rest mass.

We should note that with our identification for energy and momentum, the conservation of 
energy becomes essentially the statement that the overall density of events in the causal 
network does not change as we progress through successive spacelike surfaces. And, as we 

will discuss later, if in effect the whole hypergraph is in some kind of dynamic equilibrium, 
then we can reasonably expect that this will be the case. Expansion (or, more specifically, 
non‐uniform expansion) will lead to effective violations of energy conservation, much as it 
does for an expanding universe in the traditional formalism of general relativity [117][75].

In the previous subsection, we discussed the overall structure of spacetime, and we used the 

growth rate of the spacetime volume Ct(X) as a way to assess this. But now let us ask about 
specific values of Ct(X), complete with their “constant” multipliers. We can think of these 

multipliers as probing the local density of the causal graph. But deviations in this are what 
we have now identified as being associated with matter.

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To compute Ct(X) we ultimately need to be able to precisely count events in the causal 
graph. If the causal graph is somehow “uniform”, then it cannot contain what can be 

considered to be “matter”. In the setup we have defined, the presence of matter is effectively 

associated with “fluxes” of causal edges that reflect the non‐uniform “arrangement” of nodes 

in the causal graph. To represent this, take ρ (X) to be the “local density” of nodes in the 

causal graph. We can make a series expansion to probe deviations from uniformity in ρ (X). 
And formally we can write

ρ(X) = ρ0 ( 1 + σ δtμ δtν Tμν ...)

where the tμ are timelike vectors used in the definition of Ct and now Tμν is effectively a 

tensor that represents “fluxes of edges” in the causal graph. But these fluxes are what we 

have identified as energy and momentum, and when we think about how causal edges 

traverse spacelike and timelike hypersurfaces, Tμν turns out to correspond exactly to the 

standard energy‐momentum tensor of general relativity.

So now we can combine our formula for the effect of local density with our formula for the 

effect of curvature from the previous section to get:

1
Ct(X) = ρ0(1 + σ δti δtj Tij+ ... ) td+1(1 – δti δtj Rij + ...)

6
But if we apply the same argument as in the previous subsection, then to maintain limiting 

fixed dimension we get the condition

1
Rμν – R gμν = σ′ Tμν

2
which has exactly the form of Einsteinʼs equations in the presence of matter 

[114][115][75][116].

Just as we interpreted the curvature part of these equations in the previous subsection in 

terms of the change in area of geodesic bundles, we can interpret the “matter” part in terms 

of the change of geodesics associated with additional local connections. As an example, 
consider starting with a 2D hexagonal grid. Now imagine adding edges at each node. Doing 

this creates additional connections and additional geodesics, eventually producing some‐
thing like the hyperbolic space examples in 4.2. So what the equation says is that any such 

effect, which would lead to negative curvature, must be compensated by positive curvature 

in the “background” spacetime—just as general relativity suggests.

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8.9  Elementary Particles
Elementary particles are entities that—at least for some period—preserve their identity through 

space and time. In the context of our models, one can imagine that particles would correspond 

to structures in the hypergraph that are locally stable under the application of rules. 

As an idealized example, consider rules that operate on an ordinary graph, and have the 

property of preserving planarity. Such rules can never remove non‐planarity from a graph. 
But it is a basic result of graph theory [37][118] that any non‐planarity can always be 

attributed to one of the two specific subgraphs:

 , 

If one inserts such subgraphs into an otherwise planar graph, they behave very much as 

“particle‐like” structures. They can move around, but unless they meet and annihilate, they 

are preserved:

There are presumably analogs of this in hypergraph‐rewriting rules of the kind that appear 

in our models. Given a particular set of rules, the expectation would be that a certain set of 
local sub‐hypergraphs would be preserved by the rules. Existing results in graph theory do 

not go very far in elucidating the details.

However, there are analogs in other systems that provide some insight. Cellular automata 

provide a particularly good example. Consider the rule 110 cellular automaton [1:p32]. 
Starting from a random initial condition, the picture below shows how the system evolves to 

a collection of localized structures:

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The form of these structures is hard to determine directly from the rule. (They are a little
like hard-to-predict solutions to a Diophantine equation.) But by explicit computation one
can determine for example that rule 110 supports the following types of localized structures
[1:p292][119]

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as well as the growing structure:

There is a complex web of possible interactions between localized structures, that can at
least in some cases be interpreted in terms of conservation laws:

 , , ,

, , 

As in cellular automata, it is likely that not every one of our models will yield localized
structures, although there is reason to think that some form of conserved structure will be
more common in hypergraph rewriting than in cellular automata. But as in cellular
automata, one can expect that with a given underlying rule, there will be a discrete set of
possible localized structures, with hard-to-predict sizes and properties.

375



The particular set of localized structures will probably be quite specific to particular rules. 
But as we will discuss in the next subsection, there will o�en be symmetries that cause 

collections of similar structures to exist—or in fact force certain structures to exist.

In the previous subsection, we discussed the interpretation of energy and momentum in 

terms of additional edges in a causal graph. For particles, the expectation would be that 
there is a certain “core” structure that defines the core properties of a particle (like spin, 
charge, etc.), but that this structure is spread across a region of the hypergraph that main‐
tains the “activity” associated with energy and momentum.

It is worth noting that even in an example like non‐planarity, it is perfectly possible for 

topological‐like features to effectively be spread out across many nodes, while still maintain‐
ing their discrete character.

In the previous subsection, we discussed the potential origin of rest mass in terms of “reuse” 
of nodes in the hypergraph. Once again, this seems to fit in well with our notion of the 

nature of particles—and to make it perfectly possible to imagine both “massive” and “mas‐
sless” particles, associated with different kinds of structures in the evolving hypergraph.

In a system like the rule 110 cellular automaton, there is a clear “background structure” on 

which it is possible to identify localized structures. In some very simple cases, similar things 

happen in our models. For example, consider the rule:

{{x, y}, {y, z, u, v}} → {{x, y, z, u}, {u, v}}

The evolution of this rule yields behavior like

 , , , , , , , , ,

, , , , , , , , , , ,

, , , , , , , , , , 

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in which there is a circular “background”, with a localized “particle‐like” deformation. The 

causal graph (here generated for a larger case) also shows evidence of a particle‐like struc‐
ture on a simple grid‐like background:

But in most of our models the “background” tends to be much more complicated, and so 

such direct methods for identifying particles cannot be used. But as an alternative, one can 

consider exploring the effect of perturbations, as in 4.14. In effect, one starts the system 

with a perturbation, then sees whether the perturbation somehow decomposes into quan‐
tized elements that one can identify as “particles”. (The process is quite analogous to 

producing particles in a high‐energy collision.) 

Such quantized effects are at best rare in class 3 cellular automata, but they are a defining 

feature of class 4 cellular automata, and there is reason to believe that they will be at least 
fairly common in our models. 

The defining feature of a localized “particle‐like” structure is that it is capable of long‐range 

propagation in the system. But the presence of even short‐lived instances of particle‐like 

structures will also potentially be identifiable—though with a certain margin of error—from 

detailed properties of the hypergraph in small regions. And in the “background” evolution of 
our models, one can expect that short‐lived instances of particle‐like structures will continu‐
ally be being created and destroyed.

The process that in a sense “creates the structure of space” in our models can thus also be 

thought of as producing a “vacuum” full of particle‐like activity. And particularly when this 

is combined with the phenomenon (to be discussed in a later subsection) that pairs of 
particle‐like structures can be produced and subsequently merged in the multiway system, 
there is some definite similarity with the ubiquitous virtual particles that appear in tradi‐
tional treatments of quantum field theory.

377



8.10  Reversibility and Irreversibility
One feature of the traditional formalism for fundamental physics is that it is reversible, in 

the sense that it implies that individual states of closed systems can be uniquely evolved 

both forward and backward in time. (Time reversal violation in things like Ko particle decays 

show that the rule for going forward and backward in time can be slightly different. In 

addition, the cosmological expansion of the universe defines an overall arrow of time.) 

One can certainly set up manifestly reversible rewriting rules (like A→B, B→A) in models 

like ours. And indeed the example of cellular automata [1:9.2] tends to suggest that most 
kinds of behavior seen in irreversible rules can also be seen—though perhaps more rarely—
in reversible rules. 

But it is important to realize that even when the underlying rules for a system are not 
reversible, the system can still evolve to a situation where there is effective reversibility. One 

way for this to happen is for the evolution of the system to lead to a particular set of “attra‐
ctor” states, on which the evolution is reversible. Another possibility is that there is no such 

well‐defined attractor, but that the system nevertheless evolves to some kind of “equilibr‐
ium” in which measurable effects show effective reversibility.

In our models, there is an additional complication: the fact that different possible updating 

orders lead to following different branches of the multiway system. In most kinds of sys‐
tems, irreversible rules tend to be associated with the phenomenon of multiple initial states 

merging to produce a single final state in which the information about the initial state is lost. 
But when there is a branch in a multiway system, this is reversed: information is effectively 

created by the branch, and lost if one goes backwards.

When there is causal invariance, however, yet something different happens. Because now in 

a sense every branching will eventually merge. And what this means is that in the multiway 

system there is a kind of reversibility: any information created by a branching will always be 

destroyed again when the branches merge—even though temporarily the “information 

content” may change.

It is important to note that this kind of microscopic reversibility is quite unrelated to the 

more macroscopic irreversibility implied by the Second Law of thermodynamics. As dis‐
cussed in [1:9.3] the Second Law seems to first and foremost be a consequence of computa‐
tional irreducibility. Even when the underlying rules for a system are reversible, the actual 
evolution of the system can so “encrypt” the initial conditions that no computationally 

feasible measurement process will succeed in reconstructing them. (The idea of considering 

computational feasibility clarifies past uncertainty about what might count as a reasonable 

“coarse graining procedure”.)

378



In any nontrivial example of one of our models, computational irreducibility is essentially 

inevitable. And this means that the model will tend to intrinsically generate effective 

randomness, or in other words, the computation it does will obscure whatever simplicity 

might have existed in its initial conditions. 

There can still be large‐scale features—or particle‐like structures—that persist. But the 

presence of computational irreducibility implies that even at a level as low as the basic 

structure of space we can expect our models to show the kind of irreversibility associated 

with the Second Law. And in a sense we can view this as the reason that things like a robust 
structure for space can exist: because of computational irreducibility, our models show a 

kind of equilibrium in which the details are effectively random, and the only features that 
are computationally feasible to measure are the statistical regularities.

8.11  Cosmology, Expansion & Singularities
In our models the evolving hypergraph represents the whole universe, and the expansion of 
the universe is potentially a consequence of the growth of the hypergraph. In the minimal 
case of a model involving a single transformation rule, the growth of the hypergraph must 
be monotonic, although the rate can vary depending on the local structure of the hyper‐
graph. If there are multiple transformation rules, there can be both increase and decrease in 

hypergraph size. (Even with a single rule, there is also still the possibility—discussed below—
of effective size decrease as a result of pieces of the hypergraph becoming disconnected.)

In the case of uniform growth, measurable quantities such as length and energy would 

essentially all continually scale as the universe evolves. The core structure of particles—
embodied for example in topological‐like features of the hypergraph—could potentially 

persist even as the number of nodes “within them” increases. Since the rate of increase in 

size in the hypergraph would undoubtedly greatly exceed the measurable growth rate of 
the universe, uniform growth implies a kind of progressive refinement in which the 

length scale of the discrete structure of the hypergraph becomes ever more distant from 

any given measured length scale—so that in effect the universe is becoming an ever closer 

approximation to continuous.

In traditional cosmology, one thinks of the universe as effectively having exactly three 

dimensions of space (cf. [120]). In our models, dimension is in effect a dynamical variable. 
Possibly some of what is normally attributed to curvature in space can instead be reformu‐
lated as dimension change. But even beyond this, there is the potential for new phenomena 

associated, for example, with local change of dimension. In general, a change of dimen‐
sion—like curvature—affects the density of geodesics. Changes of dimension generated by 

an underlying rule may potentially lead to effects that for example mimic the presence of 
mass, or positive or negative energy density. (There could also be dimension‐change 

“waves”, perhaps with some rather unusual features.)

379



In our models, the universe starts from some initial configuration. It could be something 

like a single self‐loop hypergraph. Or in the multiway system it could be multiple initial 
hypergraphs. (Note that we can always “put the initial conditions into the rule” by adding a 

rule that says “from nothing, create the initial conditions”.) 

An obvious question is whether any traces of the initial conditions might persist, perhaps 

even through the whole evolution of the system. The effective randomness associated with 

computational irreducibility in the evolution will inevitably tend to “encrypt” most features 

of the initial conditions [1:9.3] to the point where they are unrecognizable. But it is still 
conceivable that, for example, some global symmetry breaking associated with the first few 

hypergraph updating events could survive—and the remote possibility exists that this could 

be visible today in the large‐scale structure of the universe, say as a pattern of density 

fluctuations in the cosmic microwave background.

Our models have potentially important implications for the early universe. If, for example, 
the effective dimension of the universe was initially much higher than 3 (as is basically 

inevitable if the initial conditions are small), there will have been a much higher level of 
causal contact between different parts of the universe than we have deduced by extrapolating 

the 3D expansion of the universe today [1:p1055]. (In effect this happens because the volume 

of the past light cone will grow like td—or perhaps exponentially with t—and not just like t3.)

As we discussed in 2.9, it is perfectly possible in our models for parts of the hypergraph to 

become disconnected as a result of the operation of the rule. But assuming that the rule is 

local (in the sense that its le�‐hand side is a connected hypergraph), pieces of the hyper‐
graph that become disconnected can never interact again. Even independent of outright 
disconnection of the spatial graph, it is also possible for the causal graph to “tear” into 

disconnected parts that can never interact again (see 6.10):

A disconnection in the causal graph corresponds to an event horizon in our system—that 
cannot be crossed by any timelike curve. (And indeed our causal graphs—consisting as they 

do of “elementary light cones knitted together”—are like microscopic analogs of the causal 
diagrams o�en used in studying general relativity.)

380



We can also ask about other extreme phenomena in spacetime. Closed timelike curves 

correspond to loops in the causal graph, and with some rules they can occur. But they do not 
represent any real form of “time travel”; they just correspond to the presence of states that 
are precisely repeated as a result of the evolution of the system. (Note that in our models, 
time effectively corresponds to the progression of computation, and has a very different 
underlying character from something like space.)

Wormholes and effective faster‐than‐light travel are not specifically excluded by the structure of 
our models, especially insofar as there can potentially be deviations in the effective local dimen‐
sionality of space. But insofar as the conditions to get general relativity as a limiting effective 

theory are satisfied, these will occur only in the circumstances where they do in that theory.

8.12  Basic Concepts of Quantum Mechanics
Quantum mechanics is a key known feature of physics, and also, it seems, a natural and 

inevitable feature of our models. In classical physics—or in a system like a cellular automa‐
ton—one basically has rules that specify a unique path of history for the evolution of a 

system. But our models are not set up to define any such unique path of history. Instead, the 

models just give possible rewrites that can be performed on hypergraphs—but they do not 
say when or where these rewrites should be applied. So this means that—like the formalism 

of quantum mechanics—our models in a sense allow many different paths of history.

There is, however, ultimately nothing non‐deterministic about our models. Although they 

allow many different sequences of updating events—each of which can be viewed as a 

different path of history—the models still completely determine the overall set of possible 

sequences of updating events. And indeed at a global level, everything about the model can 

be captured in a multiway graph [1:5.6]—like the one below—with nodes in the graph 

corresponding to states of the system (here, for simplicity, a string substitution system), and 

every possible path through the graph corresponding to a possible history.

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

AAAA AAABB AABAB ABAAB ABABA AABBA ABBAA ABABBB ABBABB AABBBB ABBBAB ABBBBA ABBBBBB

381



In the standard formalism of quantum mechanics, one usually just imagines that all one can 

determine are probabilities for different histories or different outcomes. But this has made it 
something of a mystery why we have the impression that a definite objective reality seems to 

exist. One possible explanation would be that at some level a branch of reality exists for 

every possible behavior, and that we just experience the branch that our thread of conscious‐
ness has happened to follow. 

But our models immediately suggest another, more complete, and arguably much more 

scientifically satisfying, possibility. In essence, they suggest that there is ultimately a global 
objective reality, defined by the multiway system, and it is merely the locality of our experi‐
ence that causes us to describe things in terms of probabilities, and all the various detailed 

features of the standard formalism of quantum mechanics.

We will proceed in two stages. First, we will discuss the notion of an observer in the context 
of multiway systems, and the relation of this to questions about objective reality. And having 

done this, we will be in a position to discuss ideas like quantum measurement, and the role 

that causal invariance turns out to play in allowing observers to experience definite, seem‐
ingly classical results.

So how might we represent a quantum observer in our models? The first key point is that the 

observer—being part of the universe—must themselves be a multiway system. And in 

addition, everything the observer does—and experiences—must correspond to events that 
occur in the model.

This latter point also came up when we discussed spacetime—and we concluded there that it 
meant we only needed to consider the graph of causal relationships between events. To 

characterize any given observer, we then just had to say how the observer would sample this 

causal graph. A typical example in studying spacetime is to consider an observer in an inertial 
reference frame—which corresponds to a particular foliation of the causal graph. But in 

general to characterize what any observer will experience in the course of time, we need 

some sequence of spacelike hypersurfaces that form a foliation which respects the causal 
relationships—and thus the ordering relations between events—defined by the causal graph.

But now we can see an analog of this in the quantum mechanical case. However, instead of 
considering foliations of the causal graph, what we need to consider now are foliations of 
the multiway graph:

382



A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

AAAA AAABB AABAB ABAAB ABABA AABBA ABBAA ABABBB ABBABB AABBBB ABBBAB ABBBBA ABBBBBB

In the course of time, the observer progresses through such a foliation, in effect at each step 

observing some collection of states, with certain relationships between them. A different 
observer, however, might want to sample the states differently, and might effectively define 

a different foliation.

One can potentially think of a different foliation as being a different “quantum observation 

frame” or “quantum frame”, analogous to the different reference frames one considers in 

studying spacetime. In the case of something like an inertial frame, one is effectively 

defining how an observer will sample different parts of space over the course of time. In a 

quantum observation frame one might have a more elaborate specification, involving 

sampling particular states of relevance to some measurement or another. But the key point 
is that a quantum observer can in principle use any quantum observation frame that corre‐
sponds to a foliation that respects the relationships between states defined by the multiway 

graph (and thus has a meaningful notion of time). 

In both the spacetime case and the quantum case, the slices in the foliation are indexed by 

time. But while in the spacetime case, where each slice corresponds to a spacelike hypersur‐
face that spans ordinary space, in the quantum case, each slice corresponds to what we can 

call a branchlike hypersurface that spans not ordinary space, but instead the space of states, 
or the space of branches in the multiway system. But even without knowing the details of 
this space, we can already come to some conclusions.

In particular, we can ask what observers with different quantum observation frames—and 

thus different choices of branchlike hypersurfaces—will conclude about relationships 

between states. And the point is that so long as the foliations that are used respect the order‐
ings defined by the multiway graph, all observers must inevitably come to the same conclu‐
sions about the structure of the multiway graph—and therefore, for example, the relation‐
ships between states. Different observers may sample the multiway graph differently, and 

experience different histories, but they are always ultimately sampling the same graph.

383



One feature of traditional quantum formalism is its concept of making measurements that 
effectively reduce collections of states—as exist in a multiway system—to what is basically a 

single state analogous to what would be seen in a classical single path of evolution. From the 

point of view of quantum observation frames, one can think of such a measurement as being 

achieved by sculpting the quantum observation frame to effectively pick out a single state in 

the multiway system:

AA

AAB ABA

AABB ABAB ABBA

AABBB ABABB ABBAB ABBBA

AABBBB ABABBB ABBABB ABBBAB ABBBBA

AABBBBB ABABBBB ABBABBB ABBBABB ABBBBAB ABBBBBA

AABBBBBB ABABBBBB ABBABBBB ABBBABBB ABBBBABB ABBBBBAB ABBBBBBA

AABBBBBBB ABABBBBBB ABBABBBBB ABBBABBBB ABBBBABBB ABBBBBABB ABBBBBBAB ABBBBBBBA

We will discuss this in more detail below. But the basic idea is as follows. Imagine that our 
universe is based on a simple string substitution system such as {A→AB}. If we start from a state 

AA, as in the picture above, the multiway evolution from this state immediately leads to multi‐
ple outcomes, associated with different updating events. But let us say that we just wanted some 

kind of “classical” summary of the evolution, ignoring all these different branches. 

One thing we might do is not trace individual updates, but instead just look at “generational 
states” (5.21) in which all updates that can consistently be applied together have been 

applied. And with the particular rule shown here, we then get the unique sequence of states 

highlighted above. And as we will discuss below, we can indeed consider these generational 
states as corresponding to definite (“classical‐like”) states of the system, that can consis‐
tently be thought of as potential results of measurements. 

But now let us imagine how this might work in something closer to a complete experiment. 
We are running the multiway system shown above. Multiple states are being generated. But 

384



 multiway system
at some moment we as observers notice that actually several states that have been produced 

(say ABBA and AABB) can be combined together to form a consistent generational state 

(ABBABB). But even though these states ultimately had a common ancestor, they now seem 

to be on different “branches of history”.

But now causal invariance makes a crucial contribution. Because it implies that all such 

different branches must eventually converge. And indeed a�er a couple of steps, the fully 

assembled generational state ABBABB appears in the multiway system. To us as observers 

this is in a sense the state we were looking for (it is the “result of our measurement”), and as 

far as possible, we want to use it as our description of the system. 

And by setting up an appropriate quantum observation frame, that is exactly what we can 

do. For example, as illustrated in the picture above, we can make the foliation we choose 

effectively freeze the generational state, so that in the description we use of the system, the 

state stays the same in successive slices. 

The structure of the multiway system puts constraints on what foliations we can consistently 

set up. In the case shown here, it does allow us to freeze this particular state forever, but to 

do this consistently, it effectively forces us to freeze more and more states over time. And as 

we will see later, this kind of spreading of effects in the multiway graph is closely related to 

decoherence in the standard formalism of quantum mechanics.

In what we just discussed, causal invariance is what guarantees that states the observer 

notices can consistently be assembled to form a generational (“classical‐like”) state that will 
always actually converge in the multiway system to form that state. But it is worth pointing 

out that (as discussed in [121]) strict causal invariance is not ultimately needed for a picture
like this to work. 

Recall that the observer themselves is also a multiway system. So “within their conscious‐
ness” there will usually be many “simultaneous” states. Looked at formally from the outside, 
the observer can be seen to involve many distinct states. But one could imagine that the 

internal experience of the observer would be in effect to conflate these states. 

Causal invariance ensures that branches in the multiway system will actually merge—just as 

a result of the evolution of the multiway system. But if the observer “experientially” con‐
flates states, this in effect represents an additional way in which different branches in the 

multiway system will at least appear to merge [121]. Formally, one can think of this—in 

analogy to the operation of automated theorem‐proving systems—as like the observer 

“adding lemmas” that assert the equivalence of branches, thereby allowing the system to be 

“completed” to the point where relevant branches converge. (For a given system, there is 

still the question of whether only a sufficiently  bounded number of lemmas is needed to 

achieve the convergence one wants.)

Independent of whether there is strict causal invariance or not, there is also the question of 
what kinds of quantum observation frames are possible. In the end—just like in the space‐

385



 end—just
time case—such frames reflect the description one is choosing to make of the world. And 

setting up different “coordinates”, one is effectively changing oneʼs description, and picking 

out different aspects of a system. And ultimately the restrictions on frames are computa‐
tional ones. Something like an inertial frame in spacetime is simple to describe, and its 

coordinates are simple to compute. But a frame that tries to pick out some very particular 

aspect of a quantum system may run into issues of computational irreducibility. And as a 

result, much as happens in connection with the Second Law of thermodynamics [1:9.3], 
there can still for example be elaborate correlations that exist between different parts of a 

quantum system, but no realistic measurement—defined by a computationally feasible 

quantum observation frame—will succeed in picking them out.

8.13  Quantum Formalism
To continue understanding how our models might relate to quantum mechanics, it is useful to 

describe a little more of the potential correspondence with standard quantum formalism. We 

consider—quite directly—each state in the multiway system as some quantum basis state |S>.

An important feature of quantum states is the phenomenon of entanglement—which is 

effectively a phenomenon of connection or correlation between states. In our setup (as we 

will see more formally soon), entanglement is basically a reflection of common ancestry of 
states in the multiway graph. (“Interference” can then be seen as a reflection of merging—
and therefore common successors—in the multiway graph.)

Consider the following multiway graph for a string substitution system:

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

Each pair of states generated by a branching in this graph are considered to be entangled. 
And when the graph is viewed as defining a rewrite system, these pairs of states can also be 

said to form a branch pair. 

386



Given a particular foliation of the multiway graph, we can now capture the entanglement of 
states in each slice of the foliation by forming a branchial graph in which we connect the 

states in each branch pair. For the string substitution system above, the sequence of 
branchial graphs is then:

ABA

 ABB AA , ,

AAB ABBB

AABBB AAAB

ABBA

ABABB

AAA ABAB ABBBB , ABBBBB AABA 

ABBAB

AABB

ABBBA ABAA

In physical terms, the nodes of the branchial graph are quantum states, and the graph itself 
forms a kind of map of entanglements between states. In general terms, we expect states 

that are closer on the branchial graph to be more correlated, and have more entanglement, 
than ones further away.

As we discussed in 5.17, the geometry of branchial space is not expected to be like the 

geometry of ordinary space. For example, it will not typically correspond to a finite‐dimen‐
sional manifold. We can still think of it as a space of some kind that is reached in the limit of 
a sufficiently  large multiway system, with a sufficiently  large number of states. And in 

particular we can imagine—for any given foliation—defining coordinates of some kind on it, 
that we will denote b<. So this means that within a foliation, any state that appears in the 

multiway system can be assigned a position (t, b< ) in “multiway space”.

In the standard formalism of quantum mechanics, states are thought of as vectors in a 

Hilbert space, and now these vectors can be made explicit as corresponding to positions 

in multiway space. 

But now there is an additional issue. The multiway system should represent not just all 
possible states, but also all possible paths leading to states. And this means that we must 
assign to states a weight that reflects the number of possible paths that can lead to them:

387



1
A

1
AB

1 1
AA ABB

2 2 1
AAB ABA ABBB

4 3 5 3 1
AAA AABB ABAB ABBA ABBBB

12 10 12 9 4 9 4 1
AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

In effect, therefore, each branchlike hypersurface can be thought of as exposing some linear 

combination of basic states, each one with a certain weight:

2 3

1 1, , 4 5
 1,

2 1 3

12
4

9

1 10 , , 
9

4 12

Let us say that we want to track what happens to some part of this branchlike hypersurface. 
Each state undergoes updating events that are represented by edges in the multiway graph. 
And in general the paths followed in the multiway graph can be thought of as geodesics in 

multiway space. And to determine what happens to some part of the branchlike hypersur‐
face, we must then follow a bundle of geodesics.

A notable feature of the multiway graph is the presence of branching and merging, and this 

will cause our bundle of geodesics to diverge and converge. O�en in standard quantum 

formalism we are interested in the projection of one quantum state on another < | >. In our 

388



 projection |
setup, the only truly meaningful computation is of the propagation of a geodesic bundle. But 
as an approximation to this that should be satisfactory in an appropriate limit, we can use 

distance between states in multiway space, and computing this in terms of the vectors 

ξi = (t >
i, bi ) the expected 2

 Hilbert space norm [122][123] appears: (ξ1 – ξ2)2 = ξ1 + ξ22 – 2 ξ1.ξ2.

Time evolution in our system is effectively the propagation of geodesics through the multi‐
way graph. And to work out a transition amplitude <i | S | f> between initial and final states 

we need to see what happens to a bundle of geodesics that correspond to the initial state as 

they propagate through the multiway graph. And in particular we want to know the measure 

(or essentially cross‐sectional area) of the geodesic bundle when it intersects the branchlike 

hypersurface defined by a certain quantum observation frame to detect the final state.

To analyze this, consider a single path in the multiway system, corresponding to a single 

geodesic. The critical observation is that this path is effectively “turned” in multiway space 

every time a branching event occurs, essentially just like in the simple example below:

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

If we think of the turns as being through an angle θ , the way the trajectory projects onto the 

final branchlike hypersurface can then be represented by ei θ. But to work out the angle θ for 

a given path, we need to know how much branching there will be in the region of the 

multiway graph through which it passes.

But now recall that in discussing spacetime we identified the flux of edges through spacelike 

hypersurfaces in the causal graph as potentially corresponding to energy. The spacetime 

causal graph, however, is just a projection of the full multiway causal graph, in which 

branchlike directions have been reduced out. (In a causal invariant system, it does not 
matter what “direction” this projection is done in; the reduced causal graph is always the 

same.) But now suppose that in the full multiway causal graph, the flux of edges across 

spacelike hypersurfaces can still be considered to correspond to energy.

Now note that every node in the multiway causal graph represents some event in the 

multiway graph. But events are what produce branching—and “turns”—of paths in the 

multiway graph. So what this suggests is that the amount of turning of a path in the multi‐

389



multiway
way graph should be proportional to energy, multiplied by the number of steps, or effec‐
tively the time. In standard quantum formalism, energy is identified with the Hamiltonian 

H, so what this says is that in our models, we can expect transition amplitudes to have the 

basic form ei H t—in agreement with the result from quantum mechanics.

To think about this in more detail, we need not just a single energy quantity—corresponding to 

an overall rate of events—but rather we want a local measure of event rate as a function of 
location in multiway space. In addition, if we want to compute in a relativistically invariant 
way, we do not just want the flux of causal edges through spacelike hypersurfaces in some 

specific foliation. But now we can make a potential identification with standard quantum 

formalism: we suppose that the Lagrangian density ℒ corresponds to the total flux in all 
directions (or, in other words, the divergence) of causal edges at each point in multiway space.

But now consider a path in the multiway system going through multiway space. To know 

how much “turning” to expect in the path, we need in effect to integrate the Lagrangian 

density along the path (together with the appropriate volume element). And this will give us 

something of the form ei S, where S is the action. But this is exactly what we see in the 

standard path integral formulation of quantum mechanics [124].

There are many additional details (see [121]). But the correspondence between our models 

and the results of standard quantum formalism is notable. 

It is worth pointing out that in our models, something like the Lagrangian is ultimately not 
something that is just inserted from the outside; instead it must emerge from actual rules 

operating on hypergraphs. In the standard formalism of quantum field theory, the 

Lagrangian is stated in terms of quantum field operators. And the implication is therefore 

that the structure of the Lagrangian must somehow emerge as a kind of limit of the underly‐
ing discrete system, perhaps a bit like how fluid mechanics can emerge from discrete 

underlying molecular dynamics (or cellular automata) [110]. 

One notable feature of standard quantum formalism is the appearance of complex numbers 

for amplitudes. Here the core concept is the turning of a path in multiway space; the 

complex numbers arise only as a convenient way to represent the path and understand its 

projections. But there is an additional way complex numbers can arise. Imagine that we 

want to put a metric on the full (t, x, b<) space of the multiway causal graph. The normal 
convention for (t, x) space is to have real‐number coordinates and a norm based on t2 – x2—
but an alternative is use i t for time. In extending to (t, x, b<) space, one might imagine that a 

natural norm which allows the contributions of t, x and b components to be appropriately 

distinguished would be t2 – x2 + i b2. 

390



8.14  Quantum Measurement
Above we gave a brief summary of how quantum measurement can work in the context of 
our models. Here we give some more detail. 

In a sense the key to quantum measurement is reconciling our notion that “definite things 

happen in the universe” with the formalism of quantum mechanics—or the branching 

structure of a multiway system. 

But if definite things are going to happen, what might they be?

Here we will again consider the example of a string substitution system, although the core of 
what we say also applies to the full hypergraph case. Consider the rule 

{A → AB, B → A}
We could imagine a simple “classical” procedure for evolving according to this rule, in 

which we just do all updates we can (say, based on a le�‐to‐right scan) at each step:

A AB ABA ABAAB ABAABABA ABAABABAABAAB

But in fact we know that there are many other possibilities, that can be represented by the 

multiway system:

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

AAAA AAABB AABAB ABAAB ABABA AABBA ABBAA ABABBB ABBABB AABBBB ABBBAB ABBBBA ABBBBBB

391



Most of the states that appear in the multiway system are, however, “unfinished”, in the sense 

that there are additional “independent” updates that can consistently be done on them. For 
example, with the rule {A→BA} there are 4 separate updates that can be applied to AAAA:

AAAA

A
BAAAA A A

BAAA AA A
BAA AAA A

BA

BAAAA ABAAA AABAA AAABA

But none of these depend on the others, so they can in effect all be done together, giving the 

result BABABABA.

Put another way, all of these updates involve “spacelike separated” parts of the string, so 

they are all causally independent, and can all consistently be carried out at the same time. 
As discussed in 5.21, doing all updates across a state together can be thought of as evolving a 

system in “generational steps” to produce “generational states”. 

In some multiway cases, there may be a single sequence of generational states:

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

AAAA AAABB AABAB ABAAB ABABA AABBA ABBAA ABABBB ABBABB AABBBB ABBBAB ABBBBA ABBBBBB

392



In other cases, there can be several branches of generational states:

AA

BBA ABB AAB ABA

BBBB BBAB AABB ABBB ABAB BBBA ABBA

BBABB AABBB ABABB BBBBB ABBBB BBBAB ABBAB BBBBA ABBBA

BBABBB AABBBB ABABBB BBBABB ABBABB BBBBBB ABBBBB BBBBAB ABBBAB ABBBBA BBBBBA

The presence of multiple branches is a consequence of having a mixture of spacelike and 

branchlike separated events that can be applied to a single state. For example, with the rule 

{A→AB,A→BB} the first and second updates here are spacelike separated, but the first and 

third are branchlike separated:

ABAB

A
ABBAB AB A

ABB
A
BBBAB AB A

BBB

ABBAB ABABB BBBAB ABBBB

A view of quantum measurement is that it is an attempt to describe multiway systems using 

generational states. Sometimes there may be a unique “classical path”; sometimes there may 

be several outcomes for measurements, corresponding to several generational states.

But now let us consider the actual process of doing an experiment on a multiway system—or 

a quantum system. Our basic goal is—as much as possible—to describe the multiway system 

in terms of a limited number of generational states, without having to track all the different 
branches in the multiway system.

At some point in the evolution of a string substitution system we might see a large number of 
different strings. But we can view them all as part of a single generational state if they in effect 
yield only spacelike separated events. In other words, if the strings can be assembled without 
“branchlike ambiguity” they can be thought of as forming a consistent generational state.

In the standard formalism of quantum mechanics, we can think of the states in the multiway 

system as being quantum states. The construct we form by “assembling” these states can be 

393



system  by
thought of as a superposition of the states. Causal invariance then implies that through the 

evolution of the multiway system any such superposition will then actually become a single 

quantum state. In some sense the observer “did nothing”: they just notionally identified a 

collection of states. It was the actual evolution of the system that produced the specific 

combined state.

In describing a quantum system—or a multiway system—one must in effect define coordi‐
nates, and in particular one must specify what foliation one is going to use to represent the 

progress of time. And this freedom to pick a “quantum observation frame” is critical in being 

able to maintain a view in which one imagines “definite things to happen” in the system.

With a foliation like the following, at any given time there is a mixture of different states, 
and no attempt has been made to find a way to “summarize” what the system is doing:

AA

AAB ABA

AABB ABAB ABBA

AABBB ABABB ABBAB ABBBA

AABBBB ABABBB ABBABB ABBBAB ABBBBA

AABBBBB ABABBBB ABBABBB ABBBABB ABBBBAB ABBBBBA

AABBBBBB ABABBBBB ABBABBBB ABBBABBB ABBBBABB ABBBBBAB ABBBBBBA

Consider, however, a foliation like the following:

AA

AAB ABA

AABB ABAB ABBA

AABBB ABABB ABBAB ABBBA

AABBBB ABABBB ABBABB ABBBAB ABBBBA

AABBBBB ABABBBB ABBABBB ABBBABB ABBBBAB ABBBBBA

AABBBBBB ABABBBBB ABBABBBB ABBBABBB ABBBBABB ABBBBBAB ABBBBBBA

AABBBBBBB ABABBBBBB ABBABBBBB ABBBABBBB ABBBBABBB ABBBBBABB ABBBBBBAB ABBBBBBBA

394



In this picture, generational states have been highlighted, and a foliation has been selected 

that essentially “freezes time” around a particular generational state. In effect, the observer 

is choosing a quantum observation frame in which there is a definite classical outcome for 

the behavior of the system.

“Freezing time” around a particular state is something an observer can choose to do in their 

description of the system. But the crucial point is that the actual dynamics of the evolution 

of the multiway system cause this choice to have implications.

In particular, in the case shown, the region of the multiway system in which “time is frozen” 
progressively expands. The choice the observer has made to freeze a particular state is 

causing more and more states to have to be considered as similarly frozen. In the physics of 
quantum measurement, one is used to the idea that for a quantum measurement to be 

considered to have a definite result, it must involve more and more quantum degrees of 
freedom. What we see here is effectively a manifestation of this phenomenon.

In freezing time in something like the foliation in the picture above what we are effectively 

doing is creating a coordinate singularity in defining our quantum observation frame. And 

there is an analogy to this in general relativity and the physics of spacetime. Just as we 

freeze time in our quantum frame, so also we can freeze time in a relativistic reference 

frame. For example, as an object approaches the event horizon of a black hole, its time as 

described by a typical coordinate system set up by an observer far from the black hole will 
become frozen—and just like in our quantum case, we will consider the state to stay fixed.

But there is a complicated issue here. To what extent is the singularity—and the freezing of 
time—a feature of our description, and to what extent is it something that “really happens”? 
This depends in a sense on the relationship one has to the system. In traditional thinking 

about quantum measurement, one is most interested in the “impressions” of observers who 

are in effect embedded in the system. And for them, the coordinate system they chose in 

effect defines their reality.

But one can also imagine being somehow “outside the system”. For example, one might try 

to set up a quantum experiment (or a quantum computer) in which the construction of the 

system somehow makes it natural to maintain a “frozen time” foliation. The picture below 

shows a toy example in which the multiway system by its very construction has a terminal 
state for which time does not advance:

395



XAAX

XAABX XABAX

XAABBX XABABX XABBAX

XAABBBX XABABBX XXXX XABBABX XABBBAX

XAABBBBX XABABBBX XABBABBX XABBBABX XABBBBAX

XAABBBBBX XABABBBBX XABBABBBX XABBBABBX XABBBBABX XABBBBBAX

XAABBBBBBX XABABBBBBX XABBABBBBX XABBBABBBX XABBBBABBX XABBBBBABX XABBBBBBAX

But now the question arises of what can be achieved in the multiway system corresponding 

to the actual physical universe. And here we can expect that one will not be able to set up 

truly isolated states, and that instead there will be continual inevitable entanglement. What 
one might have imagined could be maintained as a separate state will always become 

entangled with other states. 

The picture below shows a slightly more realistic multiway system, with an attempt to 

construct a foliation that freezes time:

A

AB

AA ABB

AAB ABA ABBB

AAA AABB ABAB ABBA ABBBB

AAAB AABA ABAA ABABB AABBB ABBAB ABBBA ABBBBB

AAAA AAABB AABAB ABAAB ABABA AABBA ABBAA ABABBB ABBABB AABBBB ABBBAB ABBBBA ABBBBBB

396



And what we see here is that in a sense the structure of the multiway graph limits the extent 
to which we can freeze time. In effect, the multiway system forces decoherence—or entangle‐
ment—‐just by its very structure.

We should note that it is not necessarily the case that there is just a single possible sequence 

of generational states, corresponding in a sense to a single possible “classical path”. Here is 

an example where there are four generational states that occur at a particular generational 
step. And now we can for example construct a foliation that—at least for a while—“freezes 

time” for all of these generational states:

A

AB CA

ABB AC CAB CCA

ABBB ACB ABC CABB CAC CCAB CCCA

ABBBB ACBB ABCB ABBC CABBB ACC CACB CABC CCABB CCAC CCCAB CCCCA

ABBBBB ACBBB ABCBB ABBCB ABBBC CABBBB ACCB ACBC CACBB ABCC CABCB CABBC CCABBB CACC CCACB CCABC CCCABB CCCAC CCCCAB CCCCCA

It is worth pointing out that if we try to freeze time for something that is not a proper genera‐
tional state, there will be an immediate issue. A proper generational state contains the results 

of all spacelike separated events at a particular point in the evolution of a system. So when we 

freeze time for it, we are basically allowing other branchlike separated events to occur, but 
not other spacelike separated ones. However, if we tried to freeze time for a state that did not 
include all spacelike separated events, there would quickly be a mismatch with the progress 

of time for the excluded events—or in effect the singularity of quantum observation frame 

would “spill over” into a singularity in the causal graph, leading to a singularity in spacetime.

In other words, the fact that the states that appear in quantum measurement are generational 
states is not just a convenience but a necessity. Or, put another way, in doing a quantum 

measurement we are effectively setting up a singularity in branchial space, and only if the 

states we measure are in effect “complete in spacetime” will this singularity be kept only in 

branchial space; otherwise it will also become a singularity in physical spacetime.

In general, when we talk about quantum measurement, we are talking about how an 

observer manages to construct a description of a system that in effect allows the observer to 

“make a conclusion” about what has happened in the system. And what we have seen is that 
appropriate “time‐freezing foliations” allow us to do this. And while there may be some 

397



 may
restrictions, it is usually in principle possible to construct such foliations in a multiway 

system, and to have them last as long as we want. 

But in practice, as the pictures above begin to suggest, a�er a while the foliations we have to 

construct can get increasingly complicated. In effect, what we are having to do in construct‐
ing the foliation is to “reverse engineer” the actual evolution of the multiway system, so that 
with our elaborate description we are still managing to maintain time as frozen for a particu‐
lar state, carefully avoiding complicated entanglements that have built up with other states. 

But there is a problem here. Because in effect we are asking the observer to “outcompute” the 

system itself. Yet we can expect that the evolution of the multiway system, say for one of our 
models, will usually correspond to an irreducible computation. And so we will be asking the 

observer to do a more and more elaborate computation to maintain the description they are 

using. And as soon as the computation required exceeds the capability of the observer, the 

observer will no longer be able to maintain the description, and so decoherence will be inevitable.

It is worthwhile to compare this situation with what happens in thermodynamic processes, 
and in particular with apparent entropy increase. In a reversible system, it is always in 

principle possible to recognize, say, that the initial conditions for the systems were simple 

(and “low entropy”). But in practice the actual configurations of the system usually become 

complicated enough that this is increasingly difficult  to do. In traditional statistical mechan‐
ics one talks of “coarse‐grained” measurements as a way to characterize what an observer 

can actually analyze about a system. 

In computational terms we talk about the computational capabilities of the observer, and how 

computational irreducibility in the evolution of the system will eventually overwhelm the 

computational capabilities of the observer, making apparent entropy increase inevitable [1:9.3].

In the quantum case, we now see how something directly analogous happens. The analog of 
coarse graining is the effort to create a foliation with a particular apparent outcome. But 
eventually this becomes infeasible, and—just like in the thermodynamic case—we in effect see 

“thermalization”, which we can now attribute to the effects of computational irreducibility. 

8.15  Operators in Quantum Mechanics
In standard quantum formalism, there are states, and there are operators (e.g. [125]). In our 

models, updating events are what correspond to operators. In the standard evolution of the 

multiway system, all applicable operators are in effect “automatically applied” to every state 

to generate the actual evolution of the system. But to understand the correspondence with 

standard quantum formalism, we can imagine just applying particular operators by doing 

only particular updating events.

Consider the string substitution system:

{AB → ABA, BA → BAB}

398



In this system we effectively have two operators O1 and O2, corresponding to these two 

possible updating rules. We can think about building up an operator algebra by considering 

the relations between different sequences of applications of these operators.

In particular, we can study the commutator:

[O1,O2] = O1 O2 – O2 O1

In terms of the underlying rules, this commutator corresponds to:

ABA

AB
ABAA A BA

BAB

ABAA ABAB

A BA
BABA AB AB

ABA

ABABA

At the first step, the results of applying O1 and O2 to the initial state are different, and we can 

say that the states generated form a branch pair. But then at the second step, the branch pair 

resolves, and the branches merge to the same state. In effect, we can represent this by 

saying that O1 and O2 commute, or that:

[O1,O2] = O1 O2 – O2 O1 = 0

In general, there is a close relationship between causal invariance—and its implication for 

the resolution of all branch pairs—and the commuting of operators. And given our discus‐
sion above this should not be considered surprising: as we discussed, when there is causal 
invariance, it means that all branches can resolve to a single (“classical”) state, just like in 

standard quantum formalism the commuting of operators is associated with seemingly 

classical behavior.

But there is a key point here: even if causal invariance implies that branch pairs (and similarly 

commutators) will eventually resolve, they may take time to do so. And it is this delay in 

resolution that is the core of what leads to what we normally think of as quantum effects.

Once a branch pair has resolved, there are no longer multiple branches, and a single state 

has emerged. But before the branch pair has resolved, there are multiple states, and there‐
fore what one might think of as “quantum indeterminacy”. 

In the case where a branch pair has not yet resolved, the corresponding commutator will be 

nonzero—and in a sense the value of the commutator measures the branchlike distance 

between the states reached by applying the two different updates (corresponding to the two 

different operators). 

399



In our model for spacetime, if a single event in the causal graph is connected in the causal 
graph to two different events we can ask what the spacelike separation of these events might 
be, and we might suppose that this spatial distance is determined by the speed of light c (say 

multiplied by the elementary time corresponding to traversal of the causal edge).

In thinking now about the multiway system, we can ask what the branchlike separation of 
states in a branch pair might be. This will now be a distance on a branchial graph—or 

effectively a distance in state space—and we can suppose that this distance is determined by 

ℏ. And depending on our conventions for measuring branchial distance, we might introduce 

an i, yielding a setup very much aligned with traditional quantum formalism.

Another interpretation of the non‐commuting of operators is connected to the entanglement of 
quantum states. And here we now have a very direct picture of entanglement: two states are 

entangled if they are part of the same unresolved branch pair, and thus have a common ancestor.

The multiway graph gives a full map of all entanglements. But at any particular time (correspon‐
ding to a particular slice of a foliation defined by a quantum observation frame), the branchial 
graph gives a snapshot that captures the “instantaneous” configuration of entanglements. States 
closer on the branchial graph are more entangled; those further apart are less entangled.

It is important to note that distance on the branchial graph is not necessarily correlated with 

distance on the spatial graph. If we look at events, we can use the multiway causal graph to give 

a complete map of all connections, involving both branchlike and spacelike (as well as timelike) 
separations. Ultimately, the underlying rule determines what connections will exist in the 

multiway causal graph. But just as in the standard formalism of quantum mechanics, it is 
perfectly possible for there to be entanglement of spacelike‐separated events.

8.16  Wave-Particle Duality, Uncertainty Relations, Etc.
Wave‐particle duality was an early but important concept in standard quantum mechanics, 
and turns out to be a core feature of our models, independent even of the details of particles. 
The key idea is to look at the correspondence between spacelike and branchlike projections 

of the multiway causal graph.

Let us consider some piece of “matter”, ultimately represented as features of our hyper‐
graphs. A complete description of what the matter does must include what happens on every 

branch of the multiway graph. But we can get a picture of this by looking at the multiway 

causal graph—which in effect has the most complete representation of all meaningful 
spatial and branchial features of our models.

Fundamentally what we will see is a bundle of geodesics that represent the matter, propagat‐
ing through the multiway causal graph. Looked at in terms of spacelike coordinates, the 

bundle will seem to be following a definite path—characteristic of particle‐like behavior. But 
inevitably the bundle will also be extended in the branchlike direction—and this is what 
leads to wave‐like behavior.

400



Recall that we identified energy in spacetime as corresponding to the flux of causal edges 

through spacelike hypersurfaces. But as mentioned above, whenever causal edges are 

present, they correspond to events, which are associated with branching in the multiway 

graph and the multiway causal graph. And so when we look at geodesics in the bundle, the 

rate at which they turn in multiway space will be proportional to the rate at which events 

happen, or in other words, to energy—yielding the standard E ∝ ω proportionality between 

particle energy and wave frequency.

Another fundamental phenomenon in quantum mechanics is the uncertainty principle. To 

understand this principle in our framework, we must think operationally about the process 

of, for example, first measuring position, then measuring momentum. It is best to think in 

terms of the multiway causal graph. If we want to measure position to a certain precision Δ x 

we effectively need to set up our detector (or arrange our quantum observation frame) so 

that there are O(1/Δ x) elements laid out in a spacelike array. But once we have made our 

position measurement, we must reconfigure our detector (or rearrange our quantum 

observation frame) to measure momentum instead.

But now recall that we identified momentum as corresponding to the flux of causal edges 

across timelike hypersurfaces. So to do our momentum measurement we effectively need to 

have the elements of our detector (or the pieces of our quantum observation frame) laid out 
on a timelike hypersurface. But inevitably it will take at least O(1/Δ x) updating events to 

rearrange the elements we need. But each of these updating events will typically generate a 

branch in the multiway system (and thus the multiway causal graph). And the result of this 

will be to produce an O(1/Δ x) spread in the multiway causal graph, which then leads to an 

O(1/Δ x) uncertainty in the measurement of momentum.

(Another ultimately equivalent approach is to consider different foliations, and to note for 

example that with a finer foliation in time, one is less able to determine the “true direction” 
of causal edges in the multiway graph, and thus to determine how many of them will cross a 

spacelike hypersurface.)

To make our discussion of the uncertainty principle more precise, we should consider opera‐
tors—represented by sequences of updating events. In the (t, x, b) space of the multiway causal 
graph, the operators corresponding to position and momentum must generate events that 
correspond to moving at different angles; as a result the operators do not commute.

And with this setup we can see why position and momentum, as well as energy and time, 
form canonically conjugate pairs for which uncertainty relations hold: it is because these 

quantities are associated with features of the multiway causal graph that probe distinct (and 

effectively orthogonal) directions in multiway causal space.

401



8.17 Correspondence between Relativity and
             Quantum Mechanics
One of the surprising consequences of the potential application of our models to physics is 
their implications around deep relationships between relativity and quantum mechanics. 
These are particularly evident in thinking about the multiway causal graph. As a toy model, 
consider the graph:

Timelike edges go down, but then in each slice there are spacelike and branchlike edges. A
more realistic example of the very beginning of such a graph is:

Themultiway causal graph in a sense captures in one graph both relativity and quantummechan‐
ics. Time is involved in both of them, and in ourmodels it is an essentially computational
concept, involving progressive application of the underlying rules of the system. But then
relativity is associatedwith the structure formed by spacelike and timelike edges, while quantum
mechanics is primarily associatedwith the structure formed by branchlike and timelike edges.

The spacelike direction corresponds to ordinary physical space; the branchlike direction is
effectively the space of quantum states. Distance in the spacelike direction is ordinary
spacetime distance. Distance in the branchlike direction reflects the level of quantum
entanglement between states. When we form foliations in time, spacelike hypersurfaces

402



 hypersurfaces
represent in a sense the instantaneous configuration of space, while branchlike hypersur‐
faces represent the instantaneous entanglements between quantum states.

It should be emphasized that (unlike in the idealization of our first picture above) the detailed 

structure of the spacelike+timelike component of the multiway causal graph will in practice be 

very different from that of the branchlike+timelike one. The spacelike+timelike component is 

expected to limit to something like a finite‐dimensional manifold, reflecting the characteristics 

of physical spacetime. The branchlike+timelike one potentially limits to an infinite dimen‐
sional space (that is perhaps a projective Hilbert space), reflecting the characteristics of the 

space of quantum states. But despite these substantial geometrical differences, one can expect 
many structural aspects and consequences to be basically the same. 

We are used to the idea of motion in space. In the context of our models—and of the multi‐
way causal graph—motion in space in effect corresponds to progressively sampling more 

spacelike edges in the graph. But now we can see a quantum analog: we can also have 

motion in the branchlike direction, in which, in effect, we progressively sample more 

branchlike edges, reaching more quantum states. Velocity in space is thus the analog of the 

rate at which additional states are sampled (and thus entangled).

In relativity there is a fairly well‐developed notion of an idealized observer. The observer is 

typically represented by some some causal foliation of spacetime—like an inertial reference 

frame that moves without forces acting on it. One can also define an observer in quantum 

mechanics, and in the context of our models it makes sense—as we have done above—to 

parametrize the observer in terms of a quantum observation frame that consists not of a 

sequence of spacelike hypersurfaces, but instead of a series of branchlike ones.

A quantum observation frame in a sense defines a plan for how an observer will sample 

possible quantum states—and the analog of an inertial frame in spacetime is presumably a 

quantum observation frame that corresponds to a fixed plan that cannot be affected by 

anything outside. And in general, the analog in quantum mechanics of a world line in 

relativity is presumably a measurement plan.

In special relativity a key idea is to think about comparing the perceptions of observers in 

different inertial frames. But in the context of our models we can now do the exact same thing 

for quantum observers. And the analog of relativistic invariance then becomes a statement of 
perception or measurement invariance: that in the end different quantum observers (despite 

the branching of states) in a sense perceive the same things to happen, or, in other words, that 
there is at some level an objective reality even in quantum mechanics. 

Our analogy between relativity and quantum mechanics suggests asking about quantum 

analogs of standard relativistic phenomena. One example is relativistic time dilation, in 

which, in effect, sampling spacelike edges faster reduces the rate of traversing timelike 

edges. The analog in quantum mechanics is presumably the quantum Zeno effect [126][127], 

403



in which more rapid measurement—corresponding to faster sampling of branchlike edges—
slows the time evolution of a quantum system.

A key concept in relativity is the light cone, which characterizes the maximum rate at which 

causal effects spread in spacelike directions. In our models, spacetime causal edges in effect 
define elementary light cones, which are then knitted together by the structure of the 

(spacetime) causal graph. But now in our models there is a direct analog for quantum 

mechanics, visible in the full multiway causal graph.

In the multiway causal graph, every event effectively has a cone of causal influence. Some 

of that influence may be in spacelike directions (corresponding to ordinary relativistic 

light cone effects), but some of it may be in branchlike directions. And indeed, whenever 

there are branches in the multiway graph, these correspond to branchlike edges in the 

multiway causal graph.

So what this means is that in addition to a light cone of effects in spacetime, there is also 

what we may call an entanglement cone, which defines the region affected in branchial 
space by some event. In the light cone case, the spacelike extent of the light cone is set by
the speed of light (c). In the entanglement cone case (as we will discuss below) the branch‐
like extent of the entanglement cone is essentially set by ℏ. 

As we have mentioned, the definition of time is shared between spacelike and branchlike 

components of the multiway causal graph. Another shared concept appears to be energy (or 

in general, energy‐momentum, or action). Time is effectively defined by displacement in the 

timelike direction; energy appears to be defined by the flux of causal edges in the timelike 

direction. In the relativistic setting, energy can be thought of as flux of causal edges through 

spacelike hypersurfaces; in the quantum mechanical setting, it can be thought of as a flux of 
causal edges through branchlike hypersurfaces.

An important feature of the spacetime causal graph is that it can potentially describe curved 

space, and reproduce general relativity. And here again we can now see that in our models 

there are analogs in quantum mechanics. One issue, though, is that whereas ordinary space 

is—at least on a large scale—finite‐dimensional, comparatively flat, and well modeled by a 

simple Lorentzian manifold, branchial space is much more complicated, probably in the 

limit infinite–dimensional, and not at all flat.

At a mathematical level, we are in quantum mechanics used to forming commutators of 
operators, and in many cases finding that they do not commute, with their “deviation” being 

measured by ℏ. In general relativity, one can also form commutators, and indeed the 

Riemann tensor for measuring curvature is precisely the result of computing the commuta‐
tor of two covariant derivatives. And perhaps even more analogously the Ricci scalar 

curvature gives the angle deficit for transport around a loop in spacetime.

In our context, therefore, the non‐flatness of space is directly analogous to a core phe‐
nomenon of quantum mechanics: the non‐commuting of operators. 

404



In the general relativity case, we are used to thinking about the propagation of bundles of 
geodesics in spacetime, and the fact that the Ricci scalar curvature determines the local 
cross‐section of the bundle. Now we can also consider the more general propagation of 
bundles of geodesics in the multiway causal graph. But when we look along branchlike 
directions, the limiting space we see tends to be highly connected, and effectively of high 
negative curvature. And what this means is that a bundle of geodesics can be expected to 
spread out rapidly in branchlike directions.

But this has an immediate interpretation in quantum mechanics: it is the phenomenon of 
decoherence, whereby quantum effects get spread (and entangled) across large numbers of 
quantum degrees of freedom.

In relativity, the speed of light c sets a maximum speed for the propagation of effects in space. 
In quantum mechanics, our entanglement cones in essence also set a maximum speed for the 
propagation of effects in branchial space. In special relativity, there is then a maximum speed 
defined for any observer—or, in other words, a maximum speed for motion. In quantum 
mechanics, we can now expect that there will also be a maximum speed for entanglement, or 
for measurement: it is not possible to set up a quantum observation frame that achieves a 
higher speed while still respecting the causal relations in the multiway causal graph. We will 
call this maximum speed ζ, and in 8.20 we will discuss its possible magnitude.

One may ask to what extent the correspondences between relativity and quantum mechan‐
ics that we have been discussing rely on our models. In principle, for example, one could 
imagine a kind of “multicausal continuum” that is a mathematical structure (conceivably 
related to twistor spaces [128]) corresponding to a continuum limit of our multiway causal 
graph. But while there are challenges in understanding the limits associated with our 
models, this seems likely to be even more difficult to construct and handle—and has the 
great disadvantage that it cannot be connected to explicit models that are readily amenable, 
for example, to enumeration.

8.18 Event Horizons and Singularities in Spacetime and
              Quantum Mechanics
Having discussed the general correspondence between relativity and quantum mechanics 
suggested by our models, we can now consider the extreme situation of event horizons 
and singularities.

As we discussed above, an event horizon in spacetime corresponds in our models to discon‐
nection in the causal graph: a�er some slice in our foliation in time, there is no longer 
causal connection between different parts of the system. As a result, even if the system is 
locally causal invariant, branch pairs whose products go on different sides of the disconnec‐
tion can never resolve. The only way to make a foliation in which this does not happen is 
then effectively to freeze time before the disconnection occurs.

405



When there is a true disconnection in the causal graph, there is no choice about this. But it is
also perfectly possible just to imagine setting up a coordinate system that freezes time in a
particular region of space—although it will typically take more and more effort (and energy)
to consistently maintain such a coordinate singularity as other parts of the system evolve.

But now there is an interesting correspondence with quantummeasurement. As we dis‐
cussed in 8.14, in the context of our models, one can view a quantummeasurement (or a
“collapse of the wave function”) as being associated with a foliation that freezes time for the
state that is the outcome of the measurement. In essence, therefore, quantummeasurement
corresponds to having a coordinate singularity in a particular region of branchial space.

What about an event horizon? As we saw above, one way in which an event horizon can occur
is if some branch of the multiway system simply terminates, so that in a sense time stops for it.
Another possibility is that—at least temporarily—there can be a disconnected piece in the
branchial graph. Consider for example the (causal invariant) string substitution system:

{A → BB, BBB → AA}
The multiway system for this rule is

AA

ABB BBA

BBBB

AAB BAA

ABBB BBAB BBBA BABB

AAA BBBBB

ABBA AABB BBAA BAAB

ABBBB BBBBA BBABB BBBAB BABBB

ABAA AAAB AABA BBBBBB BAAA

ABABB ABBAB ABBBA AABBB BBAAB BBABA BBBAA BAABB BABBA

ABBBBB BBBBAB AAAA BBABBB BBBABB BBBBBA BABBBB

and the branchial graph shows temporary disconnections

 , , , ,

, , , , 

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although the “spacetime” causal graph stays connected: 

One can think of these temporary disconnections in the branchial graphs as corresponding 

to isolated regions of branchial space where entanglement at least temporarily cannot 
occur—and where some pure quantum state (such as qubits) can be maintained, at least for 

some period of time. 

In some sense, one can potentially view such disconnections as being like black holes in 

branchial space. But the continued generation of branch pairs (in a potential analog to 

Hawking radiation [129]) causes the “black hole” to dissipate.

A different situation can occur when there is also disconnection in the causal graph—leading 

in our models to disconnection in the spatial hypergraph—and thus a spacetime event 
horizon. As a simple example, consider the string substitution system (starting from AA):

{AA → AAAB}
The causal graph in this case is

407



and the sequence of branchial graphs (with the standard foliation) is:

 , , ,

, , , 

What has happened here is that there are event horizons both in physical space and in 

branchial space. 

We can expect similar phenomena in our full models, and extrapolating this to a physical 
black hole what this represents is the presence of both a causal event horizon (associated 

with motion in space, propagation of light, etc.) and an entanglement event horizon (associa‐
ted with quantum entanglement). The causal event horizon will be localized in physical 
space (say at the Schwarzschild radius [130]); the entanglement event horizon can be 

considered instead to be localized in branchial space. 

It should be noted that these horizons are in a sense linked through the multiway causal 
graph, which in the example above initially has the form

408



and a�er more steps builds up the structure:

In this graph, there are both spacelike and branchlike connections, and here both of them 

exhibit disconnection, and therefore event horizons. And even though the geometrical 
structure of branchial space is very different from physical space, there are potentially 

further correspondences to be made between them. For example, while the speed of light c 

governs the maximum spacelike speed, the maximum entanglement rate υ that we intro‐
duced above governs the maximum “branchlike speed”, or entanglement rate. 

When a disconnection occurs in the spacetime causal graph (and thus the spatial hyper‐
graph), we can think of this as implying that geodesics in spacetime would have to exceed c 

in order not to be trapped. When a disconnection occurs in the branchial graph, we can 

think of geodesics having to “exceed speed υ” in order not to be trapped.

It is worth pointing out that the analog of a true singularity—and not just an event horizon—
can occur in our models if there are paths in the multiway system that simply terminate, as 

for B, BB, etc. in:

A

AA

B AAA

AB BA AAAA

AAB BAA AAAAA ABA

BB AAAB BAAA AAAAAA ABAA AABA

BAB AAAAB ABB BAAAA AAAAAAA ABAAA BBA AAABA AABAA

409



When this happens, there are many geodesics that in effect converge to a single point, like 

in spacetime singularities in general relativity. Here, however, we see that this can happen 

not only in physical space, but also in the multiway system, or, in other words, in branchial 
space. (In our systems, it is probably the case that singularities must be enclosed in event 
horizons, in the analog of the cosmic censorship hypothesis.)

Many results from general relativity can presumably be translated to our models, and can 

apply both to physical space and branchial space (see [121]). In the case of a black hole, our 

models suggest that not only may a causal event horizon form in physical space; also an 

entanglement horizon may form in branchial space. One may then imagine that quantum 

information is trapped inside the entanglement horizon, even without crossing the causal 
event horizon—with implications perhaps similar to recent discussions of resolutions to the 

black hole quantum information problem [131][132][133].

There is a simple physical picture that emerges from this setup. As we have discussed, 
quantum measurement can be thought of as a choice of coordinates that “freeze time” for 

some region in branchial space. For an observer close to the entanglement horizon, it will 
not be possible to do this. Much like an observer at a causal event horizon will be stretched 

in physical space, so also an observer at an entanglement horizon will be stretched in 

branchial space. And the result is that in a sense the observer will not be able to “form a 

classical thought”: they will not successfully be able to do a measurement that definitively 

picks the branch of the multiway system in which something fell into the black hole, or the 

one in which it did not. 

8.19  Local Gauge Invariance
An important phenomenon discussed especially in the context of quantum field theories is 

local gauge invariance (e.g. [134]). In our models this phenomenon can potentially arise as a 

result of local symmetries associated with underlying rules (see 6.12). The basic idea is that 
these symmetries allow different local configurations of rule applications—that can be 

thought of as different local “gauge” coordinate systems. 

But the collection of all such possible configurations appears in the multiway graph (and the 

multiway causal graph)—so that a local choice of gauge can then be represented by a 

particular foliation in the multiway graph. But causal invariance then implies the equiva‐
lence of foliations—and establishes local gauge invariance.

As a very simple example, consider the rule:

410



Starting from a square, this rule can be applied in two different ways:

 , 

There is similar freedom if one applies the rule twice to a larger region:

 , ,

, 

In both cases one can think of the freedom to apply the rule in different ways as being like a 

symmetry, for example characterized by the list of possible permutations of input elements. 

But now imagine taking the limit of a large number of steps. Then one can expect to apply 

the resulting aggregate rule in a large number of ways. And much as we expect the limit of 
our spatial hypergraphs to be able to be represented—at least in certain cases—as a continu‐
ous manifold, we can expect something similar here. In particular, we can think of our‐
selves as winding up with a very large number of permutations corresponding to equivalent 
rule applications, which in the limit can potentially correspond to a Lie group. 

Each different possible choice of how to apply the rule corresponds to a different event that 
is represented in the multiway graph, and the multiway causal graph:

411



But the important point is that local choices of how the rule is repeatedly applied must 
always correspond to purely branchlike connections in the multiway causal graph.

The picture is analogous to the one in traditional mathematical physics. The spatial hyper‐
graph can be thought of as a base space for a fiber bundle, then the different choices of 
which branchlike paths to follow correspond to different choices of coordinate systems (or 

gauges) in the fibers of the fiber bundle (cf. [135][136]). The connection between fibers is 

defined by the foliation that is chosen.

There is an analog when one sets up foliations in the spacetime causal graph—in which case, 
as we have argued, causal invariance leads to general covariance and general relativity. But 
here we are dealing with branchlike paths, and instead of getting general relativity, we 

potentially get gauge theories.

In traditional physics, local gauge invariance already occurs in classical theories (such as 

electromagnetism), and it is notable that for us it appears to arise from considering multi‐
way systems. Yet although multiway systems appear to be deeply connected to quantum 

mechanics, the aggregate symmetry phenomenon that leads to gauge theories in effect 
makes slightly different use of the structure of the multiway causal graph. 

But much as in other cases, we can think about geodesics—now in the multiway causal 
graph—and can study the properties of the effective space that emerges, with local phenom‐
ena (including things like commutators) potentially reflecting features of the Lie algebra.

In traditional physics an important consequence of local gauge invariance is its implication 

of the existence of fields, and gauge bosons such as the photon and gluon. In our models the 

mathematical derivations that lead to this implication should be similar. But by looking at 
the evolution of our models, it is possible to get a more explicit sense of how this works. 

Consider a particular sequence of updates with the rule shown above:

 , , , , ,

, , , , , 

At the beginning, symmetry effectively allows many equivalent updates to be made. But 
once a particular update has been made, this has consequences for which of the possible 

updates—each independently equivalent on their own—can be made subsequently. These 

“consequences” are captured in the causal relationships encoded in the multiway causal 
graph—which have not only branchlike but also spacelike extent, corresponding in essence 

to the propagation of effects in what can be described as a gauge field.

412



8.20  Units and Scales
Most of our discussion so far has focused on how the structure of our models might corre‐
spond to the structure of our physical universe. But to make direct contact between our 

models and known physics, we need to fill in actual units and scales for the constructs in our 

models. In this section we give some indication of how this might work. 
_

In our models, there is a fundamental unit of time (that we will call T) that represents the 

interval of time corresponding to a single updating event. This interval of time in a sense 

defines the scale for everything in our models. 
_ _

Given T, there is an elementary length L, determined by the speed of light c according to:
_ _
L = c T
The elementary length defines the spatial separation of neighboring elements in the 

spatial hypergraph. 
_

Another fundamental scale is the elementary energy E: the contribution of a single causal 
edge to the energy of a system. The energy scale ultimately has both relativistic and quan‐
tum consequences. In general relativity, it relates to how much curvature a single causal 
edge can produce, and in quantum mechanics, it relates to how much change in angle in an 

edge in the multiway graph a single causal edge can produce.

The speed of light c determines the elementary length in ordinary space, specifying in effect 
how far one can go in a single event, or in a single elementary time. To fill in scales for our 

models, we also need to know the elementary length in branchial space—or in effect how far 

in state space one can go in a single event, or a single elementary time (or, in effect, how far 

apart in branchial space two members of a branch pair are). And it is an obvious supposition 

that somehow the scale for this must be related to ℏ.

An important point about scales is that there is no reason to think that elementary quantities 

measured with respect to our current system of units need be constant in the history of the 

universe. For example, if the universe effectively just splits every spatial graph edge in two, 
the number of elementary lengths in what we call 1 meter will double, and so the elemen‐
tary length measured in meters will halve.

Given the structure of our models, there are two key relationships that determine scales. 
The first—corresponding to the Einstein equations—relates energy density to spacetime 

curvature, or, more specifically, gives the contribution of a single causal edge (with one 

elementary unit of energy) to the change of Vr and the corresponding Ricci curvature: 
_

G E 1
c _ ≈
4 Ld _

L2

413



(Here we have dropped numerical factors, and G is the gravitational constant, which, we 

may note, is defined with its standard units only when the dimension of space d = 3.)

The second key relationship that determines scales comes from quantum mechanics. The 

most obvious assumption might be that quantum mechanics would imply that the elemen‐
_ _

tary energy should be related to the elementary time by E ≈ ℏ/T. And if this were the case, 
then our various elementary quantities would be equal to their corresponding Planck units 

[137], as obtained with G = c = ℏ = 1 (yielding elementary length ≈ 10‐35 m, elementary time 

≈ 10‐43 s, etc.)

But the setup of our models suggests something different—and instead suggests a relation‐
ship that in effect also depends on the size of the multiway graph. In our models, when we 

make a measurement in a quantum system, we are at a complete quantum observation 

frame—or in effect aggregating across all the states in the multiway graph that exist in the 

current slice of the foliation that we have defined with our quantum frame. 

There are many individual causal edges in the multiway causal graph, each associated 

_
with a certain elementary energy E. But when we measure an energy, it will be the aggre‐
gate of contributions from all the individual causal edges that we have combined in our 

quantum frame. 

A single causal edge, associated with a single event which takes a single elementary time, 
has the effect of displacing a geodesic in the multiway graph by a certain unit distance in
branchial space. (The result is a change of angle of the geodesic—with the formation of a 

single branch pair π
 perhaps being considered to involve angle .)

2

Standard quantum mechanics in effect defines ℏ through E = ℏ ω. But in this relationship E is 

a measured energy, not the energy associated with a single causal edge. And to convert 
between these we need to know in effect the number of states in the branchial graph 

associated with our quantum frame, or the number of nodes in our current slice through the 

multiway system. We will call this number Ξ.

And finally now we can give a relation between elementary energy and elementary time:

_ ℏ
E Ξ ≈ _

T
In effect, ℏ sets a scale for measured energies, but ℏ/Ξ sets a scale for energies of individual 
causal edges in the multiway causal graph.

This is now sufficient  to determine our elementary units. The elementary length is given in 

dimension d = 3 by 

_ G ℏ 1
≈ ( )1/(d–1)

l 0‐35P m
L ≈ ≈

c3 Ξ Ξ Ξ

414



_ tP 10‐43 s
T≈ ≈

Ξ Ξ
_ EP 109 J 1019 GeV
E ≈ ≈ ≈

Ξ Ξ Ξ
where lP, tP, EP are the Planck length, time and energy.

To go further, however, we must estimate Ξ. Ultimately, Ξ is determined by the actual 
evolution of the multiway system for a particular rule, together with whatever foliation and 

other features define the way we describe our experience of the universe. As a simple 

model, we might then characterize what we observe as being “generational states” in the 

evolution of a multiway system, as we discussed in 5.21. 

But now we can use what we have seen in studying actual multiway systems, and assume 

that in one generational step of at least a causal invariant rule each generational state 

generates on average some number κ of new states, where κ is related to the number of new 

elements produced by a single updating event. In a generation of evolution, therefore, the 

total number of states in the multiway system will be multiplied by a factor κ. 

But to relate this to observed quantities, we must ask what time an observer would perceive 

has elapsed in one generational step of evolution. From our discussion above, we expect that 
the typical time an observer will be able to coherently maintain the impression of a definite 

_
“classical‐like” state will be roughly the elementary time T multiplied by the number of 
nodes in the branchlike hypersurface. The number of nodes will change as the multiway 

graph grows. But in the current universe we have defined it to be Ξ. 

Thus we have the relation
tH

Ξ ≈ κ _
Ξ T

where tH is the current age of the universe, and for this estimate we have ignored the change 

of generation time at different points in the evolution of the multiway system. 
_

Substituting our previous result for T we then get:
tH 1 1061

Ξ ≈ κ tP Ξ ≈κ Ξ

There is a rough upper limit on κ from the signature for the underlying rule, or effectively 

the ratio in the size of the hypergraphs between the right and le�‐hand sides of a rule. (For 

most of the rules we have discussed here, for example, κ ≲ 2.) The lower limit on κ is related 

to the “efficiency”  of causal invariance in the underlying rule, or, in effect, how long it takes 

branch pairs to resolve relative to how fast new ones are created. But inevitably κ > 1.

415



Given the transcendental equation
σ

Ξ = κ Ξ

we can solve for Ξ to get

Ξ = ⅇ2 W( 1 σ log(κ))
2

where W  is the product log function [138] that solves w ew = z. But for large σ log(κ) (and we 

imagine that σ ≈ 1061), we have the asymptotic result [30]:

(σ log(κ))2
Ξ ≈

4 log2( 1 σ log(κ))
2

Plotting the actual estimate for Ξ as a function of κ we get the almost identical result:

1×10117

5×10116

1×10116

5×10115

1×10115

5×10114

1.0 1.5 2.0 2.5 3.0

If κ = 1, then we would have Ξ = 1, and for κ extremely close to 1, Ξ ≈ 1 + σ (κ – 1) + ... But 
even for κ = 1.01 we already have Ξ ≈ 10112, while for κ = 1.1 we have Ξ ≈ 10115, for κ = 2 we 

have Ξ ≈ 4 × 10116 and for κ = 10 we have Ξ ≈ 5 × 10117. 

To get an accurate value for κ we would have to know the underlying rule and the statistics 

of the multiway system it generates. But particularly at the level of the estimates we are 

giving, our results are quite insensitive to the value of κ, and we will assume simply:

Ξ ≈ 10116

In other words, for the universe today, we are assuming that the number of distinct instanta‐
neous complete quantum states of the universe being represented by the multiway system 

(and thus appearing in the branchial graph) is about 10116.

416



But now we can estimate other quantities:

_ 1
elementary length L ?≈ 10‐93 m 10‐35 m Ξ‐ 2

_ 1
elementary time T ?≈ 10‐101 s 10‐43 s Ξ‐ 2

_ 1
elementary energy E ?≈ 10‐30 eV 1028 eV Ξ‐ 2

1
elementary lengths across current universe ?≈ 10120 1062 Ξ 2

3
elements in spatial hypergraph ?≈ 10358 10184 Ξ 2

elements in branchial graph Ξ ?≈ 10116 Ξ
1

overall updates of universe so far ?≈ 10119 1061 Ξ 2

individual updating events in universe so far ?≈ 10477 10245 Ξ2

_
The fact that our estimate for the elementary length L is considerably smaller than the 

Planck length indicates that our models suggest that space may be more closely approxi‐
mated by a continuum than one might expect. 

_
The fact that the elementary energy E is much smaller than the surprisingly macroscopic 

Planck energy (≈ 1019 GeV ≈ 2 GJ, or roughly the energy of a lightning bolt) is a reflection of 
the fact the Planck energy is related to measurable energy, not the individual energy associ‐
ated with an updating event in the multiway causal graph.

Given the estimates above, we can use the rest mass of the electron to make some additional 
very rough estimates—subject to many assumptions—about the possible structure of the 

electron:

1
number of elements in an electron ?≈ 1035 10‐23 Ξ 2

1
radius of an electron ?≈ 10‐81 m 10‐42 m Ξ‐ 3

1
number of elementary lengths across an electron ?≈ 1012 10‐8 Ξ 6

In quantum electrodynamics and other current physics, electrons are assumed to have zero 

intrinsic size. Experiments suggest that any intrinsic size must be less than about 10‐22 m 

[139][140]—nearly 1060 times our estimate. 

Even despite the comparatively large number of elements suggested to be within an elec‐
tron, it is notable that the total number of elements in the spatial hypergraph is estimated to 

be more than 10200 times the number of elements in all known particles of matter in the 

universe—suggesting that in a sense most of the “computational effort” in the universe is 

expended on the creation of space rather than on the dynamics of matter as we know it.

417



The structure of our models implies that not only length and time but also energy and mass 

must ultimately be quantized. Our estimates indicate that the mass of the electron is > 1036 

times the quantized unit of mass—far too large to expect to see “numerological relations” 
between particle masses. 

But with our model of particles as localized structures in the spatial hypergraph, there 

seems no reason to think that structures much smaller than the electron might not exist—
corresponding to particles with masses much smaller than the electron. 

Such “oligon” particles involving comparatively few hypergraph elements could have 

masses that are fairly small multiples of 10‐30 eV. One can expect that their cross‐sections for 

interaction will be extremely small, causing them to drop out of thermal equilibrium 

extremely early in the history of the universe (e.g. [141][142]), and potentially leading to 

large numbers of cold, relic oligons in the current universe—making it possible that oligons 

could play a role in dark matter. (Relic oligons would behave as a more‐or‐less‐perfect ideal 
gas; current data indicates only that particles constituting dark matter probably have masses 

≳ 10‐22 eV [143].) 

As we discussed in the previous subsection, the structure of our models—and specifically the 

multiway causal graph—indicates that just as the speed of light c determines the maximum 

spacelike speed (or the maximum rate at which an observer can sample new parts of the 

spatial hypergraph), there should also be a maximum branchlike speed that we call ζ that 
determines the maximum rate at which an observer can sample new parts of the branchial 
graph, or, in effect, the maximum speed at which an observer can become entangled with 

new “quantum degrees of freedom” or new “quantum information”. 

Based on our estimates above, we can now give an estimate for the maximum entanglement 
speed. We could quote it in terms of the rate of sampling quantum states (or branches in the 

multiway system)
_

1 E
_ ≈ Ξ ≈ 10102/ second
T ℏ
but in connecting to observable features of the universe, it seems better to quote it in terms 

of the energy associated with edges in the causal graph, in which case the result based on 

our estimates is:
_
E

ζ ≈ _ ≈ 1062 GeV/ second ≈ 1052 W ≈ 105 solar masses / second
T

This seems large compared to typical astrophysical processes, but one could imagine it 
being relevant for example in mergers of galactic black holes. 

418



8.21  Specific Models of the Universe
If we pick a particular one of our models, with a particular set of underlying rules and initial 
conditions, we might think we could just run it to find out everything about the universe it 
generates. But any model that is plausibly similar to our universe will inevitably show 

computational irreducibility. And this means that we cannot in general expect to shortcut 
the computational work necessary to find out what it does. 

In other words, if the actual universe follows our model and takes a certain number of 
computational steps to get to a certain point, we will not be in a position to reproduce in 

much less than this number of steps. And in practice, particularly with the numbers in the 

previous subsection, it will therefore be monumentally infeasible for us to find out much 

about our universe by pure, explicit simulation.

So how, then, can we expect to compare one of our models with the actual universe? A major 

surprise of this section is how many known features of fundamental physics seem in a sense 

to be generic to many of our models. It seems, for example, that both general relativity and 

quantum mechanics arise with great generality in models of our type—and do not depend on 

the specifics of underlying rules.

One may suspect, however, that there are still plenty of aspects of our universe that are 

specific to particular underlying rules. A few examples are the effective dimension of 
space, the local gauge group, and the specific masses and couplings of particles. The 

extent to which finding these for a particular rule will run into computational irreducibil‐
ity is not clear. 

It is, however, to be expected that parameters like the ones just mentioned will put 
strong constraints on the underlying rule, and that if the rule is simple, they will likely 

determine it uniquely.

Of all the detailed things one can predict from a rule, it is inevitable that most will involve 

computational irreducibility. But it could well be that those features that we have identified 

and measured as part of the development of physics are ones that correspond to computa‐
tionally reducible aspects of our universe. Yet if the ultimate rule is in fact simple, it is likely 

that just these aspects will be sufficient  to determine it.

In section 7 we discussed some of the many different representations that can be used for our 
models. And in different representations, there will inevitably be a different ranking of 
simplicity among models. In setting up a particular representation for a model, we are in effect 
defining a language—presumably suitable for interpretation by both humans and our current 
computer systems. Then the question of whether the rule for the universe is simple in this 

language is in effect just the question of how suitable the language is for describing physics. 

419



Of course, there is no guarantee that there exists a language in which, with our current 
concepts, there is a simple way to describe the rule for our physical universe. The results of 
this section are encouraging, but not definitive. For they at least suggest that in the represen‐
tation we are using, known features of our universe generically emerge: we do not have to 

define some thin and complicated subset to achieve this.

8.22  Multiway Systems in the Space of All Possible Rules
We have discussed the possibility that our physical universe might be described as following 

a model of the type we have introduced here, with a particular rule. And to find such a rule 

would be a great achievement, and might perhaps be considered a final answer to the core 

question of fundamental physics.

But if such a rule is found, one might then go on and ask why—out of the infinite number of 
possibilities—it is this particular rule, or, for example, a simple rule at all. And here the 

paradigm we have developed makes a potential additional suggestion: perhaps there is not just 
one rule being used a�er all, but instead in a sense all possible rules are simultaneously used.

In the multiway systems we have discussed so far, there is a single underlying rule, but 
separate branches for all possible sequences of updating events. But one can imagine a rule‐
space multiway system, that includes branches not only for every sequence of updating 

events, but also for every possible rule used to do the updating. Somewhat like with updating 

events, there will be many states reached to which many of the possible rules cannot apply. 
(For example, a rule that involves only ternary edges cannot apply to a state with only binary 

edges.) And like with updating events, branches with different sequences of rules applied 

may reach equivalent states, and thus merge. 

Operationally, it is not so difficult  to see how to set up a rule‐space multiway system. All it 
really involves is listing not just one or a few possible rules that can be used for each updat‐
ing event, but in a sense listing all possible rules. In principle there are an infinite number 

of such rules, but any rule that involves rewriting a hypergraph that is larger than the 

hypergraph that represents the whole universe can never apply, so at least at any given 

point in the evolution of the system, the number of rules to consider is finite. But like with 

the many other kinds of limits we have discussed, we can still imagine taking the limit of all 
infinitely many possible rules.

As a toy example of a rule‐space multiway system, consider all inequivalent 2 → 2 rules on 

strings of As and Bs:

{AA → AA, AA → AB, AA → BB, AB → AA, AB → AB, AB → BA}

420



We can immediately construct the rule‐space multiway graph for these rules (here starting 

from all possible length‐4 sequences of As and Bs):

AAAA

AABB

ABAB

ABBA AAAB

BAAB AABA

BABA ABAA

BBAA BAAA ABBB

BBBB BABB

BBAB

BBBA

Different branches of the rule‐space multiway system use different rules:
AB
ABBB

BABABB

AB
BABB

ABBB
BBABAB AB BABB

AAABAB BABBAB
ABBB

AAABBB ABAABB BABB
AA
ABBBAAA

AA AAB BB
AABB AAAAAB AB

AA AA
AAAB

ABAA
BB

ABAA
AAB AABB BBAB
ABB

AAAB BBAB
AAB BA

AAAB AAB BAAAAB AB
AAA

BAAAAA
ABB AAAB AABBAB

AA BAAB BAAB
BBBB AA

ABAA ABAA BB BBAB
AB
BAAA AA

AA BAABBAAB AB BAAA BAAAAA
BBBA BAA A

ABAB AAB
A

AB ABABAA AB
BAAB BAA BAAB

BBBAAAB AB
AAAABB BBBB

ABAB
AB AB

AAAA AAABAA AAAA
AB AA BAAB

AB BA ABAA AABBAA AA BBAA BAB
ABAB BBBA

BA
AABA

AA
ABA

ABAB BBAA
AB BB

AA
BBA

AAAAABA BABAAAA BAA
AABA AABABA BBAAAA BABA
ABABAA BA

AAAA AAAABA
AAAA AABAAA BABAA ABAAAAAAA AAABBA

AB
AA
ABBA

BABA
AB
AABA ABBA

AB
ABBA

421



One can include causal connections:

BABB
BBBB

ABBB
BBAB AABB

ABAB
A

BAB B B BAB AAAB
B BAB ABBB A BAAABB AA AA

A
BAAB AB AA BBBAAB BBBB

AB BAAB ABAB A AB
BB ABB ABB

BABB
BAAB AAABAB AA

AAAB ABBB
ABAAB BB

BBAB ABAA
B

AB AB
BA

B AAB A ABBB
BBAB BAB A BAB

A
ABAB

AA AA AAABB AB
BAAA

AB
B AAAA A

AB AA AAB ABAA AABBB AA A AB
AAB AB AA B

BAAB B
AA AAB

A
AB AAAB AABB AB
BAAB AA

AAAB
BAAAAB ABAB AB AB

AA AAA
A

AA AB ABA

BAABBAA AAA
B AAAAAA AA

BA AAA
BBAA AAAB AAA AA

AB ABAA

AB AA AAAB A
A AAA AA

B
BBAA ABAB AABAAAB ABBA

BB BAA
AB BA AA

ABAA B
AA AAA

AB
ABBA

BAAA AAAA
BAAA AA AA AAAA AAAA AB AA

BB ABBA
AA AAA

BB BABA
B

BBAA BBA ABA
AA BAA AAB

ABA ABA AA
BBBA

BABBAA BABABA AAAA

BBAA
AABA ABBA

BBBA BABA

But even removing multiedges, the full rule‐space multiway causal graph is complicated:

422



The “branchial graph” of the rule‐space multiway system, though, is fairly simple, at least 
a�er one step (though it is now really more of a “rule‐space graph”):

BABA BABB

BBBA
BBBB BAAA BAAB

AAAA
BBAB

ABAA

AABA
ABAB AABB

ABBA

ABBB BBAA
AAAB

At least in this toy example, we already see something important: the rule‐space multiway 

system is causal invariant: given branching even associated with using different rules, there
is always corresponding merging—so the graph of causal relationships between updating 

events, even with different rules, is always the same.

Scaling up to unbounded evolutions and unbounded collections of rules involves many 

issues. But it seems likely that causal invariance will survive. And ultimately one may 

anticipate that across all possible rules it will emerge as a consequence of the Principle of 
Computational Equivalence [1:𝕔12]. Because this principle implies that in the space of all 
possible rules, all but those with simple behavior are equivalent in their computational 
capabilities. And that means that across all the different possible sequences of rules that can 

be applied in the rule‐space multiway system there is fundamental equivalence—with the 

result that one can expect causal invariance. 

But now consider the role of the observer, who is inevitably embedded in the system, as part 
of the same rule‐space multiway graph as everything else. Just as we did for ordinary 

multiway graphs above, we can imagine foliating the rule‐space multiway graph, with the 

role of space or branchial space now being taken by rule space. And one can think of 
exploring rule space as effectively corresponding to sampling different possible descriptions 

of how the universe works, based on different underlying rules. 

But if each event in the rule‐space multiway graph is just a single update, based on a particu‐
lar (finite) rule, there is immediately a consequence. Just like with light cones in ordinary 

space, or entanglement cones in branchial space, there will be a new kind of cone that 
defines a limit on how fast it is possible to “travel” in rule space.

For an observer, traveling in rule space involves ascribing different rules to the universe, or 

in effect changing oneʼs “reference frame” for interpreting how the universe operates. (An 

“inertial frame” in rule space would probably correspond to continuing to use a particular 

423



 probably
rule.) But from the Principle of Computational Equivalence [1:𝕔12] (and specifically from the 

idea of computation universality (e.g. [1:𝕔11])) it is always possible to set up a computation 

that will translate between interpretations. But in a sense the further one goes in rule space, 
the more difficult  the translation may become—and the more computation it will require.

But now remember that the observer is also embedded in the same system, so the fundamen‐
tal rate at which it can do computation is defined by the structure of the system. And this is 

where what one might call the “translation cone” comes from: to go a certain “distance” in 

rule space, the observer must do a certain irreducible amount of computational work, which 

takes a certain amount of time. 

The maximum rate of translation is effectively a ratio of “rule distance” to “translation effort” 

(measured in units of computational time). In a sense it probes something that has been 

difficult  to quantify: just how “far apart” are different description languages, that involve 

different computational primitives? One can get some ideas by thinking about program size 

[144][145][146], or running time, but in the end new measures that take account of things, like 

the construction of sequences of abstractions, seem to be needed [147].

For our current discussion, however, the main point is the existence of a kind of “rule‐space 

relativity”. Depending on how an observer chooses to describe our universe, they may 

consider a different rule—or rather a different branch in the rule‐space multiway system—to 

account for what they see. But if they change their “description frame”, causal invariance 

(based on the Principle of Computational Equivalence) implies that they will still find a rule 

(or a branch in the rule‐space multiway system) that accounts for what they see, but it will 
be a different one.

In the previous section, we discussed equivalences between our models and other formula‐
tions. The fact that we base our models on hypergraph rewriting (or any of its many equiva‐
lent descriptions) is in a sense like a choice of coordinate system in rule space—and there 

are presumably infinitely many others we could use.

But the fact that there are many different possible parametrizations does not mean that there 

are not definite things that can be said. It is just that there is potentially a higher level of 
abstraction that can be reached. And indeed, in our models, not only have we abstracted away 

notions of space, time, matter and measurement; now in the rule‐space multiway system we 

are in a sense also abstracting away the very notion of abstraction itself (see also [2]). 

424



���������������� ���������

������������� �������� ����������
The class of models studied here represent a simplification and generalization of the
trivalent graph models introduced in [1:9] and [87] (see also [148]).

The methodology of computational exploration used here has been developed particularly
in [5][31][1]. Some exposition of the methodology has been given in [149].

The class of models studied here can be viewed as generalizing or being related to a great
many kinds of abstract systems. One class is graph rewriting systems, also known as graph
transformation systems or graph grammars (e.g. [150]). The models here are generalizations
of both the double-pushout and single-pushout approaches. Note that the unlabeled graphs
and hypergraphs studied here are different from the typical cases usually considered in
graph rewriting systems and their applications.

Multiway systems as used here were explicitly introduced and studied in [1:p204] (see also
[1:p938]). Versions of them have been invented many times, most o�en for strings, under
names such as semi-Thue systems [151], string rewriting systems [152], term rewriting
systems [65], production systems [153], associative calculi [154] and canonical systems
[153][155].

�������������������������������
An outline of applying models of a type very similar to those considered here was given in
[1:9]. Some additional exposition was given in [156][157][158]. The discussion here contains
many new ideas and developments, explored in [159].

For a survey of ultimate models of physics, see [1:p1024]. The possibility of discreteness in
space has been considered since antiquity [160][161][162][163]. Other approaches that have
aspects potentially similar to what is discussed here include: causal dynamical triangulation
[164][165][166], causal set theory [167][168][169], loop quantum gravity [170][171], pregeome-
try [172][173][174], quantum holography [175][176][177], quantum relativity [178], Regge
calculus [179], spin networks [180][181][182][183][184], tensor networks [185], superrelativity
[186], topochronology [187], topos theory [188], twistor theory [128]. Other discrete and
computational approaches to fundamental physics include:
[189][190][191][192][193][194][195][196].

The precise relationships among these approaches and references and the current work are
not known. In some cases it is expected that conceptual motivations may be aligned; in
others specific mathematical structures may have direct relevance. The latter may also be
the case for such areas as conformal field theory [197], higher-order category theory [198],
non-commutative geometry [199], string theory [200].

425



����������� �� ������� �

������������������������������
A variety of new functions have been added to the Wolfram Function Repository to directly 

implement, visualize and analyze the models defined here [201].

������������������������������������
The class of models defined here can be implemented very directly just using symbolic 

transformation rules of the kind on which the Wolfram Language [98] is based. 

It is convenient to represent relations as Wolfram Language lists, such as {1,2}. One way to 

represent collections is to introduce a symbolic operator σ that is defined to be flat (associ-
ative) and orderless (commutative):

������ SetAttributes[σ, {Flat, Orderless}]

Thus we have, for example:

������ σ[σ[a, b], σ[c]]

������ σ[a, b, c]

We can then write a rule such as

{{x, y}} → {{x, y}, {y, z}}
more explicitly as:

������ σ[{x_, y_}]⧴ Module[{z}, σ[{x, y}, {y, z}]]

This rule can then be applied using standard Wolfram Language pattern matching:

������ σ[{a, b}] /. σ[{x_, y_}]⧴ Module[{z}, σ[{x, y}, {y, z}]]

������ σ[{a, b}, {b, z$393804}]

The Module  causes a globally unique new symbol to be created for the new node z every time 

it is used:

������ σ[{a, b}] /. σ[{x_, y_}]⧴ Module[{z}, σ[{x, y}, {y, z}]]

������ σ[{a, b}, {b, z$393808}]

426



But in applying the rule to a collection with more than one relation, there is immediately an 

issue with the updating process. By default, the Wolfram Language performs only a single 

update in each collection:

������ σ[{a, b}, {c, d}] /. σ[{x_, y_}]⧴ Module[{z}, σ[{x, y}, {y, z}]]

������ σ[{a, b}, {b, z$393812}, {c, d}]

As discussed in the main text, there are many possible updating orders one can use. A 

convenient way to get a whole “generation” of update events is to define an inert form of 
collection σ1 then repeatedly replace collections σ until a fixed point is reached: 

������ σ[{a, b}, {c, d}] //. σ[{x_, y_}]⧴ Module[{z}, σ1[{x, y}, {y, z}]]

������ σ[σ1[{a, b}, {b, z$393816}], σ1[{c, d}, {d, z$393817}]]

By replacing σ1 with σ at the end, one gets the result for a complete generation update:

������ σ[{a, b}, {c, d}] //. σ[{x_, y_}]⧴ Module[{z}, σ1[{x, y}, {y, z}]] /. σ1→σ

������ σ[{a, b}, {b, z$393821}, {c, d}, {d, z$393822}]

NestList applies this whole process repeatedly, here for 4 steps:

������ evol = NestList[# //. σ[{x_, y_}]⧴ Module[{z}, σ1[{x, y}, {y, z}]] /. σ1→σ &, σ[{1, 1}], 4]

������ {σ[{1, 1}], σ[{1, 1}, {1, z$393826}], σ[{1, 1}, {1, z$393826}, {1, z$393827}, {z$393826, z$393828}],
σ[{1, 1}, {1, z$393826}, {1, z$393827}, {1, z$393829}, {z$393826, z$393828}, {z$393826, z$393830},
{z$393827, z$393831}, {z$393828, z$393832}], σ[{1, 1}, {1, z$393826}, {1, z$393827},
{1, z$393829}, {1, z$393833}, {z$393826, z$393828}, {z$393826, z$393830}, {z$393826, z$393834},
{z$393827, z$393831}, {z$393827, z$393835}, {z$393828, z$393832}, {z$393828, z$393837},
{z$393829, z$393836}, {z$393830, z$393838}, {z$393831, z$393839}, {z$393832, z$393840}]}

Replacing σ by a Graph  operator, one can render the results as graphs:

������ evol /. σ → (Graph[DirectedEdge@@@{##}] &)

������  , , , , 

427



IndexGraph creates a graph in which nodes are renamed sequentially:

������ evol /. σ → (IndexGraph[DirectedEdge@@@{##}, VertexLabels→ Automatic] &)

1 1

1 5 4 2 3
4 2 3

1
������  , 2 1 , 2 3, , 13 8 6 7 10 9 

5 6 7

4 11 12 14 15

8
16

Here is the result with a different graph layout:

������ evol /.
σ → (IndexGraph[DirectedEdge@@@{##}, GraphLayout→ "SpringElectricalEmbedding"] &)

������  , , ,

, 

Exactly the same approach works for rules that involve multiple relations. For example, 
consider the rule:

{{x, y}, {x, z}} → {{x, z}, {x, w}, {y, w}, {z, w}}
This can be run for 2 steps using:

������ NestList[# //.
σ[{x_, y_}, {x_, z_}]⧴ Module[{w}, σ1[{x, z}, {x, w}, {y, w}, {z, w}]] /.

σ1→σ &, σ[{1, 1}, {1, 1}], 2]

������ {σ[{1, 1}, {1, 1}], σ[{1, 1}, {1, w$393851}, {1, w$393851}, {1, w$393851}],
σ[{1, w$393851}, {1, w$393851}, {1, w$393852}, {1, w$393852}, {1, w$393853},
{w$393851, w$393852}, {w$393851, w$393853}, {w$393851, w$393853}]}

428



Here is the result a�er 10 steps, rendered as a graph:

������ Nest[# //.
σ[{x_, y_}, {x_, z_}]⧴ Module[{w}, σ1[{x, z}, {x, w}, {y, w}, {z, w}]] /.

σ1→σ &, σ[{1, 1}, {1, 1}], 10] /. σ → (Graph[DirectedEdge@@@{##}] &)

������

������������������������������������
As an alternative to introducing an explicit head such as σ, one can use a system-defined 

matchfix operator such as AngleBracket  (entered as <, >) that does not have a built-in 

meaning. With the definition

������ SetAttributes[AngleBracket, {Flat, Orderless}]

one immediately has for example

������ 〈a, 〈b, c〉〉

������ 〈a, b, c〉

and one can set up rules such as

������ 〈{x_, y_}, {x_, z_}〉 ⧴ Module[{w}, 〈{x, z}, {x, w}, {y, w}, {z, w}〉]

�����������������
Instead of having an explicit “collection operator” that is defined to be flat and orderless, 
one can just use lists to represent collections, but then apply rules that are defined using 

OrderlessPatternSequence:

������ {{0, 0}, {0, 0}, {0, 0}} /. {OrderlessPatternSequence[{x_, y_}, {x_, z_}, rest___]}⧴
Module[{w}, {{x, z}, {x, w}, {y, w}, {z, w}, rest}]

������ {{0, 0}, {0, w$37227}, {0, w$37227}, {0, w$37227}, {0, 0}}

429



Note that even though the pattern appears twice, /. applies the rule only once:

������ {{0, 0}, {0, 0}, {0, 0}, {0, 0}} /. {OrderlessPatternSequence[{x_, y_}, {x_, z_}, rest___]}⧴
Module[{w}, {{x, z}, {x, w}, {y, w}, {z, w}, rest}]

������ {{0, 0}, {0, w$48054}, {0, w$48054}, {0, w$48054}, {0, 0}, {0, 0}}

������������������
Yet another alternative is to use the function SubsetReplace  (built into the Wolfram Lan-
guage as of Version 12.1). SubsetReplace  replaces subsets of elements in a list, regardless of 
where they occur:

������ SubsetReplace[{a, b, b, a, c, a, d, b}, {a, b}→ x]

������ {x, x, c, x, d}

Unlike ReplaceAll (/.) it keeps scanning for possible replacements even a�er it has done one:

������ SubsetReplace[{a, a, a, a, a}, {a, a}→ x]

������ {x, x, a}

One can find out what replacements SubsetReplace  would perform using SubsetCases:

������ SubsetCases[{a, b, c, d, e}, {_, _}]

������ {{a, b}, {c, d}}

This uses SubsetReplace  to apply a rule for one of our models; note that the rule is applied 

twice to this state (Splice is used to make the sequence of lists be spliced into the collection):

������ SubsetReplace[{{0, 0}, {0, 0}, {0, 0}, {0, 0}},
{{x_, y_}, {x_, z_}}⧴ Splice[Module[{w}, {{x, z}, {x, w}, {y, w}, {z, w}}]]]

������ {{0, 0}, {0, w$55383}, {0, w$55383}, {0, w$55383}, {0, 0}, {0, w$55384}, {0, w$55384}, {0, w$55384}}

This gives the result of 10 applications of SubsetReplace :

������ Nest[SubsetReplace[{{x_, y_}, {x_, z_}}⧴ Splice[Module[{w}, {{x, z}, {x, w}, {y, w}, {z, w}}]]],
{{1, 2}, {1, 3}}, 10] // Short

������ {{1, w$55543}, {1, w$55637}, {w$55490, w$55637},705, {w$55401, w$55452}, {2, w$55394}}

430



This turns each list in the collection into a directed edge, and renders the result as a graph:

������ Graph[DirectedEdge@@@%]

������

IndexGraph can then for example be used to relabel all elements in the graph to be sequen-
tial integers.

Note that SubsetReplace  does not typically apply rules in exactly our “standard updating
order”. 

���������������
Our models do not intrinsically define updating order (see section 6), and thus allow for 
asynchronous implementation with immediate parallelization, subject only to the local 
partial ordering defined by the graph of causal relationships (or, equivalently, of data flows). 
However, as soon as a particular sequence of foliations—or a particular updating order—is 
defined, its implementation may require global coordination across the system.

���������� ����������
A visual summary of the relationships between graph types is given in [202].

������������������������� �����
Graphs obtained from particular evolution histories, with particular sequences of updating 

events.  For rules with causal invariance, the ultimate causal graph is independent of the 

sequence of updating events.

�������� ����
Hypergraph whose nodes and hyperedges represent the elements and relations in our 
models. Update events locally rewrite this hypergraph. In the large-scale limit, the hyper-
graph can show features of continuous space. The hypergraph potentially represents the 

“instantaneous” configuration of the universe on a spacelike hypersurface. Graph distances 

431



 hypersurface.
in the hypergraph potentially approximate distances in physical space.

������� �������������� ��������� ������
Graph with nodes representing updating events and edges representing their causal relation-
ships. In causal invariant systems, the same ultimate causal graph is obtained regardless of 
the particular sequence of updating events. The causal graph potentially represents the 

causal history of the universe. Causal foliations correspond to sequences of spacelike 

hypersurfaces. The effect of an update event is represented by a causal cone, which poten-
tially corresponds to a physical light cone. The translation from time units in the causal 
graph to lengths in the spatial graph is potentially given by the speed of light c.

����� ��������������������� �����
Graphs obtained from all possible evolution histories, following every possible sequence of 
updating events.  For rules with causal invariance, different paths in the multiway system 

lead to the same causal graph.

432



����� ���������� ������������ ��� �����
Graph representing all possible branches of evolution for the system. Each node represents 
a possible complete state of the system at a particular step. Each connection corresponds to 

the evolution of one state to another as a result of an updating event. The multiway graph 

potentially represents all possible paths of evolution in quantum mechanics. In a causal 
invariant system, every branching in the multiway system must ultimately reconverge.

����� ����������������� ����
Graph representing both all possible branches of evolution for states, and all causal relation-
ships between updating events. Each node representing a state connects to other states via 

nodes representing updating events. The updating events are connected to indicate their 
causal relationships. The multiway states+causal graph in effect gives complete, causally 

annotated information on the multiway evolution.

433



����� ���������� ����
Graph representing causal connections among all possible updating events that can occur in 

all possible paths of evolution for the system. Each node represents a possible updating 

event in the system. Each edge represents the causal relationship between two possible 

updating events. In a causal invariant system, the part of the multiway causal graph corre-
sponding to a particular path of evolution has the same structure for all possible paths of 
evolution. The multiway causal graph provides the ultimate description of potentially 

observable behavior of our models. Its edges represent both spacelike and branchlike 

relationships, and can potentially represent causal relations both in spacetime and through 

quantum entanglement. 

���������� ����
Graph representing the common ancestry of states in the multiway system. Each node 

represents a state of the system, and two nodes are joined if they are obtained on different 
branches of evolution from the same state. To define a branchial graph requires specifying a 

foliation of the multiway graph. The branchial graph potentially represents entanglement in 

the “branchial space” of quantum states.

434



������� ��������
I have been developing the ideas here for many years [203]. I worked particularly actively on 

them in 1995–1998, 2001 and 2004–2005 [148][1]. But they might have languished forever had it 
not been for Jonathan Gorard and Max Piskunov, who encouraged me to actively work on them 

again, and who over the past several months have explored them with me, providing extensive 

help, input and new ideas. For important additional recent help I thank Jeremy Davis, Sushma 

Kini and Ed Pegg, as well as Roger Dooley, Jesse Friedman, Andrea Gerlach, Charles Pooh, Chris 
Perardi, Toni Schindler and Jessica Wong. For recent input I thank Elise Cawley, Roger Ger-
mundsson, Chip Hurst, Rob Knapp, José Martin-Garcia, Nigel Goldenfeld, Isabella Retter, Oliver 
Ruebenkoenig, Matthew Szudzik, Michael Trott, Catherine Wolfram and Christopher Wolfram. 
For important help and input in earlier years, I thank David Hillman, Todd Rowland, Matthew 

Szudzik and Oyvind Tafjord. I have discussed the background to these ideas for a long time, with 

a great many people, including: Jan Ambjørn, John Baez, Tommaso Bolognesi, Greg Chaitin, 
David Deutsch, Richard Feynman, David Finkelstein, Ed Fredkin, Gerard ’t Hoo�, John Milnor, 
John Moussouris, Roger Penrose, David Reiss, Rudy Rucker, Dana Scott, Bill Thurston, Hector 
Zenil, as well as many others, notably including students at our Wolfram Summer Schools over 
the past 17 years. My explorations would never have been possible without the Wolfram 

Language, and I thank everyone at Wolfram Research for their consistent dedication to its 
development over the past 33 years, as well as our users for their support.

������� ������������� ����� ��
Extensive tools, data and source material related to this document and the project it 
describes are available at wolframphysicsproject.org.

This document is available in complete computable form as a Wolfram Notebook, including 

Wolfram Language input for all results shown. The notebook can be run directly in the 

Wolfram Cloud or downloaded for local use.

Specialized Wolfram Language functions developed for this project are available in the 

Wolfram Function Repository [204] for immediate use in the Wolfram Language. A tutorial 
of their use is given in [205].

The Registry of Notable Universes [8] contains results on specific examples of our models, 
including all those explicitly used in this document.

An archive of approximately 1000 working notebooks associated with this project from 1994 to the 

present is available at wolframphysicsproject.org. In addition, there is an archive of approximately 

500 hours of recorded streams of working sessions (starting in fall 2019) associated with this project.

435