#### Computational: a rule gives a successor for each state

Computational: a rule gives a successor for each state

#### Multicomputational: a rule relates multiple states

Multicomputational: a rule relates multiple states

One state is transformed into several; several states are used to determine one new state

In FindEquationalProof, one combines two theorems to get another one.... [inference rule] (“superposition”)

(“left superposition” vs. “right superposition”)

(“left superposition” vs. “right superposition”)

“Critical pair lemmas are generated by completions, superpositions, and paramodulations”

“Substitution lemmas are generated by resolutions and factorings”

#### String-based 21

String-based 21

StringJoin

In[]:=

MIMWGraph[{{n_,m_}:>{StringJoin[n,m]}},{"A","B"},3,UniqueTokens->True,Overlaps->False,VertexLabeling->True]

Out[]=

In[]:=

MIMWGraph[{{n_,m_}:>{StringJoin[n,m]}},{"A","B"},3,UniqueTokens->True,Overlaps->True,VertexLabeling->True]

Out[]=

In[]:=

MIMWGraph[{{n_,m_}:>Flatten@{StringJoin[n,m],StringTakeDrop[n,UpTo[1]]}},{"A","B"},3,UniqueTokens->True,Overlaps->False,VertexLabeling->True]

Out[]=

In[]:=

MIMWGraph[{{n_,m_}:>{StringJoin[n,m]},{n_}:>StringTakeDrop[n,UpTo[1]]},{"A","B"},3,UniqueTokens->True,Overlaps->False,VertexLabeling->True]

Out[]=

In[]:=

MIMWGraph[{{n_,m_}:>{StringJoin[n,m]},{n_}:>StringTakeDrop[n,UpTo[1]]},{"A","B"},2,UniqueTokens->True,Overlaps->False,VertexLabeling->True]

Out[]=

In[]:=

MIMWGraph[{{n_,m_}:>{StringJoin[n,m]},{n_,m_}:>StringTakeDrop[n,UpTo[1]]},{"A","B"},2,UniqueTokens->True,Overlaps->True,VertexLabeling->True]

Out[]=

#### Spacelike-separated events: one token, multiple events

Spacelike-separated events: one token, multiple events

I.e. several things can happen to one token

The most recent common ancestor is a token

The most recent common ancestor is a token

#### Branchlike-separated events: one event can lead to multiple histories

Branchlike-separated events: one event can lead to multiple histories

The most recent common ancestor is an event

#### In the WM case, each token is a hyperedge .... but there are other “identities” between hyperedges not captured by the evolution

In the WM case, each token is a hyperedge .... but there are other “identities” between hyperedges not captured by the evolution

#### Fully deduplicated:

Fully deduplicated:

Petri net analogy: Expressions are places; events are transitions

#### As soon as there is branchlike or spacelike separation of events, there is ambiguity

As soon as there is branchlike or spacelike separation of events, there is ambiguity

#### But with causal invariance, there is ultimately no ambiguity in the causal graph

But with causal invariance, there is ultimately no ambiguity in the causal graph

[ I.e. there is invariance in the causal graph, independent of the reference frame / evaluation order ]

#### Is category theory capturing the inexorable structure?

Is category theory capturing the inexorable structure?

#### A causal graph is only nontrivial if the full multihistory is a nontrivial multiway graph

A causal graph is only nontrivial if the full multihistory is a nontrivial multiway graph

Sequential always does only one update at a time ... even though multiple can be done....

### Where does inexorable structure come from?

Where does inexorable structure come from?

E.g. for CA, there is an inexorable light cone...

#### Can there be a multiway system in which there is no notion of space?

Can there be a multiway system in which there is no notion of space?

E.g. numerical multiway systems

A foliation of this says which numbers can appear “at the same time”

Is the causal graph the same as the multiway graph in this case? [And there seems to be no space; only branchlike separation]

To find places where different foliations are possible, look for branch pairs

This isn’t a valid foliation:

#### What antichains exist here? These are the possible foliations...

What antichains exist here? These are the possible foliations...

### Existence of multiple reference frames forces a certain continuity.... (?)

Existence of multiple reference frames forces a certain continuity.... (?)

Continuity must come from small variations of the foliation .... within the family of foliations......

#### Computational irreducibility + computational boundedness

Computational irreducibility + computational boundedness