# Does Co-occurrence Information Differ Qualitatively From Shannon and Algorithmic Information?

In Distributed Computation as Hierarchy Michael Manthey argues that co-occurrence of indistinguishables (critical for quantum theory) supplies spatial information that qualitatively differs from both Shannon and Algorithmic information. He offers a thought experiment called The Coin Demonstration:

The coin demonstration - Act I. A man stands in front of you with both hands behind his back, whilst you have one hand extended in front of you, palm up. You see the man move one hand from behind his back and place a coin on your palm. He then removes the coin with his hand and moves it back behind his back. After a brief pause, he again moves his hand from behind his back, places what appears to be an identical coin in your palm, and removes it again in the same way. He then asks you, “How many coins do I have?”.

The coin demonstration - Act II. The man now extends his hand and you see that there are two coins in it. [The coins are of course identical.]

The coin demonstration - Act III. The man now asks, “Where did that bit of information come from??”

The "bit" he refers to distinguishes between one of two possibilities:

1. There is only one coin.

2. There is more than one coin (that are indistinguishable except spatially when they co-occur).

In this Demonstration, the bit of information states that there is more than one coin.

He offers Answers to frequently stated Objections:

O: Whatever you do, it can be simulated on a TM.
A: You can’t ‘simulate’ co-occurrence sequentially, cf. the Coin demo.

O: But you can only check for co-occurrence sequentially - there’s always a ∆t.
A: This is a technological artifact: think instead of constructive/destructive interference - a phase difference between two wave states can be expressed in one bit.

O: Simply define a TM that operates on the two states as a whole - the “problem” disappears.
A: This amounts to an abstraction, which hierarchical shift changes the universe of discourse but doesn’t resolve the limitation, since one can ask this new TM to ‘see’ a co-occurrence at the new level. This objection thus dodges the issue.

O: Co-occurrence is primitive in Petri nets, but these are equivalent to finite state automata.
A: The phase web in effect postulates growing Petri nets, both in nodes and connections. All bets are then off.

He then uses geometric algebra to describe a concept of computation that is to space as algorithmic computation is to time in physics. He further suggests that the Linda coordination language be extended to include co and notco operations for co-occurring and not co-occurring situations respectively to implement this new computational paradigm.

It seems to me that this all boils down to a question of whether there is a qualitative difference between his notion of spatial information, and the more widely used notions of information attributed to Shannon Information Theory and Algorithmic Information Theory.

Does he adequately address the objections to his introducing a new kind of information or is his spatial notion of information reducible to Shannon and/or Algorithmic Information?

• Despite the fact that he uses the word "information" in his paper, I don't see where he gives a qualitative definition of "co-occurrence information" that you could use to say things like "the information content of a random string of $n$ symbols, each being $1$, $2$, or $3$, is $\log_2 \! 3 \cdot n,\!$" the way you can for Shannon or algorithmic information. So I would say that your question cannot be answered, because there is no qualitative concept of "co-occurrence information". Commented Jul 2, 2023 at 12:20