Two Dimensions and Beyond From: “ A New Kind of Science” by Stephen Wolfram Presented By: Hridesh Rajan

Outline Adding dimensions to CA. Turing machines. Extension of substitution systems to 2-D. Network systems and their evolution. Multiway systems. Systems based on constraints. Conclusion and remarks.

Adding Dimensions to CA: 2D Rule: Cell => black if any neighbor is black Variations of this rule gives interesting patterns e.g. Cell => black if just one of all four of its neighbors are black [on page 171 of NKS, more on page 173,174] Key Observation: Adding dimensions does not gives additional complexity to the patterns.

What About 3-D? As Wolfram observes: Some new phenomena can be seen but basic behavior remains the same. For example - rule: cell => black if any of its 6 neighbors are black produces a 3-D pattern [on page 182 of NKS] analogous to the 2-D rule on the previous slide.

Turing Machines 1D 2D 123 Three States of the 2-D Turing Machine

Behavior Exhibited by 2-D TMs Like 1-D TMs with 2-3 possible states only repetitive and nested behavior is possible. With 4-states more complex behavior but still rare. [Examples on page 185 and 186 of NKS]

Substitution Systems in 2-D Like 1-D substitution systems discussed in chapter 3 nested and repetitive behavior is observed. [Examples on Page 187,188 of NKS] Even after removing the rigid grid we don’t get complexity. [Examples on Page 189,190 of NKS] So what produces complexity? - Interaction between components - Rigid grid makes it easier to specify interaction

Network Systems Collection of nodes. Connection between these nodes. Rules to specify how the connections will change. Wolfram restricts further discussion to nodes with exactly 2 outgoing connections. [Examples of such network systems on Page 194 of NKS] Wolfram then goes on to show somewhat obvious fact that by changing the pattern of interconnection it is possible to get network system resembling multi-dimensional arrays.

Evolution of Network Systems

[This rule and more are presented on page 199 of NKS]

Behavior of This Evolution Wolfram considers several other rules including addition on new nodes, addition of new nodes + rerouting etc. [ Figure a and b on page 200] Behavior of this “evolution” also seems similar to substitution systems for the same reasons. (lack of interaction). Introducing dependency upon one hop neighbor does not leads to complex behavior.[ Figure a,b,c on page 201] However dependency on 2 hop neighbor introduces randomness in the evolution. [ Figure a,b,c on page 202]

Multiway Systems So far we have considered systems with single result state for a given state. - Multiway Systems can have more than one result state for a given state. [Non deterministic systems] - Distinctive feature is that all possible results are kept. The behavior exhibited by these systems is still simple and repetitive, however it is difficult to study these systems due to exponential growth. [Non-determinism does not increases the power.]

Systems Based on Constraints Rules are replaced by constraints that must be true. Are there simple set of constraints leading to complex behavior – Wolfram says “No” for 1-D systems. For 2-D cases there is no proof that it is not possible however examples on page 212 shows repetitive simple patterns that satisfy most of the simple constraints. Even complicating the constraints little bit does not helps in achieving complex behavior as displayed by examples on page 214 and 215.

Complex Patterns Further complicating the constraints yields few complex patterns as shown on page 219. More complex patterns can only be obtained by explicit construction. [ Example of such a pattern based on rule 30 is shown on page 221]

Conclusion “ So the fact that traditional science and math tends to concentrate on equations that operate like constraints provides yet another reason for their failure to identify the fundamental phenomenon of complexity that I discuss in this book” – Stephen Wolfram in his book “A New Kind of Science”