**SMN and System Theory**

See also related discussions.

In this discussion I will gradually map out some of the parallels between system matrix notation (SMN) and system theory. In this manner I show that SMN is a mathematical process that naturally gives rise to a system theoretic modeling paradigm.

First a table to quickly summarise the key correlations:

System Theory |
SMN |

primitive systems or observables | system states or SV elements |

single system | separate probability distributions |

Meta System Transition (MST) | Ψ operator, nested SM's and relational constructs |

collective system | collective or compound probability distributions |

actual present moment structure | state vector |

space or relational context | system matrix or interaction matrix |

time | iteration |

existence - being and doing | simulation - computation and representation |

change or behaviour | iterated system matrix or state transformation matrix |

system input | system matrix row |

system output | system matrix column |

system | group of related SV states and associated SM components |

sub/super system | relational construct (flat matrix) or nested SM's |

variable and overlapping system boundaries | variable and overlapping relational constructs |

systems experience different systems via different interaction paths | matrix mathematics, and input and output filters determine the causal network that binds systems together |

When a group of systems are formed into a super system during the system design phase their system states are interlinked within the SM so that they interact and this forms a relational construct that is a causal bond between them. If they are to respond from a common collective state space the system states must be passed through a Ψ operator to bring them into a compound state space or if they are to interact closely and manifest collective behaviour they must be strongly relating in the system matrix. Instead of relying on relational constructs within the SM one can also nest SM's where an SM element is itself an SM and the corresponding SV element is an SV. This represents sub/super system hierarchies and effectively implements private method access in C++, where the sub systems are not externally accessible other than via a system level interface because the existential space has been hierarchically partitioned. If the sub systems are integrated using a flat relational construct this is equivalent to public method access in C++, where the sub systems are externally accessible, then all systems exist within the one existential space and any system can theoretically interact with any system. Then one can have overlapping system boundaries, where a system's involvement as a sub system within a super system depends on interaction bandwidths that may vary over time and that may be shared between many systems, thus system hierarchies are not fixed and exclusive but are in fact perceptual phenomena that arise from an interpretation of the underlying network of relations. Hence different perceptual systems can perceive very different system hierarchies, compare the perspectives of a human and a neutrino that passes straight through the earth and barely interacts with it, the human says the wheel is a sub system of the vehicle but the neutino says that the whole Earth was like a very fine mist.

According to Klir the five fundamental conceptual aspects to a system are:

Klir |
SMN |

state variables | state vector elements |

range and resolution of state variables | Finite Discrete Data |

permanent behaviour | system matrix |

universe and coupling description | systemic idiom |

state space description | permutation space |

The state vector (SV) stores existential states either as probability distributions or as symbolic values. The system matrix (SM) brings every state into pairwise composition with every state. Thus in each iteration each state is potentially a function of all states thus implementing massively parallel semantic processing and self-reference. From this mathematical foundation, through iterative generation a system theoretic idiom or system algebraic space arises out of which complex systems may form. Finally the iterative multiplication of SM and SV manifests systems that exist within a world and experience their world from that present moment perspective. This dynamical space is the existential space that underlies physical space. It is a dynamical process oriented system theoretic existential space within which manifest forms may arise.

**Conclusion:**

SMN provides a model of a general finite discrete distributed computational process,

SMN provides a model of general systems,

SMN models all aspects of systems, both static such as system hierarchies and dynamic such as present moment existence.

SMN also has strong parallels with Object Oriented programming, which I will explore at a later date.