A profoundly influential idea, but
While the notion of state has been a profound influence on intellectual progress as a basis of mathematical rigor and mechanism of truth it has also been a barrier to progress. The problem is that for any behavior to which state space is being applied the behavior must conform to the demands of the state space. Put another way, reality must conform to the demands of the model instead of the model conforming to the actualities of reality; concurrent behaviors, for instance, must be sequenced. This is why computer science is a diaphanous castle in the clouds with no founding in reality.
The notion of state and state space
A state is a particular condition of differentness from a range of possible conditions of differentness of a particular manifestation of state. There may be many particular manifestations of state forming a space of manifest states. Typically the range of possible conditions of differentness are shared among manifestations of state, numeric values for instance, allowing the possibility of identical conditions of differentness among states. Condition of differentness does not uniquely individuate a state so each manifestation of state is further individuated with a unique identity in the state space which may or may not involve a notion of structure of the state space. This state space individuation of each manifestation of state does not change while the content of each state can change. A state sample is of all manifestations of state in the state space with each’s condition of differentness.
A state space must be constant
The state space remains constant with a population of state manifestations retaining their unique spatial individuation. It is the constancy of the state space that allows an external sampling agency to orient to the internal progress of the change behavior and to inspect the state of the state space state manifestation by state manifestation. There must be a referential constancy between the change behavior and the sampling agency.
The imposition
A change behavior can change the condition of differentness of state manifestations but may not change the spatial individuation of the state manifestations.
A state space must be determinate
To have a predicted state to compare to a sampled state the change behavior must be deterministic in relation to the state space.
The imposition
A change behavior cannot change states with indeterminate conditions of differentness.
A sampled state space must be stable
A state space can be reliably sampled only when all of its states are stable, i.e.,when no state is in the midst of change. A sampling agency must be able to discern from the change behavior when a stable sample can be taken or it must control the change behavior to establish a stable samplable state space. The change behavior may be halted and its internal state space inspected. Or the internal state space may be extracted from the change behavior and inspected externally. In either case the sampling must be of a stable state space relative to the change behavior.
The imposition
The change behavior must explicitly make available intervals of stable state or it must submit to an external control that can establish a stable state space. The first order appraoch is for a change behavior to be one state change “at a time” sequential in that each change is completed before the next change is begun. Halted after the completion of any state change the change behavior presents a stable state until it is restarted. A change behavior can halt itself or the external agency can halt it.
State space sampling must be coordinated
If there is a predicted state to compare to a sampled state the sampling must be specifically coordinated to the progress of the change behavior. This requires coordination between the sampling agency and the change behavior.
The imposition
A change behavior must accommodate the coordination of sampling by providing a behavior discernible by the sampling agency at the appropriate state change.
If a change behavior changes multiple states “at any time” it presents concurrent uncoordinated state change with no intrinsically discernible instants of stable state that offer opportunity for sampling and with no ordered occurrence of state changes that allows coordination of any specific state change to a specific sampled state.
The implication
The notion of state space fails in the context of concurrent change behavior. A concurrent change behavior clearly has internal manifestations of conditions of differentness and just as clearly the notion of state and state space cannot account them.
The imposition
Concurrent behavior cannot be allowed. All concurrent behavior must be mapped to sequential behavior.
Computer science has never confronted concurrency directly but has always tried to characterize it in terms of sequentiality, that convenient street lamp at the end of the dark alley discouraging one from looking into the alley.
Conforming all change behaviors to these impositions of observational needs leads to an internal samplable state space becoming viewed as endemic to all change behaviors whether the behavior might be sampled or whether it is even samplable. This internalization of state space is embodied by the Turing machine in its state machine and its paper tape. Change behaviors that do not conform, such as concurrent behaviors and neural networks do not qualify.
The notion of state space is about making sense for humans. The state space and being able to sample it is a most convenient way to externally consider, talk of, observe, control and understand a change behavior and humans explicitly constrain and form a change behavior for it to be conveniently characterizable and observable.
The notion of state space is deeply embedded in our intellectual culture and is difficult to transcend but transcended it must be. The humans must be removed from the works. There are more general ways to account change behavior that encompass concurrency and much more.
Karl Fant