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Temporal Complexity Theory

Temporal Complexity Theory would examine how the computational complexity of algorithms varies over time and conditions, offering insights into the changing nature of 'difficulty' in problem-solving. Unlike classical computational complexity theory, which largely concerns itself with worst-case or average-case scenarios for algorithms, Temporal Complexity would consider fluctuations over time and other dynamic variables. This could include computational workloads, adaptive algorithms that change strategies depending on current circumstances, or the changing structure of the data being processed.

In essence, Temporal Complexity Theory could serve as a bridge between dynamic systems theory and classical computational theory, offering a more nuanced understanding of how computational processes evolve over time. This could be particularly useful for domains like climate forecasting, where understanding the computational complexity of models in changing conditions could be invaluable.

Axiom 1: Dynamic Workload Principle

The computational workload W(t) of an algorithm is a function of time t and may be subject to fluctuations based on internal or external factors.

Axiom 2: Temporal Complexity Function

For every algorithm A, there exists a Temporal Complexity Function \( C_A(t) \) that describes how the computational complexity varies with time.

Axiom 3: Non-Stationarity

The Temporal Complexity Function  \( C_A(t) \) is non-stationary, meaning that its statistical properties can change over time.

Axiom 4: Initial State Sensitivity

The Temporal Complexity Function is sensitive to initial conditions, i.e., slight variations in the initial state can lead to significantly different complexity trajectories.

Axiom 5: Adaptability Principle

Algorithms can adapt their computational strategies over time based on current and past information, denoted by  \( A_t \), an adaptive version of A.

Axiom 6: Data Structure Dynamics

The underlying data structure upon which an algorithm operates can also change over time, denoted as D(t).

Axiom 7: Exogenous Influence

External parameters E(t) can affect the Temporal Complexity Function  \( C_A(t) \). These parameters can include hardware efficiency, parallelism, and other environmental factors.

Axiom 8: Uncertainty Principle

There is an inherent uncertainty U(t) associated with predicting the future state of  \( C_A(t) \) due to both internal and external variability.

Axiom 9: Time-Bounded Optimality

An algorithm A is considered temporally optimal if, for a given time interval [t_1​,t_2​], there exists no other algorithm B with a lower integrated Temporal Complexity  \[ \int_{{t_1}}^{{t_2}} C_B(t) \, dt \].

Axiom 10: Computational-Environmental Equilibrium

Over long time horizons, the interaction between an algorithm and its environment reaches a dynamic equilibrium  \( E_q \)​, which can be described by a set of differential equations.