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The systems studied in chaos theory are deterministic. If the initial state were known exactly, then the future state of such a system could theoretically be predicted. However, in practice, knowledge about the future state is limited by the precision with which the initial state can be measured, and chaotic systems are characterized by a strong dependence on the initial condit
This approach is not the same as that of probabilistic algorithms, but the two may be combined. For non-probabilistic, more specifically deterministic , algorithms, the most common types of complexity estimates are the average-case complexity and the almost-always complexity.
When using a probabilistic approach to design, the designer no longer thinks of each variable as a single value or number. Instead, each variable is viewed as a continuous random variable with a probability distribution. From this perspective, probabilistic design predicts the flow of variability (or distributions) through a system. [4]
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is ...
Monte Carlo simulations invert this approach, solving deterministic problems using probabilistic metaheuristics (see simulated annealing). An early variant of the Monte Carlo method was devised to solve the Buffon's needle problem , in which π can be estimated by dropping needles on a floor made of parallel equidistant strips.
Many different probabilistic numerical methods designed for ordinary differential equations have been proposed, and these can broadly be grouped into the two following categories: Randomisation-based methods are defined through random perturbations of standard deterministic numerical methods for ordinary differential equations.
One of the popular examples in computer science is the mathematical models of various machines, an example is the deterministic finite automaton (DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example ...
Deterministic algorithms are by far the most studied and familiar kind of algorithm, as well as one of the most practical, since they can be run on real machines efficiently. Formally, a deterministic algorithm computes a mathematical function ; a function has a unique value for any input in its domain , and the algorithm is a process that ...