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agapito

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All help appreciated. am

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In summary, "functions that are effectively computable in the informal sense" are functions that can be computed by an algorithm, even if that algorithm takes some time and storage.

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agapito

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All help appreciated. am

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Evgeny.Makarov

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agapito

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Evgeny.Makarov said:It would be helpful to know the context. If "effectively" means "with reasonable resources, i.e., time and storage", then it means precisely that: with reasonable resources, which is not a precise judgment. We could make it more precise, for example, by saying that the function is computable in polynomial time, or in time $O(t^5)$ on an unlimited register machine. If "effectively" means computable at all, then it means there is an something like an algorithm, which can be implemented on reasonable devices like a modern computer with unlimited memory. Here "reasonable" means in the context of computation theory, not engineering. For example, a pushdown automaton is not a reasonable device because its computational power is known to be strictly less than that of a Turing machine. Showing that a function is computable in the formal sense would involve showing that it is computable according to a precise definition of some universal device, such as Turing machines, Markov algorithms, Kleene recursive functions, etc. The Church-Turing thesis says that the informal and the formal senses of computability coincide.[/QUOTE

Thank you very much. So, if we state that a function is effectively computable in the informal sense, we are simply acknowledging the existence of an algorithm that can be used to compute it for each value of its domain. Is that correct?

Again thanks a lot for helping me out, agapito

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Evgeny.Makarov

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agapito

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Evgeny.Makarov said:

Thanks again for your patience, you have been very helpful. agapito

Formal numerical functions follow a strict set of rules and conventions, while informal numerical functions may be more flexible and less structured.

Formal numerical functions are often used to perform calculations and analyze data in a precise and reliable manner, making them important tools in scientific research.

Examples of formal numerical functions include statistical tests, mathematical models, and algorithms.

Informal numerical functions can include back-of-the-envelope calculations or estimations based on personal experience or intuition.

Formal numerical functions should be used when accuracy and precision are important, while informal numerical functions may be suitable for quick estimations or exploratory analysis.

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