# Introduction to AI

In the book Introduction to artificial intelligence Ertel asks this question (exercise 1.5(a)):

Why is a deterministic agent with memory not a function from the set of all inputs to the set of all outputs, in the mathematical sense?

I don't even know why this statement would be true! I mean, I understand that mathematical functions are "pure functions" and thus do not have "side effects." An agent with memory can be viewed as a function with side effects and thus can't be a pure math function. But what prevents from choosing as the set of all inputs the "spacetime" of input? Doing so brings back the agent with memory to a pure function which can be represented as a mathematical functions!

Am I missing something or is what I say correct?

Thanks

• I'm not sure that I am understanding your question, so I could be totally missing the point that you are trying to make. First of all, I am unable to decipher the phrase "spacetime of input". Therefore, I can only assume that the inputs to the AI are as follows: (1) Reality, as it currently exists at the time that the AI is going to be launched. ...see next comment Oct 18, 2020 at 22:30
• (2) A deterministic conclusion for what reality will be at any time point in the future. This conclusion would have to include a conclusion of exactly how the AI will be affecting reality, and (simultaneously) how reality will be affecting AI's memory and its programming. Based on these ideas, I see nothing untenable in the idea that all of the future behavior of the AI is pre-determined, therefore merely a function of reality at the time that the AI is "turned on". Oct 18, 2020 at 22:30
• spacetime would be the space of input $I$ in time $T$ $I\times T$ Oct 19, 2020 at 0:27

The behavior of a deterministic agent with memory evolves as its internal state (i.e. memory) evolves. Mathematical functions do not have an evolving internal state in this sense. A mathematical function must satisfy the requirement that every input is mapped to only one output. Consider the function $$f$$ from input set $$I$$ to output set $$O$$. $$f$$ cannot take an element of $$I$$ to more than one element of $$O$$ and still be considered a function in the mathematical sense.
However, a deterministic agent with memory state $$B$$ might map some input $$a$$ to output $$b$$ while, at a later time, that same deterministic agent with memory state $$C$$ could map input $$a$$ to output $$c$$. This is inconsistent with the mathematical notion of a function.
• Ok that I understand that. What a don't get is why the imput history could not be considered as an imput if we chose the domain to be $I\times T$ Oct 19, 2020 at 0:24