The internet includes a lot of garbage, so naturally what we're seeing from Google search summaries, for example, is best ignored IMO. The more constrained the problem space the more likely you'll get worthwhile results, but the unfiltered internet will probably never be that. In fact, even simple Google and Bing searches are corrupted by Google and Microsoft allowing what are in a way scam web sites to be listed first on search results (for example) for what are really US government programs, just by paying the required results inclusion fees. The Wall Street Journal ran an article about this a while back. My point is, is the internet easily useful in many cases for non-experts on any topic? I'm not so sure.
I think it's easy to argue that AI, even the transformer-based LLMs, can be very useful with curated data sources, and this is probably the future of practical AI assistants. When I see one for computer science, I'll probably sign up to try it. Reading a large of number of (sometimes poorly written) papers to learn a new topic is tedious. It can be done, I've done it many times, but I think an LLM with curated data sets would make it a lot more efficient.
>When I see one for computer science, I'll probably sign up to try it.
Github Copilot is pretty good. It's mostly used for programming questions, but it's ok with general theory also For example,
Q: what is the halting problem
A: The Halting Problem is a fundamental problem in computer science and the theory of computation. It was first formulated by Alan Turing in 1936. The problem is to determine, given a description of a computer program and an input, whether the program will eventually halt (stop executing) or continue to run forever.
Key Points:
- Undecidability: Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. This means there is no single program that can determine whether any given program will halt or run indefinitely.
- Implications: The halting problem is a fundamental result in the theory of computation, demonstrating the limits of what can be computed. It shows that there are well-defined problems that no algorithm can solve.
- Proof: Turing's proof involves a technique called diagonalization and a form of self-reference, showing that if such a halting algorithm existed, it could be used to construct a program that contradicts its own behavior.
Example:
Consider a program H that takes another program P and an input I and returns true if P halts on I and false otherwise. Turing's proof shows that constructing such a program H is impossible.
Formal Definition:
Given a program P and an input I, the halting problem asks whether P(I) halts. Formally, it is undecidable to determine: [ H(P, I) = \begin{cases} \text{true} & \text{if } P(I) \text{ halts} \ \text{false} & \text{if } P(I) \text{ does not halt} \end{cases} ]
This result has profound implications for computer science, particularly in understanding the limits of automated reasoning and computation.
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Naturally, you will then ask:
what is diagonalization and a form of self-reference?
A: Diagonalization and self-reference are techniques used in theoretical computer science and mathematics to prove the undecidability of certain problems, such as the Halting Problem.
Diagonalization:
Diagonalization is a method introduced by Georg Cantor to show that some infinities are larger than others. Alan Turing adapted this method to prove the undecidability of the Halting Problem.
Key Idea:
- Construct a Table: Imagine a table where each row represents a program and each column represents an input. The cell at row i and column j indicates whether program P_i halts on input I_j.
- Diagonal Elements: Consider the diagonal elements of this table (i.e., the elements where the row and column indices are the same).
- Contradiction: Construct a new program that behaves differently from all programs on the diagonal. This new program leads to a contradiction, showing that no single program can decide the halting behavior of all programs.
Self-Reference:
Self-reference involves a program or statement referring to itself. This concept is crucial in constructing paradoxes and proving undecidability.
Key Idea:
- Self-Referential Program: Create a program that takes its own description as input.
- Contradiction: Use this self-referential program to create a paradox. For example, if a program could determine whether any program halts, you could construct a program that halts if and only if it does not halt, leading to a contradiction.
Example of Diagonalization and Self-Reference in the Halting Problem:
- Assume a Halting Decider: Suppose there exists a program H(P, I) that returns true if program P halts on input I and false otherwise.
- Construct a New Program: Define a new program D that takes a program P as input and does the opposite of what H(P, P) predicts:
def D(P):
if H(P, P):
while True: # Infinite loop
pass
else:
return 0 # Halt
- Self-Reference: Consider what happens when D is given itself as input, D(D). If H(D, D) predicts that D(D) halts, then D will loop forever. If H(D, D) predicts that D(D) loops forever, then D will halt. This contradiction shows that H cannot exist.
These techniques demonstrate the inherent limitations of computation and are foundational in understanding the boundaries of algorithmic problem-solving.