Podcast
Questions and Answers
What was the decision problem posed by mathematicians in the early 20th century?
What was the decision problem posed by mathematicians in the early 20th century?
- Solving the halting problem
- Creating a program as a 'black box' to take inputs and give outputs
- Determining if given premises entail a conclusion in first-order logic (correct)
- Proving that first-order logic is decidable
Who was among the first to prove that first-order logic is not decidable?
Who was among the first to prove that first-order logic is not decidable?
- Alan Turing (correct)
- Stephen Hawking
- Isaac Newton
- Albert Einstein
What concept was difficult in Turing's proof about first-order logic?
What concept was difficult in Turing's proof about first-order logic?
- Showing no possible program could provide the answer (correct)
- Proposing the idea of a program as a 'black box'
- Proving that first-order logic is decidable
- Creating a machine 'h+' that would loop forever if given a yes answer and halt with a no answer
What did Turing propose as a 'black box' in his idea of a program?
What did Turing propose as a 'black box' in his idea of a program?
What did Turing show to be similar to the logical problem of determining if premises entail a conclusion?
What did Turing show to be similar to the logical problem of determining if premises entail a conclusion?
What did Turing prove is impossible for any machine to solve?
What did Turing prove is impossible for any machine to solve?
Study Notes
- In the early 20th century, mathematicians posed the decision problem in logic: Is there an automatic way to determine if given premises entail a conclusion in first-order logic?
- Alan Turing was among the first to prove that first-order logic is not decidable.
- Turing's proof was conceptually difficult because it required showing no possible program could provide the answer.
- Turing proposed the idea of a program as a "black box" that takes inputs and gives outputs.
- A new question arose: Given a program and input, will it halt and provide an answer or run indefinitely without halting?
- Turing showed that the logical problem of determining if premises entail a conclusion is similar to the halting problem.
- Turing proved it's impossible for any machine to solve the halting problem.
- He achieved this by creating a machine "h+" that would loop forever if given a yes answer and halt with a no answer.
- He then asked the question "Does h+ halt when given h+ as input?" which resulted in a paradox.
- Thus, Turing proved that no machine or program can solve the halting problem.
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Description
Test your knowledge about Alan Turing's proof that no machine or program can solve the halting problem, and its connection to first-order logic and the decision problem. Learn about the conceptual difficulty of proving the impossibility of determining if a program will halt and provide an answer or run indefinitely.
