Questions and Answers
What was the decision problem posed by mathematicians in the early 20th century?
Determining if given premises entail a conclusion in first-order logic
Who was among the first to prove that first-order logic is not decidable?
Alan Turing
What concept was difficult in Turing's proof about first-order logic?
Showing no possible program could provide the answer
What did Turing propose as a 'black box' in his idea of a program?
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What did Turing show to be similar to the logical problem of determining if premises entail a conclusion?
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What did Turing prove is impossible for any machine to solve?
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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.
