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Solvable and Unsolvable

Solvable and Unsolvable. Gyoung-Hwan, Hyun. Substitution Puzzle. Substitution puzzle Normal form Any puzzle can be re-expressed as a substitution puzzle Normal form principle

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Solvable and Unsolvable

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  1. Solvable and Unsolvable Gyoung-Hwan, Hyun

  2. Substitution Puzzle • Substitution puzzle • Normal form • Any puzzle can be re-expressed as a substitution puzzle • Normal form principle • Given any puzzle, we can find a corresponding substitution puzzle which is equivalent to it in the sense that given a solution of the one we can easily use it to find a solution of the other • Mathematical problems -> substitution puzzle

  3. Church-Turing Thesis • Any systematic method, we can find a corresponding Turing machine that is equivalent to it. (1936) • Turing Machine vs. Modern Computer

  4. Gödel’s Theorem • Incompleteness Theorem (1931) • Turing’s work • “no systematic method of proving mathematical theorems is sufficiently complete to settle every mathematical question, yes or no.”

  5. Unsolvable • No TM exists to solve the problem • There is no algorithm that solves the problem • Undecidable

  6. Proof of Unsolvability • Reductio ad adsurdum • Assume that a TM exists that solve this problem • Show that this assumption leads to an impossible situation • “There cannot be any systematic procedure for determining whether a puzzle be solvable or not” • To show a problem is unsolvable • Reduce the known unsolvable problem to it

  7. Unsolvable Decision Problem • Halting problem • Decision problem whether or not a puzzle comes out

  8. Halting problem • Halting problem • If a (halting decision) TM P exists,… P(T) • Returns true if T will halt • Returns false if T won’t halt • A TM Q (runs P internally) • Doesn’t halt if P returns true • Does halt if P returns false • P(Q) returns true or false? • Contradiction!

  9. Decision Problem for Puzzle • P(R,S) : rules R, starting position S • Class I : R unambiguous moves, P(R,R) comes out with the end result W • Class II : all other cases (P(R,R) does not come out, or comes out with the end result B, or else R doesn’t represent a puzzle with unambiguous moves • If K exists... • K has unambiguous moves • P(K,R) always comes out whatever R • If R is in class I, then P(K,R) has end result B • If R is in class II, then P(K,R) has end result W • P(K,K)? • K is in class I, P(K,K) has end result W • K is in class I, P(K,K) has end result B • Contradiction!

  10. Summary • Solvable and Unsolvable • Incompleteness theorem • AI? • Questions?

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