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notes17 Reductions and Rice's Theorem

notes17 Reductions and Rice's Theorem - CS 373 Theory of...

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Unformatted text preview: CS 373: Theory of Computation Gul Agha Mahesh Viswanathan Fall 2010 1 1 Reductions 1.1 Definitions and Observations Mapping Reductions Definition 1. A function f : Σ * → Σ * is computable if there is some Turing Machine M that on every input w halts with f ( w ) on the tape. Definition 2. A reduction (a.k.a. mapping reduction/many-one reduction) from a language A to a language B is a computable function f : Σ * → Σ * such that w ∈ A if and only if f ( w ) ∈ B In this case, we say A is reducible to B , and we denote it by A ≤ m B . Reductions and Recursive Enumerability Proposition 3. If A ≤ m B and B is r.e., then A is r.e. Proof. Let f be a reduction from A to B and let M B be a Turing Machine recognizing B . Then the Turing machine recognizing A is On input w Compute f ( w ) Run M B on f ( w ) Accept if M B accepts, and reject if M B rejects Corollary 4. If A ≤ m B and A is not r.e., then B is not r.e. Reductions and Decidability Proposition 5. If A ≤ m B and B is decidable, then A is decidable. Proof. Let f be a reduction from A to B and let M B be a Turing Machine deciding B . Then a Turing machine that decides A is On input w Compute f ( w ) Run M B on f ( w ) Accept if M B accepts, and reject if M B rejects Corollary 6. If A ≤ m B and A is undecidable, then B is undecidable. 2 1.2 Examples The Halting Problem Proposition 7. The language HALT = {h M,w i | M halts on input w } is undecidable....
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notes17 Reductions and Rice's Theorem - CS 373 Theory of...

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