Undecidable problems: RICE’s Theorem Undecidable problems: is empty? is regular? has size 2? This can be generalized to all non-trivial properties of Turing-acceptable languages Costas Busch - LSU

Non-trivial property: A property possessed by some Turing-acceptable languages but not all Example: : is empty? YES NO NO Costas Busch - LSU

More examples of non-trivial properties: : is regular? YES YES NO : has size 2? NO NO YES Costas Busch - LSU

A property possessed by ALL Turing-acceptable languages Trivial property: A property possessed by ALL Turing-acceptable languages Examples: : has size at least 0? True for all languages : is accepted by some Turing machine? True for all Turing-acceptable languages Costas Busch - LSU

We can describe a property as the set of languages that possess the property If language has property then Example: : is empty? YES NO NO Costas Busch - LSU

Example: Suppose alphabet is : has size 1? NO YES NO NO Costas Busch - LSU

Non-trivial property problem Input: Turing Machine Question: Does have the non-trivial property ? Corresponding language: Costas Busch - LSU

Rice’s Theorem: is undecidable Proof: Reduce (membership problem) to (the non-trivial property problem is unsolvable) Proof: Reduce (membership problem) to or Costas Busch - LSU

We examine two cases: Case 1: Examples: : is empty? : is regular? : has size 2? Costas Busch - LSU

Case 1: Since is non-trivial, there is a Turing-acceptable language such that: Let be the Turing machine that accepts Costas Busch - LSU

Reduce (membership problem) to Costas Busch - LSU

Decider for Reduction Compute Decider Given the reduction, membership problem decider Decider for Non-trivial property problem decider Reduction YES YES Compute Decider NO NO Given the reduction, if is decidable, then is decidable A contradiction! since is undecidable Costas Busch - LSU

We only need to build the reduction: Compute So that: Costas Busch - LSU

Construct from : Tape of input string yes yes Turing Machine Accept Write on tape, and accepts ? Simulate on input Costas Busch - LSU

For this we can run machine , that accepts language , with input string Turing Machine Accept yes yes Write on tape, and accepts ? Simulate on input Costas Busch - LSU

yes yes accepts does not accept Turing Machine Accept Write on tape, and accepts ? Simulate on input Prof. Busch - LSU Costas Busch - LSU 16

Therefore: accepts Equivalently: Costas Busch - LSU

Case 2: Since is non-trivial, there is a Turing-acceptable language such that: Let be the Turing machine that accepts Costas Busch - LSU

Reduce (membership problem) to Costas Busch - LSU

Decider for Reduction Compute Decider Given the reduction, membership problem decider Decider for Non-trivial property problem decider Reduction YES YES Compute Decider NO NO Given the reduction, if is decidable, then is decidable A contradiction! since is undecidable Costas Busch - LSU

We only need to build the reduction: Compute So that: Costas Busch - LSU

Construct from : Tape of input string yes yes Turing Machine Accept Write on tape, and accepts ? Simulate on input Costas Busch - LSU

yes yes accepts does not accept Turing Machine Accept Write on tape, and accepts ? Simulate on input Costas Busch - LSU

Therefore: accepts Equivalently: END OF PROOF Costas Busch - LSU