Proving Undecidability via Reductions Alice Gao Lecture 23 CS 245 - - PowerPoint PPT Presentation

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Aug 25, 2023 16 likes •140 views

Proving Undecidability via Reductions Alice Gao Lecture 23 CS 245 Logic and Computation Fall 2019 1 / 12 Outline Learning Goals A Template for Reduction Proofs Examples of Reduction Proofs Revisiting the Learning Goals CS 245 Logic and

SLIDE 1

Proving Undecidability via Reductions

Alice Gao

Lecture 23

CS 245 Logic and Computation Fall 2019 1 / 12

SLIDE 2

Outline

Learning Goals A Template for Reduction Proofs Examples of Reduction Proofs Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 2 / 12

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Learning Goals

By the end of this lecture, you should be able to:

▶ Defjne reduction. ▶ Describe at a high level how we can use reduction

to prove that a decision problem is undecidable.

▶ Prove that a decision problem is undecidable

by using a reduction from the halting problem.

CS 245 Logic and Computation Fall 2019 3 / 12

SLIDE 4

Outline

Learning Goals A Template for Reduction Proofs Examples of Reduction Proofs Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 4 / 12

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Proving that other problems are undecidable

We proved that the halting problem is undecidable. How do we prove that another problem is undecidable?

▶ We could prove it from scratch, or ▶ We could prove that it is as diffjcult as the halting problem.

Hence, it must be undecidable.

CS 245 Logic and Computation Fall 2019 5 / 12

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Proving undecidability via reductions

We will prove undecidability via reductions. Reduce the halting problem to problem 𝑄𝐶.

▶ Given an algorithm for solving 𝑄𝐶,

we could use it to solve the halting problem.

▶ If 𝑄𝐶 is decidable, then the halting problem is decidable. ▶ If the halting problem is undecidable, then 𝑄𝐶 is undecidable.

CS 245 Logic and Computation Fall 2019 6 / 12

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Proving undecidability via reductions

Theorem: Problem 𝑄𝐶 is undecidable.

Proof by Contradiction.

Assume that there is an algorithm 𝐶, which solves problem 𝑄𝐶. We will construct algorithm 𝐵, which uses algorithm 𝐶 to solve the halting problem. (Describe algorithm 𝐵.) Since algorithm 𝐶 solves problem 𝑄𝐶, algorithm 𝐵 solves the halting problem, which contradicts with the fact that the halting problem is undecidable. Therefore, problem 𝑄𝐶 is undecidable.

CS 245 Logic and Computation Fall 2019 7 / 12

SLIDE 8

Outline

Learning Goals A Template for Reduction Proofs Examples of Reduction Proofs Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 8 / 12

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Example 1 of reduction proofs

The halting-no-input problem: Given a program 𝑄 which takes no input, does 𝑄 halt? Theorem: The halting-no-input problem is undecidable.

CS 245 Logic and Computation Fall 2019 9 / 12

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Example 2 of reduction proofs

The both-halt problem: Given two programs 𝑄1 and 𝑄2 which take no input, do both programs halt? Theorem: The both-halt problem is undecidable.

CS 245 Logic and Computation Fall 2019 10 / 12

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Example 3 of reduction proofs

The exists-halting-input problem Given a program 𝑄, does there exist an input 𝐽 such that 𝑄 halts with input 𝐽? Theorem The exists-halting-input problem is undecidable.

CS 245 Logic and Computation Fall 2019 11 / 12

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Revisiting the learning goals

By the end of this lecture, you should be able to:

▶ Defjne reduction. ▶ Describe at a high level how we can use reduction

to prove that a decision problem is undecidable.

▶ Prove that a decision problem is undecidable

by using a reduction from the halting problem.

CS 245 Logic and Computation Fall 2019 12 / 12