CS 21 Decidability and Tractability Lecture 16 February

CS 21 Decidability and Tractability Lecture 16 February 10, 2021 CS 21 Lecture 16 1

Outline • Gödel Incompleteness Theorem February 10, 2021 CS 21 Lecture 16 2

Number Theory • February 10, 2021 CS 21 Lecture 16 3

Number Theory • February 10, 2021 CS 21 Lecture 16 4

Number Theory • February 10, 2021 CS 21 Lecture 16 5

Number Theory • February 10, 2021 CS 21 Lecture 16 6

Number Theory • q February 10, 2021 r CS 21 Lecture 16 7

Number Theory • true false February 10, 2021 CS 21 Lecture 16 8

Proof systems • February 10, 2021 CS 21 Lecture 16 9

Peano Arithmetic • February 10, 2021 CS 21 Lecture 16 10

Peano Arithmetic • February 10, 2021 CS 21 Lecture 16 11

Peano Arithmetic • modus ponens generalization February 10, 2021 CS 21 Lecture 16 12

Proof systems • February 10, 2021 CS 21 Lecture 16 13

Proof systems • A proof system is sound if all theorems in that proof system are true (better have this) • Peano Arithmetic (PA) is sound. true sentences = Th(N) false sentences = co-Th(N) theorems of PA February 10, 2021 CS 21 Lecture 16 14

Proof systems • A proof system is complete if all true sentences are theorems in that proof system • hope to have this (recall Hilbert’s program) true sentences = Th(N) false sentences = co-Th(N) theorems of a complete proof system February 10, 2021 CS 21 Lecture 16 15

Incompleteness Theorem: Peano Arithmetic is not complete. (same holds for any reasonable proof system for number theory) Proof outline: – the set of theorems of PA is RE – the set of true sentences (= Th(N)) is not RE February 10, 2021 CS 21 Lecture 16 16

Incompleteness Theorem • Lemma: the set of theorems of PA is RE. • Proof: – TM that recognizes the set of theorems of PA: – systematically try all possible ways of writing down sequences of formulas – accept if encounter a proof of input sentence (note: true for any reasonable proof system) February 10, 2021 CS 21 Lecture 16 17

Incompleteness Theorem • February 10, 2021 CS 21 Lecture 16 18

Incompleteness Theorem • February 10, 2021 CS 21 Lecture 16 19

Expressing computation in the language of number theory • February 10, 2021 CS 21 Lecture 16 20

Expressing computation in the language of number theory • February 10, 2021 CS 21 Lecture 16 21

Expressing computation in the language of number theory • February 10, 2021 CS 21 Lecture 16 22

Expressing computation in the language of number theory • February 10, 2021 CS 21 Lecture 16 23

Expressing computation in the language of number theory • February 10, 2021 CS 21 Lecture 16 24

Expressing computation in the language of number theory • i, k ) DIGIT(v, p i i = 0, 1, 2, 3, …, n February 10, 2021 CS 21 Lecture 16 25

Expressing computation in the language of number theory • February 10, 2021 CS 21 Lecture 16 26

Expressing computation in the language of number theory • February 10, 2021 CS 21 Lecture 16 27

Incompleteness Theorem • February 10, 2021 CS 21 Lecture 16 28

Incompleteness Theorem • Lemma: Th(N) is not RE • Proof: – reduce from co-HALT (show co-HALT ≤m Th(N)) – recall co-HALT is not RE – constructed such that M loops on w is true February 10, 2021 CS 21 Lecture 16 29

Summary • full-fledged model of computation: TM • many equivalent models • Church-Turing Thesis • encoding of inputs • Universal TM February 10, 2021 CS 21 Lecture 16 30

Summary • classes of problems: – decidable (“solvable by algorithms”) – recursively enumerable (RE) – co-RE • counting: – not all problems are decidable – not all problems are RE February 10, 2021 CS 21 Lecture 16 31

Summary • diagonalization: HALT is undecidable • reductions: other problems undecidable – many examples – Rice’s Theorem • natural problems that are not RE • Recursion Theorem: non-obvious capability of TMs: printing out own description • Incompleteness Theorem February 10, 2021 CS 21 Lecture 16 32