One-Slide Summary Godel and Computability A proof of X in a formal - PDF document

One-Slide Summary Godel and Computability • A proof of X in a formal system is a sequence of steps starting with axioms. Each step must use a valid rule of inference and the final step must be X. • All interesting logical systems are incomplete : there are true statements that cannot be proven within the system. • An algorithm is a (mechanizable) procedure that always terminates. • A problem is decidable if there exists an algorithm to solve it. A problem is undecidable if it is not possible for an algorithm to exists that solves it. • The halting problem is undecidable. Halting Problems Hockey Team #2 Outline Epimenides Paradox • Gödel's Proof Epimenides (a Cretan): • Unprovability “All Cretans are liars.” • Algorithms • Computability Equivalently: • The Halting Problem “This statement is false.” Russell’s types can help with the set paradox, but not with these. #3 #4 Gödel’s Solution Kurt Gödel • Born 1906 in Brno (now Czech Republic, then All consistent axiomatic formulations of Austria-Hungary) number theory include undecidable • 1931: publishes Über propositions. formal unentscheidbare (GEB, p. 17) Sätze der Principia Mathematica und undecidable – cannot be proven either verwandter Systeme ( On true or false inside the system. Formally Undecidable Propositions of Principia Mathematica and Related Systems ) #5 #6

Gödel’s Theorem • 1939: flees Vienna • Institute for Advanced Study, In the Principia Mathematica Princeton system, there are statements that • Died in 1978 – cannot be proven either true or convinced false. everything was poisoned and refused to eat #7 #8 Gödel’s Theorem Gödel’s Theorem All logical systems of any In any interesting rigid system, complexity are incomplete : there are statements that cannot there are statements that are true that cannot be proven be proven either true or false. within the system. #9 #10 Proof – General Idea Gödel’s Statement •Theorem: In the Principia G : This statement does not have any proof in the Mathematica system, there system of Principia are statements that cannot be Mathematica . proven either true or false. •Proof: Find such a statement! G is unprovable, but true! Why? #11 #12

Gödel’s Statement Gödel’s Proof Idea G : This statement does not have any G : This statement does not have proof in the system of PM . any proof in the system. Possibilities: If G is provable, PM would be inconsistent. 1. G is true ⇒ G has no proof If G is unprovable, PM would be incomplete. System is incomplete 2. G is false ⇒ G has a proof Thus, PM cannot be complete and consistent! System is inconsistent #13 #14 How to express “ does not have Finishing The Proof any proof in the system of PM ” • Turn G into a statement in the • What does “ have a proof of S in PM” mean? Principia Mathematica system – There is a sequence of steps that follow the inference rules that starts with the initial • Is PM powerful enough to express axioms and ends with S “ This statement does not have • What does it mean to “ not have any proof any proof in the PM system.”? of S in PM”? – There is no sequence of steps that follow the inference rules that starts with the initial axioms and ends with S #15 #16 Can we express Can PM express unprovability? “This statement” ? • There is no sequence of steps that • Yes! follows the inference rules that starts with the initial axioms and ends with S – Optional Reading: the TNT Chapter in GEB • Sequence of steps: • We can write turn every statement T 0 , T 1 , T 2 , ..., T N into a number, so we can turn “This statement does not have any proof T 0 must be the axioms T N must include S in the system” into a number Every step must follow from the previous using an inference rule #17 #18

Gödel’s Proof Generalization G : This statement does not have any proof in the system of PM . All logical systems of any complexity are incomplete: If G is provable, PM would be inconsistent. there are statements that are If G is unprovable, PM would be incomplete. true that cannot be proven PM can express G . within the system. Thus, PM cannot be complete and consistent! #19 #20 Practical Implications What does it mean for an axiomatic system to be complete and consistent? • Mathematicians will never be completely replaced by computers – There are mathematical truths that cannot Derives all true be determined mechanically statements, and no false – We can build a computer that will prove only statements starting from a true theorems about number theory, but if it finite number of axioms cannot prove something we do not know that and following mechanical that is not a true theorem. inference rules. #21 #22 Pick one: What does it mean for an axiomatic some false incomplete statements system to be complete and consistent? Derives Derives all true some, but not all true statements, and some false statements, and no false statements starting from a It means the axiomatic system is weak. statements starting from a finite number of axioms finite number of axioms and following mechanical and following mechanical Indeed, it is so weak, it cannot express: inference rules. inference rules. “This statement has no proof.” Incomplete Inconsistent Axiomatic System Axiomatic System #23 #24

Inconsistent Axiomatic System Algorithms • What’s an algorithm ? Derives A procedure that always terminates. all true • What’s a procedure ? statements, and some false A precise (mechanizable) description of statements starting from a a process. finite number of axioms and following mechanical inference rules. some false statements Once you can prove one false statement, everything can be proven! false ⇒ anything #25 #26 Computability • Is there an algorithm that solves a problem? • Computable ( decidable ) problems: – There is an algorithm that solves the problem. Are there any uncomputable – Make a photomosaic, sorting, drug discovery, problems? winning chess (it doesn’t mean we know the algorithm, but there is one) • Uncomputable ( undecidable ) problems: – There is no algorithm that solves the problem. – There might be a procedure, but it doesn’t always terminate. #27 #28 Alan Turing (1912-1954) The Halting Problem • Codebreaker at Bletchley Park Input: a specification of a – Broke Enigma Cipher – Perhaps more important than Lorenz procedure P • Published On Computable Numbers … (1936) – Introduced the Halting Problem – Formal model of computation Output: If evaluating an (now known as “Turing Machine”) application of P halts, output • After the war: convicted of homosexuality true. Otherwise, output false. (then a crime in Britain), committed suicide eating cyanide apple 5 years after Gödel’s proof! #29 #30

Examples Halting Problem Define a procedure halts? that takes a > (halts? ‘(lambda () (+ 3 3))) procedure specification and evaluates to #t #t if evaluating an application of the > (halts? ‘(lambda () procedure would terminate, and to #f if (define (f) (f)) evaluating an application of the would (f))) not terminate. #f (define ( halts? proc) … ) #31 #32 Halting Examples Halting Examples > (halts? `(lambda () > (halts? `(lambda () (define (fact n) (define ( sum-of-two-primes? n) (if (= n 1) 1 (* n (fact (- n 1))))) ;;; try all possibilities... ) (fact 7))) (define ( test-goldbach n) #t (if (not (sum-of-two-primes? n)) > (halts? `(lambda () (fact 0))) #f ; Goldbach Conjecture wrong #f (test-goldbach (+ n 2)))) > (halts? `(lambda () (test-goldbach 2)) (define (fibo n) ? (if (or (= n 1) (- n 2))) 1 (+ (fibo (- n 1)) (fibo (- n 2)))))) (fibo 100)) Goldbach Conjecture (see GEB, p. 394): #t Every even integer can be written as the sum of two primes. #33 #34 Can we define halts? ? Informal Proof (define ( paradox ) • We could try for a really long time, get (if (halts? paradox) something to work for simple examples, (loop-forever) but could we solve the problem – make it #t)) work for all possible inputs? If paradox halts, the if test is true and it evaluates to (loop-forever) - it doesn’t halt! If paradox doesn’t halt, the if test if false, and it evaluates to #t. It halts! #35 #36

How convincing is our Proof by Contradiction Halting Problem proof? Goal: Show that A is false. (define (paradox) (if (halts? ‘paradox) 1. Show X is nonsensical. (loop-forever) 2. Show that if you have A you can make X . #t)) 3. Therefore, A must not exist. If contradict-halts halts, the if test is true and it evaluates to (loop-forever) - it doesn’t halt! If contradict-halts doesn’t halt, the if test if false, and it X = paradox evaluates to #t. It halts! A = halts? algorithm This “proof” assumes Scheme exists and is consistent! Scheme is too complex to believe this...we need a simpler model of computation (in two weeks). #37 #38 Homework • Read Chapter 12 • Read Obituary • PS6 Due Monday #39