Slide 1Thales’ and Gödel’s Legacy: Proofs and Their LimitationsA Quick Recap of the Previous LectureThe Halting Problem K = {P | P(P) halts }Alan Turing (1912-1954)Computability Theory: Old VocabularyComputability Theory: New VocabularySlide 8Enumerating KSlide 10And on to newer topics*What’s a proof?Thales Of Miletus (600 BC) Insisted on Proofs!Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19E.g.: syntax for Statements in Propositional LogicE.g. recursive program to decide SSlide 22Define a function LogicSA valid proof in logic LogicSProvable Statements (a.k.a. Theorems)Example: SILLY1Example: SILLY2Example: SILLY3Example: SILLY4More Practical Example: Propositional LogicSlide 31Slide 32Super-important fact about theoremsEnumerating the Set ProvableS,LExample: Euclid and ELEMENTSExample: Peano and Peano-Arith.Slide 37Slide 38Slide 39Truths of Euclidean GeometryTruths of Natural ArithmeticTruths of JAVA Program BehaviorSlide 43Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Truth versus ProvabilitySlide 53Slide 54Slide 55Let’s prove this…JavaTruth is Not EnumerableHence: JavaTruth has no sound and complete proof systemSlide 59Slide 60Foundational CrisisSlide 62Hilbert’s Second Question [1900]Gödel’s Incompleteness TheoremFoundation FIncompletenessCONFUSEF(P)Slide 68Slide 69GODELFSlide 71So What is Mathematics?Slide 7315-251Great Theoretical Ideas in Computer ScienceThales’ and Gödel’s Legacy: Proofs and Their LimitationsLecture 27 (April 14, 2009)A Quick Recap of the Previous LectureThe Halting ProblemK = {P | P(P) halts }Is there a program HALT such that:•HALT(P) = yes, if PK•HALT(P) = no, if PK(I.e., HALT decides whether or not any given program is in K.)Alan Turing (1912-1954)Theorem: [1937]There is no program to solve the halting problemComputability Theory:Old VocabularyWe call a set of strings S* decidable or recursive if there is a program P such thatP(x) = yes, if xSP(x) = no, if xSHence, the halting set K is undecidableNo program can decide membership in KComputability Theory:New VocabularyWe call a set of strings S* enumerable or recursively enumerable (r.e.) if there is a program P such that:1.P prints an (infinite) list of strings. 2.Any element on the list should be in S.3.Each element in S appears after a finite amount of time.Is the halting set K enumerable?Enumerating KEnumerate-K { for n = 0 to forever { for W = all strings of length < n do { if W(W) halts in n steps then output W; } }}K is not decidable, but it is enumerable!Let K’ = { Java P | P(P) does not halt}Is K’ enumerable?No! If both K and K’ are enumerable,then K is decidable. (why?)And on to newer topics**(The more things change, the more they remain the same…)What’s a proof?Thales Of Miletus (600 BC)Insisted on Proofs!“first mathematician”Most of the starting theorems of geometry. SSS, SAS, ASA, angle sum equals 180, . . .AxiomsIn traditional logic, an axiom is a proposition that is not proved, but is considered to be self-evident. Its truth is taken for granted.It serves as a starting point for inferring other truths.Peano’s Axioms for ArithmeticThe Peano axioms formally define the properties of the natural numbers4. If n is a natural number and n = m, then m is also a natural number. 1. For every natural number n, n = n 2. For all natural numbers, if n = m, then m = n. 3. For all naturals if k = m and m = n then k = n.5. 0 is a natural number. 8. For all natural numbers m and n, if S(m) = S(n), then m = n. 6. For every natural number n, its “successor” S(n) is a natural number. 7. For every natural number n, S(n) ≠ 0.Peano Arithmetic (contd.)What is a proof?Intuitively, a proof is a sequence of “statements”, each of which follows “logically” from some of the previous steps. What are “statements”? What does it mean for one to follow “logically” from another?What are “statements”? What does it mean for one to follow “logically” from another?Formally, statements are strings of a decidable language S over . Intuitively, statements are strings in some language. That is, S is a subset of Σ* and there is a Java program PS(x) that outputs Yes if x is in S, and outputs No otherwise.This decidable set S is the set of “syntactically valid” strings, or “statements” of a language. Example:Let S be the set of all syntactically well formed statements in propositional logic.X X(XY) YXY(not well-formed)E.g.: syntax for Statements in Propositional LogicVariable X, Y, X1, X2, X3, …Literal Variable | VariableStatement Literal(Statement)Statement StatementStatement StatementE.g. recursive program to decide SValidProp(S) { return True if any of the following: S has the form (S1) and ValidProp(S1) S has the form (S1 S2) andValidProp(S1) AND ValidProp(S2) S has the form …..}“Statements”: some set S of strings whose membership is decidable.But what does it mean to follow “logically”? What is “logic”? Intuitively, a proof is a sequence of “statements”, each of which follows “logically” from some of the previous steps.Define a function LogicSGiven a decidable set of statements S, fix a computable “logic function”:LogicS: (S ) × S Yes/NoIf Logic(x, y) = Yes, we say that thestatement y is implied by statement x.We also have a “start statement” not in S, where LogicS(, x) = Yes will mean that our logic views the statement x as an axiom.A valid proof in logic LogicS• LogicS(, s1) = True (s1 must be an axiom of our language)• For all 1 ≤ i ≤ n-1, LogicS (sj, sj+1) = True (each statement implied by the previous one)• And finally, sn = Q (the final statement is what we wanted to prove.)A sequence s1, s2, …, sn of statements is a valid proof of statement Q in LogicS ifProvable Statements (a.k.a. Theorems)Let S be a decidable set of statements. Let L be a computable logic function.Define ProvableS,L = All statements Q in S for which there is a valid proof of Q in logic L.Example: SILLY1S = All strings over {0,1}.L = All pairs of the form: <, s>, sSProvableS,L is the set of all strings.Example: SILLY2S = All strings over {0,1}.L = <, 0> , <, 1>, and all pairs of the form: <s,s0> or <s, s1>ProvableS,L is the set of all strings.Example: SILLY3S = All strings over {0,1}.L = <, 0> , <, 11>, and all pairs of the form:
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