Each Java program has a unique and determined outcome not halting or outputting something Unless otherwise stated we will be considering programs that take no input Each Java program has an unambiguous meaning Java is a prefix free language That is no Java program is the prefix of any other 01 2 3 41 PREFIX FREE MEANS THAT THE NOTION OF A RANDOM JAVA PROGRAM IS WELL DEFINED Flip a fair coin to create a sequence of random bits Stop If the bits form a Java program P Each program gets picked with probability length of program P Java is an unambiguous prefix free language Define to be the probability that a random program halts Is the probability that a random coin sequence will describe the text of a halting program 2 halting programs p length of p 5 BERRY PARADOX The smallest natural number that can t be named in less than fourteen words List all English sentences of 13 words or less For each one if it names a number cross that number off a list of natural numbers Smallest number left is number named by the Berry Sentence As you loop through sentences you will meet the Berry sentence This procedure will not have a well defined outcome Worse In English there is not always a fact of the matter about whether or not a given sentence names a number This sentence refers to the number 7 unless the number named by this sentence is 7 BERRY PARADOX The smallest natural number that can t be named in less than fourteen words Java is a language where each program either produces nothing or outputs a unique string What happens when we express the Berry paradox in Java 6 7 8 0 9 9 1 3 41 1 01 A 1 B C 5 0 D 1 Let s call this the indexing trick E C 1 1 1 8 1 1 F 11 public class Counter public static void main String argv for int i 0 i 1000000 i System out print 01 G G H G H I 1 I 1 JK 7 1 1 I 6 When we notice a pattern we always mean something atypical So when you see a pattern in a sufficiently long string it allows you to compress it Hence incompressible strings have no pattern For example we can compress a sufficiently long Binary string with 60 1 s 1 always following 1101 ASCII Of English Language Text BERRY PARADOX The smallest natural number that can t be named in less than fourteen words I I 0 D 6 0 D 6 7 8 7 K C L L M 1 K C L L 0 D 6 7 K C L L 0 D 6 7 M 3 8D N O L L 0 C D 3 K C 4 L K C 4 L L 1 If JAVA BERRY outputs ANYTHING a real paradox would result M L N 3 K C D 8D K P L K 0 D O 3 L L L 0 D 6N L 7 C L L L L 4 L 0 K P L K 0 4 L K 8 N O M L N C 8 K C 8 K C K 1 L L 6 7 O 1 1 Q K 6 Q 1 7 R O 1 You fix any n bit foundation for mathematics Now consider that half of the strings of length m n b are incompressible Your foundation can t prove that any one of them is incompressible Random Unknowable Truths Define to be the probability that a random program halts 2 halting programs p length of p is a maximally unknowable number is the optimally compressed form of the halting oracle Let n be the first nbits of By the properties of binary representation n n 1 1 8 1 K OS K 1 8 JS9 1 S9 9 9 F 1T U V 0 E S 1 E 1 1 1 D S JS9 9 F 1T 8D 1 ED 1 Q 1 C 0D 1 0 1 O I 1 O 0 1 1 O Or else you could prove that strings longer than your axiom system were incompressible I 1 1 1 1 O O 1 C 0D 1 0 1 0D H B 1 0 R 0 0D 1 1 0 1 Reason is our most powerful tool but some truths of the mathematical world have no pattern or representation that can be reasoned about We can make a Diophantine polynomial U in 16 variables such that when X1 is fixed to k the resulting polynomial has a root iff the kth bit of Omega is 1 N N 1 0B 1 1 1 0 AND NOT AND 1 N 8 L C 0B L 8 3 colorability Circuit Satisfiability AND NOT AND K 6 7 T F Y X Output T F Y X Output T F Y X Output T X F NOT gate NOT X x y OR z NOT OR x y z x OR NOT OR y z x y z x OR NOT OR y z x y z x OR NOT OR y z x y z x OR NOT OR y z x y z x OR NOT OR How do we force the graph to be 3 colorable exactly when the circuit is satifiable y z x y z x OR NOT OR L Satisfiability of this circuit 3 colorability of this graph y z N Q A H
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