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1 Preliminaries Algorithmic Information Theory The Basics Adam Elga IAP January Preliminaries Turing machines Turing machine An idealized computing device attached to a tape each square of which is capable of holding a symbol We write a program p a nite binary string on the tap e and start the machine If the machine halts with string o written at a designated place on the tap e then o is the output Universal Turing machine A Turing machine capable of simulating any other Turing machine in the following sense Given an input that enco des the command Simulate machine k on program p a universal machine will output whatever machine k would output if given program p Denition of complexity The complexity of string s relative to machine T is dened to b e the length of the shortest program that gets T to pro duce s as output We choose some universal Turing machine U and dene the complexity of string s to b e the complexity of s relative to U How much do es the choice of universal machine matter adamphilosophersnet Lots For any string s there is a universal machine U such that the com s plexity of s relative to U is zero s 0 Not much Supp ose that U and U are b oth universal machines Then there is a constant k such that for any s the complexity of s relative to U 0 never diers from the complexity of s relative to U by more than k So 0 for increasingly long strings the dierence b etween U and U b ecomes less and less signicant Upshot this complexity measure allows us to b o otstrap a single qualitative simplicity judgment into many quantitative simplicity judgments Basic facts Strings never have complexities much greater than their lengths There is a constant k such that no strings complexity is more than k greater than its length Lowcomplexity strings are relatively rare Example of the strings of length only a tiny fraction have complexity less than A minimal program has a complexity approximately equal to its length The complexity function C is not computable Hint on how to prove this Berry paradox D The smallest number not describable in fewer than twenty syllables D describ es some number since there are only nitely many under twentysyllable descriptions But supp ose that D refers to n Then n is describable in nineteen syllables so D do esnt refer to n Contradiction Randomness Two approaches to the question of whether a string is random Consider what sort of pro cess produced the string Call the string processrandom if it was pro duced by the right sort of chancy pro cess Pay no attention to what pro duced the string Instead call the string productrandom if it is appropriately unpatterned Well fo cus on the second approach as applied to innite binary strings Think of a minimallength program for a nite string as a compressed version of the string The compressibility of a nite string s is dened to b e the length of s minus the length of a minimallength program for s Intuition an innite string that is pro ductrandom ought to have initial seg ments that arent very compressible That motivates the following denition An innite string is productrandom if and only if there is an upp er b ound B such that no initial segment of the string has compressibility greater than B Tweak require programs to b e selfdelimiting In the long run pro ductrandom sequences have just as many 1s as 0s Also pro ductrandom sequences cannot b e exploited by gambling machines Incompleteness The language of arithmetic contains logical symbols expressions that denote numbers and the addition and multiplication signs In the language of arithmetic one can make numerical assertions such as For all x for all y x y y x There exists an x such that for all y x y y y Supp ose that weve got some axioms that are such that some Turing machine prints the axioms out in order Supp ose that weve got some rules of inference for deriving consequences of the axioms rules such that there is a machine M that prints out every sentence derivable from the axioms using the rules and prints no other sen tences Question Can it b e that our axioms and rules satisfy b oth of the following conditions Every sentence derivable from the axioms on the basis of the rules is true In other words every sentence that M prints is true Every true arithmetical sentence is derivable from the axioms on the basis of the rules In other words M eventually prints every true arithmetical sentence References Gregory Chaitin Algorithmic information theory Cambridge Uni versity Press This b o ok develops algorithmic prex complex ity emphasizing its analogies with the entropy of classical information theory Chapter investigates innite random sequences Available at httpwwwcsaucklandacnzCDMTCSchaitincuppdf Ming Li and Paul Vitanyi An introduction to Kolmogorov complexity and its applications Springer A comprehensive textb o ok This b o ok has everything including extensive notes on the history of the various ideas Michiel van Lambalgen Von mises denition of random sequences recon sidered Journal of Symbolic Logic A technical article comparing various criteria for randomness Michiel van Lambalgen Algorithmic information theory Journal of Sym bolic Logic Argues against some of Chaitins claims ab out the relationship b etween algorithmic information theory and Go dels incompleteness theorems If you read anything by Chaitin you should also read this article .
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