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ALGORITHMIC INFORMATION THEORY - CHAPTER 6

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Program Size Trong chương này chúng tôi trình bày một nition de mới của chương trình kích thước phức tạp. H (AB = CD) de ned để được kích thước bit của các chương trình tự phân chia ranh giới ngắn nhất tính cho chuỗi A và B nếu được đưa ra một chương trình tự phân chia ranh giới kích thước tối thiểu để tính toán chuỗi C và D. là trường hợp trong LISP, các chương trình được yêu cầu để được tự phân chia ranh giới, nhưng thay vì đạt được điều này với các dấu ngoặc. | Chapter 6 Program Size 6.1 Introduction In this chapter we present a new definition of program-size complexity. H A B C D is defined to be the size in bits of the shortest self-delimiting program for calculating strings A and B if one is given a minimal-size self-delimiting program for calculating strings c and D. As is the case in LISP programs are required to be self-delimiting but instead of achieving this with balanced parentheses we merely stipulate that no meaningful program be a prefix of another. Moreover instead of being given c and D directly one is given a program for calculating them that is minimal in size. Unlike previous definitions this one has precisely the formal properties of the entropy concept of information theory. What train of thought led US to this definition Following Chaitin 1970a think of a computer as decoding equipment at the receiving end of a noiseless binary communications channel. Think of its programs as code words and of the result of the computation as the decoded message. Then it is natural to require that the programs code words form what is called a prefix-free set so that successive messages sent across the channel e.g. subroutines can be separated. Prefix-free sets are well understood they are governed by the Kraft inequality which therefore plays an important role in this chapter. One is thus led to define the relative complexity H A BIc D of 157 158 CHAPTER 6. PROGRAM SIZE A and B with respect to c and D to be the size of the shortest selfdelimiting program for producing A and B from c and D. However this is still not quite right. Guided by the analogy with information theory one would like H A B H A H B A A to hold with an error term A bounded in absolute value. But as is shown in the Appendix of Chaitin 1975b A is unbounded. So we stipulate instead that R A B C -D is the size of the smallest selfdelimiting program that produces A and B when it is given a minimal-size self-delimiting program for c and D. We shall show .