相关论文: The Limits of Mathematics---Third Version
This is an alternative version of the course notes in chao-dyn/9407003. The previous version is based on measuring the size of lisp s-expressions. This version is based on measuring the size of what I call lisp m-expressions, which are lisp…
A remarkable new definition of a self-delimiting universal Turing machine is presented that is easy to program and runs very quickly. This provides a new foundation for algorithmic information theory. This new universal Turing machine is…
The latest in a series of reports presenting the information-theoretic incompleteness theorems of algorithmic information theory via algorithms written in specially designed versions of LISP. Previously in this LISP code only one-character…
The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the…
I'll outline the latest version of my limits of math course. The purpose of this course is to illustrate the proofs of the key information-theoretic incompleteness theorems of algorithmic information theory by means of algorithms written in…
This is a shortened version of "The Limits of Mathematics--Course Outline & Software" (IBM Research Report RC 19324, December 1993) in which all Mathematica code has either been deleted or, if absolutely necessary, replaced by C code. The…
The present paper presents and proves a proposition concerning the time complexity of finite languages. It is shown herein, that for any finite language (a language for which the set of words composing it is finite) there is a Turing…
We start by an introduction to the basic concepts of computability theory and the introduction of the concept of Turing machine and computation universality. Then se turn to the exploration of trade-offs between different measures of…
Most work on computational complexity is concerned with time. However this course will try to show that program-size complexity, which measures algorithmic information, is of much greater philosophical significance. I'll discuss how one can…
This paper talk about the complexity of computation by Turing Machine. I take attention to the relation of symmetry and order structure of the data, and I think about the limitation of computation time. First, I make general problem named…
This is yet another version of the course notes in chao-dyn/9407003. Here we use m-expressions more aggressively to further reduce the constants in our information-theoretic incompleteness theorems. Our main theorems are: 1) an N-bit formal…
In the paper we define three new complexity classes for Turing Machine undecidable problems inspired by the famous Cook/Levin's NP-complete complexity class for intractable problems. These are U-complete (Universal complete), D-complete…
People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…
We describe the Turing Machine, list some of its many influences on the theory of computation and complexity of computations, and illustrate its importance.
Continuing the study of complexity theory of Koepke's Ordinal Turing Machines (OTMs) that was started by Rin, L\"owe and the author, we prove the following results: (1) An analogue of Ladner's theorem for OTMs holds: That is, there are…
We re-evaluate universal computation based on the synthesis of Turing machines. This leads to a view of programs as singularities of analytic varieties or, equivalently, as phases of the Bayesian posterior of a synthesis problem. This new…
Incompleteness theorems of Godel, Turing, Chaitin, and Algorithmic Information Theory have profound epistemological implications. Incompleteness limits our ability to ever understand every observable phenomenon in the universe.…
We summarize four different versions of our course notes on the limits of mathematics.
This paper considers program synthesis in the context of computational hardness, asking the question: How hard is it to determine whether a given synthesis problem has a solution or not? To answer this question, this paper studies program…
We show how to reduce a general, strictly-feasible LP problem, into a min-max problem, which can be solved by the algorithm from the third section of my thesis.