Download Computability and incompleteness. Lecture notes by Avigad J. PDF

By Avigad J.

Show description

Read or Download Computability and incompleteness. Lecture notes PDF

Similar computers books

Contemporary Computing: Third International Conference, IC3 2010, Noida, India, August 9-11, 2010. Proceedings, Part I

This quantity constitutes the refereed lawsuits of the 3rd overseas convention on modern Computing, IC3 2010, held in Noida, India, in August 2010.

Electric Words: Dictionaries, Computers, and Meanings

Using pcs to appreciate phrases is still a space of burgeoning study. electrical phrases is the 1st normal survey of and advent to the full variety of labor in lexical linguistics and corpora -- the research of such online assets as dictionaries and different texts -- within the broader fields of natural-language processing and synthetic intelligence.

Additional resources for Computability and incompleteness. Lecture notes

Sample text

As a model of computation, the lambda calculus is a rather simple calculus; the only operations are lambda abstraction and application! From these meager resources, however, it is possible to implement any computational procedure. The “if” part of the theorem is easier to see: suppose a function, f , is represented by a lambda term X. Let us describe an informal procedure to compute f . On input m0 , . . , mn−1 , write down the term Xm0 . . mn−1 . e. the twostep reductions of the original term); and keep going.

G k−1 , respectively, we need to find a term f representing f . But we can simply define f by f (x0 , . . , xl−1 ) = h(g 0 (x0 , . . , xl−1 ), . . , g k−1 (x0 , . . , xl−1 )). In other words, the language of the lambda calculus is well suited to represent composition as well. When it comes to primitive recursion, we finally need to do some work. We will have to proceed in stages. As before, on the assumption that we already have terms g and h representing functions g and h, respectively, we want a term f representing the function f defined by f (0, z) = g(z) f (x + 1, z) = h(z, f (x, z), z).

Each sm n is then just a primitive recursive function that finds a code for the appropriate Turing machine. 4 Every partial computable function has infinitely many indices. Again, this is intuitively clear. Given any Turing machine, M , one can design another Turing machine M that twiddles its thumbs for a while, and then acts like M . Throughout this chapter, we will reason about what types of things are computable. 2. COMPUTABLY ENUMERABLE SETS 59 1. Rigorously: describe a Turing machine or partial recursive function explicitly, and show that it computes the function you have in mind; 2.

Download PDF sample

Rated 4.06 of 5 – based on 15 votes

About the Author