newnote2 - Computability The study of what can/cannot be...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Computability The study of what can/cannot be done via purely mechanical means
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
History The Quest for Mechanizing Mathematics
Background image of page 2
Februray 24, 2008 © UCF EECS 3 Hilbert, Russell and Whitehead Late 1800’s to early 1900’s Axiomatic schemes Axioms plus sound rules of inference Much of focus on number theory First Order Predicate Calculus 2200 x 5 y [y > x] Second Order (Peano’s Axiom) 2200 P [[P(0) && 2200 x[P(x) P(x+1)]] 2200 xP(x)]
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Februray 24, 2008 © UCF EECS 4 Hilbert In 1900 declared there were 23 really important problems in mathematics. Belief was that the solutions of these would help address math’s complexity. Hilbert’s Tenth asks for an algorithm to find the integral zeros of polynomial equations with integral coefficients. This is now known to be impossible (In 1972, Matiyacevic showed undecidable; Martin Davis et al. contributed key ideas to showing this).
Background image of page 4
Februray 24, 2008 © UCF EECS 5 Hilbert’s Belief All mathematics could be developed within a formal system that allowed the mechanical creation and checking of proofs.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Februray 24, 2008 © UCF EECS 6 Gödel In 1931 he showed that any first order theory that embeds elementary arithmetic is either incomplete or inconsistent. He did this by showing that such a first order theory cannot reason about itself. That is, there is a first order expressible proposition that cannot be either proved or disproved, or the theory is inconsistent (some proposition and its complement are both provable). Gödel also developed the general notion of recursive functions but made no claims about their strength.
Background image of page 6
Februray 24, 2008 © UCF EECS 7 Turing (Post, Church, Kleene) In 1936, each presented a formalism for computability. Turing and Post devised abstract machines and claimed these represented all mechanically computable functions. Church developed the notion of lambda-computability from recursive functions (as previously defined by Gödel and Kleene) and claimed completeness for this model. Kleene demonstrated the computational equivalence of recursively defined functions to Post-Turing machines. Church’s notation was the lambda calculus, which later gave birth to Lisp.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Februray 24, 2008 © UCF EECS 8 More on Emil Post In the 1920’s, starting with notation developed by Frege and others in 1880s, Post devised the truth table form we all use now for Boolean expressions (propositional logic). This was a part of his PhD thesis in which he showed the axiomatic completeness of the propositional calculus. In the late 1930’s and the 1940’s, Post devised symbol manipulation systems in the form of rewriting rules (precursors to Chomsky’s grammars). He showed their equivalence to Turing machines. Later (1940s), Post showed the undecidability of
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/14/2011 for the course COT 4610 taught by Professor Dutton during the Fall '10 term at University of Central Florida.

Page1 / 230

newnote2 - Computability The study of what can/cannot be...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online