Download Assignment 6

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts

Fuzzy logic wikipedia , lookup

Gödel's incompleteness theorems wikipedia , lookup

Modal logic wikipedia , lookup

Foundations of mathematics wikipedia , lookup

Axiom of reducibility wikipedia , lookup

Structure (mathematical logic) wikipedia , lookup

History of logic wikipedia , lookup

Laws of Form wikipedia , lookup

Quantum logic wikipedia , lookup

Mathematical proof wikipedia , lookup

Catuṣkoṭi wikipedia , lookup

Halting problem wikipedia , lookup

Law of thought wikipedia , lookup

Curry–Howard correspondence wikipedia , lookup

Intuitionistic logic wikipedia , lookup

Mathematical logic wikipedia , lookup

Theorem wikipedia , lookup

History of the function concept wikipedia , lookup

Applied Logic
CS 4860 Fall 2012
Homework 6
Due Date: Thursday, November 29
Projects are due Friday, Nov 30 by 5pm.
(1) Prove these theorems in HA
(a) ∀x.(0 + x) = x
(b) ∀x, y.(x + y = y + x)
(c) Define x < y to mean ∃z.(z 6= 0 & y = x + z), prove ∀x. ∃y.(x < y) and show the
evidence term.
(2) If we apply the minimization operator to a function f (x, y) that is always positive at x, e.g.
∀y. f (x, y) 6= 0, then it does not produce a value but “diverges,” on some input x. The
domain of such a function µy.f (x, y) = 0 is {x : N | ∃y. f (x, y) = 0}.
Note, we can represent λx.µy.f (x, y) = 0 in the logic Q, say by the relation we used in
proving that MinRec functions are representable.
In HA we might try to be more precise about these functions and ask whether HA can also
prove ¬ ∃y. f (x, y) = 0 when x is not in the domain.
Sketch a proof that if HA is consistent, then it cannot prove all such theorems (and is thus
Additional problems may be added.