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Transcript
Applied Logic
CS 4860 Fall 2012
Homework 6
Due Date: Thursday, November 29
Projects are due Friday, Nov 30 by 5pm.
Questions
(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
incomplete).
Additional problems may be added.
1