Modern Treating of Properties of Indices: Their Automatic Generation
The above quotation stems from the health book “How to Live” from which were sold more than a million copies in 21 editions while Fisher was alive, and after his death went on to 90 editions and 4 million sold. It already promoted the idea of “biological
- PDF / 3,186,031 Bytes
- 47 Pages / 439.37 x 666.142 pts Page_size
- 40 Downloads / 177 Views
II. FOOD
5. Avoid overeating and over weight. 6. Avoid excess of high protein foods, such as meat, flesh foods, eggs; also excess of salt and highly seasoned foods. 7. Eat some hard, some bulky, some raw foods daily. 8. Eat slowly and taste your lood. 9. Use sufficient water mternally and externally.
III. POISONS
10. Secure thorough intestinal elimination daily. 11. Stand, sit, and walk erect. 12. Do not allow poisons and infections to enter the body. 13. Keep the teeth, gums, and tongue clean.
IV ACTIVITY 14. Work, play, rest, and sleep in moderation. 15. Breathe deeply; take deep-breathing exercises several times a day. 16. Keep serene and whole-hearted. Fisher and Fisk (1928, p.140)
3. Modern Treating of Properties of Indices: Their Automatic Generation The above quotation stems from the health book "How to Live" from which were sold more than a million copies in 21 editions while Fisher was alive, and after his death went on to 90 editions and 4 million sold. It already promoted the idea of "biological living". A couple of years ago the author heard of "automated theorem proving" with computers and he immediately thought that these methods should be applied in index theory. The following encouraging sentences ofWos (1988:247) convinced him that the present book should contain a chapter on this subject: "The beauty of a theorem from mathematics, the preciseness of an inference rule in logic, the intrigue of a puzzle, and the challenge of a game - all are present in the field of automated reasoning. The future of this field is extraordinarily bright and promising! To fully explain why this is so would require a A. Vogt et al., The Making of Tests for Index Numbers © Springer-Verlag Berlin Heidelberg 1997
70
The Making of Tests for Index Numbers
deep and complex analysis. However, we can certainly touch of some pertinent issues, the developments that will within ten years - by the year of 1996 - give us access to a very powerful and versatile automated reasoning assistant." Sections 3.1 and 3.2 are between the "traditional" and the "modem" treating of properties of indices: They are based on generating properties with the computer program MATHEMATICA. The subsequent sections 3.3, 3.4, 3.5 are based on a more sophisticated program, namely OTTER.
3.1 The Making of Reversal Tests for Indices 3.1.1 The Group of 4 Antitheses Fisher (1922) introduces for each price index its
time antithesis (2.20) 1 AT(P(-o -0 -1 -1» _ q ,p ,q ,p - p(-I -1 -0 -0) q,p,q ,p
(3.1)
and its
factor antithesis (2.21) AF(P(-O
-0 -1 -1»
q ,p ,q ,p
=
-1-1/-0-0
qp q p -1 -1)' P ,q-0 ,p,q
P(-o
(3.2)
Fisher's reversal tests (2.16) and (2.17) state that an index should be equal to the corresponding antithesis. It can be shown that these two antitheses together with the
identity antithesis AE( P(-o q ,p-0 ,q-1 ,p-1»
=P(-o -1 -1) q ,p-0 ,q,p
(3.3)
and the
simultaneous time and factor reversal antithesis AS(P(qO ,po ,qI ,pI» = AT(AF(P(qO,pO,qI ,pI»)
=AF(AT(P(qO ,po ,qI ,pI») qIpI / qOpO P(p\q\pO,qo)
(3.4)
3. Modern Treatin
Data Loading...
