COG 356. MU-system
Problem Set.
1) What, does Hofstadter claim, is one of the most central
notions running through GEB?
R: In the book,
Hofstadter claim that “Formal system” is one of the most central notions
running through GEB.
2) Who invented the sort of formal system that Hofstadter features
in his book (the sort of system that the MIU-system exemplifies), and when did
this invention take place?
R: The formal
system that Hofstadter features in his book was invented by Emil Post in
1920’s.
3) In one four-word interrogative question, state the puzzle
that is feature in this chapter.
R: “Can you
produce MU?”
4) What is the given string in the MIU-system?
R: The given
string is MI.
5) What is the goal string of the MU-puzzle?
R: The goal
string of the puzzle is MU.
6) How many rules in the MIU-system?
R: The MIU-system
has 4 rules.
7) Carefully, precisely, write down the first rule of the
MIU-system, and give two example of its use, one directly from the chapter, and
one that does not appear explicitly in the chapter.
R: Rule I: If you
possess a string whose last letter is I, you can add on a U at the end.
Example:
From MIIII, you may get MIIIIU
From MIUI, you may get MIUIU
8) Carefully, precisely, write down the second rule of the
MIU-system, and give two example of its use, one directly from the chapter, and
one that does not appear explicitly in the chapter.
R: Rule II: Suppose
you have Mx. Then you may add Mxx to your collection.
Example:
From MIU, you may get MIUIU
From MUUI, you may get MUUIUUI
9) Carefully, precisely, write down the third rule of the
MIU-system, and give two example of its use, one directly from the chapter, and
one that does not appear explicitly in the chapter.
R: Rule III: If
III occurs in one of the strings in your collection, you may make a new string
with U in place of III.
Example:
From MIII, you may get MU
From MUUIIIUU, you may get MUUUUU
10) Carefully, precisely, write down the fourth rule of the
MIU-system, and give two example of its use, one directly from the chapter, and
one that does not appear explicitly in the chapter.
R: Rule IV: IF UU
occurs inside one of your strings, you can drop it.
Example:
From UUU, you may get U
From MIUUI, you may get MII
11) What is the word used to describe strings that are
producible by the rules of a formal system from strings that have already been
produced?
R: The word used
to describe string that are producible by the rules is “Theorem.”
12) What is the technical term for the string MI in the
MIU-system?
R: The technical
term for the string MI in the MIU-system is “Axiom.”
13) In a formal system, is it more appropriate to say that
theorems are proven or that theorems are produced?
R: In a formal
system, it is more appropriate to say that theorems are merely produced
according to some rules.
14) How does Hofstadter define the term derivation?
R: Hofstadter
defines derivation as “An explicit, line-by-line demonstration of how to
produce that theorem according to the rules of the formal system.” (Page 36 –
GEB)
15) Reproduce, line by line, character by character
(including “reasons” (rule citations)) Hofstadter‘s derivation of the string
MUIIU.
R: (1) MI axiom
(2) MII from
(1) by rule II
(3) MIIII from
(2) by rule II
(4) MIIIIU from
(3) by rule I
(5) MUIU from
(4) by rule III
(6)
MUIUUIU from (5) by rule II
(7) MUIIU from
(6) by rule IV
16) Write down, line by line (Including “reasons” (rule
citations)) a derivation of the string MIIUIIU.
R: (1) MI axiom
(1) MII from
(1) by rule II
(1) MIIU from
(2) by rule I
(1) MIIUIIU from
(3) by rule I
17) On page 37, Hofstadter claims that there is a
fundamental difference between a machine and a human? What is that difference?
R: Hofstadter
claims that it is possible for a machine to act 100% unobservant and for human
this is impossible. Humans can sometimes act unobservant but not 100% like a
machine could.
18) With respect to formals systems, what is the difference
between working “inside the system” and working “outside the system?”
R: Working
“outside the system” means that something is escaping/changing a situation in
some way. The book presents the following example: Someone is reading a book
and then stop doing this and go to sleep. The person jumped off the system.
Working “inside the system” basically means: Do what you are supposed to do (or
programmed to do, if it is a machine).
19) Are there any theorems in the MIU-system that do not
start with the letter M?
R: No. All the
theorems in the MIU-system start with the letter M because all theorems must
begin with an M (Page 39 – Decision Procedures).
20) How is the previous question answered, by working within
the system or by working outside the system?
R: Outside the system.
21) What does “M-mode” refer to? What does “I-mode” refer
to?
R: M-mode:
Mechanical model, do the work to solve the puzzle like a machine
I-mode:
Intelligent mode, do the work to solve the puzzle using your
intelligence/skills or any other thing that comes in your mind.
22) Do you think that humans can work in M-mode?
R: I think humans
can work in M-mode sometimes, if you are really used to a task you may not
really pay attention in what you are doing because you already know what you
are supposed to do. Example: If you put your hand in the fire (For accident),
you will move it out without thinking.
23) Do you think that machines can work in I-mode?
R: I think in the
future machines will work in I-mode, but today in my opinion machines cannot
work in 100% I-mode. Some machines can think (AI machines), but they cannot be
compared to a human mind, not yet.
24) Two of the rules of the MIU-system are lengthening
rules. What does this mean? Two of the rules of the MIU-system are shortening
rules. What does this mean?
R: Lengthening
rules: Rules that can increase the length of your string. For example: Rules I,
II.
Shortening rules:
Rules that can decrease the length of your string. For example: Rules III, IV.
25) Define “decision procedure” with respect to a formal
system.
R: Decision
procedure is a test for theorem hood that always terminate in a finite amount
of time. If you have a decision procedure, you have a very precisely/solid
description of the nature of all theorems in the system that you are testing.
