The pq- System Strings Definition :The pq- system utilizes three symbols: p, q,
ID: 662193 • Letter: T
Question
The pq- System
Strings
Definition :The pq- system utilizes three symbols: p, q, and -. The strings of the system are strings consisting entirely of these three symbols.
Example:
The following are strings of the pq- system:
p q -
p - q - - p p - - - q q - - - - - -
- - - - - - p q q - p - - - q
Rules and Axioms
Definition
Strings that are produced according to the rules of the system are called theorems of the system. An axiom is a theorem that is assumed to be true.
The pq- system has infinitely many axioms, defined by an axiom schema:
x p - q x - is an axiom whenever x is composed of hyphens only.
The system has a single rule:
Suppose x, y, and z all stand for particular strings containing only hyphens, and suppose that x p y q z is known to be a theorem. Then x p y - q z - is a theorem.
Generate some theorems to get a feel for the system. Do your examples lead you to any conjectures about theorems of the system?
Write a program (in words in actual program) to generate axioms and theorems. Does this reveal more than generating theorems by hand? Is there an intelligent way to generate theorems?
The system is formal: it involves manipulating symbols according to rules. What meaning can you ascribe to the system?
Write-up your observations and results in 1
Explanation / Answer
pq- puzzle:
Suppose x,y and z are the string variables that stands for - , p and q respectively. Suppose xpyqz is a theorem. Then -xp-yq-z- is also a theorem.
Explanation:
--ppqq is an axiom where x is composed of hypens, y is composed of p's and z is composed of q's.
Thus, -xp-yq-z- means, ---p-pq-q- is also a theorem.
The examples does not lead to conjecture about theorems to the system.
The system accepts any string variables with any given input variables such as p, q and - for any number of times in the string. The given language is a regular language because it can be accepted over any regular set.
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