MathDB

Problem 4-I

Part of 1998 VJIMC

Problems(2)

quine in Pascal (VJIMC 1998 1.4-I)

Source:

8/1/2021
Prove that there exists a program in standard Pascal which prints out its own ASCII code. No disk operations are permitted.
Pascal
algorithm determining truth of statement (VJIMC 1998 2.4-I)

Source:

8/1/2021
Let us consider a first-order language LL with a ternary predicate Plus⁡\operatorname{Plus}. Hence (well-formed) formulas of LL are built of symbols for variables, logical connectives, quantifiers, brackets, and the predicate symbol Plus⁡\operatorname{Plus}. (∃x1)(∀x2):Plus⁡(x2,x1,x2)∧(∀x3):¬Plus⁡(x1,x3,x3)(\exists x_1)(\forall x_2):\operatorname{Plus}(x_2,x_1,x_2)\wedge(\forall x_3):\neg\operatorname{Plus}(x_1,x_3,x_3) is an example of such a formula. Recall that a formula is closed iff each variable symbol occurs within the scope of a quantifier. Show that there exists an algorithm which decides whether or not a given closed formula of LL is true for the set N\mathbb N of natural numbers ({0,1,2,…}\{0,1,2,\ldots\}) where Plus⁡(x,y,z)\operatorname{Plus}(x,y,z) is interpreted as x+y=zx+y=z.
algorithm