Download INTLOGS16 Test 2

INTLOGS16 Test 2
Prof S Bringsjord
TA Rini Palamittam
version 0331161200NY
Write your name on your answer booklets now. Thank you.
As you proceed, label each answer in your answer booklets with the appropriate ‘Qn.’
For Q1, use ‘Q1.(a)’ to label your answer to sub-question (a), etc. In general, strive
to write as legibly as possible; thanks!
Q1 For each of the following equations, first (i) write down the representation of each formula
appearing within the equation as an S-expression, then (ii) give a yes or no answer as to
whether the equation is true or not. In addition, (iii) for each of your affirmative verdicts,
provide a clear, informal proof that confirms your verdict.1
(a) {∀x(Scared(x) ↔ Small(x)), ∃x¬Scared(x)} ` ∃x¬Small(x)
(b) {¬∃x(Llama(x) ∧ Small(x)), ∀x(Small(x) ∨ Medium(x) ∨ Large(x))} `
∀x(Llama(x) → (Medium(x) ∨ Large(x))
(c) ` ∃x(Llama(x) → ∀yLlama(y))
(d) {∃x∀y(y ∈ x ↔ y 6∈ y)} ` φ ∧ ¬φ
Note: ¬(a ∈ a) is of course equivalent to a 6∈ a, and the former should be used in
Slate, where as an s-expression it’s
(not (\in a a))
(e) {∀z∃x∀y(y ∈ x ↔ (y ∈ z ∧ φ(y)))} ` ψ ∧ ¬ψ.
Note: Here, in keeping with the new notation introduced in class, φ(y) is a formula
in which y is free. In addition, we stipulate that x is not free in φ(y).
Q2 As you know, we have introduced the following numerical quantifiers: ∃=k , ∃≤k , ∃≥k , where
of course k ∈ Z + . This allows us for instance to economically represent such statements as
“There are exactly 4 dim llamas.”2 However, we haven’t discussed any new inference rules
that might make sense for one or more of these new quantifiers (we still only have rules for ∀
and ordinary ∃). Suggest in detail two new valid inference rules that you think makes sense
for one or more of the new quantifiers.
Q3 Let A be some finite alphabet of symbols {s1 , s2 , . . . , sn }. Professor Jones claims that A ∗
is a set of exactly the same size as the natural numbers N. Is she correct? Prove that your
answer is correct.
Q4 Suppose that E is a string built by concatenating the symbols that allow well-formed formulas
of first-order logic to be built. Such symbols include, of course, the five truth-functional
connectives (→, ↔, ∨, ∧, ¬), the existential and universal quantifier, etc. Can one program
a Raven-machine to decide whether or not E is in fact a well-formed formula of first-order
logic? Rigorously justify your answer.
1
Of course, informal renditions of full proofs constructed in Slate is certainly one foolproof way to proceed with
(iii), but it’s not the only way to proceed.
2
This statement would be represented by ∃=4 x (Dim(x) ∧ Llama(x)), which is an instance of this kind of formula:
=k
∃ x φ(x).
1