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MA/CSSE 474 Day 04 Announcements and Summary
Announcements:
1.
2.
3.
HW2 Due Tuesday at 8:15 AM. Late submissions not accepted. Dropbox closes at 8:20, Moodle time.
HW3 due Friday of Week 2 at 8:15 AM. Problems will be available by the end of the weekend.
HW1 solutions will appear on Moodle this afternoon..
Main ideas from today:
1. Show that ≡₃ is an equivalence relation. (a ≡₃ b iff b-a = 3k for some integer k).
reflexive:
symmetric:
transitive:
2. If L1 = {a, aa}, and L2 = {a, c, ε}, then L1 L2 =
. What is Lε?
L∅?
3. Know the meanings of Ln, L*, L+.
4. What are the two procedural ways of specifying/determining language membership?
5. Lexicographic enumeration of {w  {a, b}* : |w| is even} :
6. Can a language be uncountable?
Is the set of languages over a specific alphabet uncountable?
7. Questions about maxstring function:
maxstring( AnBn ) =
maxstring({a}*) =
(later, to check your understanding): maxstring({bna: n≥0}) =
Let INF be the set of infinite languages. Let FIN be the set of finite languages.
Are the language classes FIN and INF closed under maxstring?
8. Questions about chop function:
What is chop(AnBn)?
What is chop(AnBnCn)?
Are FIN and INF closed under chop?
9. Questions about firstchars function:
What is firstchars(AnBn)?
What is firstchars({a, b}*)?
Are FIN and INF closed under firstchars?
10. A decision problem is a problem whose answer is _____________.
11. What does <x> mean?
<x, y>?
12. Some decision problems:
a. Consecutive pair of factors
b. Halting problem
c. Web pattern matching
d. Primality testing
e. Graph connectivity
f. Protein sequence alignment
g. Multiplication as decision problem
h. Sorting as decision problem
13. Constructing one machine based on another machine
Consider the multiplication language:
INTEGERPROD = {w of the form <integer1>x<integer2>=<integer3>, where:
<integern> is any well-formed integer representation and integer3 = integer1  integer2}
Given a multiplication procedure, we can build a procedure to language recognition machine:
Given the language recognition procedure , we can build a multiplication procedure:
14. You should know the meanings of the terms entail, inference rule, first-order term, predicate, wff, free variable,
bound variable, sentence, interpretation, model, partial order, total order, infinite descending chain, wellfounded, well-ordered
15. Are these valid, satisfiable, unsatisfiable?
 x ((P(x)  Q(Smokey))  P(x)).
 (x (P(x)  (P(x))).
 x (P(x, x)).
16. Find a model for the sentence x (y (y < x)).
17. Come up with a relation R on the set S = {r ϵ rationals: 0 < r < 1} such that (S, R) is well-ordered. It does not
need to be consistent with the usual ordering. Hint: Think diagonal.