Problem Session 1

1. Problem Session 1 CS 302

2. Topics • LaTeX • Questions from Class • Extra problems

3. LaTeX • Typesetting software for math • Used by scientific community • Why should you learn it?

4. How to get LaTeX (windows) • Download MikTeX (www.miktex.org) • Download TeXnicCenter (or other IDE) (www.toolscenter.org)

5. Starting a document • \documentclass[12pt]{article} • \pagestyle{plain} • \usepackage{anysize} • \marginsize{2.5cm}{2.5cm}{1.5cm}{1.5cm} • \begin{document} • … • \end{document}

6. Math Mode • $ - begins math mode • ^ - superscript • _ - subscript • \mbox – enter text in math mode • \ - begins a command or escapes a character

7. Using the Preamble • LaTeX allows you to create your own environments • \begin{proof} • Blah blah blah • \end{proof} • There are lots provided in the preamble

8. Lists • \begin{enumerate}, \begin{itemize} • \item creates a new item • \begin{enumerate*} omits numbering

9. Formatting • \newpage • \vspace{10mm}, \hspace{10mm} • newlines

10. Including Images • \usepackage{graphicx} • \begin{figure}[!htbp] • \includegraphics[scale=.75]{hw1-9a.jpg} • \end{figure} • It probably won’t put things where you want. • Supports jpg, ps, but not gif

11. Tutorials • Links on course webpage • Lots of links on wikipedia • Search for what you want!

12. Questions From Class?

13. Example of Conceptual Power of Non-determinism • Let L be regular • Let ½(L)={x : y |x|=|y| and xy  L} • i.e., ½(L) = first halves of strings in L • Is ½(L) regular?

14. ½(L) is regular • Let M be a DFA for L • Construct M’ for ½(L) • M’ uses the “finger method” • Place left finger on q0 in M • Middle finger on qj, nondeterministic guess as to where x will lead • Right finger on qj

15. Transition Function • As characters of x are read • M’ moves left finger along transitions dictated by characters read • And simultaneously moves right finger along a non-deterministically chosen edge • which may be labeled with any character, not necessarily the one of x being read

16. Acceptance • M’ accepts x iff at the end • Left finger is on qj • Right finger is on accept state • Thus, M’ accepts x iff |x| |y| q0 qj

17. Formal Implementation • Given M=(Q,,,q0,F) • Construct M’=(Q’,,’,q0’,F) • Q’ = Q x Q x Q  {s} • Q x Q x Q = (left,middle,right) • s = new start state

18. Formal Implementation • Guess the state qj that input x will lead to • ’(s,) = {<q0,qj,qj> : qj in Q} • ’(<qi,qj,qk>, a) = • { <ql,qj,qm> : ’(qi,a)=ql and there is b in  such that (qk,b)=qm} • Note that the middle finger does not change after initial guess. • F’={ <qj,qj,qk> : qj in Q, qk in F}.

19. Extra Problems • True or false: if T is countable, then the set {S|S a subset of T, S is finite} is also countable.

20. True • Use the Subset Sum technique • T = { a1, a2, a3,…} • Add up the subscripts of each set of S • Order by sum of subscripts, then lexicographically (i.e. a set with a1 comes before a set with a2)

21. Extra Problems • Is there a bijection between the interval (0,1) on the real line and all the real numbers?

22. Yes • Tan(x-½ π)/π/2 • Project (0,1) to semi circle centered at (½, ½). Draw the ray passing through (½, ½) and intersecting the x-axis.

23. Extra Problems • Show that if we add a single 32-bit register to a DFA and that each state can store some value in it then every language this machine accepts can be accepted by a regular DFA.

24. The register adds a finite number of states to each state • Add states to the machine representing all the possible values the register could have at that state

25. Extra Problems • Create a DFA the recognizes the set of strings in which the number of 0s is divisible by 2 and the number of 1s is divisible by 3.