CSCE 355, Fall 2008matthews/Courses/C_Introduction/...Given the regular expression construct an NFA...
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Transcript of CSCE 355, Fall 2008matthews/Courses/C_Introduction/...Given the regular expression construct an NFA...
CSCE 355, Fall 2008
Foundations of Computation - Test 2November 10,2008
M Matthews
November 7,2008
1;0"'0: ~~ ~ _
~ ~ --------------
Instructions
Email: _
• Make sure your test is complete. There should be 7 pages including this cover sheet. Note there is no pagenumber 1.
• No Calculators, cell phones, or other electronic devices.
• Answer in the space provided if at all possible.
• If a question is unclear please ask early in the test.
• Good Luck!
Test2 Fall 2008
CSCE 355 Foundations of Computation
1. Relations
2
Test2 Fall 2008
(a) Given a relation R = {(a, b), (b, e), (a, a), (b, b), (e, en does it satisfy the following properties? If notindicate why the relation does not have the property.
i. reflexive
ii. transitive Nto
iii. symmetric AJ D
iv. injective
v. surjective
(b) Define the derives relation, ::::}
Test2 Fall 2008
2. Regular Expr -
(a) Given the following regular expression, a* bbb* cc*, describe in English the language it denotes.
3
(b) Is a'(a+ ba)'a' ~ (a+ba)'?Whyo,whynot? Yes be~-Jtut'V/7 tf)& ~ ~ ~ r:h chc 6eJ W ~ £,
fMMf d {# Wu A 'ti',1!fl1 ffrJJX CovJ£ju> t- v~~ D~/Ce .
3. Regular Expr - Give regular expressions that represent the following language: Strings of D's and l's such thatevery 1 is immediately followed by two D's.
(Df JOO)-;
Test2 Fall 2008 4
4. Given the regular expression construct an NFA that will accept the language denoted by the regular expression
(a) a*(b + a)*c*
(b) a*(aa + ba)*
Test2 Pall 2008
5. Given the grammar
• 8 -+ 081 I Z I E
• Z -+ OZ I E
(a) Give a parse tree for 000011
s/ I""o S I
~
o ~ ~,
::c
/\o ~
!\() ~
I
5
(b) Give a leftmost derivation for 000011 f::
~ ~~tt'~;. U1MJ ~\V~:~ ~...J- h<. Q-e+1:~.)-t
JVWltW ~ ~j) ~'jV1"t-t"1.Z>:;t S ,'v'\,c.c 6~
~\fM~ ~~ ~(~ ~ \tWttt-WtV1ttt~.
~ -:::::;>.e,P S I -==:> 0~'f, I I-=-'") D D -C l (:::::::j 0 0 0 =2:- ( \
=-;> 0 0 DO.:c (l
:-=;J 60 D~ c l \-:::;0000 {\
Test2 Fall 2008
o
6
0,1
start
6. Given the DFA give strings that distinguish states Band E.
10o
'B I ~ D -'-) ra
to, \~ -) \-\ - ~ IT
o
V1A2V\ ?i { L-ef-b'vt 1
AC ler-b'vt-.:J
0,1
Test2 Fall 2008
7
7. Non-regular proof by Pumping Lemma L = { i/..; k I' ..'. aifC z=Jorz=k}
~pt;~ It:? £+.~ !o)~ ~~ ~<€ ~~T ~ 1<\>t:JS~~ ...
-r~ dwot>~ W;1IJ ~ ~ tt b~~ G L-
~ '-Jh ~l/1MItf;--"'-'J ~ W-t (cu-
f~-t-~t-\'" ~ W::: lC J ~W~~
\\ ) \}( ~I ~ V\
G\} I ~ \ > 0
UMl0\) iJ"" WJ ~~D KOL:r G L.-::CJ w:: tt11 !::;'t'l.::. X;) =t. cv.J. \ ~ ~ t Z 11
~ K "i C, ~ ~~ ..J I ,
.s1t\L~ 1~\'> 0 "0 ~ 4; ~1 (W ~.
~ J,r ~"''D &. X~O~ It)t-'J '\~ ~ r\ tt(~\ _ n l\j}1 b~ ~ V\ c... S
cw.l +~ Y-.j:l- ~ L .lh,i'J ~:-C);- j ~ PL<Mft "j ~ J~ k ~ \M-t 'r-~~ ~
Test2 FalI 2008
8. Give a grammar for the language L = {ai~ ck I i = j or i = k}
S :y Bv:B7OG~b
D-' LV } E:
\
\
8
C-7ACc..\G
[-9 \9£ \~
Test2 Fall 2008
9. Construct a PDA for the context free language given by the grammar below
• 8 ---> 081 I Z I E
• Z ---> OZ I E
9
Test2 Fall 2008 10
10. Terminology: Explain of define
(a) What is the Pigeon hole principle?
H- ~ ~ V\ f\bt~.5· --\-..(J~ ~ ~ V\{4J~
~O\AA II'\. mi;~ ~ "'f't ~t ~ tv..-ft....yo~ CAAt. &CL.
(b) State the pumping lemma.
(c) What does it mean for a string z to distinguish two states p and q in a DFA?
~ (f) =b-) f.- r-- ~ ~ (1)) ~) 1= F\.
t> V V \ c.~ VBv5.t:.L
(d) What does it mean for a language L to distinguish two strings x and y?
JtvnllM6~ ~~ ~t>t7 PL ~'~e;.:c 6U~~
x,::z: ~!-.. w... 0~ ~ L- ~ vi t<-"en~ )