PECS5304 THEORY
OF COMPUTATION
(3-0-0)
Module – I 10 Hrs Alphabet, languages and grammars. Producton rules and der!aton o" languages. Choms# Choms#$ $ herar herarch$ ch$ o" langua languages ges.. %egular %egular gramma grammars, rs, regular regular e&pres e&presso sons ns and "nte "nte automata (determnstc and nondetermnstc). Closure and decson propertes o" regular sets. sets. Pumpng Pumpng lemma o" regular sets. 'nmaton 'nmaton o" "nte "nte automata. automata. e"t and rght lnear grammars. Module – II 12 Hrs Conte&t "ree grammars and pushdo*n automata. Choms#$ and +rebach normal "orms. Parse Parse trees, trees, Coo#, Coo#, ounge ounger, r, asam asam,, and Earl$ Earl$s s parsn parsng g algort algorthms hms.. Ambg Ambgut ut$ $ and propertes o" conte&t "ree languages. Pumpng lemma, /gdens lemma, Par#hs theorem. ete eterm rmn ns st tc c pushd pushdo* o*n n auto automa mata ta,, clos closur ure e prop proper ert tes es o" dete determ rmn ns st tc c cont conte& e&tt "ree "ree languages. Module – III 14 Hrs 1urng machnes and !araton o" 1urng machne model, 1urng computablt$ , 1$pe 0 languages. near bounded automata and conte&t senst!e languages. Prmt!e recurs!e "unctons. "unctons. Cantor and +odel numberng. numberng. Ac#ermanns Ac#ermanns "uncton, "uncton, mu-recurs mu-recurs!e !e "unctons, "unctons, recurs!eness o" Ac#ermann and 1urng computable "unctons. Church 1urng h$pothess. %ecurs!e and recurs!el$ enumerable sets.. 2n!ersal 1urng machne and undecdable problems. 2ndecdablt$ o" Post correspondence problem. ald and n!ald computatons o" 1urng machnes and some undecdable propertes o" conte&t "ree language problems. 1me comple&t$ class P, class P, P completeness. Text Text Books . 6ntroducton to Automata 1heor$, anguages and Computaton7 8.E. 9opcro"t and 8. 2llman, Pearson Educaton, 3 rd Edton. :. 6ntroducton to the theor$ o" computaton7 'chael Spser, Cengage earnng 3. 1heor$ o" computaton b$ Saradh arma, Sctech Publcaton Re!ere"#e Books . 6ntroducton to anguages and the 1heor$ o" Computaton7 'artn, 1ata 'c+ra* 9ll, 3 rd Edton :. 6ntroducton to ;ormal anguages, Automata 1heor$ and Computaton7 . rth!asan, %ama %, Pearson Educaton. 3. 1heor$ o" computer Scence (Automata anguage < computatons) .. 'shra . Chandrashe#har, P96. 4. Elements o" 1heor$ o" Computaton7 e*s, P96 5. 1heor$ o" Automata and ;ormal anguages7 Anand Sharma, a&m Publcaton =. Automata 1heor$7 asr and Srman , Cambrdge 2n!erst$ Press. 7. Introduction to Computer Theory: Daniel I.A. Cohen, Willey India, 2 nd Edition.
MCC204 - THEORY
OF COMPUTATION COMPUTATION 3-0-0) 3- 0-0)
Module- I (12 hours) Introduction of Automata, Computability, and Complexity ; Mathematical notations and terminology; inding proofs and types of proofs! inite Automata and regular languages" ormal definitions, #esigning finite automata, #eterministic finite automata, $on%deterministic finite automata, &'uialence of $As and #As, finite automata ith *%transition; regular expressions and languages, +roperties of egular languages, conersion of & to A and ice ersa! Module –II (12 hours) +ush don Automata and Context free languages" Context free grammars, #esigning context free grammar, Ambiguity in C- and its remoal, Choms.y normal form Push do*n Automata7 "ormal de"nton, graphcal notatons, anguages accepted b$ PA, E>u!alence o" PA and C;+, on-conte&t "ree languages. Module-III (12 hours) /uring Machines and Computability" ormal definition of /uring machines ith examples, -raphical notations, 0ariants of /uring machines, Church%/uring thesis, ilberts problem #ecidability, undecidability and reducibility" #ecidable languages; #ecidable problems concerning regular languages and context free languages, /he halting problem, +ost correspondence problems, 3ndecidable problems, Mapping reducibility, #ecidability of logical theories, /uring reducibility! Re#o$$e"ded Texts 1. Michael Si!e", 4Introduction to the /heory of Computation5, 6econd &dition, 2778, C&$-A-& learning India +t! 9td!, $e #elhi! 2. :ohn &! Ho#"o$% , aee Mo%&'(i < :effrey #! Ull'( , 4 Introduction to Automata Theory, Languages, and Computation5, /hird &dition, 2778, +earson &ducation Inc!, $e #elhi! Re!ere"#e Books 1! $asir 6!!=!, +!>! 6rimani 4 A text boo. on Automata /heory5, Cambridge 3niersity press India +t! 9td! 2! +eter *i(+, 4 An Introduction to Formal Languages and Automata ”, ourth &dition, 2778, $arosa +ublishing ouse, $e #elhi! ?! :ohn C! M'"%i(, 4 Introduction to Languages and the Theory of Computation , /hird &dition, 277?, /ata Mc-ra%ill (/M) +ublication +t! 9td!, $e #elhi @! /homas A! Sud', 4 Languages and Machines" An Introduction to the /heory of Computer 6cience5, /hird &dition, 277, +earson &ducation Inc!, $e
Table of problems?edt@ 9lberts t*ent$-three problems are7
Pro%le$
st
Br&e! ex'l("(t&o"
1he contnuum h$pothess (that s, there s
)t(tus
Pro!en to be mpossble to pro!e
Ye(r )ol*ed
=3
Pro%le$
Br&e! ex'l("(t&o"
)t(tus
Ye(r )ol*ed
or dspro!e *thn the Bermelo ;raen#el set theor$ *th or *thout the A&om o" Choce (pro!ded the Bermelo no set *hose cardnalt$s strctl$ bet*een
;raen#el set theor$ *th or
that o" the ntegers and that o" the real
*thout the A&om o" Choce s
numbers)
consstent, .e., contans no t*o theorems such that one s a negaton o" the other). 1here s no consensus on *hether ths s a soluton to the problem.
1here s no consensus on *hether results o" +Ddel and +enteng!e a soluton to the problem as stated b$ 9lbert. +Ddels second :nd
Pro!e that the a&oms o" arthmetc are consstent.
ncompleteness theorem, pro!ed n 3, sho*s that no proo" o" ts
3=F
consstenc$ can be carred out *thn arthmetc tsel". +enten pro!ed n 3= that the consstenc$ o" arthmetc "ollo*s "rom the *ell-"oundedness o" the ordnal ₀.
+!en an$ t*o pol$hedra o" e>ual !olume, s 3rd
t al*a$s possble to cut the "rst nto "ntel$
%esol!ed. %esult7 no, pro!ed
man$ pol$hedral peces that can be
usng ehn n!arants.
reassembled to $eld the secondF
00
Pro%le$
4th
Br&e! ex'l("(t&o"
)t(tus
Construct all metrcs *here lnes
1oo !ague to be stated resol!ed
are geodescs.
or not.?n @
Ye(r )ol*ed
%esol!ed b$ Andre* +leason, Are 5th
contnuous groups automatcall$ d""erental groupsF
dependng on ho* the orgnal statement s nterpreted. 6", ho*e!er, t s understood as an
53F
e>u!alent o" the 9lbertSmth conGecture, t s stll unsol!ed.
Partall$ resol!ed dependng on ho* the orgnal statement s nterpreted.?3@ 6n partcular, n a "urther e&planaton 9lbert proposed t*o spec"c problems7 () a&omatc treatment o" probablt$ *th lmt theorems "or 'athematcal treatment o"
=th
the a&oms o" ph$scs
"oundaton o" statstcal ph$scs and () the rgorous theor$ o" lmtng processes H*hch lead
33:00:F
"rom the atomstc !e* to the la*s o" moton o" contnuaH. olmogoro!Is a&omatcs (33) s no* accepted as standard. 1here s some success on the *a$ "rom the Hatomstc !e* to the la*s o" moton o" contnuaH. ?4@
Jth
6s a transcendental, "or algebrac a K 0,
%esol!ed. %esult7 $es, llustrated
and rratonal algebracb F
b$ +el"onds theorem or
b
35
Pro%le$
Br&e! ex'l("(t&o"
)t(tus
Ye(r )ol*ed
the+el"ondSchneder theorem.
1he %emann h$pothess (Hthe real part o" an$ non-tr!al ero o" the %emann eta Lth
"uncton s MH) and other prme number
2nresol!ed.
Partall$ resol!ed.?n :@
problems, among them +oldbachs conGecture and the t*n prme conGecture
th
;nd the most general la* o" the recproct$ theorem n an$algebrac number "eld.
;nd an algorthm to determne *hether a 0th
g!en pol$nomalophantne e>uaton *th nteger coe""cents has an nteger soluton.
th
Sol!ng >uadratc "orms *th algebrac numercal coe""cents.
%esol!ed. %esult7 mpossble, 'at$ase!chs theorem mples that there s no
J0
such algorthm.
Partall$ resol!ed.?5@
2nresol!ed.
E&tend the ronec#erNeber theorem on :th
abelan e&tensons o" theratonal numbers to an$ base number "eld.
Sol!e J-th degree e>uaton usng algebrac 1he problem *as partall$ sol!ed 3th
4th
(!arant7 contnuous)"unctons o"
b$ ladmr Arnold based on *or#
t*o parameters.
b$ Andre olmogoro!. ?n 4@
6s the rng o" n!arants o" an algebrac
%esol!ed. %esult7 no,
5J
5
Pro%le$
5th
Br&e! ex'l("(t&o"
Ye(r
)t(tus
)ol*ed
group actng on apol$nomal
countere&le *as constructed
rng al*a$s "ntel$ generatedF
b$'asa$osh agata.
%gorous "oundaton o" Schuberts enumerat!e calculus.
Partall$ resol!ed.?
citation needed @
escrbe relat!e postons o" o!als =th
orgnatng "rom a real algebrac cur!e and as lmt c$cles o" a pol$nomal !ector "eld on
2nresol!ed.
the plane.
%esol!ed. %esult7 $es, due Jth
E&press a nonnegat!e ratonal "uncton as >uotent o" sums o" s>uares.
to Eml Artn. 'oreo!er, an upper lmt *as establshed "or the
:J
number o" s>uare terms necessar$. ?
citation needed @
(a) %esol!ed. %esult7 $es (b$ arl %enhardt). (b) Ndel$ bele!ed to be resol!ed, b$ computer-asssted Lth
(a) 6s there a pol$hedron that admts onl$
proo" (b$1homas Callster 9ales).
an ansohedral tlng n three dmensonsF
%esult7 9ghest denst$ ache!ed
(b) Nhat s the densest sphere pac#ngF
b$ close pac#ngs, each *th denst$ appro&matel$ J4O, such as "ace-centered cubc close pac#ng and he&agonal close pac#ng.?n 5@?
citation needed @
(a) :L (b) L
Pro%le$
Br&e! ex'l("(t&o"
Are the solutons o" regular problems n th
the calculus o" !aratonsal*a$s necessarl$ anal$tcF
Ye(r
)t(tus
)ol*ed
%esol!ed. %esult7 $es, pro!en b$ Enno de +org and, ndependentl$ and usng d""erent
5J
methods, b$ 8ohn ;orbes ash.
%esol!ed. A sgn"cant topc o" :0th
o all !aratonal problems *th certan boundar$ condtons ha!e solutonsF
research throughout the :0th centur$, culmnatng n solutons?
citation needed @
F
"or the non-
lnear case.
Proo" o" the e&stence o" lnear d""erental :st
e>uatons ha!ng a prescrbed monodromc group
::nd
:3rd
2n"ormaton o" anal$tc relatons b$ means o" automorphc "unctons
;urther de!elopment o" the calculus o" !aratons
%esol!ed. %esult7 es or no, dependng on more e&act "ormulatons o" the problem. ?
citation
F
needed @
%esol!ed.?
citation needed @
2nresol!ed.
F