Lecture Notes 15: Closure Properties of Decidable Languages 1
Union. Both decidable and Turing recognizable languages are closed under union. - For decidable languages the proof is easy.
Closure Properties of Decidable Languages Closure Properties
Decidable languages are closed under ? °
1 Closure Properties - 1.1 Decidable Languages
Proposition 1. Decidable languages are closed under union intersection
CS 3719 (Theory of Computation and Algorithms) – Lecture 19
18 févr. 2011 1 Closure properties of semi-decidable languages. Recall that the class of regular languages is closed under union intersection
Closure Properties of Decidable and Recognizable Languages
28 oct. 2009 Recognizable. Languages. Robb T. Koether. Homework. Review. Closure. Properties of. Decidable. Languages. Intersection. Union. Closure.
Limits of Computation : HW 1 Solutions and other problems
23 févr. 2007 (a) Union: (in the textbook). (b) Concatenation: Let K L be decidable languages. The concatenation of languages K and L is the language KL = { ...
CSE 105 Theory of Computation
Theorem: A language L is decidable if and only if it is Decidable languages are closed under. –. Union. –. Intersection. –. Set Complement.
Homework 7 Solutions
Answer: For any two decidable languages L1 and L2 let M1 and M2
Tutorial 3
Prove that the class of decidable languages is closed under union concatenation and Kleene star. Solution: • Closure under union.
CSE 6321 - Solutions to Problem Set 1
Thus the intersection of two decidable languages is decidable. 3. Show that the collection of recognizable languages is closed under the following operations. 1
Closure Properties of Decidable Languages
Decidable languages are closed under ? ° * ? and complement Example: Closure under ? Need to show that union of 2 decidable L’s is also decidable Let M1 be a decider for L1 and M2 a decider for L2 decider M for L1 ?L2:On input w: Simulate M1 on w If M1 accepts then ACCEPT w Otherwise go to step 2 (because M1 has halted and rejected w)
Closure Properties of Decidable Languages
1 1 Decidable Languages Boolean Operators Proposition 1 Decidable languages are closed under union intersection and complementation Proof Given TMsM1M2that decide languagesL1 andL2 A TM that decidesL1[L2: on inputx runM1andM2onx and accept i either accepts (Similarly for intersection )
CS 373: Theory of Computation
Decidable Languages Recursively Enumerable Languages Boolean Operators Proposition Decidable languages are closed under union intersection and complementation Proof Given TMs M 1 M 2 that decide languages L 1 and L 2 A TM that decides L 1 [L 2: on input x run M 1 and M 2 on x and accept i either accepts (Similarly for intersection ) A
Decidability and Undecidability - Stanford University
Decidable Languages A language L is called decidable iff there is a decider M such that (? M) = L Given a decider M you can learn whether or not a string w ? (? M) Run M on w Although it might take a staggeringly long time M will eventually accept or reject w The set R is the set of all decidable languages L ? R iff L is decidable
Decidable and Undecidable Languages - Wellesley College
Decidable and Undecidable Languages 32-6 Warm-Up: Some Decidable Languages Show that the following languages are decidable by describing (at a high level) an algorithm that decides them (see more in Sipser 4 1) string describing a pair of (1) a deterministic finite automaton DFA and (2) an input string w
Searches related to union of decidable languages PDF
A language L is decidable if there exists a decider D such that L(D) = L R Rao CSE 322 2 Closure Properties of Decidable Languages Decidable languages are closed under ? ° * ? and complement Example: Closure under ? Need to show that union of 2 decidable L’s is also decidable Let M1 be a decider for L1 and M2 a decider for L2
What are the closing properties of decidable languages?
Closure Properties of Decidable Languages Decidable languages are closed under ?, °, *, ?, and complement Example: Closure under ? Need to show that union of 2 decidable L’s is also decidable Let M1 be a decider for L1 and M2 a decider for L2 A decider M for L1 ?L2: On input w: 1. Simulate M1 on w. If M1 accepts, then ACCEPT w.
What are decidable languages?
Decidable languages are closed under union, intersection, and complementation. Proof. Given TMs M 1, M 2that decide languages L 1, and L 2
What is Proposition 1 of the decidable languages Boolean operators?
1.1 Decidable Languages Boolean Operators Proposition 1. Decidable languages are closed under union, intersection, and complementation. Proof. Given TMs M 1, M 2that decide languages L 1, and L 2 A TM that decides L 1[L 2: on input x, run M 1and M 2on x, and accept i either accepts. (Similarly for intersection.) A TM that decides L
What are the official languages (use for official purposes of union) rules?
Short title, extent and commencement - These rules may be called the Official Languages (Use for Official Purposes of the Union) Rules, 1976. They shall extend to the whole of India, except the State of Tamil Nadu. They shall come into force on the date of their publication in the Official Gazette.
1Decidable and Undecidable Languages
The Halting Problem and
The Return of Diagonalization
CS235 Languages and Automata
Tuesday, November 23, and Wednesday, November 24, 2010Reading: Sipser 4; Kozen 31; Stoughton 5.2 & 5.3
CS235 Languages and AutomataDepartment of Computer ScienceWellesley College
Recursively Enumerable Languages
L(M) = {w | w is accepted by the
Turing Machine M}
The recursively enumerableThe recursively enumerable (r.e.) languages = the set of all languages that are the language of some Turing Machine.These are also called
Turing-acceptableand
Turin g-recognizablelanguages.RE = Recursively Enumerable
(Turing-Recognizable/Acceptable)Languages
CFL = Context-Free Languages
a n b n c n ww 32-2gg ggWe will use REto name this set.
There are many languages in
REthat are not in CFL.
ggReg = Regular Languages a*b*(a+b)*bbb(a+b)*a n b n ww RDecidable and Undecidable Languages
2Decidability and Semi-Decidability
RE = Recursively Enumerable
(Turing-Recognizable/Acceptable)Languages
A Turing Machine decidesa language if it
rejects every string it doesn't accept - i.e., it never loopsThe recursivelanguages = the set of all
languages that are decided by some TuringM hi ll l d ib d b
Dec = Recursive (Turing-Decidable)
Languages
CFL = Context-Free Languages
a n b n ww R a n b n c n ww semi-decidable+ decidableMachine = all languages described by a non-
looping TM.These are also called theTuring-decidableor
decidable languages.We will use Decto name this set.
We'll soon see examples of languages that are
in REbut not in Dec. We call these languages semi-decidable+.Reg = Regular Languages
a*b*(a+b)*bbb(a+b)*Every TM for a semi-decidable+ language halts in the accept state for strings in the language but loops for some strings not in the language.Any language outside Decis undecidable.
All semi-decidable+ languages are undecidable,
but we'll see there are undecidable languages that aren't semi-decidable+!32-3Decidable and Undecidable Languages
Dec vs. RE
accept pipe For every language Lin Dec, there is a deciding machineM that for an input string w is guaranteed to deliver a ball to either the accept pipe or reject pipe.Turing Machine Mfor
a language L in Dec accept pipe reject pipe input string w For every language Lin RE, there is an accepting machineM that for an input string w is guaranteed to deliver a ball to the acce pt pipe if w L. However, if w L, a ball mightnotTuring Machine Mfor
a language L in RE accept pipe reject pipe input string w ppp,,g be delivered to the reject pipe (Mmight loop).32-4Decidable and Undecidable Languages
3Game Plan for the Rest of this Lecture
RE = Recursively Enumerable
(Turing-Recognizable/Acceptable)Languages
Our main goal is to exhibit a language L
that's semi-decidable+: L in RE - Dec.But first:
Dec = Recursive (Turing-Decidable)
Languages
CFL = Context-Free Languages
a n b n ww R a n b n c n ww semi-decidable+ decidable1. we'll need to get some practice
describing decidable languages that involve language encodings. Then:2. we'll define a language HALT
TM that's in RE - Dec. Fi llReg = Regular Languages
a*b*(a+b)*bbb(a+b)*Finally:
3. we'll argue that there are languages
that aren't even in RE!32-5Decidable and Undecidable Languages
Language Encodings
We will consider many languages whose strings contain encodings of DFAs, FAs, NPDAs, CFGs, and TMs. Think of such encodings as Forlan-like specifications for these machines and grammars. E.g. : b aX,a -> X; X,b -> Y;
Y,a -> X; Y,b -> Y;
Za > Z; Z b > Z})"
states start states final states alphabet 32-6DFA 1
Z,a -> Z; Z,b -> Z})
transition functionSĺAB
Aĺ0A1 |%
Bĺ1B0 |%
Decidable and Undecidable Languages
4Warm-Up: Some Decidable Languages
Show that the following languages are decidable by describing (at a high level) an algorithm that decides them (see more in Sipser 4.1)ACCEPT {( DFA ) | i i L(DFA)}
string describing a pair of (1) a deterministic finite automaton DFA and (2) an input string w •ACCEPT DFA = {(32-7Decidable and Undecidable Languages
What about ACCEPT
TMACCEPT
TM = {(A Turing Machine M
UTM can acceptACCEPT TM as follows:1. Check if is a well-formed Turing Machine description.
If not, M
UTM rejects(If not, M
UTM rejects(M, w)2. If is well-formed, M
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