They are also closed under complement not part of this course. That is, if l and p are two recursively enumerable languages, then the following languages are recursively enumerable as well. Solved show that the family of linear languages is. Showing that turingrecognizable languages are closed under. Let l 1 and l 2 be two turing recognizable languages. The class of regular languages is closed under concatenation. In order to show equivalence between d and m, we need to show two things. Show that the collection of recognizable languages is closed under the following. Show that the collection of turing recognizable languages is closed under the operation of 1union 2concatenation 3star 4intersection 5 homomorphism.
The set of turingdecidable languages is closed under union, intersection, and complement. Solved show that the family of linear languages is closed. Im studying turing machines and ive already showed how turing decidable is closed for the operations of union, intersection, concatenation, complement and kleene star. The churchturing thesis states that this is a law of mathematics that a universal turing machine can, in principle, perform any calculation that any other programmable computer can. Recursively enumerable languages are closed under union, intersection, kleene star. Show that the collection of recursively enumerable turing recognizable languages is closed under the concatenation operation. Iit is possible for a tm to never reach a halting con guration. However, the recursive languages are not closed under homomorphism.
This paper first shows that 1 the class of languages recognized by olog n spacebounded twoway monte carlo turing machines is not closed under concatenation, kleene closure, and length. The other direction follows from the proof that a multitape tm is no. It is recognized by a tm u that, on input, simulates m on w step by step. Let l 1 be any context free langauge and l 2 be any regular language. Why is showing a language is turing recognizable trickier. Recursively enumerable language are closed under kleene star, concatenation, union, intersection. Cs103 handout 20 fall 2011 november 18, 2011 problem set 8. If l is a linear language, then its homomorphic image hl is regular. Showing that turingrecognizable languages are closed under union. When proving closure of the class of decidable languages under a given operation the obvious choice is an assumed decider for a given decidable language. In mathematics, logic and computer science, a formal language is called recursively enumerable also recognizable, partially decidable, semidecidable, turing acceptable or turing recognizable if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i. Kleenes theorem language ais regular i ahas a regular expression. Step 1 of 4 suppose, x and y be two turing recognizable languages that. We construct a tm m that recognizes the union of l1 and l2.
Show that the recursively enumerable languages are closed under union, intersection and kleenestar operation. Page2704 others questions and answers for other topics in. There are few more properties like symmetric difference operator, prefix operator, substitution which are closed under closure properties of regular language. Decidable languages are closed under inverse homomorphisms. Programming turing machines in problem set 7, youre asked to augment the wb6 language up to a language called wb8, which supports a finite number of unbounded counters. Next, we will show that the class of context free languages is closed under regular di erence. Theorem b let a, b y be turingrecognizable languages. By exchanging the accepting and rejecting final state of m a with each other, we. Any class of languages that is closed under difference is closed under intersection. Let abe a turing recognizable language consisting of descriptions hmiof turing machines m that are all deciders. When proving closure of the class of turing recognizable languages under a given operation the obvious choice is an assumed recognizer for a given turing recognizable language. Closure for recognizable languages turingrecognizable languages are closed under. The reason for this is that if a language is decidable, then its complement must be decidable as well. Decidable languages are not closed under homomorphism.
A language is called recognizable if it is the language of some tm. Even more, regular languages are closed under quotients with arbitrary languages. Sep, 2016 1 answer to show that the collection of turing recognizable languages is closed under the operation of a. Closure under homomorphism if l is a regular language, and h is a homomorphism on its alphabet, then hl hw w is in l is also a regular language. Show that the collection of turingrecognizable languages. Show that a turing machine with doublyin nite tape recognizes the same class of languages as an ordinary turing machine. The class of regular languages is closed under kleenestar. An re language can be accepted or recognized by turing machine which means it will enter into final state for the strings of language and may or may not enter into rejecting state for the strings which are not part of the language. Give an implementationlevel description and a formal description i. Are there languages that are recognizable but not decidable. Recursive and recursive enumerable languages in toc.
This language is not turing recognizable, and in this problem you will prove why. Here is the solution available online for textbook exercise 9. How to prove decidability or turing recognizability. Homework 1 solutions kevin matulef january 31th, 2001 problem 1. The class of turing decidable languages is closed under complementation. Thats essential to exhibit a language that is not recursively enumerable. The turingrecognizable or recursively enumerable languages are closed over homomorphism. We may assume that ais in nite since there are in nitely many decidable languages. Later, i have written a proof to show that turing recognizable languages are closed under union i am supposed to identify why closing a turing recognizable language under some operation is trickier to prove than when dealing with turing decidable languages. Both the turingrecognizable and turingdecidable languages are closed under concatenation and star hw.
Cs103 handout 20 fall 2011 november 18, 2011 problem set 8 how powerful are turing machines. Consider the following two proofs that try to show that recognizable languages are closed under complement. Let m1 be a tm for l1 and m2 a tm for l2 both may loop. For any two recognizable languages l 1 and l 2, let m 1 and m 2 be the tms that respectively recognize them. Properties of contextfree languages decision properties closure properties.
Prove that the language it recognizes is equal to the given language and that the algorithm halts on all inputs. Show that the collection of recognizable languages is closed under the operation of. Both decidable and turing recognizable languages are closed under union. Show that the collection of turing recognizable languages is closed under the operation of union. The class of regular languages is closed under union. Turing recognizable, we let machine m1 be the turing recognizer for l and m2 be the. Prove that some decidable language dis not decided by any decider msuch that hmiis in a. Theorem c let a y be a turingrecognizable language that is not turingdecidable. Recursively enumerable languages rel are closed under the following operations. To prove that a given language is turing recognizable. Cs2mj3 if you think your solution has been marked wrongly. We need to show that a turing machine with a doubly in. Both the turing recognizable and turing decidable languages are closed under concatenation and star hw. Run m 1 and m 2 alternately on w, one step at a time.
Let h be a homomorphism and l a language whose alphabet is the output language of h. The doublyin nite tape can simulate an ordinary tm by just not using the portion of its tape to the left of the input. Hw 1 solutions and other problems rajiv raman february 23, 2007 1 homework 1 1. Turing completeness is significant in that every realworld design for a computing device can be simulated by a universal turing machine. Automata and computation theory closure of turing decidable languages under kleene star closure of turing recognizable languages under kleene star here are two examples of closure proofs, both involving the kleene star operation. For any two turing recognizable languages l 1 and l 2, let m 1 and m 2, respectively, be tms that recognize them. Show that the collection of turing recognizable languages is closed under the operation of reversal. Is the class of turingrecognizable languages closed under. Show that the class of turing recognizable languages is closed under union. Pdf closure property of probabilistic turing machines.
Let l and m be languages that are recognized by algorithms a and b respectively. Since k and l are decidable languages, it follows that there exist turing machines m k and m. Give an implementationlevel description of a turing machine that decides the language. Get best help for others questions and answers in computerarchitecture page2704, stepbystep solutions, 100% plagiarism free question answers. The recognizable languages are closed under union and intersection. The set of turing decidable languages is closed under union, intersection, and complement.
Michael sipser 1 how to prove turing decidability of languages language hierarchy recognizability. Decidable languages are closed under union, intersection, and complementation. Closure properties of regular languages stanford university. It contains seven problems plus one survey question and one extra. We have to show that l1l2 is decided by some turing machine t12. Closure properties of regular languages geeksforgeeks. Theory of computation 6 homomorphisms nus computing. Basic properties of turingrecognizable languages theorem a let a, b y be turingdecidable languages. Re languages or type0 languages are generated by type0 grammars. However, the set of turing recognizable languages is not closed under complement. I have seen this question here, closure of turing recognizable languages under homomorphism but actually this question answers the question of what is the relation between homomorphism and concatenation. Let m1 be a tm for l1 and m2 a tm for l2 both may loop a tm m for l1.
Recursively enumerable languages are closed under union, intersection, kleene star, kleene plus and concatenation. Next i did some demonstrations to show how t recognizable languages are closed for union, intersection, concatenation and kleene star. Show that the collection of decidable languages is closed under the operation of 1union 2concatenation 3star. But by the previous result, the set of all languages is uncountable. How do you prove that a turing recognizable language is not closed under complementation. We showed in a previous homework that the class of turing recognizable languages is closed under union, so eq. If l1 is a regular language and l2 is a context free language, then l1.
I have written a proof to show that a turing decidable languages are closed under union amongst other things. To see why, consider the particular language l consisting of strings of the form m,w,ci, where m is a coded turing machine with binary input alphabet, w is a binary string, and c is a symbol not appearing elsewhere. Solved show that the collection of turingrecognizable. Why isnt the class of turingrecognizable languages closed. Show that this type of tm recognizes the class of turing recognizable languages. Turing machines dfa, nfa, regexp, cfg, and pdas, e. Let us consider that h be the desired homomorphism.
Decidability and undecidability stanford university. We need to show that a turing machine with a doubly infinite tape, d is equivalent to an ordinary turing. The problem we have here is that in an arbitrary string w that would be an input to t12 we do not. Turing recognizable we havent yet proved the relation between cf and td langs what are some examples in each class and not in smaller class. Showing that turing recognizable languages are closed under union. However, turing recognizable and co turing recognizable are not the same, and its this that ive decided to cover in my answer. How do you prove that a turingrecognizable language is not closed under complementation. Show that the collection of turingrecognizable languages is closed. Because we have constructed a turing machine which accepts a 8 b, the language a b is recursively enumerable. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Show that the collection of turing recognizable languages is closed under the operation of. How to prove that a turingrecognizable language is not. Rao, cse 322 3 closure for recognizable languages turing recognizable languages are closed under. We construct a tm m0that recognizes the union l 1 l 2.
Recursive turing decidable languages are closed under following kleene star, concatenation, union, intersection, complement and set difference. Let l1 be decided by a turing machine t1, and let l2 be decided by a turing machine t2. Closure of cfls under inverse homomorphism here, grammars dont help us. Closure properties of regular languages let l and m be regular languages. Let lbe a decidable language and mbe the turing machine that decides l. Homework five solution cse 355 arizona state university. Cfg is a co turing recognizable language if and only if its complement eq cfg is a turing recognizable language. Thus, the family of context free languages is not closed under di erence. Let us begin with the regular operations, under which the turing recognizable languages are indeed closed. Prove that the re languages are closed under homomorphism. Turing decidable is a subset of turing recognizable, so also countable.
Automata, computability, and complexity or, great ideas in theoretical. For any two turing recognizable language l1 and l2, let m1 and m2 be the tms that recognize them. Show that the collection of turingrecognizable languages is. The concatenation of languages k and l is the language kl xyx. I will give the proof for turing recognizable languages. We already know that regular languages are closed under complement and intersection. Properties of contextfree languages stanford university. As a consequence they are closed under arbitrary finite state transductions, like quotient k l with a regular language. That is, if l1 and l2 are recursive, then l1 l2 is recursive. That question asks two questions, one in the title is is the class of turingrecognizable languages closed under homomorphism, and the other is is my proof correct.
Prove that the language it recognizes is equal to the given language. Showing that turingrecognizable languages are closed. However, the set of turingrecognizable languages is not closed under complement. Draw a deterministic turing machine that accepts wcwcw w. Both decidable and turing recognizable languages are closed under concatenation. Do you need an answer to a question different from the above.
845 276 1301 3 122 348 1380 582 698 314 831 1099 49 1442 831 687 47 420 798 1390 1480 927 52 336 1175 665 950 115