As a consequence they are closed under arbitrary finite state transductions, like quotient k l with a regular language. Show that the class of turing recognizable languages is closed under union. The class of regular languages is closed under kleenestar. Are there languages that are recognizable but not decidable. Decidable languages are not closed under homomorphism.
Show that the collection of turing recognizable languages is closed under the operation of. Cs103 handout 20 fall 2011 november 18, 2011 problem set 8 how powerful are turing machines. Theorem b let a, b y be turingrecognizable languages. Automata, computability, and complexity or, great ideas in theoretical. Recursively enumerable language are closed under kleene star, concatenation, union, intersection. A language is called recognizable if it is the language of some tm. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Recursively enumerable languages are closed under union, intersection, kleene star. Both the turing recognizable and turing decidable languages are closed under concatenation and star hw. Show that the collection of recognizable languages is closed under the operation of. However, the set of turingrecognizable languages is not closed under complement. I will give the proof for turing recognizable languages. The class of regular languages is closed under concatenation. Let us begin with the regular operations, under which the turing recognizable languages are indeed closed.
That is, if l and p are two recursively enumerable languages, then the following languages are recursively enumerable as well. This language is not turing recognizable, and in this problem you will prove why. Is the class of turingrecognizable languages closed under. Show that this type of tm recognizes the class of turing recognizable languages. Decidability and undecidability stanford university. Show that the recursively enumerable languages are closed under union, intersection and kleenestar operation. Let m1 be a tm for l1 and m2 a tm for l2 both may loop a tm m for l1. Let l 1 and l 2 be two turing recognizable languages. Let m1 be a tm for l1 and m2 a tm for l2 both may loop. Here is the solution available online for textbook exercise 9. We showed in a previous homework that the class of turing recognizable languages is closed under union, so eq. Rao, cse 322 3 closure for recognizable languages turing recognizable languages are closed under. I have written a proof to show that a turing decidable languages are closed under union amongst other things. The recognizable languages are closed under union and intersection.
Turing decidable is a subset of turing recognizable, so also countable. Prove that the language it recognizes is equal to the given language. In order to show equivalence between d and m, we need to show two things. How do you prove that a turing recognizable language is not closed under complementation. Next, we will show that the class of context free languages is closed under regular di erence. Pdf closure property of probabilistic turing machines. Michael sipser 1 how to prove turing decidability of languages language hierarchy recognizability. Basic properties of turingrecognizable languages theorem a let a, b y be turingdecidable languages. Sep, 2016 1 answer to show that the collection of turing recognizable languages is closed under the operation of a.
Solved show that the family of linear languages is closed. To prove that a given language is turing recognizable. If l is a linear language, then its homomorphic image hl is regular. We have to show that l1l2 is decided by some turing machine t12. Show that a turing machine with doublyin nite tape recognizes the same class of languages as an ordinary turing machine. 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 concatenation of languages k and l is the language kl xyx. The turingrecognizable or recursively enumerable languages are closed over homomorphism. How to prove decidability or turing recognizability. The doublyin nite tape can simulate an ordinary tm by just not using the portion of its tape to the left of the input. 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. Both decidable and turing recognizable languages are closed under concatenation.
Let l 1 be any context free langauge and l 2 be any regular language. By exchanging the accepting and rejecting final state of m a with each other, we. Let abe a turing recognizable language consisting of descriptions hmiof turing machines m that are all deciders. Thus, the family of context free languages is not closed under di erence. Since k and l are decidable languages, it follows that there exist turing machines m k and m. Give an implementationlevel description and a formal description i. The set of turing decidable languages is closed under union, intersection, and complement.
Showing that turingrecognizable languages are closed under. 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. Even more, regular languages are closed under quotients with arbitrary languages. Next i did some demonstrations to show how t recognizable languages are closed for union, intersection, concatenation and kleene star. The set of turingdecidable languages is closed under union, intersection, and complement. Closure of cfls under inverse homomorphism here, grammars dont help us.
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. However, the recursive languages are not closed under homomorphism. Recursive turing decidable languages are closed under following kleene star, concatenation, union, intersection, complement and set difference. Consider the following two proofs that try to show that recognizable languages are closed under complement. 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. Kleenes theorem language ais regular i ahas a regular expression. Closure properties of regular languages geeksforgeeks. Theory of computation 6 homomorphisms nus computing. There are few more properties like symmetric difference operator, prefix operator, substitution which are closed under closure properties of regular language. Draw a deterministic turing machine that accepts wcwcw w. Prove that the language it recognizes is equal to the given language and that the algorithm halts on all inputs. Showing that turing recognizable languages are closed under union. Show that the collection of recursively enumerable turing recognizable languages is closed under the concatenation operation.
Decidable languages are closed under inverse homomorphisms. 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. 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. We already know that regular languages are closed under complement and intersection. Do you need an answer to a question different from the above. Both the turingrecognizable and turingdecidable languages are closed under concatenation and star hw. Turing machines dfa, nfa, regexp, cfg, and pdas, e.
Homework 1 solutions kevin matulef january 31th, 2001 problem 1. Let l1 be decided by a turing machine t1, and let l2 be decided by a turing machine t2. Showing that turingrecognizable languages are closed under union. Decidable languages are closed under union, intersection, and complementation. 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.
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. Theorem c let a y be a turingrecognizable language that is not turingdecidable. Any class of languages that is closed under difference is closed under intersection. Page2704 others questions and answers for other topics in. Properties of contextfree languages stanford university. 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. Show that the collection of turingrecognizable languages is. Show that the collection of turingrecognizable 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. Recursive and recursive enumerable languages in toc. Let h be a homomorphism and l a language whose alphabet is the output language of h.
For any two recognizable languages l 1 and l 2, let m 1 and m 2 be the tms that respectively recognize them. We may assume that ais in nite since there are in nitely many decidable languages. How do you prove that a turingrecognizable language is not closed under complementation. 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. Recursively enumerable languages rel are closed under the following operations. Cs2mj3 if you think your solution has been marked wrongly.
Show that the collection of turing recognizable languages is closed under the operation of union. Iit is possible for a tm to never reach a halting con guration. 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. 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. We need to show that a turing machine with a doubly in. For any two turing recognizable languages l 1 and l 2, let m 1 and m 2, respectively, be tms that recognize them. Why is showing a language is turing recognizable trickier. It is recognized by a tm u that, on input, simulates m on w step by step.
Turing completeness is significant in that every realworld design for a computing device can be simulated by a universal turing machine. For any two turing recognizable language l1 and l2, let m1 and m2 be the tms that recognize them. They are also closed under complement not part of this course. The other direction follows from the proof that a multitape tm is no. Show that the collection of decidable languages is closed under the operation of 1union 2concatenation 3star. However, the set of turing recognizable languages is not closed under complement. Run m 1 and m 2 alternately on w, one step at a time. 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. The reason for this is that if a language is decidable, then its complement must be decidable as well. 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. Re languages or type0 languages are generated by type0 grammars. Properties of contextfree languages decision properties closure properties. Because we have constructed a turing machine which accepts a 8 b, the language a b is recursively enumerable. How to prove that a turingrecognizable language is not.
Get best help for others questions and answers in computerarchitecture page2704, stepbystep solutions, 100% plagiarism free question answers. 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. The class of regular languages is closed under union. Prove that the re languages are closed under homomorphism. Hw 1 solutions and other problems rajiv raman february 23, 2007 1 homework 1 1. Closure for recognizable languages turingrecognizable languages are closed under. However, turing recognizable and co turing recognizable are not the same, and its this that ive decided to cover in my answer.
But by the previous result, the set of all languages is uncountable. Cs103 handout 20 fall 2011 november 18, 2011 problem set 8. Let us consider that h be the desired homomorphism. Closure properties of regular languages stanford university. Step 1 of 4 suppose, x and y be two turing recognizable languages that. Closure properties of regular languages let l and m be regular languages. Turing recognizable, we let machine m1 be the turing recognizer for l and m2 be the. Show that the collection of turing recognizable languages is closed under the operation of reversal. Give an implementationlevel description of a turing machine that decides the language. We still have to see whether or not there are recognizable languages that are not decidable, and whether or not there are languages that are not recognizable. Let l and m be languages that are recognized by algorithms a and b respectively.
That is, if l1 and l2 are recursive, then l1 l2 is recursive. Why isnt the class of turingrecognizable languages closed. The class of turing decidable languages is closed under complementation. Solved show that the family of linear languages is.
It contains seven problems plus one survey question and one extra. Prove that some decidable language dis not decided by any decider msuch that hmiis in a. We construct a tm m that recognizes the union of l1 and l2. Both decidable and turing recognizable languages are closed under union. Solved show that the collection of turingrecognizable. Recursively enumerable languages are closed under union, intersection, kleene star, kleene plus and concatenation. Thats essential to exhibit a language that is not recursively enumerable. Show that the collection of turingrecognizable languages.