For any two recognizable languages l 1 and l 2, let m 1 and m 2 be the tms that respectively recognize them. 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. Recursive turing decidable languages are closed under following kleene star, concatenation, union, intersection, complement and set difference. Both the turing recognizable and turing decidable languages are closed under concatenation and star hw. Showing that turingrecognizable languages are closed under. Closure properties of regular languages stanford university. Theorem b let a, b y be turingrecognizable languages. Cs103 handout 20 fall 2011 november 18, 2011 problem set 8 how powerful are turing machines. We construct a tm m that recognizes the union of l1 and l2.
Turing completeness is significant in that every realworld design for a computing device can be simulated by a universal turing machine. Decidable languages are closed under inverse homomorphisms. Any class of languages that is closed under difference is closed under intersection. Let m1 be a tm for l1 and m2 a tm for l2 both may loop. Homework 1 solutions kevin matulef january 31th, 2001 problem 1. Show that the collection of decidable languages is closed under the operation of 1union 2concatenation 3star. Basic properties of turingrecognizable languages theorem a let a, b y be turingdecidable languages.
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. Let l 1 be any context free langauge and l 2 be any regular language. Give an implementationlevel description and a formal description i. Hw 1 solutions and other problems rajiv raman february 23, 2007 1 homework 1 1. Why isnt the class of turingrecognizable languages closed. The problem we have here is that in an arbitrary string w that would be an input to t12 we do not. Show that the collection of turing recognizable languages is closed under the operation of reversal. Let h be a homomorphism and l a language whose alphabet is the output language of h. Show that the collection of recursively enumerable turing recognizable languages is closed under the concatenation operation.
Show that the collection of turing recognizable languages is closed under the operation of union. It contains seven problems plus one survey question and one extra. The set of turingdecidable languages is closed under union, intersection, and complement. How to prove decidability or turing recognizability. Let l1 be decided by a turing machine t1, and let l2 be decided by a turing machine t2. Rao, cse 322 3 closure for recognizable languages turing recognizable languages are closed under.
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. For any two turing recognizable language l1 and l2, let m1 and m2 be the tms that recognize them. 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. Sep, 2016 1 answer to show that the collection of turing recognizable languages is closed under the operation of a. By exchanging the accepting and rejecting final state of m a with each other, we. Both decidable and turing recognizable languages are closed under concatenation. Why is showing a language is turing recognizable trickier. Let l and m be languages that are recognized by algorithms a and b respectively. Decidability and undecidability stanford university. Prove that the language it recognizes is equal to the given language and that the algorithm halts on all inputs. If l is a linear language, then its homomorphic image hl is regular.
Get best help for others questions and answers in computerarchitecture page2704, stepbystep solutions, 100% plagiarism free question answers. 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. Closure for recognizable languages turingrecognizable languages are closed under. There are few more properties like symmetric difference operator, prefix operator, substitution which are closed under closure properties of regular language. We may assume that ais in nite since there are in nitely many decidable languages. Cfg is a co turing recognizable language if and only if its complement eq cfg is a turing recognizable language. If l1 is a regular language and l2 is a context free language, then l1. Let m1 be a tm for l1 and m2 a tm for l2 both may loop a tm m for l1. Show that a turing machine with doublyin nite tape recognizes the same class of languages as an ordinary turing machine. Closure properties of regular languages let l and m be regular languages. Theorem c let a y be a turingrecognizable language that is not turingdecidable. Let us consider that h be the desired homomorphism. However, the recursive languages are not closed under homomorphism. Show that this type of tm recognizes the class of turing recognizable languages.
Show that the collection of recognizable languages is closed under the following. In order to show equivalence between d and m, we need to show two things. Prove that the language it recognizes is equal to the given language. Step 1 of 4 suppose, x and y be two turing recognizable languages that. Is the class of turingrecognizable languages closed under. Prove that the re languages are closed under homomorphism. 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. I have written a proof to show that a turing decidable languages are closed under union amongst other things. How to prove that a turingrecognizable language is not.
Iit is possible for a tm to never reach a halting con guration. Cs103 handout 20 fall 2011 november 18, 2011 problem set 8. This language is not turing recognizable, and in this problem you will prove why. I will give the proof for turing recognizable languages. The turingrecognizable or recursively enumerable languages are closed over homomorphism.
The class of regular languages is closed under kleenestar. But by the previous result, the set of all languages is uncountable. We need to show that a turing machine with a doubly in. Im studying turing machines and ive already showed how turing decidable is closed for the operations of union, intersection, concatenation, complement and kleene star. Recursively enumerable languages are closed under union, intersection, kleene star. 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. Let us begin with the regular operations, under which the turing recognizable languages are indeed closed. Because we have constructed a turing machine which accepts a 8 b, the language a b is recursively enumerable. 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. Let l 1 and l 2 be two turing recognizable languages. However, the set of turing recognizable languages is not closed under complement. Properties of contextfree languages stanford university. Theory of computation 6 homomorphisms nus computing. Do you need an answer to a question different from the above. Automata, computability, and complexity or, great ideas in theoretical. 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. For any two turing recognizable languages l 1 and l 2, let m 1 and m 2, respectively, be tms that recognize them. Recursively enumerable language are closed under kleene star, concatenation, union, intersection. Show that the collection of turingrecognizable languages.
Show that the collection of turingrecognizable languages is closed. 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 concatenation of languages k and l is the language kl xyx. Solved show that the family of linear languages is. Properties of contextfree languages decision properties closure properties.
The class of turing decidable languages is closed under complementation. The class of regular languages is closed under union. Recursively enumerable languages are closed under union, intersection, kleene star, kleene plus and concatenation. It is recognized by a tm u that, on input, simulates m on w step by step. The recognizable languages are closed under union and intersection. We need to show that a turing machine with a doubly infinite tape, d is equivalent to an ordinary turing.
How do you prove that a turingrecognizable language is not closed under complementation. We have to show that l1l2 is decided by some turing machine t12. Decidable languages are not closed under homomorphism. 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.
That is, if l1 and l2 are recursive, then l1 l2 is recursive. 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. The reason for this is that if a language is decidable, then its complement must be decidable as well. Are there languages that are recognizable but not decidable. Homework five solution cse 355 arizona state university. Pdf closure property of probabilistic turing machines. Cs2mj3 if you think your solution has been marked wrongly. However, the set of turingrecognizable languages is not closed under complement. Draw a deterministic turing machine that accepts wcwcw w. Show that the collection of turing recognizable languages is closed under the operation of 1union 2concatenation 3star 4intersection 5 homomorphism.
Showing that turing recognizable 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. Thats essential to exhibit a language that is not recursively enumerable. Next i did some demonstrations to show how t recognizable languages are closed for union, intersection, concatenation and kleene star. Showing that turingrecognizable languages are closed under union. Since k and l are decidable languages, it follows that there exist turing machines m k and m. A language is called recognizable if it is the language of some tm. Closure of cfls under inverse homomorphism here, grammars dont help us. However, turing recognizable and co turing recognizable are not the same, and its this that ive decided to cover in my answer. Closure properties of regular languages geeksforgeeks. They are also closed under complement not part of this course. To prove that a given language is turing recognizable. Let abe a turing recognizable language consisting of descriptions hmiof turing machines m that are all deciders.
Both the turingrecognizable and turingdecidable languages are closed under concatenation and star hw. Kleenes theorem language ais regular i ahas a regular expression. Show that the collection of turingrecognizable languages is. Give an implementationlevel description of a turing machine that decides the language. Show that the recursively enumerable languages are closed under union, intersection and kleenestar operation. Both decidable and turing recognizable languages are closed under union. Michael sipser 1 how to prove turing decidability of languages language hierarchy recognizability. We construct a tm m0that recognizes the union l 1 l 2. That is, if l and p are two recursively enumerable languages, then the following languages are recursively enumerable as well. 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. 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. Learn vocabulary, terms, and more with flashcards, games, and other study tools.
Showing that turingrecognizable languages are closed. Recursively enumerable languages rel are closed under the following operations. Turing machines dfa, nfa, regexp, cfg, and pdas, e. Decidable languages are closed under union, intersection, and complementation. Re languages or type0 languages are generated by type0 grammars. Solved show that the collection of turingrecognizable. 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.
Show that the class of turing recognizable languages is closed under union. The class of regular languages is closed under concatenation. Show that the collection of turing recognizable languages is closed under the operation of. The doublyin nite tape can simulate an ordinary tm by just not using the portion of its tape to the left of the input. 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. We already know that regular languages are closed under complement and intersection. The set of turing decidable languages is closed under union, intersection, and complement.
Run m 1 and m 2 alternately on w, one step at a time. Recursive and recursive enumerable languages in toc. The other direction follows from the proof that a multitape tm is no. Let lbe a decidable language and mbe the turing machine that decides l. We showed in a previous homework that the class of turing recognizable languages is closed under union, so eq. Consider the following two proofs that try to show that recognizable languages are closed under complement. Turing decidable is a subset of turing recognizable, so also countable.
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