Nondeterministically select a nonempty leftmost part of the input xwhich has not been read yet and copy it on the second tape 3. 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. There are two equivalent major definitions for the concept of a recursive language. Turing recognizable languages are closed under union and complementation. So, if a class is not closed under an operation, we cannot say anything about the class of the resulting language of the operation it may or may not belong to the class of the operand languages. Introduction to theory of computation closure properties. The string is in l if and only if m accepts w after making at most i moves. Decidable languages are closed under union, intersection, and complementation. It rejects a string by either rejecting and halting or by never halting and running forever. Why isnt the class of turingrecognizable languages closed. We already that regular languages are closed under complement and union. Although it might take a staggeringly long time, m will eventually accept or reject w.
Show that the collection of decidable languages is closed under the operation of. Recursive and recursive enumerable languages in toc. 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. The language accepted by the machine are all strings starting with a 0. Intersection of two recursive languages are of same type. Approximately all the properties are decidable in case of finite automaton. We will show a decidable language l and a homomorphism h such that hl is undecidable. Decidable languages are not closed under homomorphism. They are in general not closed under intersection and complement.
W ew an t to pro v e that the family of con textfree languages is closed under rev ersal. Why are recursively enumerable languages not closed under. Show that the collection of turingrecognizable languages. Recursive languages are closed under the following operations. Closure under difference if l and m are regular languages, then so is l m strings in l but not m. A recursively enumerable language is accepted by a nonhalting turing machine. Show that the family of linear languages is not closed under intersection. Since regular languages are closed under union and complementation, we have il 1 and l 2 are regular il 1 l 2 is regular ihence, l 1 \l 2 l 1 l 2 is regular.
Show that the collection of turingrecognizable languages is closed under the operation of union. Show that the collection of turingrecognizable languages is closed under homomorphism. What is the collection of decidable languages closed under. No, because decidable problems are closed over complement. Closure properties of decidable languages decidable languages are closed under.
Suppose both a and the complement of a are turingrecognisable. Homework 7 solutions new jersey institute of technology. Turing decidable languages are closed under intersection and complementation. Since we can always write regular expression for any homomorphism of regular language its closed under homomorphism 6inverse homomorphism. Showing that turingrecognizable languages are closed. 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.
Closure properties of regular languages let land m be regular languages. Computer science stack exchange is a question and answer site for students, researchers and practitioners of computer science. Given a decider m, you can learn whether or not a string w. Prove that the class of decidable languages is closed under union, concatenation and kleene star. Closure properties of regular languages geeksforgeeks.
Show that the class of turingrecognizable languages is closed under c star d balance think about union solution on p. Word problems of groups, formal languages and decidability. Nonclosure under difference we can prove something more general. Statement 1 is true as we can convert every nondeterministic tm to. Feb 04, 2014 a recursively enumerable language is accepted by a nonhalting turing machine. Recall that the class of contextfree languages is closed under concatenation.
Solved show that the family of linear languages is. Is the class of turingrecognizable languages closed under. In short, closure property is applicable, only when a language is closed under an operation. That is, if l1 and l2 are recursive, then l1 l2 is recursive. In this problem, you will explore several proposed closure properties of these languages. It is not closed under homomorphism, because homomorphic images of linear conjunctive languages already constitute all recursively enumerable sets 10, 24. Cs103 handout 20 fall 2011 november 18, 2011 problem set 8. If so, then e f is a true law, and if not then the law is false notice that this is an adhoc method to decide equality of thepairs or languages.
Contextsensitive languages are closed under union, intersection, kleene star, kleene plus and concatenation. We construct the following nondeterministic 2tape turing machine m. Since k and l are decidable languages, it follows that there exist turing machines m k and m. Each of the languages below in parts a, b, c, d is of one of the following types. Similarly w e can see that for an y p ossible p osition of string vxy the resulting pump ed string is not in the language. Union, intersection, concatenation, kleene closure 5. However, the recursive languages are not closed under homomorphism. They are also closed under complement not part of this course. Let a and b be dfas whose languages are l and m, respectively. Showing that turingrecognizable languages are closed under union. Is the class of turingrecognizable languages closed under homomorphism.
Then any undecidable language l0and we know that undecidable languages exist e. Reducibility to show certain problems are not decidable or even nonre k and k. Show that the class of decidable languages is closed under. Why isnt the class of turingrecognizable languages.
Because a is recursively enumerable, there is a turing machine, t 9, which will accept a string s if and only if s. Thus, if cfls were closed under difference, they would be closed under intersection, but they are not. For any two decidable languages l 1 and l 2, let m 1 and m 2, respectively be the tms that decide them. There is clearly a contradiction somewhere in my reasoning.
Are turingrecognizable languages closed under intersection. Recall a closure property is a statement that a certain operation on languages, when applied to languages in a class e. We construct a tm m0that decides the union of l 1 and l 2. In addition, the complement ac is also turing decidable since the class of turing decidable languages is closed under complementation, so that ac is also turingrecognisable. That is, show that if l1 and l2 are decidable languages, then l1 intersection l2 is a decidable language. Union, intersection, concatenation, kleene closure re languages are not closed under. Showing that turingrecognizable languages are closed under. Given two recursively enumerable languages, a and b, we would like to show that a 8 b is recursively enumerable. We say that a class of languages f is closed under homomorphism if k. Prove that recursive languages are closed under intersection 3. Decidable languages a language l is called decidable iff there is a decider m such that.
But avoid asking for help, clarification, or responding to other answers. In theoretical computer science and formal language theory, a regular language also called a rational language is a formal language that can be expressed using a regular expression, in the strict sense of the latter notion used in theoretical computer science as opposed to many regular expressions engines provided by modern programming languages, which are augmented with features that allow. Concatenation kleene closure star operator homomorphism, and inverse homomorphism re languages are closed under. The class of regular languages is closed under homomorphism. We can form new languages via monoid homomorphisms. That is, if l and p are two recursive languages, then the following languages are recursive as well. For any two decidable languages l 1 and l 2, let m 1 and m 2 be the tms that decide them, respectively. Undecidability there are two types of tms based on halting.
This is surely a decidable language, however, any language l0is now a subset of l. Both decidable and turing recognizable languages are closed under union. Turing recognizable languages are closed under union and intersection. Dec 07, 2015 so, if a class is not closed under an operation, we cannot say anything about the class of the resulting language of the operation it may or may not belong to the class of the operand languages. Thanks for contributing an answer to computer science stack exchange.
The same characterization holds for ll contextfree languages. While these are easy to see, the following result is more dif. Solved show that the family of linear languages is closed. In addition, the complement ac is also turingdecidable since the class of turingdecidable languages is closed under complementation, so that ac is also turingrecognisable.
A recursive formal language is a recursive subset in the set of all possible words over the alphabet of the language a recursive language is a formal language for which there exists a turing machine that, when presented with any finite input string, halts and accepts if the string is in. Closure under \ proposition regular languages are closed under intersection, i. Im also not sure why the books answer for the same question for decidable languages below is not sufficient. The point is, we should not reject w just because we found a. Concatenation l1 is context free l2 is context free l1l2 is contextfree concatenation. Theory of computation 6 homomorphisms nus computing. Posted 2 years ago prove that the class of decidable languages is not closed under homomorphism. Recursively enumerable languages closed under complementation. This content was copied from view the original, and get the alreadycompleted solution here. Namely,if l is a con text free language, w ew an t to pro v e that r is also a con textfree. Need to show that union of 2 decidable ls is also decidable let m1 be a decider for l1 and m2 a decider for l2 a decider m for l1. We have seen that the regular languages are closed under common settheoretic operations. For regular languages, we can use any of its representations to prove a closure property.
Then a is obviously turingrecognisable being decidable means that there is a decider that recognises the language. 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. Here it is known that the intersection of two recursive languages is a recursive language, then cant we say that its decidable that intersection will be recursive one. Properties of contextfree languages stanford university. Recursive tms thattms that always halt, no matter accepting or nonno matter accepting or non accepting decidable problems recursively enumerable tms thattms that are guaranteed to haltare guaranteed to halt only on acceptanceonly on acceptance. That is, if and are contextfree languages, so are, and. There are few more properties like symmetric difference operator, prefix operator, substitution which are closed under closure properties of regular language.
The family of deterministic contextfree languages is closed under a homomorphism h if and only if h is either a code of bounded deciphering delay, or the images of all symbols under h are powers of the same string. A recursively enumerable language is a formal language for which there exists a turing machine or other computable function that will halt and accept when presented with any string in the language as input but may either halt and reject or loop forever when presented with a string not in the language. The contextfree languages are closed under union, concatenation and kleene closure. Use pcpto show the undecidabilityof the problem to determine if the intersection of two. F and that f is closed under inverse homomorphism if l.
We need to pick up any two cfls, say l1 and l2 and then show that the union of these languages, l1 l2 is a cfl. Show that the collection of decidable languages is closed under the operations of a. Make the final states of c be the pairs where astate is final but bstate is not. Onecounter languages the languages accepted by a onecounter automaton, i. The closure of contextfree languages maynooth university. The class of regular languages is closed under inversion. An introduction to mildly context sensitive grammar formalisms. The concatenation of languages k and l is the language kl xyx. Homomorphisms preserving deterministic contextfree languages. Regular, cfg, recursive languages real computer science. We consider a language together with the subword relation. This is known as afl abstract family of languages theory.
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