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