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