Reducibility undecidable problems from language theory pdf

Proving undecidability proofs of decidability what decidable means. We will learn about di erent types of reducibility, and. Sanchit sir is taking live sessions on unacademy plus for gate 2020 link for subscribing to the course is. If a language is in r, there is an algorithm that can decide membership in that language. Dec 07, 2015 decidable and undecidable problems on context free grammars. Polynomial veri ers and np can only enumerate a nite subset. We can ask the magic genie is the language of a certain tm.

Decidable and undecidable problems in theory of computation prerequisite turing machine a problem is said to be decidable if we can always construct a corresponding algorithm that can answer the problem correctly. Suppose we are asked to compute all the prime numbers in the range of to 2000. It explains the difficulties of computation, addressing problems that have no algorithm at all and problems that cannot be solved efficiently. The diagonalization method the halting problem is undecidable a turingunacceptable language 5. Eq tm m1,m2 m1,m2 are tms and lm1lm2 instead of setting up a reduction from a tm we can use other undecidable problems such as e tm assume towards contradiction r is a decider for eq tm construct a decider s for e tm such that on input where m is a tm 1. I suppose for a proof by contradiction that halt tm is decidable. We have already established the undecidability if atm, the problem of determining. By the churchturing thesis, any effective model of computation is equivalent in power to a turing machine. A language is polynomially veri able if it has a polynomial time veri er. Undecidable problems the problems for which we cant construct an algorithm that can answer the problem correctly in finite time are termed as undecidable problems. A veri er for a language a is an algorithm v, where a fw jv accepts hw.

More undecidable languages undecidability by rice theorem. The complement of the halting problem, denoted by hp, and dened as. Then there is an effective procedure to obtain a counter machine m such that l is a homomorphic image of lm. Today, we look at other computationally unsolvable problems. I suppose ecg returns true iff g has a eulerian cycle. Reducibility undecidable problems from language theory undecidability of the halting problem given an input of the form hm,wi, s must output accept, if m accepts w, and reject, if m loops or rejects on w. What makes some language theory problems undecidable j. An introduction to the undecidable and the intractable offers a gentle introduction to the theory of computational complexity. Undecidable problems from \\ language theory reductions via computation histories 2. Since we know atm is undecidable, we can show a new language b is undecidable if a machine that can decide b could be used to build a machine that can decide atm.

Pdf decidable and undecidable problems in schedulability. Tacas 2004, held as part of the joint european conferences on theory and. The best known such class is the class of degrees of domains of partial computable functions, the c. Introduction to the theory of computation, second edition michael sipser massachusetts institute of technology thomson course technology australia canada mexico singapore spain united kingdom united states. By solving a lot of these problems, one can become very quick in coming up with proofs for these problems on the spot. I if we solve the eulerian cycle problem, we solve the eulerian path problem. Csci 2670, spring fall 2012 introduction to theory of. Finally, we prove a variety of theorems related to the structure of d. The words language and problem can be used synonymously in theory of computation. This is all about the global theory of the turing degrees.

If a m b and b is a regular language, does that imply that a is a regular language. We can ask the magic genie is the language of a certain tm empty. We will consider the following problems as membership in languages. Decidable problems for regular languages we give algorithms for testing whether a finite automaton accepts a string, whether the language of a finite automaton is empty, and whether two finite automata are equivalent we represent the problems by languages not fas let a dfa b, wb is a dfa that accepts string w. If there is an algorithm that can decide membership in a language, that language is in r. Recall that a set of rstorder sentences sis hereditarily undecidable if there is no computable set of sentences separating sand s\ v,wherevis the set of all valid sentences in the language of s. V where v is the set of all valid sentences in the language of s.

Assume we wish to pro ve problem b to be undecidable. I to check if there is a eulerian path from s to t in g. A decision problem p is decidable if the language l of all yes instances to p is decidable for a decidable language, for each input string, the tm halts either at the accept or the reject state as depicted in the following. Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and turing degrees. A decision problem p is called undecidable if the language l of all yes instances to p is not decidable. Eqtm is the problem of testing whether two tm languages are the same. In the case of deterministic nite automata, problems like equivalence can be solved even in polynomial time. Observe that a is a contextfree language, so it is also turingdecidable. Most of the questions require unique and ingenious proofs. The halting problem the diagonalization method the halting problem is undecidable a turingunacceptable language 5. Decidable and undecidable problems in theory of computation.

Turing machines, recognizability, decidability compsci 501. The function f is called the reduction from a to b. A part of reduction automaton corresponding to a decrementation of d. In computability theory, an undecidable problem is a type of computational problem that requires a yesno answer, but where there cannot possibly be any computer program that always gives the correct answer. Use the tm m p to construct a tm m q that solves q. Convert nfa b to an equivalent dfa c, using the procedure for conversion given in theorem 1. Decidability and undecidability stanford university. A formal definition of reducibility from one language to another. Dragan, kent state university 2 we have seen that the acceptance problem for tms is undecidable. N on input b,w where b is an nfa and w is a string 1. In the theory of automata and formal languages, the undecidability of various properties. Reducibility among languages mapping reductions more.

Recall that atm acceptance problem for tms is undecidable, where. Typical approach to show l is undecidable via reduction from atm to l. A reduction is a way of converting one problem to another problem, so that the. So the problem of containment by a regular language is reducible to. A language is turingrecognizable if there exists a turing machine. The halting problem is undecidable 181 a turingunrecognizable language 183 exercises, problems, and solutions 184 5 reducibility 191 5. Undecidable languages are not recursive languages, but sometimes, they may be recursively enumerable.

Turing machines can be encoded as strings, and other turing machines can read those strings to peform \simulations. We introduce the techniques of mapping reductions, of rice theorem, and of reductions by computational histories time permitting for proving that languages are undecidablenon. What makes some language theory problems undecidable. Decidability and undecidability in toc geeksforgeeks. Hence, m p cannot exist either and p is undecidable. Since it is known that q is undecidable, m q cannot exist. Reducibility let us say a and b are two problems and a is reduced to b. If r indicates that m does not halt on w, then reject. In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yesorno answer. A language is called decidable or recursive if there is a turing machine which accepts and halts on every input string w. Prerequisite turing machine a problem is said to be decidable if we can always construct a corresponding algorithm that can answer the problem correctly.

Thus the rstorder theory of q in the language of rings is undecidable. We will first prove that a particular problem is undecidable. Today well extract some of the key ideas of those proofs and present them as general, abstract definitions and theorems. One can often show that a language l is undecidable by showing that if l is decidable, then so is atm we reduce atm to the language l.

A language is turing decidable if there exists a turing machine that decides it. In computability theory, a turing reduction also known as a cook reduction from a problem a to a problem b, is a reduction which solves a, assuming the solution to b is already known rogers 1967, soare 1987. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. There are undecidable languages over every alphabet. Reducibility a reduction is a way of converting one problem to another problem in such a way that a solution to the second problem can be. Notice that reducibility says nothing about solving a or b alone, but only about the solvability of a in the presence of a solution to b.

Schedulability analysis of the clock constraint specification language. More formally, a turing reduction is a function computable by an. Reducibility in mathematics, many problems are solved by reduction. We then employ reducibility to consider a partial ordering on the set of turing degrees, d. Cisc462, fall 2018, decidability and undecidability 1 decidability and undecidability decidable problems from language theory for simple machine models, such as nite automata or pushdown automata, many decision problems are solvable. We will use this method to prove the undecidability of other problems. The theorem presented here demonstrates that computers are limited in a fundamental way. Of course, the degree of the halting problem is the maximal c. If one of these two tm languages happens to be empty, then we are back to emptytm.

Undecidable problems for language theory reductions via computation histories mapping reducibility. By checking the production rules of the cfl we can easily state whether the language generates any strings or not. Because we have proven decidability for dfas, all we need to do is convert the nfa to a dfa. A tm recognizes a language if it accepts all and only those strings in the language a tm decides a language if it accepts all strings in the language and rejects all strings not in the language a language is called recognizable or recursively enumerable, or r. If a is undecidable and reducible to b, then what can we say about b. Language decidability a language is called decidable or recursive if there is a turing machine which accepts and halts on every input string w. Recall the reduction from eulerian path to eulerian cycle. This will either have the same language as mp, or the empty language. Computers appear to be so powerful that you may believe that all problems will eventually yield to them. Pdf we study schedulability problems of timed systems with nonuniformly recurring. Hopcroft computer science department, cornell university, ithaca, new york 14850 received september 3, 1969 in the theory of automata and formal languages, the undecidability of various. We say the language s is decidable or recursive if there is a program p such that.

Reducibility among languages mapping reductions more undecidable languages undecidability by rice theorem reductions using controlled executions steppers recompleteness sipsers book, chapter 5, sections 5. If a is undecidable and reducible to b, then b is undecidable. For an undecidable language, there is no turing machine which accepts the language and makes a decision for every input string w tm can make decision for some input string though. The field has since expanded to include the study of generalized computability and definability. It can be understood as an algorithm that could be used to solve a if it had available to it a subroutine for solving b. I construct a graph g0 that is identical to g except an additional edge between s and t. Transition function is complete, no duplicate transitions. We introduce the techniques of mapping reductions, of rice theorem, and of reductions by computational histories time permitting for proving that languages are undecidable non. Csci 2670, spring fall 2012 introduction to theory of computing department of computer science. Mapping reducibility computable functions formal definition of mapping reducibility 6. Reducibility due to time constraints we are only going to cover the first 3 pages of this chapter. In this handout, i regularly make use of two problems, namely the halting problem, denoted by hp, and dened as hp fhm. Emptytm is the problem of testing whether a tm language is empty. Readings for this lecture chapter 5 of sipser 1996, 3rd edition.

Formal languages, automata and computation reducibility. Csci 2670, spring fall 2012 introduction to theory of computing. Using reducibility to study languages decidability. Dec 07, 2016 sanchit sir is taking live sessions on unacademy plus for gate 2020 link for subscribing to the course is. If language l is decidable implies atm is decidable, then l is not decidable. Reducibility to relate the solutions of two problems if a solution to a problem b can be used to give a solution to a problem a, it seems that a cannot be harder than b e. Undecidable problems from language theory contextfree language can be shown to be undecidable with similar proofs.

The 3theory of the computably enumerable turing degrees in the language of partial orderings is undecidable. We can intuitively understand decidable problems by considering a simple example. Undecidable problems 4 eqtm hm1,m2i m1,m2 are tms and lm1 lm2 theorem. Mapping reducibility and rices theorem weve seen several undecidability proofs. W e may use reducibility to pro ve undecidability as follo ws. There is a specic problem that is algorithmically unsolvable. Language a is turing reducible to language b, written a. Introduction to the university of virginia school of.