Anand sharma, theory of automata and formal languages, laxmi publisher. Decidability and undecidability in toc geeksforgeeks. Now talking about decidability in terms of a turing machine, a problem is said to be a decidable problem if there exists a corresponding turing machine which. There are problems which are algorithmically unsolvable. That is, there are welldefined problems for which no turing machines can provide an algorithm which solves the problem. In the case of deterministic finite automata, problems like equivalence. A problem is said to be decidable if we can always construct a corresponding algorithm that can answer the problem correctly. Gate 2019 cse syllabus contains engineering mathematics, digital logic, computer organization and architecture, programming and data structures, algorithms, theory of computation, compiler design, operating system, databases, computer networks, general aptitude. In this chapter we will look at more undecidable problems. Introduction to automata theory, formal languages and computation. A decision problem p is decidable if the language l of all yes instances to p is decidable. Undecidable problems we will now discuss the notion of undecidability. A language is in r iff there is an algorithm for deciding membership in that language. The automaton sensor is the device by which data is input.
Homework assignment 5 3 post correspondence problem madness 8 points consider the post correspondence problem pcp. Such reductions and the reasoning principles they give rise to are investigated. Prerequisite turing machine a problem is said to be decidable if we can always construct a corresponding algorithm that can answer the problem correctly. We can intuitively understand decidable problems by considering a simple example. If p2 was decidable, then p1 would also be decidable but that becomes a contradiction because p1 is known to be undecidable. The problem of determining whether a string or input can be accepted by a turing machine or not is called undecidability. It explains the difficulties of computation, addressing problems that have no algorithm at all and problems that cannot be solved efficiently.
In computability theory, an undecidable problem is a type of computational problem that requires a yesno answer, but where there cannot possibly be any. Generic complexity of undecidable problems springerlink. Moreover, to clearly and accurately communicate intuitive thoughts about these subjects, a precise and welldefined terminology is required. Undecidable problem semidedidable or totally not decidable a problem that cannot be solved for all cases by any algorithm whatsoever. Computability theory deals primarily with the question of the extent to which a problem is solvable on a computer. Theory of computation emphasizes the topics such as automata, abstract models of computation, and computability. It turns out that some classical undecidable problems are, in fact, strongly. How does this proof, that the halting problem is undecidable. Hierarchy of undecidable problems and its implications. Decidable and undecidable problems in theory of computation. Definition of undecidable problem, possibly with links to more information and implementations. We show a famous result, rices theorem, that any nontrivial purely semantical property of programs undecidable.
Formal languages and automata theory, h s behera, janmenjoy nayak, hadibandhu pattnayak, vikash publishing, new delhi. Hi all, i am doin masters and studying theroy of computation. This book is intended as an upperlevel undergraduate or introductory gradu ate text in computer science theory. Theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm. This book is well known for its proof that many mathematical systems. If we can reduce an already known undecidable problem p1 to a given problem p2, then we can surely say that p2 is also undecidable. A valuable collection both for original source material as well as historical formulations of current problems. How to learn decidability and undecidabilty in theory of. The problem of decidability arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms. This section of the notes deal with computational theory.
It then delves into the important topics, including separate chapters on finite state. Im going over the proof for the halting problem in intro to the theory of computation by sipser and my main concern is about the proof below. What makes some language theory problems undecidable. The proof that the halting problem is undecidable relativizes, that is, it still works if the turing machine is given access to an oracle. The book covers the entire syllabus prescribed by anna university for be cse, jntu, hyderabad and nagpur university. Iterated function systems are used as tools to define. 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. 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. From algorithms and theory of computation handbook, pages 2419 and 2620.
It seeks to provide a comprehensive coverage of all the essential concepts of the subject. An undecidable language georgia tech computability. The words language and problem can be used synonymously in theory of computation. More undecidable problems macmillan higher education. A problem that cannot be solved for all cases by any algorithm whatsoeverequivalently, whose associated language cannot be recognized by a. What is the difference between decidable and undecidable. Computable problems you are familiar with many problems or functions that are computable or decidable, meaning there exists some algorithm that computes an answer or output to any instance of the problem or for any input to the function in a finite number of simple steps.
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. Aside from knowing what good solutions are to common problems, it is also useful to understand the theoretical aspects of computation. 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. 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. Based on research conducted from 1938 to 1952, it consists of three treatises by a prolific author who ranks among the greatest logicians of all time. We have also provided number of questions asked since 2007 and average weightage for each subject. Thus if there is any algorithm for deciding membership in the language, there must be a decider for it. That is, being able to answer the totality question for pdas would enable one to answer the acceptance question for tms. Decidable and undecidable problems about quantum automata.
The statement that the halting problem cannot be solved by a turing machine is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a turing machine. I am looking for an undecidable problem that i could give as an easy example in a presentation to the general public. 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. In the theory of automata and formal languages, the undecidability of various properties has. Introduction to the theory of computation 3rd edition edit edition. Ntms are known to be no more powerful than tms in the sense that the set of problems decidable by ntms is identical to the set of problems decidable by tms, so clearly by this definition there can be no undecidable problems in np. Computable and noncomputable problems in toc geeksforgeeks.
What are the most attractive turing undecidable problems in mathematics. 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. There are thousands of examples, so please post here only the most attractive, best examples. If mdoes accept w, then there is a computation string and so pm. Though there are no exercises, the book is suitable for use as a textbook. It also includes computational complexity, p and np completeness. Are there some problems for which selection from introduction to automata theory, formal languages and computation book. I mean easy in the sense that the mathematics behind it can be described, well, without mathematics, that is with analogies and intuition, avoiding technicalities. Pdf decidable and undecidable problems about quantum automata. On formally undecidable propositions of the principia mathematica and related systems. Pdf theory of computation notes lecture free download.
Vivek kulkarni the book begins with basic concepts such as symbols, alphabets, sets, relations, graphs, strings, and languages. An equivalent definition of np is that it consists of all problems that are decidable not just verifiable in polynomial time by a nondeterministic turing machine. On undecidable propositions of formal mathematical systems, on intuitionistic arithmetic and number theory, on the length of proofs, remarks before the princeton bicentennial conference of problems in mathematics. Consider a contextfree grammar cfg where and s is a starting. Introduction to computational theory data structures and. Major ideas from last time the universal turing machine u tm can be used as a subroutine in other turing machines.
In particular, the halting problem for turing machines with oracle access to the usual halting problem isnt solvable by turing machines with oracle access to the usual halting problem. Pdf decidable and undecidable problems about quantum. For example the automaton could have a thermometer reading temperature, a magnet reading a hard drive, an antenna reading radio waves, or anything else that would import information. Decidability and undecidability stanford university. The proof uses a reduction from the halting problem. Finite automata and regular languages, contextfree languages, turing machines and the churchturing thesis, decidable and undecidable languages and complexity theory. Lets take the halting problem that asks if it is decidable if. An example of an easy to understand undecidable problem.
Turing machines and undecidability theory of computation. An introduction to the undecidable and the intractable offers a gentle introduction to the theory of computational complexity. Computer science theory and applications pp 407417 cite as. This graduatelevel book is well known for its proof that many mathematical systemsincluding lattice theory, abstract projective geometry, and closure algebrasare undecidable. If you can figure out a systematic way an algorithm to answer the question correctly. A language is called decidable or recursive if there is a turing machine which accepts and halts on every input string w. This book explores some of the more important terminologies and questions concerning programs, computers, problems, and computation. Part of the lecture notes in computer science book series lncs, volume 4649. Suppose we are asked to compute all the prime numbers in the range of to 2000. Check out the full advanced operating systems course for free at. The decidability of the contextfree grammar depends on the decidability of the turing machine.
Lecture notes on theory of computation module bput. Computing as we know it is limited in a fundamental way. By solving a lot of these problems, one can become very quick in coming up with proofs for these problems on the spot. In the case of deterministic nite automata, problems like equivalence can be solved even in polynomial time. Over the course of this semester, we have considered many different problems, data structures and algorithms. Undecidable languages are not recursive languages, but sometimes, they may be. See also undecidable problem, np, npcomplete, solvable, tractable, computable. Undecidable problem wikimili, the free encyclopedia. Decidability for a theory concerns whether there is an effective procedure that decides whether the formula is a member of the theory or not, given an arbitrary formula in the signature of the theory.
Or, given a string of zeros and ones, is it a palindrome. Theory of computation tells us that there are some languages that cannot be recognized by a turing machine. Decidable and undecidable problems december 7, 2015 by arjun suresh 9 comments. A simple example is the integer increment operation. What are the most attractive turing undecidable problems in. Theory of computation decidable and undecidable problems. Equivalent language cannot be recognized by a turing machine that halts for all inputs. A problem that cannot be solved for all cases by any algorithm whatsoeverequivalently, whose associated language cannot be recognized by a turing machine that halts for all inputs. 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. If tm m doesnt know when its looping it cant accept or reject which is why a tm is turing recognizable for all strings, then how would could the decider h decide if m could possibly be in a loop. Cisc462, fall 2018, decidability and undecidability. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. Basically it says for some claim of computability can you detect before hand if, or if not, the claim is computable.
A decision problem p is called undecidable if the language l of all yes instances to p is not decidable. I tried to search the sol on internet but didnt find it anywhere. Undecidable problems in fractal geometry simant dube department ofmathematics, statistics, and computing science, university of new england, armidale nsw 2351, australia abstract. Sanchit sir is taking live sessions on unacademy plus for gate 2020 link for subscribing to the course is. Introduction to theory of computation download book. In this paper, a relationship between the classical theory of computation and fractal geometry is established. Some examples already appear on the wikipedia page.
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