Introduction to the theory of computation by michael. Rice theorem grammars and decidability gate questions. An introduction to the theory of computation by thomas sudkamp. Introduction to theory of computation rices theorem and closure properties sungjin im university of california, merced 04212015. Introduction to the theory of computation michael sipser. This is an introductory course on theory of computation intended for undergraduate students in computer science. Rices theorem hans huttel autumn 2007 in this note i will show a theorem which states that a very large family of problems are all undecidable. An automaton with a finite number of states is called a finite automaton.
Rice s theorem georgia tech computability, complexity, theory. Introduction to automata theory, languages and computation by j. A property about turing machines can be represented as the language of all turing machines, encoded as strings, that satisfy that property. Journal of chemical theory and computation vol 16, no 3. Prove that the problem of determining whether a given turingmachines language has property p is undecidable. Rices theorem and reductions via computation histories. It can either be recursively enumerable or not recursively enumerable. Sets, functions and other preliminaries, formal languages, finite automata, regular expressions, turing machines, contextfree languages, rice s theorem, time complexity, npcompleteness, space complexity, log space, oracle machines and turing reducibility, probabilistic complexity, approximation and optimisation. Journal of chemical theory and computation 2020, 16, 3, 19361952 article acs authorchoice publication date web. A central question asked was whether all mathematical problems can be.
Rice theorem rice theorem states that any nontrivial semantic property of a language which is recognized by a turing machine is undecidable. The intensional content of rices theorem proceedings of the 35th. For the love of physics walter lewin may 16, 2011 duration. Looking for an analogue of rices theorem in circuit complexity theory. What is the best book on the theory of computation for. Automata theory plays a major role in the theory of computation, compiler construction, artificial intelligence. Prove that the problem of determining whether a given turing machines language has property p is undecidable. This exercise was taken from the book languages and machines. This is the kind of proof used in many references like 3, 4, 2, 1. Well cover most of the material in this book, which has lots of exercises, solutions, and online supplementary material. We say a tm decide a language l if it accepts all strings in l and rejects all strings not in l. Theory of computation by vivek kulkarni, paperback. Let p be any nontrivial property of the language of a turing machine.
In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. Theory of computation archives page 2 of 2 gate cse. Topics in the theory of computation, volume 24 1st edition. Rice theorem states that any nontrivial semantic property of a language which is recognized by a turing machine is undecidable. Rices theorem georgia tech computability, complexity, theory. This means, for instance, that if a program is running in an environment where whether it acts as a virus for any definition of what a virus is depends in some nontrivial way on its behaviour, then a checker working on its code cannot in general decide whether. Computability theory is one of the oldest fields of computer science. For example, being a mathematician is a property that is possessed by some humans but not by all. Its proof by mapping reduction is surprisingly simple. Rices theorem georgia tech computability, complexity. Difference between constructive dedekind and cauchy reals in computation. Sets, functions and other preliminaries, formal languages, finite automata, regular expressions, turing machines, contextfree languages, rices theorem, time complexity, npcompleteness, space complexity, log space, oracle machines and turing reducibility, probabilistic complexity, approximation and optimisation. Develop formal mathematical models of computation that re.
A simple instructional approach for proving the nonre status of. A property, p, is the language of all turing machines that satisfy that property. Rice theorem computer science engineering cse notes edurev. In computability theory, rices theorem states that all nontrivial, semantic properties of programs are undecidable. Rices theorem applies to sets of machines which all have the same behavior that is, what they actually compute. Rices theorem says nontrivial sets of turing machines sharing a property about inputoutput behaviour are undecidable. Our result, known as rices theorem after henry rice who proved the result in 1953 2, states that if sis a nontrivial property of.
It is used to prove the undecidability of the post correspondence problem. A new reduction technique using computational histories is introduced. In most proofs of rices theorem, the halting problem, or possibly other unsoluble problem, is \reduced to an eventual algorithm that decides some non trivial property p. Rices theorem says that every nontrivial semantic property of programs. The title of this book is introduction to the theory of computation and it was written by michael sipser.
A property is nontrivial if it is neither true for every computable. As most textbooks approach the topic of undecidability. All readings come from discrete mathematics and its applications 6th edition by kenneth rosen, mcgraw hill, isbn 0072880082. We also show rices theorem 17, where the textbook proof needs slight. A semantic property is one about the programs behavior for instance, does the program terminate for all inputs, unlike a syntactic property for instance, does the program contain an ifthenelse statement. Thus, as per rices theorem the language describing any nontrivial property of turing machine is not recursive. An introduction to the theory of computer science provides readers with a mathematically sound presentation of the theory of computer science. Obviously there are also other languages which are not recursive. Acm transactions on computational logic tocl, volume 3, issue 1, january.
Rices theorem 1953 is a major stumbling block in the joint. Proof a nontrivial property is one that is possessed by some objects of a class, but not all. In that case, it would be good if you describe in more detail what sorts of contexts youre looking for. Automata theory i about this tutorial automata theory is a branch of computer science that deals with designing abstract selfpropelled computing devices that follow a predetermined sequence of operations automatically. Of course, rices theorem applies to any pure computation done by some subprogram.
A set of turing machines is, in itself, just a set of integers. We will also explore the properties of the corresponding language classes defined by these models and the relations between them. Comp 280 schedule for each class, you are to have read that days readings and written a brief essay textbook. So in that sense, rices theorem fails in this context. Indian edition published by narosa publishing house. Pdf on an extension of rices theorem and its applications in. There is a newer incarnation of the above book authored by j. Part of the lecture notes in computer science book series lncs. The theoretical concepts and associated mathematics are made accessible by a learn as you go approach that develops an intuitive understanding of the concepts through numerous examples and illustrations.
I came across the below statement in the classic text introduction to automata theory, languages, and computation by hopcroft, ullman, motwani. If you are in india and are used to indian methodologies of teaching then go for theory of computer science by klp mishra. Purchase topics in the theory of computation, volume 24 1st edition. Otherwise, introduction to automata theory, languages and computation by hopcroft and ullman is considered a standard book. Newest ricetheorem questions computer science stack.
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