An introduction to the theory of computation by thomas sudkamp. Rice theorem rice theorem states that any nontrivial semantic property of a language which is recognized by a turing machine is undecidable. For example, the set of machines which halts on input 0 is undecidable. A property is nontrivial if it is neither true for every computable. 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. Develop formal mathematical models of computation that re. We also show rices theorem 17, where the textbook proof needs slight.
A property about turing machines can be represented as the language of all turing machines, encoded as strings, that satisfy that property. Acm transactions on computational logic tocl, volume 3, issue 1, january. Any nontrivial property about the language recognized by a turing machine is undecidable. 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. Comp 280 schedule for each class, you are to have read that days readings and written a brief essay textbook. 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. It is used to prove the undecidability of the post correspondence problem. For the love of physics walter lewin may 16, 2011 duration.
Prove that the problem of determining whether a given turingmachines language has property p is 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. 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. If you are in india and are used to indian methodologies of teaching then go for theory of computer science by klp mishra. Newest ricetheorem questions computer science stack. Let p be any nontrivial property of the language of a turing machine. 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. We will also explore the properties of the corresponding language classes defined by these models and the relations between them. Theory of computation archives page 2 of 2 gate cse. 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. I suspect, though, that you want something that 1 is less trivial and 2 doesnt use oracles and perhaps 3 other conditions. I came across the below statement in the classic text introduction to automata theory, languages, and computation by hopcroft, ullman, motwani.
What is the best book on the theory of computation for. Computability theory is one of the oldest fields of computer science. In this course we will introduce various models of computation and study their power and limitations. An introduction to the theory of computer science provides readers with a mathematically sound presentation of the theory of computer science. In more formal terms, let p be a language consisting of turing machine descriptions where p. Topics in the theory of computation, volume 24 1st edition. For example, being a mathematician is a property that is possessed by some humans but not by all. 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. Rices theorem georgia tech computability, complexity, theory. Rices theorem says nontrivial sets of turing machines sharing a property about inputoutput behaviour are undecidable. Well cover most of the material in this book, which has lots of exercises, solutions, and online supplementary material.
Rice theorem computer science engineering cse notes edurev. 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. Theory of computation by vivek kulkarni, paperback. In that case, it would be good if you describe in more detail what sorts of contexts youre looking for. Thus, as per rices theorem the language describing any nontrivial property of turing machine is not recursive.
All readings come from discrete mathematics and its applications 6th edition by kenneth rosen, mcgraw hill, isbn 0072880082. Prove that the problem of determining whether a given turing machines language has property p is undecidable. Rices theorem georgia tech computability, complexity. Rice s theorem georgia tech computability, complexity, theory. Rice theorem states that any nontrivial semantic property of a language which is recognized by a turing machine is undecidable.
The title of this book is introduction to the theory of computation and it was written by michael sipser. It can either be recursively enumerable or not recursively enumerable. Purchase topics in the theory of computation, volume 24 1st edition. Proof a nontrivial property is one that is possessed by some objects of a class, but not all. Looking for an analogue of rices theorem in circuit complexity theory. Rice theorem grammars and decidability gate questions. We say a tm decide a language l if it accepts all strings in l and rejects all strings not in l. Introduction to the theory of computation by michael. Indian edition published by narosa publishing house.
Minor in computational and applied mathematics volume 24 1st edition. This is the kind of proof used in many references like 3, 4, 2, 1. A formal and constructive theory of computation programming. As most textbooks approach the topic of undecidability. Rices theorem and reductions via computation histories. A property, p, is the language of all turing machines that satisfy that property. Difference between constructive dedekind and cauchy reals in computation. A simple instructional approach for proving the nonre status of. Otherwise, introduction to automata theory, languages and computation by hopcroft and ullman is considered a standard book. This is an introductory course on theory of computation intended for undergraduate students in computer science. A set of turing machines is, in itself, just a set of integers. Rices theorem 1953 is a major stumbling block in the joint. There is a newer incarnation of the above book authored by j.
In computability theory, rices theorem states that all nontrivial, semantic properties of programs are undecidable. Introduction to automata theory, languages and computation by j. A central question asked was whether all mathematical problems can be. The intensional content of rices theorem proceedings of the 35th. Rices theorem says that every nontrivial semantic property of programs. In computability theory, rice s theorem states that all nontrivial, semantic properties of programs are undecidable.
Journal of chemical theory and computation vol 16, no 3. Rices theorem applies to sets of machines which all have the same behavior that is, what they actually compute. A new reduction technique using computational histories is introduced. Introduction to the theory of computation michael sipser. So in that sense, rices theorem fails in this context. Introduction to theory of computation rices theorem and closure properties sungjin im university of california, merced 04212015.
Automata theory plays a major role in the theory of computation, compiler construction, artificial intelligence. Of course, rices theorem applies to any pure computation done by some subprogram. Its proof by mapping reduction is surprisingly simple. 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. Pdf on an extension of rices theorem and its applications in. This exercise was taken from the book languages and machines. The proofs of major results of computability theory like rice. Journal of chemical theory and computation 2020, 16, 3, 19361952 article acs authorchoice publication date web. Obviously there are also other languages which are not recursive. Part of the lecture notes in computer science book series lncs.
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