Rices theorem part 1 for some undecidable languages any nontrivial property of the language recognizable by a turing machine recursively enumerable language is undecidable. Introduction to the theory of computation, first edition. This is the long awaited second edition of lewis and papadimitrious bestselling theory of computation text. Today, we will talk about this important theorem known as rice s theorem in the context of undecid. 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. In computability theory, the riceshapiro theorem is a generalization of rices theorem, and is named after henry gordon rice and norman shapiro. Computational and applied mathematics of computational and applied mathematics mcaam is an advanced professional degree program designed for students interested in a technical career path in industry or business. Statement of rices theorem intro to theoretical computer science. Introduction to theory of computation rices theorem and. In this substantially modified edition, the authors have enhanced the clarity of their presentation by making the. Introduction to the theory of computation some notes for cis262.
It can either be recursively enumerable or not recursively enumerable. Kleenes theorem in toc part1 a language is said to be regular if it can be represented by using a finite automata or if a regular expression can be generated for it. Introduction to the theory of computation some notes for cis262 jean gallier department of computer and information science university of pennsylvania philadelphia, pa 19104, usa email. This is both a true statement about machines and a consequence of rice s theorem. Find materials for this course in the pages linked along the left. Rice s theorem and other undecidable problemsrices theorem rice s theorem 1 iwe have shown that a number of related problems are undecidable.
Rice s theorem if p is a nontrivial property, is undecidable proof idea. Sep 04, 2016 for the love of physics walter lewin may 16, 2011 duration. This document is highly rated by computer science engineering cse students and has been viewed 143 times. Introduction to the theory of computation some notes for. Appropriate for senior and graduate level courses in computer science theory, automata, and theory of computation.
Npcompleteness example space complexity savitchs theorem reading. A central question asked was whether all mathematical problems can be. Until recently we thought that it was the only possible kind of proof. Theory of computer science rices theorem and other. Thus, as per rices theorem the language describing any nontrivial property of turing machine is not recursive. Mod06 lec33 rices theorem, linear bounded automata,properties of tm duration. Introduction introduction purpose of the theory of computation. A set of turing machines is, in itself, just a set of integers. Iinstead, we prove a much more general result, rice s theorem, which shows that a very large class. These notes are written in latex during lectures in real time, and may contain errors.
Rice theorem in automata theory rice theorem in automata theory courses with reference manuals and examples pdf. Theory of computation handout r uc davis phillip rogaway may 26, 2004 rices theorem you may. The set is obviously set of partial computable function and not empty and not all of the partial computable function then. Computability v prove by reduction this lecture shows more undecidable languages our proof is not based on diagonalization instead, we reduce the problem of. Introduction to the theory of computation michael sipser. Free computation theory books download ebooks online. Our result, known as rices theorem after henry rice who proved the result in 1953 2, states that if sis a nontrivial property of turingrecognizable languages, then the problem. 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. A hard copy of all source files must be provided on the due date. Offering an accessible approach to the topic, theory of computation focuses on the metatheory of computing and the theoretical boundaries between what various computational models can do and not dofrom the most general model, the urm unbounded register machines, to the finite automaton. Learn the skills and acquire the intuition to assess the theoretical limitations of computer programming. Rices theorem georgia tech computability, complexity, theory. A property, p, is the language of all turing machines that satisfy that property. To our knowledge most proofs of rices theorem are based on a reduction.
Faculty research interests fall in the four general areas of numerical analysis and scientific. Prove that the problem of determining whether a given turingmachines language has property p is undecidable. Rice s theorem applies to sets of machines which all have the same behavior that is, what they actually compute. Appropriate for senior and graduate level courses in computer science theory, automata and theory of computation. Introduction to the theory of computation, by michael sipser mit, 2nd edition, 2005 good articles videos. A nondeterminsitic turing machine may have several computation histories on an input. Other topics such as correctness of programs will not be treated here there just isnt enough time. Time complexity of tm halting problem rice theorem post correspondence problem. Rice theorem let p be any specific nontrivial property describing a language of a tm. The proofs of major results of computability theory like rice, rice shapiro or kleenes fixed point theorem hide more informa tion of what is usually expressed in their respective statements. Develop formal mathematical models of computation that re. This is the branch of computer science that aims to understand which problems can be solved using computational devices and how efficiently those problems can be solved. In the last century it became an independent academic discipline and was separated from mathematics. In computability theory, rices theorem states that all nontrivial, semantic properties of.
Theory and computation are crucial parts of modern chemical research since they drive and stimulate investigations by proposing testable hypotheses, as well as providing explanations for chemical observations in terms of fundamental principles. You can watch the course lecture videos on youtube. I special halting problem k i general halting problem h i halting problem on empty tape h 0 imany more results of this type could be shown. On rices theorem 3 the class of regular languages the class of turingdecidable i. 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 r uc davis phillip rogaway november 29, 2012. Computation for the kolmogorov superposition theorem. The proofs of major results of computability theory like rice, rice shapiro or kleenes fixed point theorem hide more informa tion of what is.
Rice theorem in automata theory tutorial 20 march 2020. In the literature, the proof of rice s theorem usually consists in proving an implication of the form. Models of computation basic computability theory 20. Proof a nontrivial property is one that is possessed by some objects of a class, but not all. In computability theory, rice s theorem states that all nontrivial, semantic properties of programs are undecidable. Introduction to the theory of computation computability. This model was introduced around 1963 although it was introduced earlier by post in a di erent format. In rices game, abelard is required to come up with a property of partial functions his choice and a decision procedure. The functions in kol mogorovs construction are pathological, being not di. Any nontrivial property about the language recognized by a turing machine is undecidable. Mod06 lec33 rices theorem,linear bounded automata,properties of tm duration. Prove that the problem of determining whether a given turing machines language has property p is undecidable. 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. So it largely expand the numbers of undeterminable problems. A property is nontrivial if it is neither true for every computable. Rice s theorem in computability theory, rice s theorem states that, for any nontrivial property of partial functions, no general and effective method can decide whether an algorithm computes a. The proof of rices theorem consists of a reduction from the halting prob lem. Theory of computation syllabus is subject to change home. This definition leads us to the general definition that. We shall also name the coordinates x, y, z in the usual way. An introduction robert cartwright rebecca parsons rice university this monograph is an unauthorized revision of lectures on a mathematical theory of computation by dana scott 3. Formal languages, automata models, grammars, decidability, and rice s theorem and pdf computational universality and the churchturing thesis and pdf more fractalbased algorithmic art and pdf. Notes on rices theorem stanford cs theory stanford university. Rice s theorem helps explain one aspect of the pervasiveness of undecidability. Programs all source and executable must be submitted on moodle. Theory of computation brice huang fall 2016 these are my lecture notes for the fall 2016 iteration of 18.
We shall use a righthanded coordinate system and the standard unit coordinate vectors, k. Introduction to automata, languages and computation. Obviously there are also other languages which are not recursive. Pages in category theorems in theory of computation the following 4 pages are in this category, out of 4 total.
Let p be any nontrivial property of the language of a turing machine. Class notes on theory of computation bct iii chapter 1. For example, the set of machines which halts on input 0 is undecidable. Introduction to theory of computation rice s theorem and closure properties sungjin im university of california, merced 04212015. Introduction to theory of computation rices theorem and closure properties sungjin im university of california, merced 04212015. In more formal terms, let p be a language consisting of turing machine descriptions where p. This is the kind of proof used in many references like 3, 4, 2, 1. What is the significance of rices theorem in computational.
Rices theorem if p is a nontrivial property, is undecidable proof idea. The decision problem can be expressed as the language tour tm fhmij for some input w m visits every ordinary state during its computation g tour. Dec, 2017 apr 18, 2020 rice theorem computer science engineering cse notes edurev is made by best teachers of computer science engineering cse. The sentence as it stands uses a common standard rhetorical device, and in another context something like im not talking about george, im talking about his cat.
Perhaps the theorem is more clearly stated as a game. The basic theorem relating the fundamental theorem of calculus to multidimensional in. The theory of computation can be considered the creation of models of all kinds in the field of computer science. Theory of computation r uc davis phillip rogaway november 29, 2012 rices theorem rices theorem helps explain one aspect of the pervasiveness of undecidability. Computational and applied mathematics theory of computation xv yijia chen fudan university. Probabilistic turing machines and complexity classes. Introduction to the theory of computation michael sipser gain a clear understanding of even the most complex, highly theoretical computational theory topics in the approachable presentation found only in the marketleading introduction to the theory of computation, 3e. A rejecting computation history for m on w is similar, except c l is a rejecting con. Jan 19, 2014 rice s theorem proves that any nontrivial property of a program is undeterminable.
This video is part of an online course, intro to theoretical computer science. I each cell has several variables associated with it, one for each tape symbol and state, corresponding to the possible settings of that cell. Introduction to theory of computation rices theorem. A property about turing machines can be represented as the language of all turing machines, encoded as strings, that satisfy that property. Scotts monograph uses a formulation of domains called neighborhood systems in which. Lecture notes automata, computability, and complexity. A deterministic turing machine has at most one computation history on any given input. Abelards decision procedure must accept an algorithm from his adversary, eloise, as input. This problem is obviously decidable, because there is a characteristic function for this set. Elements of the theory of computation 2nd edition by harry. For example, being a mathematician is a property that is possessed by some humans but not by all. Rice theorem computer science engineering cse notes edurev.
A property of languages is simply a set of languages. Automata and language theory, finite automata, regular expressions, pushdown automata, contextfree grammars, pumping lemmas, computability theory, turing machines, churchturing thesis, decidability, halting problem, reducibility, recursion theorem, complexity theory, time and space measures, hierarchy. Feb 23, 2015 rices theorem georgia tech computability, complexity, theory. Stokes theorem 1 chapter stokes theorem in the present chapter we shall discuss r3 only. Rice s theorem in fact has lots of useful things to say about machines and programs, albeit indirectly. It has the advantage of being closer to actual computer architecture. Elements of the theory of computation, spring 2020 course overview this course is an introduction to the theory of computation.1453 974 1014 414 441 945 822 21 1368 344 63 714 172 937 757 115 32 1051 269 371 37 79 669 221 1303 1260 853 725 1421 1346 432 31 276 941