Show how a decider for l could be used to construct a decider for an undecidable language. The proof of rices theorem consists of a reduction from the halting problem. Rice theorem computer science engineering cse notes edurev. Riceshapiro or kleenes fixed point theorem hide more informa tion of what is usually expressed in. In order to prove that fl is not computable, we will use a variation of reduction. What you refer to are syntactic properties properties of the program, such as runtime or how many variables are used. The crucial idea in the proof of rices theorem is to convert a question about membership in s into an equivalent question about whether a program halts on itself as input. This is why rices theorem is usually stated about the partial functions that the machines compute, or about the languages that they accept.
Any nontrivial property about the languagerecognized by a turing machine is undecidable. A convenient way of expressing this result is to say that. A property about turing machines can be represented as the languageof all turing machines, encoded as strings, that satisfy that property. Notes on rices theorem consider any kind of software testing problem. It is undecidable to determine whether a given turing machine accepts a nite or in nite number of inputs. Rices theorem meets program analysis bad news and how we deal with it in practice monday, december 10, 2007 reading. Rices theorem deals with semantic properties properties of the function computed by a program, e.
Undecidability by rice theorem recompleteness reductions by. Greens theorem 3 which is the original line integral. The inverse function theorem thus gives a continuously di. Rice theorem states that any nontrivial semantic property of a language which is recognized by a turing machine is undecidable. So in that sense, rices theorem fails in this context.
For example, being a mathematician is a property that is possessed by some humans but not by all. Think of all the corollaries that you can infer from rices theorem. Aspertirices theorem a rst intensional version of rices theorem. Given an integer k, define a partition of the interval 0,1 as a sequence of numbers which divide. For example, it is used for the proof of saa and hl theorems 1, 2. 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. A property, p, is the language of all turing machines that satisfy that property. The intensional content of rices theorem conference paper pdf available in acm sigplan notices 431.
It is undecidable to determine whether a given turing machine accepts a. Kozen 3234 program analysis 392 overview of todays lecture more examples of undecidability via reduction reduction arguments generalize to rices theorem. Thus, as per rices theorem the language describing any nontrivial property of turing machine is not recursive. The recursion theorem 419 by the smn theorem, there is a recursive function g such that.
Reductions and rices theorems indian statistical institute. In that case, it would be good if you describe in more detail what sorts of contexts youre looking for. Rices theorem states that ds is undecidable for a nontrivial s. A great part of its beauty lies in the simplicity of its statement. Newest ricetheorem questions computer science stack. Reductions rices theorems proof of rices theorem 2 let p be a nonmonotone property of r. Rayleigh, rice and lognormal distributions transform methods and the central limit theorem department of electrical engineering princeton university september 30, 20 ele 525. That is, the input to p is a language and the output is a truth value. Suppose towards a contradiction that for same class c the. Come up with an s, such that ds is the blank tape halting problem. Implicit function theorem 1 chapter 6 implicit function theorem chapter 5 has introduced us to the concept of manifolds of dimension m contained in rn. Rices theorem says that there is no computable method fm,p to determine, if given as input a tm m, and some nontrival property p, if the language accepted by m has property p.
As for statements sas, asa, and sss, they are considered in these books as postulates despite that in euclid 5 and many other geometry courses 3, 6 they are considered as theorems. Rices theorem proves in one clean sweep that all these problems. I suspect, though, that you want something that 1 is less trivial and 2 doesnt use oracles and perhaps 3 other conditions. Sas and ssa conditions for congruent triangles 58 proofs of theorems about congruent triangles. In social choice theory, arrows impossibility theorem, the general possibility theorem or arrows paradox is an impossibility theorem stating that when voters have three or more distinct alternatives options, no ranked voting electoral system can convert the ranked preferences of individuals into a communitywide complete and transitive ranking while also meeting a specified set of. We will only consider the case of a countable number of propositions. Statement of rices theorem intro to theoretical computer science duration. The proofs of major results of computability theory like rice. Note on rices theorem the purpose of this note is to give some details of rices theorem and its proof.
Given any machine m, string w, and nontrivial property p, we will create a new machine m. We can prove that this is true using the following technique. Taylor approximation and the delta method rice university. In the literature, the proof of rices theorem usually consists in proving an implication of the form. In mathematics, and in particular the necklace splitting problem, the hobbyrice theorem is a result that is useful in establishing the existence of certain solutions. Recall that the set of all languages, p, is uncountable.
Perplexed by rices theorem computer science stack exchange. Later we noticed an wikipedia entry containing still another proof based on the recursion theorem. To make sense of the statement of the theorem, think of a property of languages that you would like to test. A property about turing machines can be represented as the language of all turing machines, encoded as strings, that satisfy that property. It is undecidable to determine whether a given turing machine accepts only representations of prime numbers, and so on. Pdf the proofs of major results of computability theory like rice, riceshapiro or kleenes fixed point theorem hidemore information of what is. We start with a particular language for expressing programs, and a particular alphabet in which programs and other strings are written.
Suppose towards a contradiction that for same class c the language. Programs p and q are equivalent if they compute the same partial function. Let t fhmijm is a turing machinegbe the set of all turing machines. Proof by reduction suppose that we are given a language l that we believe 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. Then there are tms k and t such that lk lt and lk satisifes p but lt does not. The inverse and implicit function theorems recall that a linear map l. Here is the theorem and its proof, following the needed definition. The property p is about the language recognized by turing machinesif whenever lmln then p contains the encoding of m iff itcontains the encoding of n.
Notes on rices theorem stanford cs theory stanford university. Approximate mean and variance before, we considered the case of just estimating g with gx. The proof of the cauchy interlacing theorem will utilize a fundamental result whose proof is a basic exercise in dimension counting. The integrand in the integral over r is a special function associated with a vector. If p is a nontrivial property, then lp is undecidable. Rice theorem, the proof of the part when the empty language belongs to the property i was going through the classic text introduction to automata theory, languages, and computation by hofcroft, ullman and motwani where i came across the proof the rice theorem as shown.
To do this, we need to formulate a mathematical counterpart of the everyday notion of having a property. The set of programs computing the sorting function in. Obviously there are also other languages which are not recursive. Then there exists a unique probability p measure on irt,bt such that for all. In computability theory, rices theorem states that, for any nontrivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function. Random processes in information systems hisashikobayashi textbook. Pdf the intensional content of rices theorem researchgate. We need the technique known as reduction, where an algorithm converts instances of one problem to instances of another. I am about to remove this paragraph from the article until such time as the author chooses to repair it. The intensional content of rices theorem pearl article pdf available in conference record of the annual acm symposium on principles of programming languages 431 january 2008 with 54 reads. W v for some open neighborhoods v of a,b and w of a,0 in rm. Applying slutskys theorem with w n g0 p n y n and z n as the remainder, we have the righthand side converging to n02g0 2, and thus the desired result follows.
As a second application, we get a very short proof of rices theorem. Proof a nontrivial property is one that is possessed by some objects of a class, but not all. Rices theorem proof we will prove rices theorem by showing that, for any nontrivial property p, we can reduce the halting problem to the problem of determining if the language accepted by a turing machine has property p. Other than that twist, the proof is very similar to the proof that the halting problem is uncomputable. Gauss theorem 3 this result is precisely what is called gauss theorem in r2. In the literature, the proof of rice s theorem usually consists in proving an implication of the form. Rices theorem rices theorem helps explain one aspect of the pervasiveness of undecidability.
Assume, for the sake of contradiction, that l is decidable. Introduction to theory of computation rices theorem and closure. It can either be recursively enumerable or not recursively enumerable. Any nontrivial property about the language recognized by a turing machine is undecidable. Rices theorem proves in one clean sweep that all these problems are. The proof for larger cardinalities is essentially the same, but requires some form of the axiom of choice. Rice s theorem part 1 for some undecidable languages any nontrivial property of the language recognizable by a turing machine recursively enumerable language is undecidable. This theorem is a useful tool in determining undecidability. On the other hand, if instead hc b and hd a, then we obtain z d c fhs d ds ihsds.