Rice theorem proof pdf david

One defines an invariant called the hopf invariant which allows one to prove theorem adams. 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. First, we need to understand what problems belong to the classes p and np. This is a lightly disguised type of nonexistence claim. A formal and constructive theory of computation programming. Think of all the corollaries that you can infer from rices theorem. Since, p is nontrivial, at least one language satisfies p, i. We start with a particular language for expressing programs, and a particular alphabet in which programs and other strings are written. 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. Sn has an hstructure if and only if n 0, 1, 3 or 7. Consider the language l cde ned as follows l c fhmijlm 2cg. Acknowledgements we thank david harari for pointing out that many of the arguments in this paper hold for an arbitrary eld of characteristic not 2, and not just for global elds of characteristic not 2. We reduce hp to the set m i plm t, thereby showing that the.

Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. I am about to remove this paragraph from the article until such time as the author chooses to repair it. Assume that there was a deterministic tm h that, given m,x decides whether m halts. Connect the contours c1 and c2 with a line l which starts at a point a on c1 and ends at a point b on c2. Henry gordon rice, 1951 nontrivial means the property is true for some tms, but not for all tms. For example, it is used for the proof of saa and hl theorems 1, 2. Let c2 be a positively oriented simple closed contour entirely inside the interior of c1. We also show rices theorem 17, where the textbook proof needs slight changes. 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. Theorem, which roughly speaking describes the probability that a randomly chosen number in any interval is prime. Show how a decider for l could be used to construct a decider for an undecidable language. For the following language, prove, without using rices. We assume h is decidable and obtain a contradiction.

The main goal of the program is to take various theorems from mathematics and determine their strength. Assume that the hypothesis of the proposition holds. Any nontrivial property about the execution of any universal computing system is undecidable. Then, c1 breaks the complex plane up into two regions. Programs p and q are equivalent if they compute the same partial function. So this is a good situation for applying proof by contradiction. The theorem could be restated as there is no largest prime or there is no. Assume that the above problem were decidable and could be decided by a turing machine m. Rice s theorem deals with semantic properties properties of the function computed by a program, e. The following proof is by joel david hamkins and appeared an. A property about turing machines can be represented as the language of all turing machines, encoded as strings, that satisfy that property. I am sure you have noticed that i listed precisely your alleged counterexamples to rices theorem, which says.

Rice and riceshapiro theorems for transfinite correction. It is undecidable to determine whether a given turing machine accepts only representations of prime numbers, and so on. Obviously there are also other languages which are not recursive. Proving undecidability 5 proofs of undecidability to prove a language is undecidable, need to show there is no turing machine that can decide the language. Sas and ssa conditions for congruent triangles 58 proofs of theorems about congruent triangles. Thanks for contributing an answer to mathematics stack exchange. Our result, known as rices theorem after henry rice who proved the result in 1953 2, states that if sis a nontrivial property of. This is why rices theorem is usually stated about the partial functions that the machines compute, or about the languages that they accept. Hobbyrice theorem mathematical analysis hodge index theorem algebraic surfaces. The landes research group at rice university in houston, texas. Rices theorem, the recursion theorem, and the fixedpoint theorem cs 154 midterm in class on thursday.

Let me present rices theorem in a different way in fact, closer to the original presentation. We can prove that this is true using the following technique. The mean value theorem today, well state and prove the mean value theorem and describe other ways in which derivatives of functions give us global information about their behavior. Any nontrivial property about the language recognized by a turing machine is undecidable. It is undecidable to determine whether a given turing machine accepts a. In 1900 david hilbert 18621941 formulated 23 scientific. Then either l c is empty, or it contains the descriptions of all turing machines, or it is undecidable. Rice s theorem cannot be used to show the undecidability of these two languages. Chapter 17 proof by contradiction university of illinois. It can either be recursively enumerable or not recursively enumerable. Let c1 be a positively oriented simple closed contour. The landes research group at rice university the landes research group develops new spectroscopic tools to image chemical dynamics at interfaces at the limit of a single event. In computability theory, rice s theorem states that all nontrivial, semantic properties of programs are undecidable.

To our knowledge most proofs of rices theorem are based on a reduction. Sas and ssa conditions for congruent triangles rice university. S2n1 sn of hopf invariant 1 if and only if n 1, 2, 4, or 8. If m does not accept or doesnt halt, then do not accept x or do not. But avoid asking for help, clarification, or responding to other answers. I suspect, though, that you want something that 1 is less trivial and 2 doesnt use oracles and perhaps 3 other conditions. In the literature, the proof of rices theorem usually consists in proving. A computable extensional property of programs either holds of all programs or of none. Rice and riceshapiro theorems for transfinite correction grammars article in mathematical logic quarterly 575. Integrate over a new contour c that both begins and. Our last proof by induction in class was the binomial theorem. We also proved that the tower of hanoi, the game of moving a tower of n discs from one of three pegs to another one, is always winnable in 2n.

Proving undecidability proofs of decidability what decidable means. Hiberts program david hibert, to find a complete and consistent set of axioms for all. Proof of the extreme value theorem duke university. By kurt godel, in incompleteness theorem by formalizing the paradox. Let, w be an input in a particular instant and n is a turing machine which follows. To use rices theorem to show that a language l is not in d we must. This is why rice s theorem is usually stated about the partial functions that the machines compute, or about the languages that they accept. David ricardos theory of comparative cost advantage. From a general perspective, the proof of the main theorem relies on relating two established techniques in hyperbolic. Cutelimination theorem proof theory cybenko theorem neural networks d. Automata theory computer science university of san francisco.

We need the technique known as reduction, where an algorithm converts instances of one problem to instances of another. What you refer to are syntactic properties properties of the program, such as runtime or how many variables are used. And in december, georges gonthier, a computer scientist at microsoft research cambridge, announced a successful verification of the proof of the four color theorem, using a proof assistant called coq. On the lazarevlieb extension of the hobbyrice theorem. Perplexed by rices theorem computer science stack exchange. Here is a more carefully explained proof of ores theorem than the one given in lectures.

If p is a nontrivial property, then lp is undecidable. Given any machine m, string w, and nontrivial property p, we will create a new machine m. So in that sense, rice s theorem fails in this context. April 28, 1906 january 14, 1978 was an austrohungarianborn austrian logician, mathematician, and analytic philosopher. To make sense of the statement of the theorem, think of a property of languages that you would like to test. Generalizing rices theorem any nontrivial property about the language of a turing machine is undecidable. Then either l cis empty, or it contains the descriptions of all turing machines, or it is undecidable. If f is analytic in between and on c1 and c2, then z c1 fzdz z c2 fzdz. Proof by reduction suppose that we are given a language l that we believe is undecidable. Finally, in x5, we discuss a partial improvement to theorem 1. Most of the incorrect attempts that i have come across, are based on the misunderstanding that the notion of property is a vague notion from everyday life. Using this superresolved chemical knowledge, we create new models to understand and predict macroscale processes like protein separation and photocatalysis. In 1972 meyer and bagchi 25 conjectured a nonershov hierarchy succinctness result that, by adapting theorem 6s proof, we can confirm and improve 8 in.

So in that sense, rices theorem fails in this context. Later we noticed an wikipedia entry containing still another proof based on the recursion theorem. 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. Rice university studies and define we will show that h vanishes on c v, and that there exists a dense sub set of a,v consisting of functions g such that it follows from the proof of the theorem in bell l, as amended in bell z, that for such a g, lg 0, so 1 5 0 on a,v. Ribets theorem elliptic curves rices theorem recursion theory, computer science riceshapiro theorem computer science richardsons theorem mathematical logic riemann mapping theorem complex analysis riemann series theorem mathematical series riemanns existence theorem algebraic geometry. Assume, for the sake of contradiction, that l is decidable. 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. In that case, it would be good if you describe in more detail what sorts of contexts youre looking for. Here is the theorem and its proof, following the needed definition. Thus, as per rices theorem the language describing any nontrivial property of turing machine is not recursive. 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.

1685 555 1404 1679 1408 640 815 653 92 760 606 1332 1022 137 61 508 18 79 1647 326 998 724 841 1684 1292 296 271 1580 402 169 1531 861 684 142 1195 980 402 663 1104 785 319 590 178 128