Important undecidable problems 25 impossibility of vulnerability detection lecture 18. As we approach the centenary of his birth, this lecture offers a chance to learn more about perhaps britains most famous modern mathematician. An nphard is a problem that is at least as hard as any npcomplete problem therefore an undecidable problem can be nphard. Decision problems and code picture languages regular chain core. There are many wellknown examples of undecidable problems. Generic complexity of undecidable problems myasnikov, alexei g. As a testament to how differently things work in the quantum and classical regimes, physicists have found that a problem that is easily solved in. Other undecidable problems mississippi state university. There are thousands of examples, so please post here only the most attractive, best examples. A problem is semidecidable if there is an algorithm that says yes. A problem is nphard if an oracle for it would make solving npcomplete problems easy i. This result is in contrast with the corresponding situation for probabilistic finite automata for which it is known that strict and nonstrict thresholds both lead to.
The emptiness problem asks, given some probability 0. And some of the problems we consider turn out to be decidable or to have unknown decidability status. Classical problem becomes undecidable in a quantum setting. Download it once and read it on your kindle device, pc, phones or tablets. Construct a mapping reduction from another language already known to be undecidable to the given language. The post correspondence problem due to emil post is another undecidable problem that turns out to be a very helpful tool for proving problems in logic or in formal language theory to be undecidable.
Ideal analog computers can decide some undecidable statements. In computability theory, an undecidable problem is a type of computational problem that requires a yesno answer, but where there cannot possibly be any computer program that always gives the correct answer. What are the most attractive turing undecidable problems in mathematics. This paper tackles three algorithmic problems for probabilistic automata on finite words. Lg 2 these related problems about pdas are also undecidable. How to prove undecidability or nonturingrecognizability in.
One of the first problems suspected to be undecidable, in the second sense of the term, was the word problem for groups, first posed by max dehn in 1911, which asks if there is a finitely presented group for which no algorithm exists to determine whether two words are equivalent. What is the difference between decidable and undecidable. Here, context free grammar generates a context free language and set of all context free languages is also a set. This is often done via an intermediate step, where a ram machine with a single register is used. Highly undecidable problems for in nite computations. Hence, decidability of the knapsack problem is not preserved under direct products. For example, the free group on two generators with no relators contains within it as a subgroup the free group on a countably in. Some undecidable problems related to the herbrand theorem. To say that a problem is undecidable means that there is no way, even given unlimited resources and an infinite amount of time, that the problem can be decided by algorithmic means.
The reference grammars given for many programming languages are often ambiguous e. In each case we take a known undecidable language and reduce it to the unknown one, thereby proving that the unknown one is also undecidable. A decision problem is decidable if there exists a decision algorithm for it. A function or program f is said to be total if fx is defined for all x or similarly, if fx halts for all x. Important undecidable problems 26 solvingundecidable problems undecidable means there is no program that 1.
Two undecidable problems of analysis was written in the years 19601961. If it were decidable, the two constructions give algorithms to decide the acceptance of a string by a turing machine and the problem of whether the language accepted by a turing machine is empty. On formally undecidable propositions of principia mathematica. Find a mimimal pda in terms of number of states for a context free language. Partially decidable semidecidable and totally not decidable. Determining whether or not a function f is total is undecidable. 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. Two notions of undecidability there are two common settings in which one speaks of undecidability. An equivalent definition of np is that it consists of all problems that are decidable not just verifiable in polynomial time by a nondeterministic turing machine. Knapsack and subset sum problems in nilpotent, polycyclic, and co. Each instance of the problem has one in most cases not as easily motivated as in your example, but you can always say if you get instance x, output y since an algorithm can be judged only on the basis of correctness but hardcoding all such solutions in this way does not produce a finite procedure which is.
Still, like any undecidable problem, the word problem can only be undecidable as a question about in. M is a tm and m halts on input w proof is by reduction from atm. If problem p reduces to problem q, and p is undecidable, then q is undecidable. I am looking for an undecidable problem that i could give as an easy example in a presentation to the general public. Lecture notes on theory of computation module bput. In essence, similarly to how all npc problems are related, this whole class of undecidable problems is related. This problem is shown to be decidable for contextfree picture languages in 18 and to be npcomplete for. A context free grammar g is unambiguous iff every string s in lg has a unique leftmost derivation. Or, given a string of zeros and ones, is it a palindrome. The 5th postulate states that, given a straight line on a plane and a point on the same plane outside that line, there always exists one and only one straight line passing through that. Some examples already appear on the wikipedia page. It is also shown that for every cocontextfree group, the knapsack problem is decidable. One of the most wellknown examples of undecidable problems is the halting problem.
Is the language accepted by a tm empty, finite, regular, or context free. Thus, problem mentioned in option a is undecidable. Nonalgorithmic and approximate solutions to undecidable. What are the most attractive turing undecidable problems in. An instance of the post correspondence problem for short, pcp is given by two sequences u u. Decidable and undecidable problems on context free grammars. We will see this used extensively in the upcoming weeks. M does not halt on w does tm m halt on the empty tape. It seems that the pcp is still vary useful when considering undecidability for linear grammars. Finitely presented groups are extremely complex objects. On the other hand, at the second level of the chomsky hierarchy, most problems about context free languages accepted by pushdown automata or generated by context free grammars are undecidable. I mean easy in the sense that the mathematics behind it can be described, well, without mathematics, that is with analogies and intuition, avoiding technicalities.
A timely look at the life and mathematical work of alan turing. A valuable collection both for original source material as well as historical formulations of current problems. On the other hand, at the second level of the chomsky hierarchy, most problems about context free languages accepted by pushdown automata or generated by context free. This definition of undecidability is quite strong and it gives impressions that still there can be some kind of solutions of undecidable problems. Dec 07, 2015 decidable and undecidable problems on context free grammars.
Statement in terms of decision problems saying that problem a reduces to problem b means that, in some sense, b is equally as general or more general than a, because b can decide for a. Thus in section 1 we will consider the basic intersection problem for. Decidable and undecidable problems computer action team. So i was thinking about various undecidable problems and it occured to me that all the ones that i could think of were reducible, or in fact proven undecidiable by reducing to the halting problem. A publickey cryptosystem based on an undecidable problem. For an undecidable language, there is no turing machine which accepts the language and makes a decision for every input string w tm can make decision for some input string though. Undecidable problems for contextfree grammars liacs. In fact, we can show that any nontrivial property of the inputoutput behavior of programs is undecidable. Undecidable extensions of skolem arithmetic bes, alexis and richard, denis, journal of symbolic logic, 1998. Download bibtex we improve upon a number of recent undecidability results related to the socalled herbrand skeleton problem, the simultaneous rigid eunification problem and the prenex fragment of intuitionistic logic with equality. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. Examples of undecidable problems about turing machines. This known undecidable language can be any language for which undecidability has been proved in the textbook, in lectures, in class handouts, or in homework problems but you. We are starting to see a line of reasoning we can use to find unsolvable problems.
Relationship between nphard and undecidable problems. Decidable and undecidable problems about quantum automata. The halting problem can be used to show that other problems are undecidable. Undecidable problems we will now discuss the notion of undecidability. Reductions and undecidability csci 81 spring, 2015 kim bruce undecidable problems the problem view the language view does tm m halt on w. Try to show that the unsolvability of that problem entails the unsolvability of other problems.
Sometimes formal languages have ambiguous and unambiguous grammars. A decision problem p is called undecidable if the language l of all yes instances to p is not decidable. Please comment below if you find anything wrong in the above post. Decidability and undecidability stanford university. It from bit is undecidable inclusion of nonassociativity is a new form of nonlocality, the nonlocal property of qubits with respect to their spacetime con. So must show how a tm that decides halttm can be used to decide atm. Aug 30, 2016 heres probably the oldest known example. Ntms are known to be no more powerful than tms in the sense that the set of problems decidable by ntms is identical to the set of problems decidable by tms, so clearly by this definition there can be no undecidable problems in np. How does this proof, that the halting problem is undecidable.
We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. Use features like bookmarks, note taking and highlighting while reading on formally undecidable propositions of principia mathematica and related systems dover books on mathematics. For another survey of undecidable problems, see dav77. Ebook 102 combinatorial problems as pdf download portable. But, ambiguity is not an operation and hence we can never say that cfg is closed under ambiguity. An example of an easy to understand undecidable problem.
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