Computer Science
Halting Problem
The Halting Problem is a fundamental concept in computer science that states it is impossible to create a program that can determine whether any given program will halt or run indefinitely. This was proven by Alan Turing in 1936, and it has significant implications for the limitations of computation and the development of algorithms.
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8 Key excerpts on "Halting Problem"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 8 Halting Problem and Lambda Calculus Halting Problem In computability theory, the Halting Problem is a decision problem which can be stated as follows: Given a description of a program, decide whether the program finishes running or continues to run, and will thereby run forever. This is equivalent to the problem of deciding, given a program and an input, whether the program will eventually halt when run with that input, or will run forever. Alan Turing proved in 1936 that a general algorithm to solve the Halting Problem for all possible program-input pairs cannot exist. We say that the Halting Problem is undecidable over Turing machines. B. Jack Copeland (2004) attributes the actual term Halting Problem to Martin Davis. Formal statement The Halting Problem is a decision problem about properties of computer programs on a fixed Turing-complete model of computation. The problem is to determine, given a program and an input to the program, whether the program will eventually halt when run with that input. In this abstract framework, there are no resource limitations of memory or time on the program's execution; it can take arbitrarily long, and use arbitrarily much storage space, before halting. The question is simply whether the given program will ever halt on a particular input. For example, in pseudocode, the program while True: continue does not halt; rather, it goes on forever in an infinite loop. On the other hand, the program print Hello World! halts very soon. ________________________ WORLD TECHNOLOGIES ________________________ The Halting Problem is famous because it was one of the first problems proven algorithmically undecidable. This means there is no algorithm which can be applied to any arbitrary program and input to decide whether the program stops when run with that input.
eBook - ePubThe Most Complex Machine
A Survey of Computers and Computing
- David J. Eck(Author)
- 2018(Publication Date)
- A K Peters/CRC Press(Publisher)
But you might find a machine that makes correct predictions in a great many cases, perhaps even for all the cases that you are really interested in. In fact, if you are designing Turing machines, or writing programs, there is nothing to stop you from consciously trying to produce machines or programs whose behavior can be analyzed. The lesson that a programmer should learn from the unsolvability of the Halting Problem is not that it is impossible to write good programs, but rather that good programs don’t happen automatically. I will have more to say in Chapters 6 and 7 about techniques for writing good programs. Furthermore, our proof says nothing, one way or the other, about whether people can solve the Halting Problem. It is still conceivable that, given any Turing machine and input, you might eventually be able to decide whether or not that machine will halt on that input. However, you will not be able to write down a fixed set of rules that can be used to make the decision in all cases, since once you had done so, you could build a Turing machine that solves the Halting Problem by following those rules. (Though how you could ever be sure that you would get the right answer in all cases without writing down a set of rules, I have no idea!) 4.3.2. Other Problems. The Halting Problem is only the first of many unsolvable problems. There are many natural questions that can be asked about computers and Turing machines that cannot be answered—not, at least, by a Turing machine or a computer program. Here is a sampling of such problems. In each case, saying that the problem is unsolvable means that there is no Turing machine that can correctly answer the problem in all cases: • Given any Turing machine T, determine whether T will ever halt after being started on an empty tape. • Given a Turing machine T, determine whether there is any number n such that T will halt when started on input n. • Given a Turing machine T, determine whether it computes a function from N to N
- Clifford A. Shaffer(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
Ehalt . Thus, the problem of determining if a program will halt on no input must be unsolvable.Example 17.5 For arbitrary program P , does there exist any input for which P halts?Proof: This problem is also uncomputable. Assume that we had a function Ahalt that, when given program P as input would determine if there is some input for which P halts. We could modify our compiler (or write a function as part of a program) to take P and some input string w , and modify it so that w is hardcoded inside P , with P reading no input. Call this modified program P′ . Now, P’ always behaves the same way regardless of its input, because it ignores all input. However, because w is now hardwired inside of P′ , the behavior we get is that of P when given w as input. So, P′ will halt on any arbitrary input if and only if P would halt on input w . We now feed P′ to function Ahalt . If Ahalt could determine that P′ halts on some input, then that is the same as determining that P halts on input w . But we know that that is impossible. Therefore, Ahalt cannot exist.There are many things that we would like to have a computer do that are unsolvable. Many of these have to do with program behavior. For example, proving that an arbitrary program is “correct,” that is, proving that a program computes a particular function, is a proof regarding program behavior. As such, what can be accomplished is severely limited. Some other unsolvable problems include:- Does a program halt on every input?
- Does a program compute a particular function?
- Do two programs compute the same function?
- Does a particular line in a program get executed?
This does not mean that a computer program cannot be written that works on special cases, possibly even on most programs that we would be interested in checking. For example, some C compilers will check if the control expression for a while loop is a constant expression that evaluates to false . If it is, the compiler will issue a warning that the while loop code will never be executed. However, it is not possible to write a computer program that can check for all
eBook - PDFAlgorithms and Theory of Computation Handbook, Volume 1
General Concepts and Techniques
- Mikhail J. Atallah, Marina Blanton(Authors)
- 2009(Publication Date)
- Chapman and Hall/CRC(Publisher)
Now we can give a precise definition of what we mean by an algorithm, attempting to rule out this last situation. An algorithm is any program written in the GOTO language that has the additional property of halting on all inputs. Such programs will be called halting programs, and correspond to “total” deterministic Turing machines in Chapter 19. When we consider decision problems, which have yes/no answers, halting programs are required to end their computation with either an ACCEPT or a REJECT statement, on any input. 21.2.1 Some Computational Problems We begin by listing a collection of computational problems for which a mechanical solution can be very helpful. By a mechanical solution, we mean a step-by-step process that takes into account all possible inputs, and that can be executed without any human assistance once a certain input is provided. An algorithm is required to work correctly on all instances. We now list some problems that are fundamental either because they are inherently important or because they played a historical role in the development of computation theory [4,5]. For the first four, P stands for a program in our GOTO language, and x is a string over the input alphabet, which we fix to be { 0, 1 } . 1. Universal simulation : Given a program P and an input x to P , determine the output (if any) that P would produce on input x . 2. Halting Problem : Given P and x , output 1 (for yes) if P would halt when given input x , and 0 (for no) if P would not halt. 3. Type-0 grammar membership : Given a type-0 grammar G and a string x , determine whether x can be derived from the start symbol of G . 4. String compression : Given a string x , find the shortest program P such that when P is started with empty tape, P eventually halts with x as its output. Here “shortest” means that the total number of symbols in the program’s instructions is as small as possible.
eBook - PDFThinking About Godel And Turing: Essays On Complexity, 1970-2007
Essays on Complexity, 1970–2007
- Gregory J Chaitin(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
And being able to decide if the N th program will ever output an N th digit is a special case of Turing’s famous Halting Problem. Note that there is no problem if you place an upper bound on the time allowed for a computation. It is easy to decide if the N th program outputs an N th digit in a trillion years, all you need to do is be patient and try it and see. Turing’s Halting Problem is only a problem if there is no time limit. In other words, this is a deep conceptual problem, not a practical limitation on what we can do. And, as Turing himself points out, incompleteness is an immediate corol-lary. For let’s say we’d like to be able to prove whether individual computer programs, those that are self-contained and read no input, eventually halt or not. There can be no formal axiomatic theory for this, because if there were, by systematically running through the tree of all possible proofs, all possible deductions from the axioms using formal logic, we could always eventually decide whether an individual program halts or not, which is impossible. In my opinion this is a fundamental step forward in the philosophy of mathematics because it makes incompleteness seem much more concrete and much more natural. It’s almost a problem in physics, it’s about a machine, 324 Thinking about G¨ odel & Turing you just ask whether or not it’s going to eventually stop, and it turns out there’s no way, no general way, to answer that question. Let me emphasize that if a program does halt, we can eventually discover that. The problem, an extremely deep one, is to show that a program will never halt if this is in fact so. One can settle many special cases, even an infinity of them, but no finite set of axioms can enable you to settle all possible cases. My own work takes off from here. My approach to incompleteness follows Turing, not G¨ odel. Later I’ll consider the halting probability Ω and show that this number is wildly, in fact maximally, uncomputable and unknowable.
- Olav Lysne(Author)
- 2018(Publication Date)
- Springer Open(Publisher)
A famous postulate is that every function that is computable by any machine is computable by a Turing machine. Although this postulate has not been proven, it is widely believed to be true. In the years since its formulation, there has been no reasonable description of a machine that has been proven able to do computations that cannot be simulated by a Turing machine. Consequently, the strength of programming languages and computational concepts is often measured by their ability to simulate a 5.2 Turing and the Halting Problem 41 Fig. 5.1 A Turing machine. Based on the state of the state machine and the symbol on the tape, Ω decides the symbol to be written, Δ decides how the tape should be moved, and Π decides the new state of the machine Turing machine. If they are able to, then we can assume that anything programmable can be programmed by the language or the concept. The definition of a Turing machine and the postulation that it can simulate any other computing machine is interesting in itself. Still, its most important property, from our point of view, is that it can help us understand the limits of what a computer can do. Turing himself was the first to consider the limitations of computations, in that he proved that a computer is unable to decide if a given Turing machine terminates by reaching a halt. This is famously known as the Halting Problem and, to prove it, Turing used the liar’s paradox in much the same way as Gödel did. In a modernized form, we can state the proof as follows: assume that we have a programmed function P that, for any program U , is able to answer if U halts for all inputs. This would mean that P ( U ) returns a value of true if U halts for all inputs and false if there is an input for which U does not halt. Then we could write the following program Q : Q : if P ( Q ) then loop forever; else exit This program is a manifestation of the liar’s paradox. If Q halts, then it will loop forever and, if it does not halt, then it halts.
eBook - PDFQuantum Computation and Quantum Information
10th Anniversary Edition
- Michael A. Nielsen, Isaac L. Chuang(Authors)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
For example, the gate is a gate with one input bit and one output bit which computes the function f ( a ) = 1 ⊕ a , where a is a single bit, and ⊕ is modulo 2 addition. It is also usual to make the convention that no loops are allowed in the circuit, to avoid possible instabilities, as illustrated in Figure 3.3 . We say such a circuit is acyclic , and we adhere to the convention that circuits in the circuit model of computation be acyclic. 130 Introduction to computer science Box 3.2: The Halting Problem In Exercise 3.2 you showed that each Turing machine can be uniquely associated with a number from the list 1 , 2 , 3 , . . . . To solve Hilbert’s problem, Turing used this numbering to pose the Halting Problem : does the machine with Turing number x halt upon input of the number y ? This is a well posed and interesting mathematical problem. After all, it is a matter of some considerable interest to us whether our algorithms halt or not. Yet it turns out that there is no algorithm which is capable of solving the Halting Problem. To see this, Turing asked whether there is an algorithm to solve an even more specialized problem: does the machine with Turing number x halt upon input of the same number x ? Turing defined the halting function , h ( x ) ≡ 0 if machine number x does not halt upon input of x 1 if machine number x halts upon input of x. If there is an algorithm to solve the Halting Problem, then there surely is an al-gorithm to evaluate h ( x ). We will try to reach a contradiction by supposing such an algorithm exists, denoted by HALT( x ). Consider an algorithm computing the function TURING( x ), with pseudocode TURING(x) y = HALT(x) if y = 0 then halt else loop forever end if Since HALT is a valid program, TURING must also be a valid program, with some Turing number, t . By definition of the halting function, h ( t ) = 1 if and only if TURING halts on input of t .
eBook - ePubComplexity
A Philosophical Overview
- Nicholas Rescher(Author)
- 2020(Publication Date)
- Routledge(Publisher)
For with respect to purely theoretical problems it is clear from Turingesque results in algorithmic decision theory (ADT) that there will indeed be computer insolubilia—mathematical questions to which an algorithmic respondent will give the wrong answer or be unable to give any answer at all, no matter how much time is allowed. 4 But this is a mathematical fact which obtains of necessity so that this whole issue can be also set aside for present purposes. For in our present context of generalized problem solving (GPS) the necessitarian facts of Gödel-Church-Turing incompleteness become irrelevant. Here any search for meaningful problem-solving limitations will have to confine its attention to problems that are in principle solvable: demonstrably unsolvable problems are beside the point of present concern because an inability to do what is in principle impossible hardly qualifies as a limitation. The sort of limit that a rigid demonstration of impossibility establishes in our present context is one that bears not so much on question answering as on question askability, seeing that it makes no sense to ask for the demonstrably impossible. For present purposes, then, it is limits of capability not limits of feasibility that matter. In asking about the problem-solving limits of computers we are looking to problems that computers cannot resolve but that other problem solvers conceivably can. The limits that will concern us here are accordingly not rooted in conceptual or logico-mathematical infeasibilities of general principle but rather in performatory limitations imposed upon computers by the world’s modus operandi
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