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Size-change Termination Principle
The size-change termination principle (SCT) guarantees termination for a computer program by proving that infinite computations always trigger infinite descent in data values that are well-founded. Size-change termination analysis utilizes this principle in order to solve the universal halting problem for a certain class of programs. When applied to general programs, the principle is intended to be used conservatively, which means that if the analysis determines that a program is terminating, the answer is sound, but a negative answer means "don't know". The decision problem for SCT is PSPACE-complete; however, there exists an algorithm that computes an approximation of the decision problem in polynomial time. Size-change analysis is applicable to both first-order and higher-order functional programs, as well as imperative programs and logic programs. The latter application preceded by four years the general formulation of the principle by Lee et al. Overview The size-change t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Termination Analysis
In computer science, termination analysis is program analysis which attempts to determine whether the evaluation of a given program halts for ''each'' input. This means to determine whether the input program computes a ''total'' function. It is closely related to the halting problem, which is to determine whether a given program halts for a ''given'' input and which is undecidable. The termination analysis is even more difficult than the halting problem: the termination analysis in the model of Turing machines as the model of programs implementing computable functions would have the goal of deciding whether a given Turing machine is a total Turing machine, and this problem is at level \Pi^0_2 of the arithmetical hierarchy and thus is strictly more difficult than the halting problem. Now as the question whether a computable function is total is not semi-decidable, each ''sound'' termination analyzer (i.e. an affirmative answer is never given for a non-terminating program) is ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Program (machine)
A program is a set of instructions used to control the behavior of a machine. Examples of such programs include: *The sequence of cards used by a Jacquard loom to produce a given pattern within weaved cloth. Invented in 1801, it used holes in punched cards to represent sewing loom arm movements in order to generate decorative patterns automatically. *A computer program (software) is a list of instructions to be executed by a computer. *Barrels, punched cards and music rolls encoding music to be played by player pianos, fairground organs, barrel organ A barrel organ (also called roller organ or crank organ) is a France, French mechanical musical instrument consisting of bellows and one or more ranks of organ pipe, pipes housed in a case, usually of wood, and often highly decorated. The basic ...s and music boxes. *The automatic flute player, which was invented in the 9th century by the Banū Mūsā brothers in Baghdad, is the first known example of a programmable machin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halting Problem
In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is ''Undecidable problem, undecidable'', meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically Definable set, definable but not Computable function, computable. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program that might determine whether programs halt, that a "pathological" program exists for which makes an incorrect determination. Specifically, is the program that, when called with some input, passes its own s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE-complete
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (PSPACE, polynomial space) and if every other problem that can be solved in polynomial space can be Polynomial-time reduction, transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. Problems known to be PSPACE-complete include determining properties of regular expressions and context-sensitive grammars, determining the truth of quantified Boolean formula problem, quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization problems, and many puzzles and games. Theory A problem is defined to be PSPACE-complete if it can be solved using a polynomial amount o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Interpretation
In computer science, abstract interpretation is a theory of sound approximation of the semantics of computer programs, based on monotonic functions over ordered sets, especially lattices. It can be viewed as a partial execution of a computer program which gains information about its semantics (e.g., control-flow analysis, control-flow, data-flow analysis, data-flow) without performing all the calculations. Its main concrete application is formal static code analysis, static analysis, the automatic information extraction, extraction of information about the possible executions of computer programs; such analyses have two main usages: * inside compilers, to analyse programs to decide whether certain Optimization (computer science), optimizations or Program transformation, transformations are applicable; * for debugging or even the certification of programs against classes of bugs. Abstract interpretation was formalized by the French computer scientist working couple Patrick Cousot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undecidable Problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether an arbitrary program eventually halts when run. Background A decision problem is a question which, for every input in some infinite set of inputs, requires a "yes" or "no" answer. Those inputs can be numbers (for example, the decision problem "is the input a prime number?") or values of some other kind, such as strings of a formal language. The formal representation of a decision problem is a subset of the natural numbers. For decision problems on natural numbers, the set consists of those numbers that the decision problem answers "yes" to. For example, the decision problem "is the input even?" is formalized as the set of even numbers. A decision pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Computation
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., approximate solutions versus precise ones). The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: ''"What are the fundamental capabilities and limitations of computers?".'' In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. There are several models in use, but the most commonly examined is the Turing machine. Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to prove results, and because it represents what many consider the most powerful possible "reasonable" m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
