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Rice's Theorem
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable problem, undecidable. A ''semantic'' property is one about the program's behavior (for instance, "does the program halting problem, terminate for all inputs?"), unlike a syntactic property (for instance, "does the program contain an if-then-else statement?"). A ''non-trivial'' property is one which is neither true for every program, nor false for every program. The theorem generalizes the undecidability of the halting problem. It has far-reaching implications on the feasibility of static program analysis, static analysis of programs. It implies that it is impossible, for example, to implement a tool that checks whether any given program is correctness (computer science), correct, or even executes without error (it is possible to implement a tool that always overestimates or always underestimates, so in practice one has to decide what is less of a problem). The theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definable set, definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function (mathematics), function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of computational complexity theory ... [...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] |
Scott–Curry Theorem
In mathematical logic, the Scott–Curry theorem is a result in lambda calculus stating that if two non-empty sets of lambda terms ''A'' and ''B'' are closed under beta-convertibility then they are recursively inseparable. Explanation A set ''A'' of lambda terms is closed under beta-convertibility if for any lambda terms X and Y, if X \in A and X is β-equivalent to Y then Y \in A. Two sets ''A'' and ''B'' of natural numbers are recursively separable if there exists a computable function f : \mathbb \rightarrow \ such that f(a) = 0 if a \in A and f(b) = 1 if b \in B. Two sets of lambda terms are recursively separable if their corresponding sets under a Gödel numbering are recursively separable, and recursively inseparable otherwise. The Scott–Curry theorem applies equally to sets of terms in combinatory logic with weak equality. It has parallels to Rice's theorem in computability theorem, which states that all non-trivial semantic properties of programs are undecidable. The th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rice–Shapiro Theorem
In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, named after Henry Gordon Rice and Norman Shapiro. It states that when a semi-decidable property of partial computable functions is true on a certain partial function, one can extract a ''finite'' subfunction such that the property is still true. The informal idea of the theorem is that the "only general way" to obtain information on the behavior of a program is to run the program, and because a computation is finite, one can only try the program on a finite number of inputs. A closely related theorem is the Kreisel-Lacombe-Shoenfield-Tseitin theorem (or KLST theorem), which was obtained independently by Georg Kreisel, Daniel Lacombe and Joseph R. Shoenfield , and by Grigori Tseitin. Formal statement Rice-Shapiro theorem. Let P be a set of partial computable functions such that the index set of P (i.e., the set of indices e such that \phi_e \in P, for some fixed admissible numbering \ ... [...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] |
Reductio Ad Absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction. This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. In mathematics, the technique is called ''proof by contradiction''. In formal logic, this technique is captured by an axiom for "Reductio ad Absurdum", normally given the abbreviation RAA, which is expressible in propositional logic. This axiom is the introduction rule for negation (see ''negation introduction''). Examples The "absurd" conclusion of a ''reductio ad absurdum'' argument can take a range of forms, as these examples show: * The Earth cannot be flat; otherwise, since the Earth is assumed to be finite in extent, we would find peo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String (computer Science)
In computer programming, a string is traditionally a sequence of character (computing), characters, either as a literal (computer programming), literal constant or as some kind of Variable (computer science), variable. The latter may allow its elements to be Immutable object, mutated and the length changed, or it may be fixed (after creation). A string is often implemented as an array data structure of bytes (or word (computer architecture), words) that stores a sequence of elements, typically characters, using some character encoding. More general, ''string'' may also denote a sequence (or List (abstract data type), list) of data other than just characters. Depending on the programming language and precise data type used, a variable (programming), variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements. When a string appears lit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rice Reduction
Rice is a cereal grain and in its domesticated form is the staple food of over half of the world's population, particularly in Asia and Africa. Rice is the seed of the grass species ''Oryza sativa'' (Asian rice)—or, much less commonly, ''Oryza glaberrima'' (African rice). Asian rice was domesticated in China some 13,500 to 8,200 years ago; African rice was domesticated in Africa about 3,000 years ago. Rice has become commonplace in many cultures worldwide; in 2023, 800 million tons were produced, placing it third after sugarcane and maize. Only some 8% of rice is traded internationally. China, India, and Indonesia are the largest consumers of rice. A substantial amount of the rice produced in developing nations is lost after harvest through factors such as poor transport and storage. Rice yields can be reduced by pests including insects, rodents, and birds, as well as by weeds, and by diseases such as rice blast. Traditional rice polycultures such as rice-duck farming, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene's Recursion Theorem
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book ''Introduction to Metamathematics''. A related theorem, which constructs fixed points of a computable function, is known as Rogers's theorem and is due to Hartley Rogers, Jr. The recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions. Notation The statement of the theorems refers to an admissible numbering \varphi of the partial recursive functions, such that the function corresponding to index e is \varphi_e. If F and G are partial functions on the natural numbers, the notation F \simeq G indicates that, for each ''n'', either F(n) and G(n) are both defined and are equal, or else F(n) and G(n) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Recursive Function
In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as in a formal one. If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines (this is one of the theorems that supports the Church–Turing thesis). The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other equivalent classes of functions are the functions of lambda calculus and the functions tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Set (computability)
In computability theory, index sets describe classes of computable functions; specifically, they give all indices of functions in a certain class, according to a fixed Gödel numbering of partial computable functions. Definition Let \varphi_e be a computable enumeration of all partial computable functions, and W_ be a computable enumeration of all c.e. sets. Let \mathcal be a class of partial computable functions. If A = \ then A is the index set of \mathcal. In general A is an index set if for every x,y \in \mathbb with \varphi_x \simeq \varphi_y (i.e. they index the same function), we have x \in A \leftrightarrow y \in A. Intuitively, these are the sets of natural numbers that we describe only with reference to the functions they index. Index sets and Rice's theorem Most index sets are non-computable, aside from two trivial exceptions. This is stated in Rice's theorem: Let \mathcal be a class of partial computable functions with its index set C. Then C is computable if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
