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Call-by-push-value
In programming language theory, call-by-push-value (CBPV) is an intermediate language that embeds the call-by-value (CBV) and call-by-name (CBN) evaluation strategies. CBPV is structured as a polarized λ-calculus with two main types, "values" (+) and "computations" (-). Restrictions on interactions between the two types enforce a controlled order of evaluation, similar to monads or CPS. The calculus can embed computational effects, such as nontermination, mutable state, or nondeterminism. There are natural semantics-preserving translations from CBV and CBN into CBPV. This means that giving a CBPV semantics and proving its properties implicitly establishes CBV and CBN semantics and properties as well. Paul Blain Levy formulated and developed CBPV in several papers and his doctoral thesis. Definition The CBPV paradigm is based on the slogan "a value is, a computation does". One complication in the presentation is distinguishing type variables ranging over value types from those ...
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Call-by-value
In a programming language, an evaluation strategy is a set of rules for evaluating expressions. The term is often used to refer to the more specific notion of a ''parameter-passing strategy'' that defines the kind of value that is passed to the function for each parameter (the ''binding strategy'') and whether to evaluate the parameters of a function call, and if so in what order (the ''evaluation order''). The notion of reduction strategy is distinct, although some authors conflate the two terms and the definition of each term is not widely agreed upon. A programming language's evaluation strategy is part of its high-level semantics. Some languages, such as PureScript, have variants with different evaluation strategies. Some declarative languages, such as Datalog, support multiple evaluation strategies. The calling convention consists of the low-level platform-specific details of parameter passing. Example To illustrate, executing a function call f(a,b) may first evaluat ...
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Programming Language Theory
Programming language theory (PLT) is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of formal languages known as programming languages. Programming language theory is closely related to other fields including linguistics, mathematics, and software engineering. History In some ways, the history of programming language theory predates even the development of programming languages. The lambda calculus, developed by Alonzo Church and Stephen Cole Kleene in the 1930s, is considered by some to be the world's first programming language, even though it was intended to ''Model of computation, model'' computation rather than being a means for programmers to ''Computer programming, describe'' algorithms to a computer system. Many modern functional programming languages have been described as providing a "thin veneer" over the lambda calculus, and many are described easily in terms of it. The first programming lan ...
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Intermediate Language
An intermediate representation (IR) is the data structure or code used internally by a compiler or virtual machine to represent source code. An IR is designed to be conducive to further processing, such as optimization and translation. A "good" IR must be ''accurate'' – capable of representing the source code without loss of information – and ''independent'' of any particular source or target language. An IR may take one of several forms: an in-memory data structure, or a special tuple- or stack-based code readable by the program. In the latter case it is also called an ''intermediate language''. A canonical example is found in most modern compilers. For example, the CPython interpreter transforms the linear human-readable text representing a program into an intermediate graph structure that allows flow analysis and re-arrangement before execution. Use of an intermediate representation such as this allows compiler systems like the GNU Compiler Collection and LLVM to be u ...
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λ-calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computation based on function abstraction and application using variable binding and substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was logically consistent, and documented it in 1940. Lambda calculus consists of constructing lambda terms and performing reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A variable is a character or string representing a parameter. # (\lambda x.M): A lambda abstraction is a function definition, taking as ...
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Monad (functional Programming)
In functional programming, monads are a way to structure computations as a sequence of steps, where each step not only produces a value but also some extra information about the computation, such as a potential failure, non-determinism, or side effect. More formally, a monad is a type constructor M equipped with two operations, which lifts a value into the monadic context, and which chains monadic computations. In simpler terms, monads can be thought of as interfaces implemented on type constructors, that allow for functions to abstract over various type constructor variants that implement monad (e.g. , , etc.). Both the concept of a monad and the term originally come from category theory, where a monad is defined as an endofunctor with additional structure. Research beginning in the late 1980s and early 1990s established that monads could bring seemingly disparate computer-science problems under a unified, functional model. Category theory also provides a few formal requirem ...
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Continuation Passing Style
In functional programming, continuation-passing style (CPS) is a style of programming in which control is passed explicitly in the form of a continuation. This is contrasted with direct style, which is the usual style of programming. Gerald Jay Sussman and Guy L. Steele, Jr. coined the phrase in AI Memo 349 (1975), which sets out the first version of the programming language Scheme. John C. Reynolds gives a detailed account of the many discoveries of continuations. A function written in continuation-passing style takes an extra argument: an explicit ''continuation''; i.e., a function of one argument. When the CPS function has computed its result value, it "returns" it by calling the continuation function with this value as the argument. That means that when invoking a CPS function, the calling function is required to supply a procedure to be invoked with the subroutine's "return" value. Expressing code in this form makes a number of things explicit which are implicit in direct ...
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Algebraic Data Type
In computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of composite data type, i.e., a data type formed by combining other types. Two common classes of algebraic types are product types (i.e., tuples, and records) and sum types (i.e., tagged or disjoint unions, coproduct types or ''variant types''). The values of a product type typically contain several values, called ''fields''. All values of that type have the same combination of field types. The set of all possible values of a product type is the set-theoretic product, i.e., the Cartesian product, of the sets of all possible values of its field types. The values of a sum type are typically grouped into several classes, called ''variants''. A value of a variant type is usually created with a quasi-functional entity called a ''constructor''. Each variant has its own constructor, which takes a specified number of arguments with specified types. The set of all po ...
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Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ...
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Programming Computable Functions
In computer science, Programming Computable Functions (PCF), or Programming with Computable Functions, or Programming language for Computable Functions, is a programming language which is typed and based on functional programming, introduced by Gordon Plotkin in 1977, based on prior unpublished material by Dana Scott. It can be considered as an extended version of the typed lambda calculus, or a simplified version of modern typed functional languages such as ML or Haskell. A fully abstract model for PCF was first given by Robin Milner. However, since Milner's model was essentially based on the syntax of PCF it was considered less than satisfactory. The first two fully abstract models not employing syntax were formulated during the 1990s. These models are based on game semantics and Kripke logical relations. For a time it was felt that neither of these models was completely satisfactory, since they were not effectively presentable. However, Ralph Loader demonstrated that no effect ...
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Lambda Calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A ...
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