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Böhm Tree
In the study of denotational semantics of the lambda calculus, Böhm trees, Lévy-Longo trees, and Berarducci trees are (potentially infinite) tree-like mathematical objects that capture the "meaning" of a term up to some set of "meaningless" terms. Motivation A simple way to read the meaning of a computation is to consider it as a mechanical procedure consisting of a finite number of steps that, when completed, yields a result. In particular, considering the lambda calculus as a rewriting system, each beta reduction step is a rewrite step, and once there are no further beta reductions the term is in normal form. We could thus, naively following Church's suggestion, say the meaning of a term is its normal form, and that terms without a normal form are meaningless. For example the meanings of I = λx.x and I I are both I. This works for any strongly normalizing subset of the lambda calculus, such as a typed lambda calculus. This naive assignment of meaning is however inadequate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denotational Semantics
In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called ''denotations'') that describe the meanings of expressions from the languages. Other approaches providing formal semantics of programming languages include axiomatic semantics and operational semantics. Broadly speaking, denotational semantics is concerned with finding mathematical objects called domains that represent what programs do. For example, programs (or program phrases) might be represented by partial functionsDana S. ScottOutline of a mathematical theory of computation Technical Monograph PRG-2, Oxford University Computing Laboratory, Oxford, England, November 1970. Dana Scott and Christopher Strachey. ''Toward a mathematical semantics for computer languages'' Oxford Programming Research Group Technical Monograph. PRG-6. 1971. or by ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corrado Böhm
Corrado Böhm (17 January 1923 – 23 October 2017) was a Professor Emeritus at the University of Rome "La Sapienza" and a computer scientist known especially for his contributions to the theory of structured programming, constructive mathematics, combinatory logic, lambda calculus, and the semantics and implementation of functional programming languages. Work In his PhD dissertation (in Mathematics, at ETH Zurich, 1951; published in 1954), Böhm describes for the first time a full meta-circular compiler, that is a translation mechanism of a programming language, written in that same language. His most influential contribution is the so-called structured program theorem, published in 1966 together with Giuseppe Jacopini. Together with Alessandro Berarducci, he demonstrated an isomorphism between the strictly-positive algebraic data types and the polymorphic lambda-terms, otherwise known as Böhm–Berarducci encoding. In the lambda calculus, he established an important s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting. Example cases Logic In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Form (abstract Rewriting)
In abstract rewriting, an object is in normal form if it cannot be rewritten any further, i.e. it is irreducible. Depending on the rewriting system, an object may rewrite to several normal forms or none at all. Many properties of rewriting systems relate to normal forms. Definitions Stated formally, if (''A'',→) is an abstract rewriting system, ''x''∈''A'' is in normal form if no ''y''∈''A'' exists such that ''x''→''y'', i.e. ''x'' is an irreducible term. An object ''a'' is weakly normalizing if there exists at least one particular sequence of rewrites starting from ''a'' that eventually yields a normal form. A rewriting system has the weak normalization property or is ''(weakly) normalizing'' (WN) if every object is weakly normalizing. An object ''a'' is strongly normalizing if every sequence of rewrites starting from ''a'' eventually terminates with a normal form. An abstract rewriting system is ''strongly normalizing'', ''terminating'', ''noetherian'', or has the (str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem, the Frege–Church ontology, and the Church–Rosser theorem. He also worked on philosophy of language (see e.g. Church 1970). Alongside his student Alan Turing, Church is considered one of the founders of computer science. Life Alonzo Church was born on June 14, 1903, in Washington, D.C., where his father, Samuel Robbins Church, was a Justice of the Peace and the judge of the Municipal Court for the District of Columbia. He was the grandson of Alonzo Webster Church (1829-1909), United States Senate Librarian from 1881-1901, and great grandson of Alonzo Church, a Professor of Mathematics and Astronomy and 6t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Typed Lambda Calculus
A typed lambda calculus is a typed formalism that uses the lambda-symbol (\lambda) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus, but from another point of view, they can also be considered the more fundamental theory and ''untyped lambda calculus'' a special case with only one type. Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. Typed lambda calculi play an important role in the design of type systems for programming languages; here, typability usually captures desirable properties of the program (e.g., the program will not cause a memory a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Head Normal Form
A head is the part of an organism which usually includes the ears, brain, forehead, cheeks, chin, eyes, nose, and mouth, each of which aid in various sensory functions such as sight, hearing, smell, and taste. Some very simple animals may not have a head, but many bilaterally symmetric forms do, regardless of size. Heads develop in animals by an evolutionary trend known as cephalization. In bilaterally symmetrical animals, nervous tissue concentrate at the anterior region, forming structures responsible for information processing. Through biological evolution, sense organs and feeding structures also concentrate into the anterior region; these collectively form the head. Human head The human head is an anatomical unit that consists of the skull, hyoid bone and cervical vertebrae. The term "skull" collectively denotes the mandible (lower jaw bone) and the cranium (upper portion of the skull that houses the brain). Sculptures of human heads are generally based ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Head Normal Form
Lambda calculus is a formal mathematical system based on lambda abstraction and function application. Two definitions of the language are given here: a standard definition, and a definition using mathematical formulas. Standard definition This formal definition was given by Alonzo Church. Definition Lambda expressions are composed of * variables v_, v_, ..., v_, ... * the abstraction symbols lambda '\lambda ' and dot '.' * parentheses ( ) The set of lambda expressions, \Lambda , can be defined inductively: #If x is a variable, then x \in \Lambda #If x is a variable and M \in \Lambda , then (\lambda x . M) \in \Lambda #If M, N \in \Lambda , then (M \ N) \in \Lambda Instances of rule 2 are known as abstractions and instances of rule 3 are known as applications. Notation To keep the notation of lambda expressions uncluttered, the following conventions are usually applied. * Outermost parentheses are dropped: M \ N instead of (M \ N) * Applications are assumed to be left-a ... [...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 arbitrary programs eventually halt when run. Background A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns ''yes''. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
