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De Bruijn Indices
In mathematical logic, the De Bruijn index is a tool invented by the Netherlands, Dutch mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables. Terms written using these indices are invariant with respect to α-conversion, so the check for Lambda calculus#Alpha equivalence, α-equivalence is the same as that for syntactic equality. Each De Bruijn index is a natural number that represents an occurrence of a Variable (mathematics), variable in a λ-term, and denotes the number of Free variables and bound variables, binders that are in scope (programming), scope between that occurrence and its corresponding binder. The following are some examples: * The term λ''x''. λ''y''. ''x'', sometimes called the K combinator, is written as λ λ 2 with De Bruijn indices. The binder for the occurrence ''x'' is the second λ in scope. * The term λ''x''. λ''y''. λ''z''. ''x'' ''z'' (''y'' ''z'') (the S combinator), with De Bruijn ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic programming language families include Prolog, answer set programming (ASP) and Datalog. In all of these languages, rules are written in the form of ''clauses'': :H :- B1, …, Bn. and are read declaratively as logical implications: :H if B1 and … and Bn. H is called the ''head'' of the rule and B1, ..., Bn is called the ''body''. Facts are rules that have no body, and are written in the simplified form: :H. In the simplest case in which H, B1, ..., Bn are all atomic formulae, these clauses are called definite clauses or Horn clauses. However, there are many extensions of this simple case, the most important one being the case in which conditions in the body of a clause can also be negations of atomic formulas. Logic programming languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meta-logic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, 2001, p. 336. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived ''about'' the languages and systems that are used to express truths. Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic', University of California Press, 1973 The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory. Overview Formal language A ''formal language'' is an organized set of symbols, the symbols of which precise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-order Abstract Syntax
In computer science, higher-order abstract syntax (abbreviated HOAS) is a technique for the representation of abstract syntax trees for languages with variable binders. Relation to first-order abstract syntax An abstract syntax is ''abstract'' because it is represented by mathematical objects that have certain structure by their very nature. For instance, in '' first-order abstract syntax'' (''FOAS'') trees, as commonly used in compilers, the tree structure implies the subexpression relation, meaning that no parentheses are required to disambiguate programs (as they are, in the concrete syntax). HOAS exposes additional structure: the relationship between variables and their binding sites. In FOAS representations, a variable is typically represented with an identifier, with the relation between binding site and use being indicated by using the ''same'' identifier. With HOAS, there is no name for the variable; each use of the variable refers directly to the binding site. There are a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isabelle Theorem Prover
The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring yet supporting explicit proof objects. Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle ''Archive of Formal Proofs'' (Isabelle AFP) Isabelle was named by Lawrence Paulson after Gérard Huet's daughter. The Isabelle theorem prover is free software, released under the revised BSD license. Features Isabelle is generic: it provides a meta-logic (a weak type theory), which is used to en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information And Computation
''Information and Computation'' is a closed-access computer science journal published by Elsevier (formerly Academic Press). The journal was founded in 1957 under its former name ''Information and Control'' and given its current title in 1987. , the current editor-in-chief is David Peleg. The journal publishes 12 issues a year. History ''Information and Computation'' was founded as ''Information and Control'' in 1957 at the initiative of Leon Brillouin and under the editorship of Leon Brillouin, Colin Cherry and Peter Elias. Murray Eden joined as editor in 1962 and became sole editor-in-chief in 1967. He was succeeded by Albert R. Meyer in 1981, under whose editorship the journal was rebranded ''Information and Computation'' in 1987 in response to the shifted focus of the journal towards theory of computation and away from control theory. In 2020, Albert Mayer was succeeded by David Peleg as editor-in-chief of the journal. Indexing All articles from the ''Information and Comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |