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De Bruijn Index
In mathematical logic, the De Bruijn index is a tool invented by the 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 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 in a λ-term, and denotes the number of binders that are in 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 indices, is λ λ λ 3 1 (2 1). * The term λ''z''. (λ''y''. ''y'' (λ''x''. ''x'')) (λ''x''. ''z'' ''x'') is λ (λ 1 (λ 1) ... [...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] |
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a well-formed formula, 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 algorithm, 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 automated theorem proving, theorem provers and declarative programming languages are based on term rewriting. Example cases Logic In logic, the procedure for obtaini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nominal Techniques
Nominal techniques in computer science are a range of techniques, based on nominal sets, for handling names and binding, e.g. in abstract syntax. Research into nominal sets gave rise to nominal terms, a metalanguage for embedding object languages with name binding constructs. See also * De Bruijn index * 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'' b ... References * * {{cite journal , doi = 10.1016/j.tcs.2004.06.016 , author = Christian Urban, Andrew M. Pitts and Murdoch J. Gabbay , year = 2004 , title = Nominal unification , journal = Theoretical Computer Science , volume = 323 , issue = 1–3 , pages = 473–497 , doi-access = free Theoretical computer science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
β-reduction
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(\lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution (logic)
Substitution is a fundamental concept in logic. A substitution is a syntactic transformation on formal expressions. To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols by other expressions. The resulting expression is called a substitution instance, or instance for short, of the original expression. Propositional logic Definition Where ''ψ'' and ''φ'' represent formulas of propositional logic, ''ψ'' is a substitution instance of ''φ'' if and only if ''ψ'' may be obtained from ''φ'' by substituting formulas for symbols in ''φ'', replacing each occurrence of the same symbol by an occurrence of the same formula. For example: ::(R → S) & (T → S) is a substitution instance of: ::P & Q and ::(A ↔ A) ↔ (A ↔ A) is a substitution instance of: ::(A ↔ A) In some deduction systems for propositional logic, a new expression (a proposition) may be entered on a line of a derivation if it is a substitution instanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context. A bound variable, in contrast, is a variable that has been ''bound'' to a specific value or range of values in the domain of discourse or universe. This may be achieved through the use of logical quantifiers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
