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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Programming
Logic programming is a programming, database and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical form, representing knowledge about some problem domain. Computation is performed by applying logical reasoning to that knowledge, to solve problems in the 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'': :A :- B1, ..., Bn. and are read as declarative sentences in logical form: :A if B1 and ... and Bn. A is called the ''head'' of the rule, B1, ..., Bn is called the ''body'', and the Bi are called '' literals'' or conditions. When n = 0, the rule is called a ''fact'' and is written in the simplified form: :A. Queries (or goals) have the same syntax as the bodies of rules and are commonly written in the form: :?- B1, ..., Bn. In the simplest case of Horn clauses (or "definite" clauses), all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set .... A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Rocq (so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meta-logic
Metalogic is 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. 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. 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 precisely define it by shape and place. Such a language therefore can be defined without reference to the meanings of its expressions; it can exist before any interpretation is assigned to it—that is, befor ... [...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 name binding, 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. ... [...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 a Logic for Computable Functions (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 and implementations for code-generating, documenting, and specific support for a variety of formal methods. It can be seen as an integrated development environment (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 prov ... [...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 ... [...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 ... [...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 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 ... * Higher order abstract syntax 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
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution (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 with 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 propositional variables in ''φ'', replacing each occurrence of the same variable by an occurrence of the same formula. For example: ::''ψ:'' (R → S) & (T → S) is a substitution instance of ::''φ:'' P & Q That is, ''ψ'' can be obtained by replacing P and Q in ''φ'' with (R → S) and (T → S) respectively. Similarly: ::''ψ:'' (A ↔ A) ↔ (A ↔ A) is a substitution instance of: ::''φ:'' (A ↔ A) since ''ψ'' can be obta ... [...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 variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. 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. 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. An instance of a variable symbol is ''bound'', in contrast, if the value of that variable symbol has been bound to a specific value or range of va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
