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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] |