notebook/notes/lambda-calculus/beta-reduction.md

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β-reduction Obsidian::STEM λ-calculus
λ-calculus

Overview

Any term of form (\lambda x. M)N is called a \beta-redex. The corresponding term [N/x]M is its contractum. If and only if a term P contains an occurrence of (\lambda x. M)N and we replace that occurrence by [N/x]M, and the result is P', we say we have contracted the redex-occurrence in P, and P \beta-contracts to P' or P \,\triangleright_{1\beta}\, P'.

If and only if P can be changed to a term Q by a finite series of \beta-contractions and changes of bound variables, we say P \beta-reduces to Q, or P \,\triangleright_{\beta}\, Q.

%%ANKI Cloze \alpha-{converts} is to \beta-{reduces}. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Cloze \alpha-{conversion} is to \beta-{contraction}. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Cloze "\beta-{contracts}" refers to exactly one contraction of a redex-occurrence. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Cloze "\beta-{reduces}" refers to zero or more contractions of redex-occurrences. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Cloze {1:(\lambda x.M)N} is to a {2:\beta-redex} whereas {2:[N/x]M} is to a {1:contractum}. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic What is a \lambda-term of (\lambda x.M)N called? Back: A \beta-redex. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic What contractum corresponds to \beta-redex (\lambda x. M)N? Back: [N/x]M Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic What \beta-redex corresponds to contractum [N/x]M? Back: (\lambda x. M)N Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic What does it mean to contract a redex-occurrence of P? Back: A \beta-redex in P was replaced by its contractum. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic How do we denote "P \beta-contracts to Q"? Back: P \,\triangleright_{1\beta}\, Q Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic How do we denote "P \beta-reduces to Q"? Back: P \,\triangleright_{\beta}\, Q Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic Given \lambda-term P, is P \,\triangleright_{1\beta}\, P true? Back: No. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic Given \lambda-term P, why isn't P \,\triangleright_{1\beta}\, P true? Back: Replacing a \beta-redex in P with its contractum cannot again yield P again. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic Given \lambda-term P, is P \,\triangleright_{\beta}\, P true? Back: Yes. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic Is (\lambda x. x) \,\triangleright_{1\beta}\, (\lambda y. y) true? Back: No. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic Is (\lambda x. x) \,\triangleright_{\beta}\, (\lambda y. y) true? Back: Yes. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic In what way is \beta-contraction a stricter operation than \beta-reduction? Back: The former requires replacing a \beta-redex occurrence with its contractum. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic In what way is \beta-reduction more general than \alpha-conversion? Back: \beta-reduction involves a finite sequence of \beta-contractions and \alpha-conversions. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic How do we pronounce P \,\triangleright_{1\beta}\, Q? Back: P \beta-contracts to Q. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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%%ANKI Basic How do we pronounce P \,\triangleright_{\beta}\, Q? Back: P \beta-reduces to Q. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf.

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Bibliography