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| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| β-reduction | Obsidian::STEM | 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.
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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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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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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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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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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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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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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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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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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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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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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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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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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.
END%%
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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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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.
END%%
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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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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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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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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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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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Normal Form
A term Q which contains no \beta-redexes is called a \beta-normal form (or a term in \beta-normal form or just a \beta-nf). The class of all \beta-normal forms is called \beta-nf or \lambda\beta-nf. If a term P \beta-reduces to a term Q in \beta-nf, then Q is called a \beta-normal form of P.
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Basic
\beta-reduction terminates if and only if what?
Back: We reduce to a term in \beta-normal form.
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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Basic
Is \beta-reduction guaranteed to terminate?
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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Basic
Is \beta-reduction guaranteed to simplify?
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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Basic
How is a \beta-normal form defined?
Back: As a \lambda-term that contains no \beta-redexes.
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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Cloze
The class of {all \beta-normal forms} is called {\beta-nf/\lambda\beta-nf}.
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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Basic
What ambiguity does term "\beta-nf" introduce?
Back: It refers to a specific \beta-normal form or the class of \beta-normal forms.
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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Basic
What does it mean for term Q to be a \beta-normal form of term P?
Back: P \beta-reduces to a term Q in \beta-nf.
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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Basic
How is the class \beta-nf alternatively denoted?
Back: As \lambda\beta-nf.
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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Basic
How is the class \lambda\beta-nf alternatively denoted?
Back: As \beta-nf.
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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Basic
What is the \beta-normal form of (\lambda x. x(xy))N?
Back: N(Ny)
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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Basic
What is the \beta-normal form of (\lambda x. xx)(\lambda x. xx)?
Back: N/A.
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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Basic
How many contractions make up a \beta-reduction?
Back: Zero or more.
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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Basic
Is x(\lambda u. uv) in \beta-normal form?
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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Basic
Is (\lambda u. uv)x in \beta-normal form?
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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Basic
Why isn't x(\lambda u. uv) in \beta-normal form?
Back: N/A. It is.
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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Basic
Why isn't (\lambda u. uv)x in \beta-normal form?
Back: Because (\lambda u. uv)x is 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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Bibliography
- 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.