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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} and \beta-{reduces} is to 0 or more modifications.
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} and \beta-{contraction} is to 1 modification.
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.
END%%
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Basic
A \lambda-term of form (\lambda x.M)N is called what?
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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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Basic
Why isn't (\lambda x. x) \,\triangleright_{1\beta}\, (\lambda x. x) true?
Back: No \beta-redex was replaced 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.
END%%
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Basic
Given \lambda-term P, is P \,\triangleright_{1\beta}\, P true?
Back: Not necessarily.
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, when is P \,\triangleright_{1\beta}\, P true?
Back: When substituting a \beta-redex in P with its contractum yields 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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
Substitution is well-defined with respect to \beta-reduction. That is, if M \,\triangleright_\beta\, M' and N \,\triangleright_\beta\, N', then [N/x]M ,\triangleright_\beta, [N'/x]M'
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Basic
The proof of which implication shows "substitution is well-behaved w.r.t. \beta-reduction"?
Back: M \,\triangleright_\beta\, M' \land N \,\triangleright_\beta\, N' \Rightarrow [N/x]M \,\triangleright_\beta\, [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.
END%%
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Basic
What does Hindley et al. mean by "substitution is well-behaved w.r.t. \beta-conversion"?
Back: Substitution then \beta-reduction is congruent to \beta-reduction then substitution.
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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Cloze
{M \,\triangleright_\beta\, M' \land N \,\triangleright_\beta\, N'} \Rightarrow [N/x]M \,\triangleright_\beta\, [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.
END%%
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Basic
How does Hindley et al. describe the following implication? $M \,\triangleright_\beta\, M' \land N \,\triangleright_\beta\, N' \Rightarrow [N/x]M \,\triangleright_\beta\, [N'/x]M'$
Back: As "substitution is well-defined with respect to \beta-reduction."
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%%
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\text{-nf}). The class of all \beta-normal forms is called \beta\text{-nf} or \lambda\beta\text{-nf}. If a term P \beta-reduces to a term Q in \beta\text{-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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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Cloze
The class of {all \beta-normal forms} is called {\beta\text{-nf}/\lambda\beta\text{-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.
END%%
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Basic
What ambiguity does term "\beta\text{-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.
END%%
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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\text{-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.
END%%
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Basic
How is the class \beta\text{-nf} alternatively denoted?
Back: As \lambda\beta\text{-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.
END%%
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Basic
How is the class \lambda\beta\text{-nf} alternatively denoted?
Back: As \beta\text{-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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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Basic
Is a(\lambda u. uv)x 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.
END%%
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Basic
Why is x(\lambda u. uv) in \beta-normal form?
Back: It contains no \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.
END%%
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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.
END%%
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Basic
Why is a(\lambda u. uv)x in \beta-normal form?
Back: With parentheses, (a(\lambda u. uv))x clearly contains no \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.
END%%
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Basic
Let P \,\triangleright_\beta\, Q. How do FV(P) and FV(Q) relate to one another?
Back: FV(Q) \subseteq FV(P)
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
Suppose P \,\triangleright_\beta\, Q. When is FV(Q) \subset FV(P) true?
Back: When replacing a \beta-redex with its contractum removes a free variable.
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
\beta-reduction "loses" free variable N when it contains what \beta-redex?
Back: If x \not\in FV(M), then (\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.
END%%
As an alternative characterization, the class \beta\text{-nf} is the smallest class such that
- all atoms are in
\beta\text{-nf}; M, N \in \beta\text{-nf} \Rightarrow aMN \in \beta\text{-nf}for all atomsa;M \in \beta\text{-nf} \Rightarrow \lambda x. M \in \beta\text{-nf}
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Basic
What proposition explains how atoms relate to the definition of \beta\text{-nf}?
Back: All atoms are in \beta\text{-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.
END%%
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Basic
What proposition explains how applications relate to the definition of \beta\text{-nf}?
Back: For all atoms a, if M, N \in \beta\text{-nf}, then aMN \in \beta\text{-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.
END%%
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Basic
Given atom a and M \in \beta\text{-nf}, what application is in \beta\text{-nf}?
Back: aM
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
M, N \in \beta\text{-nf} implies what application is in \beta\text{-nf}?
Back: aMN for any atom 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.
END%%
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Basic
Given M, N \in \beta\text{-nf}, when is MN \in \beta\text{-nf}?
Back: When M is not an abstraction.
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 M, N \in \beta\text{-nf}, when is MN \not\in \beta\text{-nf}?
Back: When M is an abstraction.
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
What proposition explains how abstractions relate to the definition of \beta\text{-nf}?
Back: If M \in \beta\text{-nf}, then \lambda x. M \in \beta\text{-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.
END%%
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Basic
M \in \beta\text{-nf} implies what abstraction is in \beta\text{-nf}?
Back: \lambda 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.
END%%
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Basic
Given atom a, if M \equiv aM_1\ldots M_n and M \,\triangleright_\beta\, N, what form does N have?
Back: aN_1\ldots N_n where M_i \,\triangleright_\beta\, N_i for i = 1, \ldots, 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.
END%%
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Basic
Given atom a, if M \equiv aM_1\ldots M_n and M \,\triangleright_\beta\, N, why does N have form aN_1\ldots N_n?
Back: Since M \equiv ((\cdots((aM_1)M_2)\cdots)M_n), every \beta-redex must be in an M_i.
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
What does it mean for a \lambda-term to be a \beta\text{-nf}?
Back: The \lambda-term contains no \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.
END%%
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Basic
What does it mean for a \lambda-term to have a \beta\text{-nf}?
Back: The \lambda-term can be \beta-reduced into a term in \beta\text{-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.
END%%
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Basic
Suppose [N/x]M is a \beta\text{-nf}. Is M a \beta\text{-nf}?
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.
END%%
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Basic
Suppose [N/x]M has a \beta\text{-nf}. Does M have a \beta\text{-nf}?
Back: Not necessarily.
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%%
β-equality
We say P is \beta-equal or \beta-convertible to Q (P =_\beta Q) iff Q can be obtained from P by a finite series of \beta-contractions, reversed \beta-contractions, and changes of bound variables. That is, P =_\beta Q iff there exist P_0, \ldots, P_n (n \geq 0) such that P_0 \equiv P, P_n \equiv Q, and \forall i \leq n - 1, (P_i ,\triangleright_{1\beta}, P_{i+1}) \lor (P_{i+1} ,\triangleright_{1\beta}, P_i) \lor (P_i \equiv_\alpha P_{i+1}).
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Basic
\triangleright_\beta denotes what relation?
Back: \beta-reduction.
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
\triangleright_{1\beta} denotes what relation?
Back: \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.
END%%
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Basic
=_{\beta} denotes what relation?
Back: \beta-equality.
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-equality} is also known as {\beta-convertibility}.
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 \beta-reduction a symmetric relation?
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 \beta-equality a symmetric relation?
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.
END%%
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Cloze
{\beta-equality} is the symmetric generalization of {\beta-reduction}.
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 P to be \beta-equal to Q?
Back: Q can be obtained from P by a finite series of \beta-contractions, reversed \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 is "P is \beta-equal to Q" denoted?
Back: P =_\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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Cloze
P =_\beta Q iff \exists P_0, \ldots, P_n s.t. P_0 \equiv P, P_n \equiv Q, and \forall i \leq n - 1:
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Basic
\beta-reduction constitute what two operations?
Back: \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
\beta-equality constitute what three operations?
Back: \beta-contractions, reversed \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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Cloze
{M =_\beta M' \land N =_\beta N'} \Rightarrow [N/x]M =_\beta [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
How would Hindley et al. describe the following implication? $M =_\beta M' \land N =_\beta N' \Rightarrow [N/x]M =_\beta [N'/x]M'$
Back: As "substitution is well-defined with respect to \beta-equality."
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
If P =_\beta Q, how do P and Q's \beta-normal forms relate to one another?
Back: Either P and Q have the same \beta-normal form or neither P nor Q have a \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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Church-Rosser Theorem
If P \,\triangleright_\beta\, M and P \,\triangleright_\beta\, N, then there exists a term T such that M \,\triangleright_\beta\, T and N \,\triangleright_\beta\, T. As an immediate corollary, if P has a \beta-normal form then it it is unique modulo \equiv_\alpha.
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Basic
According to Hindley et al., what is the most quoted theorem in \lambda-calculus?
Back: The Church-Rosser theorem.
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
The Church-Rosser theorem is related to which greek-prefixed concept?
Back: \beta-reductions.
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 the Church-Rosser theorem state in terms of \triangleright_\beta?
Back: If P \,\triangleright_\beta\, M and P \,\triangleright_\beta\, N, then there exists a term T such that M \,\triangleright_\beta\, T and N \,\triangleright_\beta\, T.
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
When does a \lambda-term have zero \beta-normal forms (modulo \equiv_\alpha)?
Back: When its \beta-reductions fail to simplify.
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
When does a \lambda-term have one \beta-normal form (modulo \equiv_\alpha)?
Back: When its \beta-reductions simplify to a point of containing 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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Basic
When does a \lambda-term have two \beta-normal form (modulo \equiv_\alpha)?
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
What theorem is used to prove uniqueness of \beta-normal forms?
Back: The Church-Rosser theorem.
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
If a \lambda-term has \beta-normal forms P and Q, how do P and Q relate to one another?
Back: P \equiv_\alpha 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
What does the Church-Rosser theorem of \,\triangleright_\beta\, state in terms of confluence?
Back: \beta-reduction is confluent.
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
The following diagram is a representation of what theorem?
!
Back: The Church-Rosser theorem.
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
According to Hindley et al., what is the most important application of the Church-Rosser theorem?
Back: Showing computations in \lambda-calculus produce congruent results.
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
For a given \lambda-term P, how many \beta-normal forms does P have?
Back: Zero or one.
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 the following diagram of the Church-Rosser theorem, what do the arrows represent?
!
Back: \beta-reductions.
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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Likewise, if P =_\beta Q, then there exists a term T such that P \,\triangleright_\beta\, T and Q \,\triangleright_\beta\, T.
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Basic
What does the Church-Rosser theorem state in terms of =_\beta?
Back: If P =_\beta Q then there exists a term T such that P \,\triangleright_\beta\, T and Q \,\triangleright_\beta\, T.
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 theorem encourages giving \beta-equality its name?
Back: The Church-Rosser theorem.
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.