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λ-Calculus Obsidian::STEM λ-calculus
λ-calculus

Overview

Assume that there is given an infinite sequence of expressions called variables and a finite or infinite sequence of expressions called atomic constants, different from the variables. The set of expressions called \lambda-terms is defined inductively as follows:

  • all variables and atomic constants are \lambda-terms (called atoms)
  • if M and N are \lambda-terms, then (MN) is a \lambda-term (called application)
  • if M is a \lambda-term and x is a variable, then (\lambda x. M) is a \lambda-term (called abstraction)

If the sequence of atomic constants is empty, the system is called pure. Otherwise it is called applied.

%%ANKI Basic What does a "higher-order function" refer to? Back: A function that acts on other functions. 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%%

%%ANKI Basic How is f(x) = x - y written using \lambda-calculus? Back: \lambda x. x - y 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%%

%%ANKI Basic How is f(x, y) = x - y written using (uncurried) \lambda-calculus? Back: \lambda x y. x - y 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%%

%%ANKI Basic How do you curry expression \lambda x y. x - y? Back: \lambda x. \lambda y. x - y 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%%

%%ANKI Basic How do you uncurry expression \lambda x. \lambda y. x - y? Back: \lambda x y. x - y 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%%

%%ANKI Basic What does (\lambda x. x - y)(0) evaluate to? Back: 0 - y 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%%

%%ANKI Basic How many variables exist in a \lambda-calculus formal system? Back: An infinite number. 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%%

%%ANKI Basic How many atomic constants exist in a \lambda-calculus formal system? 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%%

%%ANKI Basic What distinguishes variables and atomic constants? Back: The latter is meant to refer to constants outside the formal system. 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%%

%%ANKI Basic What two classes of expressions does an "atom" potentially refer to? Back: Variables and atomic constants. 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%%

%%ANKI Basic What general term describes both variables and atomic constants? Back: Atoms. 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%%

%%ANKI Basic Why are variables and atomic constants called "atoms"? Back: They are not composed of smaller \lambda-terms. 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%%

%%ANKI Basic When is a \lambda-calculus considered pure? Back: When there exist no atomic constants in the system. 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%%

%%ANKI Basic When is a \lambda-calculus considered applied? Back: When there exists at least one atomic constant in the system. 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%%

%%ANKI Cloze A \lambda-calculus is either {pure} or {applied}. 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%%

%%ANKI Basic What term refers to the base case of the \lambda-term definition? Back: The atoms. 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%%

%%ANKI Basic What term refers to the inductive cases of the \lambda-term definition? Back: Application and 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%%

%%ANKI Cloze Given \lambda-terms M and N, {(MN)} is referred to as {application}. 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%%

%%ANKI Cloze Given \lambda-term M and variable x, {(\lambda x. M)} is referred to as {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%%

%%ANKI Basic Consider term (\lambda x. x)(0). Is our \lambda-calculus pure or applied? Back: Applied. 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%%

%%ANKI Basic Consider term (\lambda x. x)(y). Is our \lambda-calculus pure or applied? Back: Indeterminate. 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%%

%%ANKI Basic What three terms categorize all \lambda-terms? Back: Atoms, applications, and abstractions. 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%%

%%ANKI Basic How is a constant function returning y denoted in \lambda-calculus? Back: \lambda x. y 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%%

%%ANKI Basic How is expression MNPQ written with parentheses reintroduced? Back: (((MN)P)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%%

%%ANKI Cloze By convention, parentheses in \lambda-calculus are {left}-associative. 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%%

%%ANKI Basic How is expression \lambda x. \lambda y. MN written with parentheses reintroduced? Back: (\lambda x. (\lambda y. (MN))) 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%%

%%ANKI Cloze Expression (MN) is interpreted as applying {M} to {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%%

Bibliography