---
title: Combinators
TARGET DECK: Obsidian::STEM
FILE TAGS: combinator
tags:
  - combinator
---

## 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. Included in the atomic constants are some [[#Basic Combinators|basic combinators]]. The set of expressions called $CL$-terms is defined inductively as follows:

* all variables and atomic constants are $CL$-terms;
* if $X$ and $Y$ are $CL$-terms, then so is $(XY)$.

An **atom** is a variable or atomic constant. A **non-redex constant** is any atomic constant other than the basic combinators. A **non-redex atom** is a variable or non-redex constant. A **closed term** is a term containing no variables. A **combinator** is a closed term containing no atomic constants other than the basic combinators.

If the sequence of atomic constants is empty (besides the basic combinators), the system is called **pure**. Otherwise it is called **applied**.

%%ANKI
Basic
Who is usually attributed the creation of combinatory logic?
Back: Moses Schönfinkel.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657635-->
END%%

%%ANKI
Basic
How many variables exist in a combinatory logic 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657636-->
END%%

%%ANKI
Basic
How many atomic constants exist in a combinatory logic system?
Back: The basic combinators plus 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657637-->
END%%

%%ANKI
Basic
What distinguishes variables and atomic constants in a combinatory logic system?
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657638-->
END%%

%%ANKI
Basic
What two classes of expressions does an "atom" potentially refer to in a combinatory logic system?
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657639-->
END%%

%%ANKI
Basic
What general term refers to both variables and atomic constants in a combinatory logic system?
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657640-->
END%%

%%ANKI
Basic
Why are variables and atomic constants called "atoms" in a combinatory logic system?
Back: They are not composed of smaller $CL$-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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657641-->
END%%

%%ANKI
Basic
When is a combinatory logic system considered pure?
Back: When there exist no atomic constants in the system (besides the basic combinators).
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657642-->
END%%

%%ANKI
Basic
When is a combinatory logic system considered applied?
Back: When there exists at least one atomic constant in the system (besides the basic combinators).
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657643-->
END%%

%%ANKI
Cloze
A combinatory logic system 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657644-->
END%%

%%ANKI
Basic
What term(s) correspond to the base case of the $CL$-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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657645-->
END%%

%%ANKI
Basic
What term(s) correspond to the inductive case of the $CL$-term definition?
Back: For $CL$-terms $X$ and $Y$, $(XY)$ is a $CL$-term.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657646-->
END%%

%%ANKI
Basic
Consider $CL$-term $(S0)$. Is our combinatory logic system 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657647-->
END%%

%%ANKI
Basic
Consider $CL$-term $(SS)$. Is our combinatory logic system 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657648-->
END%%

%%ANKI
Basic
What atomic constants are permitted in a pure combinatory logic system?
Back: Just the basic combinators.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657649-->
END%%

%%ANKI
Basic
What variables are permitted in a pure combinatory logic system?
Back: Any.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657650-->
END%%

%%ANKI
Basic
What atomic constants are permitted in an applied combinatory logic system?
Back: Any.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657651-->
END%%

%%ANKI
Basic
What variables are permitted in an applied combinatory logic system?
Back: Any.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657652-->
END%%

%%ANKI
Basic
What atoms are permitted in a pure combinatory logic system?
Back: All variables and the basic combinators.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657653-->
END%%

%%ANKI
Basic
What atoms are permitted in an applied combinatory logic system?
Back: Any.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657654-->
END%%

%%ANKI
Basic
What are the non-redex constants in a combinatory logic system?
Back: Any atomic constant other than the basic combinators.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657655-->
END%%

%%ANKI
Basic
What are the redex constants in a combinatory logic system?
Back: The basic combinators.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657656-->
END%%

%%ANKI
Basic
What are the non-redex atoms in a combinatory logic system?
Back: Any variable or non-redex constant.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657657-->
END%%

%%ANKI
Basic
What are the redex atoms in a combinatory logic system?
Back: The basic combinators.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657658-->
END%%

%%ANKI
Basic
What distinguishes non-redex constants from non-redex atoms?
Back: The latter also refer to variables.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657659-->
END%%

%%ANKI
Basic
Which of non-redex constants or atoms is more general?
Back: The non-redex 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657660-->
END%%

%%ANKI
Basic
In a combinatory logic system, what is a closed term?
Back: A $CL$-term with no variables.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657661-->
END%%

%%ANKI
Basic
In a combinatory logic system, what is a combinator?
Back: A closed term with no 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657662-->
END%%

%%ANKI
Basic
In a pure combinatory logic system, what distinguishes closed terms from combinators?
Back: N/A. They are equivalent.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657663-->
END%%

%%ANKI
Basic
In an applied combinatory logic system, what distinguishes closed terms from combinators?
Back: Closed terms are permitted to have atomic constants other than the basic combinators.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657664-->
END%%

%%ANKI
Basic
Is $CL$-term $(\mathbf{S}0)$ a closed term?
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657665-->
END%%

%%ANKI
Basic
Is $CL$-term $(\mathbf{S}x)$ a closed term?
Back: No, assuming $x$ is a 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657666-->
END%%

%%ANKI
Basic
Is $CL$-term $(\mathbf{S}0)$ a combinator?
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657667-->
END%%

%%ANKI
Basic
Is $CL$-term $(\mathbf{S}x)$ a combinator?
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657668-->
END%%

%%ANKI
Basic
In what kind of combinator logic are closed terms equivalent to combinators?
Back: Pure systems.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657669-->
END%%

%%ANKI
Cloze
A {1:$CL$}-term is to {2:combinatory logic} whereas a {2:$\lambda$}-term is to {1:$\lambda$-calculus}.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657670-->
END%%

%%ANKI
Basic
What are the non-redex constants in $CL$-term $(((\mathbf{SK})((\mathbf{SK})(x)))(\mathbf{I}0))$?
Back: Just $0$.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657671-->
END%%

%%ANKI
Basic
What are the redex constants in $CL$-term $(((\mathbf{SK})((\mathbf{SK})(x)))(\mathbf{I}0))$?
Back: Each $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657672-->
END%%

%%ANKI
Basic
What are the non-redex atoms in $CL$-term $(((\mathbf{SK})((\mathbf{SK})(x)))(\mathbf{I}0))$?
Back: $x$ and $0$.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657673-->
END%%

%%ANKI
Basic
What are the redex atoms in $CL$-term $(((\mathbf{SK})((\mathbf{SK})(x)))(\mathbf{I}0))$?
Back: Each $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735413657674-->
END%%

## Basic Combinators

The combinatory logic is a notation that eliminate the need for quantified variables. We start with basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. These **basic combinators** are defined as:

* $\mathbf{S}$ (the starling); $(\mathbf{S}(f, g))(x) = f(x, g(x))$
* $\mathbf{K}$ (the kestrel); $(\mathbf{K}(a))(x) = a$
* $\mathbf{I}$ (the idiot bird); $\mathbf{I}(f) = f$

%%ANKI
Basic
How is the $\mathbf{S}$ combinator defined?
Back: As $(\mathbf{S}(f, g))(x) = f(x, g(x))$.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735403774398-->
END%%

%%ANKI
Basic
What name does Smullyan give the $\mathbf{S}$ combinator?
Back: The starling.
Reference: Smullyan, Raymond M. _To Mock a Mockingbird_. Oxford: Oxford university press, 2000.
<!--ID: 1735403774399-->
END%%

%%ANKI
Basic
How is the starling combinator defined?
Back: As $(\mathbf{S}(f, g))(x) = f(x, g(x))$.
Reference: Smullyan, Raymond M. _To Mock a Mockingbird_. Oxford: Oxford university press, 2000.
<!--ID: 1735404184954-->
END%%

%%ANKI
Basic
How is the $\mathbf{K}$ combinator defined?
Back: As $(\mathbf{K}(a))(x) = a$.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735403774400-->
END%%

%%ANKI
Basic
What name does Smullyan give the $\mathbf{K}$ combinator?
Back: The kestrel.
Reference: Smullyan, Raymond M. _To Mock a Mockingbird_. Oxford: Oxford university press, 2000.
<!--ID: 1735403774401-->
END%%

%%ANKI
Basic
How is the kestrel combinator defined?
Back: As $(\mathbf{K}(a))(x) = a$.
Reference: Smullyan, Raymond M. _To Mock a Mockingbird_. Oxford: Oxford university press, 2000.
<!--ID: 1735404184957-->
END%%

%%ANKI
Basic
How is the $\mathbf{I}$ combinator defined?
Back: As $I(f) = f$.
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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1735403774402-->
END%%

%%ANKI
Basic
What name does Smullyan give the $\mathbf{I}$ combinator?
Back: The idiot bird.
Reference: Smullyan, Raymond M. _To Mock a Mockingbird_. Oxford: Oxford university press, 2000.
<!--ID: 1735403774403-->
END%%

%%ANKI
Basic
How is the idiot bird combinator defined?
Back: As $I(f) = f$.
Reference: Smullyan, Raymond M. _To Mock a Mockingbird_. Oxford: Oxford university press, 2000.
<!--ID: 1735404184959-->
END%%

## Bibliography

* “Combinatory Logic.” In _Wikipedia_, August 25, 2024. [https://en.wikipedia.org/w/index.php?title=Combinatory_logic](https://en.wikipedia.org/w/index.php?title=Combinatory_logic&oldid=1242193088).
* 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](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
* Smullyan, Raymond M. _To Mock a Mockingbird_. Oxford: Oxford university press, 2000.