Introduction
Church Numerals
Introduction
Church Numerals
Objectives
You should be able to...
Church Numerals
Dr. Mattox Beckman
University of Illinois at Urbana-Champaign
Department of Computer Science
Introduction
Church Numerals
What is a Number?
I
The Lambda Calculus doesn’t have numbers.
I
A number n can be thought of as a potential: someday we are going
to do something n times.
2
3
4
1
2
3
f0
f1
f2
f3
=
=
=
=
\f->
\f->
\f->
\f->
\x->
\x->
\x->
\x->
x
f x
f (f x)
f (f (f x))
Prelude> let show m = m (+1) 0
Prelude> show (\f x -> f (f x))
2
Explain the form of a Church numeral
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Define some operations on Church numerals:
inc, plus, times
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Explain how to represent boolean operations: and, or, not, if
Introduction
Church Numerals
Incrementing Church Numerals, 0
I
Some Church Numerals
1
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To increment a Church Numeral, what do we want to do?
Running Example
1
finc = undefined
Introduction
Church Numerals
Incrementing Church Numerals, 1
I
To increment a Church Numeral, what do we want to do?
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First step, take the church numeral you want to increment.
Introduction
Incrementing Church Numerals, 2
Running Example
1
1
Church Numerals
Incrementing Church Numerals, 3
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To increment a Church Numeral, what do we want to do?
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First step, take the church numeral you want to increment.
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Second step, return a church numeral representing your result.
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Third step, apply f to x, m times.
To increment a Church Numeral, what do we want to do?
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First step, take the church numeral you want to increment.
I
Second step, return a church numeral representing your result.
finc = \m -> \f x -> undefined
Introduction
Church Numerals
Incrementing Church Numerals, 4
Running Example
1
I
Running Example
finc = \m -> undefined
Introduction
Church Numerals
I
To increment a Church Numeral, what do we want to do?
I
First step, take the church numeral you want to increment.
I
Second step, return a church numeral representing your result.
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Third step, apply f to x, m times.
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Finally, apply f once more to the result.
Running Example
finc = \m -> \f x -> m f x
1
finc = \m -> \f x -> f (m f x)
Introduction
Church Numerals
Adding Church Numerals
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Similar reasoning can yield addition and multiplication.
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Here is addition. Can you figure our multiplication? Hint: What
does (nf) do?
Running Example
1
fadd m n = \f x -> m f (n f x)
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In your activity you will think about how to represent booleans.
I
If a numeral represents a potential number of actions, what would a
boolean represent?