Wolfram/ Lambda

Tools for the lambda calculus

Contributed by: Nik Murzin

This paclet provides basic functionality to work with lambda expressions expressed with De Bruijn indices.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["Wolfram/Lambda"]

To load the code after installation, evaluate this code:
Needs["Wolfram`Lambda`"]

Details

De Bruijn indices are the canonical way to name variables in a lambda expression with integers referencing a lambda term situated at certain number of levels above it.

Examples

Basic Examples (5)

Create a lambda term using  (aka formal λ, \[FormalLambda] or .Lambda):

In[1]:=

$\[FormalLambda][\[FormalLambda][1[2]][\[FormalLambda][1[2]]]]$

Parse lambda from a string:

In[2]:=

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Out[2]=

In[3]:=

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Out[3]=

Generate 10 random lambda expressions with depth 3 and maximum sub-expression length of 2:

In[4]:=

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Out[4]=

Evaluate a lambda expression:

In[5]:=

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Out[5]=

In[6]:=

$InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], List[List[30.928985595703125`, 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Out[6]=

Convert lambda to different forms:

In[7]:=

$SeedRandom[2]; lambda = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], 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Out[8]=

In[9]:=

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Out[10]=

In[11]:=

Out[11]=

In[12]:=

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Out[13]=

Colorize lambda:

In[14]:=

Out[15]=

In[16]:=

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Out[16]=

Annotate lambda with smiles:

In[17]:=

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Out[17]=

Make a graphical representation of the lambda:

In[18]:=

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Out[18]=

Scope (3)

Enumerate and visualize all lambda expression up-to depth 2 and length 2:

In[19]:=

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Out[19]=

Evaluate lambdas:

In[20]:=

Out[20]=

Perform η-reduction:

In[21]:=

Out[21]=

Make lambda from a Function:

In[22]:=

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Out[22]=

Turn a random lambda expression to a Function and back:

In[23]:=

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Out[23]=

In[24]:=

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Out[24]=

In[25]:=

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Out[25]=

Make lambda from a combinator:

In[26]:=

Out[26]=

Applications (3)

Evaluate Church numerals:

In[27]:=

$churchNum[ n_Integer] := \[FormalLambda][\[FormalLambda][Nest[2, 1, n]]]$

In[28]:=

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Out[28]=

Define arithmetic functions:

In[29]:=

$z = \[FormalLambda][\[FormalLambda][1]]; s = \[FormalLambda][\[FormalLambda][\[FormalLambda][2[3[2][1]]]]]; add = \[FormalLambda][\[FormalLambda][\[FormalLambda][\[FormalLambda][ 4[2][3[2][1]]]]]]; mult = \[FormalLambda][\[FormalLambda][\[FormalLambda][3[2[1]]]]]; pred = \[FormalLambda][\[FormalLambda][\[FormalLambda][ 3[\[FormalLambda][\[FormalLambda][1[2[4]]]]][\[FormalLambda][ 2]][\[FormalLambda][1]]]]]; Y = \[FormalLambda][\[FormalLambda][2[1[1]]][\[FormalLambda][ 2[1[1]]]]]; true = \[FormalLambda][\[FormalLambda][2]]; false = \[FormalLambda][\[FormalLambda][1]]; isZero = \[FormalLambda][1[\[FormalLambda][false]][true]]; fac = Y[\[FormalLambda][\[FormalLambda][ isZero[1][s[z]][mult[1][2[pred[1]]]]]]];$

In[30]:=

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Out[30]=

Evaluate Ackermann function:

In[31]:=

$succ = \[FormalLambda][\[FormalLambda][\[FormalLambda][2[3[2][1]]]]]; ack = \[FormalLambda][ 1[\[FormalLambda][\[FormalLambda][ 1[2[2[\[FormalLambda][\[FormalLambda][2[1]]]]]]]]][succ]];$

In[32]:=

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