Wolfram/ Lambda

(1.0.2) current version: 1.0.11 »

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 (4)

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

In[1]:=

$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`, 307.0436840057373`], List[153.98200225830078`, 235.99870109558105`], List[195.`, 212.31669425964355`], List[195.`, 22.863691329956055`], List[30.928985595703125`, 117.59068870544434`], List[30.928985595703125`, 307.0436840057373`]], List[List[200.`, 410.42970085144043`], List[364.0710144042969`, 315.7036876678467`], List[241.01799774169922`, 244.65868949890137`], List[200.`, 220.97669792175293`], List[158.98200225830078`, 244.65868949890137`], List[35.928985595703125`, 315.7036876678467`], List[200.`, 410.42970085144043`]], List[List[376.5710144042969`, 320.03370475769043`], List[202.5`, 420.53370475769043`], List[200.95300006866455`, 421.42667961120605`], List[199.04699993133545`, 421.42667961120605`], List[197.5`, 420.53370475769043`], List[23.428985595703125`, 320.03370475769043`], List[21.882003784179688`, 319.1406993865967`], List[20.928985595703125`, 317.4896984100342`], List[20.928985595703125`, 315.7036876678467`], List[20.928985595703125`, 114.70369529724121`], List[20.928985595703125`, 112.91769218444824`], List[21.882003784179688`, 111.26669120788574`], List[23.428985595703125`, 110.37369346618652`], List[197.5`, 9.87369155883789`], List[198.27300024032593`, 9.426692008972168`], List[199.13700008392334`, 9.203690528869629`], List[200.`, 9.203690528869629`], List[200.86299991607666`, 9.203690528869629`], List[201.72699999809265`, 9.426692008972168`], List[202.5`, 9.87369155883789`], List[376.5710144042969`, 110.37369346618652`], List[378.1179962158203`, 111.26669120788574`], List[379.0710144042969`, 112.91769218444824`], List[379.0710144042969`, 114.70369529724121`], List[379.0710144042969`, 315.7036876678467`], List[379.0710144042969`, 317.4896984100342`], List[378.1179962158203`, 319.1406993865967`], List[376.5710144042969`, 320.03370475769043`]]]]], List[FaceForm[List[RGBColor[0.5529411764705883`, 0.6745098039215687`, 0.8117647058823529`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 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List[63.928985595703125`, 315.4997024536133`], List[200.`, 394.0606994628906`]]]]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, 15]]], StyleBox[RowBox[List["RandomLambda", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram/Lambda\"", ",", "\"Wolfram`Lambda`RandomLambda\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram/Lambda", "Wolfram`Lambda`RandomLambda"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram/Lambda", "Wolfram`Lambda`RandomLambda"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][3, 2, 10]$

Out[1]=

Evaluate a lambda expression:

In[2]:=

Out[2]=

In[3]:=

$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`, 307.0436840057373`], List[153.98200225830078`, 235.99870109558105`], List[195.`, 212.31669425964355`], List[195.`, 22.863691329956055`], List[30.928985595703125`, 117.59068870544434`], List[30.928985595703125`, 307.0436840057373`]], List[List[200.`, 410.42970085144043`], List[364.0710144042969`, 315.7036876678467`], List[241.01799774169922`, 244.65868949890137`], List[200.`, 220.97669792175293`], List[158.98200225830078`, 244.65868949890137`], List[35.928985595703125`, 315.7036876678467`], List[200.`, 410.42970085144043`]], List[List[376.5710144042969`, 320.03370475769043`], List[202.5`, 420.53370475769043`], List[200.95300006866455`, 421.42667961120605`], List[199.04699993133545`, 421.42667961120605`], List[197.5`, 420.53370475769043`], List[23.428985595703125`, 320.03370475769043`], List[21.882003784179688`, 319.1406993865967`], List[20.928985595703125`, 317.4896984100342`], List[20.928985595703125`, 315.7036876678467`], List[20.928985595703125`, 114.70369529724121`], List[20.928985595703125`, 112.91769218444824`], List[21.882003784179688`, 111.26669120788574`], List[23.428985595703125`, 110.37369346618652`], List[197.5`, 9.87369155883789`], List[198.27300024032593`, 9.426692008972168`], List[199.13700008392334`, 9.203690528869629`], List[200.`, 9.203690528869629`], List[200.86299991607666`, 9.203690528869629`], List[201.72699999809265`, 9.426692008972168`], List[202.5`, 9.87369155883789`], List[376.5710144042969`, 110.37369346618652`], List[378.1179962158203`, 111.26669120788574`], List[379.0710144042969`, 112.91769218444824`], List[379.0710144042969`, 114.70369529724121`], List[379.0710144042969`, 315.7036876678467`], List[379.0710144042969`, 317.4896984100342`], List[378.1179962158203`, 319.1406993865967`], List[376.5710144042969`, 320.03370475769043`]]]]], List[FaceForm[List[RGBColor[0.5529411764705883`, 0.6745098039215687`, 0.8117647058823529`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[44.92900085449219`, 282.59088134765625`], List[181.00001525878906`, 204.0298843383789`], List[181.00001525878906`, 46.90887451171875`], List[44.92900085449219`, 125.46986389160156`], List[44.92900085449219`, 282.59088134765625`]]]]], List[FaceForm[List[RGBColor[0.6627450980392157`, 0.803921568627451`, 0.5686274509803921`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[355.0710144042969`, 282.59088134765625`], List[355.0710144042969`, 125.46986389160156`], List[219.`, 46.90887451171875`], List[219.`, 204.0298843383789`], List[355.0710144042969`, 282.59088134765625`]]]]], List[FaceForm[List[RGBColor[0.6901960784313725`, 0.5882352941176471`, 0.8117647058823529`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[200.`, 394.0606994628906`], List[336.0710144042969`, 315.4997024536133`], List[200.`, 236.93968200683594`], List[63.928985595703125`, 315.4997024536133`], List[200.`, 394.0606994628906`]]]]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, 15]]], StyleBox[RowBox[List["EvalLambda", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram/Lambda\"", ",", "\"Wolfram`Lambda`EvalLambda\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram/Lambda", "Wolfram`Lambda`EvalLambda"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], 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Out[3]=

Convert lambda to an different expression form:

In[4]:=

$SeedRandom[0]; 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`]], List[List[30.928985595703125`, 307.0436840057373`], List[153.98200225830078`, 235.99870109558105`], List[195.`, 212.31669425964355`], List[195.`, 22.863691329956055`], List[30.928985595703125`, 117.59068870544434`], List[30.928985595703125`, 307.0436840057373`]], List[List[200.`, 410.42970085144043`], List[364.0710144042969`, 315.7036876678467`], List[241.01799774169922`, 244.65868949890137`], List[200.`, 220.97669792175293`], List[158.98200225830078`, 244.65868949890137`], List[35.928985595703125`, 315.7036876678467`], List[200.`, 410.42970085144043`]], List[List[376.5710144042969`, 320.03370475769043`], List[202.5`, 420.53370475769043`], List[200.95300006866455`, 421.42667961120605`], List[199.04699993133545`, 421.42667961120605`], List[197.5`, 420.53370475769043`], List[23.428985595703125`, 320.03370475769043`], List[21.882003784179688`, 319.1406993865967`], List[20.928985595703125`, 317.4896984100342`], List[20.928985595703125`, 315.7036876678467`], List[20.928985595703125`, 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1, 0], List[0, 1, 0]]], List[List[List[44.92900085449219`, 282.59088134765625`], List[181.00001525878906`, 204.0298843383789`], List[181.00001525878906`, 46.90887451171875`], List[44.92900085449219`, 125.46986389160156`], List[44.92900085449219`, 282.59088134765625`]]]]], List[FaceForm[List[RGBColor[0.6627450980392157`, 0.803921568627451`, 0.5686274509803921`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[355.0710144042969`, 282.59088134765625`], List[355.0710144042969`, 125.46986389160156`], List[219.`, 46.90887451171875`], List[219.`, 204.0298843383789`], List[355.0710144042969`, 282.59088134765625`]]]]], List[FaceForm[List[RGBColor[0.6901960784313725`, 0.5882352941176471`, 0.8117647058823529`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[200.`, 394.0606994628906`], List[336.0710144042969`, 315.4997024536133`], List[200.`, 236.93968200683594`], List[63.928985595703125`, 315.4997024536133`], List[200.`, 394.0606994628906`]]]]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, 15]]], StyleBox[RowBox[List["RandomLambda", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram/Lambda\"", ",", "\"Wolfram`Lambda`RandomLambda\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram/Lambda", "Wolfram`Lambda`RandomLambda"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram/Lambda", "Wolfram`Lambda`RandomLambda"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][3, 3]$

Out[5]=

In[6]:=

Out[6]=

In[7]:=

Out[7]=

Colorize lambda:

In[8]:=

Out[9]=

Annotate lambda with smiles:

In[10]:=

Out[11]=

Scope (3)

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

In[12]:=

Out[12]=

Evaluate lambdas:

In[13]:=

Out[13]=

In[14]:=

Out[14]=

Perform η-reduction:

In[15]:=

Out[15]=

Turn a random lambda expression to Function and back:

In[16]:=

Out[16]=

In[17]:=

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Out[17]=

In[18]:=

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Out[18]=

Applications (2)

Evaluate Church numerals:

In[19]:=

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

In[20]:=

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Out[20]=

Evaluate Ackermann function:

In[21]:=

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

In[22]:=

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