Wolfram/ Lambda

Tools for the lambda calculus

Contributed by: Nik Murzin, Willem Nielsen and Stephen Wolfram

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

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.

Paclet Guide

Functions for lambda expressions

Construction

RandomLambda

— generate a random lambda expression

RandomSizeLambda

— generate a random lambda with a given size

EnumerateLambdas

— enumerate lambda expressions

EnumerateSizeLambdas

— enumerate all lambdas with a given size

EnumerateLinearLambdas

— enumerate all linear lambdas of a given size

EnumerateAffineLambdas

— enumerate all affine lambdas of a given size

ChurchNumeral

— lambda terms representing Church natural numbers

ParseLambda

— parse various string forms of lambdas

Evaluation

BetaSubstitute

— perform single beta substitution of reducible terms λ[body][arg]

BetaReduce

— iterate beta substitution

BetaReducePositions

— list of positions where beta substitution is possible

BetaReductions

— list of possible single beta substitutions

BetaPositionReductions

— association with position and corresponding reduced terms

BetaReduceList

— list of iterated beta substitutions

BetaReduceSizes

— reduce lambda and return intermediate term sizes

EvalLambda

— evaluate a lambda expression with delayed substitution

BetaReduceCompiled

— compiled version of BetaReduce

BetaReduceListCompiled

— compiled version of BetaReduceList

BetaReduceSizesCompiled

— compiled version of BetaReduceSizes

Representation

LambdaConvert

— universal lambda converter into various forms

LambdaFunction

— convert lambda expression to a

Function

LambdaVariableForm

— annotate lambdas with arguments

FunctionLambda

— convert nested Function and variable form lambda to a lambda expression

TagLambda

— tag lambda with variable names

UntagLambda

— remove lambda tags

ColorizeLambda

— colorize lambdas and variables of the lambda expression

UncolorizeLambda

— remove colors from lambda

LambdaApplication

— left associative lambda form

LambdaRightApplication

— right associative lambda form

LambdaBrackets

— only brackets and parenthesis form

LambdaString

— lambda string form

LambdaBLC

—

Binary Lambda Calculus

bits

LambdaToHaskell

— Haskell source for

AProVE

FromChurchNumeral

— integer corresponding to

ChurchNumeral

lambda

Static visualization

LambdaSmiles

— graphical representation of lambda term with lines connecting variables with λs

LambdaTree

— lambda tree representation

HighlightLambdaTree

— highlight

LambdaTree

subtrees with expression positions

LambdaGraph

— lambda graph representation

LambdaLoopbackGraph

— tree graph with edges connecting each lambda and its variables

LambdaDiagram

— Tromp's lambda diagram

LambdaTreeDiagram

—

LambdaTree

on Tromp's diagram

LambdaArrayPlot

— array plot of lambda expression characters

LambdaDepthArrayPlot

— array plot of a lambda expression with depth

LambdaDepthArrayPlot3D

— array plot of multiple lambdas expressions with depth

LambdaStringDiagram

— string diagrammatic lambda object

SmoothLambdaStringDiagram

— string diagram with smoothed wires

Dynamic visualization

BetaReduceChain

— sequence of highlighted lambda terms

BetaReduceStepPlot

—

ListStepPlot

of lambda sizes with highlighted redex

BetaReduceTreeList

— list of trees for iterated beta substitutions

LambdaCausalGraph

— causal graph with beta substitution events for a single reduction sequence

LambdaCausalEvolutionGraph

— bipartite graph with events and states

LambdaMultiwayGraph

— multiway graph of lambda beta reductions

LambdaMultiwayCausalGraph

— multiway causal graph

LambdaMultiwayCausalEvolutionGraph

— multiway causal evolution graph

Query

BetaNormalQ

— true if no reductions possible

ClosedLambdaQ

— true if no free variables

LinearLambdaQ

— true if every lambda uses its argument exactly once

AffineLambdaQ

— true if every lambda uses its argument no more than once

Other

$Lambda

— main head for the lambda, by default is the formal lambda λ.

$LambdaStyles

— association with styles used for visualization

$LambdaGridStyleRules

— associated with nice grid styles for tables

$LambdaResults

— various pre-computed data

$LambdaBusyBeavers

— list of busy beavers parsed

online

$CompiledFunctions

— association with compiled functions

$LambdaCombinatorStore

—

EntityStore

with lambda combinators

Examples

Basic Examples (5)

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

In[1]:=

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

Or any symbols with name λ:

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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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[4]=

Evaluate lambda expression:

In[5]:=

$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["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"]][][\[FormalX]]$

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`, 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"]]], 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`EvalLambda"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][%]$

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`]], 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["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[8]=

In[9]:=

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In[10]:=

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In[11]:=

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Out[13]=

In[14]:=

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Out[15]=

In[16]:=

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Out[16]=

In[17]:=

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Out[17]=

In[18]:=

Out[18]=

In[19]:=

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Out[19]=

In[20]:=

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Out[20]=

In[21]:=

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Out[21]=

Colorize lambda:

In[22]:=

Out[23]=

In[24]:=

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Out[24]=

Annotate lambda with smiles:

In[25]:=

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Out[25]=

Make a graphical representation of the lambda:

In[26]:=

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Out[26]=

Scope (3)

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

In[27]:=

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Out[27]=

Evaluate lambdas:

In[28]:=

Out[28]=

Perform η-reduction:

In[29]:=

Out[29]=

Make lambda from a Function:

In[30]:=

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Out[30]=

Turn a random lambda expression to a Function and back:

In[31]:=

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Out[31]=

In[32]:=

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Out[32]=

In[33]:=

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Out[33]=

Make lambda from a combinator:

In[34]:=

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Out[34]=

Applications (3)

Evaluate Church numerals:

In[35]:=

Out[35]=

Define arithmetic functions:

In[36]:=

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