30 Y ears of Higher-Order Unification

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30 Y ears of Higher-Order Unification

G ´erard Huet

INRIA

TPHOL2002,Hampton

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35 Y ears of Higher-Order Unification

G ´erard Huet

TPPHOL2007,Somefarawayplace

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40 Y ears of Higher-Order Unification

Not me!

TPPHOLT2012,TroulesOies

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T o o m uc h is to o m uc h

\begin{gripe}

•Toomanypapersontoonarrowtopics

•Notenoughthematicmobility

•Notenoughinterdisciplinarywork

•ToomuchBrownianMotionResearch

\end{gripe}

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Mating

\begin{benediction}

•AtLICSinSantaCruz,DanaScottpreachedtheWeddingofLogicandCS

•TodayIpreachFreeLoveamongDisciplines

\end{benediction}

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Preac hing (in the desert)

Warning.Adviceintendedforstudentsandyoungscientists.

\begin{preach}

•Changeresearchtopicevery5years

•Changedisciplineatleastonceinyourcareer

•Open-mindedness

•SeekApplicationsfortheJoyofit

•Dowhatonlyyoucando

•NesoyezpasunmoutondePanurge

•Don’ttrytoownatopic

\end{preach}

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Bet w een Logic and Algebra

•LinearLogicasLogicalFoundations

•NoContraction,noWeakening

•SequentsasBagsofLiterals

•LambekSyntacticCalculus\/

•NonCommutativeLL

•NonAssociativeLL

•Pregroups

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What I attempted to do

•Acommentedbibliographyonthetopic

•Withsomereferencetoapplications

•Whathappened

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In a n utshell

•Dowek.Handbookofautomatedtheoremproving.

•Miller’spatternsandmixedprefixes

•λProlog

•DependentTypes

•Decidabilityofrestrictedmatching

•Mixingβandequations

•RewriteLogic,ExplicitSubstitutions

•OogedeMoor&Sittampalam.Matchingorder2.5.

•Loader.Higher-orderβmatchingisundecidable.

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The main results in m y PhD

•Undecidability

•Non-wellfoundedness

•Constraints

•Pre-unifiers

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One of m y T ren to slides

Non-deterministicprogrammingisnobigdeal.WhyshouldyousurrendercontroltoaPROLOGblackbox?

Thethreegoldenrulesofnon-deterministicprogramming:

•Identifywellyoursearchstatespace

•Representstatesasnon-mutabledata

•Provetermination

Thelastpointisessentialforunderstandingthegranularityandenforcingcompleteness.

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The main results in m y Do ctorat d’Etat

•CorrectdefinitionofCSU

•CSMPre-unifiers

•Second-ordermatching

•Implementation

•PrincipalUnifiersofRegularTerms

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The main op en problems at the time

•Second-orderunification

•Associativeunification

•3rdandhigherordermatching

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Where to find HO terms

•inmathematicaldescriptions

•inlogicalproofs

•incomputerprograms

•insemanticdescriptionsofnaturallanguage

•inDNApatterns?

•in...

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Applications

•Proofassistantsandtheoremprovers

•Softwareengineering

•Computationallinguistics

•Health?

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Computational Linguistics and HOU

•MontagueSemantics

•Anaphoraresolution

•Pronouns

•Focus

•Ellipsis

•Accent

•Colors

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Anaphora Resolution

•JohnlikesMaryandPeterdoestoo.

•like(j,m)∧R(p)

•R(j)=like(j,m)

•R←λx.like(x,m)

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More complex examples

•JohnlikeshiswifeandPetertoo.

•JohnsaysthathelikeshismotherandPeterdoestoo.

•JohnspenthispaycheckbutPetersavedit.

•MarywantstogotoSpainandFredwantstogotoPerubutbecauseoflimitedresources,onlyoneofthemwill.

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Sp eec h and Linguistics

•TheWatergatetapes

•Languageasaplaypenforlinguists

•TypicalpapersinCL

•Non-wellfoundednessofpublications

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Promising future in vestigations ?

•Proofnets&GoI

•ComputationalMathematics

•Constraintsengines

•Cayenne

•LiterateProgramming

•Understandingnaturallanguage

•Understandinggenes

•CognitiveScience

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