Confluence and Normalization in Reduction Systems [lecture notes]

18 Pages·2015·0.124 MB·English

by  Gert Smolka

Confluence and Normalization
in Reduction Systems
Lecture Notes
Gert Smolka
Saarland University
December16,2015
We study confluence and normalization in abstract reduction systems and
apply the results to combinatory logic and an abstract version of the lambda
beta calculus. For both systems we obtain the Church-Rosser property and a
complete normalization strategy. For an abstract weak call-by-value lambda
calculus we obtain uniform confluence, which ensures that for a given term
allmaximalreductionchainshavethesamelength. Thedevelopmentisbased
onconstructivetypetheorywithinductivepredicatesandisrealizedinCoq.
1 Introduction
TheChurch-Rossertheoremformulatesakeypropertyoflambda-calculusandother
rewriting systems. It turns out that the Church-Rosser property can be analysed in
the abstract by considering relations on an abstract type. One speaks of abstract
reductionsystemssinceone-stepreductionturnsouttobetheprimaryrelation.
Abstractreductionsystemscanbeseenasdirectedgraphs,aviewprovidingfor
strongintuitionsandgraphicalproofsketches.
The study of reduction systems profits much from inductively defined predi-
cates,withstrongnormalizationand(parallel)reductionasprominentexamples.
2 Relations
We assume a type X and call predicates X → X → Prop relations. The letters R
and S will range over relations. Inclusion and equivalence of relations are defined
1
asfollows:
R ⊆S := ∀xy.Rxy →Sxy
R (cid:155)S := R ⊆S∧S ⊆R
Reflexivity, symmetry, transitivity, and functionality of relations are defined as fol-
lows:
reflexiveR := ∀x.Rxx
symmetricR := ∀xy.Rxy →Ryx
transitiveR := ∀xyz.Rxy →Ryz→Rxz
functionalR := ∀xyz.Rxy →Rxz→y =z
Anequivalencerelationisarelationthatisreflexive,symmetric,andtransitive. We
definetheinverseandthesymmetric closureofrelations:
R− := λxy. Ryx
R↔ := λxy. Rxy ∨Ryx
Fact 1 LetR bereflexive. ThenR− andR↔ arereflexive.
Fact 2 LetR betransitive. ThenR− andR↔ aretranstive.
Composition,powers,andunionofrelationsaredefinedasfollows:
R◦S := λxz.∃y.Rxy ∧Syz
R0 := λxy.x=y
Rn+1 := R◦Rn
R∪S := λxy. Rxy ∨Sxy
Fact 3 The following statements are equivalent: R is symmetric; R− is symmetric;
R− ⊆R; R (cid:155)R−; R (cid:155)R↔.
Fact 4
a) R reflexiveiffR0 ⊆R.
b) R transitiveiffR◦R ⊆R.
c) R↔ (cid:155)R∪R−.
d) Rm+n (cid:155)Rm◦Rn.
WesaythatRisaleastrelationsatisfyingapropertypifpRandR ⊆S forevery
relation S with pS. Note that least relations satisfying a property are unique up to
equivalence.
Exercise 5 ProveRm+n (cid:155)Rm◦Rn.
Exercise 6 ProvethatRn isfunctionalifR isfunctional.
2
3 Reflexive Transitive Closure
We define reflexive transitive closure of a relation R as an inductive predicate
R∗ :X →X →Prop:
Rxx(cid:48) R∗x(cid:48)y
R∗yy R∗xy
Fact 7 (Expansion) R ⊆R∗.
Fact 8 (Transitivity) R∗ istransitive.
Proof LetR∗xy andR∗yz. WeshowR∗xz byinductiononR∗xy.
1. Letx =y. ThentheassumptionR∗yz istheclaim.
2. Let Rxx(cid:48) and R∗x(cid:48)y. We have R∗x(cid:48)z by the inductive hypothesis. The claim
followswiththesecondruledefiningstar. (cid:3)
Fact 9 (Monotonicity) IfR ⊆S,thenR∗ ⊆S∗.
Proof LetR ⊆S andR∗xy. WeshowS∗xy byinductionR∗xy.
1. Letx =y. TheclaimisS∗xx andfollowsbythefirstruledefiningstar.
2. Let Rxx(cid:48) and R∗x(cid:48)y. The Sxx(cid:48) by the assumption and S∗x(cid:48)y by the inductive
hypothesis. Theclaimfollowswiththesecondruledefiningstar. (cid:3)
WewriteR∗∗ for(R∗)∗.
Fact 10 R∗∗ ⊆R∗.
Proof LetR∗∗xy. WeshowR∗xy byinductionontheouterstarofR∗∗xy.
1. Letx =y. TheclaimisR∗xx andfollowsbythefirstruledefiningstar.
2. Let R∗xx(cid:48) and R∗∗x(cid:48)y. We have R∗x(cid:48)y by the inductive hypothesis. The claim
followsbytransitivityofR∗. (cid:3)
Fact 11 (Idempotence) R∗∗ (cid:155)R∗.
Proof FollowswithFacts7and10. (cid:3)
WeformulatetheinductionprincipleforR∗ explicitly.
Fact 12 (Star induction) LetR∗xy. Thenpx ifpy and∀ab.Rab→pb→pa.
Proof Letpy and∀ab.Rab→pb→pa. Weprovepx byinductiononR∗xy.
3
1. Letx =y. Theclaimfollowswiththefirstassumption.
2. Let Rxx(cid:48) and R∗x(cid:48)y. We have px(cid:48) by the inductive hypothesis. The claim
followswiththesecondassumption. (cid:3)
Fact 13 (Closure) R∗ isaleastreflexiveandtransitiverelationcontainingR.
Fact 14 (Power characterization) R∗xy ↔∃n.Rnxy.
Fact 15 R−∗ (cid:155) R∗−.
Fact 16 R∗↔ ⊆ R↔∗.
Exercise 17 Prove all facts stated above in Coq. Coq generates and uses a subop-
timal induction lemma for R∗xy that doesn’t treat y as a parameter. Thus the
inductivehypothesescomewithanunnecessarypremise.
Discussion
We have defined R∗ as an inductive predicate. In the literature, the reflexive tran-
sitive closure is usually defined using the power characterization stated by Fact 14
(e.g., [5, 1]). The power characterization relies on recursion and induction for num-
bers and requires no knowledge of inductive predicates. We are using an inductive
definition because we like its simplicity. It captures the notion of reflexive transi-
tive closure without the notion of numbers. Although Coq generates a suboptimal
induction lemma for the inductive definition of R∗, the inductive definition is still
more convenient to use than a power-based definition. The reader may use the ex-
amples of this section to refamiliarise himself with inductive definitions and their
realizationinCoq.
Exercise 18 FindacounterexampleforR∗↔ (cid:155) R↔∗.
Exercise 19 Prove (R∗∪S∗)∗ (cid:155) (R∪S)∗ using monotonicity and idempotence of
star. Noinductionisneeded.
Exercise 20 GivenR,defineaninductivepredicateRˆ: N→X →X →Prop:
Rxy Rˆnyz
Rˆnxx Rˆ(n+1)xz
Prove Rˆn (cid:155) Rn. Remark: In Coq, working with Rˆ rather than Rn provides for more
directinductiveproofs.
4
Exercise 21 GivenR,defineaninductivepredicateR# : X →X →Prop:
Rxy R#xy R#yz
R#xy R#xx R#xz
ProveR# (cid:155)R∗.
Exercise 22 (Impredicative Characterization)
Prove R∗xy ↔ ∀S.S0 ⊆S →R◦S ⊆S →Sxy.
NotethattheequivalencecharacterisesR∗ astheintersectionofaallreflexiverela-
tionsthatareclosedunderleft-compositionwithR.
4 Equivalence Closure
Fact 23 (Preservation of Symmetry) IfR issymmetric,thenR∗ issymmetric.
Proof LetR besymmetricandR∗xy. WeshowR∗yx byinductiononR∗xy.
1. Letx =y. TheclaimisR∗xx andfollowsbytheassumption.
2. Let Rxx(cid:48) and R∗x(cid:48)y. We have R∗x(cid:48)x by the assumption and expansion, and
R∗yx(cid:48) bytheinductivehypothesis. Theclaimfollowsbytransitivity. (cid:3)
WedefineR+ :=R◦R∗ andR≡ :=R↔∗. WecallR+ thetransitiveclosureandR≡
theequivalence closureofR.
Fact 24 R ⊆R+ ⊆R∗ ⊆R≡.
Fact 25 (Transitive Closure) R+ isaleasttransitiverelationcontainingR.
Fact 26 (Equivalence Closure) R≡ isaleastequivalencerelationcontainingR.
5 Confluence and Church-Rosser Property
Wedefinejoinability,confluence,semi-confluence,thediamondproperty,andthe
Church-Rosser propertyasfollows:
joinableRxy := ∃z.Rxz∧Ryz
diamondR := ∀xyz.Rxy →Rxz→joinableRyz
confluentR := diamondR∗
semi-confluentR := ∀xyz.Rxy →R∗xz→joinableR∗yz
Church-RosserR := ∀xy.R≡xy →joinableR∗xy
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Fact 27 Thefollowingstatementsareequivalent:
1. R isChurch-Rosser.
2. R isconfluent.
3. R issemi-confluent.
Proof Thedirections1 (cid:61)⇒2 (cid:61)⇒3arestraightforward. Weshow3 (cid:61)⇒1.
Let R be semi-confluent and R↔∗xy. We show joinable R∗xy by induction on
R↔∗xy. If x = y, the claim is trivial. Otherwise, let R↔xx(cid:48) and R↔∗x(cid:48)y. We have
R∗x(cid:48)z and R∗yz for some z by the inductive hypothesis. Case analysis on R↔xx(cid:48).
If Rxx(cid:48), the claim follows. Otherwise, we have Rx(cid:48)x. By semi-confluence of R we
obtain some u such that R∗xu and R∗zu. Thus R∗yu by transitivity. The claim
follows. (cid:3)
Fact 28 IfR satisfiesthediamondproperty,thenR issemi-confluent.
Proof Assume R satisfies the diamond property and let R∗xy and Rxz. We show
joinable R∗yz by induction on R∗xy. If x = y, the claim is trivial. Otherwise, let
Rxx(cid:48) andR∗x(cid:48)y. Bythediamondproperty,wehaveRx(cid:48)uandRzuforsomeu. By
theinductivehypothesisforR∗x(cid:48)y,wehavejoinableR∗yu. Theclaimfollows. (cid:3)
Fact 29 (Sandwich) LetR ⊆S ⊆R∗. Then:
1. R isconfluentiffS isconfluent.
2. R isconfluentifS satisfiesthediamondproperty.
Proof We have R∗ ⊆ S∗ ⊆ R∗ by monotonicity and idempotence of star. Thus
R∗ (cid:155)S∗. ThusClaim1follows. Claim2followswithFacts28and27. (cid:3)
ArelationR isstrongly confluentif
∀xyz.Rxy →Rxz→(R∗yz∨∃u.R∗yu∧Rzu)
Fact 30 Everyrelationsatisfyingthediamondpropertyisstronglyconfluent.
Fact 31 Everystronglyconfluentrelationisconfluent.
Proof By Fact 27 it suffices to show that a strongly confluent relation is semi-
confluent. ThisfollowswithaninductionsimilartotheoneusedforFact28. (cid:3)
Exercise 32 Show that star preserves the diamond property. That is, if R satisfies
thediamondproperty,thenR∗ satisfiesthediamondproperty.
Exercise 33 Provethatthefollowingstatementsareequivalent:
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1. diamondR.
2. ∀xyzn.Rxy →Rnxz→∃u.Rnyu∧Rzu.
3. ∀xyzmn.Rmxy →Rnxz→∃u.Rnyu∧Rmzu.
Exercise 34 Provethatthefollowingstatementsareequivalent:
1. ∀xyz.Rxy →Sxz→∃u.Syu∧Rzu.
2. ∀xyzn.Rxy →Snxz→∃u.Snyu∧Rzu.
3. ∀xyzmn.Rmxy →Snxz→∃u.Snyu∧Rmzu.
Exercise 35 ProvethatarelationR isChurch-RosserifandonlyifR≡ (cid:155)R∗◦R∗−.
6 Normal Forms
Wedefinereducibility,normality,andnormal forms:
reducibleRx := ∃y.Rxy x is R-reducible
normalRx := ¬reducibleRx x is R-normal
R⇓xy := R∗xy ∧normalRy y is R-normal form of x
Fact 36 Letx beR-normal. Then:
1. IfR∗xy,thenx =y.
2. IfRnxy,thenn=0andx =y.
Fact 37 LetR beconfluent. Then:
1. R⇓ isfunctional. Thatis,nox hasmorethanoneR-normalform.
2. R⇓xy iffR≡xy andy isR-normal.
3. Ifx andy areR-normal,thenx =y iffR≡xy.
4. Ifx andy areR-normal,then¬R≡xy iffx ≠y.
WesaythatR isclassicalifeveryx iseitherR-reducibleorR-normal.
7 Triangle Method
We now introduce the triangle method, which provides for elegant confluence
proofs for lambda calculus and combinatory logic. The triangle method also pro-
videsareductionstrategythatfindsanormalformwheneverthereisone.
We start with a definition. A function ρ : X → X is a triangle operator for R if
Ry(ρx)wheneverRxy.
7
Fact 38 Letρ beatriangleoperatorforR. Then:
1. R satisfiesthediamondproperty.
2. Ifx isreducible,thenR2x(ρx).
3. IfR isreflexive,thenRx(ρx).
Fact 39 Letρ beatriangleoperatorforR. Then:
1. IfRxy,thenR(ρx)(ρy).
2. IfRxy,thenR(ρnx)(ρny).
3. IfR∗xy,thenR∗y(ρnx)forsomen.
Afunctionρ :X →X isanormalizerforRifitsatisfiesthefollowingproperties:
1. R∗x(ρx).
2. Ifx ⇓R y,thenρnx =y forsomen.
Fact 40 Let ρ be a normalizer for R. Then y is an R-normal form of x if and only
ify isR-normalandy =ρnx forsomen.
Theorem 41 (Triangle) Let R ⊆S ⊆R∗, S be reflexive, and ρ be a triangle operator
forS. ThenR isconfluentandρ isanormalizerforR.
Proof Confluence of R follows with Facts 29 and 38. We have R∗ (cid:155) S∗ by the
assumption R ⊆ S ⊆ R∗ and monotonicity and idempotence of ∗. That ρ is a
normalizerforR nowfollowswiththereflexivityofS andFact39. (cid:3)
Exercise 42 We call x R-quasi-normal if x = y whenever Rxy. We call y an R-
quasi normal form of x if R∗xy and y is quasi-normal in R. Prove the following
claims:
a) Lety andz beR-quasinormalformsofx. Theny =z ifR isconfluent.
b) Ify isanR-normalformofx,theny isanR∗-quasi-normalformofx.
c) Let ρ be a triangle operator for R and y be an R-quasi-normal form of x. Then
ρy =y andy =ρnx forsomen.
8 Confluence of Combinatory Logic SK
Figure 1 shows the definition of the combinatory logic SK. We show that the reduc-
tion relation (cid:31) is confluent and give a normalizer for (cid:31). The proof is based on the
trianglemethodandsandwichesarelationknownasparallelreduction. Thedefini-
tionsofparallelreduction(cid:29)andthetriangleoperatorρ appearinFigure2. Aterm
iscalledaredexifithastheformKs orSstu.
8
s,t ::= x|K|S|st (x :N)
s (cid:31)s(cid:48) t (cid:31)t(cid:48)
Kst (cid:31)s Sstu(cid:31)su(tu) st (cid:31)s(cid:48)t st (cid:31)st(cid:48)
Figure1: DefinitionofSK
s (cid:29)s(cid:48) s (cid:29)s(cid:48) t (cid:29)t(cid:48) u(cid:29)u(cid:48) s (cid:29)s(cid:48) t (cid:29)t(cid:48)
Kst (cid:29)s(cid:48) Sstu(cid:29)s(cid:48)u(cid:48)(t(cid:48)u(cid:48)) s (cid:29)s st (cid:29)s(cid:48)t(cid:48)
ρ(Kst) := ρs
ρ(Sstu) := (ρs)(ρu)((ρt)(ρu))
ρ(st) := (ρs)(ρt) ifst notaredex
ρ(s) := s ifs notanapplication
Figure2: ParallelreductionandtriangleoperatorforSK
Informally,wemayunderstandaparallelreductions (cid:29)tasatwophaseprocess:
First one chooses a collection of redexes in s and then one reduces these redexes
one after the other following an innermost strategy. For this to work it is essential
thataredexisnotdestroyedifitsconstituentsarereduced. Thisisclearlythecase
ofSK.
The notion of parallel reduction is elegantly captured by its inductive definition
inFigure2. Definingparallelreductionformallyfollowingtheinformalexplanation
and not using an inductive definition is not an easy exercise. The inductive def-
inition of parallel reduction and its use for confluence proofs is due to Tait and
Martin-Löf.
Fact 43 (Triangle) ρ isatriangleoperatorfor(cid:29).
Proof Let s (cid:29) t. We prove t (cid:29) ρs by induction on s following the case analysis
in the definition of ρ. If s is not an application, then s = t and the claim is trivial.
If s is an application but not a redex, then s (cid:29) t is obtained with a non-redex rule
for(cid:29). Hencetheclaimiseithertrivialorfollowswiththeinductivehypothesis. We
considerthecases =Ks s . Thecases =Ss s s issimilar.
1 2 1 2 3
Let s = Ks s . We show t (cid:29) ρs . If t is obtained with the rule for K, then
1 2 1
s (cid:29) t and the claim follows with the inductive hypothesis. If t is obtained with
1
the reflexivity rule, we have t = Ks s and the claim follows with the inductive
1 2
9
s,t ::= n|st |λs (n:N)
s (cid:31)s(cid:48) t (cid:31)t(cid:48) s (cid:31)s(cid:48)
(λs)t (cid:31)βst st (cid:31)s(cid:48)t st (cid:31)st(cid:48) λs (cid:31)λs(cid:48)
Figure3: Definitionofabstractλβ
hypothesis. Ift isobtainedwiththecompatibilityruleforapplications,wehavet =
Ks1(cid:48)s2(cid:48) ands1 (cid:29)s1(cid:48) ands2 (cid:29)s2(cid:48). Theclaimfollowswiththeinductivehypothesis. (cid:3)
Fact 44 (Compatibility) Lets (cid:31)∗ s(cid:48) andt (cid:31)∗ t(cid:48). Thenst (cid:31)∗ s(cid:48)t(cid:48).
Proof Byinductionont (cid:31)∗ t(cid:48) withnestedinductionons (cid:31)∗ s(cid:48). (cid:3)
Fact 45 (Sandwich) (cid:31)⊆(cid:29)⊆(cid:31)∗.
Proof The first inclusion follows by induction on (cid:31). The second inclusion follows
byinductionon(cid:29)usingcompatibility(Fact44). (cid:3)
Theorem 46 (SK) (cid:31)isconfluentandρ isanormalizerfor(cid:31).
Proof FollowswithTheorem41andFacts43and45. (cid:3)
9 Confluence of Abstract λβ
Figure3definesanabstractversionofthelambdabetacalculuswecallabstractλβ.
The definition assumes a function β that given two terms yields a term. We will
need one assumption about β to show confluence of abstract λβ. The proof will
be based on the triangle method and will exhibit a normalizer for abstract λβ. The
definitionsofparallelreduction(cid:29)andthetriangleoperatorρ appearinFigure4.
Fact 47 (Compatibility) Lets (cid:31)∗ s(cid:48). Thenλs (cid:31)∗ λs(cid:48).
Proof Byinductionons (cid:31)∗ s(cid:48). (cid:3)
Fact 48 (Compatibility) Lets (cid:31)∗ s(cid:48) andt (cid:31)∗ t(cid:48). Thenst (cid:31)∗ s(cid:48)t(cid:48).
Proof Byinductionont (cid:31)∗ t(cid:48) withnestedinductionons (cid:31)∗ s(cid:48). (cid:3)
Fact 49 (Reflexivity) s (cid:29)s.
Proof Byinductionons. (cid:3)
10