The correctness of Newman s typability algorithm and some of its extensions

Transcription

1 The coectness of Newman s typability algoithm an some of its extensions Heman Geuves a,b, Robbet Kebbes a a Institute fo Computing an Infomation Science Faculty of Science, Rabou Univesity Nijmegen, The Nethelans b Faculty of Mathematics an Compute Science Einhoven Univesity of Technology, The Nethelans Abstact We stuy Newman s typability algoithm [14] fo simple type theoy. The algoithm oiginates fom 1943, but was left unnotice until [14] was ecently eiscovee by Hinley [10]. The emakable thing is that it ecies typability without computing a type. We give a moen pesentation of the algoithm (also a gaphical one), pove its coectness an show that it implicitly oes compute the pincipal type. We also show how the typing algoithm can be extene to othe type constuctos. Finally we show that Newman s algoithm actually inclues a unification algoithm. Keywos: Simply type lamba calculus, Unification, Typing algoithms 1. Intouction A type checking algoithm fo simple type theoy solves the poblem whethe an untype λ-tem M can be given a type accoing to the ules of simple type theoy. As usual, we istinguish between the type checking poblem: M : σ (oes the tem M have type σ?) an the type synthesis poblem: M :? (compute a type fo the tem M). It is a classic esult that fo simple type theoy, these poblems ae both eciable [9, 12]. The solution amounts to computing the pincipal type of M an then to check whethe the given type σ is an instance of it. All known type checking algoithms fo λ-calculus compute a pincipal type, using a unification algoithm. It is ha to imagine that one coul o something funamentally iffeent. Howeve, thee is a (seemingly) iffeent way to ecie typing, ue to Newman, aleay in 1943 [14], an ecently eviewe by Hinley [10]. The stategy of Newman s algoithm is vey iffeent fom pesent ay algoithms, because it only ecies whethe a tem is typable an oes not compute its pincipal type. This looks stange, because the most common way to ecie whethe a λ-tem is typable is tying to assign a type to it. aesses: heman@cs.u.nl (Heman Geuves), mail@obbetkebbes.nl (Robbet Kebbes) Pepint submitte to Theoetical Compute Science June 22, 2011

2 In this pape, we escibe an analyze Newman s algoithm (Section 3) an we show that it (implicitly) oes compute a pincipal type. The way it computes it is actually quite close to othe constaint base typing algoithms fo simple type theoy, fo example Wan s algoithm [21]. We pove that Newman s algoithm is coect an iscuss the coesponence with Wan s algoithm (Section 4). Befoe oing that, we escibe Newman s algoithm iectly as a manipulation of the tem-gaph that epesents the λ-tem (Section 2). Following the metho escibe by Newman, we manipulate the tem-gaph using a sot of tem-gaph euction an if no futhe euctions ae possible, we check whethe a cetain elation on the noes is cyclic. The oiginal tem is typable if an only if the elation is acyclic. The coesponence between the gaphical esciption an Newman s is left implicit, but will be clea to the eae. It is moe inteesting to exten the gaphical esciption to inclue othe type constuctions, like pouct types an weakly polymophic types (Section 5). An impotant pat of a typing algoithm is to solve unification poblems. In the pesentation by Wan [21], this is nicely single out: one fist ceates a set of equations, which ae then solve by a (stana) unification algoithm (fo example Robinson s unification algoithm [19]). Looking back at Newman s algoithm, it can be obseve that Newman oes something simila: he ceates a set of equations, which is then moifie. To emphasize this, we also give a esciption of unification à la Newman (Section 6) Histoical backgoun Newman s stating point was Quine s poposal [18] fo a new logical system calle New Founations in Newman evelope an algoithm to ecie typability fo that system, but it was also able to ecie typability fo othe type systems, fo example the Pincipia Mathematica an Chuch s simply type λ-calculus [5]. The latte connection is not etaile in Newman s pape, maybe because Chuch s pape ha only appeae ecently an thee was moe inteest in Quine s New Founations at the time. To be pecise, Newman actually uses a system that is now bette known as λ-calculus à la Cuy [3], whee thee is no type infomation in the λ-tem; λ-calculus à la Chuch has type infomation in the tem an then the issue of typability is simple. In 1944 Chuch eviewe Newman s pape [6], but his eview was moe like a summay of Newman s pape concluing with: The eae s fist impession of Newman s pape may be that the machiney intouce is heavy in compaison with the esults obtaine. The value of the pape is in fact ifficult to estimate at pesent, as this will epen on the extent to which esults obtaine in the futue by Newman s methos justify the weight of machiney. We coul say that Newman was ahea of his time, since thee was no actual nee fo a typing algoithm until the 1950s an 1960s. But at that point type theoists seem to have ignoe o fogotten about Newman s wok an invente thei own algoithm [10]. 2

3 1.2. Simple type theoy an Newman s type system A type λ-calculus can be pesente à la Chuch o à la Cuy [3, 7]. In Chuch-style, a boun vaiable is type in the λ-abstaction, fo example λx : α.λy : α β.y x : α (α β) β. Futhemoe, a fee vaiable is type in the context, fo example x : α λy : α β.y x : (α β) β. Altenatives to this syntax ae foun in the liteatue, fo example whee fee vaiables cay thei type as an annotation instea of in the context, fo example λy : α β.y x α : (α β) β. O, if we also let boun vaiables cay thei type as an annotation: λy α β.y α β x α : (α β) β. The cucial point is that type infomation fo the vaiables is fixe an can be compute via a simple lookup. Theefoe, type checking an type-synthesis ae tivial: types can be simply ea off fom the tem. In this pape a system à la Cuy is stuie because most functional pogamming languages eal with this an that is also what Newman s algoithm is about. Now the vaiables o not cay any type infomation an the question is whethe, given an untype λ-tem M, we can ecie whethe M is typable (an then give a type fo M) o M is not typable. In this section we summaize the main efinitions an intouce notations fo the est of the pape. Fo an extensive iscussion concening these notions see, fo example [3, 7]. Definition 1.1. The tems in λ à la Cuy (typically N, M, P,... ) ae inuctively efine as follows. Λ ::= Va (ΛΛ) (λva.λ) Hee Va anges ove vaiables (typically x, y, z,... ). The types of λ (typically σ, τ, ϕ,... ) ae inuctively efine as follows. Type ::= TVa (Type Type) Hee TVa anges ove type vaiables (typically α, β, γ,... ). A context (typically Γ o ) is a finite set of vaiable eclaations x : σ (x Va, σ Type), whee all vaiables ae istinct. We wite (x) to enote the type that assigns to x. The notions of fee vaiables FV(M) of a tem M, an boun vaiables BV(M) of a tem M, ae efine as usual, whee λ bins vaiables. Similaly, we use the notion of fee type vaiables FTV(σ) of a type σ, to enote the set of type vaiables occuing in σ. We now give the euction ules fo assigning types to tems in simple type theoy. This is the moen way of inuctively escibing the set of well-type tems of a type. Newman oes not explicitly efine the well-type tems. Definition 1.2. A λ -typing jugment Γ M : ρ enotes that a λ-tem M has type ρ in context Γ. The eivation ules fo eiving such a jugment ae as follows. 3

4 x : σ Γ Γ x : σ (a) Vaiable Γ M : σ τ Γ N : σ Γ MN : τ (b) Application Γ, x : σ P : τ Γ λx.p : σ τ (c) Abstaction We wite M if no context Γ an type ρ exist such that Γ M : ρ. Definition 1.3. The Type Synthesis (TSP) o Type Assignment Poblem (TAP) is given a λ-tem M an a context Γ to fin a type ρ such that Γ M : ρ. Newman puts a estiction on the λ-tems: he only consies tems M fo which FV(M) BV(M) = an fo evey x BV(M) thee is exactly one λx-abstaction in M. So the boun an fee vaiables ae istinct an no vaiable has a uplicate bine. Nowaays, this estiction goes une the name of Baenegt convention [4] which states that, wheneve we consie a numbe of λ-tems, we may always assume that all fee vaiables ae iffeent fom the boun ones an that all boun vaiables ae istinct. We can safely assume this, because we can always ename boun vaiables to get a tem that is α-equivalent to it. In the time Newman wote his aticle, these issues ha not yet been claifie completely, so Newman simply only consiee tems that satisfy the Baenegt convention. It shoul be note, howeve that the way Newman pesente his algoithm, the actual names of vaiables o eally matte Pincipal types an most geneal unifies Well-known typing algoithms, like Hinley-Milne s [9, 12] o Wan s [21], ecie whethe a tem is typable by computing its pincipal type. Also, these algoithms ely on the computation of a most geneal unifie. While we postpone the iscussion of Wan s algoithm until Section 4.2, we will efine the notion of a most geneal unifie an a pincipal type now. Definition 1.4. Given a set of type equations E = {ρ 1 σ 1,..., ρ n σ n } an a substitution δ then δ is a unifie of E, notation δ = E, if δ(ρ) = δ(σ) fo each ρ σ E. We wite = E (E is not unifiable) if no substitution δ exists such that δ = E. A substitution δ is the most geneal unifie of E, notation δ = mgu E, if δ = E an fo all τ, if τ = E then thee exists a substitution ν such that τ = ν δ. (Any othe unifie of E is an instance of δ.) Definition 1.5. Given a λ-tem M, a context Γ an a type τ, then Γ, τ is a pincipal pai of M if: 1. Γ M : τ 2. M : ρ = ( σ : TVa Type. σ(γ) ρ = σ(τ)) Definition 1.6. Given a close λ-tem M an a type τ, then τ is a pincipal type of M iff, τ is a pincipal pai of M. 4

5 The goal of a typing algoithm is to compute a pincipal type fo a close tem an a pincipal pai fo an open tem. Fo λ, pincipal types (espectively pincipal pais) exist an can be compute. By efinition a pincipal pai (espectively pincipal type) is unique up to isomophism. Hee Γ, τ an, ρ ae isomophic, notation Γ, τ =, ρ, if thee exist substitutions σ 1 an σ 2 such that = σ 1 (Γ), ρ = σ 1 (τ), Γ = σ 2 ( ) an τ = σ 2 (ρ) Some basic notions fom ewiting In this pape we will be using the notions of confluence an stong nomalization, not so much fo the simple type theoy, but fo auxiliay elations that will be efine in what follows. We use the stana teminology of [20, 4, 1], which we ecall hee. Definition 1.7. Let be a binay elation on a set A an let enote the tansitive eflexive closue of. The elation satisfies the iamon popety if fo evey a, b, c A, if a b an a c, then thee is a A such that b an c. The elation is confluent if satisfies the iamon popety. It is locally confluent if fo evey a, b, c A, if a b an a c, then thee is a A such that b an c. An element a A is in -nomal fom if thee is no b A fo which a b. The elation is stongly nomalizing o teminating if fo evey element a A thee is no infinite -sequence stating fom a. Lemma 1.8. (Newman s Lemma fo abstact ewiting systems [13]). Let be a binay elation on a set A. If is stongly nomalizing an locally confluent, then is confluent. 2. Newman gaphs We efine Newman s algoithm fo λ. It is eive fom [14] an [10], although the pesentation is funamentally iffeent. Fist of all, we use a tem-gaph notation fo λ-tems as inicate in the following efinition. Definition 2.1. Given an untype λ-tem M, we efine the tem-gaph of M, Tgaph(M) in the usual way. We eplace an application by an a λ-abstaction λx.m by a λ-noe whee we also have a pointe to x as a shae leaf, as inicate in Figue 1 in the left an mile awing 1. 1 This is also known as a DAG (Diecte Acyclic Gaph) in which all occuences of vaiables ae shae [2]. 5

6 T( λx.m) = λ T(M N) λ T(M) T(M) T(N) x x Figue 1: Tanslating a tem to a tem-gaph. As an example of a tanslation of a tem to a tem-gaph, we have epicte the tem-gaph of λx.x(λy.x) on the ight in Figue 1. Now we tanslate the tem-gaph of the tem M, Tgaph(M), into the Newman gaph of M, Ngaph(M). Of couse, we coul have efine Ngaph(M) iectly fom M, but the pesent efinition moe clealy emphasizes the elation between the two. Definition 2.2. Given an untype λ-tem M with Tgaph(M), the Newman gaph of M, Ngaph(M) is efine as epicte in the figues below. In the tanslation we ignoe an λx infomation in the noes. Instea we give all noes a unique label (typically U, V, X, Y, Z,... ), except fo the vaiable noes fo which we use the vaiable name itself as a label. Futhemoe, we eplace eges by labele aows an emove othes. λ λw x V The intuitive meaning of the aows shoul be clea: X Y iff the omain of the type of X is the type of Y X Y iff the ange of the type of X is the type of Y 6

7 (In simple type theoy, when σ = ρ τ, then ρ is calle the omain of σ an τ the ange of σ.) We will now euce the Newman gaph. A euction step is pefome by joining two conguent noes, so we fist efine when two noes X an Y ae conguent. Definition 2.3. Given a Newman gaph G, the conguence elation X G Y between noes X an Y is inuctively efine as follows. (A1) If U X an U Y, then X G Y (A2) If U X an U Y, then X G Y (B) If X U, X V an Y U, Y V, then X G Y The intuitive meaning of X G Y is the tems epesente by X an Y have the same type. (To be pecise: the tems epesente by X an Y have the same type if the tem epesente by G is typable. In case the tem epesente by G is not typable, the tems epesente by X an Y may not be typable eithe. Note that in any case X G Y is well efine.) Combine with the intuitive unestaning of an, this claifies Definition 2.3. Fo example clause (A1) of Definition 2.3 then eas: if the omain of the type of U is both the type of X an the type of Y, then X an Y have the same type. When witing X G Y we will often suppess G, an wite X Y. The efinition can be epicte gaphically as follows. X (A1) U Y = X Y X (A2) U Y = X Y (B) X Y U an X Y V = X Y Remak 2.4. Although Newman also consies clause (B) in his efinition coesponing to X G Y, it shoul be note that fo type checking, only clauses (A1) an (A2) ae neee. So, Theoem 2.8 also hols if we estict to the euction steps that aise fom joining noes that ae equal accoing to clauses (A1) an (A2). This will be poven in Section 3, Coollay Definition 2.5. A euction step on Newman gaphs, G 1 G 2, is efine by joining two noes X an Y fom G 1 fo which we have X Y. So, if X Y, 7

8 we join the noes an all incoming an outgoing noes ae eiecte to the new joint noe, that we name eithe X o Y. We teat an example to show the efinitions at wok. Example 2.6. Consie the λ-tem M λx. (λy. y x) (x (λz. z)). We epict its tem-gaph Tgaph(M) an its Newman gaph Ngaph(M). λ V W λ Y x Z y x z y z We now join conguent noes, using the conguence elation as efine in Definition 2.3. In the fist step, we obseve the two conguent noes X an y, which we inicate by a box. We join them an then we obseve the conguent noes V an Y, that we join. The est of the example shoul be self-explanatoy: in evey step we join the conguent noes an we inicate the two conguent noes that we join in the next step. U U V V W X W Y y x Z z Y X x Z z 8

9 U W Y X x Z z W Y x Z U z In the final gaph of the example, no euction is possible anymoe. The cucial point is now that it contains a cycle an theefoe accoing to Newman the oiginal tem is not typable. Definition 2.7. Given a Newman gaph G, the gaph G is in nomal fom if no istinct noes X an Y of G ae conguent. Moeove the gaph G is statifie in case the tansitive closue of contains no cycles. The euction is stongly nomalizing, simply because the numbe of noes eceases, so we will always fin a nomal fom. That the nomal fom of a Newman gaph is unique, up to enaming of the labels, will be poven in Section 3. In Section 4 we will also pove that the nomal fom of Ngaph(M) is statifie if an only if M is typable in λ. This is poven by showing how the euction of Newman gaphs implicitly keeps tack of the type infomation of the tem an thus computes the pincipal type of a tem. Theoem 2.8. The euction is stongly nomalizing an confluent 2. Moeove, the nomal fom of the Newman gaph of M is statifie if an only if M is typable in λ. Poof. The fist is by Lemma 3.10 an 3.15, the secon by Theoem Newman s algoithm can be extene to othe type constuctions in simple type theoy, like pouct types an weakly polymophic types. This will be shown in Section 5. Fo now, we inicate how we can use Newman s algoithm to ea off the pincipal type of M fom the nomal fom of the Newman gaph of M. This will be etaile in the Section 4, but an example shoul be vey explanatoy. Example 2.9. Consie the λ-tem P λx.x(λy.x(λz.y)). If we euce the Newman gaph of P to nomal fom we obtain the gaph on the left, whee U coespons to the oot noe of the tem P. 2 See Definition 1.7 fo a pecise efinition of these notions. 9

10 x U V W ((α α) α) α U α V x (α α) α W α α The gaph contains no cycles. We compute the type at noe U by labeling all noes without outgoing eges by a type vaiable. In the gaph above this is just noe V that we label by α. Then we constuct the types at all noes in the obvious way: noe x has type (α α) α because points to a noe with label α α an points to a noe labele with α. So, the pincipal type of the tem P is ((α α) α) α. One may suspect that a nomal fom always has a simple cycle, involving only one o two noes. Howeve, this is not the case: if one consies the Newman gaph of the tem N λx.x(λy.y(λz.x)), one can obseve that only one euction step is possible an then a nomal fom is obtaine with a cycle of length 4 (hence N is not typable). 3. Newman s algoithm fo λ In this section we pesent Newman s algoithm in a way close to Newman s oiginal pesentation [14]. Newman has escibe his algoithm in a vey abstact an geneal way in oe to apply it to vaious type systems. Howeve, we ae meely inteeste in λ, theefoe we will specialize ou efinitions fo the case of λ. Ou esciption of Newman s algoithm has a lot in common with Hinley s [10], howeve ou esciption is moe extensive an we will pove some basic popeties. Newman s algoithm is basically a ewite system. It stats with a set of equations pesenting the gaph of a λ-tem. Fom this set of equations a elation is geneate which is use to ewite the equations. This pocess is iteate until no futhe ewiting steps ae possible. We pove that Newman s algoithm is stongly nomalizing an confluent fo the case of λ. Definition 3.1. A scheme of a λ-tem (typically S, S,... ) is a finite set of equations of the following shape. Name Name Name Name λname.name Hee Name anges ove tem names (typically U, V, X, Y, Z,... ) an vaiables (typically x, y, z,... ). Moeove, let TemNames(S) enote the set of tem names occuing in S an Names(S) the set of names occuing in S. 10

11 Definition 3.2. Given a λ-tem M, the scheme S(M) is compute simultaneously with the entance E(M) of the scheme using the following algoithm. M S(M) E(M) x x MN {Z E(M) E(N)} S(M) S(N) Z λx.p {Z λx.e(p )} S(P ) Z In the computation of S(M), the new tem names Z shoul be chosen in such a way that they ae fesh. That is, we have to make sue that: in the application case, Z / TemNames(S(M)) TemNames(S(N)), in the application case, TemNames(S(M)) TemNames(S(N)) =, in the λ case, Z / TemNames(S(P )). Example 3.3. The scheme S = S(M) of the λ-tem M λfx.f(fx) is: U λf.v V λx.w W fz Z fx Definition 3.4. Given a scheme S, the elations S, S Name Name ae inuctively efine as follows. 1. If Z MN then M S N an M S Z 2. If Z λx.p then Z S x an Z S P When witing X S Y o X S Y we will often suppess S, an wite X Y o X Y, espectively. Note that the equations coespon to the eges of the tem gaph an the elations an coespon to the eges an of the Newman gaph escibe in the Section 2. Example 3.5. The elations an fo the λ-tem λfx.f(fx) ae: U f U V V x V W f Z f W f x f Z Definition 3.6. Given a scheme S, a binay elation S Name Name is efine as S := S S. Definition 3.7. Given a scheme S, a binay elation S Name Name is inuctively efine as follows. (A1) If U X an U Y, then X S Y (A2) If U X an U Y, then X S Y (B) If X U, Y U, X V an Y V, then X S Y As usual, when witing X S Y o X S Y we will often suppess S, an wite X Y o X Y, espectively. Newman also inclue the following clauses in his efinition of X Y. 11

12 (C1) If X MN an Y MN, then X Y (C2) If X λx.p an Y λx.p, then X Y Howeve, as the following lemma inicates, these clauses can be omitte fo the specific case of λ. Lemma 3.8. If one of the following popeties hol we have X Y. 1. X MN an Y MN 2. X λx.p an Y λx.p Poof. Suppose that X MN an Y MN, then by clause (A): M N M X M N M Y = X Y Suppose that X λx.p an Y λx.p, then by clause (B): X x X P Y x Y P = X Y Newman infomally escibes the meaning of X Y as X has the same type as Y, which means that X an Y eceive the same type accoing to the typing ules of λ. He escibes the meaning of X Y as X has a highe type than Y, that means that the type that X eceives contains the type that Y eceives as a subtem. Definition 3.9. An η-euction step 3 X:=Y on schemes, S 1 S 2, is efine by eplacing X by Y in all equations of S 1 povie that X Y an X S Y. ν Multiple steps ae enote by S 1 S 2 whee ν is a substitution. A scheme S is in η-nomal fom if no η-euction steps ae possible. The following Lemma is also poven by Newman [14]. Lemma The notion of η-euction is stongly nomalizing. That is, any η-euction path leas to a nomal fom. Poof. The numbe of iffeent tem names euces at each η-euction step. Because a scheme consists of finitely many equations it is immeiate that η-euction leas to a nomal fom in finitely many steps. Definition A cycle in a scheme S is a sequence X 1,... X n Name fo which: X 1 X 2... X n 1 X n X n X 1 A scheme S is statifie if it contains no cycles. 3 This notion is something completely iffeent fom the well-known η-euction in the λ-calculus: λx.mx η M povie that x / FV(M). 12

13 The situation with espect to the scheme an the elations efine fom it is as follows, whee an aow inicates the geneation of one entity fom anothe. λ-tem M Scheme S Relations S, S Relation S Afte an η-euction step, we obtain a new scheme S, fo which we eefine the elations S an S an theeby the elation S. Afte having epeate this pocess finitely many times we obtain a nomal fom S f by Lemma Accoing to Newman M is typable iff S f is statifie. In Section 2 we have efine euction on Newman gaphs athe than tem gaphs. Likewise, it is also possible to pefom η-euction on the elations S an S. This way we o not have to keep tack of the equations of the scheme. Howeve, this esults in loss of infomation because it is not possible to estoe an abitay scheme fom the elations S an S. Fo statification, as efine in Definition 3.11, this is howeve no poblem. Example Consie the scheme S = S(M) of the λ-tem M f(fx). W fz Z fx f Z f W f x f Z Afte one step of η-euction, S Z:=x S, the scheme S is obtaine. W fx x fx f x f W f x Finally a nomal fom S f is obtaine by S W :=x S f. f x x fx f x Thee ae no cycles, thus we conclue that S f is statifie. The following lemma coespons to what Newman calls an ae peseve une any homomophic change of lettes. Lemma If S ν S, then fo all X, Y Name we have: X S Y = νx S νy X S Y = νx S νy Poof. This lemma follows fom the obsevation that an epen meely on the elative positions of the tem names in a scheme S. 13

14 A := B S 1 B := C A := B S 1 C := B A := B S 1 B := C A := B S 1 C := D S S 3 = S4 A := C S 2 C := B S C := B S 3 S 2 A := B S B := C S 3 S 2 A := C S C := D S 3 S 2 A := B (a) (b) (c) () Figue 2: Cases consiee in the poof of Lemma Example Consie the scheme S = S(M) of the λ-tem M λx.xx. W λx.v V xx W x W V x x x V This scheme contains a cycle. By Lemma 3.13 we conclue that if we euce S to a nomal fom S f this scheme will contain a cycle as well, thus S f is not statifie eithe. Note that the pevious example oes not claim that S f is unique, but only that in whateve nomal fom S f we en up with, S f is not statifie. We now pove that S f is unique up to isomophism, whee S 1 an S 2 ae isomophic, notation S 1 = S2, if thee exist substitutions σ 1 an σ 2 such that S 2 = σ 1 (S 1 ) an S 1 = σ 1 (S 2 ). The following Lemma is also poven by Newman [14]. Lemma The notion of η-euction is locally confluent up to enaming of tem names. That is given a scheme S an euctions S ν S 1 an S δ S 2, ν thee ae schemes S 3 an S 4 an euctions S δ 1 S 3 an S 2 S 4 such that S 3 = S4. Poof. Consie the following cases. 1. If ν an δ ae equal then S 1 = S 2 an no futhe euction is neee. 2. If ν an δ ae each othe s invese (ν = [A := B] an δ = [B := A]) then S 1 = S2 an no futhe euction is neee. 3. If ν = [A := B] an δ = [A := C] then thee ae euctions S 1 S 3 an S 2 S 4 such that S 3 = S4 as shown in Figue 2a. These euctions ae vali accoing to Lemma Fo the othe cases thee ae euctions S 1 S 3 an S 2 S 3 as shown in Figue 2b-2. These euctions ae vali accoing to Lemma These ae all the possible cases because thee ae two cases with two iffeent tem names involve, thee cases with thee iffeent tem names involve an only one case with fou iffeent tem names involve. The following Theoem was also poven by Newman [14]. Theoem Each scheme S has a nomal fom (hencefoth S f ) which is unique up to isomophism. 14

15 Poof. The notion of η-euction is stongly nomalizing by Lemma 3.10 an locally confluent by Lemma 3.15, thus using Newman s Lemma fo abstact ewiting systems (Lemma 1.8) η-euction is confluent. Hence the nomal fom S f of S is unique up to isomophism. Coollay The following popeties hol fo each scheme S. 1. Whethe S f is statifie is inepenent of the oe of η-euction. 2. The numbe of steps in an η-euction path fom S to S f is fixe an moeove less o equal to the numbe of names in S. Anothe inteesting obsevation is that the (B) clause in Definition 3.7 can be omitte. That is, given a scheme S, in oe to etemine whethe S f is statifie we o not nee to pefom any euction steps by clause (B). We intouce the following efinition befoe we pove this esult. Definition Let S A S enote an η-euction step by clause (A1) o (A2) an let S B S enote an η-euction step by clause (B). Lemma If S 1 B S 2 A S 3 then S 1 A S 4 fo some S 4. Poof. Let X S1 U, Y S1 U, X S1 V, Y X:=Y S1 V an let S 1 B S 2 be the esulting B -step. Now we have to consie the following cases. 1. Thee exists a Z U such that X S1 Z o Y S1 Z. Then we can pefom an A -step in S 1, namely [U := Z]. 2. Thee exists a Z V such that X S1 Z o Y S1 Z. Then we can pefom an A -step in S 1, namely [V := Z]. 3. Thee oes not exist a Z as escibe in the peceing cases. Then the step S 2 A S 3 oes not epen on the substitution of Y fo X, so we can o the same A -step in S 1. Coollay We have the following popeties. 1. A B -step cannot ceate an A -step. That is, if S is in A -nomal fom an S B S, then S is in A -nomal fom. 2. The euction B can be postpone. That is, given a scheme S, we have S A S B S f fo some S. Poof. The fist follows immeiately fom Lemma Fo the secon: evey scheme S euces to a (unique) scheme in nomal fom S f (Theoem 3.16) with a euction sequence of fixe length (Coollay 3.17). By Lemma 3.19, we can move all A -steps to the beginning of the euction sequence. Lemma Given a scheme S that is in A -nomal fom an let S B S, then S contains a cycle if S contains a cycle. Poof. Let X S U, Y S U, X S V, Y S V an let S X:=Y B S be the esulting B -step. Let C be the cycle in S. We now exhibit a cycle in S by istinguishing the following cases. 15

16 1. The cycle C inclues Y. Since S is in A -nomal fom we know by Coollay 3.20 that S is in A -nomal fom. Hence thee is no Z U such that Y S Z o Z V such that Y S Z, theefoe C inclues W S Y S U o W S Y S V fo some name W. So in S we eithe have W S Y o W S X. In the fist case we ae finishe: C is also a cycle in S. In the secon case we have X S U o X S V, so we can eaange C into a cycle in S by passing though X instea of Y. 2. The cycle C oes not inclue Y. Then C aleay exists in S. Coollay Given a scheme S an let S A S such that S is in A -nomal fom, S f is statifie iff S is statifie. Poof. The nomal fom of scheme S is the same as the one of S, which is S f. Due to Coollay 3.20, we have S B S f an each scheme in this euction sequence is in A -nomal fom. By Lemma 3.21, we fin that fo each S 1 A S 2 in this euction sequence, S 1 contains a cycle iff S 2 contains a cycle. Thus, S f is statifie iff S is statifie. 4. Computing a type using Newman s algoithm In this section we pove that Newman s algoithm is coect an we exten it to esult in a pincipal type. Futhemoe, we compae it to Wan s algoithm an his coectness poof [21] Coectness of Newman s algoithm To pove sounness an completeness with espect to typing we efine a set of type equations fo each scheme. Fistly, we pove that the most geneal unifie of these equations gives ise to a pincipal type. Seconly, we pove that pefoming η-euction while keeping tack of the pefome substitutions coespons to computing a most geneal unifie. Definition 4.1. Given a scheme S, a set of type equations E(S) is inuctively efine as follows. 1. If Z MN then M N Z E(S) 2. If Z λx.p then Z x P E(S) Hee X is a fesh type vaiable fo each tem name X. Fact 4.2. Given a scheme S an X, Y, Z Name, we have: X Y X Z X Y Z E(S) Definition 4.3. Given a λ-tem M, efine Γ M := {x : x x FV(M)}. Lemma 4.4. Given a λ-tem M an a substitution σ such that σ = E(S(M)), we have σ(γ M ) M : σ(e(m)). Poof. This popety is poven by inuction on the stuctue of M. 16

17 (va) Now E(x) = x. The esult is immeiate because x : σ(x) x : σ(x) fo each substitution σ. (app) Now S(MN) = {E(MN) E(M) E(N)} S(M) S(N). So we have σ = E(S(M)) an σ = E(S(N)) an theefoe by the inuction hypothesis σ(γ M ) M : σ(e(m)) an σ(γ N ) N : σ(e(n)). Moeove we have σ(e(m)) = σ(e(n)) σ(e(m N)) hence by the application ule an weakening σ(γ MN ) MN : σ(e(mn)). (λ) Simila to the peceing case. Lemma 4.5. Given a eivable typing jugment M : ρ, thee exists a substitution σ such that σ = E(S(M)), σ(γ M ) an ρ = σ(e(m)). Poof. This popety is poven by inuction on the type eivation M : ρ. (va) Let x : ρ such that x : ρ. Now E(x) = x an S(x) =. Define σ = [x := ρ]. Then we have σ = E(S(x)). Also σ(γ x ) an ρ = σ(e(m)), so we ae one. (app) Let MN : ρ, then we have M : δ ρ an N : δ. Now E(MN) = Z an S(MN) = {Z E(M) E(N)} S(M) S(N). By the inuction hypothesis we obtain substitutions σ N an σ M. We efine σ as follows. σ(y ) = σ M (Y ) σ(y ) = σ N (Y ) σ(z) = ρ σ(α) = α if Y Names(S(M)) if Y Names(S(N)) othewise This substitution is well efine because (a) The tem M N satisfies the Baenegt convention an theefoe all boun vaiables in M an N ae fesh. (b) The tem names assigne to the subtems of M an N ae isjoint (by Definition 3.2). (c) By the inuction hypothesis we have σ M (x) = (x) fo all vaiables x FV(M) an σ N (x) = (x) fo all vaiables x FV(N). By the inuction hypothesis we have σ = E(S(M)), σ = E(S(N)) an σ(e(m)) = δ ρ = σ(e(n)) σ(z), so σ = E(S(MN)). Moeove, we have σ(γ MN ) an ρ = σ(e(mn)), so we ae one. (λ) Let λx.p : δ ρ, then we have, x : δ P : ρ. Now E(λx.P ) = Z an S(λx.P ) = {Z λx.e(p )} S(P ). By the inuction hypothesis we obtain a substitution σ P. We efine σ as follows. σ(y ) = σ P (Y ) if Y Names(S(P )) σ(x) = δ σ(z) = δ ρ σ(α) = α othewise 17

18 By the inuction hypothesis we obtain that σ = E(S(P )) an σ(z) = δ ρ = σ(x) σ(e(p )), hence σ = E(S(λx.P )). Moeove, we have σ(γ λx.p ) an δ ρ = σ(e(λx.p )), so we ae one. Coollay 4.6. The equations à la Newman ae coect. That is, given a λ-tem M an its scheme S = S(M), we have 1. If σ = mgu E(S) then σ(e(m)), σ(γ M ) is a pincipal pai of M. 2. If = E(S) then M. Poof. Immeiate fom Lemma 4.4 an 4.5. Now that we have shown that the fist pat of Newman s algoithm actually consists of the geneation of type equations, we will pove that pefoming η-euction coespons to computing a most geneal unifie. Fist we efine a substitution Ŝ fo each scheme S that is statifie an in nomal fom. Also, we pove that this is in fact the most geneal unifie of E(S). Definition 4.7. Given a scheme S that is statifie an in A -nomal fom, the substitution Ŝ : TVa Type is efine as follows. Ŝ(α) = {Ŝ(β) Ŝ(γ) α if α β γ E(S) othewise Lemma 4.8. Given a scheme S that is statifie an in A -nomal fom, the substitution Ŝ is well efine. Poof. In oe to show that Ŝ is well efine we have to pove that the efinition of Ŝ is unambiguous an total. Fo the fist conition we have to show that fo each Z thee is at most one equation Z X Y E(S). Let us suppose the contay, Z X Y E(S) an Z A B E(S), now by Fact 4.2 we have X A an Y B. So a contaiction is obtaine because S is in A -nomal fom. Moeove by Fact 4.2 an statification we know that no cycles in the type equations exist, hence Ŝ is total. Lemma 4.9. Given a scheme S that is statifie an in A -nomal fom, we have Ŝ = mgu E(S). Poof. We have Ŝ(X) = Ŝ(A) Ŝ(B) fo each X A B E(S) by efinition, hence Ŝ = E(S). So it emains to pove that Ŝ is a most geneal unifie of E(S). Theefoe suppose that we have a substitution δ such that δ = E(S). Now we shoul efine a substitution σ such that σ Ŝ = δ. We efine σ := δ. We pove that δ(ŝ(x)) = δ(x) fo all X Name by well-foune inuction ove S (this is allowe because S is statifie). So suppose that we have δ(ŝ(y )) = δ(y ) fo all Y Name such that X Y. Now we pove that δ(ŝ(x)) = δ(x) by istinguishing the following cases. 18

19 1. X A B E(S). Then we have δ(ŝ(a)) = δ(a) an δ(ŝ(b)) = δ(b) by the hypothesis. Hence we have δ(ŝ(x)) = δ(x) as shown below. δ(ŝ(x)) = δ(ŝ(a)) δ(ŝ(b)) = δ(a) δ(b) = δ(x) 2. X A B / E(S). Then we have δ(ŝ(x)) = δ(x). Lemma Given a scheme S that is not statifie, we have = E(S). Poof. If S is not statifie, E(S) contains a cycle an theefoe = E(S). So fa we have poven two pats of the coectness of Newman s algoithm: the equations E(S) ae coect an Ŝf is the most geneal unifie of E(S f ). So it emains to show how we can use Newman s algoithm to compute a most geneal unifie of E(S). As intouce befoe, we will o this by keeping tack of the substitutions while pefoming η-euction. Because these substitutions eplace names fo names instea of types fo type vaiables we intouce the following efinition. Definition Given a substitution ν : Name Name, the substitution ν : TVa TVa is efine as ν(x) = ν(x). In the emaine of this section we will pove that if S ν S f then Ŝf ν is the most geneal unifie of E(S) iff S f is statifie. This finally leas to a coectness poof of Newman s algoithm. Lemma If S ν S an δ = E(S ) then δ ν = E(S). Poof. Immeiate because δ = ν(e(s)) implies δ ν = E(S). Lemma If S X:=Y S an δ = E(S) then δ(x) = δ(y ). Poof. We pove this esult by istinguishing the following cases. (A1) Z X V, Z Y W E(S). Now we have δ(z) = δ(x) δ(v ) an δ(z) = δ(y ) δ(w ). So δ(x) δ(v ) = δ(y ) δ(w ) an theefoe δ(x) = δ(y ). (A2) Simila to the peceing case. (B) X V W, Y V W E(S). Now we have δ(x) = δ(v ) δ(w ) an δ(y ) = δ(v ) δ(w ) an theefoe δ(x) = δ(y ). Coollay If S ν S an δ = E(S) then δ ν = δ. Poof. Immeiate fom Lemma Lemma If S ν S an δ = E(S) then δ = E(S ). Poof. We have to pove that we have δ(νx) δ(νa) δ(νb) fo each equation X A B E(S). By assumption we have δ(x) δ(a) δ(b) so by Coollay 4.14 we ae one. 19

20 Theoem Given a scheme S an let S ν S f, we have: 1. If S f is statifie then Ŝf ν = mgu E(S). 2. If S f is not statifie then = E(S). Poof. By Lemma 4.9 we have Ŝf = E(S f ) an by Lemma 4.12 we obtain that Ŝ f ν = E(S). So fo the fist popety it emains to pove that Ŝf ν is a most geneal unifie of E(S). Theefoe let us suppose that we have a substitution δ such that δ = E(S). By Lemma 4.15 we have δ = E(S f ) an by Lemma 4.9 we obtain a substitution σ such that δ = σ Ŝf. Now it emains to pove that δ = σ Ŝf ν, but by Coollay 4.14 we have δ = δ ν, so we ae one. To pove the secon popety let us suppose that we have a substitution δ such that δ = E(S), then by Lemma 4.15 we have δ = E(S f ). But now we obtain a contaiction with Lemma 4.10 because S f is not statifie. Theoem Newman s algoithm is soun an complete with espect to typing. That is, given a λ-tem M, its scheme S = S(M) an let S ν S f, we have: 1. If S f is statifie then σ(e(m)), σ(γ M ), whee σ = Ŝf ν, is a pincipal pai of M. 2. If S f is not statifie then M. Poof. Immeiate fom Coollay 4.6 an Theoem Compaison with the algoithm of Wan The geneation of type equations as efine in Definition 4.1 an the statements of Lemma 4.4 an 4.5 look quite simila to Wan s algoithm an his coectness poof [21], espectively. Howeve we show that thee ae some notable iffeences between Wan s metho an ous (boowe fom Newman). Wan escibes his algoithm with a skeleton an an action table. Given a close tem M one stats with an empty set of equations E an a set of goals G consisting of one initial goal (, M, α) whee α is a fesh type vaiable. While the set of goals G is non-empty one picks a goal g fom G, eletes it fom G, an uses the following action table to geneate new goals an equations. g SG(g) EQ(g) (Γ, x, τ) τ Γ(x) (Γ, λx.m, τ) (Γ; x : α 1, M, α 2 ) τ α 1 α 2 (Γ, M P, τ) (Γ, M, α τ), (Γ, P, α) The newly geneate goals SG(g) ae ae to the set of goals G an the newly geneate equations EQ(g) ae ae to the set of equations E. This pocess is epeate until the set of goals G is empty. The coectness of Wan s algoithm states that δ = mgu E iff δ(α) is a pincipal type of M. So we obseve at least the following impotant iffeences between Wan s metho an ous. 20

21 1. Wan s algoithm esults in a set of type equations of the shape σ τ whee σ, τ Type. In ou metho the geneate equations ae of the shape α β γ whee α, β, γ TVa. 2. Wan s algoithm woks fo tems that o not satisfy the Baenegt convention. Theefoe Wan s algoithm keeps tack of the fee vaiables explicitly by caying a context aoun. 3. Wan s algoithm takes a type vaiable α as its input. Fo the geneate equations E we have δ = mgu E iff δ(α) is a pincipal type of M. Ou algoithm esults in a type vaiable E(M) an a set of equations E(M) such that δ = mgu E(S(M)) iff δ(e(m)) is a pincipal type of M. So, Wan s algoithm geneates equations top-own while ou algoithm geneates equations bottom-up. One coul ty to moify Wan s action table in such a way that it esults in type equations of the equie shape 4 in the following way. g SG(g) EQ(g) (Γ, x, α) α Γ(x) (Γ, λx.m, α) (Γ; x : α 1, M, α 2 ) α α 1 α 2 (Γ, M P, α) (Γ, M, α 1 ), (Γ, P, α 2 ) α 1 α 2 α Howeve, in the case of a vaiable an equation of the shape α β is geneate. Unfotunately, ue to Wan s top-own appoach we cannot take these equations into account in the application an λ cases. 5. Extension with othe type constuctions The typing algoithm à la Newman can be extene to inclue othe type constuctos. Fist we show how to inclue pouct types. This is a pope extension of Newman s metho, because the algoithm shoul now also fail ue to conflicting type constuctos, a case Newman oes not eal with. This equies a efinement of the notion of being statifie. Othe simple extensions, like sum types can be ae in a simila way. Seconly, we show how to a weak polymophism, which involves univesal quantification ove type vaiables (but only on the top level). This is moe inteesting, because it is basically only elevant if we consie tems insie a context, in which the vaiables have weakly polymophic types Pouct types We fist look into pouct types, so we efine the ules of the system. 4 Fo claity of pesentation we geneate type equations instea of tem equations that have to be tanslate into type equations. 21

22 Definition 5.1. The system λ is extene with pouct types as follows. Fistly, we exten the tems to inclue paiing an the pojections. Λ ::= Va Λ, Λ π 1 Λ π 2 Λ (ΛΛ) (λva.λ) Seconly, we exten the simple types with pouct types. Type ::= TVa (Type Type) (Type Type) Thily, we exten Definition 1.2 to inclue the following ules. Γ M : σ Γ N : τ Γ M, N : σ τ (a) Paiing Γ M : σ τ Γ π 1 M : σ (b) Fist Pojection Γ M : σ τ Γ π 2 M : τ (c) Secon Pojection Now we exten the tee of a tem an the gaph of a tem to inclue paiing an pojection constuctions. Definition 5.2. We exten the tanslation of tems to tees an the tanslation of tees to Newman gaphs as epicte in the figue. We wite Tgaph(M) fo the tee we obtain an Ngaph(M) fo the Newman gaph we obtain. In the constuction of the Newman gaph, we again choose a fesh label fo each noe. T( π i M) = π i T(<M, N>) = <,> T(M) T(M) T(N) <,> V 1 2 π 1 W π 2 W 1 2 The intuition will be clea again: X 1 Y iff X is a pouct type whose fist component type is Y X 2 Y iff X is a pouct type whose secon component type is Y. 22

23 We now exten the notion of conguence between noes to inclue the new aows in the gaph. Definition 5.3. The notion of conguence G between noes in a Newman gaph of Definition 2.3 is extene by aing the following inuctive clauses. 1. If U 1 X an U 1 Y, then X G Y 2. If U 2 X an U 2 Y, then X G Y 3. If X 1 U, X 2 V an Y 1 U, Y 2 V, then X G Y The notion of euction of Definition 2.5 is immeiately extene: we can join noes X an Y in case X Y, whee the conguence elation X Y is now the one of Definition 5.3. A gaph is in nomal fom in case no futhe euctions ae possible. The only notion that eally changes is that of a gaph being statifie. Definition 5.4. Given a Newman gaph G in the system extene with poucts, then G is statifie in case the following conitions hol. 1. The elation 1 2 contains no cycles. 2. The elation 1 2 contains no conflict. Hee we efine a conflict as a noe X that has an outgoing aow l1 an an outgoing aow l2 such that l 1 {1, 2} an l 2 {, }. Note that the new case in the efinition of statifie coespons to the case of conflicting function symbols in the unification algoithm. In the type system with only, we have only one function symbol, so unification can only fail ue to the occus check (the cyclicity in the efinition of statifie). In the pesence of poucts, one can also fail ue to conflicting type constuctos. Lemma 5.5. The euction is stongly nomalizing an confluent. Moeove, the nomal fom of the Newman gaph of M is statifie if an only if M is typable in λ extene with poucts. Example 5.6. Consie the λ-tem M λx.x(π 1 x). The nomal fom of its Newman gaph is epicte below an one can obseve that it is not statifie, ue to a conflict in the noe x. x U V 1 W Just as we have one fo aow types in example 2.9, one can now a type infomation to the noes in a statifie Newman gaph, to compute the type of the oiginal tem. We will not etail that hee. 23

24 5.2. Weak polymophism In this section we teat the λ-calculus with weak polymophism (hencefoth λ2w) an give a typing algoithm à la Newman. The weak polymophism means that the types of ou system ae type schemes, of the fom α.σ, whee σ is a simple type. Definition 5.7. The weakly polymophic types ae of the following fom Type ω ::= TVa.Type ω Type The fee type vaiables, FTV(σ), of a weakly polymophic type ae inuctively efine as follows. FTV(α) = {α} FTV(σ τ) = FTV(σ) FTV(τ) FTV( α.σ) = FTV(σ)\{α} A context Γ is now a sequence of eclaations of the fom x : σ, whee σ is a weakly polymophic type. Definition 5.8. A λ2w-typing jugment Γ M : ρ enotes that a λ-tem M has type ρ in context Γ. The eivation ules fo eiving such a jugment ae as follows. x : σ Γ Γ x : σ (c) Vaiable Γ M : σ τ Γ N : σ Γ MN : τ () Application Γ, x : σ P : τ τ Type Γ λx.p : σ τ (e) Abstaction Γ M : σ α F T V (Γ) Γ M : α.σ (f) Genealization Γ M : α.σ τ Type Γ M : σ[α := τ] (g) Instantiation Example 5.9. Consie a eivation of x : β.β β xx : α.α α. We abbeviate Γ = x : β.β β. Γ x : β.β β Γ x : β.β β Γ x : (α α) α α Γ x : α α Γ xx : α α Γ xx : α.α α Note that we cannot abstact ove x, because it has a weakly polymophic type. So the tem λx.xx is not typable in λ2w. To exten the notion of Newman gaph to inclue tems of λ2w, we only have to look at the polymophic vaiables in the context, as the tem stuctue oes not change. So, we efine the tanslation of a eclaation x : α.σ into a Newman gaph. Fist we efine the tanslation of a simple type into a Newman gaph. (We coul o this via fist efining the tee of the type, but we skip that step now.) 24

25 T(σ τ) = U U V T(σ) T(τ) α β Figue 3: Tanslating a type to a Newman gaph. Definition Given a type σ Type, we efine the Newman gaph of this type, Ngaph(σ) by inuction as escibe in Figue 3, whee we choose a fesh label (U in the pictue) an we shae all ientical type vaiables in one leaf. As an example we inicate the tanslation of α β α in Figue 3. Hee we see the shaing of noe α. Definition Given a context Γ = x 1 : α 1.σ 1,..., x n : α n.σ n an an untype λ-tem M with FV(M) {x 1,..., x n }, we efine the Newman gaph of M as in Definition 2.2, with the poviso that: 1. Evey occuence of a fee vaiable x i is labele uniquely an eplace by the Newman gaph of its type, Ngaph(σ i [ α i := β]) whee x i : α i.σ i Γ an β consists of fesh type vaiables. 2. All occuences of fee type vaiables in the types of x 1,..., x n ae shae. To see what the efinition means exactly, we teat an example. Example Let us consie the λ-tem M x y x in the context Γ = x : α.(α β) α, y : γ.(β γ) γ. Then the Newman gaph of M in Γ is as follows. We see that the gaph of the type of x occus twice, because x occus twice in the tem M. Futhemoe, the fee type vaiable β is shae by all the thee type tees, of the fist occuence of x, of y, an of the secon occuence of x. 25

26 The conguence elation (Definition 2.3) oes not change, an neithe oes the euction on Newman gaphs (Definition 2.5). The nomal fom of the Newman gaph of a λ2w-tem is again unique up to enaming of labels an we have the following Lemma. Lemma The euction is stongly nomalizing an it is confluent. Moeove, the nomal fom of the Newman gaph of M is statifie if an only if M is typable in λ2w. We can also ea off the type fom the Newman gaph in nomal fom. As we ae in λ2w, we want to genealize as many type vaiables as possible. This can be one by keeping tack of the vaiables that ae fixe in the context, like β in the example context of Example To show this, we look at the tem Q (λy. y x) (x (λz. z)) in the context Γ = x : α.α α. Remak that this tem is closely elate to the tem M λx. (λy. y x) (x (λz. z)) that we have consiee in Example 2.6 an which is not typable in λ. Example Let us consie the λ-tem Q (λy. y x) (x (λz. z)) in the context Γ = x : α.α α. We aw its Newman gaph an euce it to nomal fom. In boxes an in cicles we inicate the conguent noes that we join in a couple of lage steps. V Y U y x x' α Z z V U Y α x' 26

27 The gaph in nomal fom contains no cycle, so we conclue that the tem is typable. We can again ea off the type by annotating the noes without outgoing eges with type vaiables an constucting the type fo the noe U. We conclue that the pincipal type of the tem in λ2w is: α.α α. 6. Unification à la Newman In the pevious sections we have seen that Newman s algoithm consists of the following steps. 1. Ceate a set of equations fom a tem. 2. Using these tem equations, efine elations coesponing to type constaints. Fom these elations a conguence elation is ceate. 3. Rewite the equations by eplacing a name by anothe conguent one; estat fom (2) until no moe ewites ae possible. Newman ewites the equations, but we have seen that we can also ewite the elations, an basically foget about the equations afte step (1). In ou gaphical epesentation we have efomulate Newman s algoithm as follows. 1. Ceate a set of equations fom a tem an fom that a set of elations coesponing to type constaints. 2. Define fom these elations a conguence elation on names. 3. Rewite the elations by eplacing a name by anothe conguent one; estat fom (2) until no moe ewites ae possible. Ou elations fo eciing typing ae an fo λ (Section 2) an, 1,, an 2 fo λ extene with poucts (Section 5.1). These elations coespon to type constaints of the shape α β γ an α β γ. As shown in Section 4, the phases of Newman s algoithm ae compaable to the typing algoithm of Wan, whee a set of equations is ceate, which is then solve by a unification algoithm. We have seen that the equations of Wan ae constucte top-own, while Newman ceates the elations bottomup. Howeve, on anothe tack, Newman s algoithm is vey close to what happens in Wan s algoithm, because the steps (2) an (3) in the esciption above actually o unification! In this section we will make this pecise an show that Newman s metho gives ise to a unification algoithm an actually a quite efficient one. It is in set-up vey close to what is calle unification of tem DAGs (Diecte Acyclic Gaphs) in [16, 2]. Befoe going into the efinitions we give an example that shoul claify the connection with unification. Example 6.1. Suppose we have a fist oe language with two function symbols: f of aity 2 an h of aity 1. Now we want to fin a unifie fo the one-membe set of equations E = {f(x, h(v)) f(h(y), x)}. We intouce names fo all subtems an eplace E by the equivalent set of equations E = {A f(x, B), B h(v), A f(c, x), C h(y)}. We now ewite E by eplacing a symbol X by Y in case the set of equations contains two equations of the shape U f(..., X,...) an U f(..., Y,...), whee X 27

28 x f1 f2 B h v A f2 f1 C h y A f2 f2 f2 f1 f1 [x:=c] [B:=C] [v:=y] B C C h h h h v y v y A A f1 C h y f2 Figue 4: Unification algoithm á la Newman gaphically. an Y ae in the same position in the agument list of f. The ewiting goes as follows. A f(x, B) B h(v) A f(c, x) C h(y) A f(c, C) C h(v) C h(y) [x:=c] [v:=y] A f(c, B) B h(v) A f(c, C) C h(y) { A f(c, C) C h(y) } [B:=C] If we collect the substitutions we have pefome along the way, we obtain the most geneal unifie [x := h(y), v := y] of E. In the example we see that we neve substitute a compoun tem, but only names fo names. This pevents a size blow up that one coul encounte in a simple-mine oinay unification algoithm. We also give a gaphical epesentation of the unification, base on the same example. Example 6.2. Let us consie the set of equations E = {f(x, h(v)) f(h(y), x)} again. We now eplace each tem by a Newman gaph, which is a DAG obtaine by ceating fo evey subtem f(t, q) a new noe U with f1 pointing to the noe of subtem t an f2 pointing to the noe of subtem q. We shae vaiable noes. Futhemoe, we epesent an equation s by joining the two top noes of the tems an s. Fo E, this pouces the Newman gaph epicte in Figue 4. We now efine pecisely how unification à la Newman woks. Let a signatue be given with function symbols with fixe aity (typically f, g,... ) an fist oe tems ove this signatue (typically t, q,... ). We fist efine fo evey tem t a set of equations of the fom U t, whee U is a tem name that names the subtem q of t. Definition 6.3. Given a fist oe tem t, the set of Newman equations Neqns(t) is efine as follows. 1. Ceate a fesh tem name (typically U, V, X, Y, Z) to name each nonvaiable subtem of t. 28