Barendregt, H.P. (Henk)

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Discriminating coded lambda terms Dedicated in frien.dship to Co1· Baayen, o·n the occasion of hi.s retir·e·merit Henk Barendregt Computing Science Institute Catholic University Nijmegen Department of Software Technology CW/, Amsterdam A coding for a (type-free) lambda term A1 is a lambda term r A1 7 in normal form such that M (and its parts) can be reconstructed from r l\:f7 in a lambda definable way. Kleene[1936] defined a coding I lvI 7 K and a self-iriterpreter EK E.1.\0 such that 'vlv/EA0 EKr A,f7 K == M. In this style one can construct a discriminator ~ K E1\0 such that true false ( = Aa~y .a.~) if .J.'ftvf = N; (= Axy.y) else. (1) (2) The terms EK and l:::,. K are complicated. They depend on the lambda defin-ability of functions on the integers dealing with coded syntactic properties. Inspired by a construction of P. de Bruin (see Barendregt [1991]) Mogensen [1992] constructed a different coding r .lvf7 and an efficient self-interpreter EEA0 such that (3) This construction does not use an encoding of syntax as numbers but directly as lambda terms. This results in a much less complex E. Mogensen's construction was simplified even further in Bohm et al. [1994]. In this paper we construct a simple discriminator Cl.EA0 such that true if !v1 =et N; false else. (4) Note that in (1) and ( 4) the statement is only about closed lambda terms, while that in (2) and (3) is about all lambda terms. It will become clear why this is so. 141 • 1. INTRODUCTION Tl1e 1nost ir11portant r1otatior1s for tl1e type-free lc1111bcla C'.alc:l1lus will be giver1 here. Bac~kgrou11d can be f'ound i11 Barend1·egt, [1984]. 1.1. DEFINI,..rION. Var·ia1Jles and t.er111s ()f t,he l}t111l)(lct C'.~l.l(:ult1s are (iefi1·1ecl by the followir1g al)stract sy11tax. var term a I var' var I term term I ...\ var term NorrA'l.,ION. (i) Af,N, ... ,P,Q, ... l'clilge o,rer· ...\-ter·111s. Tl1e letters x,y,z, . .. range over var·ialJles. Note that, tl1e "rariables are { a, a,', a'', . .. , a<·ri,), .. . } . (ii) A is tl1e set of la111bda t.er·rns. FV ( .1.'•f) is t.he set of f1·ee vc:1riables of J\,f.Tl1e set of closed tern1s is A0 == {1\tfEAIFV(M) = 0}. (iii) Tl1e relatio11 = de11otes syntac:tic equality; tl1e relatior1 ==,t denotes syn-tact,ic equality up to a cl1ange of 11c:1.r11es of the bou11d variables. For exa111ple (iv) Tl1e relatio11 = de11otes p-cor1vertibilit,y, axio111c1tized l)y (>..x.lvf)N = lvf[;r :== N]. He1·e [x := n] denotes substitutio11 of N in the free oc~curre11c~es of 2-~. E.g. (x(>..x.;r))[x: :==a]= a(>..x.x). (v) 1N is the set of 11atural n11mbers. For ·nEJN the terms Cn = >..Jx.f'ix, where J0 x = x arid fri+ 1x = f(f 12·x), de11ote the so c:alled Cht1rch 1111merals. Note that the c1 ,, are disti11ct norn1al for111s; he11ce by the Churcl1-Rosser tl1eoren1. A la111bda term can be see11 as a11 executable: the redexes want to be eval-uated. In this sense a nor111al forn1 is not exect1tal)le anymore. For a lan1bda term M its code r M 7 is a nor1nal for1n such that lvf is reconstructible from M. Kleene [1936] defined a code r Af 7K essentially as follows. 1.2. DEFINITION. (i) By induction 011 t,r1e structure of M we define #A1. #(a(n)) #(PQ) #(>..x.P) < 0, n >; < 1, < #(P)~ #(Q) >>; < 2,< #(x),#(P) >>. Here < -, > denotes a recursive pairing function on 1l'J with the rec11rsive projections (-)o, ( · ) 1: 142 (ii) The map r_ 7 K : A >A is defined by rM7K ==C#M· Note that for all MEA the term r M 7 K is in normal forn1. Moreover, rM7K == rN7K =} M = N. 1.:3. PROPOSITION. There is no lambda term Q such that for all A1 E1\ (o) one has PROOF. S11.ppose Q exists. Then for I = .\x.x 011e has Q(II) = But also Q(II) ==QI== r1 7 K == c#I == C<2,<#(x),#(x)>>· Hence< 1, < #(1), #(1) >>==< 2, < #(x), #(x) >>, a cor1t,radic~tio11. ■ In spite of this fact that the 'qt1ote' Q does 11ot exists, tl1e i11ve1·se 'evaluatior1' E can be constructed. 1.4. THEOREM (Klee11e [1936]). There exists an EK EA0 such that fo1· all A1EA0 one has PROOF. See Klee11e [1936] or Barendregt [1984], theoren1 8.1.16. ■ The self-interpreter E can work only for closed terms M (or terms having at most a fixed finite set of free variables). The reason is tl1at if eKr Af7K == M, then FV(M) C FV(EKr M 7 K) = FV(EK)_ Therefore if EK is closed, then the A1 have to be closed as well. This causes one difficulty in the construction of EK. The closed terms do not forrn a co11text-free language. Kleene solved this problem by constructir1g E first for the set of con1binatory terms C0 built from the basis {K, S} using applicat.ion only; then the real self-interpreter· can be obtained by tra11slations between A0 and C0 • A different construction of a self-interpreter was given by a for111er student of mine, usi11g ideas fro1n denotational se1nantics. 1.5. THEOREM (P. de Bruin). There exists an E0 EA0 such that for all M EA and all FEA one has (5) (simultaneous substitution), where {x1, ... , Xri} == FV(M). 143 PROOF. By the representability of con1putable fu11ctior1s and the fixed point theorem there is a term E0 EA0 such that Eorx7K F EorPQ7K F EorAx.P7 K F where Frrx7• ;:1;] = F; with p~rx7 p~ry7 pr x~7K; ' F(Eor p7K F)(E(lrQ7K F); K ..\.x.(E(lr P 7 F[r x71 ,:1:] ), x· ' Note that F~ can be written as GFx, with G closed. By induction on the structure of M EA one can show that the staten1ent holds. ■ 1.6. COROLLARY. There exists ari EdB EA0 sucli that for all lvf EA0 one has er ]\;f7K == Jvf. PROOF (P. de Bruin). We can take E <i B = A 171, • E () ,,,i I . Indeed, for closed terms M it follows from ( 5) that EdBr ,A,f7 == Ear Af71 == j\;f_ ■ 2. REPRESENTING DATA TYPES After seeing the r11ethod of P. de Brui11, l\1ogenser1 [1992] gave ar1 improved ver-sion of it by representing data types directly (i.e. not using t,l1e r1atural nun1bers) in lan1bda calculus as done in e.g. Bohn1 and Berarducci [1985]. Tl1is approach was improved later by Bohrn et al. (1994] by co11structing a r1ew rep1~esentatior1 of data types into type-free la111bda calct1lus. This new representation will be treated in a slightly modified form in this section. 2.1. DEFINITION. Write (M1, .. - , lv1ri) Az. z Nf 1 ... lv1ri; AX1 ... Xrl .• Xi; Note that U '~l. i true false U2. 1, U2 2· ( M 1 , . • • , _M.n) Uf'-true P Q falsePQ P· ' Q. In particular we have (M) == Az.zM and ( ) = Ax.x == I. Now we define the notion of lists inspired by the language LISP, McCarthy et al. (1961]. 144 2.2. DE:f .. lNITION. (i) \\Trit,e nil cons car ( ) ~ A;l;"Y. (.1·, 11) ~ Q , .. , (U~)~ (ii) Define So for exa111plt") cdr null·1-• [ ] [ 1\f 1 , ... , 1\fr1 + 1] 2 (U2)~ ( >~ cons 1\f 1 [ .. l\1!2, ... ~ .i\f,,+1]-( .. J\11, 1\!2, 1\f:1] == (1\f1, (j\12, (1\1:i, { )))). (I11 Ba,re11dregt) [1984] t,l1is t<-~1·111 is writ,tt:~11 as [11\11, 1\12 • 1\l:_J. I]. 1-\t tl1t:· t,i111t~ <)f ,vrit,i11g t.l1at, boc)k we dicl r1c)t, yt~t set:~ tl1e t1st~ful11ess <)f tt)r1r1i11<1,t.ir1g tl list, ,vitl1 a SJ)ecial c:011stru<•t,or.) N (>t<-? t.l1at, car ( cons PCJ) -· cdr ( cons PQ) null·, nil • null·t( cons PQ) true: ' false. 2.3. PRO POSITION. Th,t:·r·e ei~ists la1n.b(la d(~firi(iblt~ fii,11.cti<)'lt.9 ( ) 1 :.;·11,cli tlitzt for· 1 < i < 1·1, one lia.s PROOF. Take (l)1 (l)i+l car l~ ( cdr l)i- ■ 2.4. D EFINI'l"'ION. An (algebraic) .c;ig1i<it,ur·e s co11sists <>f cl r1u1r1l)er ·11 E L.'\J ( t,hot1gl1t of as tl1e list of' s:yTr11l)ols [f 1, ... , f 11 ]) t.()g<~t,11<:~1· ,vit,l1 <t list. c>f 1111111b<~r~s [.~1, ... , s,,] (thougl1t of as t.he arity of t;l·1e respe<~tive !1 's). \Ve \Vrit.e s:::::: [,91, ... , t'{,l]. For exa111ple a field l1as sigr1at.t1re s = [2~ 2~ 1, 1, (), O] ( t·,l1ought of· cts t.l1E~ ctrit,ies of tl1e fu11ctic)nsy111bc)ls [+, x,-, - 1 ,(), lJ; s<> f 1 = +,f2 = x <:~t,c~ett~r,1). 2.5. D EFlNITION. If s is a sig11att1r·e t,l1e11 term·", t.l1e st:.t <:lf lt::'r;rz .• '; <)j ."-;ig'ricLt·u1·(:1 s, is ciefi11ed as follows. ~l~Evar ~1.·Eterms; t 1 , . . . , t s i E term s f'i ( t 1 , . . . , t ,., i ) E term s . 145 For exa1nple ir1 the signat,ure of fields the ter111 J1(f1(;r,f~i(f2(Y,f,1(z)}).,f0) is 11s11ally written as x - yz- 1 + 1. 2.6. DEFINITION. Let, s == [s1, ... , .9 1;,] be a signat1.1re. (i) A lambda iriterpretation, of .c; is a list of '(~or1st,ri1c:tc>rs' C;1, ... , C,1. EA. (ii) Let. C 1, ... , C 11 be a larn t)da interpret.atic>11 c)f s. The11 \Ve defir1e a 1nap as follows. T: term .«i ,A Tx T1i(t1~•··,t1Ji> X . ' where [Tt 1 , ••• , Tts.] is tl1e list operation on la1nbda t,er111s defi11ed ir1 2.2. l Exa1nple. The signatt1re of bina·111 trees is (0, 2]. The terrr1 t == f2(f2(f1,f1), / 1) denotes a simple tree and t' == J2 (f1 , f 2 (f1 , f 1 )) its 111iror in1age. Can we find a la1nbda interpretatio11 for this signat.ure in such a way tl1at 111irroring bec<)n1es lan1bda definable, i.e. for some FEA0 011e has FTt = Tt' ~? The followi11g result,, due t.o Bohm et al. [1994], will affir111 this. We present the result in a r11odified form that ,vill be useful for §4. 2.7. THEOREM. For ever·y algebr·aic! S'ignat11,re s == [s 1 , ... ,s,1 ] there exists a lambda interpretation C 1 , ... , Cri such that the f ollou,irig hold. (i) \fA1 ... A-ri :3F 1 < i < ri. (6) (ii) The C1 , ..• ,Cri only depend on n., n.ot ori tlie [s1, ... ,.Sri]- Iri (6) ·we can take F = ( ( A 1 , ... , Ari)) . PROOF. Define C Ji = .Xle.e uii l ( e). (i) Given A 1, ... , Ari, we try wl1ether F = ( ( A 1, ... , A 11 )) works. Indeed, { { A 1 , ... , Ari)) (CJ i [Tt 1 , ••• , Tt $ i ] ) C Ji [Tt 1 , ••• , Tt.,.] (A1, ... , A.,i) 'I, ( A 1, ... , Ari) U i [Tt 1 , ••• , Tt s. ] ( ( A 1, ... , A,,i)) 't Ai[Tt 1 , ••• , Ttsi]F. (ii) By the construction. ■ 2.8. COROLLARY. Lets = [s1, . .. , s 11 ] be ari algebmic signature. Let C1, ... , Cri be the lambda interpretation of s constructed in theorerri 2. 7. Then for all B 1 ... Bn there exists an F such that 1 < i < ri. (7) 146 PROOF. Let B1, ... , Bn be given. Define Ai = Al.Bi(l)1 ... (l)si. The11 Then the F for tl1e Ai found in theorem 2. 7 is the F satisfying (7). ■ Now we can program the function that '1nir·rors' trees. In the sig11ature [0,2] fo1· binary trees let T11 leaf tree Aab. T12 ( a,b) By corollary 2.8 there exists an F sucl1 tl1at Fleaf F(treeab) 3. A SIMPLE SELF-INTEllPRETER leaf; tree(f b) (fa). In Mogensen [1992] a si1nple coding and self-interpreter for larr1bda ter111s is defined, using the fact that data types ( ter1n algebras of a sig11at ure s) have a lan1bda interpretation. The method was simplified by Bohm et al. [1994] by making use of their lambda representatior1 of algebraic signatures given in §2. 3.1. DEFINITION. Let s be the signature [l, 2, l]. Define const == C11 = Ale.eUf l(e); app == C12 = Ale.eU~l(e); abs == C13 = Ale.eU~l(e). 3.2. DEFINITION. For M EA define r M 7 as follows. const [x]; app [r p7, rQ7]; abs [Ax.r P 7 ]. Note that FV(r M 7 ) = FV(M). 3.3. THEOREM (Mogensen [1992]). There exists an EEA0 such that VMEA ErM 7 = M. PROOF (B''ohm et al. [1994]). By corollary 2.8 there exists a term EEA0 such that E( const [p]) E{ app [p, q]) E( abs [p]) 147 p; (Ep)(Eq); Ax.E(px). er .1·7 er /'.l(,7 er ,,\J"·. p 7 ( Er 1·> 7 ) ( Er (J -, ) : \ "' er- 1:> 7 AJ. ~ . • E -·"' ( { >.l J. ( l) 1, >-.l f. J (I} 1 (, J ( l) 2), :\l J .1·. J ( ( l) 1 ~r·))). canst app abs ---1"'l1t'.) rest1lti11g self-i11t,(!>r1>rt"t.er t.l1t:\11 l><~)<~(>Illf•s E11 - ((K, S, C)). fiert\ K = ✓\ .. 1:y.:1·, S -· >-.~ryz.3·z(yz) i111d C = ,,\.1~11z .. 1~(z.lJ). f""(-)f r·t\ct.-,<:>11s <:.•f l111ifc>r111its· \\'(• 11:tv(• givE:~11 t,l1e d<~fi.11it,io11 <>f canst , app il.Il{l abs ,tS ir1 :3.1. Tl1is will 1>t~ ust·~f11l i11 §,t. 4. ..~ SI !vf PI .. E: DISC-~RifviIN i\'l"Oll Ir1 this sec:tio11 wt~ will c~o11st,r11<·t, ll si111plE:· t,<)r111 tiisc:ri111i11itt.ir1g l>etWE"(~Il c·c>c·lc~d cl()sed larr1l)da terr11. Tl1e clisc·:1·i111i11at i<>r1 is f~Vt~11 r11<)<i lllc, <~-('(> 11 \r(:)t•sic.)Il. Fc)I' <)µe11 tern1s disc·ri111ir1at.io11 is J)ossil>lf:l c>11ly for tl1e <::oclir1g r 7 K <)f l~l(><\Il(~. true false if T! == 'T11. ~ f illL• #) , ., J-t."'.> l_,, • (ii) Tht~re exi.'St,.;; ti te1"r11. and EA0 suc~Ji thcit and true true an.d true false and false true and false false true: • false~ false; false. PROOF. (i) By" tl1e represe11tal)ilit,y c>t· the rec~ur·sive fu11ct,io11s. ( ii) Take and = Altb.a true b. ■ 148 .:1.2. P1ic>PC>SI"I~ION. 11l1t:'rY:: t:;.:ti.,t:, <l lit"1·rr1 b(-=,1\1) .si1.<"l1. tl,,it (1ir1"'itiri!J b,i j{Jr bc,i) t>rit' lias t5}N .ri,~ ••• false~ false~ false: ' and ( l, ri r J>7r· J>' 7) ( /J~, r (J-,r- ('J' 7 _} false; false: false~ s:.: ( r f> -, [ . . ] ) . r / )f -~, r . , l ) <'·ri-t-l . · ~l' ::::::: C 11 _. ( - lX :== C,,, ... P ROOI-..... \Ve i11t;r<)<i u(·:e t:l1e follo\vir1g a(l }·1(·,c· 11c)t.atit,r1. ( i) L(~t A 1 ....... 4 ·,i E 1\. Tt1f~Il we W'ri t.E:' -- [ [ r11 l , . · · , .. 4 1 11] • [ ,;,:\ 21 ' . . . ' 1\ 21~.] '\ • • • If tl == 3 we n1ay writ,e [ ~4ij] as No\v defir1e b = A·ritt'. bfN ( l) l ( t' ) 1 Atd!At' d'·ri!! false false ,;411 A12 .. 41:.) A21 A22 .. 423 • .. 431 A32 ,;4~J:; false a.nd ( cl ( t ) 1 ( t' ) 1 rt ) ( tl ( f ) 2 ( t' ) 2 11 ) false false false cl ( t 11.) ( t' 1i) ( S + 12 ) I tt 11, ,v l1ere S + la111l)da defir1es tl1e st1C:(~t·ssc>1· f1111(·t,it)11. 1"'l1is b st1t isfit~s t.r1t~ SI)t~<~ific~it-t,ior1. ■ 4.3. PROPOSITION. Fo,r· all NI, ~'/\f I E i\ sttl~li tli<it FV ( 1\f 1\f') ~ { ~: 1, ... , .1· ,, } ci1i<l for· s·ubstittL,tiori.'> * = [x1 := Ci: 1 ] • •• [.r,.i :== CJ. 1.] 'U-'itlz h:1. # h:J (f<>r· l < i < j < 11) orie has f 01· p > ki (J or· cill 1 < ·i :::; 11) tli<1t true false 14t} if l\l =(t 1\1'; ,t;;) l s "=: 1.....-- ~,. \...- • PROOF"'. By inductior1 on the st,ru<~tt1re of A,f, i11 ea.c:l1 c_:a~e r11aking dist.ir1c~t.ior·1s accordi11g to t.l1e structure of A1f'. We treat. fou1· i11st.ru<~ti\'e <:ases. Case 1\1[ = x,Jvf' = x'. Then w}1ere :z; = Xi, :i;' = :l~i'. This is trllt\ or false dt~[>t~ncli11g c->r1 \'.\'l1t~t.l1er :i_: = x' (so i = i') or· x :j. :.r:' (soi f- i'). Case Af = x,fvl' = P'Q'. ,.fl1e11 false. and (6'1)r p7r P'7*)('5,>rQ7rQ'7*) =111 and (true/ false)( true/ false) true/ false, as it should (== true 011ly if PQ = P'Q' i.f~. if l)ot.11 P = P' c111cl Q = Q'). Case J\,,f = .-\x.P,lvf' = >,..x'.P'. The11 81,+i(r P 7 [x := Cp]}(' P' 7 [:r~' := Cp]) * 8p+I r P 71 P'[x' := x] 7 [x :== cp] * bp+I r p7r P'[x' :== x],*'' witl1 *' == *[x :== cv] beir1g an adn1issible substitut.io11. So true if P =o P'[x' :== x}; false else. Now A1 =0: A1f 1 iff >..x.P =c\: A:r:'.P' (=c~ .Ax.P'[x:1 := x]) iff P =tt: P'[:i;' := x]. Hence we are done. ■ 4.4. COROLLARY. Write 6. = 50 • Then /01· all lvf, A·f' EA0 one has true false PROOF. I1111nediate fron1 the propositio11. ■ if A1 =a !vl'; else. Note tl1at this corollary cannot hold for arbitrary M., Af' EA. For exa111ple, it is impossible to discri1ninate r x 7 and r :i;' 7 . I11deed take x~ :/= x' arid rr1ake a substitution: _6r x 7r x'7 == false, a contradictio11. REFERENCES BARENDREGT' H.P. [1984] The Lambda Calculus, its Syntax and Semantics, revised edition, Studies in Logic and the Foundations of Mathematics, North-Holland, Amster-dam. [1991] Theoretical pearls: Self-interpretation in lambda calculus, J. Funct. Pro-gramming, 1(2), 229-233. BOHM, C., and A. BERARDUCCI [1985] Automatic synthesis of typed ..\-programs on term algebras, Theor. Com-put. Sci. 39, 135-154. BOHM, C., A. PIPERNO and S. GUERRINI [1994] ..\-definitions of function(al)s by normal forms, in: ESOP '94, (ed. D. San-nella), LNCS 788, Springer, Berlin, 135-149 CHURCH, A. [1941] The calculi of lambda conversion, Princeton University Press, Princeton. Reprinted by Kraus reprint corporation, New York, 1965. KLEENE, 8.C. [1936] ..\-definability and recursiveness, Duke Math. J. 2, 340-353. McCARTHY, J., P.W. ADAMS, D.J. EDWARDS, T.P. HART and M.I. LEVIN [1962] LISP 1.5 Programmer',s manual, MIT Press. MOGENSEN, T . ..tE. [1992] Efficient self-interpretation in lambda calculus, J. Funct. Programming, 2(3), 345-364. 151 

Discriminating coded lambda terms (Dedicated in friendship to Cor Baayen on the occasion of his retirement)

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