On lambda-term skeletons, with applications to all-term and random-term generation of simply-typed closed lambda terms

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1 On mbd-trm sktons, with ppitions to -trm nd rndom-trm gnrtion of simpy-typd osd mbd trms Pu Tru Dprtmnt of Computr Sin nd Enginring Unirsity of North Txs ABSTRACT Lmbd trms in d Bruijn nottion r Motzkin trs (so d binry-unry trs) with indis t thir s ounting up to bindr mong th mbds on th pth to th root bing thir s. Dfin th skton of mbd trm s th Motzkin tr obtind by rsing th d Bruijn indis bing thir s. Thn, gin Motzkin tr, on n sk if it is th skton of t st on osd mbd trm. Mor intrstingy, on n sk th sm qustion for simpy-typd osd trms. A typ-hoding Motzkin tr is on for whih it xists simpy-typd osd trm hing it s its skton. A typ-rping Motzkin tr is on for whih no simpy-typd osd trm xists hing it s its skton. W study som of thir sttisti proprtis with fous on th rti dnsity of rious sss of trms. As gnriztion of ffin mbd trms, w introdu k- oord mbd trms obtind by bing thir unry nods with ountrs for th ribs thy bind. This ds to study thir k-oord sktons nd thir sttisti proprtis. W dsign drti impmnttion of Rémy s gorithm nd dfin nw bijtion from binry trs to 2-oord Motzkin trs from whr w dri simpy-typd osd mbd trms by dorting thm with d Bruijn indis. Th rsuting squnti nd pr rndom simpy-typd osd mbd trm gnrtors produ trms bo sizs of 1000 nd 2000, n ordr of mgnitud bo th bst priousy known rsuts. Th ppr is struturd s itrt Proog progrm to fiitt n siy rpib, onis nd drti xprssion of our onpts nd gorithms. KEYWORDS simpy-typd osd mbd trms, Motzkin trs, bijtions btwn dt typs, gnrtion of trms of gin siz, gnrtion of rndom mbd trms, typ infrn, unifition, ogi ribs, pr gorithms 1 INTRODUCTION Lmbd trms, in d Bruijn nottion [9], n b sn s Motzkintrs buit of unry mbd nods, binry ppition nods nd * Prmission to mk digit or hrd opis of prt or of this work for prson or ssroom us is grntd without f proidd tht opis r not md or distributd for profit or ommri dntg nd tht opis br this noti nd th fu ittion on th first pg. Copyrights for third-prty omponnts of this work must b honord. For othr uss, ontt th ownr/uthor(s). PPDP 2017, 2017 Copyright hd by th ownr/uthor(s). 978-x-xxxx-xxxx-x/YY/MM... $0.00 DOI: /nnnnnnn.nnnnnnn ribs t thir s, bd with d Bruijn indis pointing towrd thir mbd bindr. This brings up som ntur qustions bout th undrying tr strutur of spifi fmiis of mbd trms. Cn w ssify mbd nods by inrting th funtion from indis t th s to thir bindrs? Wi this rsut in intrsting bijtions to simpr dt typs,.g., mmbrs of th Ctn fmiy of ombintori objts? Cn w sing out Motzkin trs s sktons, tht, bus of som obsrb strutur, n or nnot hod gin fmiy of trms? W wi s tht th nswrs r trii for som fmiis,.g., in th sns tht ny suh skton n b dortd to t st on mbd trm tht is mmbr of th fmiy, but it suggsts instigting if it n xpos som intrsting strutur proprtis for othrs, ding to gorithms onnting with ss tht r known to b notoriousy hrd. This brings us to th min fous of this ppr, th s of simpytypd trms nd som of thir sub-fmiis ik ffin or inr trms. Dspit th nishing symptoti dnsity of simpy-typd mbd trms [15], thir -trm nd rndom-trm gnrtion hs bn spdd-up signifinty by th us of Proog-bsd gorithms tht intr gnrtion nd typ-infrn stps [23, 5]. Howr, th strutur of simpy-typd mbd trms hs so fr spd hnding by nyti mthods. Bsi ombintori proprtis ik ounts for trms of gin siz h bn obtind so fr ony by gnrting trms, or, s in [5], by mimiking thir xhusti gnrtion with rursi strutur tht, whi omitting th tu mbd trms, kps th typ-infrn mhnism intt. Ths diffiutis suggst to strt by ying down som mpiri bkground, hoping tht it wi hp us rn mor bout th strutur of simpy-typd mbd trms, nd s prti outom, push furthr -trm nd rndom-trm gnrtion, inuding i bttr priztion opportunitis. A usfu distintion n b md btwn mbd onstrutors tht bind ribs nd thos tht do not. Among othr bnfits, distinguishing thm wi mk th nysis of inr nd ffin trms simpr nd put thir sktons, th 2-oord Motzkin trs, in bijtion with th w-known Ctn fmiy of ombintori objts. This bijtion wi so nb fst gorithms for gnrting rndom binry trs to b trnsmigrtd into 2-Motzkin trs nd thn mbd trms. Th sprtion of th gnrtion pross into stgs ows wi ow on-th fy od gnrtion tht spds-up both squnti nd pr prossing. Th rst of th ppr is orgnizd s foows. Stion 2 dsribs nw bijtion btwn binry trs nd 2-oord Motzkin trs. 1

2 Stion 3 disusss th s of osd, inr nd ffin mbd trms. Stion 4 introdus typ-hoding nd typ-rping Motzkin trs nd studis som mpiri proprtis rtd to thir dnsity nd growth. Stion 5 dsribs rndom-trm gnrtor ombining uniformy rndom 2-Motzkin-tr gnrtor nd its dortion into simpy-typd mbd trm, i Rmy s gorithm nd bijtion to binry trs. Stion 6 dsribs pr gorithm for spding up rndom trm gnrtors nd iustrts thir prformn gins. Stion 7 oriws rtd work nd stion 8 onuds th ppr. Th ppr is struturd s itrt Proog progrm to fiitt n siy rpib, onis nd drti xprssion of our onpts nd gorithms. Th od xtrtd from th ppr is ib t: tru/rsrh/2017/rp.pro, tstd with SWI-Proog [24] rsion A BIJECTION BETWEEN 2-COLORED MOTZKIN TREES AND BINARY TREES A Motzkin tr (so d binry-unry tr) is rootd ordrd tr buit from binry nods, unry nods nd f nods. A k- oord Motzkin tr is obtind by bing its unry nods with oors from st of k mnts. W dfin 2-oord Motzkin trs (shorty 2-Motzkin trs) s th fr gbr gnrtd by th onstrutors /0, /1, r/1 nd /2. W dfin mbd trms in d Bruijn form s th fr gbr gnrtd by th onstrutors /1, r/1 nd /2 with s bd with ntur numbrs (nd sn s wrppd with th onstrutor /1 whn onnint). Thus, w n s mbd trms in d Bruijn form s Motzkin trs with s bd with ntur numbrs. W intrprt th bs s pointing to thir mbd bindr on pth to th root of th tr. If h f rhs i this its d Bruijn indx t st on unry onstrutor, w th trm osd, othrwis w it pin. W obsr tht th onstrutors mrking mbds my h t st on d Bruijn indx pointing to thm or h non. Thus, w ssify our unry onstrutors into: binding mbds, tht r rhd by t st on d Bruijn indx (dnotd /1) fr mbds tht nnot b rhd by ny d Bruijn indx, dnotd r/1. W n think bout ths s 2-oord mbd trms. W dfin th 2-oord Motzkin skton of mbd trm (shorty skton) s th tr obtind by rsing th d Bruijn indis bing thir s. It is w-known tht 2-Motzkin trs r ountd by th Ctn numbrs nd sr bijtions btwn thm to mmbrs of th Ctn fmiy of ombintori objts h bn idntifid in th pst [10]. W wi introdu hr nw on tht is dfind indutiy in omposition wy, bsd on mpping btwn sm tr omponnts on th two sids. W dsrib binry trs s th fr gbr gnrtd by th onstrutors /0 nd /2. Binry trs r w known mmbr of th Ctn fmiy of ombintori objts. Our bijtion n b sn s onnting ny othr mmbr of this fmiy to 2-oord Motzkin trs (from now on, 2-Motzkin trs). W dfin th bijtion btwn non-mpty binry trs nd 2- Motzkin trs simpy by noding h of th nods /0, /1, r/1 nd /2 by uniqu sm binry tr s shown by th rrsib bidirtion prdit t mot/2, with th binry tr s its first rgumnt nd th 2-Motzkin tr s its sond. t_mot((,),). t_mot((x,),(a)):-x=(_,_),t_mot(x,a). t_mot((,y),r(b)):-y=(_,_),t_mot(y,b). t_mot((x,y),(a,b)):-x=(_,_),y=(_,_), t_mot(x,a), t_mot(y,b). Proposition 1. Th prdit t mot/2 dfins bijtion btwn non-mpty binry trs nd Motzkin trs. PROOF. It foows by strutur indution by obsring tht th 4 uss or i disjoint unifition pttrns th 4 possib tr shps mthd on-to-on on th two sids. Exmp 1. W iustrt Th bidirtion Proog prdit t mot/2 with th two trs so shown in Fig. 1, togthr with two rgr trs on th right sid, twinnd in simir wy, Motzkintr on th ft, binry tr on th right.?- t_mot(bintr,((),r())),t_mot(bintr,mottr). BinTr = (((, ), ), (, (, ))), MotTr = ((), r()). On n tst it so with input from th foowing simp binry tr gnrtor t(n,t) whih, gin ntur numbr N rturns tr X of siz N, ssuming siz dfinition tht ounts h intrn nod s 1. t(n,x):-t(x,n,0). t(,n,n). t((a,b),sn,n3):-su(n1,sn),t(a,n1,n2),t(b,n2,n3). Not th us of th bidirtion su/2 buit-in, whih so tsts for bing rgr thn 0, whn working s prdssor. 3 CLOSED, AFFINE AND LINEAR TERMS W n dfin mbd trm in d Bruijn form s Motzkin tr dortd with ntur numbrs t its s. With siz dfinition (ssumd hr), tht gis 2 units to binry onstrutors, 1 unit to unry onstrutors nd 0 units to th s of th tr, mbd trm nd its skton n b, onninty, sn s hing th sm siz, in ft orrsponding (up to onstnt ftor) to its hp rprsnttion in th runtim systm of progrmming ngugs w know of. Smntiy, th bs r undrstood s pointing to unry nod sn s mbd bindr on th pth to th root, strting with 0 for th osst on. W dfin th skton of mbd tr in d Bruijn form s th 2-Motzkin tr obtind by rsing th d Bruijn indis t its s. Thus mbd trm is buit with th onstrutors /2 rprsnting ppitions, /1 nd r/1 rprsnting mbd nods nd ntur numbrs mrking s (possiby wrppd s /1 nods, whn onnint). 2

3 two twinnd trs of siz 4 r two twinnd trs siz 10 r D((SL,R,A),(L,R,A)):-su(L,SL). rd((l,sr,a),(l,r,a)):-su(r,sr). D((L,R,SA),(L,R,A)):-su(A,SA). W wi strt with gnrtors for osd, ffin nd inr trms. As nyti mthods r known for omputing ounts for osd trms s w s osd ffin nd inr trms [16], w wi fous hr on som simp proprtis of thir sktons nd on thir ffiint gnrtors. 3.1 Cosd mbd trms A mbd trm in d Bruijn form is osd, if for h of its d Bruijn indis, thr is mbd bindr to whih it points, on th pth to th root of th tr rprsnting th trm. W Motzkin tr osb if it is th skton of t st on osd mbd trm. It immdity foows tht: r Proposition 2. If Motzkin tr is skton of osd mbd trm thn it xists t st on mbd bindr on h pth from th f to th root. Thr r sighty mor unosb Motzkin trs thn osb ons s siz grows: r r r numbr of osb trms of sizs 0,1,2,... : 0,1,1,2,5,11,26,65,163,417,1086,2858,7599,20391,55127,150028,410719,... numbr of unosb trms of sizs 0,1,2,... : 1,0,1,2,4,10,25,62,160,418,1102,2940,7912,21444,58507,160544,442748,... W s n opn probm to study wht hppns to thm symptotiy. Figur 1: Th 2-oord Motzkin trs to non-mpty binry trs bijtion A 2-Motzkin tr is buit with /2 rprsnting binry nods, /1 nd r/1 rprsnting unry nods nd /0 stnding for f nods. Thus w omput skton by rping th d Bruijn indis t th s of mbd trm with th onstnt /0. Whn gnrting trs of gin siz, with sr nod onstrutors, it mks sns to h sprt ountrs for h. Th prdit sum to/2 mintins suh ountrs for nods of typs /1, r/1 nd /2. sum_to(n,(l,r,a),(0,0,0)):-n>=0, btwn(0,n,a2),0=:=a2/\1,a is A2>>1, LR is N-A2, btwn(0,lr,l), R is LR-L. Th prdits (suggstiy nmd) D/2, rd/2 nd D/2 dfin sing stps onsuming on ib unit of siz for h of th orrsponding onstrutors. Not th us of th bidirtion buit-in prdit su/2 tht omputs in this s th prdssor of ntur numbr nd fis ftr rhing Cosd ffin mbd trms An ffin mbd trm hs on or zro ribs bound by h mbd onstrutor. Proposition 3. If 2-Motzkin tr with n binry nods is skton of n ffin mbd trm, thn it hs xty n + 1 unry nods, with t st on on h pth from th root to its n + 1 s. This suggst gnrtors tht sprt unry nd binry nod ounts for th sktons nd nfor this onstrint on thir rspti sizs. Th prdit flm/2, foows osy th on dsribd in dti in [21], xpt tht it hnds /1 nd r/1 s sprt ss. flm(n,t):-sum_to(n,hi,lo), hs_nough_mbds(hi), flinlm(t,[],hi,lo). hs_nough_mbds((l,_,a)):-su(a,l). Th prdit hs nough mbds/1 is usd to xprss th onstrint tht th numbr of ppition nods /2 shoud b on mor thn th numbr of /1 onstrutors (in bijtion with th s thy bind). Th prdit flinlm/4 is dfind i Dfinit Cus Grmmrs (DCGs) tht npsut th onsumption of 3

4 th siz units 1. It uss th prdit subst nd ompmnt of/3 to dirt h mbd bindr on ithr ft or right pth t n ppition nod. flinlm((x),[x])-->[]. flinlm((x,a),vs)-->d,flinlm(a,[x Vs]). flinlm(r(a),vs)-->rd,flinlm(a,vs). flinlm((a,b),vs)-->d, {subst_nd_ompmnt_of(vs,as,bs)}, flinlm(a,as), flinlm(b,bs). subst_nd_ompmnt_of([],[],[]). subst_nd_ompmnt_of([x Xs],NwYs,NwZs):- subst_nd_ompmnt_of(xs,ys,zs), p_mnt(x,ys,zs,nwys,nwzs). p_mnt(x,ys,zs,[x Ys],Zs). p_mnt(x,ys,zs,ys,[x Zs]). Ersur of d Bruijn indis turns 2-oord mbd trm into 2-oord Motzkin tr. tomotsk((_),). tomotsk((x),(y)):-tomotsk(x,y). tomotsk((_,x),(y)):-tomotsk(x,y). tomotsk(r(x),(y)):-tomotsk(x,y). tomotsk((x,y),(a,b)):-tomotsk(x,a),tomotsk(y,b). Th prdits fskgn/2 nd inskgn/2 trnsform th gnrtor for mbd trms into gnrtors for thir sktons. fskgn(n,s):-flm(n,t),tomotsk(t,s). inskgn(n,s):-inlm(n,t),tomotsk(t,s). Th mutist of sktons is trimmd to st of uniqu sktons using SWI-Proog s distint/2 buit-in. fsk(n,t):-distint(t,fskgn(n,t)). insk(n,t):-distint(t,inskgn(n,t)). 3.3 Cosd inr mbd trms Proposition 4. If Motzkin tr with n binry nods is skton of inr mbd trm, thn it hs xty n + 1 unry nods, with on on h pth from th root to its n + 1 s. inlm(n,t):-n mod 3=:=1, sum_to(n,hi,lo),hs_no_unusd(hi), flinlm(t,[],hi,lo). hs_no_unusd((l,0,a)):-su(a,l). Not th us of th prdit hs no unusd/1 tht xprsss, quit onisy, th onstrints tht r/1 nods shoud not our in th trm nd tht th st of /1 nods shoud b in bijtion with th st of s. It is immdit tht osd ffin nd inr trms r w-typd. Th unry nods of th sktons of ffin trm n b sn s hing 1 Funtion progrmmrs might noti hr th nogy with th us of monds npsuting stt hngs with onstruts ik Hsk s do nottion. 2 oors, /1 nd r/1. This suggst to instigt nxt th s of k-oord trms. 3.4 K-oord simpy-typd osd mbd trms As ntur gnriztion drid from k-oord Motzkin trs, w dfin k-oord mbd trm hing s its mbd onstrutor /1 bd with th numbr of ribs tht it binds. Thus n ffin trm is 2-oord mbd trm. Th prdit kcoordcosd/2 gnrts trms whi prtitioning mbd bindrs in k-oord sss. It works by inrmnting th ount of f ribs mbd binds, in bktrkb wy by using sussor rithmti with th dpst nod kpt s fr ogi rib t h stp. kcoordcosd(n,x):-kcoordcosd(x,[],n,0). kcoordcosd((i),vs)-->{nth0(i,vs,v),in_r(v)}. kcoordcosd((k,a),vs)-->, kcoordcosd(a,[v Vs]), {os_r(v,k)}. kcoordcosd((a,b),vs)-->, kcoordcosd(a,vs), kcoordcosd(b,vs). (SX,X):-su(X,SX). -->,. in_r(x):-r(x),!,x=s(_). in_r(s(x)):-in_r(x). os_r(x,k):-r(x),!,k=0. os_r(s(x),sk):-os_r(x,k),(sk,k). Not so th DCG-mhnism tht ontros th intndd siz of th trms i th (onninty nmd) prdits /2 nd /2 tht drmnt ib siz by 1 nd rsptiy 2 units. Exmp 2. 3-oord mbd trms of siz 3, xhibiting oors 0,1,2.?- kcoordcosd(3,x). X = (0, (0, (1, (0)))) ; X = (0, (1, (0, (1)))) ; X = (1, (0, (0, (2)))) ; X = (2, ((0), (0))). Gin tr with n ppition nods, th ounts for k-oord mbds in it must sum up to n + 1. Thus w n gnrt binry tr nd thn dort it with mbds stisfying this onstrint. Not tht th onstrint hods for subtrs, rursiy. W it s n opn xprimnt to find out if this mhnism n rdu th mount of bktrking nd rt trm gnrtion. 3.5 Typ infrn for k-oord trms Th study of th ombintori proprtis of simpy-typd mbd trms is notoriousy hrd. In prti, th two most striking things whn infrring typs tht on might noti r non-monotoniity: rossing mbd inrss th siz of th typ, whi rossings n ppition nod trims it down 4

5 grmnt i unifition (with ours hk) btwn th typs of h rib undr mbd. W wi foow th intring of trm gnrtion, hking for osdnss nd typ infrn stps shown in [19], but nhn it to so idntify ribs ord by h mbd bindr. In ft, gin th surjti funtion f : V L tht ssoits to h f rib in osd mbd trm its mbd bindr, on n omput th st f 1 () for h L, xprssing whih ribs r mppd to h bindr. Exmp 3. W iustrt two 2-oord simpy typd trms with mbd nods shown s /2 onstrutors hing s thir ft hid muti-wy tr oting th st of ribs tht it binds. W p th infrrd typ s th right hid of root bd with :. X Y X : : Y Z U Z }. simpytypdcoord((vs,a),s->t,vss)-->, simpytypdcoord(a,t,[vs:s Vss]), {osbindr(vs)}. simpytypdcoord((a,b),t,vss)-->, simpytypdcoord(a,(s->t),vss), simpytypdcoord(b,s,vss). Not tht ddtobindr/2 dds h f undr bindr to th opn nd of th ist of rib/typ pirs ist, osd by osbindr/1. ddtobindr(ps,p):-r(ps),!,ps=[p _]. ddtobindr([_ Ps],P):-ddToBindr(Ps,P). osbindr(xs):-ppnd(xs,[],_),!. Exmp 4. Som trms of siz 5 gnrtd by th prdit simpytypdcoord/3 nd thir typs.?- simpytypdcoord(5,trm,typ). Trm = ([], ([], ([], ([], ([A], (A)))))), Typ = (B->C->D->E->F->F) ;... Trm = ([A, B], (([], (A)), ([], (B)))), Typ = (C->C) ;... Trm = ([A, B], (([], ([], (A))), (B))), Typ = (C->D->C) ;... W h notid tht both rg nd mximum numbr of oors of mbd trms grow ry sowy with siz. Fig. 2 omprs on og-s th growths of simpy typd osd trms nd thir osd ffin trms subst. As for d Bruijn trms, w n dfin X Y X As in [19], our typ infrn gorithm nsurs tht ribs undr th sm bindr gr on thir typ i unifition with ours hk, to oid formtion of ys in th typs, rprsntd s binry trs with intrn nods ->/2 nd ogi ribs s s. simpytypdcoord(n,x,t):-simpytypdcoord(x,t,[],n,0). Y Z U V Z simpy typd osd trms s. ffin osd trms (og. s) simpy typd osd trms ffin osd trms siz simpytypdcoord((x),t,vss)-->{ mmbr(vs:t0,vss), unify_with_ours_hk(t,t0), ddtobindr(vs,x) Figur 2: Counts of simpy typd osd trms nd ffin osd trms by inrsing sizs th Motzkin sktons of k-oord mbd trms by rsing th first 5

6 rgumnt of th /2 nd /1 onstrutors. W n so dfin th k-oord Motzkin sktons of ths trms by rping th rib ists in rgumnt 1 of /2 onstrutors by thir ngth nd by rsing th rgumnts of th /1 onstrutors. Th prdit tosks/3 omputs th (k-oord) Motzkin sktons by msuring th ngth of th ist of ribs for h bindr. tosks((_),,). tosks((vs,a),(k,cs),(s)):-ngth(vs,k),tosks(a,cs,s). tosks((a,b),(ca,cb),(sa,sb)):- tosks(a,ca,sa), tosks(b,cb,sb). W obtin gnrtors for sktons nd k-oord sktons by ombining th gnrtor simpytypdcoord with toskton. gntypdsks(n,cs,s):-gntypdsks(n,_,_,cs,s). g. oors nd typ sizs g. oors g. typ sizs gntypdsks(n,x,t,cs,s):- simpytypdcoord(n,x,t), tosks(x,cs,s) trm siz typbcosks(n,cs):-gntypdsks(n,cs,_). Figur 3: Growth of oors nd typ sizs typbsks(n,s):-gntypdsks(n,_,s). W n gnrt (but quit inffiinty) th st of typb sktons from th mutist of sktons by using th buit-in distint/2 tht trims dupit soutions. W wi risit ffiint gnrtion in th nxt stion. mx oors mx typ siz sowtypbcosk(n,cs):- distint(cs,typbcosks(n,cs)). 80 sowtypbsk(n,s):- distint(s,typbsks(n,s)). W dfin th typ siz of simpy typd trm s th numbr of rrow nods -> its typ ontins, s omputd by th prdit tsiz/2. tsiz(x,s):-r(x),!,s=0. tsiz((a->b),s):-tsiz(a,sa),tsiz(b,sb),s is 1+SA+SB. Now tht w n ount, for gin trm siz, how mny k-oord trms xists, on might sk if w n sy somthing bout th sizs of thir typs. Figur 3 shows th signifinty sowr growths of th rg numbr of oors of oord trms s. th rg siz of thir typs, with possib og-s orrtion btwn thm. W most oorfu trm of gin siz trm tht rhs th mximum numbr of oors. Fig. 4 show th rtion btwn th numbr of oors of most oorfu trm nd mximum siz rhd by th typ of suh trm. Fig. 5 shows th rtion btwn th rgst typ sizs th most oorfu trms of gin siz n ttin nd th mximum possib typ siz of thos trms. W n obsr tht th rgst most oorfu trms rh th rgst possib typ siz for gin trm siz, most of th tim, but s Fig. 5 shows, thr r xptions trm siz Figur 4: Coors of most oorfu trm s. its mximum typ siz W s n opn probm to pro or dispro tht thr s trm siz suh tht for rgr trms, th most oorfu suh trms rh th rgst typ siz possib. 6

7 80 60 mx oorfu typ mx typ ([_ Vs],V,N1,N3):-su(N1,N2),(Vs,V,N2,N3). On oud so unfod this into onjuntion of disjuntions, to possiby ry on th sris of SAT-sor to tst if th rsuting CNF is stisfib, but w s n opn xprimnt to work out th dtis of suh unfodings nd study thir fftinss. For now, t us obsr tht this ds to ry ffiint mthod for tsting if skton is typb or not, by tsting if ths qutions h soution. typ siz untypbsk(n,sk):-gneqs(n,x,_,eqs), not(eqs),tomotsk(x,sk). typbsk(n,sk):-gneqs(n,x,_,eqs), on(eqs),tomotsk(x,sk) trm siz Figur 5: Lrgst typ siz of most oorfu trm s. rgst typ siz 4 TYPE-HOLDING AND TYPE-REPELLING MOTZKIN TREES A typ-hoding Motzkin tr is on for whih it xists simpy-typd osd trm hing it s its skton. A typ-rping Motzkin tr is on for whih no simpy-typd osd trm xists hing it s its skton. To ffiinty gnrt ths sktons w wi spit th gnrtion of mbd trms in two stgs. Th first stg wi gnrt th unifition qutions tht nd to b sod for typ infrn s w s th rdy to b fid out mbd trs. It is onnint to tuy gnrt on th fy th od to b xutd in th sond stg. This s to th sond stg to just us Proog s mt to tit this od. gneqs(n,x,t,eqs):-gneqs(x,t,[],eqs,tru,n,0). gneqs((i),v,[v0 Vs],Es1,Es2)-->{dd_q(Vs,V0,V,I,Es1,Es2)}. gneqs((a),(s->t),vs,es1,es2)-->,gneqs(a,t,[s Vs],Es1,Es2). gneqs((a,b),t,vs,es1,es3)-->, gneqs(a,(s->t),vs,es1,es2), gneqs(b,s,vs,es2,es3). dd_q([],v0,v,0,es,es):-unify_with_ours_hk(v0,v). dd_q([v1 Vs],V0,V,I,(([V0,V1 Vs],V,0,I),Es),Es). Th sond stg od is ompty xprssd s onjuntion of /3 prdits, h nforing th onstrint tht typ rib V orrsponding to d Bruijn indx I unifis with th typ otd so fr in th typ rib t position I on th ist Vs. (I,Vs,V):-(Vs,V,0,I). ([V0 _],V,N,N):-unify_with_ours_hk(V0,V). Not th us of ngtion s w s th buit-in on usd to dtt th xistn of soution. An intrsting probm is to find out if thr r sktons tht hod xty on typ. In wy, tht impis tht th thir typ nd st of bindings woud b pr-dtrmind ompty by th undrying Motzkin tr. Th prdit uniquytypbsk/2 omputs ffiinty suh sktons by gnrting th dortion od Eqs tht w onstrin to h xty on soution whn titd. uniquytypbsk(n,sk):- gneqs(n,x,_,eqs),suds_on(eqs),eqs, tomotsk(x,sk). suds_on(g):-findnsos(2,_,g,sos),!,sos=[_]. Not th us of th SWI-Proog buit-in findnsos/4 tht ffiinty dtts th xistn of xty on soution to our unifition qutions. Fig. 6 shows th xistn of uniquy typb trms of for rg nough sizs but so sms to indit tht upwrds jumps hppn for trms of n odd siz. Not tht on n so sk simir numbr of uniquy typb sktons siz Figur 6: 3 Growth of th numbr of uniquy typb sktons 7

8 ) typ-hoding typ-hoding sktons typ-rping sktons b) typ-rping skton ounts (og. s) skton siz Figur 8: typ-hoding s. typ-rping sktons Figur 7: Typ-hoding, typ-rping Motzkin trs of siz 8 qustion bout th ss of osd mbd trms, whih is iky to b trtb by nyti mthods. In Fig. 7 w show 2 typ-hoding nd 2 typ rping Motzkin trs. Fig. 8 nd th tb in Fig. 9 ompr ounts of typ rping nd typ hoding Motzkin sktons for trm sizs up to 20. Fig. 10 shows th simir growth rts of typ-hoding nd typrping sktons. Fig. 11 omprs ounts for Motzkin trs nd thir subst md of typ-hoding sktons for inrsing tr sizs. Fig. 12 omprs, on og-s, ounts of simpy typd osd trms nd thir sktons. 5 AN APPLICATION TO RANDOM LAMBDA TERM GENERATORS As s study, putting sr of our onpts togthr, w dsign drti impmnttion of Rémy s gorithm tht proids rndom binry trs of spifid siz. W thn us our bijtion to Motzkin trs to dort to simpy-typd osd mbd trms. trm siz typ hoding sktons typ rping sktons Figur 9: Numbr of typ-hoding nd typ rping sktons 5.1 Risiting Rémy s gorithm, drtiy Rémy s origin gorithm [18] grows binry trs by grfting nw s with qu probbiity for h nod in gin tr. An gnt produr impmnttion is gin in [14] s gorithm R, by using dstruti ssignmnts in n rry rprsnting th tr. Whi on oud mut it on top of produr or drti 8

9 skton growth rts typ-hoding sktons typ-rping sktons skton siz simpy typd osd trms nd thir sktons (og. s) simpy typd osd trms typ-hoding sktons siz Figur 10: Growth rts of typ-hoding s. sktons typ-rping Figur 12: Counts of simpy typd osd trms nd thir sktons by inrsing sizs Motzkin trs nd typ-hoding sktons (og. s) Motzkin trs typ-hoding sktons siz Figur 11: Counts for Motzkin trs nd typ-hoding sktons for inrsing trm sizs mution of updtb rrys (.g., with nb strg/3 in SWI- Proog), w wi dsign hr drti impmnttion Trs r onntd grphs: t s buid thm s sts of dgs. First, s trs r (onntd) grphs, on n rprsnt thm s sts of dgs. W wi us ogi ribs to b thir nds rprsnting ithr intrn or f nods. W woud so b h dg s ft or right to indit thir position rti to nod in th binry tr. Thus ft dg originting in A with trgt B wi b rprsntd s (ft,a,b). W strt with ist of two dgs from root nod A rturnd by th prdit rmy init/1. rmy_init([(ft,a,_),(right,a,_)]). Th rndom hoi of th dgs (or th non-dtrministi on, by rping hoi of/2 with its ommntd out trnti) is gnrtd by th prdit ft or right/2 s: ft_or_right(i,j):-hoi_of(2,di), ft_or_right(di,i,j). hoi_of(n,k):-k is rndom(n). % hoi_of(n,k):-n>0,n1 is N-1,btwn(0,N1,K). ft_or_right(0,ft,right). ft_or_right(1,right,ft). W n grow nw dg by spitting n xisting dg in two i th prdit grow/3: grow((lr,a,b), (LR,A,C),(I,C,_),(J,C,B)):- ft_or_right(i,j). Not tht sing us dfins grow/3, indpndnty of th ft or right dnoting th rtion of th dg to its sour nod A. It dds thr nw dgs orrsponding to rgumnts 2, 3 nd 4 nd rmos on, rprsntd s its first rgumnt. Not so, tht ontrry to Rémy s origin gorithm, our tr grows downwrd s nw dgs r insrtd t th trgt of xisting ons, though this is iky to b n rbitrry hoi. Th nw nod C, onntd to A by inhriting th typ LR of (LR,A,B) wi b md to point to nw f i th dg 9

10 (I,C, ) nd to th tr bow nod B i th dg (J,C,B). Th ft / right hoi mong I nd J, is don by ft or right(i,j). This ds us th th bsi stp of th gorithm, whr st of dgs Es is rwrittn s st of nw dgs NwEs s gin by th prdit rmy stp/4. To oid omputing th siz L of th st Es w mintin it by dding 2=3-1 nods, s on nod is rmod nd 3 r ddd t gin stp. Not tht w pik n dg rndomy mong th L ib by ing th hoi of/2, oprtion proidd by rmy stp1, tht nigts th ist to th point whr th rwriting stp grow/3 hppns. rmy_stp(es,nwes,l,nwl):- NwL is L+2,hoi_of(L,Di), rmy_stp1(di,es,nwes). rmy_stp1(0,[u Xs],[X,Y,Z Xs]):-grow(U, X,Y,Z). rmy_stp1(d,[a Xs],[A Ys]):-D>0, D1 is D-1, rmy_stp1(d1,xs,ys). Th prdit rmy oop itrts or rmy stp unti th dsird 2K siz is rhd, in K stps s w grow by 2 dgs t h stp. Not so tht th initi 2 dgs r ddd whn K=1 by ing rmy init. rmy_oop(0,[],n,n). rmy_oop(1,es,n1,n2):-n2 is N1+2,rmy_init(Es). rmy_oop(k,nwes,n1,n3):-k>1,k1 is K-1, rmy_oop(k1,es,n1,n2), rmy_stp(es,nwes,n2,n3). Exmp 5. Th gnrtion of rndom ist of dgs of siz 4:?- rmy_oop(2,edgs,0,n). Edgs = [(ft, A, B), (right, A,C), (right,c,d), (ft,c,e)], N = From sts of dgs to trs s Proog trms. Th fin stp, unshing th powr of ogi ribs, xtrts Proog trm rprsnting th binry tr from th ist of dgs bd with unbound ribs. Th prdit bind nods/2 itrts or dgs, nd for h intrn nod it binds it with trms proidd by th onstrutor /2, ft or right, dpnding on th typ of th dg. Not tht, gin th ordr-indpndn of th binding of ogi ribs, th sm trm is buit indpndnty of th ordr of th dgs. Nxt, th prdit bind f binds th rmining unbound nods with th onstnt /0 bing th f nods. Corrtnss foows from th ft tht nod is f if nd ony if it rmins unbd whn th sour of n dg is mrkd with th /2 onstrutor, i., if it is not th sour of n dg. Not tht w us mpist to itrt or ists nd to ppy prdit to thir orrsponding mnts. bind_f((_,_,lf)):-lf=->tru;tru. Th prdit rmy trm/2 puts th two min stps togthr. rmy_trm(k,b):-rmy_oop(k,es,0,_),bind_nods(es,b). Exmp 6. Th gnrtion of rndom trm with 4 intrn nods s w timings for rgr rndom trs.?- rmy_trm(4,t). T = ((, ), ((, ), )).?- tim(rmy_trm(1000,_)). % 526,895 infrns, CPU in sonds (85% CPU, Lips) Whi th gorithm hnds firy rg trms in rsonb tim, w do not im tht its rg prformn is inr, ik in th s of Knuth s produr impmnttion, gin tht it tks tim proportion to th siz of th st of dgs to pik th on to b xpndd. Not howr, tht on n impro its xptd O(N 2 ) prformn with tr rprsnttion of th st of dgs to O(Nog(N)) or n to mortizd O(N) with dynmiy growing rry rprsnttion using rbitrry rity ompound trms s ontinrs. 5.2 Gnrting rndom simpy typd trms W obtin 2-Motzkin tr gnrtor from th binry tr gnrtor i our bijtion. mot_gn(n,m):-n>0,rmy_trm(n,c),t_mot(c,m). By strting from our rndom gnrtor for Motzkin trs, or, if on prfrs uniform distribution for gin siz, by using Botzmnn smpr s th on utomtiy gnrtd by [4], on n dort it to mbd trms in d Bruijn nottion simpy by bing its s with d Bruijn indis, inditing thir bindr s th numbr of /1 onstrutors nountrd on th pth to th root of th tr. W intr th dortion pross with ry rjtion of typs tht do not unify or d Bruijn indis tht d to trms tht r not osd. As intrsting siz dfinitions dpnd mosty on th wight w tth to th d Bruijn indis, w ustomiz th od to pug-in siz dfinition of our hoi. In ft, on n us sttistis from r progrms to mimi ny distribution of th d Bruijn indis. Th prdit inchoi/4 piks n mnt T0 mong th K mnts of th ist Ts, with th sm probbiity for h. inchoi(k,ts,i,t0):-k>0,i is rndom(k),nth0(i,ts,t0). Th prdit xpchoi dos th sm thing with probbiitis dying xponntiy th frthr th mnt is from th bginning of th ist. bind_nods([],). bind_nods([x Xs],Root):-X=(_,Root,_), mpist(bind_intrn,[x Xs]), mpist(bind_f,[x Xs]). bind_intrn((ft,(a,_),a)). bind_intrn((right,(_,b),b)). xpchoi(k,ts,i,t):-k>0,n is 2ˆ(K-1), R is rndom(n),n1 is N>>1, xpchoi1(n1,r,ts,t,0,i). xpchoi1(n,r,[x _],Y,I,I):-R>=N,!,Y=X. xpchoi1(n,r,[_ Xs],Y,I1,I3):-N1 is N>>1,su(I1,I2), xpchoi1(n1,r,xs,y,i2,i3). 10

11 Our rndom simpy-typd osd mbd trm gnrtor prods by dorting 2-oord Motzkin tr, with rtris triggrd by fiur of bing ithr osd or not w-typd. Th prdit dorttypd/3 dds d Bruijn indis to 2- oord Motzkin tr, but quiky fis if it is not osd or not w-typd. dorttypd(m,x,t):-dorttypd(m,x,t,0,[]). dorttypd(,(i),t,k,ts):- inchoi(k,ts,i,t0), % <= ny siz dfinition! unify_with_ours_hk(t,t0). dorttypd((x),(a),(s->t),n,ts):-su(n,sn), dorttypd(x,a,t,sn,[s Ts]). dorttypd(r(x),r(a),(_->t),n,ts):- dorttypd(x,a,t,n,ts). dorttypd((x,y),(a,b),t,n,ts):- dorttypd(x,a,(s->t),n,ts), dorttypd(y,b,s,n,ts). Rtris, for gnrting rndom trms of siz N with MxI rndom 2-oord Motzkin trs trid, nd mxj dortion ttmpts for h, is xprssd s th prdit rntypd/7. On suss, it rturns rndom trm X of typ T s w s th numbr I of rtris for Motzkin trs nd th numbr J of rtris of dortion ttmpts. rntypd(n,mxi,mxj,x,t,i,j):- btwn(1,mxi,i), mot_gn(n,mot), btwn(1,mxj,j), dorttypd(mot,x,t),!. Exmp 7. Running th rndom simpy-typd trm gnrtor.?- rntypd(400,1000,200). % sonds... rg trm nd its typ hr... stps(791*107),ntsiz(1174),hpsiz(400),typ_siz(19) Not tht w obtin trms of signifinty rgr siz thn with th Botzmnn smpr of [5], in xhng for giing up uniformity during th dortion stp, whi prsring it for th gnrtion of 2-oord Motzkin trs. On oud so dpt this dortion gorithm to work on top of Botzmnn smpr for Motzkin trs. 6 A PARALLEL GENERATOR FOR RANDOM SIMPLY-TYPED CLOSED LAMBDA TERMS An importnt onsidrtion whn dsigning pr gorithms for irrgur dt-typs (trs in our s) is to nsur tht od bning hppns ffiinty nd without prohibiti ommunition osts. Howr, spding-up rndom gnrtion is surprisingy sy nd ffti, gin tht on n simpy strt s mny indpndnt srh prosss s ib thrds. Th prdit prtypd/7 ows pssing fw prmtrs to fin-tun th srh, N=hp siz of th trm, MxI, MxI = numbr of rtry stps pssd to rntypd/7, X, T = trm nd its typ, I, J = numbr of rtris ndd to find soution. Othrwis, w ry on SWI-Proog s first soution/3 buit-in prdit, tht gin n nswr spifition, runs ist of gos, dirtd hr i n option to ontinu dspit of som thrds trminting without finding n nswr. prtypd(n,mxi,mxj,x,t,i,j):- Rs=[X,T,I,J], proog_fg(pu_ount,mxthrds), G=rnTypd(N,MxI,MxJ,X,T,I,J), ngth(gos,mxthrds), mpist(=(g),gos), first_soution(rs,gos,[on_fi(ontinu)]). Exmp 8. Running th pr rndom simpy-typd trm gnrtor on 44 CPU / 88 hypr-thrd mhin.?- prtypd(600,3000,600). % sonds... rg trm nd its typ... stps(591*355),ntsiz(2014),hpsiz(600),typ_siz(1244) This brings us to rndom simpy-typd trms of ntur siz 2000 (or hp-rprsnttion siz 600), n ordr of mgnitud bo [5]. 7 RELATED WORK Sr pprs xist tht dfin bijtions btwn 2-Motzkin trs nd mmbrs of th Ctn fmiy of ombintori (.g., in [10]), typiy i dpth-first wks in trs onntd to Motzkin, Dyk or Shrödr pths. Howr, w h not found ny simp nd intuiti bijtion tht onnts omponnts of th two fmiis, or on tht onnts dirty binry trs nd 2-Motzkin trs, ik th on shown in this ppr. Th ssi rfrn for mbd uus is [2]. Vrious instns of typd mbd ui r oriwd in [3]. Th ombintoris nd symptoti bhior of rious sss of mbd trms r xtnsiy studid in [13, 7]. Distribution nd dnsity proprtis of rndom mbd trms r dsribd in [8]. Gnrtion of rndom simpy-typd mbd trms nd its ppitions to gnrting funtion progrms from typ dfinitions is ord in [11]. Th gnrtion nd ounting of ffin nd inr mbd trms is xtnsiy ord in [16], with imits for ounting rgr thn in this ppr rhb using ffiint rurrn formus. Thir symptoti bhior, in rtion with th BCK nd BCI ombintor systms, s w s bijtions to ombintori mps r studid in [6]. Asymptoti dnsity proprtis of simp typs (orrsponding to tutoogis in minim ogi) h bn studid in [12] with th surprising rsut tht most ssi tutoogis r so intuitionisti ons. In [17] typ-dirtd mhnism for th gnrtion of rndom trms is introdud, rsuting in mor risti (whi not uniformy rndom) trms, usd sussfuy in disoring som bugs in th Gsgow Hsk Compir (GHC). A sttisti xportion of th strutur of th simp typs of mbd trms of gin siz in [22] gis inditions tht som typs frqunt in humn-writtn progrms r mong th most frqunty infrrd ons. Gnrtors for osd simpy-typd mbd trms, s w s thir norm forms, xprssd s funtion progrmming gorithms, r gin in [13], drid from ombintori rurrns. Howr, 11

12 thy r signifinty mor ompx thn th ons dsribd hr in Proog, nd imitd to trms up to siz 10. Rémy s gorithm [18], produry impmntd s gorithm R in [14], hs gnrtd signifint numbr of ttmpts to dpt it to uniformy gnrt simir dt typs. Among thm w mntion [1] whr it is so shown how diffiut it is to nsur uniformity. For uniform gnrtion of rbitrry dt-typs spifid by ontxtfr grmmr, th Botzmnn smpr gnrtor of [4] stnds out, s it tuy produs ffiint Hsk progrms gnrting uniformy trms of n xptd siz or bo. In th unpubishd drft [20] w h otd sr mbd trm gnrtion gorithms writtn in Proog nd oring mosty d Bruijn trms nd omprssd d Bruijn rprsnttion. Among thm, w h ord inr, ffin inr trms s w s trms of boundd unry hight nd in th binry mbd uus noding. 8 CONCLUSIONS Contrry to osd, inr nd ffin mbd trms (s w s sr othr sss of trms subjt to simir onstrints) th strutur of simpy-typd trms hs so fr spd pris hrtriztion. Whi th fous of th ppr is mosty mpiri, it hs unwrppd som nw obsrbs tht highight intrsting sttisti proprtis. In wy, th nw onpts introdud ino bstrtion mhnisms tht forgt proprtis of th diffiut ss of simpy-typd osd mbd trms to r quin sss tht r iky to b sir to grsp with nyti toos. Among thm, k-oord trms subsum inr nd ffin trms nd r iky to b usb to fintun rndom gnrtors to mor osy mth oor-distributions of mbd trms rprsnting r progrms. Th xistn of typ-rping sktons shows tht thr r Motzkin trs tht nnot b sktons of simpy-typd osd trms. This suggsts xporing th dsign of ffiint gorithms buit on oiding sm typ-rping sktons stord in dtbs. Th nw, intuiti bijtion btwn binry trms nd 2-oord Motzkin trms nd th gnrtion gorithms ntrd on th distintion btwn fr nd binding mbd onstrutors hs bn usfu to rt gnrtion of ffin nd inr trms. In ombintion with our drti impmnttion of Rémy s gorithm, it hs so hpd produ rndom simpy-typd trms of siz s rg s mor thn 1000 nods. Our rndom trm gnrtion gorithms turnd out to b sy to priz, rsuting on mhin with rg numbr of prossor to produ simpy-typd osd trms of sizs bo 2000 nods. Lst but not st, w h shown tht ngug s simp s sid-fft-fr Proog, with imitd us of impur fturs nd mt-progrmming, n hnd gnty ompx ombintori gnrtion probms, whn th synrgy btwn sound unifition, bktrking nd DCGs is put t work. ACKNOWLEDGEMENT This rsrh hs bn supportd by NSF grnt W thnk th prtiipnts of th CLA 2017 workshop ( progrmm.htm) for iuminting disussions nd thir ommnts on our prsnttion oring th min ids of this ppr. REFERENCES [1] A. Bhr, O. Bodini, nd A. Jquot. Ext-siz Smping for Motzkin Trs in Linr Tim i Botzmnn Smprs nd Hoonomi Spifition. In M. E. Nb nd W. Szpnkowski, ditors, 2013 Prodings of th Tnth Workshop on Anyti Agorithmis nd Combintoris (ANALCO), pgs SIAM, [2] H. P. Brndrgt. Th Lmbd Cuus Its Syntx nd Smntis, oum 103. North Hond, risd dition, [3] H. P. Brndrgt. Lmbd ui with typs. In Hndbook of Logi in Computr Sin, oum 2. Oxford Unirsity Prss, [4] M. Bndkowski. Botzmnn-brin Softwr (Hsk stk modu), pubishd troniy t [5] M. Bndkowski, K. Grygi, nd P. Tru. Botzmnn smprs for osd simpytypd mbd trms. In Y. Lirr nd W. Th, ditors, Prti Aspts of Drti Lngugs - 19th Intrntion Symposium, PADL 2017, Pris, Frn, Jnury 16-17, 2017, Prodings, oum of Ltur Nots in Computr Sin, pgs Springr, [6] O. Bodini, D. Grdy, nd A. Jquot. Asymptotis nd rndom smping for BCI nd BCK mbd trms. Thorti Computr Sin, 502: , [7] R. Did, K. Grygi, J. Kozik, C. Rffi, G. Thyssir, nd M. Zion. Asymptotiy most λ-trms r strongy normizing. Prprint: rxi: mth. LO/ , [8] R. Did, C. Rffi, G. Thyssir, K. Grygi, J. Kozik, nd M. Zion. Som proprtis of rndom mbd trms. Logi Mthods in Computr Sin, 9(1), [9] N. G. d Bruijn. Lmbd uus nottion with nmss dummis, too for utomti formu mnipution, with ppition to th Churh-Rossr Thorm. Indgtions Mthmti, 34: , [10] E. Dutsh nd L. W. Shpiro. A bijtion btwn ordrd trs nd 2-motzkin pths nd its mny onsquns. Disrt Mthmtis, 256(3): , [11] B. Ftshr, K. Cssn, M. H. Pk, J. Hughs, nd R. B. Findr. Mking rndom judgmnts: Automtiy gnrting w-typd trms from th dfinition of typ-systm. In Progrmming Lngugs nd Systms - 24th Europn Symposium on Progrmming, ESOP 2015, Hd s Prt of th Europn Joint Confrns on Thory nd Prti of Softwr, ETAPS 2015, London, UK, Apri 11-18, Prodings, pgs , [12] A. Gnitrini, J. Kozik, nd M. Zion. Intuitionisti s. Cssi Tutoogis, Quntitti Comprison. In M. Miun, I. Sgntto, nd F. Hons, ditors, Typs for Proofs nd Progrms, Intrntion Confrn, TYPES 2007, Ciid d Friui, Ity, My 2-5, 2007, Risd Std Pprs, oum 4941 of Ltur Nots in Computr Sin, pgs Springr, [13] K. Grygi nd P. Lsnn. Counting nd gnrting mbd trms. J. Funt. Progrm., 23(5): , [14] D. E. Knuth. Th Art of Computr Progrmming, Voum 4, Fsi 4: Gnrting A Trs History of Combintori Gnrtion (Art of Computr Progrmming). Addison-Wsy Profssion, [15] Z. Kostrzyk nd M. Zion. Asymptoti dnsitis in ogi nd typ thory. Studi Logi, 88(3): , [16] P. Lsnn. Quntitti spts of inr nd ffin osd mbd trms. CoRR, bs/ , [17] M. H. Pk, K. Cssn, A. Russo, nd J. Hughs. Tsting n optimising ompir by gnrting rndom mbd trms. In Prodings of th 6th Intrntion Workshop on Automtion of Softwr Tst, AST 11, pgs 91 97, Nw York, NY, USA, ACM. [18] J.-L. Rémy. Un proédé itértif d dénombrmnt d rbrs binirs t son ppition à ur génértion étoir. RAIRO - Thorti Informtis nd Appitions - Informtiqu Théoriqu t Appitions, 19(2): , [19] P. Tru. On Uniform Rprsnttion of Combintors, Arithmti, Lmbd Trms nd Typs. In E. Abrt, ditor, PPDP 15: Prodings of th 17th intrntion ACM SIGPLAN Symposium on Prinips nd Prti of Drti Progrmming, pgs , Nw York, NY, USA, Juy ACM. [20] P. Tru. A ogi progrmming pyground for mbd trms, ombintors, typs nd tr-bsd rithmti omputtions. CoRR, bs/ , [21] P. Tru. On Logi Progrmming Rprsnttions of Lmbd Trms: d Bruijn Indis, Comprssion, Typ Infrn, Combintori Gnrtion, Normiztion. In E. Ponti nd T. C. Son, ditors, Prodings of th Sntnth Intrntion Symposium on Prti Aspts of Drti Lngugs PADL 15, pgs , Portnd, Orgon, USA, Jun Springr, LNCS [22] P. Tru. On Typ-dirtd Gnrtion of Lmbd Trms. In M. D Vos, T. Eitr, Y. Lirr, nd F. Toni, ditors, 31st Intrntion Confrn on Logi Progrmming (ICLP 2015), Thni Communitions, Cork, Irnd, Spt CEUR. ib onin t [23] P. Tru. A hiking trip through th ordrs of mgnitud: Driing ffiint gnrtors for osd simpy-typd mbd trms nd norm forms. CoRR, bs/ , [24] J. Wimkr, T. Shrijrs, M. Trisk, nd T. Lgr. SWI-Proog. Thory nd Prti of Logi Progrmming, 12:67 96,

13 APPENDIX Bijtion btwn ntur numbr nd binry tr rithmtis from [19] % bijtion btwn N x N nd N+ ons(i,j,c) :- I>=0,J>=0, D is mod(j+1,2), C is 2ˆ(I+1)*(J+D)-D. % inrs bijtion btwn N+ nd N x N dons(k,i1,j1):-k>0,b is mod(k,2),kb is K+B, dydiv(kb,i,j), I1 is mx(0,i-1),j1 is J-B. % dydi ution of KB nd rsidu dydiv(kb,i,j):-i is sb(kb),j is KB // (2ˆI). % bijtion btwn N nd st of binry trs n(,0). n((a,b),k):-n(a,i),n(b,j),ons(i,j,k). % inrs bijtion btwn th st of binry trs nd N t(0,). t(k,((a,b))):-k>0,dons(k,i,j),t(i,a),t(j,b). % prity of th ntur numbr ssoitd to tr prity(,0). prity((_,),1). prity((_,(x,xs)),p1):-prity((x,xs),p0),p1 is 1-P0. % img of n in N+ n_((_,xs)):-prity(xs,1). % img of odd in N+ odd_((_,xs)):-prity(xs,0). Sussor nd prdssor prdits in binry tr rithmti, from [19] Ths prdits r omptib with th dfinitions of rithmti oprtions in N, i.., if th img of tr is n thn th img of its sussor is n+1. p1(0,x,(,(y,xs)),(x,(sy,xs))):-s(y,sy). p1(0,x,((y,ys),xs),(x,(,(py,xs)))):-p((y,ys),py). p1(1,,(x,xs),(sx,xs)):-s(x,sx). p1(1,(x,ys),xs, (,((PX,Xs)))):-p((X,Ys),PX). Shifting th binry trs to Motzkin tr bijtion to inud mpty binry trs is hid with th prdits t2mot/2 nd mot2t/2. t2mot(c,m):-s(c,suc),t_mot(suc,m). mot2t(m,c):-t_mot(suc,m),p(suc,c). Rnking nd unrnking of 2-oord Motzkin trs Rnking of 2-oord Motzkin trs i this bijtion is dfind s rnk(m,n):-mot2t(m,c),n(c,n). Unrnking n thn b dfind s: unrnk(n,m):-t(n,c),t2mot(c,m). Th prdits rnk nd unrnk work s shown bow:?- btwn(0,15,n),unrnk(n,m),rnk(m,n1), N==N1, % tsts ssrtion tht it is bijtion writn(n -> M),fi;n. 0-> 1->r() 2->() 3->(,) 5->r(()) 6->(r()) 7->(r(),) 8->r((,)) 9->r(r(r())) 10->(,r()) 11->(,()) 13->r((r())) 14->(()) 15->((),) tru. % sussor s(,(,)). s((x,),(x,((,)))):-!. s((x,xs),z):-prity((x,xs),p),s1(p,x,xs,z). s1(0,,(x,xs),(sx,xs)):-s(x,sx). s1(0,(x,xs),xs,(,(px,xs))):-p((x,xs),px). s1(1,x,(,(y,xs)),(x,(sy,xs))):-s(y,sy). s1(1,x,(y,xs),(x,(,(py,xs)))):-p(y,py). % prdssor p((,),). p((x,(,)),(x,)):-!. p((x,xs),z):-prity((x,xs),p),p1(p,x,xs,z). 13