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

Transcription

1 On Type-hoding and type-repeing ambda-term skeetons, with appications to a-term and random-term generation of simpy-typed cosed ambda terms Pau Tarau Department of Computer Science and Engineering Uniersity of North Texas CLA 2017 Research supported by NSF grant Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

2 Motiation simpy-typed cosed ambda terms enjoy a number of nice properties, but the study of their combinatoria properties is notoriousy hard the two most striking things when inferring types: non-monotonicity: crossing a ambda increases the size of the type, whie crossings an appication node trims it down agreement ia unification (with occurs check) between the types of each ariabe under a ambda as a methodica work-around: can we unwrap some of new obserabes that highight interesting statistica properties? going deeper in the structure of the term than what their CF-syntax reeas we need abstraction mechanisms that forget properties of the difficut cass (simpy-typed cosed ambda terms) to reea equiaence casses that might be easier to grasp k-coored Motzkin skeetons bijections with simper things: from binary trees to 2-coored Motzkin trees Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

3 Outine 1 A bijection between 2-coored Motzkin trees and non-empty binary trees 2 Motzkin skeetons for cosed, affine and inear terms 3 K-coored ambda terms and type inference 4 Type-hoding and type-repeing skeetons 5 An appication to random ambda term generators 6 Reated work 7 Concusions Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

4 Making ambda terms (somewhat) more coorfu 2-coored Motzkin trees: the free agebra generated by the constructors /0, /1, r/1 and a/2 we spit ambda constructors into 2 casses: binding ambdas, /1 that are reached by at east one de Bruijn index free ambdas, r/1, that cannot be reached by any de Bruijn index a side note: free ambda terms wi hae no say on terms being cosed, but they wi impact on which cosed terms are simpy typed ambda terms in de Bruijn form: as the free agebra generated by the constructors /1, r/1 and a/2 with eaes abeed with natura numbers (and seen as wrapped with the constructor /1 when conenient) 2-coored Motzkin skeeton of a ambda term: the tree obtained by erasing the de Bruijn indices abeing their eaes Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

5 A bijection between 2-coored Motzkin trees and non-empty binary trees Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

6 A bijection between 2-coored Motzkin trees and non-empty binary trees binary trees: the free agebra generated by e/0 and c/2 bijections between the Cataan famiy of combinatoria objects and 2-coored Motzkin trees : they exist but they inoe artificia constructs (e.g., depth first search on rose-trees, indirect definition of the mapping) a reason to find a new one! In Proog we need one reation for f and f 1 : cat_mot(c(e,e),). cat_mot(c(x,e),(a)):-x=c(_,_),cat_mot(x,a). cat_mot(c(e,y),r(b)):-y=c(_,_),cat_mot(y,b). cat_mot(c(x,y),a(a,b)):-x=c(_,_),y=c(_,_), cat_mot(x,a), cat_mot(y,b). one can incude the empty tree e by an arithmetization mechanism that defines successor and predecessor (e.g., our PPDP 15 paper) Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

7 The 2-Motzkin to non-empty binary trees bijection at work two twinned trees of size 4 c c c e e e c e c e e a r two twinned trees size 10 c c c c c e c e e e c c c e e e c c e e e e e r r a r a r r Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

8 Motzkin skeetons for cosed, affine and inear terms Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

9 Generating 2-Motzkin trees size of constructors is their arity a=2, =1, r=1, =0 in fact, this size definition matches the size of their heap representation generate a the spits of the size N into 2*A+L+R where A=size of a/2 etc. define predicates to consume size units as needed: Dec(c(SL,R,A),c(L,R,A)):-succ(L,SL). rdec(c(l,sr,a),c(l,r,a)):-succ(r,sr). adec(c(l,r,sa),c(l,r,a)):-succ(a,sa). Proposition If a Motzkin tree is a skeeton of a cosed ambda term then it exists at east one ambda binder on each path from the eaf to the root. Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

10 Cosabe and uncosabe Motzkin trees there are sighty more uncosabe Motzkin trees than cosabe ones as size grows: cosabe: 0,1,1,2,5,11,26,65,163,417,1086,2858,7599,20391,55127,150028,410719,... uncosabe: 1,0,1,2,4,10,25,62,160,418,1102,2940,7912,21444,58507,160544,442748,... Possiby easy open probem: what happens to them asymptoticay? Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

11 Generating cosed affine terms aflam(n,t):-sum_to(n,hi,lo),has_enough_ambdas(hi),aflinlam(t,[],hi,lo) has_enough_ambdas(c(l,_,a)):-succ(a,l). aflinlam((x),[x])-->[]. aflinlam((x,a),vs)-->dec,aflinlam(a,[x Vs]). aflinlam(r(a),vs)-->rdec,aflinlam(a,vs). aflinlam(a(a,b),vs)-->adec,{subset_and_compement_of(vs,as,bs)}, aflinlam(a,as), % a ambda cannot go aflinlam(b,bs). % to both branches!!! subset_and_compement_of([],[],[]). subset_and_compement_of([x Xs],NewYs,NewZs):- subset_and_compement_of(xs,ys,zs), pace_eement(x,ys,zs,newys,newzs). pace_eement(x,ys,zs,[x Ys],Zs). pace_eement(x,ys,zs,ys,[x Zs]). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

12 Linear terms, skeetons of affine and inear terms inlam(n,t):-n mod 3=:=1, sum_to(n,hi,lo),has_no_unused(hi), aflinlam(t,[],hi,lo). has_no_unused(c(l,0,a)):-succ(a,l). Proposition If a Motzkin tree with n binary nodes is a skeeton of a inear ambda term, then it has exacty n + 1 unary nodes, with one on each path from the root to its n + 1 eaes. affine: 0,1,2,3,9,30,81,242,838,2799,9365,33616,122937, inear: 0,1,0,0,5,0,0,60,0,0,1105,0,0,27120,0,0, Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

13 K-coored ambda terms and type inference Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

14 K-coored cosed ambda terms kcooredcosed(n,x):-kcooredcosed(x,[],n,0). kcooredcosed((i),vs)-->{nth0(i,vs,v),inc_ar(v)}. kcooredcosed((k,a),vs)-->, % <= the K-coored ambda binder kcooredcosed(a,[v Vs]), {cose_ar(v,k)}. kcooredcosed(a(a,b),vs)-->a, kcooredcosed(a,vs), kcooredcosed(b,vs). (SX,X):-succ(X,SX). % <= the size-consuming /2 and a/2 predicates a-->,. inc_ar(x):-ar(x),!,x=s(_). % count new ariabe under the binder inc_ar(s(x)):-inc_ar(x). cose_ar(x,k):-ar(x),!,k=0. % cose and conert to ordinary numbers cose_ar(s(x),sk):-cose_ar(x,k),succ(k,sk). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

15 Exampes of k-coored cosed ambda terms 3-coored ambda terms of size 3, exhibiting coors 0,1,2.?- kcooredcosed(3,x). X = (0, (0, (1, (0)))) ; X = (0, (1, (0, (1)))) ; X = (1, (0, (0, (2)))) ; X = (2, a((0), (0))). in a tree with n appication nodes, the counts of k-coored ambdas must sum up to n + 1 we can generate a binary tree and then decorate it with ambdas satisfying this constraint the constraint hods for subtrees, recursiey open experiment : can this mechanism reduce the amount of backtracking and acceerate term generation? Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

16 Two 3-coored ambda trees and their types : Z U Z a Y X Y X : Z V U Z a Y X Y X Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

17 Type inference for k-coored terms simpytypedcoored(n,x,t):-simpytypedcoored(x,t,[],n,0). simpytypedcoored((x),t,vss)-->{ member(vs:t0,vss), unify_with_occurs_check(t,t0), addtobinder(vs,x) }. simpytypedcoored((vs,a),s->t,vss)-->, simpytypedcoored(a,t,[vs:s Vss]), {cosebinder(vs)}. simpytypedcoored(a(a,b),t,vss)-->a, simpytypedcoored(a,(s->t),vss), simpytypedcoored(b,s,vss). addtobinder(ps,p):-ar(ps),!,ps=[p _]. addtobinder([_ Ps],P):-addToBinder(Ps,P). cosebinder(xs):-append(xs,[],_),!. Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

18 Counts for cosed 2-coored simpy-typed terms (i.e., affine) simpy typed cosed terms s. affine cosed terms (og. scae) simpy typed cosed terms affine cosed terms size Figure: Simpy typed cosed terms and affine cosed terms by increasing sizes Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

19 Type-hoding and type-repeing skeetons Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

20 Skeetons of cosed simpy-typed terms toskes((_),,). toskes((vs,a),(k,cs),(s)):-ength(vs,k),toskes(a,cs,s). toskes(a(a,b),a(ca,cb),a(sa,sb)):-toskes(a,ca,sa),toskes(b,cb,sb). generators for skeetons and k-coored skeetons by combining the generator simpytypedcoored with toskeeton gentypedskes(n,cs,s):-gentypedskes(n,_,_,cs,s). gentypedskes(n,x,t,cs,s):-simpytypedcoored(n,x,t),toskes(x,cs,s). typabecoskes(n,cs):-gentypedskes(n,cs,_). typabeskes(n,s):-gentypedskes(n,_,s). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

21 Are coors growing proportiona to the og of their type-sizes? 8 ag. coors and type sizes 6 4 ag. coors ag. type sizes term size Figure: Growth of coors and type sizes Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

22 a most coorfu term: reaches the maximum number of coors max coors max type size term size Figure: Coors of a most coorfu term s. its maximum type size Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

23 Does a most coorfu term reach max type size? type size 40 max coorfu type max type term size Figure: Largest type size of a most coorfu term s. argest type size Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

24 Type-hoding and type-repeing Motzkin trees a type-hoding Motzkin tree is one for which it exists a simpy-typed cosed term haing it as its skeeton a type-repeing Motzkin tree is one for which no simpy-typed cosed term exists haing it as its skeeton to efficienty generate these skeetons we wi spit the generation of ambda terms in two stages the first stage wi generate the unification equations that need to be soed for type inference as we as the ready to be fied out ambda trees it is conenient to actuay generate on the fy the code to be executed in the second stage the second stage wi just use Proog s metaca to actiate this code Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

25 Generating the executabe equation set we generate a ready to run conjunction of unification constraints geneqs(n,x,t,eqs):-geneqs(x,t,[],eqs,true,n,0). geneqs((i),v,[v0 Vs],Es1,Es2)-->{add_eq(Vs,V0,V,I,Es1,Es2)}. geneqs((a),(s->t),vs,es1,es2)-->,geneqs(a,t,[s Vs],Es1,Es2). geneqs(a(a,b),t,vs,es1,es3)-->a, geneqs(a,(s->t),vs,es1,es2), geneqs(b,s,vs,es2,es3). % soe this equation as it can either succeed once, or fai add_eq([],v0,v,0,es,es):-unify_with_occurs_check(v0,v). % <== add_eq([v1 Vs],V0,V,I,(e([V0,V1 Vs],V,0,I),Es),Es). e(i,vs,v):-e(vs,v,0,i). e([v0 _],V,N,N):-unify_with_occurs_check(V0,V). e([_ Vs],V,N1,N3):-succ(N1,N2),e(Vs,V,N2,N3). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

26 Efficient generation of type-hoding and type-repeing skeetons we soe the equations for 2-coored terms to aoid backtracking Motzkin trees: 1,1,2,4,9,21,51,127,323,835,2188 Type equation trees: 0,1,1,1,5, 9,17, 55,122,289, 828 we can generate efficienty type-hoding and type-repeing skeetons type repeing: Eqs hae no soution (their negation succeeds) type hoding: Eqs hae at east one soution (no need to compute a) untypabeske(n,ske):-geneqs(n,x,_,eqs),not(eqs),tomotske(x,ske). typabeske(n,ske):-geneqs(n,x,_,eqs),once(eqs),tomotske(x,ske). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

27 Are there uniquey typabe skeetons? we compute efficienty such skeetons by generating the decoration code Eqs that we constrain to hae exacty one soution when actiated uniqueytypabeske(n,ske):- geneqs(n,x,_,eqs),succeeds_once(eqs),eqs, tomotske(x,ske). succeeds_once(g):-findnsos(2,_,g,sos),!,sos=[_]. not ery many: 0,1,0,0,2,0,1,7,1,13,34,20,100,226,234 open probem: proe that they exist for a sizes Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

28 Growth of uniquey typabe skeetons 800 number of uniquey typabe skeetons Figure: 3 Growth of the number of uniquey typabe skeetons size Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

29 3 type-hoding and 3 type-repeing Motzkin trees a a a a a a a a a a a a a a a Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

30 Growth of the number of skeetons is simiar on a og-scae 10 6 skeeton counts (og. scae) type-hoding skeetons type-repeing skeetons skeeton size Figure: type-hoding s. type-repeing skeetons up to size 20 Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

31 Counts of type repeing and type hoding Motzkin skeetons term size type hoding skeetons type repeing skeetons Figure: Number of type-hoding and type repeing skeetons Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

32 Growth rates of type-hoding and type-repeing skeetons 5 4 skeeton growth rates 3 2 type-hoding skeetons type-repeing skeetons skeeton size Figure: The simiar growth rates of type-hoding and type-repeing skeetons Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

33 Motzkin trees s. their subset of type-hoding skeetons 10 6 Motzkin trees and type-hoding skeetons (og. scae) Motzkin trees type-hoding skeetons size Figure: Counts for Motzkin trees and type-hoding skeetons for increasing term sizes Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

34 Counts of simpy typed cosed terms s. their skeetons simpy typed cosed terms and their skeetons (og. scae) simpy typed cosed terms type-hoding skeetons size Figure: Counts of simpy typed cosed terms and their skeetons by increasing sizes Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

35 An appication to random ambda term generators Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

36 An appication to random ambda term generators Rémy s agorithm: exact size uniformy random binary trees binary trees - bijection to 2-coored Motzkin trees decorating 2-coored Motzkin trees to ambda terms Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

37 Reisiting Rémy s agorithm, decaratiey émy s origina agorithm [7] grows binary trees by grafting new eaes with equa probabiity for each node in a gien tree an eegant procedura impementation is gien in [?] as agorithm R, by using destructie assignments in an array representing the tree we wi work with edges instead of nodes and graft new edges at each step a two stage agorithm: first sets of edges, then trees represented as Proog terms some magic with ogic ariabes whie the agorithm handes terms with thousands of nodes its aerage performance is sighty aboe inear as it takes time proportiona to the size of the set of edges to pick the one to be expanded Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

38 Same code for a-terms and random term generation initia set of two edges remy_init([e(eft,a,_),e(right,a,_)]). same code for a-term or random-term generation except for defining a random choice of the edges or a backtracking one, by repacing choice_of/2 with its commented out aternatie eft_or_right(i,j):-choice_of(2,dice),eft_or_right(dice,i,j). choice_of(n,k):-k is random(n). % choice_of(n,k):-n>0,n1 is N-1,between(0,N1,K). eft_or_right(0,eft,right). eft_or_right(1,right,eft). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

39 The grafting step we can grow a new edge by spitting an existing edge in two grow(e(lr,a,b), e(lr,a,c),e(i,c,_),e(j,c,b)):-eft_or_right(i,j). we add three new edges corresponding to arguments 2, 3 and 4 we remoe one, represented as the first argument contrary to Rémy s origina agorithm, our tree grows downward as new edges are inserted at the target of existing ones Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

40 Finding the spot where we graft chose the grafting point s position remy_step(es,newes,l,newl):-newl is L+2, choice_of(l,dice), remy_step1(dice,es,newes). perform the graft remy_step1(0,[u Xs],[X,Y,Z Xs]):-grow(U, X,Y,Z). remy_step1(d,[a Xs],[A Ys]):-D>0, D1 is D-1, remy_step1(d1,xs,ys). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

41 The iteration to desired size The predicate remy_oop iterates oer remy_step unti the desired 2K size is reached, in K steps we grow by 2 edges at each step, 2K edges tota remy_oop(0,[],n,n). remy_oop(1,es,n1,n2):-n2 is N1+2,remy_init(Es). remy_oop(k,newes,n1,n3):-k>1,k1 is K-1, remy_oop(k1,es,n1,n2), remy_step(es,newes,n2,n3). an exampe of output?- remy_oop(2,edges,0,n). Edges = [e(eft, A, B), e(right, A,C), e(right,c,d), e(eft,c,e)], N = 4. Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

42 From sets of edges to trees as Proog terms the predicate bind_nodes/2 iterates oer edges, and for each interna node it binds it with terms proided by the constructor c/2, eft or right, depending on the type of the edge bind_nodes([],e). bind_nodes([x Xs],Root):-X=e(_,Root,_), mapist(bind_interna,[x Xs]), mapist(bind_eaf,[x Xs]). bind an interna node with constructor c/2 bind_interna(e(eft,c(a,_),a)). bind_interna(e(right,c(_,b),b)). bind a eaf node with the constant /0 bind_eaf(e(_,_,leaf)):-leaf=e->true;true. Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

43 Running the agorithm the predicate remy_term/2 puts the two main steps together remy_term(k,b):-remy_oop(k,es,0,_),bind_nodes(es,b). uniformy random generation of a random term with 4 interna nodes and timings for a arge tree:?- remy_term(4,t). T = c(c(e, e), c(c(e, e), e)).?- time(remy_term(1000,_)). 526,895 inferences, CPU in 0.078s (7,995,978 Lips) we obtain a 2-Motzkin tree generator from the binary tree generator ia the bijection gien by the (bi-directiona) predicate cat_mot/2 mot_gen(n,m):-n>0,remy_term(n,c),cat_mot(c,m). this is unikey to be a uniformy random generator as /1 nodes coer a coors except coor 0 coered by r/1, and, ia the bijection, the tota count is a Cataan number rather than a Motzkin number Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

44 Decorating Motzkin trees to ambda terms if one wants uniformy random Motzkin trees a Botzmann samper or one based on ia hoonomic equations can be used we can mimic (actuay in a stronger way) the ideas behind the natura size that faors ariabes coser to their binders one can buid for each ist of binders from a eaf to the root, a distribution that decays exponentiay with each step nat2probs(0,[]). nat2probs(n,ps):-n>0,sum is 1-1/2^N,Last is 1-Sum,Inc is Last/N, make_probs(n,inc,1,ps). at each step, the probabiity to continue further is reduced to haf make_probs(0,_,_,[]). make_probs(k,inc,p0,[p Ps]):-K>0,K1 is K-1,P1 is P0/2, P is P1+Inc, make_probs(k1,inc,p1,ps). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

45 The decoration agorithm gien a Motzkin tree, we decorate each eaf /0 by turning it into a natura number representing a de Bruijn index the aue of the de Bruijn index is determined for each eaf independenty by waking up on the chain of ambda binders with decaying probabiities decorate(,0,(x))-->[x]. % free ariabe decorate(,n,k)-->{n>0,nat2probs(n,ps),wak_probs(ps,0,k)},[]. decorate((x),n,(y))-->{n1 is N+1},decorate(X,N1,Y). decorate(r(x),n,r(y))-->decorate(x,n,y). decorate(a(x,y),n,a(a,b))-->decorate(x,n,a),decorate(y,n,b). Pain ambda terms are generated as foows: pain_gen(n,t,freevars):-mot_gen(n,b),decorate(b,0,t,freevars,[]). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

46 Retrying for cosed terms to ensure that a term is cosed, we restarts unti the ist of free ariabes is empty we aso returning the number of retries cosed_gen(n,t,i):- Lim is 100+2^min(N,24),try_cosed_gen(Lim,0,I,N restarts are managed by the predicate try_cosed_gen/5, which, when the decorated term is not cosed, tries generating a new term try_cosed_gen(lim,i,j,n,t):- I<Lim, ( mot_gen(n,b),decorate(b,0,t,[],[])*->j=i ; I1 is I+1, try_cosed_gen(lim,i1,j,n,t) ). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

47 Exampe of generation of a random cosed term random cosed ambda terms obtained by decorating motzkin trees.?- cosed_gen(10,t,i). T = (((r(r(r(a((2), 2))))))),?- cosed_gen(2000,_,i). I = 3. with a size definition that counts ariabes as being of size 0 or 1, these ambda terms are actuay of exacty the same size as their Motzkin tree skeetons Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

48 Generating random simpy typed terms as before, we decorate the 2-Motzkin trees with de Bruijn indices we intereae the decoration process with eary rejection of types that do not unify or de Bruijn indices that ead to terms that are not cosed interesting size definitions depend mosty on the weight we attach to the de Bruijn indices we customize the code to pug-in a size definition of our choice in fact, one can use statistics from rea programs to mimic any distribution of the de Bruijn indices inchoice(k,ts,i,t0):-k>0,i is random(k),nth0(i,ts,t0). expchoice(k,ts,i,t):-k>0,n is 2^(K-1), R is random(n),n1 is N>>1, expchoice1(n1,r,ts,t,0,i). expchoice1(n,r,[x _],Y,I,I):-R>=N,!,Y=X. expchoice1(n,r,[_ Xs],Y,I1,I3):-N1 is N>>1,succ(I1,I2), expchoice1(n1,r,xs,y,i2,i3). Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

49 The random simpy-typed cosed ambda term generator decoratetyped(m,x,t):-decoratetyped(m,x,t,0,[]). decoratetyped(,(i),t,k,ts):- inchoice(k,ts,i,t0), % <= pug-in here the size definition! unify_with_occurs_check(t,t0). decoratetyped((x),(a),(s->t),n,ts):-succ(n,sn), decoratetyped(x,a,t,sn,[s Ts]). decoratetyped(r(x),r(a),(_->t),n,ts):- decoratetyped(x,a,t,n,ts). decoratetyped(a(x,y),a(a,b),t,n,ts):- decoratetyped(x,a,(s->t),n,ts), decoratetyped(y,b,s,n,ts). rantyped(n,maxi,maxj,x,t,i,j):- between(1,maxi,i), mot_gen(n,mot),%ppp(i=mot), between(1,maxj,j), decoratetyped(mot,x,t),!. Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

50 Exampe of random simpy typed term of natura size 8 r a r r 1 0 U V U X Z Y X steps(1*2) natsize(8)+heapsize(7)+tsize(6) Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

51 Running the random simpy-typed term generator Can we generate terms of natura size 1000??- rantyped(400). % 11,982,837 inferences, CPU in seconds (99% CPU, big boring term here... steps(64*43) natsize(486)+heapsize(399)+tsize(146)?- rantyped(400). % 122,666,661 inferences, CPU in seconds (99% CPU, big boring term here... steps(644*10) natsize(1162)+heapsize(399)+tsize(21) Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

52 Reated work Bacher, A., Bodini, O., Jacquot, A.: Exact-size Samping for Motzkin Trees in Linear Time ia Botzmann Sampers and Hoonomic Specification. In Nebe, M.E., Szpankowski, W., eds.: 2013 Proceedings of the Tenth Workshop on Anaytic Agorithmics and Combinatorics (ANALCO), SIAM (2013) Tarau, P.: A hiking trip through the orders of magnitude: Deriing efficient generators for cosed simpy-typed ambda terms and norma forms. CoRR abs/ (2016) Bendkowski, M., Grygie, K., Tarau, P.: Botzmann sampers for cosed simpy-typed ambda terms. In Lierer, Y., Taha, W., eds.: Practica Aspects of Decaratie Languages - 19th Internationa Symposium, PADL 2017, Paris, France, January 16-17, 2017, Proceedings. Voume of Lecture Notes in Computer Science., Springer (2017) Deutsch, E., Shapiro, L.W.: A bijection between ordered trees and 2-motzkin paths and its many consequences. Discrete Mathematics 256(3) (2002) Lescanne, P.: Quantitatie aspects of inear and affine cosed ambda terms. CoRR abs/ (2017) Tarau, P.: On Logic Programming Representations of Lambda Terms: de Bruijn Indices, Compression, Type Inference, Combinatoria Generation, Normaization. In Pontei, E., Son, T.C., eds.: Proceedings of the Seenteenth Internationa Symposium on Practica Aspects of Decaratie Languages PADL 15, Portand, Oregon, USA, Springer, LNCS 8131 (June 2015) Rémy, J.L.: Un procédé itératif de dénombrement d arbres binaires et son appication à eur génération aéatoire. RAIRO - Theoretica Informatics and Appications - Informatique Théorique et Appications 19(2) (1985) Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

53 Concusions we hae introduced abstraction mechanisms that forget properties of the difficut cass of simpy-typed ambda terms, to reea interesting structura properties k-coored terms subsume inear and affine terms and are ikey to be usabe to fine-tune random generators to more cosey match coor-distributions in ambda terms representing rea programs type-repeing skeetons might be usefu for efficient agorithms buit on aoiding a sma type-repeing skeetons stored in a database the new bijection between binary terms and 2-coored Motzkin terms and the generation agorithms centered on the distinction between free and binding ambda constructors has been usefu to acceerate generation of affine and inear terms and random terms ia Rémy s agorithm the toos used: a anguage as simpe as (mosty) Horn Cause Proog can hande eeganty combinatoria generation probems when the synergy between sound unification, backtracking and DCGs is put at work Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54

54 Questions? sides: draft paper: code: Picasso: Questions tempt you to te ies, particuary when there is no answer. Pau Tarau (Uniersity of North Texas) Lambda-term skeetons and simpy-typed terms CLA / 54