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Lambda terms and maps, formally (II) Catherine Dubois 1 and Alain Giorgetti 2 1 Samovar, ENSIIE, Evry, France 2 FEMTO-ST, Univ. Bourgogne Franche-Comt e, Besan con, France Computational Logic and Applications Paris, 24-25 May 2018 Dubois

  1. Introduction Blooms λ terms Maps Conclusion Exhaustive generation with Prolog bloom(b0,0). bloom(b1(S,I,B),N) :- N > 0, M is N-1, bloom(B,M), in(S,0,1), Imax is 3*N-2, in(I,0,Imax). bloom(b2(B1,B2),N) :- N > 0, M is N-1, in(N1,0,M), bloom(B1,N1), N2 is M-N1, bloom(B2,N2). bloom( B , N ) holds iff B is a bloom with N unary and binary nodes (size) Dubois & Giorgetti Lambda terms and maps, formally (II) 11 / 59

  2. Introduction Blooms λ terms Maps Conclusion Exhaustive generation with Prolog bloom(b0,0). bloom(b1(S,I,B),N) :- N > 0, M is N-1, bloom(B,M), in(S,0,1), Imax is 3*N-2, in(I,0,Imax). bloom(b2(B1,B2),N) :- N > 0, M is N-1, in(N1,0,M), bloom(B1,N1), N2 is M-N1, bloom(B2,N2). bloom( B , N ) holds iff B is a bloom with N unary and binary nodes (size) � Exhaustive generator for free: ?- bloom(B,2). ?- bloom(B,0). B = b1(1, 4, b1(1, 1, b0)) . B = b0 ; B = b1(1, 3, b1(1, 1, b0)) ; B = b1(1, 2, b1(1, 1, b0)) ; ?- bloom(B,1). B = b1(1, 1, b1(1, 1, b0)) ; B = b1(1, 1, b0) ; ... B = b1(1, 0, b0) ; B = b2(b1(1, 1, b0), b0) ; B = b1(0, 1, b0) ; B = b2(b1(1, 0, b0), b0) ; B = b1(0, 0, b0) ; B = b2(b1(0, 1, b0), b0) ; B = b2(b0, b0) ; ... Dubois & Giorgetti Lambda terms and maps, formally (II) 11 / 59

  3. Introduction Blooms λ terms Maps Conclusion Property checking by counting :- compile(measure). swipl library written by Valerio Senni [ , Senni, GASCom’12] Dubois & Giorgetti Lambda terms and maps, formally (II) 12 / 59

  4. Introduction Blooms λ terms Maps Conclusion Property checking by counting :- compile(measure). swipl library written by Valerio Senni [ , Senni, GASCom’12] ◮ First numbers ?- iterate(0,6,bloom). � 1,5,60,1105,27120,828250,30220800 (in less than 1 minute) Blooms are counted by the OEIS sequence A062980 Dubois & Giorgetti Lambda terms and maps, formally (II) 12 / 59

  5. Introduction Blooms λ terms Maps Conclusion Property checking by counting :- compile(measure). swipl library written by Valerio Senni [ , Senni, GASCom’12] ◮ First numbers ?- iterate(0,6,bloom). � 1,5,60,1105,27120,828250,30220800 (in less than 1 minute) Blooms are counted by the OEIS sequence A062980 ◮ The size of a bloom is its number of unary and binary nodes size(b0,0). size(b1(_,_,B),NP1) :- size(B,N), NP1 is N+1. size(b2(B1,B2),N) :- size(B1,N1), size(B2,N2), N is N1+N2+1. bloom12([B,N],N) :- bloom(B,N), size(B,N). ?- iterate(0,6,bloom12). � same sequence Dubois & Giorgetti Lambda terms and maps, formally (II) 12 / 59

  6. Introduction Blooms λ terms Maps Conclusion Property-based testing in Coq: QuickChick ◮ QuickChick: plugin for Coq for testing conjectures (B. Pierce, C. Hrit ¸cu, L. Lampropoulos, Z. Paraskevopoulou, https://github.com/QuickChick , 2014) - inspired by QuickCheck for Haskell (2000) - similar tools exist for Agda, Isabelle, PVS, FoCaLiZe . . . ◮ Lemma L: ∀ x: T, precondition x → conclusion x where precondition and conclusion : T → Prop are logical predicates ◮ How to test lemma L (before proving)? 1. Define a random generator gen T of values of type T that satisfy the property precondition 2. Prove that the random generator is correct � we test it 3. Turn conclusion into a Boolean function conclusionb : T → bool 4. Run the test on, e.g., 10000 test cases: QuickCheck (forAll gen_T (fun x ⇒ conclusionb x) 5. To reduce the trusted base, prove that conclusion and conclusionb are semantically equivalent Dubois & Giorgetti Lambda terms and maps, formally (II) 13 / 59

  7. Introduction Blooms λ terms Maps Conclusion Random generation of blooms with QuickChick Random generator of values of type (bloom n) (follows the inductive definition of the type bloom ) Program Fixpoint gen_bloom_n (n : nat) {measure n}: G (bloom n) := match n with 0 ⇒ returnGen b0 | S m ⇒ oneof ... [ (bindGen ’ (choose (0, m)) (fun e1 H ⇒ liftGen2 (@b2 m e1 H) (gen_bloom_n e1) (gen_bloom_n (m - e1 )))); (bindGen ’ (choose (0, 3*m+1)) (fun i H ⇒ genBool (fun s ⇒ bindGen liftGen (@b1 m s i _) (gen_bloom_n m)))) ] end. Dubois & Giorgetti Lambda terms and maps, formally (II) 14 / 59

  8. Introduction Blooms λ terms Maps Conclusion Random generator validation How to validate the random generator? Dubois & Giorgetti Lambda terms and maps, formally (II) 15 / 59

  9. Introduction Blooms λ terms Maps Conclusion Random generator validation How to validate the random generator? By testing some properties, like the ones checked with Prolog (e.g., the size of a bloom is its number of unary and binary nodes) QuickCheck (sized (fun n ⇒ forAll (gen_bloom_n n) (fun t ⇒ beq_nat (size n t) n))). +++ Passed 10000 tests (0 discards) sized : iterates over different values for n Dubois & Giorgetti Lambda terms and maps, formally (II) 15 / 59

  10. Introduction Blooms λ terms Maps Conclusion Random generator validation How to validate the random generator? By testing some properties, like the ones checked with Prolog (e.g., the size of a bloom is its number of unary and binary nodes) QuickCheck (sized (fun n ⇒ forAll (gen_bloom_n n) (fun t ⇒ beq_nat (size n t) n))). +++ Passed 10000 tests (0 discards) sized : iterates over different values for n Properties also proven in Coq (easy, by induction) Lemma bloom_size: ∀ n (t : bloom n), size n t = n. Dubois & Giorgetti Lambda terms and maps, formally (II) 15 / 59

  11. Introduction Blooms λ terms Maps Conclusion Blooms in the middle λ terms Blooms ATM 2 λ x bloomlam bloomatm2 b 1 (? , ? , b 0 ) @ lambloom atm2bloom λ y x y Dubois & Giorgetti Lambda terms and maps, formally (II) 16 / 59

  12. Introduction Blooms λ terms Maps Conclusion Blooms in the middle, with rooted trivalent maps Rooted trivalent maps (RTM): all vertices have degree 3 Dubois & Giorgetti Lambda terms and maps, formally (II) 17 / 59

  13. Introduction Blooms λ terms Maps Conclusion Blooms in the middle, with rooted trivalent maps Rooted trivalent maps (RTM): all vertices have degree 3 λ terms Blooms RTM λ x bloomlam bloomrtm b 1 (1 , 0 , b 0 ) @ lambloom rtmbloom λ y x y Dubois & Giorgetti Lambda terms and maps, formally (II) 17 / 59

  14. Introduction Blooms λ terms Maps Conclusion Outline Introduction 1 Blooms 2 Lambda Semantics 3 Map Semantics 4 Summary and Conclusion 5 Dubois & Giorgetti Lambda terms and maps, formally (II) 18 / 59

  15. Introduction Blooms λ terms Maps Conclusion CLLT: Coq definition Inductive lam : Set := Abs : variable → lam → lam | App : lam → lam → lam | Var : variable → lam. Definition is_closed t := (fv t) = [ ]. Definition is_lin t := noDup (uvar t) ∧ nbAbs t = nbVar t Definition is_cllt t := distinctabs t ∧ is_closed t ∧ is_lin t. A CLLT is a λ -term closed with distinct bound variables such that the list of used variables has no duplicates and such that the number of abstractions is equal to the number of variables Dubois & Giorgetti Lambda terms and maps, formally (II) 19 / 59

  16. Introduction Blooms λ terms Maps Conclusion CLLT: Coq definition Inductive lam : Set := Abs : variable → lam → lam | App : lam → lam → lam | Var : variable → lam. Definition is_closed t := (fv t) = [ ]. Definition is_lin t := noDup (uvar t) ∧ nbAbs t = nbVar t Definition is_cllt t := distinctabs t ∧ is_closed t ∧ is_lin t. A CLLT is a λ -term closed with distinct bound variables such that the list of used variables has no duplicates and such that the number of abstractions is equal to the number of variables Redundant, but may help for proving and can be checked with Prolog Dubois & Giorgetti Lambda terms and maps, formally (II) 19 / 59

  17. Introduction Blooms λ terms Maps Conclusion CLLT: Coq random generator (gen cllt n rk fvar) generates a λ -term with n abstractions named rk , rk +1, . . . and free variables in fvar ◮ may fail (when n = 0 and | fvar | > 0) ◮ uses the QuickChick combinator backtrack We test the outputs have distinct bound variables, no duplicates in the used variables, no free variables and as many abstractions as variable terms (fun n ⇒ QuickCheck (sized forAll (gen_cllt n 0 []) is_clltb )). ++ Passed 10000 tests (0 discards) is clltb : Boolean function that verifies its argument is a CLLT (proved as equivalent with the logical predicate is cllt ) Dubois & Giorgetti Lambda terms and maps, formally (II) 20 / 59

  18. Introduction Blooms λ terms Maps Conclusion CLLT: Coq theorems Application preserves closed linear terms (with distinct variables) Lemma app_cllt: ∀ t1 t2 , is_cllt t1 → is_cllt t2 → disjoint (abvar t1) (abvar t2) → is_cllt (App t1 t2). Dubois & Giorgetti Lambda terms and maps, formally (II) 21 / 59

  19. Introduction Blooms λ terms Maps Conclusion From blooms to lambda terms b 0 λ x . x @ L ′ L b 2 B B ′ where B � L and B ′ � L ′ s = true s = false λ v λ v L [ v M ] i L [ M v ] i @ @ i i v v M M b 1 s i B where B � L and L | i = M Dubois & Giorgetti Lambda terms and maps, formally (II) 22 / 59

  20. Introduction Blooms λ terms Maps Conclusion From blooms to lambda terms: Prolog For generation and counting bloomlam(B,L,V) holds iff L is the lambda semantics of the bloom B , with V as first (highest) variable ◮ Table of correspondence ◮ Check by counting that the lambda terms L are closed and linear bl([B,L],N) :- bloom(B,N), bloomlam(B,L,N), closed(L), uv(L,U), nodup(U), eq(L). :- iterate(0,6,bl). � 1,5,60,1105,27120,828250,30220800 ( https://oeis.org/A062980 ) Dubois & Giorgetti Lambda terms and maps, formally (II) 23 / 59

  21. Introduction Blooms λ terms Maps Conclusion From blooms to lambda terms: Coq ◮ Definition Fixpoint bloomlam_aux: ∀ n : nat , bloom n → variable → option lam := ... Definition bloomlam n t := bloomlam_aux n t n. Dubois & Giorgetti Lambda terms and maps, formally (II) 24 / 59

  22. Introduction Blooms λ terms Maps Conclusion From blooms to lambda terms: Coq ◮ Definition Fixpoint bloomlam_aux: ∀ n : nat , bloom n → variable → option lam := ... Definition bloomlam n t := bloomlam_aux n t n. ◮ Random property-based testing with QuickChick: the λ term obtained from a bloom is a closed linear λ term QuickCheck (sized (fun n ⇒ forAll (gen_bloom_n n) (fun t ⇒ boolopt is_clltb (bloomlam n t)))). +++ Passed 10000 tests (0 discards) Dubois & Giorgetti Lambda terms and maps, formally (II) 24 / 59

  23. Introduction Blooms λ terms Maps Conclusion From blooms to lambda terms: Coq ◮ Definition Fixpoint bloomlam_aux: ∀ n : nat , bloom n → variable → option lam := ... Definition bloomlam n t := bloomlam_aux n t n. ◮ Random property-based testing with QuickChick: the λ term obtained from a bloom is a closed linear λ term QuickCheck (sized (fun n ⇒ forAll (gen_bloom_n n) (fun t ⇒ boolopt is_clltb (bloomlam n t)))). +++ Passed 10000 tests (0 discards) We get a new random CLLT generator by combining the bloom generator and the bloomlam function Dubois & Giorgetti Lambda terms and maps, formally (II) 24 / 59

  24. Introduction Blooms λ terms Maps Conclusion Outline Introduction 1 Blooms 2 Lambda Semantics 3 Map Semantics 4 Summary and Conclusion 5 Dubois & Giorgetti Lambda terms and maps, formally (II) 25 / 59

  25. Introduction Blooms λ terms Maps Conclusion Map semantics of blooms λ terms Blooms RTM λ x bloomlam bloomrtm b 1 (1 , 0 , b 0 ) @ lambloom rtmbloom λ y x y Dubois & Giorgetti Lambda terms and maps, formally (II) 26 / 59

  26. Introduction Blooms λ terms Maps Conclusion From blooms to RTM, visually (1/3) M 1 M 2 B 1 → B 2 → b 0 → M 1 M 2 b 2 B 1 B 2 → Dubois & Giorgetti Lambda terms and maps, formally (II) 27 / 59

  27. Introduction Blooms λ terms Maps Conclusion From blooms to RTM, visually (2/3) M B → M b 1 0 0 B → Dubois & Giorgetti Lambda terms and maps, formally (II) 28 / 59

  28. Introduction Blooms λ terms Maps Conclusion From blooms to RTM, visually (2/3) M M B → B → M M b 1 1 0 B → b 1 0 0 B → Dubois & Giorgetti Lambda terms and maps, formally (II) 28 / 59

  29. Introduction Blooms λ terms Maps Conclusion From blooms to RTM, visually (3/3) M B → M b 1 S I B → For I > 0 The splitted dart is chosen from I (rank in depth-first traversal) and S (orientation) Dubois & Giorgetti Lambda terms and maps, formally (II) 29 / 59

  30. Introduction Blooms λ terms Maps Conclusion How to formalize maps? ◮ As topological objects? � � ◮ As combinatorial objects (transitive couples of permutations on half-edges, modulo edge renaming)? [Dubois, , Zeilberger, CLA’15] ◮ As an inductive structure constructed by edge and vertex additions? � ◮ How to discover such a structure? Dubois & Giorgetti Lambda terms and maps, formally (II) 30 / 59

  31. Introduction Blooms λ terms Maps Conclusion Rooted map analysis and synthesis Analysis by root edge removal T. R. S. Walsh and A. B. Lehman. Counting rooted maps by genus I. J. Comb. Theory, Ser. B, 1972. Dubois & Giorgetti Lambda terms and maps, formally (II) 31 / 59

  32. Introduction Blooms λ terms Maps Conclusion Rooted map analysis and synthesis Analysis by root edge removal Theorem (Rooted map synthesis) Any rooted map M with e ( M ) edges is either the vertex map (no edges) or can be uniquely obtained by adding an edge ◮ either between two rooted maps M 1 and M 2 s.t. e ( M ) = 1 + e ( M 1 ) + e ( M 2 ) , ◮ or to a rooted map M 1 with e ( M 1 ) = e ( M ) − 1 edges in 2 e ( M 1 ) + 1 distinct ways. T. R. S. Walsh and A. B. Lehman. Counting rooted maps by genus I. J. Comb. Theory, Ser. B, 1972. Dubois & Giorgetti Lambda terms and maps, formally (II) 31 / 59

  33. Introduction Blooms λ terms Maps Conclusion Rooted map analysis and synthesis Analysis by root edge removal Theorem (Rooted map synthesis) Any rooted map M with e ( M ) edges is either the vertex map (no edges) or can be uniquely obtained by adding an edge ◮ either between two rooted maps M 1 and M 2 s.t. e ( M ) = 1 + e ( M 1 ) + e ( M 2 ) , ◮ or to a rooted map M 1 with e ( M 1 ) = e ( M ) − 1 edges in 2 e ( M 1 ) + 1 distinct ways. C. Dubois, A. Giorgetti, and R. Genestier. Tests and proofs for enumerative combinatorics. In TAP’16 , volume 6792 of LNCS , pages 57–75. Springer, 2016. ◮ Formal proofs in Coq T. R. S. Walsh and A. B. Lehman. ◮ Definitions and conjectures checked Counting rooted maps by genus I. ◮ Prolog-based bounded-exhaustive testing J. Comb. Theory, Ser. B, 1972. Dubois & Giorgetti Lambda terms and maps, formally (II) 31 / 59

  34. Introduction Blooms λ terms Maps Conclusion Root edge removal/addition Removal (analysis): → deletion → contraction Dubois & Giorgetti Lambda terms and maps, formally (II) 32 / 59

  35. Introduction Blooms λ terms Maps Conclusion Root edge removal/addition Removal (analysis): → deletion → contraction Addition (synthesis): ← drawing ← stretching Dubois & Giorgetti Lambda terms and maps, formally (II) 32 / 59

  36. Introduction Blooms λ terms Maps Conclusion Root edge removal/addition Removal (analysis): → deletion → contraction Addition (synthesis): ← drawing ← stretching ◮ Works well for planar maps [Tutte, 68] and maps with any genus [Walsh & Lehman, 72] Dubois & Giorgetti Lambda terms and maps, formally (II) 32 / 59

  37. Introduction Blooms λ terms Maps Conclusion Root edge removal/addition Removal (analysis): → deletion → contraction Addition (synthesis): ← drawing ← stretching ◮ Works well for planar maps [Tutte, 68] and maps with any genus [Walsh & Lehman, 72] ◮ More difficult for maps with degree constraints: deletion reduces vertex degrees, contraction increases them! ◮ Deletion only (for trivalent maps): Consider all the families of maps with vertex degrees 3 or less? ◮ With contraction: Also consider families of maps with arbitrary vertex degrees? Dubois & Giorgetti Lambda terms and maps, formally (II) 32 / 59

  38. Introduction Blooms λ terms Maps Conclusion Root edge removal/addition Removal (analysis): → deletion → contraction Addition (synthesis): ← drawing ← stretching ◮ Works well for planar maps [Tutte, 68] and maps with any genus [Walsh & Lehman, 72] ◮ More difficult for maps with degree constraints: deletion reduces vertex degrees, contraction increases them! ◮ Deletion only (for trivalent maps): Consider all the families of maps with vertex degrees 3 or less? ◮ With contraction: Also consider families of maps with arbitrary vertex degrees? ◮ Here, a solution for trivalent maps, with only four families of maps Dubois & Giorgetti Lambda terms and maps, formally (II) 32 / 59

  39. Introduction Blooms λ terms Maps Conclusion Method for rooted trivalent maps 1. Map decomposition/analysis by root edge removal 2. Inverse constructions: map synthesis by edge addition 3. Mutually inductive families of maps formalized as (Prolog) terms 4. Validation by counting (OEIS) 5. Interpretation of constructors as operations on combinatorial maps, represented by permutations Dubois & Giorgetti Lambda terms and maps, formally (II) 33 / 59

  40. Introduction Blooms λ terms Maps Conclusion Method for rooted trivalent maps 1. Map decomposition/analysis by root edge removal 2. Inverse constructions: map synthesis by edge addition 3. Mutually inductive families of maps formalized as (Prolog) terms 4. Validation by counting (OEIS) 5. Interpretation of constructors as operations on combinatorial maps, represented by permutations ◮ Done for rooted maps [DGG16] ◮ Here for rooted trivalent maps ◮ Steps 1-3 only for one edge removal/addition Dubois & Giorgetti Lambda terms and maps, formally (II) 33 / 59

  41. Introduction Blooms λ terms Maps Conclusion Method for rooted trivalent maps 1. Map decomposition/analysis by root edge removal 2. Inverse constructions: map synthesis by edge addition 3. Mutually inductive families of maps formalized as (Prolog) terms 4. Validation by counting (OEIS) 5. Independent inductive characterization of each map family by incremental formal composition of up to 3 constructions 6. Interpretation of constructors as operations on combinatorial maps, represented by permutations ◮ Done for rooted maps [DGG16] ◮ Here for rooted trivalent maps ◮ Steps 1-3 only for one edge removal/addition Dubois & Giorgetti Lambda terms and maps, formally (II) 33 / 59

  42. Introduction Blooms λ terms Maps Conclusion Method for rooted trivalent maps 1. Map decomposition/analysis by root edge removal 2. Inverse constructions: map synthesis by edge addition 3. Mutually inductive families of maps formalized as (Prolog) terms 4. Validation by counting (OEIS) 5. Independent inductive characterization of each map family by incremental formal composition of up to 3 constructions 6. Validation by counting (OEIS) at each intermediate step 7. Interpretation of constructors as operations on combinatorial maps, represented by permutations ◮ Done for rooted maps [DGG16] ◮ Here for rooted trivalent maps ◮ Steps 1-3 only for one edge removal/addition Dubois & Giorgetti Lambda terms and maps, formally (II) 33 / 59

  43. Introduction Blooms λ terms Maps Conclusion Global picture split 2 ATM 2 ATM 22 split unary handle isthmic uouter ATM 1 RTM uinner mty ◮ ATM 1 : univalent root vertex (degree 1), other vertices of degree 3 ◮ ATM 22 : ordered pair of roots, two distinct bivalent root vertices Dubois & Giorgetti Lambda terms and maps, formally (II) 34 / 59

  44. Introduction Blooms λ terms Maps Conclusion Construction of ATM 2 split ATM 2 RTM Dubois & Giorgetti Lambda terms and maps, formally (II) 35 / 59

  45. Introduction Blooms λ terms Maps Conclusion Construction of ATM 2 split ATM 2 RTM ◮ For E = 0 , from the vertex map mty = ◮ Example (for E = 4 ) from Dubois & Giorgetti Lambda terms and maps, formally (II) 35 / 59

  46. Introduction Blooms λ terms Maps Conclusion Construction of ATM 2 split ATM 2 RTM ◮ For E = 0 , from the vertex map mty = ◮ Example (for E = 4 ) from atm2(split(M),E) :- E > 0, D is E-1, rtm(M,D). Dubois & Giorgetti Lambda terms and maps, formally (II) 35 / 59

  47. Introduction Blooms λ terms Maps Conclusion Construction of ATM 1 ATM 2 unary ATM 1 Example from atm1(unary(M),E) :- E > 1, D is E-1, atm2(M,D). Dubois & Giorgetti Lambda terms and maps, formally (II) 36 / 59

  48. Introduction Blooms λ terms Maps Conclusion Construction of ATM 22 split 2 ATM 2 ATM 22 Add a (thus bivalent) vertex at the middle of an edge: 7 5 6 -1 -5 -3 -4 -7 -6 1 -2 3 4 From the ATM 2 2 get 1st root 2nd root for I = 3 and 1st root 2nd root for I = -2 atm22(split2(I,M),E) :- E > 0, D is E-1, atm2(M,D), Dopp is -D, in(I,Dopp,D), I =\= 0. Dubois & Giorgetti Lambda terms and maps, formally (II) 37 / 59

  49. Introduction Blooms λ terms Maps Conclusion Construction of RTM: Isthmic case isthmic ◮ Binary isthmic construction of one RTM from two ATM 2 Dubois & Giorgetti Lambda terms and maps, formally (II) 38 / 59

  50. Introduction Blooms λ terms Maps Conclusion Construction of RTM: case 2 first root E X E - X second root J J ATM 22 handle RTM Dubois & Giorgetti Lambda terms and maps, formally (II) 39 / 59

  51. Introduction Blooms λ terms Maps Conclusion Construction of RTM: case 3 → ATM 1 uouter RTM The root face is unary Dubois & Giorgetti Lambda terms and maps, formally (II) 40 / 59

  52. Introduction Blooms λ terms Maps Conclusion Construction of RTM: case 4 → ATM 1 uinner RTM The inner face (incident to the opposite of the root) is unary Dubois & Giorgetti Lambda terms and maps, formally (II) 41 / 59

  53. Introduction Blooms λ terms Maps Conclusion Construction of trivalent maps: 5 cases ATM 2 ATM 22 handle isthmic uouter ATM 1 RTM uinner mty Dubois & Giorgetti Lambda terms and maps, formally (II) 42 / 59

  54. Introduction Blooms λ terms Maps Conclusion Construction of trivalent maps: 5 cases rtm(mty,0). ATM 2 ATM 22 handle isthmic uouter ATM 1 RTM uinner mty Dubois & Giorgetti Lambda terms and maps, formally (II) 42 / 59

  55. Introduction Blooms λ terms Maps Conclusion Construction of trivalent maps: 5 cases rtm(mty,0). ATM 2 ATM 22 rtm(isthmic(M2,U2),E) :- E > 2, D is E-1, in(E1,1,D), atm2(M2,E1), E2 is D-E1, E2 > 0, atm2(U2,E2). handle isthmic uouter ATM 1 RTM uinner mty Dubois & Giorgetti Lambda terms and maps, formally (II) 42 / 59

  56. Introduction Blooms λ terms Maps Conclusion Construction of trivalent maps: 5 cases rtm(mty,0). ATM 2 ATM 22 rtm(isthmic(M2,U2),E) :- E > 2, D is E-1, in(E1,1,D), atm2(M2,E1), E2 is D-E1, E2 > 0, atm2(U2,E2). handle isthmic rtm(uouter(M),E) :- E > 0, D is E-1, atm1(M,D). uouter rtm(uinner(M),E) :- E > 0, ATM 1 RTM D is E-1, atm1(M,D). uinner mty Dubois & Giorgetti Lambda terms and maps, formally (II) 42 / 59

  57. Introduction Blooms λ terms Maps Conclusion Construction of trivalent maps: 5 cases rtm(mty,0). ATM 2 ATM 22 rtm(isthmic(M2,U2),E) :- E > 2, D is E-1, in(E1,1,D), atm2(M2,E1), E2 is D-E1, E2 > 0, atm2(U2,E2). handle isthmic rtm(uouter(M),E) :- E > 0, D is E-1, atm1(M,D). uouter rtm(uinner(M),E) :- E > 0, ATM 1 RTM D is E-1, atm1(M,D). uinner mty rtm(handle(M),E) :- E > 0, D is E-1, atm22(M,D). Dubois & Giorgetti Lambda terms and maps, formally (II) 42 / 59

  58. Introduction Blooms λ terms Maps Conclusion Construction of trivalent maps: 5 cases rtm(mty,0). ATM 2 ATM 22 rtm(isthmic(M2,U2),E) :- E > 2, D is E-1, in(E1,1,D), atm2(M2,E1), E2 is D-E1, E2 > 0, atm2(U2,E2). handle isthmic rtm(uouter(M),E) :- E > 0, D is E-1, atm1(M,D). uouter rtm(uinner(M),E) :- E > 0, ATM 1 RTM D is E-1, atm1(M,D). uinner mty rtm(handle(M),E) :- E > 0, D is E-1, atm22(M,D). Validation by counting: 1, 5, 60, 1105, 27120, 828250, 30220800, in less than 1 minute ( http://oeis.org/A062980 ) Dubois & Giorgetti Lambda terms and maps, formally (II) 42 / 59

  59. Introduction Blooms λ terms Maps Conclusion Where formalization helps split 2 ATM 2 ATM 22 Elimination of intermediate maps split by composition of constructors unary handle isthmic uouter ATM 1 RTM uinner mty Dubois & Giorgetti Lambda terms and maps, formally (II) 43 / 59

  60. Introduction Blooms λ terms Maps Conclusion Where formalization helps split 2 ATM 2 ATM 22 Elimination of intermediate maps split by composition of constructors 1. Composition of split and unary handle unary , detailed isthmic 2. Elimination of ATM 22 3. Elimination of ATM 1 4. Elimination of ATM 2 uouter ATM 1 RTM 5. Put everything together uinner mty Dubois & Giorgetti Lambda terms and maps, formally (II) 43 / 59

  61. Introduction Blooms λ terms Maps Conclusion Step 1: Composition of split and unary Since the family ATM 2 is characterized by atm2(split(M),E) :- E > 0, D is E-1, rtm(M,D). the clause atm1(unary(M),E) :- E > 1, D is E-1, atm2(M,D). can be replaced with ◮ atm1(unary(split(M)),E) :- E > 1, D is E-1, atm2(split(M),D). by case analysis, ◮ atm1(unary(split(M)),E) :- E > 1, D is E-1, D > 0, C is D-1, rtm(M,C). by unfolding, and ◮ atm1(balloon(M),E) :- E > 1, C is E-2, rtm(M,C). by folding, with balloon = split;unary (+ variable elimination) ◮ Validation by counting: again http://oeis.org/A062980 Dubois & Giorgetti Lambda terms and maps, formally (II) 44 / 59

  62. Introduction Blooms λ terms Maps Conclusion New global picture balloon(M) = unary(split(M)) split 2 ATM 2 ATM 22 split isthmic unary handle balloon uouter ATM 1 RTM uinner mty Dubois & Giorgetti Lambda terms and maps, formally (II) 45 / 59

  63. Introduction Blooms λ terms Maps Conclusion Step 2: Elimination of ATM 22 split 2 ATM 2 ATM 22 hook isthmic split handle balloon uouter ATM 1 RTM uinner mty hook = split 2 ;handle Dubois & Giorgetti Lambda terms and maps, formally (II) 46 / 59

  64. Introduction Blooms λ terms Maps Conclusion Elimination of ATM 22 , formally With hook(I,M) = handle(split2(I,M)) and atm22(split2(I,M),E) :- E > 0, D is E-1, atm2(M,D), Max is 2*D, in(I,1,Max). the clause rtm(handle(M),E) :- E > 0, D is E-1, atm22(M,D). becomes rtm(hook(I,M),E) :- E > 1, C is E-2, atm2(M,C), Max is 2*C, in(I,1,Max). Dubois & Giorgetti Lambda terms and maps, formally (II) 47 / 59

  65. Introduction Blooms λ terms Maps Conclusion Step 3: Elimination of ATM 1 ATM 2 hook isthmic split balloon uo, ui uouter ATM 1 RTM uinner mty uo = balloon;uouter and ui = balloon;uinner rtm(uo(M),E) :- E > 2, B is E-3, rtm(M,B). rtm(ui(M),E) :- E > 2, B is E-3, rtm(M,B). Dubois & Giorgetti Lambda terms and maps, formally (II) 48 / 59

  66. Introduction Blooms λ terms Maps Conclusion Step 4: Elimination of ATM 2 ATM 2 hook isthmic split h uo, ui i RTM mty h(I,M) = hook(I,split(M)) i(M1,M2) = isthmic(split(M1),split(M2)) rtm(h(I,M),E) :- E > 2, B is E-3, rtm(M,B), Max is 2*B+2, in(I,1,Max). rtm(i(M1,M2),E) :- E > 2, B is E-3, in(E1,0,B), rtm(M1,E1), E2 is B-E1, rtm(M2,E2). Dubois & Giorgetti Lambda terms and maps, formally (II) 49 / 59

  67. Introduction Blooms λ terms Maps Conclusion Inductive definition of trivalent maps ◮ All together rtm(mty,0). rtm(i(M1,M2),E) :- E > 2, B is E-3, in(E1,0,B), rtm(M1,E1), E2 is B-E1, rtm(M2,E2). rtm(uo(M),E) :- E > 2, B is E-3, rtm(M,B). rtm(ui(M),E) :- E > 2, B is E-3, rtm(M,B). rtm(h(I,M),E) :- E > 2, B is E-3, rtm(M,B), Max is 2*B+2, in(I,1,Max). Dubois & Giorgetti Lambda terms and maps, formally (II) 50 / 59

  68. Introduction Blooms λ terms Maps Conclusion Inductive definition of trivalent maps ◮ All together rtm(mty,0). rtm(i(M1,M2),E) :- E > 2, B is E-3, in(E1,0,B), rtm(M1,E1), E2 is B-E1, rtm(M2,E2). rtm(uo(M),E) :- E > 2, B is E-3, rtm(M,B). rtm(ui(M),E) :- E > 2, B is E-3, rtm(M,B). rtm(h(I,M),E) :- E > 2, B is E-3, rtm(M,B), Max is 2*B+2, in(I,1,Max). ◮ Unary cases merged ( K = I+2 ) rtm(mty,0). rtm(b(M1,M2),E) :- E > 2, B is E-3, in(E1,0,B), rtm(M1,E1), E2 is B-E1, rtm(M2,E2). rtm(u(K,M),E) :- E > 2, B is E-3, rtm(M,B), Max is 2*B+3, in(K,0,Max). Dubois & Giorgetti Lambda terms and maps, formally (II) 50 / 59

  69. Introduction Blooms λ terms Maps Conclusion Inductive definition of trivalent maps ◮ All together rtm(mty,0). rtm(i(M1,M2),E) :- E > 2, B is E-3, in(E1,0,B), rtm(M1,E1), E2 is B-E1, rtm(M2,E2). rtm(uo(M),E) :- E > 2, B is E-3, rtm(M,B). rtm(ui(M),E) :- E > 2, B is E-3, rtm(M,B). rtm(h(I,M),E) :- E > 2, B is E-3, rtm(M,B), Max is 2*B+2, in(I,1,Max). ◮ Unary cases merged ( K = I+2 ) rtm(mty,0). rtm(b(M1,M2),E) :- E > 2, B is E-3, in(E1,0,B), rtm(M1,E1), E2 is B-E1, rtm(M2,E2). rtm(u(K,M),E) :- E > 2, B is E-3, rtm(M,B), Max is 2*B+3, in(K,0,Max). Dubois & Giorgetti Lambda terms and maps, formally (II) 50 / 59

  70. Introduction Blooms λ terms Maps Conclusion Inductive definition of trivalent maps Let n ( M ) = e ( M ) / 3 be the size of a rooted trivalent map M rtm(mty,0). rtm(b(M1,M2),N) :- N > 0, Nm1 is N-1, in(N1,0,Nm1), rtm(M1,N1), N2 is Nm1-N1, rtm(M2,N2). rtm(u(K,M),N) :- N > 0, Nm1 is N-1, rtm(M,Nm1), Max is 6*Nm1+3, in(K,0,Max). Dubois & Giorgetti Lambda terms and maps, formally (II) 51 / 59

  71. Introduction Blooms λ terms Maps Conclusion Inductive definition of trivalent maps Let n ( M ) = e ( M ) / 3 be the size of a rooted trivalent map M rtm(mty,0). rtm(b(M1,M2),N) :- N > 0, Nm1 is N-1, in(N1,0,Nm1), rtm(M1,N1), N2 is Nm1-N1, rtm(M2,N2). rtm(u(K,M),N) :- N > 0, Nm1 is N-1, rtm(M,Nm1), Max is 6*Nm1+3, in(K,0,Max). Theorem (Rooted trivalent map by size) A rooted trivalent map M of size n ( M ) is either the vertex map (size 0 ) or can be uniquely obtained by adding an edge ◮ between two rooted trivalent maps M 1 and M 2 such that n ( M ) = n ( M 1 ) + n ( M 2 ) + 1 , ◮ or to a rooted trivalent map M 1 with n ( M 1 ) = n ( M ) − 1 , in 6 n ( M 1 ) + 4 distinct ways. Dubois & Giorgetti Lambda terms and maps, formally (II) 51 / 59

  72. Introduction Blooms λ terms Maps Conclusion Inductive definition of trivalent maps Let n ( M ) = e ( M ) / 3 be the size of a rooted trivalent map M rtm(mty,0). rtm(b(M1,M2),N) :- N > 0, Nm1 is N-1, in(N1,0,Nm1), rtm(M1,N1), N2 is Nm1-N1, rtm(M2,N2). rtm(u(K,M),N) :- N > 0, Nm1 is N-1, rtm(M,Nm1), Max is 6*Nm1+3, in(K,0,Max). Theorem (Rooted trivalent map by size) A rooted trivalent map M of size n ( M ) is either the vertex map (size 0 ) or can be uniquely obtained by adding an edge ◮ between two rooted trivalent maps M 1 and M 2 such that n ( M ) = n ( M 1 ) + n ( M 2 ) + 1 , ◮ or to a rooted trivalent map M 1 with n ( M 1 ) = n ( M ) − 1 , in 6 n ( M 1 ) + 4 distinct ways. Back to blooms! Dubois & Giorgetti Lambda terms and maps, formally (II) 51 / 59

  73. Introduction Blooms λ terms Maps Conclusion From blooms to RTM, formally (1/3) M 1 M 2 B 1 → B 2 → b 0 → M 1 M 2 b 2 B 1 B 2 → bloomrtm(b0,mty). bloomrtm(b2(B1,B2),isthmic(split(M2),split(M1))) :- bloomrtm(B1,M1), bloomrtm(B2,M2). Dubois & Giorgetti Lambda terms and maps, formally (II) 52 / 59

  74. Introduction Blooms λ terms Maps Conclusion From blooms to RTM, formally (2/3) M M B → B → M M b 1 1 0 B → b 1 0 0 B → bloomrtm(b1(0,0,B),uouter(unary(split(M)))) :- bloomrtm(B,M). bloomrtm(b1(1,0,B),uinner(unary(split(M)))) :- bloomrtm(B,M). Dubois & Giorgetti Lambda terms and maps, formally (II) 53 / 59

  75. Introduction Blooms λ terms Maps Conclusion From blooms to RTM, formally (3/3) M B → M b 1 S I B → bloomrtm(b1(0,I,B),handle(split2(K,split(M)))) :- I > 0, nb01(B,N), K is 3*N-1-I, bloomrtm(B,M). bloomrtm(b1(1,I,B),handle(split2(K,split(M)))) :- I > 0, nb01(B,N), K is -(3*N-1-I), bloomrtm(B,M). Dubois & Giorgetti Lambda terms and maps, formally (II) 54 / 59

  76. Introduction Blooms λ terms Maps Conclusion Map semantics checked by counting br([B,M],N) :- bloom(B,N), bloomrtm(B,M). :- iterate(0,6,br). � 1, 5, 60, 1105, 27120, 828250, 30220800 ( http://oeis.org/A062980 ) Not sufficient: several other interpretations also hold! � labeled structures [Dubois, , Zeilberger, CLA’15] Dubois & Giorgetti Lambda terms and maps, formally (II) 55 / 59

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