Three termination problems. Patrick Dehornoy. Laboratoire de Mathématiques Nicolas Oresme, Université de Caen

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1 Three termintion prolems Ptrick Dehornoy Lortoire de Mthémtiques Nicols Oresme, Université de Cen

2 Three termintion prolems Ptrick Dehornoy Lortoire Preuves, Progrmmes, Systèmes Université Pris-Diderot Three unrelted termintion prolems : prtil specific nswers known, ut no glol understnding: cn some generl tools e useful?

3 Pln : 1. The Polish Algorithm for Left-Selfdistriutivity 2. Hndle reduction of rids 3. Suword reversing for positively presented groups

4 1. The Polish Algorithm for Left-Selfdistriutivity 2. Hndle reduction of rids 3. Suword reversing for positively presented groups

5 The y prolem I A i-term rewrite system (????) The ssocitivity lw (A): x (y z) = (x y) z,... nd the corresponding Word Prolem: Given two terms t, t, decide whether t nd t re A-equivlent. A trivil prolem: t, t re A-equivlent iff ecome equl when rckets re removed. (Right-) Polish expression of term: t 1 t 2 for t 1 t 2 (no rcket needed) Exmple: In Polish, ssocitivity is xy z = xy z. Definition. The Polish Algorithm for A: strting with two terms t, t (in Polish): - while t t do - p := first clsh etween t nd t (pth letter of t pth letter of t ) - cse type of p of - vrile vs. lnk : return NO; - lnk vs. vrile : return NO; - vrile vs. vrile : return NO; - vrile vs. : pply A + to t; (t 1 t 2 t 3 t 1 t 2 t 3 ) - vs. vrile : pply A + to t ; (t 1 t 2 t 3 t 1 t 2 t 3 ) - return YES.

6 The y prolem II Rememer : in Polish, ssocitivity is ( xy z xy z. Exmple: t = x (x (x x)), t = ((x x) x) x, i.e., in Polish, t 0 = xxxx t 0 = xx x x t 1 = xx xx t 1 = xx x x t 2 = xx x x t 2 = xx x x So t 2 = t 2, hence t 0 nd t 0 re A-equivlent. Theorem. The Polish Algorithm works for ssocitivity. (In prticulr, it termintes.)

7 The rel prolem I Left-selfdistriutivity ( (LD) : x (y z) = (x y) (x z), xy z i.e., in Polish, compre with ssocitivity xy xz Polish Algorithm: the sme s for ssocitivity. ( xy z xy x Exmple: t = x ((x x) (x x)), t = (x x) (x (x x)), i.e., in Polish, t 0 = xxx xx t 0 = xx xxx t 1 = xx xx xxx t 1 = xx xxx (= t 0 ) t 2 = xx xx xxx (= t 1 ) t 2 = xx xx xx t 3 = xx xx xxx (= t 2 ) t 3 = xx xx xx xx t 4 = xx xx xx xx t 4 = xx xx xx xx (= t 3 ) So t 4 = t 4, hence t 0 nd t 0 re LD-equivlent.

8 The rel prolem II Conjecture. The Polish Algorithm works for left-selfdistriutivity. Known. (i) If it termintes, the Polish Algorithm works for left-selfdistriutivity. (ii) The smllest counter-exmple to termintion (if ny) is huge.

9 1. The Polish Algorithm for Left-Selfdistriutivity 2. Hndle reduction of rids 3. Suword reversing for positively presented groups

10 The y prolem I A true (ut infinite) rewrite system. Alphet:,,A,B (think of A s n inverse of, etc.) Rewrite rules: - A ε, A ε, B ε, B ε (so fr trivil: free group reduction ) - A B, BA BA, A B, AB AB, nd, more generlly, - i A B i, B i A BA i, A i i B, AB i A i B for i 1. Aim: otin word tht does not contin oth nd A. Exmple: w 0 = AAA w 1 = BAA w 2 = BBA w 3 = BBB, word without A

11 The y prolem II Theorem. The process termintes in qudrtic time. Proof: (Length does not increse, ut could cycle.) Associte with the sequence of reductions rectngulr grid (qudrtic re). For the exmple: w 0 = AAA w 1 = BAA w 2 = BBA w 3 = BBB drw the grid:

12 The rel prolem This is the rid hndle reduction procedure; so fr: cse of 3-strnd rids; now: cse of 4-strnd rids (cse of n strnd rids entirely similr for every n). Alphet:,,c,A,B,C. Rewrite rules: - A ε, A ε, B ε, B ε, cc ε, Cc ε, (s ove) - for w in {,c,c} or {B,c,C} : wa φ (w), Aw φ A(w), with φ (w) otined from w y B nd B BA, nd φ A(w) otined from w y B nd B AB, - for w in {c} or {C} : wb φ (w), Bw φ B(w), with φ (w) otined from w y c Cc nd C CBc, nd φ B(w) otined from w y c cc nd C cbc. Remrk. i A (B) i B i : extends the 3-strnd cse.

13 The rel prolem Exmple: cababcba BcBBABCBA BcBABCBA BcBBCBA BCcBCBA BCCcCBA BCCBA BCCA BCC Termintes: the finl word does not contin oth nd A (y the wy: contins neither nor A, nd not oth nd B.) Theorem. Hndle reduction lwys termintes in exponentil time (nd id. for n-strnd version). Experimentl evidence. It termintes in qudrtic time (for every n).

14 Brids A 4-strnd rid digrm = 2D-projection of 3D-figure: isotopy = move the strnds ut keep the ends fixed: isotopic to rid := n isotopy clss represented y 2D-digrm, ut different 2D-digrms my give rise to the sme rid.

15 Brid groups Product of two rids: := Then well-defined with respect to isotopy), ssocitive, dmits unit: = nd inverses: isotopic to rid = rid rid d For ech n, the group B n of n-strnd rids (E.Artin, 1925).

16 Artin presenttion of B n Artin genertors of B n: = σ 1 σ 2 σ 3 σ 1 1 Theorem (Artin): The group B n is generted y σ 1,..., σ n 1, j σi σ j σ i = σ j σ i σ j for i j = 1, suject to σ i σ j = σ j σ i for i j 2. σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 σ 1 σ 3 σ 3 σ 1

17 Hndle reduction A σ i -hndle: i + 1 i Reducing hndle: Hndle reduction is n isotopy; It extends free group reduction; Terminl words cnnot contin oth σ 1 nd σ 1 1. Theorem. Every sequence of hndle reductions termintes.

18 1. The Polish Algorithm for Left-Selfdistriutivity 2. Hndle reduction of rids 3. Suword reversing for positively presented groups

19 The y prolem I This time: truly true rewrite system... Alphet:,,A,B (think of A s n inverse of, etc.) Rewrite rules: - A ε, B ε ( free group reduction s usul, ut only one direction) - A A, B B. ( reverse + ptterns into + ptterns ) Aim: trnsforming n ritrry signed word into positive negtive word. Exmple: BBA BBA BA B.

20 The y prolem II Theorem. It termintes in qudrtic time. Proof: (ovious). Construct reversing grid: Cler tht reversing termintes with qudrtic time upper ound (nd liner spce upper ound). Oviously: id. for ny numer of letters.

21 The rel prolem I Exmple 2: Sme lphet:,,a,b Rewrite rules: - A ε, B ε (free group reduction in one direction) - A BA, B AB. ( reverse + into +, ut different rule) Agin: trnsforms n ritrry signed word into positive negtive word. Termintion? Not cler: length my increse... Exmple: BBA BBBA BBA ABBA ABB AB A.

22 The rel prolem III Reversing grid: sme, ut possily smller nd smller rrows. Theorem. Reversing termintes in qudrtic time (in this specific cse). Proof: Return to the y cse = find (finite) set of words S tht includes the lphet nd closed under reversing. v for ll u, v in S, exist u, v in S s.t. reversing grid u u Here: works with S = {,,,}. v

23 Mny more rel prolems Alwys like tht? Not relly... Exmple 3: Alphet,,A,B, rules A ε, B ε, plus A......BABA, B......ABAB. {z } {z } {z } {z } mletters mletters mletters mletters Here : terminting in qudrtic time nd liner spce Exmple 4: Alphet,,A,B, rules A ε, B ε, plus A A, B BA Strt with B: B BA BA BAA BAA BAAA BAAA BAAAA w Here : non-terminting wa 2 wa 2 Exmple 5: Alphet,,A,B, rules A ε, B ε, plus B 2 2, B BABAB 2 AB 2 ABA, Here : terminting in cuic time nd qudrtic spce

24 Reversing: connection with monoids nd groups Wht re we doing? We re working with semigroup presenttion nd trying to represent the elements of the presented group y frctions. A semigroup presenttion: list of genertors (lphet), plus list of reltions, e.g., {,}, plus { = }. monoid, = +, group, =. Definition. Assume (A, R) semigroup presenttion nd, for ll s t in A, there is exctly one reltion s... = t... in R, sy sc(s, t) = tc(t, s). Then reversing is the rewrite system on A A ( copy of A, here : cpitlized letters) with rules ss ε nd st C(s, t)c(t, s) for s t in A. Reversing does not chnge the element of the group tht is represented; if it termintes, every element of the group is frction fg 1 with f, g positive. Exmple 1 = reversing for the free Aelin group:, = ; Exmple 2 = reversing for the 3-strnd rid group:, = ; Exmple 3 = reversing for type I 2 (m + 1) Artin group:,... {z } =... {z } ; m+1 m+1 Exmple 4 = reversing for the Bumslg Solitr group:, 2 = ; Exmple 5 = reversing for the ordered group:, = 2 2.

25 Reversing: questions The only known fcts: - reduction to the y cse termintion; - self-reproducing pttern non-termintion; - if reversing is complete for (A, R), then it is terminting iff ny two elements of the monoid A R + dmit common right-multiple. Question. Wht cn YOU sy out reversing?

26 References For the Polish Algorithm: P. Dehornoy, Brids nd selfdistriutivity, Progress in mth. vol 192, Birkhüser 2000 (Chpter VIII) O. Deiser, Notes on the Polish Algorithm, (Technishe Universität München) For Hndle Reduction of rids: P. Dehornoy, with I. Dynnikov, D. Rolfsen, B. Wiest, Brid ordering, Mth. Surveys nd Monogrphs vol. 148, Amer. Mth. Soc (Chpter V) For reversing ssocited with semigroup presenttion: P. Dehornoy, with F. Digne, E. Godelle, D. Krmmer, J. Michel, Foundtions of Grside Theory, sumitted dehornoy/ (Chpter II)...venez u groupe de trvil du vendredi!