A new approach to collapsing words Alessandra Cherubini Andrzej Kisielewicz Pavel Gawryochowski Brunetto Piochi

Basic definitions and first results A= finite deterministic automaton w*w* df A (w)=|Q|-|Q  w| A is n-compressible iff df A (w)≥n for some w  *, w is an n-compressing word for A w  * is an n-collapsing word iff it is an n-compressing word for each n-compressible automaton over . For each alphabet  and n  N n-collapsing words over  exist (Sauer and Stone) Each n-collapsing word is an n-synchronizing word.

Basic definitions and first results (cont.) n-collapsing words are n-full. An n-compressible automaton A is proper iff |w| ≥n for each n-compressing word w for A. A word w  * is n-collapsing iff it is n-full and n- compressing for each proper n-compressible automaton over . It is decidable whether a word is n-collapsing or not: check all n-compressible automanton with less then 3(n-1)|w|+n+1 states (Petrov) A characterization of collapsing words for n=2 was given in terms of indices of a family of finited generated subgroups associated to each word in some finitely generated free groups (Ananichev, Vokov and I).

Proper 2- compressible automata Notation: a  is of type {x,y}/z (x,y,z  Q) if x  a= y  a and {z}= Q-Q  a An automaton A is proper 2-compressible iff it has letters acting both as non-permutation and as permutation transformations, all non-permutations are of type {x,y}/x and one of the following occurs: all non permutations are of the same type {1,2}/1, and the group of permutations fixes neither {1} nor {1,2} (MONO1); there is 1  Q such that all non-permutations are of type {1,x}/1 for some x  Q, two types exist and the group of permutations does not fix {1} (MONO2); there are 1,2  Q such that all non-permutations are of type {1,2}/1 and {1,2}/2, and the group of permutations does not fix {1,2} (STEREO).

A new more combinatorial approach (P  ) is a partition of  encoding the roles of the letters on the automaton), where P is a special block (representing permutations) and  is partitionned either in B 2,…,B k, k≥2 (DB-partition) or in B 1,B 2 (3DB-partition) Let w  * and let (P  ) be a partition of , then w=  0  1 ….  n-1  n  n where  0,  n  P*,  i  P + (1  i<n),  j  + (1  j  n) B i ={a  of the type {1,i}/i } B 1 ={a  of the type {1,2}/1}, B 2 ={a  of the type {1,2}/2}.

A new approach (systems on permutations) (P,  ) a partition of , w=  0  1 ….  n-1  n  n (P,  ) DB-partition   w (P,  ) denotes the system of equations on permutations of the form 1  i  {1,h} (1  i<n) where the first letter of  i+1 is in B h (P,  ) 3DB-partition   ’ w (P,  ) denotes the system of the equations of the form j  i  {12} (1  i<n) with j=1,2 according with the last letter of  i-1 being in B 1 or in B 2 An assignement of permutations to letters of P which satisfies all the equations is a solution of the system. A solution of  w (P,  ) is trivial if all permutations fix 1, and in case of  w (P,B 2 ) also if all permutations fix {1,2}. A solution of  ’ w (P,  ) is trivial if all permutations fix {1,2}.

A new approach (characterization) A word w   is 2-collapsing iff it is 2-full and the following conditions hold: for any DB-partition (P,  ),  w (P,  ) has no nontrivial solution for any 3DB-partition (P,  ),  ’ w (P,  ) has no nontrivial solution General idea of the proof: To compute the deficency of a word w proceeding letter by letter and observing if and how the deficency increases. A non trivial solution of the system gives an example of automaton which is not compressed by w.

An example  ={a,b,c} w=aba 2 c 2 bab 2 acbabcacbcb DB-partition (P,B 2 ) with P={a}  a fixes either 1 or {1,2} DB-partition (P,B 2,B 3 ) with P={a}  a fixes 1 3DB-partition with P={a}  a fixes {1,2} 1  a 2  {1,2}, 1  a  {1,2}, 1  a  {1,2}, 1  a  {1,3} 1  a  {1,2}, 2  a  {1,2}

An example (continued) w=aba 2 c 2 bab 2 acbabcacbcb is 2-collapsing!! DB-partition with P={a,b}  a,b fix either 1 or {1,2} 1  bab 2 a  {1,2}, 1  bab  {1,2}, 1  a  {1,2}, 1  b  {1,2} 1  a=1, 1  b=1 1  a=2, 1  b=2 1  a=1, 1  b=2  2  ab  {1,2}, 2  ab 2 a  {1,2} hence if 2  a=2 then 2  b=2, if 2  a=x  {1,2}, then x  b=1 and the contradiction 2  a=x  {1,2}, 1  a=2, 1  b=1…..

The new approach vs the old one The old approach is elegant and can be translated in an automaton based algorithm to recognize if a word is 2-collapsing but it did not allow to settle the natural complexity problem, to answer a few questions about the possibility of simplifying the characterization, to generalize the characterization to n- collapsing words.

The new approach vs the old one The new approach allows to solve the complexity problems, to answer in negative to the questions on simplification giving examples of words compressing each type of compressing automata and not the remaining ones and we hope that it allows also to characterize n- collapsing words and to calculate further values of c(n,k)=|w| where w is a shortest n-collapsing word on k letter alphabet.

Computational complexity For a 2-element alphabet , checking whether a word w  * is 2-collapsing may be done in polynomial time with respect to |w|. (Pribavkina) The problem of recognizing 2-collapsing words over a fixed alphabet with more than 2 letter is co-NP-complete. The general problem of recognizing n-collapsing words is co-NP-complete. The general problem of recognizing 2-synchronizing words is co-NP-complete (Gawryochowski-Kisieliewicz)

A sketch of the proof |  |=3, difficult case: partition with |P|=2  Permutation Condition Problem INSTANCE: {u 1,u 2,…u s }, u i  {a,b} + PROBLEM: Does the system of permutations 1  u i  {1,2}, i=1,2,…,s, have a non trivial solution? Converted in a problem on coloring a binary tree with distinguished nodes T(u 1,u 2,…u s ). Assuming that going to the left child represents applying permutation a and going to the right child represents applying permutation b, all vertices represent words in a natural way. T(u 1,u 2,…u s ) is the minimal tree in which all words u i are represented.

A sketch of the proof Distiguished nodes of T(u 1,…u s ): nodes representing u i and the root. 1-2 coloring of T(u 1,…u s ): a coloring of vertices with integers so that distinguished nodes are colored 1 or 2 and the root is colored 1 Coherent coloring: (left) right childrens of nodes with the same color have the same color and conversely Trivial coloring: coloring whose colors are only 1 and 2 T(a,b,bab,bab 2 a) A non coherent coloring of T(a,b,bab,bab 2 a)

A sketch of the proof The system of permutations 1  u i  {1,2}, i=1,…,s, has a non trivial solution iff T(u 1,…u s ) has a nontrivial coherent 1-2 coloring (provided that all the words of length 2 are represented in T(u 1,…u s )).  NC-coloring problem INSTANCE: A binary tree T=T(u 1,…u s ) with distinguished nodes PROBLEM: Does T have a non trivial coherent 1-2 coloring?  3SAT can be reduced polinomially to the above problem.

Open problems The language of 2-collapsing words over a given alphabet  is context sensitive, and also the language of 2-collapsing word on a 2-letter language is not context free, are these languages contained in some intermediate family of languages? Values of c(n,k) for general n and k and list of the shortest n-collapsing words on an alphabet of k letters (up to now only c(2,2)=8, c(2,3)=21 and the list of all collapsing words on a 3-letter alphabet are known). Characterization of n-collapsing words for n>2 (up to now we have only the characterization of proper 3- compressible automata).