Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words
aa r X i v : . [ c s . F L ] J u l Completely Reachable Automata, PrimitiveGroups and the State Complexity of the Set ofSynchronizing Words
Stefan Hoffmann
Informatikwissenschaften, FB IV, Universit¨at Trier, Universit¨atsring 15, 54296 Trier,Germany, [email protected]
Abstract.
We give a new characterization of primitive permutationgroups tied to the notion of completely reachable automata. Also, weintroduce sync-maximal permutation groups tied to the state complex-ity of the set of synchronizing words of certain associated automata.Furthermore, we state several sufficient criteria which give automata forwhich the state complexity of the set of synchronizing words is maximal.One such criterion applies to a family of automata from the literaturefor which this was only conjectured. Hence, we solve an open open andgive a wealth of additional automata with this property. Lastly, we define k -reachable groups in analogy with synchronizing groups. Keywords: finite automata · synchronization · completely reachable au-tomata · primitive permutation group · state complexity A deterministic semi-automaton is synchronizing if it admits a reset word, i.e.,a word which leads to some definite state, regardless of the starting state. Thisnotion has a wide range of applications, from software testing, circuit synthe-sis, communication engineering and the like, see [20, 23]. The famous ˇCern´yconjecture [8] states that a minimal synchronizing word has at most quadraticlength. We refer to the mentioned survey articles for details. An automaton iscompletely reachable, if for each subset of states we can find a word which mapsthe whole state set onto this subset. This is a generalization of synchronizability,as a synchronizing word maps the whole state set to a singleton set. The classof completely reachable automata was formally introduced in [5], but alreadyin [9, 15] such automata appear in the results. The time complexity of decid-ing if a given automaton is completely reachable is unknown. A sufficient andnecessary criterion for complete reachability of a given automaton in terms ofgraphs and their connectivity is known [6], but it is not known if these graphscould be constructed in polynomial time. A special case of the general graphconstruction, which gives a sufficient criterion for complete reachability [5], isknown to be constructible in polynomial time [10]. The size of a minimal au-tomaton accepting a given regular language is called the state complexity of thatlanguage. The set of synchronizing words of a given automaton is a regular ideallanguage whose state complexity is at most exponential in the size of the origi-nal automaton [15, 16]. The ˇCern´y family of automata [8], a family of automatayielding a quadratic lower bound for the length of shortest synchronizing words,
S. Hoffmann is completely reachable, but also the set of synchronizing words has maximalstate complexity [15, 16]. These properties are shared by many families of au-tomata that are also slowly synchronizing [1, 2, 15, 16]. The notion of primitivepermutation groups could be traced back to work by Galois [17] on the solu-bility of equations by radicals. Nowadays, it is a core notion of the theory ofpermutation groups [7].
Outline and our Contribution:
In Section 2 we give definitions and stateknown results we will need. Then, in Section 3, we give a general criterion forcompletely reachable automata to deduce that the set of synchronizing words hasmaximal state complexity, which yields a polynomial time decision procedure. InSection 4, we introduce a new characterization of primitive permutation groups,motivated by work on synchronizing and completely reachable automata and ondetecting properties of permutation groups by functions [3, 4, 5, 9]. We also relatethis to the notion of the state complexity of the set of synchronizing words. TheˇCern´y family of automata, the first given family yielding a lower bound for thelength of synchronizing words, is completely reachable and its set of synchroniz-ing words has maximal state complexity. But these properties are also shared bya wealth of different slowly synchronizing automata [1, 2, 15, 16]. Motivated bythis, we introduce the class of sync-maximal permutation groups and show thatthey fit between the 2-homogeneous and the primitive groups. In Section 5, weapply the results from Section 3 to give more specific sufficient criteria. We applythese to solve an open problem; we confirm a conjecture concerning the maxi-mal state complexity of the set of synchronizing words for a family of automatafrom [16]. Lastly, in Section 6 we introduce k -reachable groups motivated by ourinvestigations and the definition of synchronizing groups [4]. We show that foralmost all k , only the symmetric and alternating groups are k -reachable. General Notions:
Let Σ “ t a , . . . , a k u be a finite set of symbols, called an alphabet . The set Σ ˚ denotes the set of all finite sequences, i.e., of all words orstrings. The finite sequence of length zero, or the empty word , is denoted by ε . Weset Σ ` “ Σ ˚ zt ε u . For a given word w P Σ ˚ , we denote by | w | its length . Subsetsof Σ ˚ are called languages . With N “ t , , , . . . u we denote the set of naturalnumbers, including zero, and N “ N zt u . By Z “ t . . . , ´ , ´ , ´ , , , , , . . . u we denote the integers. For n ą r n s “ t , . . . , n ´ u and r s “ H . If a, b P Z and b ‰
0, by a mod b we denote the unique number 0 ď r ă b with a “ qb ` r for some q P Z . For some set X by P p X q we denote the power set of X , i.e, the set of all subsets of X . Every function f : X Ñ Y induces a functionˆ f : P p X q Ñ P p Y q by setting ˆ f p Z q : “ t f p z q | z P Z u . Here, we will denote thisextension also by f . Let k ě
1. A k -subset Y Ď X is a finite set of cardinality k .For functions f : A Ñ B and g : C Ñ B , the functional composition f g : C Ñ A is the function p f g qp x q “ f p g p x qq , i.e., the function on the right is applied first . Automata-Theoretic Notions: A finite deterministic and complete automa-ton will be denoted by A “ p Σ, Q, δ, s , F q with δ : Q ˆ Σ Ñ Q the statetransition function, Q a finite set of states, s P Q the start state and F Ď Q the In group theory usually the other convention is adopted, but we stick to the conven-tion most often seen in formal language theoryompletely Reachable Automata, Primitive Groups and State Complexity 3 set of final states. The properties of being deterministic and complete are impliedby the definition of δ as a total function. The transition function δ : Q ˆ Σ Ñ Q could be extended to a transition function on words δ ˚ : Q ˆ Σ ˚ Ñ Q bysetting δ ˚ p s, ε q : “ s and δ ˚ p s, wa q : “ δ p δ ˚ p s, w q , a q for s P Q , a P Σ and w P Σ ˚ . In the remainder we drop the distinction between both functionsand will also denote this extension by δ . For S Ď Q and w P Σ ˚ , we write δ p S, w q “ t δ p s, w q | s P S u and δ ´ p S, w q “ t q P Q | δ p q, w q P S u . The languageaccepted by A “ p Σ, S, δ, s , F q is L p A q “ t w P Σ ˚ | δ p s , w q P F u . A language L Ď Σ ˚ is called regular if L “ L p A q for some finite automaton A . For a lan-guage L Ď Σ ˚ and u, v P Σ ˚ we define the Nerode right-congruence with respectto L by u ” L v if and only if @ x P Σ : ux P L Ø vx P L. The equivalence classfor some w P Σ ˚ is denoted by r w s ” L : “ t x P Σ ˚ | x ” L w u . A language isregular if and only if the above right-congruence has finite index, and it couldbe used to define the minimal deterministic automaton A L “ p Σ, Q, δ, r ε s ” L , F q with Q : “ tr w s ” L | w P Σ ˚ u , δ pr w s ” L , a q : “ r wa s ” L for a P Σ , w P Σ ˚ and F : “ tr w s ” L | w P L u . It is indeed the smallest automaton accepting L interms of states, and we will refer to this construction as the minimal automa-ton [11] of L . The state complexity of a regular language is defined as the numberof Nerode right-congruence classes. We will denote this number by sc p L q . Let A “ p Σ, Q, δ, s , F q be an automaton. A state q P Q is reachable , if q “ δ p s , u q for some u P Σ ˚ . We also say that a state q is reachable from a state q if q “ δ p q , u q for some u P Σ ˚ . Two states q, q are distinguishable , if there ex-ists u P Σ such that |t δ p q, u q , δ p q , u qu X F | “
1. An automaton for a regularlanguage is isomorphic to the minimal automaton if and only if all states arereachable and distinguishable [11]. A semi-automaton A “ p Σ, Q, δ q is like anordinary automaton, but without a designated start state and without a set offinal states. Sometimes we will also call a semi-automaton simply an automatonif the context makes it clear what is meant. Also, definitions without explicitreference to a start state and a set of final states are also valid for automata.Let A “ p Σ, Q, δ q be a finite semi-automaton. A word w P Σ ˚ is called syn-chronizing if δ p q, w q “ δ p q , w q for all q, q P Q , or equivalently | δ p Q, w q| “ p A q “ t w P Σ ˚ | | δ p Q, w q| “ u . The power automaton (for synchroniz-ing words) associated to A is P A “ p Σ, P p Q q , δ, Q, F q with start state Q , finalstates F “ tt q u | q P Q u and the transition function of P A is the transitionfunction of A , but applied to subsets of states. Then, as observed in [22], theautomaton P A accepts the set of synchronizing words, i.e., L p P A q “ Syn p A q .As for t q u P F , we also have δ pt q u , x q P F for each x P Σ ˚ , the states in F couldall be merged to a single state to get an accepting automaton for Syn p A q . Also,the empty set is not reachable from Q . Hence sc p Syn p A qq ď | Q | ´ | Q | and thisbound is sharp [15, 16]. We call A completely reachable if for any non-empty S Ď Q there exists a word w P Σ ˚ with δ p Q, w q “ S , i.e., in the power au-tomaton, every state is reachable from the start state. When we say a subset ofstates in A is reachable, we mean reachability in P A . The state complexity ofSyn p A q is maximal, i.e., sc p Syn p A qq “ | Q | ´| Q | , if and only if all subsets S Ď Q with | S | ě Q , and all thesestates are distinguishable in P A . For strongly connected automata , i.e., those forwhich all pairs of states are reachable from each other, the state complexity ofSyn p A q is maximal iff A is completely reachable and all S Ď Q with | S | ě P A (note here F “ tt q u | q P Q u , hence |tt q u , S u X F | “ S. Hoffmann
Transformations and Permutation Groups:
Let n ě
0. Denote by S n the symmetric group on r n s , the group of all permutations of r n s . A permutationgroup (of degree n ) is a subgroup of S n . For n ą
1, the alternating group is theunique subgroup of size n ! { S n , see [7]. The orbit of an element i P r n s fora permutation group G is the set t g p i q | g P G u . A permutation group G over r n s is primitive , if it preserves no non-trivial equivalence relation on r n s , i.e.,for no non-trivial equivalence relation „Ď r n s ˆ r n s we have p „ q if and onlyif g p p q „ g p q q for all g P G and p, q P r n s . A permutation group G over r n s iscalled k -homogeneous for some k ě
1, if for any two k -subsets S, T of Q , thereexists g P G such that g p S q “ T . A transitive permutation group is the sameas a 1-homogeneous permutation group. Note that here, all permutation groupswith n ď n ą n “ G over r n s is called k -transitive for some k ě p p , . . . , p k q , p q , . . . , q k q P r n s k , there exists g P G such that p g p p q , . . . , g p p k qq “ p q , . . . , g k q . By T n we denote the set of all mapping on r n s .A submonoid of T n for some n is called a transformation monoid . If the set U is a submonoid (or a subgroup) of T n (or S n ) we denote this by U ď T n (or U ď S n ). For a set A Ď T n (or A Ď S n ) we denote by x A y the submonoid (or thesubgroup) generated by A . Let A “ p Σ, Q, δ q be an automaton and for w P Σ ˚ define δ w : Q Ñ Q by δ w p q q “ δ p q, w q for all q P Q . Then, we can associate with A the transformation monoid of the automaton T A “ t δ w | w P Σ ˚ u , where wecan identify Q with r n s for n “ | Q | . Seeing the letters as transformations of thestate set, T A “ x Σ y . The rank of a map f : r n s Ñ r n s is the cardinality of itsimage. For a given automaton, seeing a word as a transformation of its state set,the rank of the word is the rank of this transformation. Known Results:
We need the following result from [9].
Proposition 1. [9] Let A “ p Σ, Q, δ q be a finite automaton with n states.Suppose we have two letters a, b P Σ such that a has rank n ´ and b is apermutation with a single orbit. Choose s, t P Q and ă d ă | Q | such that δ p Q, a q “ Q zt s u , | δ ´ p t, a q| “ and δ p s, b d q “ t . If d and n are coprime, thenfor every non-empty set S Ď Q of size k , there exists a word w S of length atmost n p n ´ k q such that δ p Q, w S q “ S . The next result appears in [3] and despite it was never clearly spelled out byRystsov himself, it is implicitly present in arguments used in [19].
Theorem 1. (Rystsov) [3, 19] A permutation group on r n s is primitive if andonly if for every map f : r n s Ñ r n s of rank n ´ , the transformation monoid x G Y f y contains a constant map. In [5] a sufficient criterion for complete reachability was given. It is based onthe following graph construction associated to a transformation monoid or anautomaton.
Definition 1. [5] Let A “ p Σ, Q, δ q be a semi-automaton. Then, we define thegraph Γ p A q “ p Q, E q with vertex set Q and edge set E “ tp p, q q | D w P Σ ˚ : p R δ p Q, w q , | δ ´ p q, w q| “ , w has rank | Q | ´ u . The trivial equivalence relations on r n s are r n s ˆ r n s and tp x, x q | x P r n su .ompletely Reachable Automata, Primitive Groups and State Complexity 5 For transformation monoids M ď T n , a similar definition Γ p M q applies. The construction was extended in [6] to give a sufficient and necessary crite-rion. In [10] it was shown that Γ p A q could be computed in polynomial time. Theorem 2. [5] Let A “ p Σ, Q, δ q . Then A is completely reachable, if Γ p A q “p Q, E q is strongly connected. If we know that a given semi-automaton is completely reachable, we can decidein polynomial time if the state complexity of the set of synchronizing words ismaximal. The key to this result is the next lemma.
Lemma 1.
Let A “ p Σ, Q, δ q be a completely reachable semi-automaton with n states. Then sc p Syn p A qq “ n ´ n if and only if all -sets of states are pairwisedistinguishable in P A . Hence, we only need to check for all pair states t p, q u with p ‰ q in the powerautomaton if they are all distinguishable to each other. This could be done inpolynomial time. Corollary 1.
Let A “ p Σ, Q, δ q be a completely reachable semi-automaton with n states. Then we can decide in polynomial time if sc p Syn p A qq “ n ´ n . p A q An automaton is synchronizing, precisely if its transformation monoid containsa constant mapping. Hence, the next result is a strengthening of Theorem 1.
Theorem 3.
A finite permutation group G ď Sym pr n sq with n ě is primi-tive, if and only if for every transformation f : r n s Ñ r n s of rank n ´ , thetransformation semigroup x G Y t f uy is completely reachable.Proof. First, suppose G is a permutation group on r n s that is not primitive.Then n ě
3. Let π be a non-trivial partition of r n s respected by G . Take anytwo distinct points p, q in a class of π and define a map f : r n s Ñ r n s by letting f p x q “ " q if x “ p,x if x ‰ p. Then f is a transformation of rank n ´ Ω { π . Hence the transformation monoid generated by G and f acts on Ω { π exactly as does G . In particular, if A “ g pr n sq for g P x G Y t f uy ,then | A | ě | Ω { π | , so that no subsets of cardinality strictly less than | Ω { π | arereachable.Conversely, let G be a primitive permutation group on Ω . Take any transfor-mation monoid M generated by G and a transformation of rank n ´
1. If p p, q q isan edge of the graph Γ p M q , then for each g P G , the pair p g p p q , g p q qq also con-stitutes an edge of Γ p M q . Since G is transitive, we see that every element in r n s has an outgoing edge in Γ p M q . This clearly implies that Γ p M q has a directedcycle, and by the definition of Γ p M q this cycle is not a loop. Assume Γ p M q is S. Hoffmann not strongly connected. The partition of Γ p M q into strongly connected compo-nents induces a partition π of r n s which is nontrivial, since Γ p M q has a directedcycle which is not a loop. As G preserves the edges of Γ p M q , if we have a pathbetween any two vertices, we also have a path between their images. Hence G respects π , which is not possible as G is primitive by assumption. So Γ p M q must be strongly connected. Then M is completely reachable by Theorem 2. [\ Remark 1.
For finite permutation groups, Theorem 1, Theorem 2, Higman’s orbitalgraph characterization of primitivity and the fact that, for finite orbital graphs, con-nectedness implies strongly connectedness (please see [7] for these notions) could beused to give another proof of Theorem 3.
Because our main motivation comes from the theory of automata, let us statea variant of Theorem 3 formulated in terms of automata.
Corollary 2.
Let n ě . Suppose G “ x g , . . . , g k y ď Sym pr n sq . Then G isprimitive if and only if for every transformation f : r n s Ñ r n s of rank n ´ , thesemi-automaton A “ p Σ, Q, δ q with Σ “ t g , . . . , g k , f u , Q “ r n s and δ p m, g q “ g p m q for g P Σ is completely reachable. Let A “ p Σ, Q, δ q be a strongly connected semi-automaton. Then A iscompletely reachable, if Syn p A q has maximal state complexity. Proposition 2.
Let G “ x g , . . . , g k y ď S n be a permutation group and f : r n s Ñ r n s a non-permutation. Set Σ “ t g , . . . , g k , f u and A “ p Σ, r n s , δ q with δ p m, g q “ g p m q for m P r n s and g P Σ . If n ą and sc p Syn p A qq “ n ´ n , then G is transitive and A completely reachable.Remark 2. For n “
2, if G only contains the identity transformation, then addingany non-permutation gives an automaton such that the set of synchronizing words hasstate complexity two, but it is not completely reachable nor is G transitive. Also notethat the assumption sc p Syn p A qq “ n ´ n implies that f must have rank n ´
1, forotherwise sets of size n ´ Definition 2.
A permutation group G “ x g , . . . , g k y ď S n is called sync-maximal , if for any map f : r n s Ñ r n s of rank n ´ , for the automaton A “ p Σ, r n s , δ q with Σ “ t g , . . . , g k , f u and δ p m, g q “ g p m q for m P r n s and g P Σ , we have sc p Syn p A qq “ n ´ n , As written, the definition involves a specific set of generators for G . But theresulting transformation monoids are equal for different generators and we canwrite one set of generators in terms of another. So reachability of subsets anddistinguishability of subsets is preserved by a change of generators. Hence, thedefinition is actually independent of the specific choice of generators for G . Thismight be different if we are concerned with the length of shortest words to reachcertain subsets, but that is not part of the definition. Proposition 3.
Every sync-maximal permutation group is primitive.Remark 3.
By case analysis, note that for n ď n ď Next, we relate the condition that the set of synchronizing words has maximalstate complexity to the notion of 2-homogeneity. But, as shown in Example 1,we do not get a characterization of 2-homogeneity similar to Theorem 3.
Proposition 4. If G ď S n is -homogeneous, then G is sync-maximal. The next example shows that the converse of Proposition 4 does not hold.
Example 1.
Let g “ p q and G “ x g y . Then, as it is a cycle of prime length, G is primitive. So, by Theorem 3, for any f : r n s Ñ r n s the transformation semigroup x G, f y is completely reachable. Also, we show sc p Syn p A qq “ n ´ n . But G is not2-homogeneous. We have two orbits on the 2-sets: A “ tt , u , t , u , t , u , t , u , t , uu and B “ tt , u , t , u , t , u , t , u , t , uu . Let f : r n s Ñ r n s be any map of rank n ´
1. Without loss of generality, we can assume f p q “ f p q . Then, two distinct 2-sets are distinguishable, if one could be mapped to t , u , but not the other, as a final application of f gives that one is mapped to asingleton, but not the other. First, note that all 2-sets in A are distinguishable, as foreach t x, y u P A we find a unique 0 ď k ă | A | such that g k pt z, v uq “ t , u if and onlyif t z, v u “ t x, y u for each t z, v u P A , as g permutes A . If we have any t x, y u P B suchthat f pt x, y uq P A , then all sets in B are distinguishable. For if t z, u u P B there existsa unique g k with 0 ď k ă | B | , g k pt z, u uq “ t x, y u and t z , u u g k ‰ t x, y u for each other t z , u u P B ztt z, u uu . As t , u R B and f is injective on t , , , u and on t , , , u , f isinjective on B . Hence f p g k pt z , u uq ‰ f p g k pt z, u uqq for the previously chosen t z, u u P B and t z , u u P B ztt z, u uu . Now, choose g l such that g l p f p g k pt z, u uqqq “ t , u . In anycase, i.e., whether f p g k pt z , u uqq is in B or in A , we have g l p f p g k pt z , u uqqq ‰ t , u .So, all 2-sets in A are distinguishable and all 2-sets in B . That a 2-set from A isdistinguishable from any 2-set in B is clear, as we can map the 2-set from A to t , u by a power of g , and the one from B would not be mapped to t , u . Lastly, we showthat we must have some t x, y u P B with f pt x, y uq P A , which gives the claim. Considerthe sets t , u , t , u , t , u and t , u from B and suppose their images are all containedin B , i.e., tt f p q , f p qu , t f p q , f p qu , t f p q , f p quu Ď B . But in B at most two sets sharean element, hence |t f p q , f p q , f p qu| ď
2, which is not possible as f has rank n ´ f p q “ f p q . So, some image of these three sets must be in A . [\ Here, we take a closer look at automata over a binary alphabet. Our first lemmais a result of how the letters act on automata with at least n ą n ´
1. In particular, it applies to completelyreachable automata for n ą n ´ Lemma 2.
Let Σ “ t a, b u be a binary alphabet and A “ p Σ, Q, δ q a finite semi-automaton with n ą states. Then, the following conditions are equivalent: S. Hoffmann
1. every subset of size n ´ is reachable,2. exactly one letter acts as a cyclic permutation with a single orbit and theother letter has rank n ´ .Remark 4. In Lemma 2, we need n ą
2. For let A “ pt a, b u , t p, q u , δ q with δ p p, a q “ δ p q, a q “ q and δ p p, b q “ δ p q, b q “ p . Then A is completely reachable, but no letteracts as a non-trivial permutation. As transitive permutation groups of prime degree, i.e., those generated by acycle of prime length, are primitive, by Theorem 3, the non-permutation letterfrom Lemma 2 obeys no further restriction than collapsing exactly two states,i.e., any such letter will give a completely reachable automaton. In this sense,Lemma 2 states the form of these automata as precisely as possible, at least ifthe number of states is a prime number.
Corollary 3.
Let Σ be a binary alphabet and A “ p Σ, Q, δ q a semi-automatonwith n ą states. If sc p Syn p A qq “ n ´ n , then A is completely reachable.Remark 5. The statement of Corollary 3 does not hold for automata with two states.For let A “ pt a, b u , t p, q u , δ q with δ p p, a q “ q “ δ p q, a q and δ p p, b q “ p, δ p q, b q “ q . ThenSyn p A q “ t b m au | m ě , u P t a, b u ˚ u . So sc p Syn p A qq “
2. But A is not completelyreachable, as for w P t a, b u ˚ we have δ pt p, q u , w q ‰ t p u . Proposition 5.
Let t a, b u Ď Σ and suppose A “ p Σ, Q, δ q has n states. Assumethe letter b permutes the states with a single orbit and the letter a has rank n ´ .Then all -sets are distinguishable in P A , if we can find a state q P Q and anumber d ą coprime to n such that for each ă m ă n we either have δ p q, b m a q “ δ p q, ab n ` m ´ d q (1) or, but only in case m is not divisible by d , δ p q, ab r q “ δ p q, b m a q or δ p q, b m ab r q “ δ p q, a q (2) for some number ď r ă n divisible by d . In the formulation of Proposition 5, we have r ą
0, as in this case m ‰ δ p q, b m ab r q “ δ p q, a q is equivalent with δ p q, ab n ´ r q “ δ p q, b m a q ,as for any states s, t P Q and 0 ď k ă n we have δ p s, b k q “ t if any onlyif δ p t, b n ´ k q “ s , as δ p s, b n q “ s . With Proposition 5, we can solve an openproblem from [16]. For n ą
5, define the automata K n “ p Σ, r n s , δ q , introducedin [16], with δ p i, b q “ i ` i P t , . . . , n ´ u , and δ p n ´ , b q “ δ p i, a q “ i ` i P t , . . . , n ´ u , δ p n ´ , a q “ , δ p n ´ , a q “ , δ p , a q “ . Please see Example 2 for an illustration of this automata family. In [16], it wasconjectured that sc p Syn p K n qq “ n ´ n for every odd n ą
5. With Proposition 5,together with Proposition 1 and Lemma 1, we can confirm this. Note that 0 ă m ă n implies δ p q, b m q ‰ q . Also note that we added n on the righthand side to account for values d ą
1. In Proposition 7 we only subtract one fromthe exponent of b , which is always non-zero and strictly smaller than n , and so we donot needed this “correction for the b -cycle” in case of resulting negative exponents. I slightly changed the numbering of the states with respect to the action of the letter a compared to [16].ompletely Reachable Automata, Primitive Groups and State Complexity 9 Proposition 6.
Let n ą be odd. Then we have sc p Syn p K n qq “ n ´ n . Also, let us state another sufficient criterion for maximal state complexity ofthe set of synchronizing words. It resembles Proposition 5, but it only requiresthat we can reduce the “cyclic distance” for all states which are no more than t n { u ` a along these states as in theprevious proposition (which was necessary to encompass examples like K n ), butonly that it reduces the distance as stated. Proposition 7.
Let Σ “ t a, b u and suppose A “ p Σ, Q, δ q has n states and iscompletely reachable with the letter a having rank n ´ and the letter b permutingthe states with a single orbit. Then sc p Syn p A qq “ n ´ n if we can find a state q P Q such that for all ď m ď t n { u ´ we have δ p q, b m ` a q “ δ p q, ab m q . (3) Example 2.
Please see Figure 1 for the automata families. The automata C n givesthe ˇCern´y family, the automata L n , V n , F n and K n where introduced in [1, 15, 16].There, except for K n , it was established that in each case (for F n only if n is odd and n ą
3) the set of synchronizing words has maximal state complexity. Note that ourresults, namely Proposition 5, together with Proposition 1 and Lemma 1 also give thisresult. Additionally, to illustrate that the scope of our results is much wider, with U n we introduce a new family of automata for n ě
2, which is completely reachable andsc p Syn p U n qq “ n ´ n . Set U n “ pt a, b u , r n s , δ q , where δ p i, b q “ i ` i P t , . . . , n ´ u , and δ p n ´ , b q “ δ p i, a q “ n ´ ` i mod n for i P t , . . . , n ´ u , δ p , a q “ n ´ . Then, for all 0 ď m ă n , we have δ p , b m ` a q “ δ p , ab m q . So, by Proposition 7, wehave sc p Syn p U n qq “ n ´ n . k -Reachable Permutation Groups A permutation group G ď S n is called synchronizing , if for any non-permutationthe transformation monoid x G Y t f uy contains a constant map. This notionwas introduced in [4]. For further information on synchronizing groups and itsrelation to the ˇCern´y conjecture, see the survey [3]. In [3], the question wasasked to detect properties of permutation groups by functions. Theorem 3 andTheorem 1 are in this vain. Note that every synchronizing group is primitive [4],but not conversely [18]. By Theorem 3, primitive groups have the property thatif we add any function of rank n ´ n ´ k , k ą
0, areall conceivable subsets, i.e., those of size n ´ k, n ´ k, . . . and so on, reachable?And what groups do we get, if we assume this property? Definition 3.
A permutation group over the finite set r n s with n ą is called k -reachable , if for any map f : r n s Ñ r n s of rank n ´ k all subsets of cardinality n ´ k, n ´ k, . . . , n ´ p r n { k s ´ q ¨ k are reachable, i.e., we have some transformation in the transformation monoidgenerated by G and f which maps r n s to any such set. n ´ n ´ . . .b a, b b ba a a a C n n ´ n ´ . . .b b a, b a, ba a L n n ´ n ´ . . .b a, b a, b a, ba V n n ´ n ´ . . .b b b baa aa a F n , n ą n ´ n ´ . . .b a, b b a, ba, baa K n , n ą n ´ n ´ . . .b b b bbaa aaa U n , n ě Fig. 1.
Families of automata whose sets of synchronizing words have maximal statecomplexity. Please see Example 2 for explanation.
By Theorem 3, the 1-reachable group are precisely the primitive groups. Alsonote that p n ´ q -reachable is the same as transitivity. Proposition 8. A k -reachable permutation group is k -homogeneous. The reverse implication does not hold in the previous proposition. For exam-ple, we find 1-homogeneous, i.e., transitive groups, which are not 1-reachable,i.e., primitive by Theorem 3. By a result of Livingstone and Wagner [12, 14], for5 ď k ď n { k -homogeneous permutation group of degree n is k -transitive, andfor k ď n { k -homogeneous permutation group is also p k ´ q -homogeneous.As 2-homogeneity implies synchronizability [3], combined with the fact that k -homogeneity is equivalent with p n ´ k q -homogeneity, a k -reachable group for any1 ă k ă n ´ Proposition 9. A k -reachable permutation group of degree n for ă k ă n ´ is synchronizable. For k “ we have precisely the primitive permutation groups,and for k “ n ´ precisely the permutation groups which are transitive in theiraction. For k P t , , u the non- k -transitive but k -homogeneous groups where deter-mined by Kantor [13]. A list of all possible k -transitive groups of finite degreefor k ě k ě k -reachability for all 1 ď k ď n , for n ě IBLIOGRAPHY 11
Proposition 10.
If a permutation group of degree n ě is k -reachable for all ď k ď n ´ , then it is either the symmetric group or the alternating group. Or to be more specific.
Proposition 11.
If a permutation group of degree n is k -reachable for ď k ď n ´ , then it is either the symmetric or the alternating group. Note that the classification of the k -transitive groups of finite degree for k ą We have given a new characterization for primitive permutation groups. In ananalogous way, with the property that the set of synchronizing words has maxi-mal state complexity, we have introduced the class of sync-maximal permutationgroups. We have shown that the sync-maximal permutation groups are primi-tive and that 2-homogenous groups are sync-maximal, but not conversely. Moreresults on the structure of these permutation groups would be highly interestingand might be the goal of future investigations. Also, for future investigations,their relation to the synchronizing groups, as introduced in [4], is of interest,as synchronizing groups lie strictly between the 2-homogenous and primitivegroups [3, 4, 18]. Also, we took a closer look at completely reachable automataover binary alphabets. We have given sufficient conditions for completely reach-able automata over binary alphabets to also have the property that their sets ofsynchronizing words have maximal state complexity. Our conditions yield thatthis property is true for a family of automata for which this was previously onlyconjectured. Lastly, we introduced k -reachable permutation groups. But for most k these do not give any groups beside the symmetric and alternating groups. Amore close investigation and characterization of these groups for k P t , , u isstill open. Acknowledgement:
I thank my supervisor, Prof. Dr. Henning Fernau, for givingvaluable feedback, discussions and research suggestions concerning the content of thisarticle. I also thank Prof. Dr. Mikhail V. Volkov for introducing our working group tothe idea of completely reachable automata at a joint workshop in Trier in the springof 2019, from which the present work draws inspiration. Lastly, the argument in theproof of Theorem 3, which closely resembles a proof from [4], was communicated to meby an anonymous referee of a considerable premature version of this work. I therebysincerely thank the referee for this and other remarks.
Bibliography [1] Ananichev, D.S., Gusev, V.V., Volkov, M.V.: Slowly synchronizing automata anddigraphs. In: Hlinen´y, P., Kucera, A. (eds.) Mathematical Foundations of Com-puter Science 2010, 35th International Symposium, MFCS 2010, Brno, CzechRepublic, August 23-27, 2010. Proceedings. Lecture Notes in Computer Science,vol. 6281, pp. 55–65. Springer (2010)[2] Ananichev, D.S., Volkov, M.V., Gusev, V.V.: Primitive digraphs with large ex-ponents and slowly synchronizing automata. Journal of Mathematical Sciences (3), 263–278 (2013)[3] Ara´ujo, J., Cameron, P.J., Steinberg, B.: Between primitive and 2-transitive: Syn-chronization and its friends. EMS Surveys in Math. Sciences (2), 101–184 (2017)2 S. Hoffmann[4] Arnold, F., Steinberg, B.: Synchronizing groups and automata. Theor. Comput.Sci. (1-3), 101–110 (2006)[5] Bondar, E.A., Volkov, M.V.: Completely reachable automata. In: Cˆampeanu, C.,Manea, F., Shallit, J. (eds.) Descriptional Complexity of Formal Systems - 18thIFIP WG 1.2 International Conference, DCFS 2016, Bucharest, Romania, July5-8, 2016. Proceedings. Lecture Notes in Computer Science, vol. 9777, pp. 1–17.Springer (2016)[6] Bondar, E.A., Volkov, M.V.: A characterization of completely reachable automata.In: Hoshi, M., Seki, S. (eds.) Developments in Language Theory - 22nd Interna-tional Conference, DLT 2018, Tokyo, Japan, September 10-14, 2018, Proceedings.Lecture Notes in Computer Science, vol. 11088, pp. 145–155. Springer (2018)[7] Cameron, P.J.: Permutation Groups. London Mathematical Society Student Texts,Cambridge University Press (1999)[8] ˇCern´y, J.: Pozn´amka k homog´ennym experimentom s koneˇcn´ymi automatmi.Matematicko-fyzik´alny ˇcasopis (3), 208–216 (1964)[9] Don, H.: The ˇCern´y conjecture and 1-contracting automata. Electr. J. Comb. (3), P3.12 (2016)[10] Gonze, F., Jungers, R.M.: Hardly reachable subsets and completely reachableautomata with 1-deficient words. Journal of Automata, Languages and Combina-torics (2-4), 321–342 (2019)[11] Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, andComputation. Addison-Wesley Publishing Company (1979)[12] Huppert, B., Blackburn, S.: Finite Groups III. Grundlehren der mathematischenWissenschaften, Springer-Verlag Berlin Heidelberg (1982)[13] Kantor, W.M.: k-homogeneous groups. Mathematische Zeitschrift (4), 261–265(Dec 1972)[14] Livingstone, D., Wagner, A.: Transitivity of finite permutation groups on un-ordered sets. Mathematische Zeitschrift (5), 393–403 (Oct 1965)[15] Maslennikova, M.I.: Reset complexity of ideal languages. CoRR abs/1404.2816 (2014)[16] Maslennikova, M.I.: Reset complexity of ideal languages over a binary alphabet.Int. J. Found. Comput. Sci. (6-7), 1177–1196 (2019)[17] Neumann, P.M.: The Mathematical Writings of ´Evariste Galois. Heritage of Eu-ropean Mathematics, European Mathematical Society (2011)[18] Neumann, P.M.: Primitive permutation groups and their section-regular parti-tions. Michigan Math. J. , 309–322 (2009)[19] Rystsov, I.K.: Estimation of the length of reset words for automata with simpleidempotents. Cybernetics and Systems Analysis (3), 339–344 (May 2000)[20] Sandberg, S.: Homing and synchronizing sequences. In: Broy, M., Jonsson, B.,Katoen, J.P., Leucker, M., Pretschner, A. (eds.) Model-Based Testing of ReactiveSystems. LNCS, vol. 3472, pp. 5–33. Springer (2005)[21] Solomon, R.: A brief history of the classification of the finite simple groups. Bul-letin of the American Mathematical Society (3), 315–352 (2001)[22] Starke, P.H.: Eine Bemerkung ¨uber homogene Experimente. Elektronische Infor-mationsverarbeitung und Kybernetik (later Journal of Information Processing andCybernetics) (4), 257–259 (1966)[23] Volkov, M.V.: Synchronizing automata and the ˇCern´y conjecture. In: Mart´ın-Vide,C., Otto, F., Fernau, H. (eds.) Language and Automata Theory and Applications,2nd Int. Conference, LATA. LNCS, vol. 5196, pp. 11–27. Springer (2008)IBLIOGRAPHY 13 Here we collect some proofs not given in the main text.
Let A “ p Σ, Q, δ q be a completely reachable semi-automaton with n states. Then sc p Syn p A qq “ n ´ n if and only if all -sets of states are pairwisedistinguishable in P A .Proof. In P A , for the singleton sets we have, for each w P Σ ˚ , that | δ pt q u , w q| “
1. Hence, no two singleton sets could be distinguished in P A . Two states thatcould not be distinguished are also called equivalent, and could be merged togive an automaton accepting the same language [11]. Looking at the definition of P A , we find, for completely reachable A , that sc p Syn p A qq “ n ´ n if and onlyif all non-empty non-singleton set, i.e., those with at least two distinct elements,are distinguishable in P A . We will show that the latter condition is equivalentto the fact that we could distinguish all 2-sets of states.If sc p Syn p A qq “ n ´ n , then in particular all 2-sets are distinguishable. Con-versely, assume all 2-subsets of Q are distinguishable. Now, we proceed induc-tively. Suppose the statement is true for k ě
2. Let
A, B be two non-empty non-singleton sets with max t| A | , | B |u ď k `
1. We can assume max t| A | , | B |u “ k ` A and B .(i) | B | ă | A | “ k ` | A | ě
3. Choose two distinct 2-sets t q , q u , t q , q u Ď A . By hypoth-esis, the 2-sets are distinguishable, so we find a word w P Σ such that oneis mapped to a singleton, but not the other. Then | δ p A, w q| ă | A | . And as | δ pt q , q , q , q u , w q| ě
2, the set δ p A, w q is not a singleton set. If δ p B, w q is asingleton, then w distinguishes A and B . Otherwise 2 ď | δ p B, w q| ď | B | ď k and we can apply the induction hypothesis to δ p A, w q and δ p B, w q . So, if u distinguishes these two sets, then wu distinguishes A and B .(ii) | B | “ | A | “ k ` w P Σ ˚ such that 2 ď | δ p A, w q| ď k .If δ p B, w q is a singleton, than both sets are distinguished. Otherwise, if | δ p B, w q| “ k `
1, we continue with δ p A, w q and δ p B, w q as in case (i). If2 ď | δ p B, w q| ď k , then we apply the induction hypothesis.Working our way up to k “ | Q | , we see that we can distinguish all non-emptynon-singleton subsets of Q in P A . As A is assumed to be completely reachable,the minimal automaton has 2 n ´ n states, i.e., sc p Syn p A qq “ n ´ n . [\ Let A “ p Σ, Q, δ q be a completely reachable semi-automaton with n states. Then we can decide in polynomial time if sc p Syn p A qq “ n ´ n .Proof. We essentially use the table filling minimization algorithm from [11]. Notethat, in general, for an automaton A “ p Σ, Q, δ, s , F q , two states t p, q u aredistinguishable if and only if A set is called a singleton set , if it contains precisely one element.4 S. Hoffmann |t p, q u X F | “
1, or2. there exists x P Σ such that t δ p p, x q , δ p q, x qu are distinguishable.This observation directly gives the algorithm, which is called table filling as wecan tabulate the ` | Q | ˘ pairs and successively mark pairs, beginning with the pairsfulfilling the first condition above, until no pair changes anymore. By associatingwith each pair, which is not distinguished so far, a list of pairs that lead intothis pair, which is updated accordingly with condition two above, such that eachpair is contained in at most | Σ | many lists, a algorithm running in time O p| Q | q is possible. I refer to [11] for details.But the essential observation for our purpose is that the start state is ir-relevant for this algorithm, and that the algorithm could independently run onsubsets S Ď Q such that δ p S, x q Ď S for each x P Σ .Now suppose A “ p Σ, Q, δ q is completely reachable. Construct the automa-ton P “ p Σ, Q , δ, s , F q with F “ tt q u | q P Q u , s P Q arbitrary and Q “ tt q, q u | q, q P Q u , i.e., we only take the 2-subsets and the singleton setsof Q as states. Note that, as | δ p S, w q| ď | S | for any S Ď Q , the semi-automaton iscomplete and we have δ p Q , x q Ď Q . Then, run the table filling algorithm on P to find out if the pairs are distinguishable in P . And, by the previous remarks,a pair state t p, q u is distinguishable in P if and only if it is distinguishable in P A . [\ Let G “ x g , . . . , g k y ď S n be a permutation group and f : r n s Ñ r n s a non-permutation. Set Σ “ t g , . . . , g k , f u and A “ p Σ, r n s , δ q with δ p m, g q “ g p m q for m P r n s and g P Σ . If n ą and sc p Syn p A qq “ n ´ n , then G is transitive and A completely reachable.Proof. Suppose n ą
2. As sc p Syn p A qq “ n ´ n , in P A the sets of size n ´ . Hence f must have rank n ´
1. Suppose, without loss of generality,that f pr n sq “ t , . . . , n ´ u . Let A Ď r n s be any set of size n ´
1. So, if we couldreach A in A , it must be by a word containing at least one f , for otherwise itwould only permute r n s . Hence, we have g P G and some h P T A such that A “ g p f p h pr n sqqq . As | A | “ n ´ | f p h pr n sq| “ n ´
1. But for any B Ď r n s ,if | f p B q| “ n ´
1, then f p B q “ t , . . . , n ´ u . Hence A “ g pt , . . . , n ´ uq . Butthen, as g is bijective, g pt uq “ r n sz A . As A was chosen arbitrarily, we could map0 to any other element, i.e., G is transitive. So, if any singleton set is reachable,all singleton sets must be reachable from each other, i.e., the automaton A isstrongly connected, and maximal state complexity of Syn p A q implies that A iscompletely reachable. [\ Every sync-maximal permutation group is primitive.Proof.
By our definition, every permutation group of degree n ď This argument only works for n ą
2. If n “
2, as singletons sets are not distin-guishable, if the state complexity is maximal, the singleton sets do not need to bereachable from Q . Also see Remark 2.IBLIOGRAPHY 15 with [4]. If n ą
2, then, by Proposition 2, the automaton is completely reachable.Hence, by Theorem 3, the group G is primitive. Alternatively, by noting thatsc p Syn p A qq “ n ´ n implies that the automaton A has at least one synchroniz-ing word, i.e., the transformation monoid admits a constant map. Then invokingTheorem 1 gives primitivity. [\ If G ď S n is -homogeneous, then G is sync-maximal.Proof. Assume n ą
1, otherwise the statement is trivially true. Suppose G “x g , . . . , g k y is 2-homogeneous and let f : r n s Ñ r n s be of rank n ´
1. With-out loss of generality, assume f p q “ f p q . As every 2-homogeneous permuta-tion group is primitive [4], by Theorem 3, the semi-automaton A is completelyreachable. By Lemma 1, it is enough to show that all 2-sets are distinguish-able. Let t q , q u , t q , q u Ď Q be two distinct 2-sets. By 2-homogeneity of G ,we find g P G such that g pt q , q uq “ t , u . Then g pt q , q uq ‰ t , u . Hence, | f p g pt q , q uqq| “ | f p g pt q , q uqq| “
2, and the function f g could be writtenas a word over Σ . [\ Let Σ “ t a, b u be a binary alphabet and A “ p Σ, Q, δ q a finite semi-automaton with n ą states. Then, the following conditions are equivalent:1. every subset of size n ´ is reachable,2. exactly one letter acts as a cyclic permutation with a single orbit and theother letter has rank n ´ .Proof. Let us denote the rank of a function f : r n s Ñ r n s by rk p f q . Note thatrk p f g q ď min t rk p f q , rk p g qu (4)for functions f, g : r n s Ñ r n s . Set n “ | Q | . First, assume that every subset of size n ´ a , must have rank | Q | ´
1. For, if not,then any word w P t a, b u ˚ has rank | Q | , if it is composed out of permutations,or a rank strictly smaller than | Q | ´ | Q | ´ S “ δ p Q, a q . If rk p b q ď | Q | ´ T ‰ S of size | Q | ´ p b q “ | Q | ´ T “ δ p Q, b q . Write Q “ t s u Y S “ t t u Y T with s, t P Q . Let R Ď Q with | R | “ | Q | ´ R R t
S, T u . Then t r, s u Ď R .But, inductively, for every w P t a, b u ˚ , we have either s R δ p Q, w q , if w ends with a , or t R δ p Q, w q , if w ends with b . So, R is not reachable. Hence rk p b q “ n ,i.e., b permutes Q . Note that every set R ‰ S of size | Q | ´ R “ δ p Q, ab n q for some n ě
1, as the condition R ‰ S implies s P R . Now,suppose the permutation given by b has two distinct orbits A, B Ď Q . Then oneorbit, say A , is contained entirely in S , i.e. A Ď S . But then, A Ď δ p S, b m q foreach m ě
0. So, by the previous observations, subsets R of size | Q | ´ A would not be reachable. Hence, b has only a single orbit.Now, conversely assume that the letter a has rank n ´ b permutes the states in a single cycle. Set S “ δ p Q, a q and write Q “ S Y t s u .Let R Ď Q be any subsets of size n ´ r P Q z R . Then, δ p s, b m q “ r forsome m ě
0. But then, as b is bijective, δ p S, ab m q “ R . Hence, every subset ofsize n ´ [\ Let Σ be a binary alphabet and A “ p Σ, Q, δ q a semi-automatonwith n ą states. If sc p Syn p A qq “ n ´ n , then A is completely reachable.Proof. If sc p Syn p A qq “ n ´ n , then in P A all non-empty non-singleton sets arereachable from the start state Q and at least one singleton set must be reachable.In particular, all subsets of size n ´ [\ Let t a, b u Ď Σ and suppose A “ p Σ, Q, δ q has n states. Assumethe letter b permutes the states with a single orbit and the letter a has rank n ´ .Then all -sets are distinguishable in P A , if we can find a state q P Q and anumber d ą coprime to n such that for each ă m ă n we either have δ p q, b m a q “ δ p q, ab n ` m ´ d q (1) or, but only in case m is not divisible by d , δ p q, ab r q “ δ p q, b m a q or δ p q, b m ab r q “ δ p q, a q (2) for some number ď r ă n divisible by d .Proof. Choose the notation for q P Q and r, d ą n ą
1. Note that δ p q, b d a q “ δ p q, a q , where δ p q, b d q ‰ q , as d is coprime to n . Hence, as a has rank n ´
1, it is injective on Q zt q u . Choose 0 ď s ă n such that δ p q, a q “ δ p q, b s q . Then note that, as δ p q, b m a q “ δ p q, b s ` n ` m ´ d q or δ p q, b m a q “ δ p q, b s ` r q or δ p q, b m a q “ δ p q, b s ` n ´ r q , the set of these equationstogether with the state δ p q, a q completely determine the action of the letter a .Let t p , q u , t s , t u Ď Q be two distinct 2-sets. We want to show that wecan distinguish them in P A . First, let u P b ˚ be such that δ p q , u q “ q . Byapplying u to both subsets of Q , we can assume that q “ q with respect todistinguishability. Write p “ δ p q , b m q for some 0 ă m ă n . Define, for i ě t p i u , t q i u , t s i u and t t i u by x i ` “ δ p x i , ab n ´ s q , x P t p, q, s, t u where p , q o , s , t are given as above. Also set 0 ď m i ă n with δ p q i , b m i q “ p i .Then, we have the following.Claim 1: For all i ě q i “ q . Proof of Claim 1.
It is q “ q by definition. Then, inductively assuming q i “ q , as δ p q, a q “ δ p q, b s q , we have q i ` “ δ p q i , ab n ´ s q “ δ p q, ab n ´ s q “ δ p q, b n q “ q . [\ Note that 0 ă m ă n implies δ p q, b m q ‰ q . Also note that we added n on the righthand side to account for values d ą
1. In Proposition 7 we only subtract one fromthe exponent of b , which is always non-zero and strictly smaller than n , and so we donot needed this “correction for the b -cycle” in case of resulting negative exponents.IBLIOGRAPHY 17 Claim 2: Let i ě |t p i , q i u| “ |t s i , t i u| “
2. If p i ` “ q i ` ,then s i ‰ t i ; and if s i ` “ t i ` , then p i ` ‰ q i ` . Proof of Claim 2. As b is a permutation of Q and a has rank n ´
1, thetransformation given by the word ab n ´ s has rank n ´
1. So, as preciselyone pair is collapsed, we have that if t p i , q i u ‰ t s i , t i u and both are 2-subsets, then t p i ` , q i ` u ‰ t s i ` , t i ` u , as only |t p i ` , q i ` , s i ` , t i ` u| Pt , , u is possible because at most one 2-set could be collapsed. Notethat, as by assumption t p , q u ‰ t s , t u , this gives inductively t p i , q i u ‰t s i , t i u for all i with |t p i , q i u| “ |t s i , t i u| “
2. As |t p i , q i u| “ |t s i , t i u| “ t p i , q i u ‰ t s i , t i u and as at most onepair could be collapsed the claim follows. [\ Claim 3: There exists some i ą p i “ q i “ q . Proof of Claim 3.
First, suppose we have some i ě |t p i , q i u| “ a , Equation (2) applies, i.e., we have δ p q i , ab r q “ δ p q i , b m i a q or δ p q i , b m i ab r q “ δ p q i , a q . Write r “ md for some m ą δ p q i , ab r q “ δ p q i , b m i a q “ δ p p i , a q .Then p i ` “ δ p p i , ab n ´ s q “ δ p q i , ab n ´ s ` r q “ δ p q i ` , b r q “ δ p q i , b r q ,using q i “ q , by Claim (i), and δ p q, ab n ´ s q “ q . Then δ p p i ` , a q “ δ p q, b r a q . As r ” p mod n q , Equation (1) must apply, and, as 0 ă r ă n , we find δ p p i ` , a q “ δ p q, ab n ` r ´ d q , which equals δ p q, ab r ´ d q as r ´ d ě
0. So, p i ` “ δ p p i ` , ab n ´ s q “ δ p q, ab n ´ s ` r ´ d q “ δ p δ p q, ab n ´ s , b r ´ d q “ δ p q, b r ´ d q . Continuing in-ductively for m ´ p i ` ` m “ δ p q, b r ´ mr q “ q “ q i ` ` m . (ii) Suppose δ p q i , b m i ab r q “ δ p q i , a q .As p i “ δ p q i , b m i q , we have δ p p i , ab r q “ δ p q i , a q . Hence, for 0 ď j ă n such that δ p p i , ab j q “ q , we have δ p q i , ab j q “ δ p δ p p i , ab r q , b j q “ δ p δ p p i , ab j q , b r q “ δ p q, b r q . Then we can write r “ md and proceedexactly as in Case (i), to find δ pt q, δ p q, b r qu , p ab n ´ s q m q “ t q u . So that δ pt δ p p i , a q , δ p p i , ab r qqu , b j p ab n ´ s q m qq “ t q u .Otherwise, after reading the letter a in the process of constructing thesequences p i and q i , only Equation (1) applies. As d is coprime to n , wefind k ě m ´ kd ” p mod n q , i.e., m ´ kd ` ln “ k ě l (as 0 ď m ă n , we have l ď k , for l ą k would imply ln ą kd , andso we could not have ln ` m “ kd ). But note that l ď Equation (1) and using k ě l , p k “ δ p p , p ab n ´ s q k q“ δ p q , b m p ab n ´ s q k q r p “ δ p q , b m qs“ δ p q, b m p ab n ´ s q k q [ q “ q ] “ δ p q, b m ab n ´ s p ab n ´ s q k ´ q“ δ p q, ab n ` m ´ d b n ´ s p ab n ´ s q k ´ q [Equation (1)] “ δ p q, b n ` m ´ d p ab n ´ s q k ´ q [ δ p q, ab n ´ s q “ q ] “ δ p q, ab n ` m ´ d b n ´ s p ab n ´ s q k ´ q ... “ δ p q, b p k ´ q n ` m ´p k ´ q d ab n ´ s q“ δ p q, b kn ` m ´ kd q“ δ p q, b kn ´ ln q“ δ p q, b p k ´ l q n q “ q “ q k . So, the word w “ p ab n ´ s q k maps both states t p , q u to the same state q . [\ The Claims (2) and (3) imply that we have some word w P p ab n ´ s q ˚ whichcollapses precisely one 2-set. For, by Claim (3), we must have some smallest i ą |t p i ´ , q i ´ u| “ |t s i ´ , t i ´ u| “ p i “ q i , or s i “ t i ,but not both could be equal by Claim (2) and the minimality of i . [\ Let n ą be odd. Then we have sc p Syn p K n qq “ n ´ n .Proof. First, we will show, using Proposition 1, that the automata K n , for odd n ą
5, are completely reachable. Then, we will show, using Proposition 5, thatall 2-subsets of states are distinguishable in the power automaton P K n . WithLemma 1, this would then give sc p Syn p K n qq “ n ´ n .1. For n ą K n are completely reachable.We have two letters, the letter a has rank n ´ b is a cyclicpermutation of all the states. Also δ p Q, a q “ Q zt n ´ u , δ ´ p , a q “ t , u and δ p n ´ , b q “
3. If n is odd, then n and 4 are coprime. We have listed theprerequisites of Proposition 1, hence applying it gives that K n is completelyreachable.2. For n ą K n all 2-sets are distinguishable in P K n .Let q “
0. Then δ p q, ba q “ “ δ p q, ab n ´ q “ δ p q, ab n ` ´ q . For m P t , . . . , n ´ u , we have δ p , b m q “ m and δ p q, b m a q “ m ` “ δ p , b m ´ q “ δ p q, ab m ´ q “ δ p q, ab n ` m ´ q . IBLIOGRAPHY 19
The value m “ n ´ δ p q, b n ´ ab q “ δ p , b q “ “ δ p q, a q . And lastly, for m “ n ´
1, we have δ p q, b n ´ a q “ q “ δ p q, ab n ´ ´ q . So, with d “ r “
2, for odd n , as then n ´ d ,and with q “
0, the prerequisites of Proposition 5 are fulfilled and give theclaim.So, both statements taken together with Lemma 1 yield sc p Syn p K n qq “ n ´ n . [\ Let Σ “ t a, b u and suppose A “ p Σ, Q, δ q has n states and iscompletely reachable with the letter a having rank n ´ and the letter b permutingthe states with a single orbit. Then sc p Syn p A qq “ n ´ n if we can find a state q P Q such that for all ď m ď t n { u ´ we have δ p q, b m ` a q “ δ p q, ab m q . (3) Proof.
Let q P Q be the state from the statement. Note that, as δ p q, ba q “ δ p q, a q and a has rank n ´
1, the letter a acts injective on Q zt q u . Also, on all states δ p q, b k q with 0 ď k ď t n { u the action of a is determined by the state δ p q, a q .We will show that all 2-sets of states are distinguishable in P A . By Lemma 1,this will give our claim. Let t s, t u , t p, r u Ď Q be two distinct 2-sets. Choose m , m ą δ p s, b m q “ t or δ p t, b m q “ s (5)and δ p p, b m q “ r or δ p r, b m q “ p. (6)As δ p t, b m q “ s if and only if δ p s, b n ´ m q “ t , and similarly for t p, r u , wehave 0 ă m , m ď t n { u . We will do induction on min t m , m u . Without lossof generality, assume m ď m and δ p s, b m q “ t . Choose 0 ď k ă n such that δ p s, b k q “ q . Then, by assumption, δ p s, b k ` m a q “ δ p s, b k ab m ´ q . We distinguishtwo cases.(i) Suppose s R t p, r u . Then q R δ pt p, r u , b k q . As a acts injective on Q zt q u , wehave | δ pt p, r u , b k a q| “
2. If m “ δ p s, b k a q “ δ p s, b k ab m ´ q “ δ p s, b k ` m a q “ δ p t, b k a q . Hence | δ pt s, t u , b k a q| “ t s, t u and t p, r u are distinguished in P A by b k a . If m ą
1, then δ p s, b k ab m ´ q “ δ p s, b k ` m a q “ δ p t, b k a q . As0 ă m ´ ă t n { u , we have |t δ p s, b k a q , δ p t, b k a qu| “
2. Hence, for the two2-sets t δ p s, b k a q , δ p t, b k a qu and t δ p p, b k a q , δ p r, b k a qu the minimal powers of b that map them to each other, i.e., fulfill the cor-responding Equations (5) and (6), have strictly smaller exponents thanmin t m , m u . So, we can use our induction hypothesis, implying that someword u P Σ ˚ maps one set to a singleton set, but not the other. Then b k au would distinguish t s, t u and t p, r u . (ii) Suppose s P t p, r u . Without loss of generality, assume δ p p, b m q “ r . If s “ p , then m “ m would imply r “ t , which is excluded as both 2-setsare assumed to be distinct. Hence, as m “ min t m , m u , we have m ă m .If m “
1, then, as in case (i), we have | δ pt s, t u , b k a q| “
1, but δ p q, b q R t δ p p, b k q , δ p r, b k qu . As δ p q, b q “ δ p r, b k q would imply, as b is a permutation, that δ p p, b q “ r . But, as p “ s , δ p p, b k q “ q , and δ p p, b m q “ r is minimal with 1 ă m ă t n { u . Hence δ p p, b q “ r would contradict the minimality of m .So, as a acts injective on Q zt δ p q, b qu , as it only collapses t q, δ p q, b qu , wehave |t δ p p, b k q , δ p r, b k qu| “
2. So, the word b k a distinguishes t s, t u and t p, r u .Now suppose m ą
1. Then, as in case (i), δ p s, b k ab m ´ q “ δ p t, b k a q and |t δ p s, b k a q , δ p t, b k a qu| “
2. Similarly, as 1 ă m ă n , we get |t δ p p, b k a q , δ p r, b k a qu| “