An Infinite Game over ω-Semigroups tfs07

Table of Contents An Infinite Game over ω-Semigroups . . . . . . . . . . . . . . . . . . . . . . . . 1 J ´ er ´ emie Cabessa, Jacques Duparc ii Stefan Bold, Benedikt L ¨ owe, Thoralf R ¨ asch, Johan van Benthem ( eds. ) Infinite Games Papers of the conference “Foundations of the Formal Sciences V” held in Bonn, November 26-29, 2004 An Infinite Game over ω-Semigroups J ´ er ´ emie Cabessa, Jacques Duparc  University of Lausanne Centre Romand de logique, histoire et philosophie des sciences ch. de la Colline 12 CH-1015 Lausanne Switzerland E-mail: jeremie.cabessa@unil.ch, jacques.duparc@unil.ch Abstract. Jean- ´ Eric Pin introduced the structure of an ω-semigroup in [PerPin04] as an algebraic counterpart to the concept of automaton reading infinite words. It has been well studied since, specially by Carton, Perrin [CarPer97] and [CarPer99], and Wilke. We introduce a reduction relation on subsets of ω-semigroups defined by way of an infinite two-player game. Both Wadge hierarchy and Wagner hier- archy become special cases of the hierarchy induced by this reduction relation. But on the other hand, set theoretical properties that occur naturally when study- ing these hierarchies, happen to have a decisive algebraic counterpart. A game theoretical characterization of basic algebraic concepts follows. 1 Introduction This work comes from an interaction between classical game theory, and the algebra of automata theory, which rests on the following main facts. In case of finite words, a well-known correspondence between an automaton and a finite semigroup exists: from any finite automaton A Received: ; In revised version: ; Accepted by the editors: 2000 Mathematics Subject Classification. PRIMARY SECONDARY.  This line is for acknowledgments. c  2006 Kluwer Academic Publishers. Printed in The Netherlands, pp. 1–16. 2 J ´ ER ´ EMIE CABESSA, JACQUES DUPARC recognizing a regular language L, one can build a finite semigroup S A recognizing (in an algebraic way) the same language, and vice-versa [PerPin04]. Moreover, this correspondence generalizes in case of infi- nite words. Indeed, for that purpose, J ´ E. Pin introduced the structure of ω-semigroup [PerPin04] as an algebraic counterpart to the concept of an automaton on infinite words. More precisely, he proved the equivalence between a finite B¨uchi automaton and a finite ω-semigroup. This paper presents a game theoretical study of the structure of ω- semigroup, leading to an expected new foundation of the Wagner hierar- chy, but also to promising general set theoretical, and algebraic results. 2 Preliminaries We recall that a relation R is a preorder if it is reflexive, and transitive. It is a partial order if it is reflexive, transitive, and antisymmetric. And it is an equivalence relation if it is reflexive, transitive, and symmetric. Given a set A (called the alphabet), we respectively denote by A ∗ , A + , A ω , the sets of finite words over A, non empty finite words over A, and infinite words over A. We set A ∞ = A ∗ ∪ A ω , and the empty word is denoted by . Given two words u and v (u finite), we write uv for the concatenation of u and v, u ⊆ v for ”u is an initial segment of v”, v  n for the restriction of v to its n first letters. Given X ⊆ A ∗ , and Y ⊆ A ∞ , we set: XY = {xy : x ∈ X ∧y ∈ Y }, X ∗ = {x 1 · · · x n : n ≥ 0 ∧ x 1 , . . . , x n ∈ X}, X + = {x 1 · · · x n : n > 0 ∧ x 1 , . . . , x n ∈ X}, and X ω = {x 0 x 1 x 2 · · · : ∀ n ≥ 0, x n ∈ X}. The class of ω-rational subsets of A ∞ is the smallest class of subsets of A ∞ containing the finite subsets of A ∞ , and closed under finite union, finite product, and both operations X → X ∗ , and X → X ω . A semigroup (S, ·) is a set S equipped with an associative operation from S × S into S. A morphism of semigroups is a map φ from a semi- group S into a semigroup T such that ∀ s 1 , s 2 ∈ S, φ(s 1 s 2 ) = φ(s 1 )φ(s 2 ) holds. A monoid is a set equipped with an associative operation, and an identity element. If S is a semigroup, S 1 denotes S if S is a monoid, and S ∪ {1} otherwise (with the operation of S completed as follows: 1 · s = s · 1 = s , ∀s ∈ S). A group G is a monoid such that every element has an inverse, i.e. ∀ s ∈ G ∃ s −1 ∈ G s.t. s −1 · s = s · s −1 = 1. For any set A, the set A ω is a topological space equipped with the product topology of the discrete topology on A. The basic open sets of AN INFINITE GAME OVER ω-SEMIGROUPS 3 A ω are of the form W A ω , where W ⊆ A ∗ . Given a topological space E, the class of Borel subsets of E is the smallest class containing the open sets, and closed under countable union, and complementation. Let F ⊆ 2 ω , F is a flip set [And03] iff ∀ x, y ∈ 2 ω (∃! k ∈ ω (x(k) = y(k))) → (x ∈ F ↔ y ∈ F ). We use the fact that a flip set cannot be Borel (as it doesn’t satisfy the Baire property). Let Σ be a set, and A ⊆ Σ ω . The Gale-Stewart game G(A) [GalSte53] is a two-player infinite game with perfect information where players take turn playing letters from Σ. Player I begins. After ω moves, they produce an infinite word α ∈ Σ ω . Player I wins iff α ∈ A. A play of this game is illustrated below. I : x 0 x 2 · · · · · · · · · x n x n+2 · · · · · · · · ·      II : x 1 · · · · · · x n+1 · · · · · · Let Σ A , Σ B be two sets, and A ⊆ Σ A ω , B ⊆ Σ B ω . The Wadge game W(A, B) [Wad72] is a two-player infinite game with perfect informa- tion, where player I is in charge of subset A, and player II is in charge of subset B. Players take turn playing letters from Σ A and Σ B , respectively. Player I begins. Player II is allowed to skip provided he plays infinitely many letters; player I is not. After ω moves, player I and II have respec- tively produced two infinite words α ∈ Σ A ω , and β ∈ Σ B ω . Player II wins in W(A, B) iff (α ∈ A ↔ β ∈ B). A play of this game is illus- trated below. (A) I : a 0 a 1 · · · · · · after ω moves −→ α = a 0 a 1 a 2 · · ·    (B) II : b 0 b 1 · · · · · · after ω moves −→ β = b 0 b 1 b 2 · · · 3 ω-semigroups J ´ E. Pin introduced the structure of an ω-semigroup [PerPin04] in order to give an algebraic counterpart to the notion of automaton reading infi- nite words. He showed the equivalence between a finite B¨uchi automaton and a finite ω-semigroup in the following sense: – For any finite B¨uchi automaton A recognizing the language L(A), one can build a finite ω-semigroup S A recognizing (in an algebraic sense) the same language L(A). 4 J ´ ER ´ EMIE CABESSA, JACQUES DUPARC – For any finite ω-semigroup S recognizing the language L(S), one can build a finite B¨uchi automaton recognizing the same language L(S). Definition 3.1. [PerPin04] An ω-semigroup is an algebra consisting in two components, S = (S + , S ω ), and equipped with the following oper- ations: • A binary operation defined on S + and denoted multiplicatively. • A mapping S + × S ω → S ω called mixed product, that associates with each pair (s, t) ∈ S + × S ω an element st of S ω . • A surjective mapping π S : S + ω → S ω called infinite product. Moreover, these three operations must satisfy the following properties: 1. S + equipped with the binary operation is a semigroup, 2. ∀ s, t ∈ S + ∀ u ∈ S ω s(tu) = (st)u, 3. the infinite product π S is ω-associative, meaning that for every strictly increasing sequence of integers (k n ) n>0 , and for every sequence (s n ) n∈ω ∈ S + ω , we have π S (s 0 s 1 · · · s k 1 −1 , s k 1 · · · s k 2 −1 , . . .) = π S (s 0 , s 1 , s 2 , . . .), 4. ∀ s ∈ S + ∀ (s n ) n∈ω ∈ S + ω sπ S (s 0 , s 1 , s 2 , . . .) = π S (s, s 0 , s 1 , s 2 , . . .). Intuitively, an ω-semigroup is just a semigroup equipped with a suit- able infinite product. It is finite precisely when S + is finite. Otherwise it is infinite. A subset X ⊆ S ω is called an ω-subset. We focus on those subsets in the sequel. Definition 3.2. Let S = (S + , S ω ), T = (T + , T ω ) be two ω-semigroups. A morphism of ω-semigroups from S into T is a pair φ = (φ + , φ ω ), where φ + : S + −→ T + is a morphism of semigroups, and φ ω : S ω −→ T ω is a mapping preserving the infinite product, i.e. for every sequence (s n ) n∈ω of elements of S + , one has φ ω  π S (s 0 , s 1 , s 2 , . . .)  = π T  φ + (s 0 ), φ + (s 1 ), φ + (s 2 ), . . .  . Example 3.3. Let A be an alphabet. The ω-semigroup A ∞ = (A + , A ω ) AN INFINITE GAME OVER ω-SEMIGROUPS 5 equipped with the usual concatenation is the free ω-semigroup over al- phabet A. It is free in the sense that, for any ω-semigroup S = (S + , S ω ), any function f from A into S + can uniquely be extended to a morphism of ω-semigroups ¯ f = (f + , f ω ) from A ∞ into S [CarPer97]. We do this by setting f + : A + −→ S + defined by f + (a 0 a 1 · · · a n ) = f (a 0 )f(a 1 ) · · · f (a n ) , with a i ∈ A (∀i ≤ n), and f ω : A ω −→ S ω defined by f ω (a o a 1 a 2 · · · ) = π S (f(a 0 ), f(a 1 ), f(a 2 ), . . .) , with a i ∈ A (∀i). So, sets of ω-words, in other words sets of reals, are the less constraint ones with regard to the algebraic structure. In order to state further results, we put the following topology on ω-subsets: Definition 3.4. Let S = (S + , S ω ) be any ω-semigroup, and X ⊆ S ω , we set: X is a basic open if and only if π −1 S (X) is an open of S + ω where S + ω is equipped with the product topology of the discrete topology on S + . Remark 3.5. For any ω-semigroup S = (S + , S ω ), the infinite product π S is a continuous function by definition of the previous topology. Remark 3.6. At first glance, the topology defined by taking sS ω = def {st : t ∈ S ω } as a basic open set (for any s ∈ S + ) would look much nicer. Unfortunately, this topology is much too weak for our purpose. Indeed, with this particular topology, in case S + is a group, Borel subsets of S ω come down to the empty set and the whole space; the reason being that, given sS ω any basic open set, then S ω = ss −1 S ω ⊆ sS ω , meaning that sS ω = S ω . We certainly need much more than that as we’ll see in the last section. 4 An infinite game over ω -semigroups In this section, we define a reduction relation between ω-subsets by use of an infinite two-player game over ω-semigroups. We then state some general properties of this reduction relation in order to characterize the set hierarchy that it generates. 6 J ´ ER ´ EMIE CABESSA, JACQUES DUPARC 4.1 Definitions Definition 4.1. Let S = (S + , S ω ), T = (T + , T ω ) be two ω-semigroups, and X, Y be two ω-subsets of S ω and T ω , respectively. The infinite two- player game SG(X, Y ) is defined as follows: player I is in charge of subset X, player II is in charge of subset Y . Players I and II alternately play elements of S + and T + ∪ {}, respectively. Player I begins, player II is allowed to skip its turn (by playing ) provided he plays infinitely many moves, otherwise he loses the play. Player I cannot skip its turn. After ω moves, players I and II have respectively produced two infinite sequences s 0 , s 1 , . . ., and t 0 , t 1 , . . .. A play of this game is illustrated below. (X) I : s 0 s 1 · · · · · · after ω moves −→ s 0 , s 1 , s 2 , . . .    (Y ) II : t 0 t 1 · · · · · · after ω moves −→ t 0 , t 1 , t 2 , . . . The winning condition is the following: player II wins in SG(X, Y ) if and only if π S (s 0 , s 1 , . . .) ∈ X ⇔ π T (t 0 , t 1 , . . .) ∈ Y where π S and π T are the infinite products of S and T respectively, and π T (t 0 , . . . , t n−1 , , t n , . . .) = def π T (t 0 , . . . , t n−1 , t n , . . .), meaning that the skipping moves of II are not considered in the infinite product. A strategy for player II is a mapping σ : S + + → T + ∪{}. A strategy for player I is defined similarly. A winning strategy for a player (w.s.) is a strategy such that the player always wins when using it. We can now define the following reduction relation: X ≤ SG Y ⇔ def II has a w.s. in SG(X, Y ) and of course X < SG Y ⇔ def X ≤ SG Y but Y ≤ SG X X ≡ SG Y ⇔ def X ≤ SG Y and Y ≤ SG X Following the terminology of Wadge games, we set that: AN INFINITE GAME OVER ω-SEMIGROUPS 7 • an ω-subset X is self-dual (s.d.) iff X ≡ SG X c where X c stands for the complement of X. Otherwise, we say that X is non-self-dual (n.s.d.); • an ω-subset X is initializable iff there exists Y such that X ≡ SG Y and Y ≡ SG s −1 Y , ∀ s ∈ S + where s −1 Y = {x ∈ S ω : x = π S (u 1 , u 2 , . . .) ∧ π S (s, u 1 , u 2 , . . .) ∈ Y }. From a playful point of view, a player in charge of a initializable set X in the SG-game never loses his playful strength during the play. Indeed, for any position s ∈ S + that he reaches, he remains as strong as at the beginning, when being in charge of the whole subset X. Example 4.2. Let S = (S + , S ω ) be any ω-semigroup, and X ⊆ S ω , with X = ∅, S ω . • The relation ∅ ≤ SG X holds. Indeed, we give a w.s. for player II in the game SG(∅, X). At the end of the play, the infinite product of any infinite sequence played by I obviously doesn’t belong to ∅. So the w.s. for II simply consists in playing in order to be outside X at the end of the play (possible, as X = S ω ). • Similarly, the relation S ω ≤ SG X holds. The w.s. for II in the game SG(X, S ω ) consists in in playing in order to be inside X at the end of the play (possible, as X = ∅). • The relation ∅ ≤ SG S ω holds. Indeed, at the end of the play, the infinite product of any infinite sequence played by I doesn’t belong to ∅, and the infinite product of any infinite sequence played by II belongs to S ω , so that II cannot win against I in any case. • Similarly, the relation S ω ≤ SG ∅ holds, as there is no possible w.s. for II in the game SG(S ω , ∅). This shows that the empty set and the whole space are non-self-dual sets, since no one is equivalent to its complement. Moreover, any other set reduces to both of them. 8 J ´ ER ´ EMIE CABESSA, JACQUES DUPARC 4.2 Properties of the SG-relation Not using yet any determinacy principle for this game, one cannot say much of the SG-relation, except that it is a partial ordering with no par- ticular interesting properties. However, Martin’s Borel Determinacy re- sult [Mar75] easily induces Borel Determinacy for SG-games. As it is the case with the Wadge ordering, this property turns the SG-relation into a much more interesting one. Theorem 4.3. (Martin) Let Σ be a set. If A is a Borel subset of Σ ω , then G(A) is determined. Corollary 4.4. (SG-Borel Determinacy) Let S = (S + , S ω ), T = (T + , T ω ) be two ω-semigroups, and X ⊆ S ω , Y ⊆ T ω be two Borel ω-subsets. Then SG(X, Y ) is determined. Proof. We define a Borel subset Z ⊆ (S + ω ∪ T + ω ∪ {}) ω such that a player P has a w.s. in G(Z) iff the same player P has a w.s. in SG(X, Y ). Let p 1 and p 2 be the following continuous projections from (S + ∪ T + ∪ {}) ω into (S + ∪ T + ∪ {}) ω defined by p 1 (u 0 u 1 u 2 u 3 . . .) = u 0 u 2 u 4 . . ., and p 2 (u 0 u 1 u 2 u 3 . . .) = u 1 u 3 u 5 . . Let X  , X  , Y  , Y  ⊆ (S + ∪ T + ∪ {}) ω be defined by X  = {α = u 0 u 1 u 2 . . . : π S (u 0 , u 2 , u 4 , . . .) ∈ X} = p −1 1 (π −1 S (X)) X  = {α = u 0 u 1 u 2 . . . : π S (u 0 , u 2 , u 4 , . . .) ∈ X c } = p −1 1 (π −1 S (X c )) Y  = {α = u 0 u 1 u 2 . . . : π T (u 1 , u 3 , u 5 , . . .) ∈ Y } = p −1 2 (π −1 T (Y )) Y  = {α = u 0 u 1 u 2 . . . : π T (u 1 , u 3 , u 5 , . . .) ∈ Y c } = p −1 2 (π −1 T (Y c )) By continuity of the functions p 1 , p 2 , π S , π T , these sets are all Borel, and we conclude by taking Z = (X  ∩ Y  ) ∪ (X  ∩ Y  ).  Similarly to the Wadge ordering, and as a consequence of Borel de- terminacy for these games, come the following interesting results. The first one is an immediate consequence of determinacy. The second one is a corollary of the first one: it states that, for this partial ordering ≤ SG , the antichains have length at most two. The third one is a result from Martin and Monk establishing the wellfoundness of this ≤ SG -relation on Borel ω-subsets. [...]... assume that there exists an infinite sequence of ω-semigroups {Si = (Si,+ , Si,ω )}i∈ω , and an infinite strictly SG An+ 1 implies that both An ≤SG An+ 1 and Ac ≤SG An+ 1 hold, meaning that player I has n 0 1 w.s σn and σn in both games SG (An , An+ 1 ) and SG(Ac , An+ 1 ), respecn α(k) tively... in the SG -game is allowed to skip his turn, provided he plays infinitely many letters, otherwise he loses (3) There exists an ω-semigroup T = (T+ , Tω ) and a Borel ω-subset Y ⊆ Tω such that T+ is a monoid, and X ≡SG Y Proof (sketch) (1) ⇒ (2) : We show that we can assume without loss of generality that any player in charge of X in the SG -game can skip his turn, provided he plays infinitely many letters... the SG -game is allowed to erase his moves, provided he plays infinitely many letters, otherwise he loses (3) There exists an ω-semigroup T = (T+ , Tω ) and a Borel ω-subset Y ⊆ Tω such that T+ is a group, and X ≡SG Y Proof (sketch) (3) ⇒ (2) : We show that we can assume without loss of generality that any player in charge of X in the SG -game can erase his moves, provided he plays infinitely many letters... consider ω many SG-games linked this way: in the first game, α(0) player I applies strategy σ0 to II’s play Since it is a strategy for I, it gives the first letter a0 before II has ever played anything, but then, apply0 α(0) ing σ0 means to know II’s first move a1 Precisely, II copies I’s moves 0 α(1) in the second game, in which I applies the w.s σ1 And so on for every α(n) game This means, in game number... algebraic properties Important algebraic notions can be expressed in a natural game theoretical way by use of the SG -game These results militate in favor of developing the use of game theoretical tools in algebra The two following propositions give a game theoretical approach of the algebraic concepts of monoid and group Proposition 5.5 Let S = (S+ , Sω ) be any ω-semigroup, and X ⊆ Sω be any Borel ω-subset... Finite Rank, Journal of Symbolic Logic 66 (2001) no 1, p 56–86 Jacques Duparc, Wadge Hierarchy and Veblen Hierarchy Part II : Borel Sets of Infinite Rank Journal of Symbolic Logic, to appear Jacques Duparc, Mariane Riss The Missing Link for ω-Rational Sets, Automata, and Semigroups, International Journal of Algebra and Computation (2003), to appear Gale, Stewart Infinite Games with Perfect Information, Annals... By identifying b and the identity element, i.e by setting the monoid T+ = (S+ ∪{b})+ = (S+ ∪{1})+ , Tω = (S+ ∪{b})ω = (S+ ∪{1})ω , ¯ and by taking Y = Y b ⊆ Tω , one gets the result Proposition 5.6 Let S = (S+ , Sω ) be any ω semigroup, and X ⊆ Sω be any Borel subset The following conditions are equivalent: (1) X ≤SG s−1 X, ∀ s ∈ S+ (i.e X is initializable) AN INFINITE GAME OVER ω-SEMIGROUPS 15 (2)... by φ of a basic open set is a basic open set As φ and πS0 are continuous, so is ψ Consider B = ψ −1 (A0 ) By construction of these chained games, we notice that if α and α only differ by one position (i.e ∃! i s.t α(i) = α (i)), then α ∈ B ⇔ α ∈ B This means that B is a flip set, and it is Borel as ψ is continuous, a contradiction AN INFINITE GAME OVER ω-SEMIGROUPS 11 Remark 4.8 Quotienting Borel ω-subsets... hierarchy among languages recognized by Muller automata called the Wagner hierarchy [Wag79] This hierarchy has height ω ω and actually coincides with the restriction of the Wadge hierarchy to ω-rational languages In other words, it is the hierarchy induced by the following ordering on Muller automata: A ≤W B iff the language recognized by A is the inverse image of the language AN INFINITE GAME OVER ω-SEMIGROUPS. .. first, second and fourth cases come from the very definition The third case comes by the previous proposition, and by the obvious fact that A ≤SG B ⇔ Ac ≤SG B c holds, for any ω-subset A and B Proposition 4.7 (Martin, Monk) The partial ordering SG A1 >SG >SG An >SG An+ 1 >SG