The Alternation Hierarchy for the Theory of Μ-Lattices

The alternation hierarchy for the theory of µ-lattices

Luigi Santocanale BRICS† [email protected]

Abstract. We examine the alternation hierarchy problem for the theory of µ-lattices and give it a solution by showing that the hierarchy is strict.

The alternation hierarchy problem is at the core of the definition of categories of µ-algebras [11, 12] which we resume as follows. For a given equational theory T, we let T0 be the category of its partially ordered models and order preserving morphisms. Out of T0 we can select objects and morphisms so that all the “desired” least prefix-points exist and are preserved, this process giving rise to a category S1. The desired least prefix-points are those needed to have models of an iteration theory [3], which we denote S1, where the dagger operation is interpreted as the least prefix-point. If we use as a selection criterion the existence of greatest postfix-points and their preservation, we obtain a category P1 and a corresponding iteration theory P1. We let T1 be the intersection of S1 and P1, let T1 be the theory determined by T1, and repeat the process out of T1 and T1. The iteration of this process leads to construct categories Sn, Pn, Tn for arbitrary positive numbers n and the category of µ-T-algebras is defined to be the inverse limit of the corresponding diagram of inclusions; the alternation hierarchy problem asks whether this process stops after a finite number of steps, i.e. whether the category of µ-T-algebras is equivalent to a category among Sn, Pn, Tn for some bounded n. The main contribution presented here is theorem 1.6 stating that the alternation hierarchy for the theory of µ-lattices is strict, i.e. that there is no such number. The alternation hierarchy for the propositional µ-calculus has recently been shown to be strict in several cases [1, 4, 10]; together with open problems on fix- point free polynomials in free lattices [5], these results have challenged us to the hierarchy problem for the theory of µ-lattices; in particular we were interested in understanding whether the explicit characterisation of free µ-lattices [13, 14] could be of help. It is our opinion that µ-algebras are algebraic objects suitable to generalise the role of iteration theories in the context of the theorisation of interactive computation. This statement is exemplified by the consideration of free µ-lattices. These have been characterised by means of games and strategies and provide a natural semantics for a class of simple interactive systems recursively built up †Basic Research in Computer Science, University of Aarhus, Ny Munkegade, building 540, DK - 8000 Arhus˚ C, Denmark. from a few primitives: internal and external choices - the lattice operations - and internal and external iterations - the least and greatest fix-point operations. The order theoretic point of view, which we adopt here, identifies two such systems G and H if there are winning strategies for one chosen player in compound games of communication hom G, H and hom H, G . The analysis of different strategies, in the spirit of categorical proof theory and of the bicompletion of categories [6, 7, 8], is probably a more appropriate setting for modelling interactive computation; this kind of study is under way and will possibly lead to an explicit characterisation of free bicomplete categories with enough initial algebras and terminal coalgebras of functors. However, we can still ask whether the order theoretic identification is too degenerate and wonder whether every system is equivalent to another one where the number of alternations between internal and external iterations is bounded by a fixed positive integer - this is the translation of the hierarchy problem in the language of systems. A negative answer to this order theoretic problem implies also that a categorical identification is not degenerate. We have been able to answer negatively to the above question by means of a fortunate coincidence of order theoretic ideas with categorical ideas. There are games A for which the copycat strategy, which plays the role of the identity, is the unique strategy in hom A, A . These games impose strong conditions on the structure of games H equivalent to A, it can be shown that they are good representatives of their equivalence class as far as we are concerned with their alternation complexity. We introduce the category of µ-lattices together with the hierarchy problem in section 1 and sketch the ideas used in the proof of strictness in section 2.

1 The theory of µ-lattices and the hierarchy

Let P be a partially ordered set and let φ : P - P be an order preserving function. Recall that the least prefix-point of φ, whenever it exists, is an element µz.φ(z) of P such that φ( µz.φ(z) ) ≤ µz.φ(z) and such that, if φ( p ) ≤ p, then µz.φ(z) ≤ p. The greatest postfix-point of φ is defined dually and is denoted by νz.φ(z).

- Deﬁnition 1.1 The set of terms Λω and the arity-function a :Λω N are deﬁned by induction as follows:

1. Vn ∈ Λω and a(Vn)= n, for n ≥ 0.

2. Wn ∈ Λω and a(Wn)= n, for n ≥ 0.

3. If φi ∈ Λω, a(φi) = ki, for i = 1,...,n, φ ∈ Λω, a(φ) = n, then φ ◦ (φ1,...,φn) ∈ Λω and a(φ ◦ (φ1,...,φn)) = Pi=1,...,n ki.

2 4. If φ ∈ Λω, a(φ) = n + 1, then µs.φ ∈ Λω and a(µs.φ) = n, for s = 1,...,n + 1.

5. If φ ∈ Λω, a(φ) = n + 1, then νs.φ ∈ Λω and a(νs.φ) = n, for s = 1,...,n + 1.

Deﬁnition 1.2 Let L be a lattice, we deﬁne a partial interpretation of terms n - φ ∈ Λω, a(φ)= n, as order preserving functions |φ| : L L.

1. | Vn |(l1,...,ln) = Vi=1,...,n li . 2. As in 1, but substituting each symbol V with the symbol W.

3. Let φ ∈ Λω be such that a(φ)= n and for i =1,...,n let φi ∈ Λω be such that a(φi) = ki. Suppose |φ| and |φi| are deﬁned. In this case we deﬁne |φ ◦ (φ1,...,φn)| to be:

|φ ◦ (φ1,...,φn)|(l1,...,lk)

= |φ|( |φ1|(l − ,...,l + ),..., |φn|(l − ,...,l + ) ), k1 k1 kn kn

− i−1 + i + n where ki =1+ Pj=1 kj , ki = Pj=1 kj and k = kn = Pj=1 kj . Other- wise |φ ◦ (φ1,...,φn)| is undeﬁned.

4. Let φ ∈ Λω be such that a(φ) = n + 1. Suppose that |φ| is defined and n let s ∈{1,...,n +1}. If for each vector (l1,...,ln) ∈ L the least prefix- point of the order preserving function |φ|(l1,...,ls−1,z,ls,...,ln) exists, then we define |µs.φ| to be:

|µs.φ|(l1,...,ln) = µz.|φ|(l1,...,ls−1,z,ls,...,ln).

Otherwise |µs.φ| is undefined. 5. As in 4, but substituting each symbol µ with the symbol ν, and the word least prefix-point with the word greatest postfix-point.

Deﬁnition 1.3 A lattice L is a µ-lattice if the interpretation of terms φ ∈ Λω is a total function. Let L1,L2 be two µ-lattices, an order preserving function - n f : L1 L2 is a µ-lattice morphism if the equality |φ|◦ f = f ◦ |φ| holds for all φ ∈ Λω such that a(φ)= n. We shall write Lω for the category of µ-lattices.

Definition 1.4 We define classes of terms Σn, Πn, Λn ⊆ Λω, for n ≥ 0. We set Σ0 = Π0 = Λ0, where Λ0 is the least class which contains Wn and Vn, n ≥ 0, and which is closed under substitution (rule 3 of definition 1.1). Suppose that Σn and Πn have been defined. We define Σn+1 to be the least class of terms which contains Σn ∪ Πn and which is closed under substitution and the µ-operation (rule 4 of definition 1.1). Similarly, we define Πn+1 to be the least class of terms which contains Σn ∪ Πn and which is closed under substitution and the ν-operation (rule 5 of definition 1.1). We let Λn = Πn+1 ∩ Λn+1 and observe that Λω = Sn≥0 Σn = Sn≥0 Πn = Sn≥0 Λn.

3 Definition 1.5 We say that a lattice is a Σn-model if for every φ ∈ Σn such k - that a(φ) = k |φ| : L L is defined. Let L1,L2 be two Σn-models, an - order preserving function f : L1 L2 is a morphism of Σn-models if for k every φ ∈ Σn as above the equality f ◦ |φ| = |φ|◦ f holds. We let Sn be the category of Σn-models and morphisms of Σn-models. We define in a similar way a Πn-model, a morphism of Πn-models and the category Pn,aΛn-model, a morphism of Λn-models and the category Ln.

Clearly L0 is the category of lattices and we have inclusion of categories

S1 Sn Oo _ Oo _ _?? _?? ?/O ?/O . . . o ? _ . . . o ? _ L0 o L1 o Ln−1 o Ln o Lω _ O _ O _?? ?_ ? ?/O ?/O P1 Pn

The alternation hierarchy problem for the theory of µ-lattices can be stated in the following way: is there a number n ≥ 0 and a category Cn among ⊂ - Sn, Pn, Ln such that the inclusion functor Lω Cn is an equivalence of categories? If such a Cn exists, then Cn = Lω, since if P is a partially ordered set which is order-isomorphic to a µ-lattice, then it is itself a µ-lattice; as a consequence for every m>n all the Sm, Pn and Ln are equal to Lω.

Theorem 1.6 The alternation hierarchy for the theory of µ-lattices is strict, i.e. there is no positive integer n such that Ln = Lω.

Proof. For every n ≥ 0, we exhibit a sub-Λn-model Jn,P of the free µ-lattice JP over the partially ordered set P . Using methods developed in [13, 14] Jn,P can be shown to be the free Λn-model over the partially ordered set P , in ⊂ - particular it is generated by P and the inclusion in,P : Jn,P JP preserves the generators. If Ln = Lω, then in,P has to be an isomorphism; however, we prove in 2.10 that in,P is a proper inclusion for every n ≥ 0 if P contains an antichain of cardinality six.

The alternation hierarchy problem for a class K of µ-lattices can be stated as follows. Let Kω be the quasi-variety generated by the class of µ-lattices K in Lω, by which we mean the closure of the full sub-category determined by objects in K under products, sub-objects and regular epis. Similarly let Kn be the quasi- ⊂ - variety generated by the class K in Ln. The inclusion functors Lω Ln ⊂ - restrict to inclusions Kω Kn and the problem is to determine whether there exists a number n ≥ 0 such that the above inclusion is an equivalence. The above theorem has the following consequence.

Theorem 1.7 The alternation hierarchy for the class of complete lattices is strict.

4 Proof. Let K be the class of complete lattices. Since every free µ-lattices can be embedded in a complete lattice by a morphism of µ-lattices, as proved in [13, 14], then Kω = Lω. Similarly Kn = Ln, since Jn,P is the free Λn-model, so that free Λn-models can be embedded into complete lattices. However Ln 6= Lω.

2 Free µ-lattices and free Λn-models

Definition 2.1 A partial game G is a tuple hG0, G1,g0,ǫ,Wσi where hG0, G1,g0i is a pointed graph (of positions and moves, with a chosen initial - position), ǫ : G0 {0, σ, π} is a colouring (of positions by players), and Wσ is a set of infinite paths in hG0, G1i (i.e. the infinite plays which are wins for ′ ′ player σ). We require that if ǫ(g) = 0, then { g | g → g } = ∅ and write XG for ′ the set { x ∈ G0 | ǫ(x)=0 }. We say that a game G is bipartite if g → g implies ′ ǫ(g) 6= ǫ(g ). A partial game G is in the class TB if and only if hG0, G1,g0i is a 1 finite tree with back edges and moreover γ ∈ Wσ if and only if ǫ(rγ ) = π. If G ∈ TB , we denote by R(G) the set of vertexes which are returns of hG0, G1,g0i. We let J be the class of partial games G ∈ TB satisfying the following condition: if r ∈ R(G) then there exists a unique back edge P (r) → r and a unique move r → S(r).

Deﬁnition 2.2 Let P be a partially ordered set. A game over P is a pair - hG, λi where G ∈ TB is a partial game and λ : XG P is a valuation in P . We write J (P ) for the class of games over P hG, λi such such that G ∈ J . We use the notation G for a game over P hG, λi, leaving in the background the valuation λ.

A relation on the class of games over P is defined by means of the following con- struction. If G and H are two such games, the communication game hom G, H is played at the same time on the two boards. Mediator is a team composed by player π on G and player σ on H; in order to move, he must wait for both his opponents, player σ on G and player π on H, to exhaust their moves; on the other hand, as soon as one of the opponents can move, then this opponent must move. Mediator’s goal is either to reach a pair of positions (x, y) ∈ XG × XH such that λ(x) ≤ λ(y) or to win on at least one board. This game is essentially 1 A tree with back edges is a pointed graph hG0,G1,g0i such that G1 ⊆ G0 × G0 and with the property that, for every vertex g ∈ G0, there exists a unique simple path γg from g0 to g. ′ In this case, we say that an edge τ : g → g is a forward edge if γg ⋆ τ = γg′ and that it is a back edge otherwise. A vertex r ∈ G0 is called a return if there exists a back edge t → r. If γ is an infinite path in hG0,G1,g0i, there exists a unique return rγ which is visited infinitely often and which is of minimal height, the height of a vertex being the length of the unique simple path from the root to the vertex. Similarly, if γ is a proper cycle, then there exists a unique return rγ of minimal height lying on γ.

5 the one described in [2, 9] and is pictured as follows:

// // // // // // σ : / : π − σ : / : π G // H // / /

Deﬁnition 2.3 Let G, H be games over P , we stet that G ≤ H if and only if mediator has a winning strategy in the game hom G, H .

The following were the main results achieved in [14].

Theorem 2.4 The above relation is always a preorder which is decidable if the order of P is decidable. Denote by JP the antisymmetric quotient of the preordered class J (P ). Then JP is a µ-lattice which is free over P .

If G, H ∈ J (P ), we write G ≡ H if G ≤ H and H ≤ G and let G be the equivalence class of G. JP is therefore the set of those equivalence classes; it - - comes with an embedding ηP : P JP with the usual universal property: if f : P - L is an order preserving function to a µ-lattice, then there exists a unique morphism of µ-lattices f˜ such that f˜◦ ηP = f. We recall the concepts needed to prove that the above relation is a preorder. We proved that G ≤ G by exhibiting the copycat strategy in hom G, G , which is played as follows. From a pair of positions of the form (g,g) it is always the case that only one opponent has to move. When he stops moving, if he does, mediator copies all the moves played by that opponent on the other board until the play reaches again a position of the form (g′,g′). We proved that if G ≤ H and H ≤ K then G ≤ K, by describing a new game hom G,H,K , a generalisation of the game hom G, H since it can be described as follows. It is played at the same time on the three boards; mediator is a team composed by π on G, σ and π on H and σ on K; in order to move, he must wait for both his opponents (player σ on G and player π on K) to exhaust their moves; on the other hand, as soon as one of the opponents can move, then this opponent must move. Mediator’s goal is either to reach a triple of positions (x,y,z) ∈ XG × XH × XK such that λ(x) ≤ λ(y) ≤ λ(z) or to win on G or on K. The game is pictured as follows:

// // // // // // // // // σ : / : π − σ : / : π − σ : / : π G // H // K // / / /

6 Let θ be a sequence of moves in the game hom G,H,K , i.e. a path in the underlying graph. We can forget about moves occurred on two boards and project this sequence on the third; we shall use the notation θG,θH ,θK for those projections. The above game has the following two properties. Given two winning strategies R on hom G, H and S on hom H,K , there exists a winning strategy R||S on hom G,H,K ; this strategy is described by saying that R is used on the boards G, H where where S is used on the boards H,K. Moreover, given a winning strategy T on hom G,H,K , there exists a winning strategy T\H on hom G, K : this strategy is described by saying that mediator in hom G, K keeps for himself what happens in H. We shall make use later (2.8) of the following observation: when mediator is playing according to a winning strategy in the game hom G, H there is a sort of game going on between player σ of G and player π of H; a similar remark is valid also for the game hom G,H,K . If H is equal to G and mediator is using the copycat strategy, such game can be identiﬁed with G. The ﬁrst step toward the main result is a combinatorial characterisation of elements of the free µ-lattice which are representable by terms in Λn.

Deﬁnition 2.5 Let G be a game over P , a chain C in G is a totally ordered subset {r0 <...

1. ǫ(ri) 6= ǫ(ri+1), for i =0,...,k − 1,

2. for i = 0,...,k − 1, there is a proper cycle γ of G such that rγ = ri and ri+1 lies on γ.

We write C ⊏ G if C is a chain in G and let L(G) be the number max{ card C | C ⊏ G }.

Deﬁnition/Proposition 2.6 For every G ∈ JP we let LG be the number

LG = min{ n | L(H)= n , H ∈ G } and deﬁne Jn,P = { G ∈ JP | LG ≤ n }.

The set Jn,P isaΛn-model free over P , i.e. it comes with a speciﬁed embedding - - ηn,P : P Jn,P with the usual universal property with respect to order preserving functions with codomain a Λn-model. Moreover the inclusion in,P : ⊂ - Jn,P JP is a morphism of Λn-models and the equality ηP = in,P ◦ ηn,P holds.

The second step is to ﬁnd good representatives to equivalence classes.

7 Deﬁnition 2.7 We say that a game A over P is synchronising if it is bipartite and the only winning strategy for mediator in the game hom A, A is the copycat strategy.

Since a strategy in the game hom A, A is (morally) an endomorphism of A, and the copycat strategy plays the role of the identity, we can also think of a synchronising game as being a sort of an asymmetric object. We observe that if A ∈ J (P ) and L(A) = 0, then A is synchronising if and only if it is in normal form as a free lattice term [5, 15].

Proposition 2.8 Let G ∈ JP be such that there exists a synchronising game over P An satisfying An ≤ G and G ≤ An. If L(An)= n, then also LG = n.

Let An be such a game. We prove the proposition on the one hand by con- n n n n n n structing a game A• ∈ J (P ) such that A ≤ A• ≤ A and L(A• )= L(A ). On the other hand, if H ∈ G, then An ≤ H ≤ An and we fix two winning strategies R and S for mediator in the games hom An, H and hom H, An respectively. It can be shown that if mediator has a winning strategy in the game hom G, H , then he has also a bounded memory winning strategy; therefore we shall suppose that R and S are bounded memory strategies, i.e. finite combinatorial objects. Since An is synchronising, we can identify the game between opponents, which results when mediator is playing according to any winning strategy in hom An,H,An , with the game An itself. The choice of an infinite path γ in An based at the root gives rise to a deterministic strategy for the opponents in the game hom An,H,An . We reformulate the definition of a winning infinite path in the game hom An,H,An - by saying that opponents win as soon as they can oblige mediator to visit a chain in H of cardinality n - and prove that opponents have a winning strategy in this new game under the assumption that mediator is playing with the strategy R||S. The proof depends on the following key lemma.

Lemma 2.9 There exists a positive number K ≥ 0, determined by the bounded memory strategies R and S, with the following property. Let γ be a proper cycle of An and let θ be a path of hom An,H,An , played according to the strategy K R||S, such that θAn = γ . It is possible to ﬁnd a factorisation

θ = θ0 ⋆ Θ ⋆θ1

k such that ΘH and ΘAn are proper cycles and ΘAn = γ , with 1 ≤ k ≤ K.

Moreover, ǫ(rΘH )= ǫ(rγ ).

The lemma can be understood as follows: if opponents play enough time along a cycle γ of An, then they can force mediator to play along a proper cycle of H

8 of the same colour as γ. The lemma can be used to prove by induction another lemma, the interpretation of which is the desired statement: if opponents play enough time and in the right way along a chain C of An, then they can force mediator to play along a chain of H of the same cardinality and colour as C. This lemma completes the proof of proposition 2.8.

The last step is to provide a bunch of good representatives.

Theorem 2.10 If P contains an antichain {a0,a1,a2,b0,b1,b2}, then the in- ⊂ - clusion im,P : Jm,P JP is proper.

In order to prove theorem 2.10 we construct a game W n such that L(W n)= n, where n>m. Using the fact that {a0,a1,a2,b0,b1,b2} is an antichain and a few combinatorial properties, this game is shown to be synchronising. It is deﬁned as follows.

• The set of position is

{ gj, wj | j =0,..., 6n − 1 }

and the initial position is g0. • The set of forward edges is

{ gj → gj+1 | j =0,..., 6n − 2 }∪ { gj → wj | j =0,..., 6n − 1 }.

• The set of back edges is

{ g3(2n−k)−1 → g3k | k =0,...,n − 1 }.

• ǫ(wj ) = 0, for i = 1,..., 6n − 1 and ǫ(gj) = Qj mod2, where Q0 = σ and Q1 = π.

• Eventually, λ(wj )= aj mod3, if j < 3n and λ(wj )= bj mod3, if j ≥ 3n.

n n By proposition 2.8 LW• = n>m, so that W• 6∈ Jm,P .

The origins of the games W n can be traced back to Philip Whitman’s proof that the free lattice over three generators is not complete [16]. His result – also documented in the monography [5] – can be used to show that the inclusion ⊂ - i0,X : J0,X JX is proper when card X ≥ 3, where we recall that J0,X coincides with the free lattice on the set X.

9 References

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