Algebraic Tools for the Product of Overlapping Tiles E. Dubourg - - PowerPoint PPT Presentation

Algebraic Tools for the Product of Overlapping Tiles

• E. Dubourg

joint work with D. Janin

LaBRI

March 7, 2014

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

Research context

1 Development of a language theory for inverse monoids.

A monoid S is an inverse monoid when for any x ∈ S, there exists a unique x−1 so that xx−1x = x and x−1xx−1 = x.

2 Quasi-recognizable languages of tiles.

(using MacAlister’s inverse monoid)

3 Closure under product and restricted product.

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

The Monoid of Tiles

Example (Tiles (a, bcb, ab) and (bc, b, abc)−1 = (bcb, ¯ b, babc)) a bcb ab bc b abc T (A): inverse monoid of overlapping tiles (i.e. birooted words1): product, neutral element 1 = (1, 1, 1), absorbing element 0. Example (the product (a, bcb, ab)(bc, b, abc)−1) a bcb ab bc b abc = a bc babc When these conditions are no met, uv = 0.

1D.B. McAlister, Inverse semigroups which are separated over a

subsemigroup, Trans. Amer. Math. Soc., vol. 182, pp. 85-117 (1973)

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

Natural Order over Tiles

Definition (Left and right-projections) a bcb ab For any u = a bcb ab uL = and uR = a bcb ab Remark uuL = u = uRu. Definition (Natural order over tiles1) For any u, v ∈ T (A), u ≤ v when u = vuL or u = uRv. Example u1 u2 u3 ≤ u0 u1 u2 u3 u4

1K.S.S. Nambooripad, The natural partial order on a regular semigroup,

• Proc. Edinburgh Math. Soc., vol. 23, pp.249–260, (1983)
• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

The monoid of Tiles: Remarkable Elements

Subunits U(T (A)) = {u ∈ T (A) | u ≤ 1} = {(u1, u2, u3) ∈ T (A) | u2 = 1} Elements of the set of subunits U(T (A)) are idempotents. abcd bc abcd bc = abcd bc Maximal elements A∗ ≃ maximal positive tiles ; (A∗)−1 ≃ maximal negative tiles u0 u0 ≃ u0 ; u−1 ≃ We use the embedding u0 → (1, u0, 1). Left and right-projections

uL = min{v ∈ U(T (A)) | uv = u}, uR = min{v ∈ U(T (A)) | vu = u}.

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

Definability of Languages of Tiles

Theorem (MSO-definability) A language of tiles is MSO-definable iff it is a finite union of languages of the form ULVW R, with U, V and W being regular languages of A∗. Fact Recognizability by morphisms in finite monoids collapses over tiles1. Definition (Quasi-recognizability) A language L ⊆ T (A) is quasi-recognizable, i.e. L ∈ Q-REC, when there exists an adequate premorphism ϕ : T (A) → S, with S an E-ordered monoid, so that L = ϕ−1(ϕ(L)).

• 1D. Janin, On languages of one-dimensional overlapping tiles, Int. Conf. on

Current Thrends in Theo. and Prac. Comp. Science (SOFSEM). LNCS, vol. 7741, pp. 244-256

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

E-ordered Monoids

This definition comes from M. Lawson’s and V. Gould’s work over Ehresmann’s inverse semigroups: Definition A finite monoid S equipped with a preorder stable by product is an E-ordered monoid when S possesses a minimum 0. is an order over U(S), and U(S) is a ∧-semilattice with product as ∧. For any x ∈ S, left and right projections xL and xR are defined. These projections are monotonic: if x y then xR yR. Right and left semi-congruence induced by projections: (xy)L = (xLy)L and (xy)R = (xyR)R.

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

Adequate Premorphisms

A premorphism is a monotonic mapping ϕ : T (A) → S so that S is an E-ordered monoid, ϕ(1) = 1, for any u, v ∈ T (A) with u ≤ v, ϕ(u) ϕ(v), ϕ(uv) ϕ(u)ϕ(v). It is adequate when it preserves left and right-projections: ϕ(uR) = ϕ(u)R, ϕ(uL) = f (u)L, disjoint products: for u = (u1, u2, 1) and v = (1, v2, v3), ϕ(uv) = ϕ(u)ϕ(v).

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

Closure properties of Quasi-recognizability

Fact The class of languages Q-REC is closed under ∪ and ∩. Let L1, L2 ⊆ T (A) be languages quasi-recognized respectively by ϕ1 : T (A) → S1 and ϕ2 : T (A) → S2, both are quasi-recognized by ϕ1, ϕ2 : T (A) → S1 × S2 u → (ϕ1(u), ϕ2(u)). Question Is the class of languages Q-REC closed under product ? Counterexemple {(1, a2n, 1) | n ∈ N} · {(1, a2n, 1)−1 | n ∈ N} ∈ Q-REC However, the answer is yes for languages of positive tiles.

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

The Monoid of Positive Tiles

Definition A tile is positive when its input occurs before its output. Positive non-zero tiles can therefore be seen as elements of T +(A) = A∗ × A∗ × A∗ ∪ {0}, with the product being simpler. Example (the product (u1, v1, w1)(u2, v2, w2)) u1 v1 w1 u2 v2 w2 It is the concatenation of their roots with matching conditions: u2 is a suffix of u1v1 or u1v1 is a suffix of u2, w1 is a prefix of v2w2 or v2w2 is a prefix of w1.

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

The restricted product

Definition (Restricted product) u • v is defined when uL = vR, and in this case u • v = uv. Lemma (Preservation of the restricted product) For any adequate premorphism ϕ, we have ϕ(u • v) = ϕ(u) • ϕ(v). We will show closure under restricted product, then express the product from the restricted product: Fact If the restricted product preserves quasi-recognizability over languages of positive tiles, then the product does too. L1L2 =

• (A∗)LL1(A∗)R • L2
• ∪
• L1 • (A∗)LL2(A∗)R

∪

• (A∗)LL1 • L2(A∗)R

∪

• L1(A∗)R • (A∗)LL2
• And Q-REC is closed by product with (A∗)L or (A∗)R and by ∪.
• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

The Monoid of Restricted Decompositions

For any E-ordered monoid S, we define the set Dr(S) by Dr(S) = {X ∈ P(S × S) | ∃c ∈ S, (c, cL) ∈ X, (cR, c) ∈ X, ∀(x, y) ∈ X, x • y = c} We define the product ∗ from S × S to P(S × S) by (x, x′) ∗ (y, y′) = {(x(x′yy′)R, xLx′yy′), (xx′yy′R, (xx′y)Ly′)}. We extend ∗ to Dr(S) in a point-wise manner X ∗ Y =

• (x,x′)∈X

(y,y′)∈Y

(x, x′) ∗ (y, y′). Dr(S) is an E-ordered monoid preordered by defined by X Y iff for any x ∈ X, there exists y ∈ Y so that x y.

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

Premorphism ψ

Let ϕ : T +(A) → S be an adequate premorphism, we define ψ : T +(A) → Dr(S) u → {(ϕ(u1), ϕ(u2)) ∈ S × S | u = u1 • u2} ψ is an adequate premorphism. Lemma For any L1 = ϕ−1(ϕ(X1)) and L2 = ϕ−1(ϕ(X1)), L1 • L2 = ψ−1(ψ({X ∈ Dr(S) | X ∩ (X1 × X2) = ∅})). Since for any L1, L2 quasi-recognized by respectively ϕ1 and ϕ2, both are recognized by ϕ1, ϕ2, Corollary The product of two quasi-recognizable languages of positive tiles is quasi-recognizable.

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

To be continued

Extension to the recognition of Kleene’s ∗ over quasi-recognizable languages of positive tiles ? Extension to the non-linear case : birooted trees ? b a c b a c a a b c First-order logic definability ?

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles

Thank you for your attention.

• E. Dubourg

Algebraic Tools for the Product of Overlapping Tiles