The code problem for directed figures

(1)

DOI:10.1051/ita/2011005 www.rairo-ita.org

THE CODE PROBLEM FOR DIRECTED FIGURES

Micha l Kolarz

¹

Abstract. We consider directed figures defined as labelled polyominoes with designated start and end points, with two types of catenation operations. We are especially interested in codicity verification for sets of figures, and we show that depending on the catenation type the question whether a given set of directed figures is a code is decidable or not. In the former case we give a constructive proof which leads to a straightforward algorithm.

Mathematics Subject Classification.68R15, 68R99.

Introduction

Variable-length word codes,i.e., subsetsXof a monoid such that every product of the elements decomposes uniquely overX, have been studied by many authors and extensive literature exists in this subject, includinge.g.the well-known mono- graph [3]. Much less is known about the codicity in two dimensions. Some authors have extended word codes to trees (see [6]) or polyominoes (e.g.[1,2]); decidability of the codicity testing problem varies in these cases. However, it is undecidable in the case of both standard and labelled polyominoes (cf. [8,9]).

The interest in two-dimensional codes is natural in the context of various disci- plines that use picture encodings,e.g.[4]. In the present paper we study pictures composed of labelled polyominoes, equipped with start and end points; we call them directed figures. Catenation of directed figures is defined with an optional merging function. We characterize decidability of the question whether a given set of directed figures is a code in both situations (with or without the merging function). In the decidable cases we give proofs that lead to simple algorithms.

Keywords and phrases.Directed ﬁgures, variable-length codes, codicity veriﬁcation, Sardinas-Patterson algorithm.

1 Institute of Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Krak´ow, Poland;mkol@smp.if.uj.edu.pl

Article published by EDP Sciences c EDP Sciences 2011

(2)

In Section 1 we define directed figures and related operations. Then, in Sec- tion2, the notion of a code is introduced in the context of catenation with and without the merging function. The decidability question for some specific cases is also solved here. Section3 is concerned with the codicity verification for both kinds of codes, giving algorithms in typical cases. We end with some plans for future study.

1. Preliminaries

Let Σ be a ﬁnite, nonempty alphabet. Foru= (u_x, u_y)∈Z², atranslation in Z²by vectoruis denoted byτ_u,

τ_u:Z²(x, y)→(x+u_x, y+u_y)∈Z².

For a setV ⊆Z² and an arbitrary functionf :V →X it obviously induces τ_u:P(Z²)V → {τ_u(v)|v∈V} ∈P(Z²),

τ_u:X^V f → f ◦τ_−u∈X^τ^u^(V⁾.

Definition 1.1 (Directed ﬁgure). LetD⊆Z²be ﬁnite and non-empty,b, e∈Z² and l :D → Σ. A quadruple f = (D, b, e, l) is called adirected figure (over Σ) with

domain dom(f) = D,

start point begin(f) = b,

end point end(f) = e,

labelling function label(f) = l, translation vector tran(f) = e−b.

In addition we deﬁne theempty directed figure εas (∅,(0,0),(0,0),∅).

Note that we make no additional assumptions about the relative placement of domain, start point and end point of the ﬁgure and connectedness of the domain.

Example 1.2. Directed ﬁgures with graphical representation (a circle marks the start point and a diamond marks the end point of the ﬁgure).

ε i

({(0,0),(1,1)},(0,0),(1,0),{(0,0)→a,(1,1)→c}) aic

({(0,0)},(0,2),(2,0),{(0,0)→a}) a i

(3)

The set of all directed ﬁgures over Σ is denoted by Σ. Two directed ﬁgures x, y areequal if there existsu∈Z²such that

y= (τ_u(dom(x)), τ_u(begin(x)), τ_u(end(x)), τ_u(label(x))).

Thus, we actually consider ﬁgures up to a translation.

Definition 1.3 (Catenation). Let x= (D_x, b_x, e_x, l_x) andy = (D_y, b_y, e_y, l_y) be directed ﬁgures. IfD_x∩τ_x_e_−y_b(D_y) =∅,catenation ofxandy is deﬁned as

x◦y= (D_x∪τ_x_e_−y_b(D_y), b_x, τ_x_e_−y_b(e_y), l), where

l(z) =

l_x(z) forz∈D_x,

τ_x_e_−y_b(l_y)(z) forz∈τ_x_e_−y_b(D_y).

IfD_x∩τ_x_e_−y_b(D_y)=∅, catenation ofxandy is not deﬁned.

Example 1.4. Catenation of two ﬁgures.

aib

c ◦ ai

c =

aib c a

c

Example 1.5. Catenation of the following figures is not defined (point labeled by cin the first figure overlaps with point labeled byb in the second figure).

aib c

ai b c

Definition 1.6(m-catenation). Letx= (D_x, b_x, e_x, l_x) andy= (D_y, b_y, e_y, l_y) be directed ﬁgures. m-catenation ofxandyw.r.t. amerging function m: Σ×Σ→Σ is deﬁned as

x◦my= (D_x∪τ_x_e_−y_b(D_y), b_x, τ_x_e_−y_b(e_y), l), where

l(z) =

⎧⎨

⎩

l_x(z) forz∈D_x\τ_x_e_−y_b(D_y), τ_x_e_−y_b(l_y)(z) forz∈τ_x_e_−y_b(D_y)\D_x, m(l_x(z), τ_x_e_−y_b(l_y)(z)) forz∈D_x∩τ_x_e_−y_b(D_y).

Example 1.7. π₁-catenation of two ﬁgures (π₁ denotes projection on the ﬁrst argument).

aib c ◦π1

ai

b c =

aib c a

c

(4)

Observe that◦ is associative. ◦_mis associative if and only if mis associative.

So for associativem, Σ_m= (Σ,◦_m) is a monoid (which is never free).

Abusing this notation, we also write X (resp. X_m) to denote the set of all ﬁgures that can be composed by◦catenation (resp. ◦_mm-catenation) from ﬁgures inX⊆Σ. When some results hold for◦catenation as well as for◦_mm-catenation, we use•catenation symbol, and “x•y” should then be read as “x◦y(resp. x◦_my)”.

In the same way we use “x∈X_•” to denote “x∈X (resp. x∈X_m)”.

From now on letm be an arbitrary associative merging function.

2. Codicity

2.1. Codes

In this subsection we show that in general it is not decidable wether a given set is a code w.r.t. ◦ catenation.

Definition 2.1(Code). X⊆Σis a code if for anyx∈Xthere exists only one sequencex₁, . . . , x_k∈X such thatx=x₁◦. . .◦x_k.

Example 2.2. { ai , ai} ⊆ {a} is a code.

Example 2.3. {w= aia, x= aia , y= aai , z=

ai

a } ⊆ {a} is not a code sincew◦x=y◦z= aai

a a

The following consideration is based on ideas presented in [2] (Sects. 3 and 4).

Let Σ ={a}. For h, h_N, h_E, h_S, h_W ∈ Z+ such that h_N, h_E, h_S, h_W ≤ hand b, e∈ {N, E, S, W} (N as north,E as east,S as south andW as west) we deﬁne a directed hooked square DHS_h(h_N, h_E, h_S, h_W)^b_e to be a directed ﬁgure f ∈ Σ with:

dom(f) = B\(H_N⁻∪H_E⁻∪H_S⁻∪H_W⁻)∪(H_N⁺∪H_E⁺∪H_S⁺∪H_W⁺),

begin(f) =

⎧⎪

⎪⎨

⎪⎪

⎩

(0, h+ 2) ifb=N, (h+ 2,0) ifb=E, (0,−h−2) ifb=S, (−h−2,0) ifb=W,

end(f) =

⎧⎪

⎪⎨

⎪⎪

⎩

(0, h+ 3) ife=N, (h+ 3,0) ife=E, (0,−h−3) ife=S, (−h−3,0) ife=W,

(5)

Figure 1. DHS₄(1,2,3,4)^N_E - full and reduced graphical representation.

where

B = {(x, y)|x, y∈ {−h−2, . . . , h+ 2}},

H_N⁻ = {(−1, y)|y∈ {h+ 2−h_N, . . . , h+ 2}} ∪ {(0, h+ 2−h_N)}, H_E⁻ = {(x,1)|x∈ {h+ 2−h_E, . . . , h+ 2}} ∪ {(h+ 2−h_E,0)}, H_S⁻ = {(1, y)|y∈ {−h−2, . . . ,−h−2 +h_S}} ∪ {(0,−h−2 +h_S)}, H_W⁻ = {(x,−1)|x∈ {−h−2, . . . ,−h−2 +h_W}} ∪ {(−h−2 +h_W,0)},

H_N⁺ = {(1, y)|y∈ {h+ 3, . . . , h+ 3 +h_N}} ∪ {(0, h+ 3 +h_N)}, H_E⁺ = {(x,−1)|x∈ {h+ 3, . . . , h+ 3 +h_E}} ∪ {(h+ 3 +h_E,0)}, H_S⁺ = {(−1, y)|y∈ {−h−3−h_S, . . . ,−h−3}} ∪ {(0,−h−3−h_S)}, H_W⁺ = {(x,1)|x∈ {−h−3−h_W, . . . ,−h−3}} ∪ {(−h−3−h_W,0)},

i.e. f is a square with hooks on each side (seee.g. Fig.1).

Observe that forx= DHS_h(h_N, h_E, h_S, h_W)^b_eandx = DHS_h(h_N, h_E, h_S, h_W)^b_e

catenationx◦x is deﬁned if and only ifematches tob, i.e.:

e = N and b = S or

e = E and b = W or

e = S and b = N or

e = W and b = E

andh_e=h_b.

(6)

a_i₁ x b_x

I_i - ◦

a_i₂ x x

x - ◦. . .◦

a_i_ri x x

x -

[x_i[^W_E= DHS(a_i₁, x, b_x, I_i)^W_E ◦DHS(a_i₂, x, x, x)^W_E ◦. . .◦DHS(a_i_ri, x, x, x)^W_E a_i₁

x x_i

x - ◦

a_i₂ x x

x - ◦. . .◦

a_i_ri x x

x -

]x_i[^W_E= DHS(a_i₁, x, x_i, x)^W_E ◦DHS(a_i₂, x, x, x)^W_E ◦. . .◦DHS(a_i_ri, x, x, x)^W_E a_i₁

x x

x - ◦

a_i₂ x x

x - ◦. . .◦

a_i_ri e e_x_i x

@ R

]x_i]^W_S = DHS(a_i₁, x, x, x)^W_E ◦DHS(a_i₂, x, x, x)^W_E ◦. . .◦DHS(a_i_ri, e, e_x_i, x)^W_S Figure 2. Basic-ﬁgures forx_i=a_i₁· · ·a_i_ri.

Now we encode the Post problem in a set of directed ﬁgures over Σ = {a}.

The Post problem can be stated as follows:

Let A = {a₁, . . . , a_p} be a ﬁnite alphabet, x₁, . . . , x_k, y₁, . . . , y_k ∈ A⁺ such that x_i = y_i (for i ∈ {1, . . . , k}). One cannot decide if there exists i₁, . . . , i_n ∈ {1, . . . , k},n≥2, such thatx_i₁· · ·x_i_n=y_i₁· · ·y_i_n.

We describe a set of directed ﬁguresXsuch that the Post problem has a solution if and only ifX is a code.

Consider the following set:

H=

i∈{1,...,k}

{x_i, y_i, e_x_i, y_y_i, I_i} ∪ {a_i|i∈ {1, . . . , p}} ∪ {x, y, x, y, b_x, b_y, e},

where I_i are additional elements related to each pair (x_i, y_i) from corresponding Post Problem. Seth=|H|= 5k+p+ 7. We can define a bijection betweenH and {1, . . . , h}, so from now on, each element ofH is identified with its image by this bijection. Since h is fixed in our consideration we write DHS(h_N, h_E, h_S, h_W)^b_e instead of DHS_h(h_N, h_E, h_S, h_W)^b_e.

For eachx_i∈X we deﬁne basic-figures [x_i[,]x_i[ and ]x_i] (Fig.2); these ﬁgures will be used to encode the word x_i standing at the beginning (we call it begin solution figure), in the middle (middle solution figure) and at the end (end solution figure) of a solution of the Post problem, respective.

In addition we defineannex-figures(Figs.3–6); first three types of these figures will be used to convey information about solution of the Post problem from north to south, from north to west and from east to west respective, last one will convey no information.

(7)

x_i x x_i

x -

x_i x x_i

x

M_x[i, N.S]^W_E = DHS(x_i, x, x_i, x)^W_E M_x[i, N.S]^E_W = DHS(x_i, x, x_i, x)^E_W e_x_i

e e_x_i x

e_x_i e e_x_i x @R

E_x[i, N.S]^N_W = DHS(e_x_i, e, e_x_i, x)^N_W E_x[i, N.S]^W_S = DHS(e_x_i, e, e_x_i, x)^W_S Figure 3. Annex-ﬁgures for passing information from north to south.

x_i x x x_i -

x_i x x x_i

M_x[i, N.W]^W_E = DHS(x_i, x, x, x_i)^W_E M_x[i, N.W]^E_W = DHS(x_i, x, x, x_i)^E_W e_x_i

e e e_x_i

e_x_i e e e_x_i

@ R

E_x[i, N.W]^N_W = DHS(e_x_i, e, e, e_x_i)^N_W E_x[i, N.W]^W_S = DHS(e_x_i, e, e, e_x_i)^W_S Figure 4. Annex-ﬁgures for passing information from north to west.

In the same way we deﬁne ﬁgures for “y-part” of the Post problem, replacing the letterxby the lettery.

Let X be the set of all defined figures (6k basic-figures and 32k+ 2 annex- figures, 16kfor each parts: “x-part” and “y-part”). Observe that there exists no half-plain of integer values anchored in (0,0) (i.e. {v ∈ Z² |u·v >0} for some u∈Z², where· denotes the usual dot product) containing all translation vectors of the figures we have defined.

Proposition 2.4. If the Post problem has a solution thenXis not a code.

Proof. Leti₁, . . . , i_n∈ {1, . . . , k} be a solution of the Post problem,i.e.

x_i₁· · ·x_i_n =y_i₁· · ·y_i_n.

(8)

x x_i x x_i -

x x_i x x_i

M_x[i, E.W]^W_E = DHS(x, x_i, x, x_i)^W_E M_x[i, E.W]^E_W = DHS(x, x_i, x, x_i)^E_W x

e_x_i e e_x_i -

x e_x_i e e_x_i

E_x[i, E.W]^W_E = DHS(x, e_x_i, e, e_x_i)^W_E E_x[i, E.W]^E_W = DHS(x, e_x_i, e, e_x_i)^E_W b_x

x_i b_x I_i

b_x x_i b_x I_i @R

BM_x[i, E.W]^E_S = DHS(b_x, x_i, b_x, I_i)^E_S BM_x[i, E.W]^N_E = DHS(b_x, x_i, b_x, I_i)^N_E b_x

e_x_i e I_i

b_x e_x_i e I_i @R

BE_x[i, E.W]Ê_S = DHS(b_x, e_x_i, e, I_i)Ê_S BE_x[i, E.W]^N_E = DHS(b_x, e_x_i, e, I_i)^N_E Figure 5. Annex-figures for passing information from east to west.

Consider following directed ﬁgures:

wx₁ = [x_i₁[^W_E◦]x_i₂[^W_E◦. . .◦]x_i_n−1[^W_E◦]x_i_n]^W_S ,

wx_j = E_x[i_n, N.S]^N_W◦N_x[]^E_W ◦. . .◦N_x[]^E_W

|xn|−2times

◦ N_x[]^E_W ◦. . .◦N_x[]^E_W

|xn−1|−1times

◦M_x[i_n−1, N.E]^E_W ◦ . . .

N_x[]^E_W ◦. . .◦N_x[]^E_W

|xj+1|−1times

◦M_x[i_j+1, N.E]^E_W ◦

N_x[]^E_W ◦. . .◦N_x[]^E_W

|xj|−1times

◦M_x[i_j, N.E]^E_W ◦

M_x[i_j, E.W]^E_W ◦. . .◦M_x[i_j, E.W]^E_W

|xi1···x_ij−1|−1times

◦BMx[i_j, E.W]^E_S

(for evenj < n),

(9)

x x x

x -

x x x

x

N_x[]^W_E = DHS(x, x, x, x)^W_E N_x[]Ê_W = DHS(x, x, x, x)Ê_W Figure 6. Annex-figure which pass no information.

wx_j = BM_x[i_j, E.W]^N_E◦M_x[i_j, E.W]^W_E ◦. . .◦M_x[i_j, E.W]^W_E

|xi1···x_ij−1|−1times

◦ M_x[i_j, N.E]^W_E ◦N_x[]^W_E ◦. . .◦N_x[]^W_E

|xj|−1times

◦ M_x[i_j+1, N.E]^W_E ◦N_x[]^W_E ◦. . .◦N_x[]^W_E

|xj+1|−1times

◦

. . .

M_x[i_n−1, N.E]^W_E ◦N_x[]^W_E ◦. . .◦N_x[]^W_E

|xn−1|−1times

◦ N_x[]^W_E ◦. . .◦N_x[]^W_E

|xn|−2times

◦E_x[i_n, N.S]^W_S

(for oddj < n),

wx_n = E_x[i_n, N.W]^N_W◦E_x[i_n, E.W]^E_W ◦. . .◦E_x[i_n, E.W]^E_W

|xi1···x_in|−2times

◦BE_x[i_n, E.W]^E_S

(ifnis even),

wx_n = BE_x[i_n, E.W]^N_E◦E_x[i_n, E.W]^W_E ◦. . .◦E_x[i_n, E.W]^W_E

|xi1···xin|−2times

◦E_x[i_n, N.W]^W_S

(ifnis odd).

In the same way we deﬁne ﬁgureswy₁, . . . , wy_n.

It is easy to see thatwx₁◦. . .◦wx_n =wy₁◦. . .◦wy_n ⊆X. HenceXis not a

code.

Example 2.5. Consider

Σ = {a, b},

X = (x₁, x₂, x₃) = (b, ab, bab), Y = (y₁, y₂, y₃) = (ba, abb, b).

(10)

b_x e_x₃ e I₃ @R

x e_x₃ e e_x₃ -

e_x₃ e e e_x₃

@ R b_x

x₁ b_x I₁

x x₁ x x₁

x₁ x x x₁

x x x

x

x x x

x

e_x₃ e e_x₃ x a

b_x I₂

b x x

-

b x x₁

x -

b

x x

a

x

b e e_x₃

@ R

Figure 7. “X”-tiling off.

b_y e_y₃ e I₃ @R

y e_y₃ e e_y₃ -

e_y₃ e e e_y₃

@ R b_y

y₁ b_y I₁

y y₁ y y₁

y₁ y y y₁

y y y

y

e_y₃ e e_y₃ y a

b_y I₂

b

y

b y y

-

b

y₁ y

a y y

-

b e e_y₃ y

@ R

Figure 8. “Y”-tiling off.

We havex₂x₁x₃=y₂y₁y₃. Figuref with two different tiling with elements ofX is presented in Figures7 and8 (where thick arrows show the flow of information through annex-figures).

Proposition 2.6. IfXis not a code then the related Post problem has a solution.

Proof. Letf be a ﬁgure of minimal size (w.r.t. to the size of its domain) which admits two tilings with elements ofX, i.e. there existf₁, . . . , f_p, g₁, . . . , g_q ∈X such thatf₁=g₁ andf₁◦. . .◦f_p=g₁◦. . .◦g_q.

(11)

Consider directed hooked squares tilingf (these are annex-ﬁgures and squares of which basic-ﬁgures are built). Letdbe the westmost among the northmost of them. We have following possibilities:

Case 1: d∈

z∈{x,y}

j∈{1,...,k}{E_z[j, N.S]^N_W, E_z[j, N.W]^N_W, E_z[j, E.W]^N_W}.

Sincedis the westmost among the northmost of all squares tilingf, it cannot have north and west neighbour squares,i.e. squares hooked to it at north side and west side, respective. This implies thatf =d, which contradicts the deﬁnition of the double tiling off.

Case 2: d∈

z∈{x,y}

j∈{1,...,k}{E_z[j, N.S]^W_S , E_z[j, N.W]^W_S }.

Since d has no north and west neighbours, north and west hooks of d are uniquely determined byf. Each of listed figures is uniquely determined by its north and west hooks. This implies that dis also uniquely determined byf. dhas no west neighbour and it has the start point at its west side, which implies that it must be the first one in sequence of figures whose catenation givesf,i.e. d=f₁=g₂. Then eitherf =d(contradiction as previously), orf=f₂◦. . .◦f_p=g₂◦. . .◦g_q is a smaller figure with two tilings, which contradicts the minimality off.

Case 3: d∈

z∈{x,y}

j∈{1,...,k}{Mz[j, N.S]Ê_W, M_z[j, N.W]Ê_W, M_z[j, E.W]Ê_W} ∪ {N_x[]Ê_W, N_y[]Ê_W}.

As in Case 1, d is uniquely determined byf. d has no west neigbour and it has the end point at its west side, which implies that it must be the last one in sequence of ﬁgures whose catenation givesf,i.e. d=f_p=g_q. Then eitherf =d (contradiction as previously) orf =f₁◦. . .◦f_p−1=g₁◦. . .◦g_q−1is smaller ﬁgure with two tilings, which contradicts the minimality off.

Case 4: d∈

z∈{x,y}

j∈{1,...,k}{BM_z[j, E.W]^N_E, BE_z[j, E.W]^N_E}.

Now dmust be the first one in tiling since it has the start point at its north side and it is the northmost in tiling. Observe that there exists no square with e-hook at the north side. This implies that BM_z[j, E.W]^N_E and BE_z[j, E.W]^N_E (z ∈ {x, y}) cannot be the first elements of two different tiling sequences of f. This implies thatdis uniquely determined byf andd=f₁ =g₁. Contradiction as inCase 2.

Case 5: d∈

z∈{x,y}

j∈{1,...,k}{BMz[j, E.W]^E_S, BE_z[j, E.W]^E_S}

As inCase 4, d is uniquely determined by f. If d =BE_z[j, E.W]Ê_S (for z ∈ {x, y}) thend is the last element of a tiling sequence. Contradiction as in Case 3. If d = BM_z[j, E.W]Ê_S (for z ∈ {x, y}) then (for some i ∈ {1, . . . , p}) f = f₁◦f₂◦. . .◦f_i−1◦d◦f_i+1◦. . .◦f_p, where f₁ ∈ {Mz[j, E.W]Ê_W, M_z[j, N.W]Ê_W} andf₂=. . .=f_i−1=M_z[j, E.W]Ê_W. Contradiction as inCase 1.

This leads us to a conclusion that:

Case 6: Directed hooked squaredis a part of a basic-ﬁgure. In particulardis a “ﬁrst part” off₁andg₁.

Now it is easy to observe the following properties off’s tiling:

(1) If f₁ is a ﬁgure that encodes one of the words from X, then all f_i (i ∈ {1, . . . , p}) are ﬁgures encoding “x-part” of the related Post problem (since

(12)

there is no figure that links a figure from “x-part” with a figure from “y- part”). In the same way, iff₁ encodes a word fromY, thenf_iencodes “y- part” of the Post problem. Similar statement is true forg_i(i∈ {1, . . . , q}).

(2) First “row” of figures in tiling is a sequence of middle solution figures (may be empty) which is ended by an end solution figure (that ends row) and may be started with a begin solution figure.

(3) Sequence of middle solution figures from the first row implies that in the tiling, leftmost column’s hooks (I_j hooks of some BM and BE annex- figures) correspond to the sequence of indices of words encoded by those figures.

This leads us to simple observation, that the only possible two tilings of f are tilings of the form deﬁned in the proof of Proposition2.4. This implies that the

3. Codicity verification

Results on codicity verification in this section are based on our previous work [5], where they were obtained in the context ofm-catenation with an additional re- striction for the figures. Considerations presented here work for both types of catenation, with no constraints on the relative position of domain and both begin and end points of figures.

(13)

Let us consider a situation, when for a given X = {x₁, . . . , x_n} ⊆ Σ there exists a vectorτ such

∀x∈X :τ·tran(x)>0.

Without loss of generality we can assume that

∀x∈X : begin(x) = (0,0)

and

∠(R₋^π₂(τ),tran(x₁))≤∠(R₋^π₂(τ),tran(x₂))≤. . .≤∠(R₋^π₂(τ),tran(x_n)),

where∠denotes an angle between two wectors, andR_φ denotes a rotation by φ.

Now choose constantsr_E, r_N, r_W, r_S >0 such that the vectors

τ_E = r_Eτ,

τ_N = r_NR^π₂(tran(x_n)), τ_W = −rWτ,

τ_S = r_SR₋^π₂(tran(x₁))

deﬁne a “bounding area” for ﬁgures inX,i.e.,

∀x∈X : dom(x)∪ {end(x)} ⊆

u∈{τE,τN,τWτS}

{HP(u,begin(x))},

where foru, v∈Z² HP(u, v) denotes half-plane{w∈Z²|u·(w−(v+u))≤0}

(see Fig.9).

Forx∈X_• deﬁne

CE⁺(x) = HP(τ_s,end(x))∩HP(τ_n,end(x))∩HP(τ_w,end(x)), CE⁻(x) = Z²\CE⁺(end(x)),

CW⁺(x) =

v

{v+ (CE⁺(end(x))∩HP(τ_e,end(x)))}, CW⁻(x) = Z²\CW⁺(end(x)),

(14)

• -• v

u

v+u

Figure 9. HP(u, v). The half-plain contains integer grid points lying on vertical line and to the left side of that line (the region marked by horizontal lines).

where the union in the deﬁnition of CW⁺(x) is taken overv∈Z²lying within an angle spanned by vectors−τ(x1) and−τ(xn) (see Figs. 10and11).

Immediately from deﬁnition we have following properties:

Proposition 3.1. Letx, y ∈X_•. Labels ofCE⁻(x)cannot be changed or defined inx•y:

u∈CE⁻(x)∩dom(x) ⇒ label(x)(u) = label(x•y)(u), u∈CE⁻(x)\dom(x) ⇒ u∈dom(x•y).

Proposition 3.2. Let x∈X_•. Labels ofCW⁻(x)are not defined in x:

u∈CW⁻(x)⇒u∈dom(x).

Proposition 3.3.

∀x, y ∈X_•: CE⁺(x•y)⊆CE⁺(x).

Proposition 3.4.

∀x, y∈X_•: CW⁺(x)⊆CW⁺(x•y).

For x, y ∈ X_• we deﬁnie a configuration as a pair (x, y). We say that for x, y ∈X_•conﬁguration (x, y) is asuccessor of (x, y) and write (x, y)≺(x, y) if for somez∈X either x =x•z or y =y•z. By≺^∗ we denote the transitive closure of≺. Obviously we have

X is not a code (resp. m-code)⇔ ∃x, y∈X, z∈X_•:x=y, (x, y)≺^∗(z, z).

(15)

@@

• - τ_E AA AKτ_N τ_W

τ_S

HP(τ_E,begin(x))

@@

@@ AA

AA AA

AA A AA

AA AA

AA A AA

AA AA

AA AA AA

AA AA

A AA

AA AA

AA

@@

τ_S

HP(τ_N,begin(x))

@@

τ_S

HP(τ_W,begin(x))

@@

τ_S HP(τ_S,begin(x))

Figure 10. Half-planes HP(τ,begin(x)) (τ ∈ {τ_E, τ_N, τ_W, τ_S});

the black dot denotes the start point ofx.

Our goal is either to ﬁnd a conﬁguration (x, y) such that

(x, y)≺. . .≺(z, z) (2)

(16)

@@

•

@@

@

@@

CW⁺(x)

CE⁺(x)

Figure 11. CW⁺(x) and CE⁺(x) regions; the black dot denotes the end point ofx.

– thenX is not a code (resp. m-code), or to prove that such a conﬁguration does not exist – then X is a code (resp. m-code). A conﬁguration satisfying (2) is called aproper configuration.

Proposition 3.5. If(x, y) is proper and(x, y)≺(x, y), then (x, y)is proper.

Proof. Obvious.

Let ρ= max

x∈X{minimal natural numbernsuch that B(begin(x), n)∩dom(x)=∅}, where foru= (u_x, u_y)∈Z² andn∈NB(u, n) denotes a ball on integer grid with centeruand radiusr,i.e.,

B(u, n) ={(v_x, v_y)∈Z²| |u_x−v_x|+|u_y−v_y| ≤n} (see Fig.12).

Proposition 3.6. If(x, y) is proper, then

B(end(x), ρ)∩(CW⁺(y)∪CE⁺(y))=∅, B(end(y), ρ)∩(CW⁺(x)∪CE⁺(x))=∅.

Proof. See deﬁnitions of CW⁺() and CE⁺().

(17)

a

i a b c

b

ai b c

Figure 12. Directed ﬁgures and minimal balls (marked with thick lines).

Proposition 3.7. If(x, y) is proper, then

label(x)|CE⁻(x)∩CE⁻(y)≡label(y)|CE⁻(x)∩CE⁻(y).

Proof. See deﬁnition of CE⁻() and Proposition3.1.

Notice that we do not need all of the information contained in conﬁgurations, just those labellings that can be changed by future catenations. By Proposi- tion3.7, instead of (x, y) we can consider areduced configuration deﬁned as a pair (π_RC(x, y), π_RC(y, x)) where

π_RC(z, z) = (end(z),label(z)|dom(z)\(CE⁻(z)∩CE⁻(z))).

Now Proposition3.5implies that we need only consider configurations where the span alongτ_eis bounded by|τ_e|,i.e.,|τ_e·(end(x)−end(y))| ≤ |τ_e|², since no single figure advances end(x) or end(y) by more than |τ_e|. Moreover, Proposition 3.6 restricts the perpendicular span (in the direction ofR₋^π₂(τ_e)). Hence the number of reduced configurations, up to translation, is finite.

This leads us to the following result:

Theorem 3.8. Let X⊆Σ be such that there existsu∈Z²

x∈X

{tran(x)} ⊆ {v∈Z²|u·v >0}. (3)

It is decidable whetherX is a code (resp. m-code).

In particular, form-catenation we have:

Corollary 3.9. It is decidable whether a given finite setX⊆Σ is am-code.

Proof. If for anyu∈Z²condition (1) holds then, from Theorem2.11,X is not an m-code. On the other hand, if there existsu∈Z² such that condition (3) holds,

from Theorem3.8, we can verify ifX is anm-code.