Vector Addition System Reachability Problem: A Short Self-Contained Proof

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Vector Addition System Reachability Problem: A Short Self-Contained Proof

Jérôme Leroux

To cite this version:

Jérôme Leroux. Vector Addition System Reachability Problem: A Short Self-Contained Proof. Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2010, 6 (3), pp.1-25.

�10.2168/LMCS-6(3:22)2010�. �hal-00645446�

A Short Self-Contained Proof

Jérôme Leroux¹

LaBRI, Université de Bordeaux, CNRS leroux@labri.fr

Abstract. The reachability problem for Vector Addition Systems (VASs) is a central problem of net theory. The general problem is known to be decidable by algorithms exclusively based on the classical Kosaraju-Lambert-Mayr-Sacerdote- Tenney decomposition (KLMTS decomposition). Recently from this decomposi- tion, we deduced that a final configuration is not reachable from an initial one if and only if there exists a Presburger inductive invariant that contains the initial configuration but not the final one. Since we can decide if a Preburger formula denotes an inductive invariant, we deduce from this result that there exist check- able certificates of non-reachability in the Presburger arithmetic. In particular, there exists a simple algorithm for deciding the general VAS reachability prob- lem based on two semi-algorithms. A first one that tries to prove the reachability by enumerating finite sequences of actions and a second one that tries to prove the non-reachability by enumerating Presburger formulas. In this paper we provide the first proof of the VAS reachability problem that is not based on the KLMST decomposition. The proof is based on the notion of production relations, inspired from Hauschildt, that directly proves the existence of Presburger inductive invari- ants.

1 Introduction

Vector Addition Systems (VASs) or equivalently Petri Nets are one of the most popular formal methods for the representation and the analysis of parallel processes [1]. Their reachability problem is central since many computational problems (even outside the realm of parallel processes) reduce to the reachability problem. Sacerdote and Tenney provided in [9] a partial proof of decidability of this problem. The proof was completed in 1981 by Mayr [7] and simplified by Kosaraju [4] from [9,7]. Ten years later [5], Lambert provided a further simplified version based on [4]. This last proof still remains difficult and the upper-bound complexity of the corresponding algorithm is just known to be non-primitive recursive. Nowadays, the exact complexity of the reachability prob- lem for VASs is still an open-problem. Even the existence of an elementary upper-bound complexity is open. In fact, the known general reachability algorithms are exclusively based on the Kosaraju-Lambert-Mayr-Sacerdote-Tenney (KLMST) decomposition.

⋆This version extends the POPL’2011 paper with additional figures and examples. Some classes of sets get more intuitive names like the polytope conic sets, the polytope periodic sets, and the Petri sets that are now called the definable conic sets, the asymptotically definable periodic sets, and the almost semilinear sets.

Recently [6] we proved thanks to the KLMST decomposition that Parikh images of languages accepted by VASs are semi-pseudo-linear, a class that extends the Presburger sets. An application of this result was provided; we proved that a final configuration is not reachable from an initial one if and only if there exists a forward inductive invariant definable in the Presburger arithmetic that contains the initial configuration but not the final one. Since we can decide if a Presburger formula denotes a forward inductive invariant, we deduce that there exist checkable certificates of non-reachability in the Presburger arithmetic. In particular, there exists a simple algorithm for deciding the general VAS reachability problem based on two semi-algorithms. A first one that tries to prove the reachability by enumerating finite sequences of actions and a second one that tries to prove the non-reachability by enumerating Presburger formulas.

In this paper we provide a new proof of the reachability problem that is not based on the KLMST decomposition. The proof is based on the production relations inspired by Hauschildt [3] and it proves directly that reachability sets are almost semilinear, a class of sets introduced in this paper that extend the class of Presburger sets and contained in the class of semi-pseudo-linear sets. In particular this paper provides a more precise characterization of the reachability sets of VASs.

Outline of the paper: Section 2 provides notations and classical definitions. Sec- tion 3 and Section 4 introduce classes of sets used in the sequel : definable conic sets and vector spaces in the first one and asymptotically definable periodic sets, Presburger sets, and almost semilinear sets in the second one. Section 5 and Section 6 show that is sufficient to prove that the reachability relation of a Vector Addition system is an almost semilinear relation in order to deduce the existence of forward inductive invari- ants definable in the Presburger arithmetic proving the non-reachability. In Section 7 we introduce the class of Vector Addition Systems and the central notion of production re- lations. We show in the next Section 8 that these relations are asymptotically definable periodic. In Section 9 we prove that the reachability relation of a Vector Addition Sys- tem is an almost semilinear relation. Finally in Section 10 we combine all the previous results to deduce the decidability of the Vector Addition System reachability problem based on Presburger inductive invariants.

2 Notations

We introduce in this section notations and classical definitions used in this paper.

We denote byN,N_>0,Z,Q,Q_≥0,Q_>0 the set of natural numbers, positive inte- gers, integers, rational numbers, non negative rational numbers, and positive rational numbers. Vectors and sets of vectors are denoted in bold face. Theith component of a vectorv ∈ Q^d is denoted byv(i). We introduce||v||^∞ = max1≤i≤d|v(i)|where

|v(i)|is the absolute value ofv(i). The total order≤overQis extended component- wise into an order≤over the set of vectorsQ^d. The addition function+is also ex- tended component-wise overQ^d. Given two setsV₁,V₂⊆Q^dwe denote byV₁+V₂ the set {v₁ +v₂ | (v1,v₂) ∈ V₁ ×V₂}, and we denote by V₁ −V₂ the set {v₁−v₂ | (v1,v₂) ∈ V₁ ×V₂}. In the same way givenT ⊆ QandV ⊆ Q^d we letTV={tv|(t,v)∈T×V}. We also denote byv₁+V₂andV₁+v₂the sets

{v₁}+V₂ andV₁+{v₂}, and we denote bytVandTvthe sets{t}VandT{v}. In the sequel, an empty sum of sets included inQ^ddenotes the set reduced to the zero vector{0}.

A (binary) relationRoverQ^dis a subsetR⊆Q^d×Q^d. The composition of two relationsRandSis the relation denoted byR◦Sand defined as usual by the following equality:

R◦S= [

y∈Q^d

(x,z)∈Q^d×Q^d|(x,y)∈R ∧ (y,z)∈S

The reflexive and transitive closure of a relationR is denoted by R^∗. In this paper, notions introduced over the sets are transposed over the relations by identifyingQ^d×Q^d withQ^2d.

An order⊑over a setSis said to be well if for every sequence(sn)n∈Nof elements s_n ∈S we can extract a sub-sequence that is non-decreasing for⊑, i.e. there exists a strictly increasing sequence(nk)k∈Nof natural numbers in(N,≤)such that(snk)k∈N

is non decreasing for⊑. A minimal element of an ordered set(S,⊑)is an elements∈S such that for everyt∈Tthe relationt⊑simpliess=t. Given a setY ⊆Swe denote bymin⊑(Y)the set of minimal elements of the ordered set(Y,⊑). Let us recall that if (S,⊑)is well ordered thenX = min⊑(Y)is finite and for everyy ∈ Y there exists x∈Xsuch thatx⊑y.

Let us consider an order⊑over a setS. We introduce the component-wise extension of⊑over the set of vectorsS^ddefined bys⊑tifs(i)⊑t(i)for everyi∈ {1, . . . , d}. Lemma 2.1 (Dickson’s Lemma). The ordered set(S^d,⊑)is well for every well or- dered set(S,⊑).

Example 2.2. The set(N,≤)is well ordered. Hence(N^d,≤)is also well ordered. The set(Z,≤)is not well ordered.

3 Definable Conic Sets

A conic set is a setC⊆Q^dsuch that0∈C,C+C⊆Cand such thatQ_≥0C⊆C.

A conic setCis said to be finitely generated if there exists a finite sequencec₁, . . . ,c_k of vectorsc_j∈Csuch thatC=Q_≥0c₁+· · ·+Q_≥0c_k.

Definition 3.1. A conic setCis said to be definable if it is definable inFO (Q,+,≤,0).

In this section definable conic sets are geometrically characterized thanks to the vector spaces and the topological closure.

Example 3.2. Fig. 1 depicts examples of finitely generated conic sets and (non finitely generated) definable conic sets. The conic setC ={(c1, c2)∈Q²_≥0 | √

2c2 ≤c1}is not definable.

Fig. 1. The finitely generated conic set Q≥0(1,1) +Q≥0(1,0) and the definable conic set {(0,0)} ∪ {(c¹, c²)∈Q²>0|c²≤c¹}

A vector space is a setV ⊆ Q^d such that0 ∈ V,V+V ⊆ Vand such that QV ⊆ V. LetX ⊆ Q^d. The following set is a vector space called the vector space generated byX.

V=







k

X

j=1

λjx_j|k∈Nand(λj,x_j)∈Q×X







This vector space is the minimal for inclusion among the vector space that containsX.

Note that the vector spaceV generated by a conic setC satisfies the equalityV = C−C. Let us recall that every vector spaceVis generated by a finite setXwith at mostdvectors. The rankrank(V)of a vector spaceVis the minimal natural number r∈ {0, . . . , d}such that there exists a finite setXwithrvectors that generatesV. Note thatrank(V)≤rank(W)for every pair of vector spacesV⊆W. Moreover, ifVis strictly included inWthenrank(V)<rank(W).

Example 3.3. Vector spacesVincluded inQ²satisfyrank(V)∈ {0,1,2}. Moreover these vectors spaces can be classified as follows :rank(V) = 0if and only ifV={0}, rank(V) = 1if and only ifV=Qvwithv∈Q²\{0}, andrank(V) = 2if and only ifV=Q².

The (topological) closure of a setX ⊆ Q^d is the setXof vectorsr ∈ Q^d such that for everyǫ∈Q>0there existsx∈Xsatisfying||r−x||^∞ < ǫ. A setXis said to be closed ifX=X. Note thatXis closed and this set is the minimal for inclusion among the closed sets that containX. Let us recall that a vector spaceVis closed and the closure of a conic set is a conic set. Since the classical topological interior of a conic setCis empty when the vector space generated byCis not equal toQ^d(the conic set is degenerated), we introduce the notion of interior ofCrelatively to the vector space V=C−C. More precisely, a vectorc∈Cis said to be in the interior ofCif there existsǫ∈Q_>0such thatc+v∈Cfor everyv ∈C−Csatisfying||v||^∞ < ǫ. We denote byint(C)the set of interior vectors ofC. Let us recall thatint(C)is non empty for every conic setC, andC₁ =C₂if and only ifint(C1) = int(C2)for every conic setsC₁,C₂.

Example 3.4. LetX= (1,5)×(1,5). ThenX= [1,5]×[1,5](see Fig. 2).

Fig. 2. SetsX= (1,5)×(1,5)andX= [1,5]×[1,5]

The following lemma characterizes the finitely generated cones.

Lemma 3.5 (Duality). LetV⊆Q^dbe a vector space. A conic setC⊆Vis finitely generated if and only if there exists a sequence(h_j)1≤j≤kof vectorsh_j∈V\{0}such that:

C=

k

\

j=1

( v∈V|

d

X

i=1

h_j(i)v(i)≥0 )

Moreover in this case the following equality holds if and only ifVis the vector space generated byC:

int(C) =

k

\

j=1

( v∈V|

d

X

i=1

h_j(i)v(i)>0 )

Proof. This is a classical result of duality [10]. ⊓⊔

h¹ h²

Fig. 3. A picture of the duality lemma 3.5

Example 3.6. Let us introduce the whole vector spaceV=Q²and the finitely gener- ated conic setC=Q_≥0(1,1) +Q_≥0(1,0). Fig. 3 shows thatC=T

j∈{1,2}{v∈V| Pd

i=1h_j(i)v(i)≥0}whereh₁= (0,2)andh₂= (2,−2).

Lemma 3.7. The topological closure of a set definable inFO (Q,+,≤,0)is a finite union of finitely generated conic sets.

Proof. Let X ⊆ Q^d be a set definable inFO (Q,+,≤,0). Since this logic admits quantification elimination we deduce that there exists a quantifier free formula in this logic that denotesX. Hence there exists a finite sequence(Aj)1≤j≤kof finite setsAj⊆ Q^d× {>,≥}such thatX=Sk

j=1X_jwhere:

X_j= \

(h,#)∈Aj

(

x∈Q^d|

d

X

i=1

h(i)x(i)#0 )

We can assume without loss of generality thatX_jis non empty. Moreover ifk= 0the proof is immediate sinceX= ∅. So we can assume thatk ≥1. Let us introduce the following setR_j:

R_j = \

(h,#)∈Aj

(

x∈Q^d|

d

X

i=1

h(i)x(i)≥0 )

Lemma 3.5 shows thatR_jis finitely generated. Thanks to Lemma 3.5, we deduce that R=Sk

j=1R_j is closed. We are going to prove thatX=R. SinceX_j ⊆R_j we get X⊆R. AsRis closed we deduce thatX⊆R. Let us prove the converse inclusion.

Letr∈R. There existsj∈ {1, . . . , k}such thatr∈R_j. SinceX_jis non empty, there existsx_j ∈X_j. Asr_j ∈R_j andx_j ∈X_jwe deduce thatr_j+Q_>0x_j ⊆X_j. Hence r_j ∈X_jand we have proved the other inclusionR⊆X. ThereforeXis a finite union of finitely generated conic sets since it is equal toR. ⊓⊔ Theorem 3.8. A conic setC ⊆ Q^d is definable if and only if the conic setC∩Vis finitely generated for every vector spaceV⊆Q^d.

Proof. Let us first consider a definable conic setC⊆Q^d, letVbe a vector space, and let us prove thatXis finitely generated whereX = C∩V. SinceXis definable in FO (Q,+,≤,0), Lemma 3.7 shows thatX=Sk

j=1C_jwhereC_jis a finitely generated conic sets. Moreover, asXis non empty we deduce thatk ≥1. AsXis a conic set we deduce that Pk

j=1C_j ⊆ X. Moreover, as0 ∈ C_j for everyj, we deduce that C_j⊆Pk

j=1C_jfor everyj. ThusX=Pk

j=1C_jand we have proved thatXis finitely generated.

Conversely, we prove by induction overrthat the conic sets C ⊆ Q^d such that rank(C−C) ≤ rand such that the conic setC∩V is finitely generated for every vector spaceV ⊆ Q^d are definable. The caser = 0is immediate since in this case C={0}. Let us assume the induction proved for an integerr∈Nand let us consider a conic set C ⊆ Q^d such that rank(C−C) ≤ r+ 1and such that the conic set C∩Vis finitely generated for every vector spaceV ⊆ Q^d. We introduce the vector spaceW = C−C. SinceC =C∩VwithV =Q^d, we deduce thatCis finitely generated. Lemma 3.5 shows that there exists a finite sequence(h_j)1≤j≤k of vectors h_j∈W\{0}such that the following equality holds:

C=

k

\

j=1

(

x∈W|

d

X

i=1

h_j(i)x(i)≥0 )

Sinceint(C) = int(C)we get the following equality:

int(C) =

k

\

j=1

(

x∈W|

d

X

i=1

h_j(i)x(i)>0 )

In particularint(C)is definable inFO (Q,+,≤,0,1). Asint(C)⊆C⊆Cwe deduce the following decomposition whereW_j ={w∈W|Pd

i=1h_j(i)w(i) = 0}: C= int(C)∪

k

[

j=1

(C∩W_j)

Observe that h_j ∈ W\W_j and in particular W_j is strictly included in W. Thus rank(Wj) <rank(W) ≤ r+ 1. Note thatC_j = C∩W_j is a conic set such that rank(Cj−C_j)≤rank(Wj)≤rand such thatC_j∩Vis a finitely generated conic set for every vector spaceV. Thus by inductionC_jis definable inFO (Q,+,≤,0,1).

We deduce thatCis definable. We have proved the induction. ⊓⊔ Example 3.9. Observe that the conic setC = {(c1, c2) ∈ Q²_≥0 | √

2c2 ≤ c1}is not finitely generated. Let us considerV = Q²and observe thatC∩V = Cand since C =Cwe deduce thatC∩Vis not finitely generated. Theorem 3.8 shows thatCis not definable.

4 Presburger Sets And Almost Semilinear Sets

In this section we introduce the Presburger sets and the almost semilinear sets.

A periodic set is a subsetP⊆Z^dsuch that0∈Pand such thatP+P⊆P. A periodic setPis said to be finitely generated if there exists a finite sequencep₁, . . . ,p_k of vectorsp_j ∈Psuch thatP=Np₁+· · ·+Np_k (see Fig. 4). A subsetS⊆Z^dis called a Presburger set if it can be denoted by a formula in the Presburger arithmetic FO (Z,+,≤,0,1). Let us recall [2] that a subsetS⊆Z^dis Presburger if and only if it is semilinear, i.e. a finite union of setsb+Pwhereb∈Z^dandP⊆Z^d is a finitely generated periodic set. The class of almost semilinear sets is obtained by weakening the finiteness property of the periodic setsP.

Definition 4.1. A periodic setPis said to be asymptotically definable if the conic set Q_≥0Pis definable.

Remark 4.2. Every finitely generated periodic setPis asymptotically definable since in this caseQ_≥0Pis a finitely generated conic set and in particular a definable conic set.

Example 4.3. The periodic setP = {(p1, p2)∈ N² | √

2p2 ≤ p1}is not asymptot- ically definable since Q_≥0P = {(c1, c2) ∈ N² | √

2c2 ≤ c1}is not definable (see example 3.9).

p(2)

p(1)

Fig. 4. The finitely generated periodic setP=N(1,1) +N(2,0) p(2)

p(1) p(1) + 1≤2p(2)

p(2)≤p(1)

Fig. 5. An asymptotically definable periodic set.

Example 4.4. The periodic setP = {p ∈ N² | p(2) ≤ p(1) ≤ 2^p(2)−1} is rep- resented in Figure 5. Observe thatQ_≥0P = {0} ∪ {c ∈ Q²_>0 | p(2) ≤ p(1)}is a definable conic set. ThusPis an asymptotically definable periodic set.

The following lemma shows that the class of asymptotically definable periodic sets is stable by finite intersections.

Lemma 4.5. We have(Q≥0P1)∩(Q≥0P2) =Q≥0(P1∩P2)for every periodic sets P1,P2⊆Z^d.

Proof. Observe thatP₁ ⊆ Q_≥0P₁ andP₂ ⊆ Q_≥0P₂. HenceP₁∩P₂ ⊆ Cwhere C= (Q≥0P₁)∩(Q≥0P₂). AsCis a conic set we deduce thatQ_≥0(P1∩P₂)⊆C.

For the converse inclusion. Let c ∈ C. Sincec ∈ Q_≥0P₁, there existsλ1 ∈ Q_≥0 such thatc∈ λ1P₁. Symmetrically there existsλ2 ∈ Q_≥0 such thatc ∈ λ2P₂. Let n1, n2 ∈ N_>0 such thatn1λ1 ∈ Nandn2λ2 ∈ N. Letn = n1n2 and observe that nc∈n2(n1λ1)P1⊆P1sinceP1is a periodic set. Symmetricallync∈P2. We have proved thatnc∈P1∩P2. Thusc∈Q≥0(P1∩P2)and we get the other inclusion. ⊓⊔

Definition 4.6. An almost semilinear set is a subsetX⊆Z^dsuch that for every Pres- burger setS ⊆ Z^dthe setX∩Sis a finite union of setsb+Pwhereb ∈ Z^dand P⊆Z^dis an asymptotically definable periodic set.

Fig. 6. An asymptotically definable periodic set that is not almost semilinear.

Example 4.7. Let us consider the periodic setP = {(0,0)} ∪ {(2ⁿ,1) | n ∈ N} ∪ ((1,2) +N²)depicted in Fig.6. Observe thatQ_≥0Pis the definable conic set{(0,0)} ∪ Q_≥0×Q_>0. Note thatPis not almost semilinear sinceP∩(N× {1}) ={(2ⁿ,1)|n∈ N}can not be decomposed as a finite union of setsb+Pwhereb∈Z^dandP⊆Z^d is an asymptotically definable periodic set.

The class of almost semilinear sets is included in the class of Presburger sets. The strict inclusion will be proved strict as a direct consequence of a stronger result proved in this paper. In fact the reachability relation of a Vector Addition System is proved to be almost semilinear and we know that in general such a relation is not Presburger.

5 Linearizations

The linearization of a periodic setP ⊆ Z^d is the periodic setlin(P)defined by the following equality:

lin(P) = (P−P)∩Q_≥0P

Lemma 5.1. The linearization of an asymptotically definable periodic set is finitely generated.

Proof. Let Vbe the vector space generated byPand let us introduce the conic set C = Q_≥0P. Note that Q_≥0P ⊆ V and since V is closed we get C ⊆ V. As Q_≥0Pis a definable conic set we deduce that C is finitely generated. Hence there existsc1, . . . ,c_k ∈ Csuch thatC = Q_≥0c1+· · ·+Q_≥0c_k. As c_j ∈ C ⊆ V = Q_≥0P−Q_≥0P, by replacingc_jby a vector inN_>0c_jwe can assume thatc_j∈P−P for everyj∈ {1, . . . , k}.

We introduce the following setR:

R=







r∈P−P|r=

k

X

j=1

λjc_j λj∈Q 0≤λj<1







We observe that every vectorr ∈ Rsatisfies ||r||^∞ ≤ s wheres = Pk

j=1||c_j||^∞. HenceR⊆ {−s, . . . , s}^dand we deduce thatRis finite.

LetLbe the periodic set generated by the finite setR∪ {c₁, . . . ,c_k}. Since this finite set is included inlin(P)we deduce thatL ⊆lin(P). Let us prove the converse

inclusion. Letx∈lin(P). Sincex∈C, there exists a sequence(µj)1≤j≤k of rational elementsµj ∈ Q_≥0 such thatx = Pk

j=1µjc_j. Let us introducenj ∈ Nsuch that λ_j =µ_j−n_jsatisfies0≤λ_j <1. Letr=Pk

j=1λ_jc_j. Asr=x−Pk

j=1n_jc_jwe getr∈P−P. Thusr∈R. Fromx=r+Pk

j=1n_jc_jwe getx∈L. We have proved thatlin(P)is the finitely generated periodic setL. ⊓⊔ We observe that if the intersection(b1+P₁)∩(b2+P₂)is empty whereb₁,b₂∈ Z^dandP₁,P₂⊆Z^dare two asymptotically definable periodic sets then the intersection (b1+ lin(P1))∩(b2+ lin(P2))may be non empty (see Example 5.3). In this section we show that a dimension is strictly decreasing.

Let us first introduce our definition of dimension. The dimensiondim(X)of a non- empty setX⊆Z^dis the minimal integerr∈ {0, . . . , d}such that there existsk∈N>0, a sequence(bj)1≤j≤kof vectorsb_j ∈Z^d, and a sequence(Vj)1≤j≤kof vector spaces V_j ⊆Q^dsuch thatrank(Vj)≤rand such thatX⊆Sk

j=1b_j+V_j. The dimension of the empty set is defined bydim(∅) =−1.

In the reminder of this section we prove the following Theorem 5.2. All the other results or definitions introduced in this section are not used in the sequel.

Theorem 5.2. Letb1,b2 ∈ Z^d and letP1,P2be two asymptotically definable peri- odic sets such that the intersection(b1+P1)∩(b2+P2)is empty. The intersection X= (b1+ lin(P1))∩(b2+ lin(P2))satisfies:

dim(X)<max{dim(b1+P1),dim(b2+P2)}

Example 5.3. Sets introduced in this example are depicted in Fig. 7. Let us introduce the asymptotically definable periodic setsP1 ={p∈N²|p(2)≤p(1)≤2^p(2)−1} andP2=N(1,0) +N(3,−1). We considerb1= (0,0)andb2= (7,2). We observe that the intersection ofb1+P1 andb2+P2is empty. Note that the intersectionX ofb₁+ lin(P₁)andb₂+ lin(P₂)satisfiesX={(7,2),(10,1),(13,0)}+N(1,0). In particular we havedim(X) = 1whereasdim(b₁+lin(P₁)) = dim(b₂+lin(P₂)) = 2.

Fig. 7. A figure for Theorem 5.2 and Example 5.3.

We first characterize the dimension of a periodic set.

Lemma 5.4. LetVbe the vector space generated by a periodic setP. Thenrank(V) = dim(P).

Proof. LetPbe a periodic set and let us first prove by induction overk ∈ N_>0that for every sequence(Vj)1≤j≤kof vector spacesV_j ⊆Q^d, the inclusionP⊆Sk

j=1V_j implies that there existsj ∈ {1, . . . , k} such that P ⊆ V_j. The casek = 1is im- mediate. Assume the property proved for an integerk ∈ N_>0and let us assume that P ⊆ Sk+1

j=1V_j. If P ⊆ V_k+1 the property is proved. So we can assume that there existsp∈ P\Vk+1. Let us prove thatP ⊆Sk

j=1V_j. We considerx∈ P. Observe that if x 6∈ Vk+1 then x ∈ Sk

j=1V_j. So we can assume thatx ∈ Vk+1. We ob- serve thatp+nx ∈ Pfor everyn ∈ Nsince the setPis periodic. We deduce that there existsj ∈ {1, . . . , k+ 1}such thatp+nx ∈ V_j. Naturally this integerj de- pends onn. However, since{1, . . . , k+ 1}is finite whereasNis infinite, there exists j ∈ {1, . . . , k+ 1}andn < n^′ inNsuch thatp+nxandp+n^′xare both inV_j. AsV_j is a vector space, we deduce thatn^′(p+nx)−n(p+n^′x)is inV_j. Hence p∈V_j. Asp6∈V_k+1we deduce thatj6=k+ 1. AsV_jis a vector space we deduce that(p+n^′x)−(p+nx)∈V_j. Hencex∈V_j. We have proved thatx∈Sk

j=1V_j. ThusP ⊆Sk

j=1V_j and by induction there existsj ∈ {1, . . . , k}such thatP⊆V_j. We have proved the induction.

Now, let us prove the lemma. We consider a periodic setPand we letVbe the vector space generated by this set. SinceP⊆Vwe deduce thatdim(P)≤rank(V).

For the converse inclusion, sincePis non empty we deduce thatP⊆Sk

j=1b_j+V_j wherek ∈ N_>0,b_j ∈ Z^d andV_j ⊆ Q^d is a vector space such that rank(Vj) ≤ dim(P). Let us consider the setJ ={j∈ {1, . . . , k} |b_j ∈V_j}and let us prove that P⊆S

j∈JV_j. Letp∈Pandn∈N. Sincenp∈Pthere existsj∈ {1, . . . , k}such thatnp∈b_j+V_j. Hence there existsj ∈ {1, . . . , k}andn < n^′inNsuch thatnp andn^′pare both inb_j+V_j. AsV_jis a vector space we deduce thatn^′p−np∈V_j. Thusp∈V_j. Moreover asb_j ∈np−V_j⊆V_jwe deduce thatj∈J. We have prove the inclusionP⊆S

j∈JV_j. From the previous paragraph we deduce that there exists j ∈ J such thatP ⊆ V_j. By minimality of the vector space generated byPwe get V ⊆V_j. Hencerank(V)≤rank(Vj). Sincerank(Vj)≤dim(P)we have proved

the inequalityrank(V)≤dim(P). ⊓⊔

Next we prove a separation property.

Lemma 5.5. LetC_≤ andC_≥ be two finitely generated conic sets that generates the same vector spaceVand such that the vector space generated byC_≤∩C_≥is strictly included inV. Then there exists a vectorh∈V\{0}such that for every#∈ {≤,≥}, we have:

C_#⊆ (

v∈V|

d

X

i=1

h(i)v(i)#0 )

Proof. Lemma 3.5 shows that there exists two finite setsH_≤,H_≥included inV\{0} such that:

C_#= \

h∈H#

( v∈V|

d

X

i=1

h(i)v(i)≥0 )

int(C_#) = \

h∈H#

( v∈V|

d

X

i=1

h(i)v(i)>0 )

Assume by contradiction that the intersectionint(C≤)∩int(C≥)is non empty and let cbe a vector in this set. Observe that there existsǫ∈Q>0such thatc+v∈C≤∩C≥

for everyv ∈Vsuch that||v||^∞ < ǫ. We deduce that the vector space generated by C≤∩C≥containsVand we get a contradiction.

We deduce that the following intersection is empty whereH=H≤∪H≥

\

h∈H

( v∈V|

d

X

i=1

h(i)v(i)>0 )

Farkas’s Lemma [10] shows that there exists a non-zero functionf : H → Q_≥0 such thatP

h∈Hf(h)h=0. Let us introducea=P

h∈H≥f(h)handb=P

h∈H\H≥f(h)h.

Assume by contradiction thata = 0. Sincea+b= 0we deduce thatb= 0. Asf is not the zero function, there exists h ∈ H such thatf(h) 6= 0. Note that either h ∈ H≥ orh ∈ H\H≥. In the first case we deduce thatint(C≥)is empty and in the second case we deduce thatint(C≤)is empty. Since both cases are impossible we get a contradiction. Thusa66=0. For everyc∈int(C≥)we havePd

i=1a(i)c(i)≥0.

Since the set {c ∈ Q^d | Pd

i=1a(i)c(i) ≥ 0} is closed we deduce that for every c∈ int(C≥) =C_≥the same inequality holds. Now let us considerc∈int(C≤). In this casePd

i=1b(i)c(i)≥0. Sincea+b=0we getPd

i=1a(i)c(i)≤0. We deduce

that this inequality holds for everyc∈C_≤. ⊓⊔

Remark 5.6. The previous Lemma 5.5 is wrong if we remove the finitely generated condition on the conic setsC_≤ andC_≥. In fact let us consider the conic setsC_≤ = {x ∈ Q²_≥0 | x(1) ≤ √

2x(2)}andC_≥ ={x ∈ Q²_≥0 | x(2) ≥ √

2x(2)}. Observe thatC_≤∩C_≥ ={0}. Hence the vector space generated by the intersection is strictly included inQ². However there does not exist a vectorh∈Q²\{0}satisfying the sepa- ration property required by Lemma 5.5. This problem can be overcome by introducing the vector spaces ofR^d. We do not introduce this extension to simplify the presentation.

We can now provide a proof for Theorem 5.2. We consider two vectorsb1,b2∈Z^d and two periodic setsP₁,P₂⊆Z^dsuch that(b₁+P₁)∩(b₂+P₂) =∅. We introduce the intersectionX= (b₁+ lin(P₁))∩(b₂+ lin(P₂)). Observe that ifXis empty the theorem is proved. So we can assume that there exists a vectorbin this intersection.

Let us denote byV₁ andV₂the vector spaces generated byP₁andP₂. Lemma 5.4 shows thatrank(Vj) = dim(Pj)and fromdim(bj +P_j) = dim(Pj)we deduce thatdim(bj +P_j) = rank(Vj). AsXis included inb+VwhereV =V₁∩V₂,

we deduce that if V is strictly included in V_j for one j ∈ {1,2} thendim(X) ≤ rank(V)<rank(V_j) = dim(b_j+P_j)and the theorem is proved. So we can assume thatV₁=V₂=V. Let us consider the conic setsC₁ =Q_≥0P₁andC₂=Q_≥0P₂. SinceP₁andP₂are asymptotically definable periodic sets, we deduce thatC₁andC₂ are finitely generated conic sets. Note thatC₁,C₂⊆V. We introduce the intersection C=C₁∩C₂.

Assume by contradiction that the vector space generated byCis equal toV. Let us consider a vectorcin the interior ofC. The characterization given by Lemma 3.5 shows that in this caseint(C) = int(C1)∩int(C2). Sinceint(Cj) = int(Q_≥0P_j)we deduce thatc∈(Q_≥0P1)∩(Q_≥0P2). Lemma 4.5 shows thatc∈Q_≥0(P1∩P2). By replacingcbe a vector inN_>0cwe can assume thatc∈P₁∩P₂.

Let us prove that there existsk1 ∈ Nsuch thatb+k1c ∈ b1+P1. Fromb ∈ b1+ lin(P1)we deduce that there existsp1,p^′₁ ∈P1such thatb=b1+p1−p^′₁. Since−p^′₁is in the vector space generated byCandcis in the interior of C, there existsn1∈Nlarge enough such thatn1c+ (−p^′₁)∈C₁. Hence there existsn^′₁∈N_>0 such thatn1n^′₁c−n^′₁p^′₁ ∈P₁. Thusn1n^′₁c−p^′₁ ∈(n^′₁−1)p^′₁+P₁ ⊆P₁. Hence b+k1c∈b₁+P₁withk1=n1n^′₁.

Symmetrically we deduce that there existsk2∈Nsuch thatb+k2c∈b₂+P₂. We have proved thatb+ (k1+k2)c∈(b₁+P₁)∩(b₂+P₂)and we get a contradiction since this intersection is supposed to be empty.

We deduce that the vector space generated byCis strictly included inV. Lemma 5.5 shows that there exists a vectorh∈V\{0}such that:

C₁⊆ (

v∈V|

d

X

i=1

h(i)v(i)≥0 )

C₂⊆ (

v∈V|

d

X

i=1

h(i)v(i)≤0 )

By replacinghby a vector inN_>0hwe can assume thath∈Z^d. Now let us consider x∈X. Sincex−b1∈C1we deduce thatPd

i=1h(i)(x(i)−b1(i))≥0and sincex− b₂∈C₂we deduce thatPd

i=1h(i)(x(i)−b₂(i))≤0. We introduce the integersz1= Pd

i=1h(i)b1(i)andz2=Pd

i=1h(i)b2(i). We have proved thatXcan be decomposed into a finite union of slicesX=Sz2

z=z1X_zwhere:

X_z= (

x∈X|

d

X

i=1

h(i)x(i) =z )

Let us prove thatdim(Xz)<rank(V). IfX_zis empty the relation is immediate.

IfX_zis non empty let us considerx∈X_zand observe thatX_z⊆x+Wwhere:

W= (

v∈V|

d

X

i=1

h(i)v(i) = 0 )

Note thath ∈ V\W. We deduce thatWis strictly included inVand in particular rank(W)<rank(V). Hencedim(X_z)<rank(V).

FromX = Sz2

z=z1X_z and dim(Xz) < rank(V) for every z, we deduce that dim(X)<rank(V)and the theorem is proved.

6 Presburger Invariants

Given a relationRoverZ^dand two setsX,Y ⊆Z^dwe introduce the forward image post_R(X)and the backward imagepre_R(Y)defined by the following equalities:

(post_R(X) =S

x∈X{y∈Z^d|(x,y)∈R} pre_R(Y) =S

y∈Y{x∈Z^d|(x,y)∈R}

We say that a setX⊆Z^dis a forward invariant forRifpost_R(X)⊆Xand we say that a setY⊆Z^dis a backward invariant forRifpre_R(Y)⊆Y. In the reminder of this section we prove the following Theorem 6.1. All the other results or definitions introduced in this section are not used in the sequel.

Theorem 6.1. LetR^∗be a reflexive and transitive almost semilinear relation overZ^d and letX,Y⊆Z^dbe two Presburger sets such thatR^∗∩(X×Y)is empty. There exists a partition ofZ^dinto a Presburger forward invariant that containsXand a Presburger backward invariant that containsY.

We first prove the following lemma.

Lemma 6.2. The setspost_R(X)andpre_R(Y)are almost semilinear for every almost semilinear relationR⊆Z^d×Z^dand for every Presburger setsX,Y⊆Z^d

Proof. Let us first prove that post_R(X)is an almost semilinear set. We consider a Presburger setS ⊆ Z^d. Observe that X×Sis a Presburger relation. Since R is an almost semilinear relation we deduce thatR∩(X×S)can be decomposed into a finite unionSk

j=1(aj,b_j) +Rjwithk∈N,(aj,b_j)∈Z^d×Z^dandRjis an asymptotically definable periodic relation. We deduce that post_R(X)∩S = Sk

j=1b_j +P_j where P_j = {v ∈ Z^d | ∃(u,v) ∈ R_j}. SinceR_j is a periodic relation we deduce thatP_j is a periodic set. Moreover sinceQ_≥0R_j is definable we deduce thatC_j ={v∈Q^d |

∃(u,v)∈Q_≥0R_j}is definable. Let us prove thatQ_≥0P_j =C_j. By construction we haveP_j ⊆ C_j. SinceC_j is conic we deduce thatQ_≥0P_j ⊆ C_j. For the converse inclusion letv ∈ C_j. There existsu ∈ Q^d such that(u,v) ∈ Q_≥0R_j. Hence there existsλ∈Q_≥0such that(u,v)∈λR_j. Let us considern∈N_>0such thatnλ_j ∈N and observe that(nu, nv)∈ (nλ)Rj ⊆R_j sinceR_jis periodic. Thusnv ∈ P_j and we have proved thatv∈Q_≥0P_j. HenceQ_≥0P_j =C_jis a definable conic set and we have proved thatpost_R(X)is an almost semilinear set. Frompre_R(Y) = post_R−1(Y) withR⁻¹ = {(y,x) |(x,y) ∈R}we deduce thatpre_R(Y)is an almost semilinear

set. ⊓⊔

Now, let us prove Theorem 6.1. We consider a reflexive and transitive almost semi- linear relation R^∗. We introduce the notion of separators. A separator is a couple (X,Y)of Presburger sets such that the intersectionR^∗ ∩(X×Y)is empty. Since R^∗is reflexive, the intersectionX∩Yis empty. The Presburger setD=Z^d\(X∪Y) is called the domain of (X,Y). We observe that a separator (X,Y) with an empty domain is a partition of Z^d such that X is a Presburger forward invariant andY is a Presburger backward invariant. In particular Theorem 6.1 is obtained thanks to the following Lemma 6.3 with an immediate induction.

Lemma 6.3. Let(X0,Y0)be a separator with a non-empty domainD0. There exists a separator(X,Y)with a domainDsuch thatX₀ ⊆ X,Y₀ ⊆ Yanddim(D) <

dim(D₀).

Proof. We first observe that a couple(X,Y)of Presburger sets is a separator if and only ifpost_R∗(X)∩pre_R∗(Y) =∅if and only ifpost_R∗(X)∩Y=∅if and only if pre_R∗(Y)∩X=∅.

Since R^∗ is an almost semilinear relation we deduce thatpost_R∗(X0)is an al- most semilinear set. AsD₀ is a Presburger set, we deduce thatpost_R∗(X0)∩D₀ = Sk

j=1b_j+P_j whereb_j ∈Z^d andP_j ⊆ Z^dis an asymptotically definable periodic set. We introduce the following Presburger set:

S=

k

[

j=1

b_j+ lin(Pj)

Observe thatpost_R∗(X0)∩D₀⊆S. We deduce that the setY=Y₀∪(D0\S)is such thatpost_R∗(X0)∩Y=∅. Hence(X0,Y)is a separator.

Symmetrically, sinceR^∗is an almost semilinear relation we deduce thatpre_R∗(Y) is an almost semilinear set. AsD₀is a Presburger set, we deduce thatpre_R∗(Y)∩D₀= Sn

l=1c_l+Q_lwherec_l∈Z^dandQ_l⊆Z^dis an asymptotically definable periodic set.

We introduce the following Presburger set:

T=

n

[

l=1

c_l+ lin(Q_l)

Observe thatpre_R^∗(Y)∩D0⊆T. We deduce that the setX=X0∪(D0\T)is such thatpre_R^∗(Y)∩X=∅. Hence(X,Y)is a separator.

Let us introduce the domainDof(X,Y). We have the following equality where Z_j,l= (b_j+ lin(P_j))∩(c_l+ lin(Q_l)):

D=D₀∩( [

1≤j≤k 1≤l≤n

Z_j,l)

As(X,Y)is a separator we deduce thatpost_R∗(X)∩pre_R∗(Y)is empty. Asb_j+Pj⊆ post_R∗(X0)⊆post_R∗(X)andc_l+Q_l⊆pre_R∗(Y)we deduce that the intersection

(b_j+P_j)∩(c_l+Q_l)is empty. Theorem 5.2 shows thatdim(Z_j,l)<max{dim(b_j+ P_j),dim(c_l+Q_l)}. Sinceb_j+P_j ⊆D₀andc_l+Q_l⊆D₀we deduce thatdim(b_j+ P_j) ≤ dim(D₀)anddim(c_l +Q_l) ≤ dim(D₀). We have proved thatdim(D) <

dim(D0). ⊓⊔

7 Vector Addition Systems

In this section we introduce the Vector Addition Systems, the production relations and a well order over the set of runs of Vector Addition Systems.

A Vector Addition System (VAS) is a finite subsetA ⊆Z^d. A marking is a vector m∈N^d. The semantics of vector addition systems is obtained by introducing for every wordw=a₁. . .a_kof vectorsa_j ∈Athe relation−→^w over the set of markings defined byx −→^w y if there exists a wordρ = m₀. . .m_k of markingsm_j ∈ N^d such that (x,y) = (m0,m_k)andm_j = m_j−1+a_j for everyj ∈ {1, . . . , k}. The wordρis unique and it is called the run fromxtoylabeled byw. The markingxis called the source ofρand it is denoted bysrc(ρ), and the markingyis called the target ofρand it is denoted bytgt(ρ). The set of runs is denoted byΩ.

The reachability relation is the relation denoted by −→^∗ over the set of markings defined byx−→^∗ yif there exists a wordw∈ A^∗such thatx−→^w y. In the sequel we often used the fact thatx−→^w yimpliesx+v−→^w y+vfor everyv∈N^d.

The production relation of a markingm ∈ N^d (see Fig. 8) is the relation −→^{∗ m} overN^d defined by r −→^{∗ m} sifm+r −→^∗ m+s. The production relation of a run ρ=m₀. . .m_kis the relation−→^∗ρdefined by the following composition:

−∗

→^ρ=−→^{∗ m}0 ◦ · · · ◦−→^{∗ m}k

m m+r

m+s

0

Fig. 8. The production relation of a markingm.

Example 7.1. The production relation−→^{∗ m}withm=0is the reachability relation.

The following Lemma 7.2 shows that−→^{∗ ρ} seens as a subset ofZ^2dis periodic for every runρas a composition of periodic relations (see Fig. 9). Note that in Section 8 we prove that these periodic relations are asymptotically definable.

Lemma 7.2. The relation−→^{∗ m}is periodic.

Proof. Let us assume thatr1 ∗

−→^m s1andr2 ∗

−→^ms2. Sincer1 ∗

−

→^m s1we deduce that r1+r2 ∗

−

→^m s1+r2. Moreover, sincer2 ∗

−→^ms2we deduce thatr2+s1 ∗

−

→^ms2+s1.

Thereforer₁+r₂−→^∗ms₁+s₂. ⊓⊔

m

m+r¹ m+s1

0

m

m+r² m+s²

0

m

m+r¹+r² m+s¹+r² m+s¹+s²

0

Fig. 9. Production relations are periodic.

We introduce a well order over the set of runs based on the following Lemma 7.3 Lemma 7.3. The following inclusion holds for every runρ:

(src(ρ),tgt(ρ))+−→^{∗ ρ} ⊆ −→^∗

Proof. Assume thatρ= m₀. . .m_k withm_j ∈ N^d, and let(r,s)be a couple in the production relation −→^{∗ ρ}. Since this relation is defined as a composition, there exists a sequence(vj)0≤j≤k+1 of vectorsv_j ∈ N^d satisfying the following relations with v0=randvk+1 =s:

v₀−→^{∗ m}0 v₁· · ·v_k −→^∗mkv_k+1

We introduce the vectora_j =m_j−m_j−1for everyj ∈ {1, . . . , k}. Sincem_j−1−→^aj m_jwe deduce thatm_j−1+v_j −→^aj m_j+v_j. Moreover, asv_j −→^{∗ m}j v_j+1, there exists a wordwj ∈A^∗ such thatm_j+v_j −−→^wj m_j+v_j+1. We deduce that the following relation holds:

m₀+v₀−−−−−−−−−−→^{w0a1w1...akwk} m_k+v_k+1

Therefore(m0,m_k) + (v0,v_k+1)is in the reachability relation. ⊓⊔ We introduce the orderover the set of runs defined byρ ρ^′ if the following inclusion holds:

(src(ρ^′),tgt(ρ^′))+−→^{∗ ρ′} ⊆ (src(ρ),tgt(ρ))+−→^{∗ ρ}

In the reminder of this section we prove the following theorem. All the other results or definitions introduced in this section are not used in the sequel.

Theorem 7.4. The orderis well.

The orderis proved well thanks to the Higmann’s Lemma. We first recall this lemma. Let us consider an order⊑over a setS. We introduce the order⊑^∗ over the set of words overSdefined byu⊑^∗ vwhereu=s1. . . skwithsj∈Sif there exists a sequence(tj)1≤j≤k withtj ∈ S andsj ⊑tj and a sequence(wj)0≤j≤k of words wj∈S^∗such thatv=w0t1w1. . . tkwk.

Lemma 7.5 (Higmann’s Lemma). The ordered set (S^∗,⊑^∗) is well for every well ordered set(S,⊑).

We associate to every runρ=m0. . .m_kthe wordα(ρ) = (a1,m1). . .(ak,m_k) wherea_j=m_j−mj−1. Note thatα(ρ)is a word over the alphabetS=A×N^d. We introduce the order⊑over this alphabet by(a,m)⊑(a^′,m^′)ifa=a^′andm≤m^′. SinceAis a finite set and≤is a well order overN^d, we deduce that⊑is a well order overS. From the Higmann’s lemma, the order⊑^∗ is well overS^∗. We introduce the well orderover the set of runs defined byρρ^′ifα(ρ)⊑^∗α(ρ^′),src(ρ)≤src(ρ^′) andtgt(ρ)≤tgt(ρ^′). The following lemma provides a useful characterization of this order.

Lemma 7.6. Letρ =m₀. . .m_k be a run and letρ^′ be another run. We haveρρ^′ if and only if there exists a sequence (vj)0≤j≤k+1 of vectors in N^d such that ρ^′ = ρ^′₀. . . ρ^′_kwhereρ^′_jis a run fromm_j+v_jtom_j+v_j+1.

Proof. We introduce the sequence(aj)1≤j≤kdefined bya_j=m_j−m_j−1. Assume first thatρρ^′.

Sinceα(ρ)⊑^∗ α(ρ^′)we getα(ρ^′) = w0(a₁,m^′₁)w1. . .(a_k,m^′_k)wk wherew_j ∈S^∗ andm^′_j ≥ m_j. We introduce the sequence(v_j)0≤j≤k+1 defined byv₀ = src(ρ^′)− src(ρ),v_k+1 = tgt(ρ^′)−tgt(ρ)andv_j =m^′_j−m_jfor everyj∈ {1, . . . , k}. Observe thatv_j ∈N^dfor everyj∈ {0, . . . , k+ 1}. We deduce thatρ^′ can be decomposed into ρ^′=ρ^′₀. . . ρ^′_kwhereρ^′_jis the run fromm_j+v_jtom_j+v_j+1such thatα(ρ^′_j) =w_j. Conversely let(vj)0≤j≤k+1be a sequence of vectors inN^dsuch thatρ^′=ρ^′₀. . . ρ^′_k whereρ^′_jis a run fromm_j+v_jtom_j+v_j+1. We deduce that we have the following equality wherem^′_j=m_j+v_janda^′_j ∈A:

α(ρ^′) =α(ρ^′₀)(a^′₁,m^′₁)α(ρ^′₁). . .(a^′_k,m^′_k)α(ρ^′_k)

Observe thata^′_j = tgt(ρ^′_j−1)−m^′_j= (mj+v_j)−(mj−1+v_j)and in particulara^′_j= a_j. We deduce thatα(ρ) ⊑^∗ α(ρ^′). Moreover, sincesrc(ρ) ≤ src(ρ^′)andtgt(ρ) ≤

tgt(ρ^′)we deduce thatρρ^′. ⊓⊔

Sinceis a well order, the following lemma shows thatis a well order. We have proved Theorem 7.4.

Lemma 7.7. ρρ^′impliesρρ^′.

Proof. Assume that ρ = m₀. . .m_k. Lemma 7.6 shows that there exists a sequence (vj)0≤j≤k+1of vectors inN^dsuch thatρ^′=ρ^′₀. . . ρ^′_kwhereρ^′_jis a run fromm_j+v_j tom_j+v_j+1. Lemma 7.3 shows that(src(ρ^′_j),tgt(ρ^′_j))+−→^∗ρ^′_j⊆−→^∗ .