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arXiv:1609.00214v1 [cs.FL] 1 Sep 2016
Separability of Reachability Sets of Vector
Addition Systems
Lorenzo Clemente1, Wojciech Czerwi´nski1, S lawomir Lasota1, and Charles
Paperman2
1University of Warsaw
2University of T¨ubingen
Abstract. Given two families of sets Fand G, the Fseparability prob-
lem for Gasks whether for two given sets U, V ∈ G there exists a set
S∈ F, such that Uis included in Sand Vis disjoint with S. We con-
sider two families of sets F: modular sets S⊆Nd, defined as unions of
equivalence classes modulo some natural number n∈N, and unary sets.
Our main result is decidability of modular and unary separability for
the class Gof reachability sets of Vector Addition Systems, Petri Nets,
Vector Addition Systems with States, and for sections thereof.
1 Introduction
In this paper we mainly investigate separability problems for sets of vectors
from Nd. We say that a set Uis separated from set Vby a set Sif U⊆Sand
V∩S=∅. For two families of sets Fand G, the F-separability problem for G
asks for two given sets U, V ∈ G whether Uis separated from Vby some set
from F. Concretely, we consider Fto be modular sets or unary sets, and Gto
be reachability set of Vector Addition Systems, or generalizations thereof.
Motivation. The separability problem is a classical problem in theoretical com-
puter science. It was investigated most extensively in the area of formal lan-
guages, for Gbeing the family of all regular word languages. Since regular lan-
guages are effectively closed under complement, the F-separability problem is
a generalization of the F-characterization problem, which asks whether a given
language belongs to F. Indeed, L∈ F if and only if Lis separated from its
complement by some language from F. Separability problems for regular lan-
guages attracted recently a lot of attention, which resulted in establishing the
decidability of F-separability for the family Fof separators being the piecewise
testable languages [2,22] (recently generalized to finite ranked trees [5]), the lo-
cally and locally threshold testable languages [21], the languages definable in
first order logic [24], and the languages of certain higher levels of the first order
hierarchy [23], among others.
Separability of nonregular languages attracted little attention till now. The
reasons for this are twofold. First, for regular languages one can use standard al-
gebraic tools, like syntactic monoids, and indeed most of the results have been ob-
tained with the help of such techniques. Second, some strong intractability results
have been known already since 70’s, when Szymanski and Williams proved that
regular separability of context-free languages is undecidable [25]. Later Hunt [10]
generalized this result: he showed that F-separability of context-free languages
is undecidable for every class Fwhich is closed under finite boolean combina-
tions and contains all languages of the form wΣ∗for w∈Σ∗. This is a very
weak condition, so it seemed that nothing nontrivial can be done outside regular
languages with respect to separability problems. Furthermore, Kopczy´nski has
recently shown that regular separability is undecidable even for languages of vis-
ibly pushdown automata [12], thus strengthening the result by Szymanski and
Williams. On the positive side, piecewise testable separability has been shown
decidable for context-free languages, languages of Vector Addition Systems (VAS
languages), and some other classes of languages [3]. This inspired us to start a
quest for decidable cases beyond regular languages.
To the best of our knowledge, beside [3], separability problems for VAS lan-
guages have not been investigated before.
Our contribution. In this paper, we make a substantial step towards solving
regular separability of VAS languages. Instead of VAS languages themselves
(i.e., subsets of Σ∗), in this paper we investigate their commutative closures,
or, alternatively, subsets of Ndrepresented as reachability sets of VASes, VASes
with states, or Petri nets. A VAS reachability set is just the set of configurations
of a VAS which can be reached from a specified initial configurations. Towards a
unified treatment, instead of considering separately VASes, VASes with states,
and Petri nets, we consider sections of VAS reachability sets (abbreviated as VAS
sections below), which turn out to be expressive enough to represent sections
of VASes with states and Petri nets, and thus being a convenient subsuming
formalism. A section of a set of vectors X⊆Ndis the set obtained by first fixing
a value for certain coordinates, and then projecting the result to the remaining
coordinates. For example, if Xis the set of pairs {(x, y)∈N2|xdivides y},
then the section of Xobtained by fixing the first coordinate to 3 is the set
{0,3,6,...}. It can be easily shown that VAS sections are strictly more general
than VAS reachability sets themselves, and they are equiexpressive with sections
of VASes with states and Petri nets.
We study the separability problem of VAS sections by simpler classes, namely,
modular and unary sets. A set X⊆Ndis modular if there exists a modulus n∈N
s.t. Xis closed under the congruence modulo non every coordinate, and it is
unary if there exists a threshold n∈Ns.t. it is closed under the congruence
modulo nabove the threshold non every coordinate. Clearly, VAS sections are
more general than both unary and modular sets, and unary sets are more general
than modular sets. Moreover, unary sets are tightly connected with commutative
regular languages, in the sense that the Parikh image3of a commutative regular
language is a unary set, and vice versa, the inverse Parikh image of a unary set is
a commutative regular language. As our main result, we show that the modular
3The Parikh image of a language of words L⊆ {a1,...,ak}is the subset of Nk
obtained by counting occurrences of letters in L.
2
and unary separability problems are decidable for VAS sections (and thus for
sections of VASes with states and Petri nets). Both proofs use similar techniques,
and invoke two semi-decision procedures: the first one (positive) enumerates
witnesses of separability, and the second one (negative) enumerates witnesses
of nonseparability. A separability witness is just a modular (or unary) set, and
verifying that it is indeed a separator easily reduces to the VAS reachability
problem. Thus, the hard part of the proof is to invent a finite and decidable
witness of nonseparability, i.e., a finite ob ject whose existence proves that none
of infinitely many modular (resp. unary) sets is a separator. Our main technical
observation is that two nonseparable VAS reachability sets always admit two
linear subsets thereof that are already nonseparable.
From our result, thanks to the tight connection between unary sets and com-
mutative regular languages mentioned above, we can immediately deduce decid-
ability of regular separability for commutative closures of VAS languages, and
commutative regular separability for VAS languages. This constitutes a first step
towards determining the status of regular separability for languages of VASes.
Related research. Choffrut and Grigorieff have shown decidability of separability
of rational relations by recognizable relations in Σ∗×Nd[1]. Rational subsets
of Ndare precisely the semilinear sets, and recognizable (by morphism into a
monoid) subsets of Ndare precisely the unary sets. Thus, by ignoring the Σ∗
component, one obtains a very special case of our result, namely decidability
of the unary separability problem for semilinear sets. Moreover, since modular
sets are subsets of Ndwhich are recognizable by a morphism into a monoid
which happens to be a group, we also obtain a new result, namely, decidability
of separability of rational subsets of Ndby subsets of Ndrecognized by a group.
From a quite different angle, our research seems to be closely related to
the VAS reachability problem. Leroux [15] has shown a highly nontrivial re-
sult: the reachability sets of two VASes are disjoint if, and only if, they can be
separated by a semilinear set. In other words, semilinear separability for VAS
reachability sets is equivalent to the VAS (non-)reachability problem. This con-
nection suggests that modular and unary separability are interesting problems
in themselves, enriching our understanding of VASes. Finally, we show that VAS
reachability reduces to unary separability, thus the problem does not become
easier by considering the simpler class of unary sets as opposed to semilinear
sets. For modular separability we have a weaker complexity lower bound, i.e.
ExpSpace-hardness, by a reduction from control state reachability for VASSes.
2 Preliminaries
Vectors. By Nand Zwe denote the set of natural and integer numbers, respec-
tively. For a vector u= (u1,...,ud)∈Zdand for a coordinate i∈ {1,...,d}, we
denote by u[i] its i-th component ui. The zero vector is denoted by 0. The order
≤and the sum operation + naturally extend to vectors pointwise. Moreover,
if n∈Z, then nu is the vector (nu1,...,nud). These operations extend to sets
3
element-wise in the natural way: For two sets of vectors U, V ⊆Zdwe denote by
U+Vits Minkowski sum {u+v|u∈U, v ∈V}. For a (possibly infinite) set of
vectors S⊆Zd, let Lin(S) and Lin≥0(S) be the set of linear combinations and
non-negative linear combinations of vectors from S, respectively, i.e.,
Lin(S) = {a1v1+...+akvk|v1,...,vk∈S, a1,...,ak∈Z},and
Lin≥0(S) = {a1v1+...+akvk|v1,...,vk∈S, a1,...,ak∈N}.
When the set S={v1,...,vk}is finite, we alternatively write Lin(v1,...,vk)
instead of Lin({v1,...,vk}), and similarly for Lin≥0(v1,...,vk).
Modular, unary, linear, and semilinear sets. Two vectors x, y ∈Zdare n-modular
equivalent, written x≡ny, if, for all i∈ {1,...,d}, we have x[i]≡y[i] mod n.
Moreover, two non-negative vectors x, y ∈Ndare n-unary equivalent, written
x∼
=ny, if x≡nyand x[i]≥n⇐⇒ y[i]≥nfor all i∈ {1,...,d}. A d-
dimensional set S⊆Ndis modular if there exists a number n∈N, s.t. Sis a
union of n-modular equivalence classes. Unary sets S⊆Ndare defined similarly
w.r.t. n-unary equivalence classes.
A set S⊆Ndis linear if it is of the form S={b}+Lin≥0(p1,...,pk) for
some base b∈Ndand some periods p1,...,pk∈Nd. A set is semilinear if it
is a finite union of linear sets. Note that a modular set is also unary (since ∼
=n
is finer than ≡n), and that unary set is in turn a semilinear set, which can be
presented as a finite union of linear sets in which all the periods are parallel to
the coordinate axes, i.e., they have exactly one non-zero entry.
Separability. For S, U, V ⊆Nd, we say that Sseparates Ufrom Vif U⊆S
and V∩S=∅. The set Sis also called a separator of U, V . For a family Fof
sets, we say that Uis Fseparable from Vif Uis separated from Vby a set
S∈ F. In this paper, the set of separators Fwill be the modular sets and the
unary ones. Since both classes are closed under complement, the notion of F
separability is symmetric: Uis Fseparable from Viff Vis Fseparable from U.
Thus we use also a symmetric notation, in particular we say that Uand Vare
Fseparable instead of saying that Uis Fseparable from V. For two families
of sets Fand G, the Fseparability problem for Gasks whether two given sets
U, V ∈ G are Fseparable. In this paper we mainly consider two instances of F,
namely modular sets and unary sets, and thus we speak of modular separability
and unary separability problems, respectively.
Vector Addition Systems. Ad-dimensional Vector Addition System (VAS) is a
pair V= (s, T ), where s∈Ndis the source configuration and T⊆fin Zdis the
set of finitely many transitions. A partial run ρof a VAS V= (s, T ) is a sequence
(v0, t0, v1),(v1, t1, v2),...,(vn−1, tn−1, vn)∈Nd×T×Nd
such that for all i∈ {0,...,n−1}we have vi+ti=vi+1. The source of this partial
run is the configuration v0and the target of this partial run is the configuration
4
vn, we write source(ρ) = v0,target(ρ) = vn. The labeling of ρis the sequence
t0...tn−1∈T∗, we write label(ρ) = t0...tn−1. For a sequence α∈T∗and a
partial run ρsuch that label(ρ) = α,source(ρ) = uand target(ρ) = vwe
write uα
−→ vto denote this unique partial run. A partial run ρof (s, T ) with
source(ρ) = sis called a run. The set of all runs of a VAS Vis denoted as
Runs(V). The reachability set Reach(V) of a VAS Vis the set of targets of
all its runs; the sets Reach(V) we call VAS reachability sets in the sequel. The
family of all VAS reachability sets we denote as Reach(VAS).
Example 1. Consider a VAS V= (s, T ), for a source configuration s= (1,0,0)
and a set of transitions T={(−1,2,1),(2,−1,1)}. One easily proves that
Reach(V) = {(a, b, c)∈N2|a+b=c+ 1 ∧a−b≡1 mod 3}.
Vector Addition Systems with states. Ad-dimensional VAS with states (VASS)
is a triple V= (s, T , Q), where Qis a finite set of states,s∈Q×Ndis the source
configuration and T⊆fin Q×Zd×Qis a finite set of transitions. Similarly as
in case of VASes, a run ρof a VASS V= (s, T, Q) is a sequence
(q0, v0, s0, q1, v1),...,(qn−1, vn−1, sn−1, qn, vn)∈Q×Nd×Zd×Q×Nd
such that (q0, v0) = sand for all i∈ {0,...,n−1}we have (qi, si, qi+1)∈Tand
vi+si=vi+1. We write target(ρ) = (qn, vn). The reachability set of a VASS
Vin state qis
Reachq(V) = {v∈Nd|(q, v) = target(ρ) for some run ρ}.
The family of all such reachability sets of all VASSes we denote as Reach(VASS).
Example 2 (cf. [8]). Let Vbe a 3-dimensional VASS with two states, pand p′,
the source configuration (p, (1,0,0)), and four transitions:
(p, (−1,1,0), p),(p, (0,0,0), p′),(p′,(2,−1,0), p′),(p′,(0,0,1), p).
Then Reachp(V) = {(a, b, c)∈N3|1≤a+b≤2c}.
3 Sections
VAS reachability sets are central for this paper. However, in order to make this
family of sets more robust, we prefer to consider the slightly larger family of
sections of VAS reachability sets. The intuition about a section is that we fix
values on a subset of coordinates in vectors, and collect all the values that can
occur on the other coordinates. For a vector u∈Ndand a subset I⊆ {1,...,d}
of coordinates, by πI(u)∈N|I|we denote the I-projection of u, i.e., the vector
obtained from uby removing coordinates not belonging to I. The projection
extends element-wise to sets of vectors S⊆Nd, denoted πI(S). For a set of
5
vectors S⊆Nd, a subset I⊆ {1,...,d}, and a vector u∈Nd−|I|, the section of
Sw.r.t. Iand uis the set
secI,u (S) := πI({v∈S|π{1,...,d}\I(v) = u})⊆N|I|.
We denote by SecReach(VAS) the family of all sections of VAS reachability
sets, which we abbreviate as VAS sections below. Similarly, the family of all
sections of VASS-reachability sets we denote by SecReach(VASS).
Example 3. Consider the VAS Vfrom Example 1. For I={1,2}and u= 7 ∈N1
we have
secI,u (Reach(V)) = {(0,8),(3,5),(6,2)}.
Note that in a special case of I={1,...,d}, when uis necessarily the empty
vector, secI,u (S) = S. Thus Reach(VAS) is a subfamily of SecReach(VAS),
and likewise for VASSes. We argue that VAS sections are a more robust class
than VAS reachability sets. Indeed, as shown below VAS sections are closed
under positive boolean combinations, which is not the case for VAS reachability
sets.
Reachability sets of VASes are a strict subfamily of reachability sets of VASes
with states, which in turn are a strict subfamily of sections of reachability sets
of VASes. However, when sections of reachability set are compared, there is
no difference between VASes and VASes with states, which motivates consider-
ing sections in this paper. These observations are summarized in the following
propositions:
Proposition 4. Reach(VAS) (Reach(VASS) (SecReach(VAS).
Proof. In order to prove strictness of the first inclusion, consider the VASS V
from Example 2. The reachability set Reachp(V) is not semilinear; on the other
hand the reachability sets of of 3-dimensional VASes are always semilinear [8].
Now we turn to the second inclusion. It is folklore that for a d-dimensional
VASS Vwith nstates and mtransitions one can construct a (d+n+m)-
dimensional VAS V′simulating V. Among the new coordinates, ncorrespond
to states and mto transitions. For a transition t= (q, v, q′) of Vthere are two
transitions in V′: the first one subtracts 1 on the coordinate corresponding to
state qand adds 1 on the coordinate corresponding to t; the second one subtracts
1 on the coordinate corresponding to t, adds 1 on the coordinate corresponding
to q′, and adds von the original dcoordinates. Finally, if (q0, v0) is the initial
configuration of V, then the initial configuration of V′is a copy of v0on the
original ddimensions, equals 1 on the coordinate corresponding to q0, and equals
0 on the rest of the new coordinates. Then the reachability set Reachq(V) equals
the section of Reach(V′) obtained by fixing the coordinate corresponding to q
to 1 and all other new coordinates to 0.
For strictness of the second inclusion, apply the above-mentioned transfor-
mation to the VASS Vfrom Example 2, in order to obtain a 9-dimensional VAS
V′. The section of Reach(V′) that fixes the second original coordinate to 0, the
coordinate corresponding to state pto 1, and all the other new coordinates to
6
0 is S:= {(a, b)∈N2|0≤a≤2b}. This 2-dimensional set is not semilinear,
while reachability sets of 2-dimensional VASSes are always semilinear [8]. Thus
Sis not a 2-dimensional VAS reachability set.
Proposition 5. SecReach(VAS) = SecReach(VASS).
Proof. One inclusion is obvious, since VASSes are more general than VASes,
and the same holds when taking sections. For the other directions, consider a
VASS Vand a section thereof S:= secI ,v (Reachq(V)). Reconsider the folk-
lore construction of a VAS V′that simulates V(cf. the proof of the previous
Proposition 5). The section of the reachability set of Reach(V′) that fixes the
coordinate corresponding to qto 1, all the other new coordinates to 0, and all
the original coordinates not belonging to the set Ias in vector v, equals S.⊓⊔
Remark 6. In the similar vein one shows that reachability sets of Petri nets
include Reach(VAS) and are included in Reach(VASS). Therefore, as long as
sections are considered, there is no difference between VASes, Petri nets, and
VASSes. In consequence, our results apply not only to VASes, but to all the
three models.
We conclude this section by proving a closure property of VAS sections.
Proposition 7. The family of VAS sections is closed under positive boolean
combinations.
Proof. We only sketch the proof. For closure under union, we just use nondeter-
minism to guess which VAS to run. Dealing with sections is straightforward since
1) we can assume w.l.o.g. that sections are done w.r.t. the 0 vector, 2) by padding
coordinates we can assume that the two input VASes have the same dimension,
and 3) by reordering coordinates we can guarantee that the coordinates that are
projected away appear all together on the right (the same simplifying assump-
tions will be made in Sections 6 and 7; cf. the details just before Lemma 27).
For closure under intersection, we proceed under similar assumptions, and the
intuition is to run the first VAS forward in two identical copies, and then to run
backward the second VAS only in the second copy, using a section to make sure
that the second VAS is accepting, and then project away the second copy. ⊓⊔
4 Results
As our main technical contribution, we prove decidability of the modular and
unary separability problems for the class of sections of VAS reachability sets.
Theorem 8. The modular separability problem for VAS sections is decidable.
Theorem 9. The unary separability problem for VAS sections is decidable.
7
The proofs are postponed to Sections 5–7. Furthermore, as a corollary of Theo-
rem 9 we derive decidability of two commutative variants of the regular separa-
bility of VAS languages (formulated in Theorems 10 and 11 below).
To consider languages instead of reachability sets, we need to assume that
transitions of a VAS are labeled by elements of an alphabet Σ, and thus every
run is labeled by a word over Σobtained by concatenating labels of consecutive
transitions of a run. We allow for silent transitions labeled by ε, i.e., transitions
that do not contribute to the labeling of a run. The language L(V) of a VAS V
contains labels of those runs of Vthat end in an accepting configuration. Our
results work for several variants of acceptance; for instance, for a given fixed
configuration v0,
–we may consider a configuration vaccepting if v≥v0(this choice yields so
called coverability languages); or
–we may consider a configuration vaccepting if v=v0(this choice yields
reachability languages).
The Parikh image of a word w∈Σ∗, for a fixed total ordering a1< . . . < ad
of Σ, is a vector in Ndwhose ith coordinate stores the number of occurrences
of aiin w. We lift the operation element-wise to languages, thus the Parikh
image of a language L, denoted pi(L), is a subset of Nd. Two words w, v over Σ
are commutative equivalent if their Parikh images are equal. The commutative
closure of a language L⊆Σ∗, denoted cc(L), is the language containing all
words w∈Σ∗commutative equivalent to some word v∈L. A language Lis
commutative if it is invariant under commutative equivalence, i.e., L=cc(L).
Unary sets of vectors are exactly the Parikh images of commutative regular
languages; reciprocally, commutative regular languages are exactly the inverse
Parikh images of unary sets. Note that a commutative language is uniquely
determined by its Parikh image.
As a corollary of Theorem 9 we deduce decidability of the following two
commutative variants of the regular separability of VAS languages:
–commutative regular separability of VAS languages: given two VASes V, V ′,
decide whether there is a commutative regular language Rthat includes
L(V) and is disjoint from L(V′);
–regular separability for commutative closures of VAS languages: given two
VASes V, V ′, decide whether there is a regular language Rthat includes
cc(L(V)) and is disjoint from cc(L(V′)).
Theorem 10. Commutative regular separability is decidable for VAS languages.
Indeed, given two VASes V, W one easy constructs another two VASes V′, W ′
s.t. pi(L(V)) is a section of Reach(V′), and similarly for W′. By the tight cor-
respondence between commutative regular languages and unary sets, we observe
that L(V) and L(W) are separated by a commutative regular language if, and
only if, their Parikh images pi(L(V)) and pi(L(W)) are separated by a unary
set, which is is decidable by Theorem 9.
8
Theorem 11. Regular separability is decidable for commutative closures of VAS
languages.
Similarly as above, we reduce to unary separability of VAS reachability sets
(which is decidable once again by Theorem 9), which is immediate once one
proves the following crucial observation.
Lemma 12. Two commutative languages L, K ⊆Σ∗are regular separable if,
and only if, their Parikh images are unary separable.
Proof. We start with the “if ” direction. Let pi(K) and pi(L) be separable by
some unary set U⊆Nd. Let S={w∈Σ∗|pi(w)∈U}. It is easy to see that S
is (commutative) regular since Uis unary, and that Sseparates Kand L.
Now we turn to the “only if” direction. Let Kand Lbe separable by a regular
language S, say K⊆Sand S∩L=∅. Let Mbe the syntactic monoid of Sand
ωbe its idempotent power, i.e., a number such that for every m∈Mit holds
mω=m2ω. In particular, for every word u∈Σ∗we have
uvωw∈L⇐⇒ uv2ωw∈S; (1)
in other words, one can substitute vωby v2ωand vice versa in every context. Let
Σ={a1,...,ad}. For u= (u1,...,ud)∈Nddefine a word wu=au1
1···aud
d. For
every u, v ∈Ndsuch that u∼
=ωv, by repetitive application of (1) we get wu∈S
iff wv∈S. As Kis commutative and K⊆S, we have wu∈Sfor all u∈pi(K);
similarly, we have wv6∈ Sfor all v∈pi(L). Therefore for all u∈pi(K), v∈pi(L)
we have u6∼
=ωv. Let U={x∈Nd| ∃y∈pi(K)x∼
=ωy}. The set Useparates pi(K)
and pi(L) and, being a union of ∼
=ωequivalence classes, it is unary. ⊓⊔
5 Modular and unary separability of linear sets
The rest of the paper is devoted to the proofs of Theorems 8 and 9. In this section
we prove that modular separability of linear sets is decidable4, and provide a
condition on linear sets that makes modular separability equivalent to unary
separability. The two results, stated in Lemmas 16 and 19 below, respectively,
are used in Sections 6 and 7, where the proofs of Theorems 8 and 9 are completed.
Linear combinations modulo n.We start with some preliminary results from lin-
ear algebra. For n∈N, let Lin≥0
n(v1,...,vk) be the closure of Lin≥0(v1,...,vk)
modulo n, i.e.,
Lin≥0
n(v1,...,vk) = {v∈Nd| ∃u∈Lin≥0(v1,...,vk)v≡nu}.
Similarly one defines Linn(v1,...,vk) be the closure of Lin(v1,...,vk) modulo
n. Observe however that Linn(v1,...,vk) = Lin≥0
n(v1,...,vk). Indeed, if v≡n
l1v1+...+lkvkfor l1,...,lk∈Zthen v≡n(l1+Nn)v1+...+ (lk+N n)vkfor
any N∈N.
4While decidability follows from [1] and is thus not a new result, we provide here
another simple proof to make the paper self-contained.
9
Lemma 13. Lin(v1,...,vk) = Tn>0Lin≥0
n(v1,...,vk).
Proof. The left-to-right inclusion is immediate: for any n∈Nwe have
Lin(v1,...,vk)⊆Linn(v1,...,vk) = Lin≥0
n(v1,...,vk).
For the right-to-left inclusion we take an algebraic perspective, and treat
S:= Lin(v1, . . . , vk) as a subgroup of Zdgenerated by F={v1,...,vk}. Let I
be the set of all dunit vectors in Zd. For every n∈N≥0, let nZddenote the
subgroup of Zdgenerated by nI, and let Snbe the subgroup of Zdgenerated by
F∪(nI). In algebraic terms, our obligation is to show that
\
n∈N≥0
Sn⊆S. (2)
Let G:= Zd/S be the quotient group and consider the quotient group ho-
momorphism h:Zd→G. It is legal, as every subgroup of an abelian group
is normal, thus we can consider a quotient with respect to it. We have thus
ker(h) = {x∈Zd|h(x) = 0G}=S, where 0Gis the zero element of G. Now (2)
is equivalent to
h\
n∈N≥0
Sn={0G},
which will immediately follow, once we manage to show
\
n∈N≥0
h(Sn) = {0G}.
Observe that h(Sn) = h(nZd), for every n∈N≥0, and hence we may equally
well demonstrate:
\
n∈N≥0
h(nZd) = {0G}.(3)
The group G, being a finitely generated abelian group, is isomorphic to the direct
product of a finite group G1(let lbe its order, i.e., the number of its elements)
and G2=Zk, for some k∈N(see for instance Theorem 2.2, p. 76, in [9]). For
showing (3), consider an element g∈Gwhich belongs to h(nZd) for all n∈N≥0,
and its two projections g1and g2in G1and G2, respectively. As g∈h(lZd), then
necessarily g1=l·g′for some g′∈G1, and since the order of every element
divides the order of the group l, we have g1= 0G1. Similarly, we deduce that
g2= 0G2; indeed, this is implied by the fact that for every n∈N≥0,g2=ng′
for some g′∈G2. Thus g= 0Gas required. ⊓⊔
Modular separability. In the rest of the paper, we heavily rely on the following
straightforward characterization of modular separability:
10
Proposition 14. Two sets U, V ⊆Ndare modular separable if, and only if,
there exists n∈Nsuch that for all u∈U,v∈Vwe have u6≡nv.
Proof. If U, V are separable by some n-modular set, then for all u∈U, v ∈V
we have u6≡nv. On the other hand, if for all u∈U, v ∈Vwe have u6≡nv, then
the modular set S={s∈Nd| ∃u∈Us≡nu}separates Uand V.⊓⊔
Lemma 15. Two linear sets {b}+Lin≥0(P)and {c}+Lin≥0(Q)are not mod-
ular separable if, and only if, b−c∈Lin(P∪Q).
Proof. Let L={b}+Lin≥0(P) and M={c}+Lin≥0(Q), with P={p1,...,pm}
and Q={q1, . . . , qn}. First we show the “if” direction. By Proposition 14, it is
enough to show that, for every n∈N, there exist two vectors u∈Land v∈M
s.t. u≡nv. Fix an n∈N. By assumption, we have b−c∈Lin(P∪Q), and thus
c−b∈Lin(P∪Q) = Lin(P∪ −Q). By Lemma 13, c−b∈Lin≥0
n(P∪ −Q), i.e.,
there exist δ∈Lin≥0(P) and γ∈Lin≥0(Q) such that c−b≡nδ−γ. Thus, if
we take u=b+δand v=c+γwe clearly have u−v= (b−c) + (δ−γ)≡n
(b−c) + (c−b) = 0, and thus u≡nv.
For the “only if” direction, assume that Land Mas above are not modular
separable. By Proposition 14, for every n≥0 there exist vectors un∈Land
vn∈Ms.t. un≡nvn. By definition, un=b+δnand vn=c+γn, for some
δn∈Lin≥0(P) and γn∈Lin≥0(Q). Since un≡nvn, we have b−c≡nγn−δn∈
Lin(P∪Q), and thus b−c∈Lin≥0
n(P∪Q). Since nwas arbitrary, by Lemma 13
we have b−c∈Lin(P∪Q), as required. ⊓⊔
Since the condition in the lemma above is effectively testable being an instance
of solvability of systems of linear Diophantine equations, we get the following
corollary:
Corollary 16. Modular separability of linear sets is decidable.
Remark 17. Since linear Diophantine equations are solvable in polynomial time,
we obtain the same complexity for modular separability of linear sets. This
observation however will not be useful in the sequel.
Unary separability. We start with a characterization of unary separability, which
is the same as Proposition 14, with unary equivalence ∼
=nin place of modular
equivalence ≡n. (Recall that unary equivalence is modular equivalence “above a
threshold”, i.e., u∼
=nvholds for two vectors u, v ∈Ndif, for every component
1≤i≤d, either u[i] = v[i]≤n, or u[i], v[i]≥nand u[i]≡nv[i].)
Proposition 18. Two sets U, V ⊆Ndare unary separable if, and only if, there
exists n∈Nsuch that, for all u∈Uand v∈V, we have u6∼
=nv.
We say that a set of vectors U⊆Ndis diagonal if, for every threshold x∈N,
there exists a vector u∈Uwhich is strictly larger than xin every component. Let
I⊆ {1,...,d}be a set of coordinates. Two set of vectors U, V ⊆Ndare I-linked
if there exists a sectioning vector u∈Nd−|I|s.t. π{1,...,d}\I(U) = π{1,...,d}\I(V) =
{u}and πI(U), πI(V) are diagonal. The sets U, V are linked if they are I-linked
for some I⊆ {1,...,d}.
11
Lemma 19. Let U, V ⊆Ndbe two linked linear sets. Then, Uand Vare unary
separable if, and only if, they are modular separable.
Proof. Let Uand Vbe two linked linear sets. One direction is obvious since
modular separability implies unary separability. For the other direction, let U
and Vbe modular nonseparable, and we show that they are unary nonsep-
arable either. By Lemma 14, there exists a sequence of vectors un∈Uand
vn∈Vs.t. un≡nvn. We construct a new sequence u′
n∈Uand v′
n∈V
s.t. u′
n∼
=nv′
n, which will then show that Uand Vare not unary separa-
ble by Lemma 18. Since Uand Vare linked, there exist a set of coordinates
I⊆ {1,...,d}and a sectioning vector for the remaining coordinates u∈Nd−|I|
s.t. 1) π{1,...,d}\I(U) = π{1,...,d}\I(V) = {u}and 2) πI(U), πI(V) are diagonal. In
particular, by 1) the two sequences unand vnproject to uon the complement
of I, i.e., π{1,...,d}\I(un) = π{1,...,d}\I(vn) = {u}. Moreover, for any n∈N, since
πI(un)∈πI(U), and the latter set is diagonal by 2), there exists an increment
δn∈N|I|s.t. πI(un)≤πI(un) + δn∈πI(U). Moreover, since Uis a linear set,
δncan be chosen to have its components multiple of n. Let u′
nbe πI(un) + δnon
coordinates I, and uon the remaining ones. By the choice of δn,u′
n≡nun, and,
moreover, u′
nis larger than non coordinates I. The vector v′
ncan be constructed
similarly from vn. We thus have u′
n∼
=nv′
n, since on coordinates Iboth u′
nand
v′
nare above n, and on the remaining coordinates they are equal to u.⊓⊔
Remark 20. The unary separability problem is decidable for linear sets, as shown
in [1], but we will not need this fact in the sequel. Moreover, it will follow from
our stronger decidability result about the more general VAS reachability sets
stated in Theorem 9 (since linear sets are included in VAS reachability sets).
6 Modular separability of VAS sections
In this section we prove Theorem 8, and thus provide an algorithm to decide
modular separability for VAS reachability sets. Given two VAS sections Uand V,
the algorithm runs in parallel two semi-decision procedures: one (positive) which
looks for a witness of separability, and another one (negative) which looks for a
witness of nonseparability. Directly from the characterization of Proposition 14,
the positive semi-decision procedure simply enumerates all candidate moduli
n∈Nand checks whether u6≡nvfor all u∈Uand v∈V. The latter condition
can be decided by reduction to the VAS (non)reachability problem [20,17].
Lemma 21. For two VAS sections Uand Vand a modulus n∈N, it is decidable
whether there exist u∈Uand v∈Vs.t. u≡nv.
Proof. Recall that Uis obtained from the reachability set of a VAS by fixing
values ¯uon some coordinates, and projecting to the remaining coordinates; and
likewise Vis obtained, by fixing values ¯von some coordinates. We modify the
two VASes by allowing each non-fixed coordinate to be decremented by n, and
we check whether the two thus modified VASes admit a pair of reachable vectors
u, v that agree on fixed coordinates with ¯uand ¯v, respectively, and on all the
non-fixed coordinates are equal and smaller than n.⊓⊔
12
It remains to design the negative semi-decision procedure, which is the non-
trivial part. In Lemma 27, we show that if two VAS reachability sets are not
modular separable, then in fact they already contain two linear subsets which
are not modular separable. In order to construct such linear witnesses of non-
separablity, we use the theory of well quasi orders and some elementary results
in algebra, which we present next.
The order on runs. A quasi order (X, 4) is a well quasi order (wqo) if for every
infinite sequence x0, x1,...∈Xthere exist indices i, j ∈N, i < j, such that xi4
xj. It is folklore that if (X, 4) is a wqo, then in every infinite sequence x0, x1,... ∈
Xthere even exists an infinite monotonically non-decreasing subsequence xi14
xi24.... We will use Dickson’s and Higman’s Lemmas to define new wqo’s on
pairs and sequences. For two quasi orders (X, ≤X) and (Y, ≤Y), let the product
(X×Y, ≤X×Y) be ordered componentwise by (x, y)≤X×Y(x′, y′) if x≤Xx′
and y≤Yy′. By Dickson’s Lemma [4], if both (X , ≤X) and (Y, ≤Y) are wqos,
then (X×Y, ≤X×Y) is a wqo too. As a corollary of Dickson’s Lemma, if two
quasi orders (X, ≤1) and (X, ≤2) on the same domain are wqos, then the quasi
order defined as the conjunction of ≤1and ≤2is a wqo too. For a quasi order
(X, ≤), let (X∗,≤∗) be quasi ordered by the subsequence order ≤∗, defined as
x1x2···xk≤∗y1y2...ymif there exist 1 ≤i1< . . . < ik≤msuch that xj≤yij
for all j∈ {1, . . . , k}. By Higman’s Lemma [7], if (X, ≤) is a wqo then (X∗,≤∗)
is a wqo too.
By considering the finite set of transitions Twell quasi ordered by equality,
we define the order ≤1on triples Nd×T×Ndcomponentwise as (u, s, u′)≤1
(v, t, v′) if u≤v,s=t, and u′≤v′, which is a wqo by Dickson’s Lemma. We
further extend ≤1to an order Eon runs by defining, for two runs ρand σin
(Nd×T×Nd)∗,ρEσif ρ≤1
∗σand target(ρ)≤target(σ).5Here, ≤1
∗is
the extension of ≤1to sequences, and thus a wqo by Higman’s Lemma, which
implies that Eis itself a wqo by the corollary of Dickson’s Lemma.
Proposition 22. Eis a well quasi order.
Lemma 23. Let ρ,ρ1, and ρ2be runs of a VAS s.t. ρEρ1, ρ2. There exists a
run ρ′s.t. ρEρ′and target(ρ′)−target(ρ) = (target(ρ1)−target(ρ)) +
(target(ρ2)−target(ρ)).
Proof. The proof is almost identical to the proof of Proposition 5.1. in [18]. Let
the VAS be (s, T ), and let ρ=v0
t0
−→ v1
t1
−→ · · · tn−1
−→ vn, where v0=s. Then ρi,
for i∈ {1,2}is of the form
ρi=v0
ρi
0
−→ v0+δi
0
t0
−→ v1+δi
0
ρi
1
−→ v1+δi
1
t1
−→ v1+δi
2
ρi
2
−→ · · ·
ρi
n−1
−→ vn−1+δi
n−1
tn−1
−→ vn+δi
n−1
ρi
n
−→ vn+δi
n,
5A weaker version of this order not considering target configurations was defined
in [11].
13
where for all i∈ {1,2}and j∈ {0,...,n}we have δi
j≥0. Thus by letting ρ′:=
ρ1
0ρ2
0t0ρ1
1ρ2
1t1ρ1
2ρ2
2···ρ1
n−1ρ2
n−1tn−1ρ1
nρ2
nwe clearly have a run v0
ρ′
−→ vn+δ1
n+δ2
n
which indeed looks like
v0
ρ1
0
−→ v0+δ1
0
ρ2
0
−→ v0+δ1
0+δ2
0
t0
−→ v1+δ1
0+δ2
0
ρ1
1
−→ v1+δ1
1+δ2
0
ρ2
1
−→ v1+δ1
1+δ2
1
t1
−→ v2+δ1
1+δ2
1
ρ1
2
−→ · · · tn−1
−→ vn+δ1
n−1+δ2
n−1
ρ1
n
−→ vn+δ1
n+δ2
n−1
ρ2
n
−→ vn+δ1
n+δ2
n.
This finishes the proof of Lemma 23. ⊓⊔
We formulate an immediate but useful corollary:
Corollary 24. Let ρ0, ρ1,...,ρkbe runs of a VAS s.t., for all i∈ {1,...,k},
ρ0Eρi, and let δi:= target(ρi)−target(ρ0)≥0. For any δ∈Lin≥0(δ1,...,δk),
there exists a run ρs.t. ρ0Eρand δ=target(ρ)−target(ρ0).
We conclude this part by showing that any (possibly infinite) subset of Zd
can be overapproximated by taking linear combinations of a finite subset thereof.
This will be important below in order to construct linear sets as witnesses of
nonseparability.
Lemma 25. For every (possibly infinite) set of vectors S⊆Zd, there exist
finitely many vectors v1,...,vk∈Ss.t. S⊆Lin(v1,...,vk).
Proof. Treat Zdas a freely finitely generated abelian group, and consider the
subgroup Lin(S) of Zdgenerated by S, i.e., the subgroup containing all linear
combinations of finitely many elements of S. We use the following result in
algebra: every subgroup of a finitely generated abelian group is finitely generated
(see for instance Corollary 1.7, p. 74, in [9]). By this result applied to Lin(S) we
get a finite set of generators F⊆Lin(S) s.t. Lin(F) = Lin(S). Every element
of Fis a linear combination of finitely many elements of S. Thus let v1,...,vk
be all the elements of Sappearing as a linear combination of some element
from F. Then clearly F⊆Lin(v1,...,vk), and thus S⊆Lin(S) = Lin(F)⊆
Lin(Lin(v1,...,vk)) = Lin(v1,...,vk), as required. ⊓⊔
Remark 26. In fact one can show that the generating set Fhas at most dele-
ments. However, no upper bound on kfollows, and even for d= 1 the number of
vectors kcan be arbitrarily large. Indeed, let p1,...,pkbe different prime num-
bers, let ui= (p1·...·pk)/piand S={u1,...,uk}. Then for every i∈ {1,...,k},
the number uiis not a linear combination of numbers uj,j6=i, as uiis not di-
visible by pi, while all the others are. Therefore we need all the elements of Sin
the set {v1,...,vk}.
14
Modular nonseparability witness. We now concentrate on the negative semi-
decision procedure. Let U, V ⊆Ndbe two VAS sections:
U=secI, ¯u(RU)⊆Ndand V=secJ,¯v(RV)⊆Nd,
where RU⊆NdUand RV⊆NdVare the reachability sets of the two VASes
WUand WV, and I⊆ {1,...,dU}and J⊆ {1,...,dV}with |I|=|J|=dare
projecting coordinates, and ¯u∈NdU−d,¯v∈NdV−dare two sectioning vectors.
Observe that by padding coordinates we can assume w.l.o.g. that the two
input VASes have the same dimension d′=dU=dV. Furthermore, we can also
assume w.l.o.g. that ¯u= ¯v= 0. Indeed, one can add an additional coordinate,
such that for performing any transition it is necessary that this coordinate is
nonzero and a special, final transition, which causes the additional coordinate
to be equal zero and subtracts ¯u(or ¯v) from the other coordinates. The result of
adding this gadget is that now we can assume ¯u= ¯v= 0, but the section itself
does not change.
Finally, by reordering coordinates we can guarantee that the coordinates that
are projected away appear on the same positions in both VASes, i.e., I=J. With
these assumptions, we observe that modular separability of sets U, V ⊆Ndis
equivalent to modular separability of sets U′, V ′⊆Nd′, defined as U, V but
without projecting onto the subset Iof coordinates:
U′={v∈RU|π{1,...,d′}\I(v) = 0}V′={v∈RV|π{1,...,d′}\I(v) = 0}.
We call the set U′(resp. V′) the expansion of U(resp. V).
We say that a linear set L={b}+Lin≥0(p1,...,pk)⊆Nd′is a U-witness if
WUadmits runs ρ, ρ1,...,ρksuch that
b=target(ρ)∈U′
b+pi=target(ρi)∈U′for i∈ {1,...,k}
ρEρifor i∈ {1,...,k}.
(4)
Analogously one defines V-witnesses, but with respect to WV.
Lemma 27. For two VAS sections U, V ⊆Nd, the following conditions are
equivalent:
1. U, V are not modular separable;
2. the expansions U′, V ′of U, V are not modular separable;
3. there exist linear subsets L⊆U′,M⊆V′that are not modular separable;
4. there exist a U-witness Land a V-witness Mthat are not modular separable.
Proof. Equivalence of points 1 and 2 follows by the definition of expansion.
Point 4 implies 3, as a U-witness is necessarily a subset of the expansion U′
by Corollary 24. Point 3 implies 2, since if two sets are separable, also subsets
thereof are separable (moreover, the separator remains the same). It remains to
show that 2 implies 4.
15
Let U′, V ′⊆Nd′be the expansions of two VAS sections U, V ⊆Nd, as above,
and assume that they are not modular separable. We construct two linear sets
L, M ⊆Nd′constituting a U-witness and a V-witness, respectively. By Propo-
sition 14, there exists an infinite sequence of pairs of reachable configurations
(u0, v0),(u1, v1),... ∈U′×V′s.t. un≡nvnfor all n∈N. By taking an ap-
propriate infinite subsequence we can ensure that even un≡n!vnfor all n∈N.
Let us fix for every n∈Nruns ρnand σnsuch that un=target(ρn) and
vn=target(σn). Since Eis a wqo by Proposition 22, we can extract a mono-
tone non-decreasing subsequence, and thus we can ensure that even ρ0Eρ1E···
and σ0Eσ1E· · · . Here we use the fact that un≡n!vnin the original sequence,
and thus un≡ivnfor every i∈ {1,...,n}, consequently the new subsequence
still has un≡nvnfor all n∈N. For all n∈N, let δn:= un−u0and γn:= vn−v0,
and consider the set of corresponding differences Sinf := {δn−γn|n∈N}. By
Lemma 25, there exists a finite subset thereof S:= {δi1−γi1,...,δik−γik}such
that Sinf ⊆Lin(S), and thus there exist two finite subsets P:= {δi1,...,δik}
and Q:= {γi1, . . . , γik}such that
Sinf ⊆Lin(P−Q)⊆Lin(P)−Lin(Q)⊆Lin≥0
n(P)−Lin≥0
n(Q),(5)
where the last inclusion follows from Lemma 13. Let the two linear sets Land
Mbe defined as
L:= {u0}+Lin≥0(P),and
M:= {v0}+Lin≥0(Q).
By the construction, Lis a U-witness and MaV-witness. It thus only remains
to show that Land Mare not modular separable. For any n, by Eq. 5 we have
δn−γn≡nδ′
n−γ′
nfor some δ′
n∈Lin≥0(P) and γ′
n∈Lin≥0(Q). Consider now
the two new infinite sequences u′
1, u′
2,· · · ∈ Land v′
1, v′
2,· · · ∈ Mdefined, for
every n≥1, as u′
n:= u0+δ′
nand v′
n:= v0+γ′
n. Then,
u′
n−v′
n= (u0+δ′
n)−(v0+γ′
n)
= (u0−v0) + (δ′
n−γ′
n) (by def. of δ′
n, γ′
n)
≡n(u0−v0) + (δn−γn)
= (u0+δn)−(v0+γn)
=un−vn≡n0 (by def. of un, vn),
and thus u′
n≡nv′
n. This, thanks to the characterization of Proposition 14,
implies that Land Mare not modular separable. ⊓⊔
Remark 28. Note that a modular nonseparability witness exists even in the case
when the two reachability sets U, V have nonempty intersection. In this case,
it is enough to consider two runs ρ0and σ0ending up in the same configura-
tion target(ρ0) = target(σ0), and considering the linear sets L:= M:=
{target(ρ0)}.
16
Using the characterization of Lemma 27, the negative semi-decision proce-
dure enumerates all pairs L, M , where Lis a U-witness and Mis a V-witness
and checks whether Land Mare modular separable, which is decidable due to
Lemma 15. Note that enumerating U-witnesses (and V-witnesses) amounts of
enumerating finite sets of runs {ρ, ρ1,...,ρk}satisfying (4).
Remark 29. It is also possible to design another negative semi-decision procedure
using Lemma 27. This one enumerates all linear sets Land M(not necessarily
only those in the special form of U- or V- witnesses) and checks whether they are
modular separable and included in Uand V, respectively. While this procedure
is conceptually simpler than the one we presented, we now need the two extra
inclusion checks L⊆Uand M⊆V. Indeed, U- and V-witnesses were designed in
such a way that the two inclusions above hold by construction and do not have
to be checked. The problem whether a given linear set is included in a given
VAS reachability is decidable [14], however we chose to present the previous
semi-decision procedure in order to be self contained.
7 Unary separability of VAS sections
We now embark on the proof of Theorem 9. It goes along the lines of the proof of
Theorem 8, but with some details more complicated, thus we only concentrate
on explaining the necessary adjustments. As before, the positive semi-decision
procedure enumerates all n∈Nand checks whether the ∼
=n-closures of the two
reachability sets are disjoint, which is effective thanks to the following fact:
Lemma 30. For two VAS sections Uand Vand n∈N, it is decidable whether
there exist u∈Uand v∈Vsuch that u∼
=nv.
This can be proved in a way similar to Lemma 21, with the adjustment that we
allow on every coordinate a decrement by nonly if the value is above 2n.
The negative semi-decision procedure enumerates nonseparability witnesses,
along the same lines as in the case of modular separability. The following crucial
lemma is an exact copy of Lemma 27, except that “modular” is replaced by
“unary”:
Lemma 31. For two VAS sections U, V ⊆Nd, the following conditions are
equivalent:
1. U, V are not unary separable;
2. the expansions U′, V ′of U, V are not unary separable;
3. there exist linear subsets L⊆U′,M⊆V′that are not unary separable;
4. there exist a U-witness Land a V-witness Mthat are not unary separable.
Proof. We only concentrate on showing that 2 implies 4. Assume that the expan-
sions U′and V′are not unary separable, for two sections Uand Vrepresented
as (recall the simplifying assumptions about VAS sections from Section 6)
U=secI,0(RU)⊆Ndand V=secI ,0(RV)⊆Nd,
17
where RU, RV⊆Nd′are the reachability sets of two VASes and I⊆ {1,...,d′}
with |I|=dare projecting coordinates. Since U′and V′are not unary separable,
by Proposition 18, there exists an infinite sequence of pairs of reachable config-
urations (u0, v0),(u1, v1),... ∈U′×V′s.t. un∼
=nvnfor all n∈N. It means
that for every n∈Nthere exist runs ρnand σnin the two VASes ending up in
reachable configurations un:= target(ρn)∈RUand vn:= target(σn)∈RV.
Define δn:= un−u0and γn:= vn−v0for all n∈N. Since Eis a wqo, by
reasoning as in the proof of Lemma 27, we can assume w.l.o.g. that ρ0Eρ1E···,
and similarly for the σi’s.
Since un∼
=nvn, the two sequences u0≤u1≤ · · · and v0≤v1≤ · · · are
unbounded on the same set of coordinates. Let F⊆ {1,...,d′}be this set; note
that F⊆I. By eliminating a sufficiently long prefix of these two sequences,
we can further assume that bounded coordinates are in fact constant, and again
from un∼
=nvnit follows that this constant is the same vector for both sequences.
Consequently,
π{1,...,d′}\F(u0) = π{1,...,d′}\F(v0),and (6)
∀n∈Nπ{1,...,d′}\F(δn) = π{1,...,d′}\F(γn) = 0.(7)
By proceding as in the proof of Lemma 27, there exist two finite sets P:=
{δi1,...,δik}and Q:= {γi1,...,δik}such that the linear sets L:= {u0}+
Lin≥0(P)⊆Uis a U-witness, the linear set M:= {v0}+Lin≥0(Q)⊆Vis a
V-witness, and L, M are not modular separable. It remains to show that Land
Mare not unary separable either. While unary nonseparability is a stronger
property than modular nonseparability in general, by Lemma 19 the two condi-
tions are in fact equivalent when the two sets are linked. We make use of the set
Fas chosen before, and we show that Land Mare F-linked. Indeed, if j∈F
then w.l.o.g. we may assume that the two sequences πj(u0)< πj(u1)< . . .
and πj(v0)< πj(v1)< . . . are strictly increasing. Thus, πj(δn), πj(γn)> n
for every n∈N, which implies that πF(L) and πF(M) are diagonal. On the
other hand, if j∈ {1,...,d′} \ F, from properties (6) and (7) above, we have
π{1,...,d}\F(L) = π{1,...,d}\F(M) = {π{1,...,d}\F(u0)}. Thus Land Mare indeed
F-linked. ⊓⊔
8 Final remarks
We have shown decidability of modular and unary separability for sections of
VAS reachability sets, which include (sections of) reachability sets of VASes with
states and Petri nets. As a corollary, we have derived decidability of regular
separability of commutative closures of VAS languages, and of commutative
regular separability of VAS languages. The decidability status of the regular
separability problem for VAS languages remains an intriguing open problem.
Complexity. Most of the problems shown decidable in this paper are easily shown
to be at least as hard as the VAS reachability problem. In particular, this applies
18
to unary separability of VAS reachability sets, and to regular separability of
commutative closures of VAS languages. Indeed, for unary separability, it suffices
to notice that a configuration ucannot reach a configuration vif, and only if,
the set reachable from ucan be unary separated from the singleton set {v}, also
a VAS reachability set. When the separator exists, it can be taken to be the
complement of {v}itself, which is unary.
While the problem of modular separability is ExpSpace-hard, we do not
know whether it is as hard as the VAS reachability problem. The hardness can be
shown by reduction from the control state reachability problem in VASSes, which
is ExpSpace-hard [19]. For a VASS Vand a target control state qthereof, we
construct two new VASSes V0and V1, which are copies of Vwith one additional
coordinate, which at the beginning is zero for V0and one for V1. We also add one
new transition from control state q, which allows V1to decrease the additional
coordinate by one. One can easily verify that the two VASS reachability sets
definable by V0and V1are modular separable if, and only if, the control state q
is not reachable in V, which finishes the proof of ExpSpace-hardness.
The unarity and modularity characterization problems. Closely related problems
to separability are the modularity and unarity characterization problems: is a
given section of a VAS reachability set modular, resp., unary? We focus here
on the unarity problem, but the other one can be dealt in the same way. De-
cidability of the unarity problem would follow immediately from Theorem 9,
if sections of VAS reachability sets were (effectively) closed under complement.
This is however not the case. Indeed, if the complement of a VAS reachability
set is a section of another VAS reachability set, then both sets are necessarily a
section of a Presburger invariant [15], hence semilinear. But we know that VAS
reachability sets can be non-semilinear, and thus they are not closed under com-
plement. However, the unarity problem can be shown to be decidable directly,
at least for VAS reachability sets, by using the following two facts: first, it is
decidable if a given VAS reachability set Uis semilinear (see the unpublished
works [6,13]); second, when a VAS reachability set is semilinar, a concrete rep-
resentation thereof as a semilinear set is effectively computable [16]. Indeed, if a
given Uis not semilinear, it is not unary either; otherwise, compute a semilinear
representation, and check if it is unary. The latter can be checked directly, or
can be reduced to unary separability of semilinear sets (since semilinear sets are
closed under complement, as discussed above).
Acknowledgements We thank Maria Donten-Bury for providing us elegant proofs
of Lemmas 13 and 25.
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