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Definable sets in the weak Presburger arithmetic
∗
CHRISTIAN CHOFFRUT
LIAFA, Université Paris 7 & CNRS,
2, pl. Jussieu – 75251 Paris Cedex – 05, France
E-mail: cc@liafa.jussieu.fr
www. liafa. jussieu. fr/ ~cc
ACHILLE FRIGERI
Dipartimento di Matematica, Politecnico di Milano & LIAFA, Université Paris 7
via Bonardi, 9 – 20133 Milano, Italia
E-mail: achille.frigeri@polimi.it
We show the following: given a relation defined by a first order formula on the
structure hZ; +, <i, it is recursively decidable whether or not it is first order
definable in the structure hZ; +i.
Keywords: Presburger arithmetic, arithmetical definability.
Introduction
Presburger arithmetic is the fragment of arithmetic concerning the integers
with addition and order. Presburger’s supervisor considered the decidability of this fragment too modest a result to deserve a Ph.D. degree and
he accepted it only as a Master’s Thesis in 1928. Looking at the number
of citations, we may say that history revised this depreciative judgment
long ago. There still remains, at least as far as we can see, some confusion
concerning the domain of the structure: Z or N? with or without the order relation? (the main popular mathematical web sites disagree on that
respect). The original paper deals with the additive group of positive and
negative integers with no binary relation, but in a final remark of the original communication the author asserts that the same result, to wit quantifier
elimination, holds on the structure of the “whole” integers, i.e., the natural
∗ Partially
...
...
supported by . . .
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numbers with the binary relation <. In ?, which is the main reference on
the subject, Presburger arithmetic is defined as the elementary theory of
integers with equality, addition, having 0 and 1 as constant symbols and
< as binary predicate, see also ?. On the other hand, the majority of the
“modern” papers referring to Presburger arithmetic is concerned with the
natural numbers where order relation is unnecessary as it is first order expressible.
The origin of the present work is the simple remark that concerning the set
of integers Z, the binary relation matters. Here we study the decidability
of the definability in the structure hZ; +i for a given relation defined in
hZ; +, <i. We show that it is indeed recursively decidable and we prove this
result by revisiting the notion of linear subsets introduced by Ginsburg and
Spanier? in the sixties.
Despite of its simplicity, this arithmetic is central in many areas of theoretical and applied computer science. From a theoretical point of view, it
has many surprising properties: 1) it admits quantifiers elimination?,?,? and
therefore it is decidable, 2) given a formula on the expansion of the structure
obtained by adding the function which to each integer assigns the maximal
power of 2 which divides it, it is decidable whether or not it is definable
by a Presburger formula over N (Cobham-Semënov theorem? , improved
in ? with a polynomial time algorithm), and 3) Presburger arithmetic is
self-definable (i.e. there is Presburger definable criterion for definability? ).
Moreover, there is a strong and old connection between language theory,
Presburger definable sets and rational relations on Z and N dating back to
the sixties?,?,? . The concept is also widely used in many application areas,
such as program analysis and model-checking and more specifically timed
automata: roughly speaking, the main idea is that we can describe an infinite system with unbounded integer variables using Presburger formulas as
guards? (see also the introduction of ? for some historical remarks on the
role of Presburger arithmetic in the development of theoretical and applied
computer science).
1. Preliminaries
1.1. Variants of Presburger arithmetic
As observed above, a source of confusion is the lack of agreement in
the definition of Presburger arithmetic itself. We make the convention of
calling weak Presburger arithmetic the structure Z W = hZ; +; 0, 1i originally studied in ?, while with Z we mean the (standard) Presburger arith-
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metic hZ; =, <; +; 0, 1i. The positive Presburger arithmetic is the structure
N = hN; =; +; 0, 1i and we observe that in this case the < predicate (as
restriction of the order on Z to N) is already definable in N . All these three
structures are decidable in the sense that given a closed formula, it is recursively decidable whether or not it holds. In particular Z W and Z admit
quantifier elimination in the augmented languages with the additional unary
functional symbol − and the (recursive) set of binary functional symbols
(≡m )m∈N\{0,1} , having the usual meaning of opposite and modulo, while
for N it suffices to add the binary functional symbols (<m )m∈N\{0} , where
x <m y if and only if x < y ∧ x ≡m y ?,? .
1.2. Logical definability
Here we are concerned with the definability issue. We recall that given
a logical structure D with domain D and a first order formula on this
structure, say φ(x1 , . . . , xn ) where x1 , . . . , xn is the set of free variables,
the n-ary relation R defined by φ is the set of n-tuples (a1 , . . . , an ) such
that φ holds true when the variable xi is assigned the value ai , i.e., R =
{(a1 , . . . , an ) ∈ Dn | D |= φ(a1 , . . . , an )}.
Example 1.1. E.g., the formula (x1 + x2 = 0) ∨ (x1 = x2 + 1) defines, in
the structure Z, the union of a point and of a line in the discrete plane. 2. N-linear and Z-linear sets
2.1. Some notations
The free abelian monoid and the free abelian group on k generators are
respectively identified with Nk and Zk with the usual additive structure.
The addition is extended from elements to subsets: if X, Y ⊆ Nk (resp.
X, Y ⊆ Zk ), X + Y ⊆ Nk (resp. X + Y ⊆ Zk ) is the set of all sums x + y
where x ∈ X and y ∈ Y . It might be convenient to consider the elements
of Nk and Zk as vectors of the Q-vector space Qk .
Given v in Nk or in Zk , the expression Nv represents the subset of all
vectors nv where n range over N. This expression can be extended to Zv in
a natural way whenever v is in Zk . Thus Zu + Zv represents the subgroup
generated by the vectors u and v.
2.2. Linear sets
The following discussion requires (the adaptation of) few definitions. The
symbol K stands either for N or for Z when concerning the free abelian
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group Zk or for N when concerning the free abelian monoid Nk .
Definition 2.1. A subset of Zk (resp. Nk ) is K-linear if it is of the form
a+
n
X
Kbi ,
a, bi ∈ Zk (resp. Nk ), i = 1, . . . , n.
(1)
i=1
It is K-simple if the bi ’s are linearly independent as vectors of Qk . It is
K-semilinear if it is a finite union of K-linear sets and K-semisimple if it
is a finite disjoint union of K-simple sets. A subset of Zk is Z-quasisimple
Pn
if it is of the form A + i=1 Zbi , where A is a finite set such that for all
a, a0 ∈ A the vector (a − a0 ) belongs to the Q-vector space spanned by the
bi ’s.
Example 2.1. The subset Z(1, 0) ∪ Z(0, 1) is Z-semilinear. It is also Nsemilinear and N-semisimple (it is equal to the union of N(1, 0), (−1, 0) +
N(−1, 0), N(0, 1) and (0, −1) + N(0, −1)). The subset {(0, 1), (1, 0)} +
Z(2, 0)+Z(0, 2) is Z-quasilinear. The subset {(0, 1, 0), (1, 1, 1)}+Z(2, 0, 0)+
Z(0, 2, 0) is Z-semilinear but not Z-quasilinear.
Ginsburg and Spanier proved? the following result for Nk , but it can readily
be seen to hold for Zk .
Theorem 2.1. Given a subset X of Nk (resp. Zk ) the following assertions
are equivalent:
(i) X is first order definable in N (resp. Z);
(ii) X is N-semilinear;
(iii) X is N-semisimple.
Example 2.2. The binary relation of Example 1.1 is the union of the two
Z-linear subsets:
{(1, 1)} ∪ Z(1, −1).
Clearly, a finite union of Z-linear subsets is also a finite union of N-linear
subsets but the converse does not hold, e.g., a moment’s reflection will
convince the reader that the subset N is not expressible as a finite union
of Z-linear subsets. Still every Z-linear set is Z W -definable. Indeed, given
Pn
X = a + i=1 Zbi , then x = (x1 , . . . , xk ) ∈ X if, and only if, the following
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(i)
formula holds (where bj means the i-th component of the vector bj ):
P(x) = ∃z1 . . . ∃zn
k
^
(xi = a(i) +
i=1
n
X
(i)
zj b j ) .
(2)
j=1
Would the family of finite unions of Z-linear subsets by chance capture the
notion of subsets which are definable in the structure Z W ? This is not the
case, since this family is not closed under taking the complement. Actually
we will see that the Boolean closure of the family of Z-linear subsets is
precisely the class of Z W definable sets, see Theorem 4.1.
3. Properties of Z-linear sets
Our decidability result is based on the equivalence between definable subsets in the structure Z W and the Boolean closure of the Z-linear subsets.
This characterization is obtained, in particular, by proving that the class
of Z-linear sets enjoys many properties such as closure under finite sum,
projection, direct product, and, more interestingly, intersection.
3.1. Closure properties
We start with an elementary technical result.
Lemma 3.1. Let S be a Z-simple set in Zk . Then, for every v ∈ Zk , S +Zv
is Z-simple.
Pn
Proof. We argue by induction and assume that S = i=1 Zbi , for some
bi ∈ Zk such that {b1 , . . . , bn } is a linearly independent set of vectors in
Zk . We may further assume that {b1 , . . . , bn , v} is linearly dependent since
Pn
otherwise we are done. We have hv ∈
i=1 Zbi for some h ∈ N which
implies v ∈ R = hb1 /h, . . . , bn /hi as a lattice in (Z/h)k . In particular R is
a free abelian group on n generators and S + Zv is a free subgroup of R.
Pn
There exist c1 , . . . , cn ∈ (Z/h)k such that S + Zv = hc1 , . . . , cn i = i=1 Zci
(see [?, Theorem 4, (I,§10)]). But S + Zv ⊆ Zk , so that ci ∈ Zk , for every
1 ≤ i ≤ k. Finally, if the vectors ci ∈ Zk are independent in (Z/h)k , they
are certainly free in Zk and so S + Zv is a Z-simple set.
As a consequence we have the following two Corollaries which show a first
important departure from the N-linear and Z-linear sets.
Corollary 3.1. Every Z-linear set is Z-simple.
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Corollary 3.2. The family of Z-simple set is closed under projection, direct
product and finite sum.
Pn
The dimension of a simple set a + i=1 Zbi is the integer n and the dimension of a union of simple sets is the maximum dimension of these sets. E.g.,
the dimensions of the example in 2.2 is equal to 1 and those of example 2.1
are equal to 2.
In order to prove the closure under intersection, we recall a classical and
useful theorem of linear algebra (for each integer n, GLn (Z) represents the
group of all n × n invertible matrices with entries in Z).
Theorem 3.1 (Smith normal form, [?, p. 74]). Let A ∈ Zm×n be a
matrix of rank s. Then there exist two matrices U ∈ GLm (Z) and V ∈
GLn (Z), such that
D0
A0 = U AV =
0 0
where D = (dij ) ∈ Zs×s is an integer diagonal matrix such that dii divides
djj for 1 ≤ i ≤ j ≤ s.
Proposition 3.1. Let A ∈ Zm×n and b ∈ Zm . The set S of solutions in
Zn of the linear system Ax = b is an effective Z-simple set of dimension
equal to n − s where s is the rank of A.
Proof. If A0 = U AV is the Smith normal form of A, then the given system
is equivalent to U AV V −1 x = U b. Put V −1 x = y and U b = b0 . The set
Pm
of solutions of A0 y = b0 is the Z-simple set S 0 = c + i=s+1 Zei , where
cj = b0j /ajj for 1 ≤ j ≤ s and cj = 0 for s < j ≤ n and the ei are the vectors
of the canonical basis. Observe that the matrix V −1 has integer entries, thus
y is a vector with integer entries and therefore the system admits solutions
in Zn if and only if ajj divides b0j for 1 ≤ j ≤ s. The vectors ĉ = V c
and êi = V ei have integer entries and the V ei ’s are linearly independent
because V is unimodular (i.e. |V | = ±1). Since equality S = V S 0 holds, we
Pm
have S = ĉ + i=s+1 Zêi . The effectiveness should be clear, knowing that
obtaining the Smith normal form is effective? .
This leads us to the main property of this section.
Theorem 3.2. The intersection of two simple subsets of dimension n and
m is a simple subset of dimension less than or equal to min{n, m}. In
particular the family of Z-simple sets is closed under finite intersection.
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Pn
Pm
Proof. Let P = a + i=1 Zbi and Q = c + j=1 Zdj be two Z-simple
sets in Zk and assume without loss of generality that n ≤ m. Consider
Pn
Pn+m
the linear system i=1 bi xi − i=n+1 di−m xi = c − a and let S be the
set of solutions in Zm+n obtained by applying Theorem 3.1, with A =
[b1 , . . . , bn , −d1 , . . . , −dm ]. Observe that its rank is greater than or equal to
m so that S is Z-simple with dimension is ` ≤ n. Its projection on the first n
P`
coordinates S 0 = πn (S) is Z-linear: S 0 = u+ i=1 Zvi , for some u ∈ Zn and
vi ∈ Zn , i = 1, . . . , `. Now, considering for example the first n components
Pn
of S, P ∩ Q can be written as {a + i=1 xi bi | (x1 , . . . xn ) ∈ S 0 }. Define
B as the k × n matrix whose columns vectors are b1 , . . . , bn and V as the
n × ` matrix whose columns vectors are v1 , . . . , v` . Then the intersection
can be obtained as a composition of two affine transformations φ : Zn → Zk
defined by α 7→ a + αB and ψ : Z` → Zn defined by β 7→ u + βV (the
matrices operate to the left on row vectors)
P ∩ Q = φ(ψ(Z` )) = a + (u + Z` V )B = (a + uB) + Z` (V B).
This prove that P ∩ Q is Z-linear, and via Corollary 3.1, that it is actually
Z-simple.
3.2. Quasisimple sets
The following two results deal with quasisimple sets which are slight generalizations of simple sets, see Definition 2.1. They will be used in the next
section.
Proposition 3.2. Let S, T be Z-simple sets in Zk . Suppose dim S = dim T
and T ⊂ S. Then X = S \ T is a Z-quasisimple set of basis T .
Pn
Pn
Proof. Let S = a + i=1 Zbi and T = c + i=1 Zdi , with a, c, bi , di ∈ Zk .
Pn
By T ⊆ S, we have c = a + i=1 γi bi , for some γi ∈ Z, so S = c +
Pn
Pn
Pn
Pn
i=1 (−γi )bi +
i=1 Zbi = c+
i=1 (Z−γi )bi = c+
i=1 Zbi . After possibly
translating both subsets, we may suppose without loss of generality that
Pn
Pn
c = 0, so that S = i=1 Zbi and T = i=1 Zdi . Consider the isomorphism
φ which maps the subgroup generated by the bi ’s onto the free group Zn .
Then φ maps T onto a subgroup of Zn of finite index and Zn \ φ(T ) is a
(finite) union of cosets, so T = φ−1 (Zn \ φ(T )) is a finite union of linear
Pn
Pn
sets of the form g + i=1 Zdi where g ∈ i=1 Zbi .
In the same spirit we have the following:
Proposition 3.3. Let S a finite union of Z-simple sets of dimension k in
Zk . Then S is Z-quasisimple.
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Proof. Let us suppose S =
Sm
i=1 Xi ,
(i)
bj ’s, 1 ≤
where Xi = a(i) +
Pk
j=1
(i)
Zbj is a Z-
simple set. Since all vectors
j ≤ k, are linearly independent, they
generate a subgroup of Zk of finite index, in particular for every Xi there
exists a morphism φi of Zk into a finite group Gi and a subset Fi ⊆ Gi such
that Xi = φ−1 (Fi ). Let φ be the morphism of Zk into the direct product
G = G1 × · · · × Gm defined by φ(x) = (φ1 (x), . . . , φm (x)) and let H be
the set of m-tuples (g1 , . . . , gm ) ∈ G for which gi belongs to Fi for some
1 ≤ i ≤ m. Then S = φ−1 (H) which shows that S is a union of cosets of
the kernel of φ. As such, it is Z-quasisimple set of dimension k.
The last result of this paragraph concerns a weak condition of equality of
two quasisimple sets of maximum dimension. Intuitively, it means that it
suffices for two quasisimple sets to be equal almost everywhere (in some
precise sense) in order to be equal.
Proposition 3.4. Two quasilinear subsets of maximum dimension k are
equal if, and only if, for all integers n there exists an hypercube [α1 , α1 +
n] × · · · × [αk , αk + n] ⊆ Zk on which they agree.
Proof. The condition is clearly necessary. In order to prove that it is sufficient, consider a morphism φ : Zk → G onto some finite group recognizing
two Z-semilinear subsets X and Y , i.e., φ−1 φ(X) = X and φ−1 φ(Y ) = Y .
Let n be a common multiple of degrees of the k elements φ(1, 0, . . . , 0),
φ(0, 1, . . . , 0), . . . , φ(0, . . . , 0, 1). Then a vector (v1 , . . . , vk ) belongs to X if,
and only if, so does the vector ([v1 ], · · · , [vk ]) where [vi ] is the remainder of
the division of vi by n.
Corollary 3.3. Let X and Y be two quasilinear subsets of maximum dimension k and let Z be a finite union of linear subsets of dimension less
than k. Then X = Y if, and only if, X \ Z = Y \ Z.
4. Definable sets in weak Presburger arithmetic
4.1. An algebraic characterization of definable subsets
The Boolean closure of the linear sets enjoys some properties which we
will take advantage of when characterizing the definable sets in Z W . In
particular, condition (iii) of the following Lemma is useful when designing
a decision procedure. ¿From now on, “linear” and “simple” means Z-linear
and Z-simple.
Lemma 4.1. The following families of subsets of Zk are equal:
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(i) the Boolean closure of linear sets;
TJ
(ii) the family of all finite unions of the form S ∩ j=1 T j where S, Tj are
Z-simple;
SJ
(iii) the family of all finite unions of the form S \
j=1 Tj where S, Tj
are Z-simple with Tj ⊂ S, and dim Tj < dim S.
Proof. Clearly the families (ii) and (iii) are included in the Boolean closure of the linear sets. The family (ii) contains all linear sets (take J = 0)
and all complements of linear sets (take S = Zk for S and J = 1). FurTJ
thermore the complement of a subset of the form S ∩ j=1 T j is a finite
union of linear sets and complements of linear set, thus the complement of
a finite union of such subsets is a finite union of intersections of linear sets
and complements of linear sets. Since the intersection of linear sets is again
linear, this shows that the family (ii) is closed under complement, and thus
coincides with the Boolean closure.
We now show that each subset of the form (ii) is equivalent to a union
S
of subsets of the form (iii). Indeed, because of equality S \ ( i Ti ) =
S
S \ ( 1≤i≤r (S ∩ Ti )), without loss of generality we may assume that
all Ti ’s are Z-simple subsets of S. Assume further that the p ≤ r first
subsets Ti ’s have the same dimension as S, with p = 0 if no Ti has
the same dimension as S. Then by repeatedly applying Lemma 3.2 to
S
S
S
S \( i Ti ) = (. . . ((S \T1 )\T2 ) . . . Tp )( i>p Ti ) we may transform S \( i Ti )
into a finite union of subsets of the form required.
Theorem 4.1. The family of Z W -definable sets is the Boolean closure of
the family of Z-linear sets.
Proof. Clearly, all linear sets are Z W -definable, see expression (2). Let
us verify the converse. Using the quantifier elimination result, all Z W definable subsets belong to the Boolean closure of the relations which are
solutions of a system of linear equations or of modular equations of the type
Pn
i=1 ai xi ≡m b. In the former case, this is a consequence of Proposition
3.1. As for the latter case, let us proceed by induction on n. Let n = 1,
then we must solve the equation ax ≡m b. If b is not a multiple of the
greatest common divisor d of a and m then there is no solution. Otherwise a/d has an inverse modulo m/d and the set of solutions is defined by
Pn−1
(a/d)−1 (b/d) + Z(m/d). Now if n > 1, then set t = i=1 ai xi . The con
Pn
Wm−1
gruence i=1 ai xi ≡m b is equivalent to j=0 t ≡m j ∧ an xn ≡m (b − j) ,
and so by the induction hypothesis, we may conclude.
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4.2. A special case
The case studied here can be considered as the top level case in the recursive
procedure of the next paragraph. The following notation is useful. Given
Pm
a N-simple set S = a + i=1 Nbi , we denote by S Z the Z-simple set a +
Pm
i=1 Zbi .
Theorem 4.2. Let X ⊆ Zk be a Z-definable set of the form
X=T ∪
m
[
Yi ,
i=1
Pk
(i)
(i)
where Yi = a(i) + j=1 Nbj for some linearly independent vectors bj (i.e.
it is a N-simple set of dimension k) and T is a finite union of N-simple sets
of dimension less than k. Then X is Z W -definable if, and only if, it can be
decomposed as S ∪ (P \ R), where:
Pk
S
S
(j)
(1) P = 1≤i≤m (a(i) + j=1 Zbj ) = 1≤i≤m YiZ ;
(2) R and S are Z W -definable sets which are included in a finite union of
Z-simple sets of dimension less than k;
(3) R ⊆ P and S ∩ P = ∅.
Proof. Clearly the condition is sufficient. Suppose X is Z W -definable and
express it as in (ii) of Lemma 4.1. Isolate all the simple sets of dimension
k. By the set-theoretic equality
!
[
[
[
[ E i \ Fi =
Ei \
(Ej \ Fj ) ,
Fi \
i
i
i
j6=i
if the Ei ’s are simple sets of maximal dimension, by Proposition 3.3 their
union is a quasisimple set and we may write
X=S∪
A+
k
X
Zdj \ R ,
j=1
where A ⊆ Zk is finite, the vectors dj are linearly independent, S and R are
Z W -definable sets included in a finite union of Z-simple sets of dimension
less than k. Now we prove the following, which means that the symmetric
difference of the two subsets is included in a finite union of linear subsets
of dimension less than k:
m
[
i=1
a(i) +
k
X
j=1
(i) Zbj
∼ A+
k
X
j=1
Zdj .
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Pk
Clearly, if we take a vector in A + j=1 Zdj not belonging to R ∪ T
(which is contained in R ∪ T Z , that is, in a finite union of Z-simple sets
of dimension less than k), it must belong to one of the Yi , and thus to
Sm
Pk
(i)
(i)
+ j=1 Zbj . Conversely, consider one of the sets Yi , namely
i=1 a
Pk
Y = a + j=1 Nbj (we drop the upper indices to simplify the notation),
Pk
then we show Y Z ⊆ A + j=1 Zdj except for a union of Z-simple sets
of dimension less than k. We observe that S is in particular Z-definable,
S
so S = 1≤i≤m Si , where every Si is a N-simple set of dimension less
than k. For each Si , let Ji ⊆ I = {1, . . . , k} be the maximal ordered set
of indices 1 ≤ j1 < · · · < jp ≤ k, such that Si is not parallel to the
vectors bj , j ∈ Ji , and let πi : Zk → Z|Ji | be the projection defined by
πi ((xj )j∈I ) = ((xj )j∈Ji ). So, we can define Sbi = πi−1 (Si ), if Ji 6= I, and
Sbi = ∅ otherwise. E.g., if S consists of the unique simple subset
(1, 0, 1) + N(0, 2, 2) + N(3, 0, 0) = {(1 + 3p, 2n, 2n + 1) | n, p ∈ N},
then it is parallel to the vector (1, 0, 0) and we have Sb = {(p, 2n, 2n + 1) |
p ∈ Z, n ∈ N}. We want to show by induction on i that the subset
!
[
c
Wi = (Zb1 + · · · + Zbi−1 + Nbi + · · · + Nbk ) \
Sj
1≤j≤m
P
is included in A+ 1≤j≤k Zdj . This is clear for i = 1. Now assume 1 < i ≤ k
and consider a vector v ∈ Wi . Since it does not belong to any subset Sj
parallel to bi , the subset v + Nbi is included in A + Zd1 + · · · + Zdk except
maybe for finitely elements which are the possible intersections of this linear
set with the subsets Sj parallel to none of the vectors b1 , . . . , bk . This implies
(k)
(1)
for sufficiently large n, v + nbi = an + λn d1 + · · · + λn dk . Since the set
A is finite, the elements an ∈ A start to repeat and for some integer r the
following holds for all integers m and 0 ≤ r0 < r:
v + (mr + r0 )bi = ar0 + mλ(1) d1 + · · · + mλ(k) dk .
For all integers `r + j ∈ Z where 0 ≤ j < r we have v + (`r + j)bi =
aj + `λ(1) d1 + · · · + `λ(k) dk which shows that v + Zbi ⊆ A + Zd1 + · · · + Zdk .
Because of Corollary 3.3, X can be written as S ∪ (P \ R), where S and
R are Z W -definable sets which are included is simple sets of dimension less
than k and which are computable from T and form the Yi ’s. To end the
proof we must verify the last item; indeed, we have:
S ∪ P \ (R \ S) = S ∪ (P ∩ (R ∪ S)) = S ∪ (P ∩ S) ∪ (P \ R) = S ∪ P \ R,
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and observe that since R is included in P so is R \ S which shows that we
may assume S ∩ R = ∅. Furthermore, we have
S ∪ P \ R = (S ∩ P ) ∪ (S \ P ) ∪ (P ∩ R) = S \ P ∪ (P ∩ (S ∪ R)).
Since S ∩ R = ∅ holds we have S ∪ R = R which shows that without loss
of generality we may suppose that S ∩ P = ∅ holds. This implies
X \ P = S, P \ X = R.
4.3. The procedure
We recall our problem. Given a subset X ⊆ Zk which is Z-definable, i.e.,
specified as a finite union of N-simple sets, decide whether or not it is
actually Z W -definable and, in the affirmative case, give a representation
as Boolean combination of Z-simple sets. We cannot directly use Theorem
4.2, because it requires that one of the simple sets in the specification have
dimension k. So we proceed as follow:
(1) Let X ⊆ Zk be the union of the sets
Xi = a(i) +
Ji
X
(i)
Nbj ,
1 ≤ i ≤ m.
j=1
PJ
Consider all the affine subspaces of Qk , Hi = ha(i) + j=1 Qbj ∩ Zk ,
and suppose (after possibly changing some indices) H1 , . . . , Hr , r ≤ m,
are the maximal (for the inclusion) elements of the collection H = {Hi |
1 ≤ i ≤ m}.
(2) Compute all the sets Hi0 = Hi ∩ Zk and observe that they are ZW PJ
definable. Indeed for a subset of the form H = a + j=1 Qbj ∩
Zk , where a, bj ∈ Zk , there exist k − J linear equations F1 (y) =
0, . . . , Fk−J (y) = 0 whose set of solutions is exactly the subspace generated by the vectors bj . Then H is defined by the Presburger formula
∃y (x = y + a) ∧ (F1 (y) = 0) ∧ . . . ∧ (Fk−J (y) = 0) .
Clearly, X is Z W -definable if, and only if, all the intersections Yi =
X ∩ Hi0 are Z W -definable.
(3) For all 1 ≤ i ≤ r, consider an isomorphism τi : Hi → QJi such that
τi (Hi0 ) = ZJi . Such isomorphism clearly exists and moreover it can be
expressed, relatively to Hi0 , in the structure Z W , so that τi (Yi ) is again
a Z W -definable set if, and only if, Yi is Z W -definable.
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(4) Finally we can apply Theorem 4.2 to τi (Yi ), since it has at least one
component of maximal dimension J, obtaining the sets Pi , Si , and
Ri that satisfy all conditions of 4.2. Now Yi = τi−1 (Si ) ∪ τi−1 (Pi ) ∩
τi−1 (Ri ) , and observe that the inverse images of Pi , Si , and Ri by τ
again satisfy the three conditions, in particular τi−1 (Pi ) is Z-simple.
(5) Now apply the procedure recursively to the sets τi−1 (Si ) and τi−1 (Ri ).
(6) It only remains to observe that if we fail to apply Theorem 4.2, this
means that the set X actually was not Z W -definable.