LINEAR EQUATIONS FOR UNORDERED DATA VECTORS IN

. Following a recently considered generalisation of linear equations to unordered-data vectors and to ordered-data vectors, we perform a further generalisation to data vectors that are functions from k-element subsets of the unordered-data set to vectors of integer numbers. These generalised equations naturally appear in the analysis of vector addition systems (or Petri nets) extended so that each token carries a set of unordered data. We show that nonnegative-integer solvability of linear equations is in nondeterministic exponential time while integer solvability is in polynomial time.


Introduction.
Diophantine linear equations.The solvability problem for systems of linear Diophantine equations is defined as follows: given a finite input set of d-dimensional integer vectors I = {v 1 . . .v m } ⊆ Z d , and a target vector v ∈ Z d , we ask if there is a solution (n 1 . . .n m ) such that (1.1) Restricting solutions n 1 . . .n m to the set Z of integers or the set N of nonnegative integers, we may speak of Z-solvability and N-solvability, respectively.The former problem is in P-Time, while the latter is equivalent to integer linear programming, a well-known NP-complete problem [Kar72].This paper is a continuation of a line of research that investigates generalisations of the solvability problem to data vectors [HLT17,HL18] i.e. on a high level of abstraction, to vectors indexed by orbit-finite sets instead of finite ones (for an introduction to orbit-finite sets, also known as sets with atoms, see [BKL11,BKLT13]).In the simplest setting, given a fixed countable infinite set D of data values, a data vector is a function a : D → Z d .Addition and scalar multiplication are defined pointwise i.e. (a+a )(α) = a(α) + a (α) for every α ∈ D. The solvability problems over input sets I of data vectors are defined analogously, with the important difference that the input set I of data vectors is the closure, under data permutations, of a finite set of data vectors.Given such an infinite, but finite up to data permutation set I, and a target data vector v, we ask if there are data vectors v 1 . . .v m ∈ I and numbers n 1 . . .n m , such that the equality (1.1) is satisfied.
Finally, let v β (β) = 2 and v β (x) = 0 for x = β, where β ∈ D is some fixed data value.On the one hand, this instance admits a Z-solution, since v β is presentable as for any two different data values δ, ε different than β.On the other hand, there is no N-solution, as there is no similar presentation of v β in terms of data vectors v δε that uses nonnegative coefficients.Simply, every vector that is a sum of data vectors from the family I must be strictly positive for at least two data values.Furthermore, if v β (β) = 3 instead of 2, then there is no Z-solution, too.Indeed, notice that α∈D w(α) is always an even number if w is a sum of data vectors from the family I.
The above simple example is covered by the theory developed in [HLT17].Here, we extend the results from [HLT17] to data vectors in [D]  The set I is closed under data permutations, indeed for any data permutation π and any v γδε ∈ I the data vector v γδε • π = v π −1 (γ)π −1 (δ)π −1 (ε) ∈ I. Finally, we want to know if I and the following target data vector v γδ admits a Z-solution v γδ (x) = 6 if x = {γ, δ} 0 otherwise 11:3 (v γδ is a single edge with weight 6).The answer is yes, but it is not trivial, where α, β, and ε are any data values different than γ, δ.
The idea behind the above sum is presented in Figure 2.
First we construct a gadget g δαε as presented on the left.Blue color denotes addition and orange subtraction of an edge.
Next, we use such gadgets combined with triangles in the following way g γδα + g δγα + 2v αγδ = v γδ .If we ask about N-solvability then the answer is no.If we add triangles then the number of nonzero edges will be greater than 2, so there is no way of reaching v γδ .But, what if we change 6 to 3? The answer is postponed to Section 3. 1.1.Related work and our contribution.The above-discussed, simplest extension of the Z-solvability problem to data vectors of the form D − → Z d is in P-Time, and the N-solvability is NP-complete [HLT17].Further known results concern the more general case of ordered data domain D [HL18].For ordered data we assume that the set of data forms a dens linear order and the set of data permutations is restricted to the set of order preserving bijections D → D. In the ordered data case the Z-solvability problem remains in P-Time, while the complexity of the N-solvability is equivalent to the reachability problem of vector addition systems with states (VASS), or Petri nets, and hence Ackermann-complete [Ler21,CO21].The increase of complexity caused by the order in data is thus remarkable.An example of a result that builds on top of [HLT17] is [GSAH19], where the continuous reachability problem for unordered data nets is shown to be in P-Time.The question if the continuous reachability problem is solvable for the ordered data domain remains open.It is particularly interesting due to [BFHRV10], where the coverability problem in timed data nets [AN01] is proven to be interreducible with the coverability problem in ordered data nets.
In this paper, we perform a further generalisation to k-element subsets of unordered data i.e. we consider data vectors of the form [D] k − → Z d , where [D] k stands for the k-element subsets of D. We prove two main results: first, for every fixed k ≥ 1 the complexity of the Z-solvability problem again remains polynomial.Second, we present a NExp-Time algorithm for the N-solvability problem.This is done by an improvement of techniques developed in [HLT17].Namely, we nontrivially extend Theorems 11 and 15 from [HLT17].
• To address N-solvability, we reprove Theorem 11 from [HLT17] in a more general setting (the proof is slightly modified).Next, we combined it with new idea to obtain a reduction to the (easier) Z-solvability, witnessing a nondeterministic exponential blowup.
• Our approach to Z-solvability is an extension of Theorem 15 from [HLT17].Precisely, if we reformulate the Z-solvability question in terms of (weighted) hypergraphs, then there is a natural way to lift the characterisation of Z-solvability proposed in Theorem 15 [HLT17].The main contribution of this paper is a new tool-box developed to prove the lifted theorem 15 from [HLT17].The new characterisation is easily checkable in polynomial time, resulting in the algorithm.
No analogous of Theorems 11 and 15 from [HLT17] appear in [HL18] or are otherwise known, thus we are pessimistic about applicability of the studied here approach in case of the ordered data domain.
As we comment in Conclusions, we believe that elaboration of the techniques of this paper allows also tackling the case of tuples of unordered data.
Motivation.Our motivation for this research is two-fold.On the one hand, from a foundational research perspective, our results are a part of a wider research program aiming at lifting computability results in finite-dimensional linear algebra to its orbitfinite-dimensional counterpart.Up to now, research has been focused on understanding solvability [HLT17,HL18], but there are other natural questions about definitions of bases, dimension, linear transformations, etc.On the other hand, we are interested in the analysis of systems with data and solvability may be a useful tool.We highlight three areas where understanding of N/Z-solvability may be crucial for further development.
Unordered Data nets reachability/coverability [LNO + 08].The model can be seen as a special type of Coloured Petri nets [Jen98].Data nets are an extension of Petri nets where every token caries a tuple of data values.In addition, each transition is equipped with a Boolean formula that connects data of tokens that are consumed and data of tokens that are produced (for unordered data the formula may use = and =).To fire a transition we take a valuation satisfying the formula and according to it we remove and produce tokens.
Example 1.3.Consider the following simple net with 3 places and one transition.The initial marking has 3 tokens each with two data values.
then we may fire the transition and get the new marking.
If we had chosen another valuation x = α, y = β, z = β, u = γ, x = α, u = γ then the formula would hold, but we would lack tokens that can be consumed so we could not fire the transition with this valuation.
The reachability and coverability questions can be formulated as usual for Petri nets.It is not hard to imagine that some workflow or a flow of data through a program can be modelled with data nets.Unfortunately, in this richer model the reachability and coverability problems are undecidable [Las16] already for k = 2.For k = 1 i.e. data vectors in D − → Z d , the status of reachability is unknown and coverability is decidable but known to be Ackermannhard [LT17].This is not a satisfying answer for engineers and in this case we should look for over and under approximations of the reachability relation, or for some techniques that will help in the analysis of industrial cases.One of the classic over-approximations of the Petri nets reachability relation is so-called integer reachability or Marking Equation in [DE95] Lemma 2.12, where the number of tokens in some places may go negative during the run.It can be encoded as integer programming and solved in NP.Its analogue for data nets can be stated as N-solvability over an input set of data vectors, where the set is closed under data permutations.
Here, we should mention that the integer reachability is a member of a wider family of algebraic techniques for Petri nets.We refer to [STC96] for an exhaustive overview of linear-algebraic and integer-linear-programming techniques in the analysis of Petri nets.The usefulness of these techniques is confirmed by multiple applications including, for instance, recently proposed efficient tools for the coverability problem of Petri nets [GLS16,BFHH16].
π-calculus.Another formalism close to unordered data nets are ν-nets (unordered data nets additionaly equipped with an operation of creating a new datum that is not present in the current configuration of the net; in other words, a transition may force creation of data that are globally fresh/unique).In [Ros10], Rosa-Velardo observes that they are equivalent to so-called multiset rewriting with name binding systems which, as he showed, are a formalism equivalent to π-calculus.Thus, one may try to transfer algebraic techniques for data nets to π-calculus.This is a long way, but there is no possibility to start it without a good understanding of integer solutions of linear equations with data.
Here, it is worth mentioning that in π-calculus we use constructs like c y .P which send a datum y trough the channel c, thus the sending operation is parameterized with a pair of data values i.e. the name of the channel and the data value.Thus, one cannot expect that already existing results for data vectors D − → Z d [HLT17] will be sufficient and we will need at least theory for D 2 − → Z d , for example, to count messages that are sent and received.In fact even D 2 − → Z d may be not enough.In [Ros10], the fundamental concept for the encoding are derivatives, which essentially are terms of bounded depth labelled with data values.In his encoding, Rosa-Velardo represent each derivative with a token, so each token has to carry all data that are needed to identify the derivative.For a given process definition, he produces a ν-net with tokens with bounded but arbitrary high number of data values.Thus, if one wants to use linear algebra with data to describe some properties of the produced ν-net, he will have to work with data vectors in D k → Z d , for k grater than 2.
Parikh's theorem.Finally, we may try to lift Parikh's theorem from context-free grammars and finite automata to context-free grammars with data [CK98] and register automata [KF94].It is not clear to what extent it is possible but there are some promising results [HJLP21].If we want to use this lifted Parikh's theorem then we have to work with semilinear sets with data and be able to check things like membership or nonemptiness of the intersection.Here, one more time techniques to solve systems of linear equations with data will be inevitable.
Outline.In Section 2 we introduce the setting and define the problems.Next, in Section 3 we provide the polynomial-time procedure for the Z-solvability problem: the hypergraph reformulation and an effective characterisation of hypergraph solvability.In Section 4 for pedagogical reasons we present the proof of the characterisation if data vectors are restricted to [D] 2 − → Z d .After this, in Sections 5, 6, 7, 8, 9, 10 we provide the full proof of the characterisation.The proof follows the same steps as the proof of the case [D] 2 − → Z d , but is much more involved at the technical level.Next, in Section 11 we present a reduction from N-to Z-solvability.Finally, Section 12 concludes this work.
2. Linear equations with data.
In this section, we introduce the setting of linear equations with data and formulate our results.For a gentle introduction of the setting, we start by recalling classical linear equations.
Let Z and N denote integers and nonnegative integers, respectively.Classical linear equations are of the form a 1 x 1 + . . .+ a m x m = a, where x 1 . . .x m are variables (unknowns), and a 1 . . .a m ∈ Z are integer coefficients.For a finite system U of such equations over the same variables x 1 . . .x m , a solution of U is a vector (n 1 . . .n m ) ∈ Z m such that the valuation x 1 → n 1 . .., x m → n m satisfies all equations in U.It is well known that integer solvability problem (Z-solvability problem) i.e. the question whether U has a solution (n 1 . . .n m ) ∈ Z m , is decidable in P-Time.In the sequel we are often interested in nonnegative integer solutions (n 1 . . .n m ) ∈ N m , but one may consider also other solution domains than N.It is well known that the nonnegative-integer solvability problem (N-solvability problem) of linear equations i.e. the question whether U has a nonnegative-integer solution, is NP-complete (for hardness see [Kar72]; NP-membership is a consequence of [Pot91]).The complexity remains the same for other natural variants of this problem, for instance, for inequalities instead of equations (a.k.a.integer linear programming).The K-solvability problem (where K ∈ {Z, N}) is equivalently formulated as follows: for a given finite set of coefficient vectors I = {v 1 . . .v m } ⊆ Z d and a target vector v ∈ Z d (we use bold font to distinguish vectors from other elements), check whether v is an K-Sums of I i.e. (2.1) The dimension d corresponds to the number of equations in U. Data vectors.Linear equations can be naturally extended with data.In this paper, we assume that the data domain D is a countable infinite set, whose elements are called data values.The bijections ρ : D → D are called data permutations.For a set X and k ∈ N, by [X ] k we denote the set of all k-element subsets of X (called k-sets in short).Data permutations lift naturally to k-sets of data values: ρ({α 1 . . .α k }) = {ρ(α 1 ) . . .ρ(α k )}.
Fix a positive integer k ≥ 1.A data vector is a function v : [D] k → Z d such that v(x) = 0 ∈ Z d for all but finitely many x ∈ [D] k .(Again, we use bold font to distinguish data vectors from other elements.)We call the numbers k and d the arity and the dimension of v, respectively.
The vector addition and scalar multiplication are lifted to data vectors pointwise: It is the natural lift of normal function composition.For a set I of data vectors we define A data vector v is said to be a K-permutation sum of a finite set of data vectors I if (we deliberately overload the symbol K-Sums( ) and use it for data vectors, while in (2.1) it is used for (plain) vectors) (2.2) We investigate the following decision problems (for K ∈ {Z, N}): K-solvability input: A finite set I of data vectors and a target data vector v, all of the same arity and dimension.output: Is v a K-permutation sum of I?
The insightful reader may notice that, in the motivating examples we consider K-sums of an input set of data vectors that is closed under data permutations.Here, we switch to the K-solvability problem, which is defined by K-permutation sum.But observe that, to formalise expressibility by K-sums as a problem we have to provide a finite representation of the input set.That is why as an input to the problem we take a finite set of data vectors I, and we consider K-sums of its closure under data permutations i.e. of the set Perm(I).This is why we consider K-permutation sums instead of pure K-sums.
For complexity estimations we assume binary encoding of numbers appearing in the input to all decision problems discussed in this paper.Our main results are the following complexity bounds: Theorem 2.1.For every fixed arity k ∈ N, the Z-solvability problem is in P-Time.(The dependency on k is exponential).
Theorem 2.2.For every fixed arity k ∈ N, the N-solvability problem is in NExp-Time.
For the special case of the solvability problems when the arity k = 1, the P-Time and NP complexity bounds, respectively, have been shown in [HLT17].Thus, according to Theorem 2.1, in the case of Z-solvability the complexity remains polynomial for every k > 1.In the case of N-solvability the NP-hardness carries over to every k > 1, and hence a complexity gap remains open between NP and NExp-Time.

Proof of Theorem 2.1.
We start by reformulating the problem in terms of (undirected) weighted uniform hypergraphs (Lemma 3.2 below).
Fix a positive integer k ≥ 1.By a k-hypergraph we mean a pair H = (V, µ) where V ⊂ D is a finite set called vertices and µ : [V ] k → Z d is a weight function (when k is not relevant we skip it and write a hypergraph).As before, we call the numbers k and d the arity and the dimension of H, respectively.When k = 2, we speak of graphs instead of hypergraphs.
(Note however that the (hyper)graphs we consider are always weighted, with weights from Z d .) Because vertices are data then instead of usual u, v for vertices we will use Greek letters α, β, γ, δ, ε . ... Also, for a hypergraph we denote the set of its vertices V by The set of hyperedges is then defined as When α ∈ e for α ∈ V and e ∈ Edges(H), we say that the vertex α is adjacent with the hyperedge e.The degree of α is the number of hyperedges adjacent with α.We call vertices of degree 0 isolated.Two hypergraphs are isomorphic if there is a bijection between their sets of vertices that preserves the value of the weight function.Two hypergraphs are equivalent if they are isomorphic after removing their isolated vertices.For a family H of hypergraphs, by Eq(H) we denote the set of all hypergraphs equivalent to ones from H. If H = {H} then instead of Eq({H}) we write Eq(H).
Scalar multiplication and addition are defined naturally for hypergraphs.First, for c ∈ Z and a hypergraph ) and H = (V, µ) of the same arity k and dimension d, we first add isolated vertices to both hypergraphs to make their vertex sets equal to the union W ∪ V , thus obtaining G = (W ∪ V, µ ) and H = (W ∪ V, µ ) with the accordingly extended weight functions µ , µ : [(W ∪ V )] k → Z d , and then define G + H def = (W ∪ V, µ + µ ).Using these operations we define K-sums of a family H of hypergraphs of the same arity and dimension (again, we overload the symbol K-Sums( ) further and use it for hypergraphs): We say that a hypergraph H is a K-sum of H up to equivalence if H is a K-sum of hypergraphs equivalent to elements of H: H ∈ K-Sums(Eq(H)).
Example 3.1.We illustrate the Z-sums in arity k = 2 i.e. using graphs.Consider the following graph G consisting of 3 vertices and 2 edges: Let x, y ∈ Z d be arbitrary vectors.Here, there are two examples of graphs which can be presented as a Z-sum of {G} up to equivalence, using a sum of two graphs equivalent to G: and a difference of two such graphs: We are now ready to formulate the hypergraph Z-sum problem, to which Z-solvability is going to be reduced: Proof.The reduction encodes each data vector v by a hypergraph H = (V, µ), where and µ is the restriction of v to [V ] k .In this way, a set I of data vectors and a target data vector v are transformed into a set of H hypergraphs and a target hypergraph H such that v is a Z-permutation sum of I if, and only if, H is a Z-sum of H up to equivalence.
Thus, from now on, we concentrate on solving the hypergraph Z-sum problem.One may ask why we perform such a reduction.The reasons are of pedagogical nature, namely graphs are more convenient for examples and proofs by pictures.
As the next step, we formulate our core technical result (Theorem 3.4).In the theorem we state that the hypergraph Z-sum problem is equivalent to a local Z-sum problem, defined in the following paragraph.It is easy to design a polynomial time algorithm for the local Z-sum problem.This gives us the proof of Theorem 2.1.Let H = (V, µ) be a hypergraph.For a set X ⊂ D we consider {e ∈ Edges(H) | X ⊆ e} -the set of all hyperedges that include X -and define the weight of X as the sum of weights of all these edges: In particular when X ⊆ V then Λ X (H) = 0. Further, Λ ∅ (H) is the sum of weights of all hyperedges of H.When X = {α}, then Λ X (H) is the sum of weights of all hyperedges adjacent with α (we write Λ α instead of Λ {α} ).Finally, when the cardinality of X equals k i.e. |X | = k, Λ X (H) = µ(X ) is the weight of the hyperedge X .
Example 3.3.Let x, y, z ∈ Z d be arbitrary vectors.As an illustration, consider the graph G on the left below, with the weights of chosen subsets of its vertices listed on the right: Weights of sets are important as they form a family of homomorphisms from hypergraphs to Z d .Namely, for any subset of vertices X we have that This allows us to design a partial test for the hypergraph Z-sum problem.If there is a set of vertices X such that Λ X (H) cannot be expressed as a Z-sum of vectors in Λ X (Eq(H)) then H ∈ Z-Sums(Eq(H)).This motivates the next definition.
We say that a k-hypergraph H is locally a Z-sum of a family H of hypergraphs if for every (3.1)Note that we only consider hypergraphs H ∈ H, and we do not need to consider hypergraphs H ∈ Eq(H) equivalent to ones from H as Theorem 3.4.The following conditions are equivalent, for a finite set H of hypergraphs and a hypergraph H, all of the same arity and dimension: (1) H is a Z-sum of Eq(H); (2) H is locally a Z-sum of H.
Before we embark on proving the result (in the next section) we first discuss how it implies Theorem 2.1.Recall that the arity k is fixed.Instead of checking if H is a Z-sum of Eq(H), the algorithm checks if H is locally a Z-sum of H. Observe that the condition (3.1) amounts to solvability of a (classical) system of linear equations.Let V be the set of vertices of the hypergraph H. Therefore, the algorithm tests Z-solvability of a system of the corresponding linear equations, d for every subset X ⊆ V of the cardinality at most k.The number of equations is exponential in k, but due to fixing k it is polynomial in the input hypergraphs H and H. Thus, Theorem 2.1 is proved once we prove Theorem 3.4.
As 6 = 3 + 3 and 6 = 2 + 2 + 2 and 6 = 1 + 1 + 1 + 1 + 1 + 1 we see that the target is locally a Z-sum of the triangle.Hence the target is a Z-sum of the triangle up to equivalence.Moreover, if we change 6 to a smaller positive number then the target graph will not be a Z-sum of the triangle up to equivalence.For example if, we change 6 to 3, then v γδ is not a Z-sum of the triangles as Λ δ (v γδ ) = 3 is not divisible by 2 i.e. the weight of any single vertex of the triangle.

Proof of Theorem 3.4 (the case for
which implies, for some hypergraphs H i ∈ H equivalent to H i , and subsets As X was chosen arbitrarily, this shows that H is locally a Z-sum of H. 11:11 The proof of the converse implication 2 =⇒ 1 is more involved.The case for arity k = 1 is considered in [HLT17].Here, for pedagogical reasons, we provide a simplified version of the proof for arity k = 2 i.e. for (undirected) graphs.Consequently, we speak of edges instead of hyperedges.We recall that the graphs we consider are actually Z d -weighted graphs.
Let H be a graph and assume that H is locally a Z-sum of H.We are going to demonstrate that H is equivalent to a Z-sum of Eq(H).Since we work with arity k = 2, the assumption amounts to the following conditions, for every vertex α ∈ V ert(H) and every edge e ∈ Edges(H): Proof.Indeed, due to the assumption (4.1) and due the following equality (note that the symbol Z-Sums( ) applies to vectors on the left, and to graphs on the right) there is a graph We proceed in two steps.We start by defining a class of particularly simple graphs, called H-simple graphs, and argue that H is a Z-sum of these graphs (Lemma 4.4 below).Then we prove that every H-simple graph is representable as a Z-sum of H up to equivalence (Lemma 4.10 below).We may compose these two lemmas because of Lemma 4.2.By this we get that H is a Z-sum of H up to equivalence.Lemma 4.2.For a family of k-hypergraphs G and k-hypergraphs G, G 1 . . .G l , all of the same dimension, If G 1 . . .G l ∈ Z-Sums(Eq(G)) and G ∈ Z-Sums(Eq({G 1 . . .G l })) then G ∈ Z-Sums(Eq(G)).

Z-Sums(Eq(Z-Sums(Eq(G)))) = Z-Sums(Eq(G)).)
Proof.This is simply because of three trivial facts: (1) Eq(G i + G j ) ⊆ Eq(G i ) + Eq(G j ) where G i , G j are any two k-hypergraphs of the same dimension, and the second plus is the Minkowski sum.(2) Z-Sums(F) = Z-Sums(Z-Sums(F)), for any F a family of k-hypergraphs of the same dimension.
(3) Eq(F) = Eq(Eq(F)), for any F a family of k-hypergraphs of the same dimension.Now,

Z-Sums(Eq(H)).
The inclusion in the opposite direction is trivial.We do not specify names of vertices, as we will consider these graphs up to equivalence.We call both types of graphs simple graphs. Let and Proof.Let V be the set of vertices of the hypergraph H.The proof of the lemma is done in steps.
We will use it to further simplify our problem, as H ∈ Z-Sums(Eq(S

and only if, G + H ∈ Z-Sums(Eq(S •
H ∪ S •−• H )). Proof of the claim.Let α, α be two vertices not in V and let F ⊆ Eq(S • H ) be the family of all S • a simple graphs (depicted below) where a = Λ β (H) for β ∈ V .
where the second equality reflects the fact that every edge has two ends and the last equality is due to Claim 4.1.As Λ x is a homomorphism we get that Λ x (H + G) = 0 for every x ∈ V ∪ {α, α }.
Because of the previous claim w.l.o.g we may restrict our self to the following case.
Claim 4.6.We assume that H has the following property for every β ∈ V , it holds that Λ β (H) = 0.
Here, there is one issue that should be discussed.Suppose H = H + G as in Claim 4.5.The issue is that S ).Thus, we do not loose generality because of the proposed restriction.
We will use it to further simplify our problem, as H ∈ Z-Sums(Eq(S The main property of the sequence H i is that H i > H i+1 in some well-founded quasi-order on graphs.Suppose that, there is ALG an algorithm, that takes as an input H i−1 and produces the next G i ∈ Eq(S •−• H ). The precondition of the algorithm ALG is that H i−1 has more than 3 nonisolated vertices.Now, due to the well-foundedness of the quasi order, the sequence G i is finite.Further, the graph H last has at most 3 nonisolated vertices because of the precondition of ALG.To this end, we need to define the order on graphs and provide the algorithm ALG.
Order on graphs.We assume an arbitrary total order < on vertices V .We lift the order to an order on edges {α, β}.We define it as the lexicographic order on pairs (α, β) satisfying α > β.Finally, the order is extended to a quasi-order on graphs: G < G iff e < e , where e and e are the largest edges in G and G , respectively.
The algorithm ALG.Suppose {α, β} is the largest edge in the graph H i and that H i has at least 4 nonisolated vertices.Observe that Λ α (H and by the definition of edge simple graphs Λ ε (S •−• H ) = 0 for every vertex ε.As a consequence there must be at least one vertex γ ∈ {α, β} such that (α, γ) is an edge in H i .As H i has at least 4 nonisolated vertices, there is δ ∈ {γ, α, β} a nonisolated vertex in H i .We define G i+1 as follows Observe that edges (α, γ), (β, δ), (γ, δ) are smaller than the edge (α, β) and After usage of the claim we get a graph H def = H+G .We know that H ∈ Z-Sums(Eq(S The graph H has at most 3 nonisolated vertices, Λ ∅ (H ) = 0, and Λ α (H ) = 0 for any vertex α.

Representation of H-simple graphs
As the second step, we prove that every H-simple graph is representable as a Z-sum of H up to equivalence (Lemma 4.10).We start with two preparatory lemmas.Proof.Let V be the set of vertices of the graph G .Suppose e = {α, β} ⊆ V , Λ e (G) = a, and let γ, δ / ∈ V be two additional fresh vertices outside of V .We consider three graphs G 1 , G 2 , G 12 which differ from G only by: • replacing α with γ (in case of G 1 ), • replacing β with δ (in case of G 2 ), • replacing both α and β with γ and δ, respectively (in case of G 12 ).Clearly all the three graphs are equivalent to G. We claim that G − G 1 − G 2 + G 12 yields: that is a graph equivalent to S •−• a .Indeed, the above graph operations cancel out all edges nonadjacent to the four vertices α, β, γ, δ, as well as all edges adjacent to only one of them.The two horizontal a-weighted edges originate from G and G 12 , while the two remaining vertical ones originate from −G 1 and −G 2 .Lemma 4.9.Let α be a vertex of a graph G, and let Λ α (G) = a.Then S • a ∈ Z-Sums(Eq({G})).Proof.Let V be a set of vertices of the graph G. Suppose a graph G differ from G only by replacing α with a fresh vertex β / ∈ V .The graph G − G has the following shape (with blue square vertices representing the set V \ {α} = {γ 1 , γ 2 . . .γ n }):

− vn
Relying on Lemma 4.8, we add to the graph G − G the following graphs equivalent to This results in collapsing all blue vertices into one: Since a = v 1 + . . .+ v n , the resulting graph is equivalent to S • a , as required.Lemma 4.10.Suppose H is locally a Z-sum of H, then every H-simple graph is a Z-sum of H up to equivalence: S • H ⊆ Z-Sums(Eq(H})) and S •−• H ⊆ Z-Sums(Eq(H})).
Proof.We have to prove that for any a ∈ V H (V H is defined in Equation 4.6) it holds that S • a ∈ Z-Sums(Eq(H)) and that for any b ∈ E H (E H is defined in Equation 4.5) it holds that S •−• b ∈ Z-Sums(Eq(H)).We prove only S • a ∈ Z-Sums(Eq(H)) as the second proof is the almost same.Because of Lemma 4.2 we know that

Z-Sums(Eq(Z-Sums(Eq(H)))) = Z-Sums(Eq(H)).
Thus, because of Lemma 4.9 it is sufficient to prove that there is a graph G a ∈ Z-Sums(Eq(H)) such that Λ α (G a ) = a for some α ∈ V ert(G a ).
As a ∈ V H we know that where z i ∈ Z and a i is Λ β i (H) for some β i ∈ V ert(H).By the assumption (4.2), for every i we know we may use the equivalence relation and rewrite it as follows Λ α is a homomorphism, so so we define The construction of elements of S •−• H relays on Lemma 4.8 instead of Lemma 4.9.
Combining Lemmas 4.4 and 4.10 we know that H is a Z-sum of H-simple graphs, each of which in turn is a Z-sum of H up to equivalence.By Lemma 4.2, H is a Z-sum of H up to equivalence, as required.The proof of Theorem 3.4, for graphs, is thus completed.Now, we are ready to prove the implication 2 =⇒ 1 from Theorem 3.4 in full generality.
The proof is split into five parts.We want to mimic the approach presented in Section 4, thus we need to generalise a few things: (1) In Section 6 we build an algebraic background to be able to solve systems of equations that in the general case correspond to Equations 4.7.
(2) In Section 7 we introduce simple hypergraphs that generalise graphs defined in Definition 4.3.(3) In Section 8 we show that our target hypergraph is a Z-sum of simple hypergraphs.This generalises Lemma 4.4.(4) In Section 9 we prove that simple hypergraphs are Z-sums of Eq(H).This generalises Subsection Representation of H-simple graphs i.e.Lemmas 4.8, 4.9, and 4.10.(5) Finally, in Section 10 we complete the proof of Theorem 3.4.

Reduction matrices.
In this section we introduce a notion of reduction matrices and prove a key lemma about their rank, Lemma 6.3.They are 0, 1 matrices related to adjacency matrices of Kneser graphs.We recall that by x-set we mean a set with x elements.(1) 4, 3, 1 = up to a permutation of rows and columns.
The below lemma expresses the important property of reduction matrices.
The proof relies on a result from the spectral theory of Kneser graphs.
Definition 6.4.Let A be an a-set.The Kneser graph K a,c is the graph whose vertices are c-element subsets of A, and where two vertices x, y are adjacent if, and only if, x ∩ y = ∅.
The adjacency matrix of a graph (X , E) is a X × X -matrix M such that M [x, y] = 1 if there is an edge between x and y and 0 otherwise.Theorem 6.5 [GR01, Theorem 9.4.3].All eigenvalues of the adjacency matrix for a Kneser graph are nonzero.
Corollary 6.6.The rank of the adjacency matrix for a Kneser graph is maximal.
Proof of Lemma 6.3.We relabel columns of 2k + 1, k + 1, k in the following way.If a column is labelled with the set B ⊂ A we relabel it to A \ B. Now, both rows and columns are labelled with k-subsets of A. We denote the relabelled matrix by M .On the one hand, observe that if a k-set C is a subset of a (k + 1)-set and M [C, A \ B] = 0.So we see that M is the adjacency matrix of the Kneser graph K 2k+1,k .But , because of Corollary 6.6, it has maximal rank, and the same holds for M as relabelling of columns does not change the rank of the matrix.7. Simple hypergraphs.
Definition 7.1.Let H be a k-hypergraph and V be the set of its vertices.We say that it is m-isolated if for any subset X ⊆ V such that |X | ≤ m it holds that Λ X (H) = 0. Definition 7.2.Let G be a k-hypergraph and V be its set of vertices.Suppose 0 ≤ m ≤ k and a ∈ Z d .We call G (m, a)-simple if there exist pairwise disjoint sets A, B, C such that This definition is hard so we analyse it using examples, and explain the required properties.Also, observe that contrary to S • a and S •−• a (m, a)-simple hypergraphs are not fully specified up to isomorphism.Further, it is not clear that they exist for all (m, a).In the example below we comment on this as well.the Property 4 is responsible for edges with nonzero weights and 5 for the edges with the weight 0. Property 6 says that the weight of ∅ and weights of single vertices are 0.
Remark 7.4.Note that (m, a)-simple hypergraphs are not defined uniquely.For example, the (0, a)-simple hypergraph from the example above is not fully defined, as for any x and y we may choose appropriate z = a − x − y.The existence of all simple hypergraphs is proven later (we show how to construct them).
In the following we justify our design.The most important property of simple hypergraph is the last one.It implies the following lemma: Lemma 7.5.Let G be a k-hypergraph and S m a be an (m, a)-simple k-hypergraph.Then for any a ).Proof.Indeed, Λ X is a homomorphism and Λ X (S m a ) = 0 because of Property 6 of Definition 7.2.Property 6, by this lemma, allows for the following structure of the proof of Theorem 3.4.We gradually simplify the target hypergraph by adding (i, a)-simple hypergraphs for growing i.In the step i we start from an (i − 1)-isolated hypergraph.Using (i, a)-simple hypergraphs we simplify it to a hypergraph with Λ X (H) = 0 for all X ⊂ V with exactly i vertices.The property 6 guaranties that while we perform the step i, we do not ruin our work from previous steps i.e. we reach an i-isolated hypergraph.Eventually, we reach a hypergraph with all the weights equal to 0 i.e. the empty hypergraph.Now, let us justify the design of Properties 4 and 5.We already mentioned that using (i, a)-simple hypergraphs we want to reduce to 0 all weights of sets in [V ] i ; thus it is good to keep weights on the level i as simple as possible i.e. a, 0, −a.
The designs of Properties 1-3 is a trade-off between two things: • we want to have simple hypergraphs as small as possible in terms of number of vertices and number of nonzero hyperedges, • for the proof of Theorem 3.4, we need that simple hypergraphs, from which we construct the target hypergraph H, may be constructed from hypergraphs in the family H.Now, that Properties 1-6 are commented, we may go back to the proof of the Theorem 3.4.We introduce two other notions to work with families of simple hypergraphs.
Definition 7.6.Let G be a family of k-hypergraphs, for every G ∈ G we denote the set of its vertices by V G .A family of k-hypergraphs S is a simplification of the family G if for every 0 ≤ m ≤ k and every Definition 7.7.The family S is self-simplified if S is a simplification of S. For a family of k-hypergraphs H, a family S is H-simplified if S is self-simplified and is a simplification of H.
Remark 7.8.To produce a H-simplified family, it suffices to design an algorithm alg(S) that produces a simplification of its input.The following family is H-simplified i∈N alg i+1 (H) where alg i is i-th iteration of the alg.
Note that the produced family is not necessarily finite.
Example 7.9.The picture below presents an example of an {H}-simplified family G. z ∈ Z and the sets A, B, C are marked with colours.Like in Definition 7.6 for all a in Z-Sums({x The family is infinite.Depicted three types of graphs represent (0, z(x 1 +x 2 +x 3 ))-simple hypergraphs, (1, a)-simple hypergraphs, (2, y)-simple hypergraphs.To see that the family is self-simplified we observe the few following facts: Indeed, Λ ∅ (G) = 0 for G of the second or the third type of graphs.So the family contains all required (0, )-simple graphs (triangles).
for any G of the third type of graphs and any α ∈ V G .
-Λ α (G) = 0 for G of the second type of graphs and α being the middle vertex.
. This is also easy to check.
Lemma 7.10.Let H be a k-hypergraph and S be an H-simplified family.Suppose G ∈ Z-Sums(Eq(S)) then the family S is {H + G}-simplified.
Proof.As S is self-simplified, we only need prove that S is a simplification of {H + G}.Let We have to prove that S contains an (m, g)-simple hypergraph.Suppose that Observe that it is sufficient to prove that S contains an (m, a i )-simple hypergraph for each a i .This is because S is self-simplified, precisely.If in S there are (m, a 1 )-simple and (m, a 2 )-simple hypergraphs then The fact that (m, a i )-simple hypergraphs are elements of S is easy.Suppose that a i = Λ X (H + G).Observe that Λ X (H + G) = Λ X (H) + Λ X (G).One more time, as S is self-simplified it is sufficient to show that in S there are (m, Λ X (H))-simple and (m, Λ X (G))-simple hypergraphs.The first one is in S as S is H-simplified.To show that the second is an element of S we have to use the fact that G ∈ Z-Sums(Eq(S)), then

Expressing H with simple hypergraphs
Our goal in this section is to prove Theorem 8.2.Definition 8.1.Let H ∈ Z-Sums(Eq(H)).We say that V supports H for the family H if there is a solution to the following equation One should think that V includes all vertices of the elements of the sum.Theorem 8.2.Let H be a k-hypergraph and V be its set of vertices.Further, let S be an {H}-simplified family of hypergraphs.Then H ∈ Z-Sums(Eq(S)).Moreover, if |V | > 2k − 1 then V supports H for the family S.
The proof of this theorem requires a few definitions and lemmas stated below.We start with them and then we prove the theorem while proofs of lemmas are postponed.
Let us recall (Definitiont 7.1) that, a k-hypergraph H is m-isolated if for any subset then H is equivalent to the empty hypergraph i.e. it is a union of isolated vertices.Definition 8.4.Let H be a k-hypergraph and V be its set of vertices.We say that H is pre m-isolated if the following two conditions are satisfied: Lemma 8.6.Let H be an m-isolated k-hypergraph and V be its set of vertices.Suppose S is an {H}-simplified family of hypergraphs.Then there is a hypergraph G ∈ Z-Sums(Eq(S)), such that Proof of Theorem 8.2.We assume that |V | > 2k − 1, as otherwise we extend H with a few isolated vertices.Suppose we have a sequence of hypergraphs G i ∈ Z-Sums(Eq(S)) which satisfies following properties.
• For every j ≤ k the hypergraph H − j i=0 G i is j-isolated.• For every i ≤ k the set V supports G i for the family S. We define G as k i=0 G i .Now, observe that • G ∈ Z-Sums(Eq(S)) as each G i is in Z-Sums(Eq(S)).
• V supports G for the family S as V supports G i for the family S for each i ≤ k.
• H = G up to equivalence.Because of the first property we know that H − G is a k-isolated k-hypergraph, and every k-isolated k-hypergraph is the empty hypergraph, because of Remark 8.3.Thus, if we have the sequence G i then we are done.Now, we show how the sequence G i may be constructed.The construction is via induction on i.As S is an {H}-simplified family of hypergraphs then there is G 0 ∈ Eq(S) and V ert(G 0 ) ⊂ V such that H − G 0 is 0-isolated.This creates the induction base.
For the inductive step we reason as follows.First we observe that because of Lemma 7.10 the family S is H − i j=0 G j -simplified.Thus, we may use Lemma 8.6 for the hypergraph H − i j=0 G j and the family S.As a consequence we get G i+1 ∈ Z-Sums(Eq(S)) such that (H − i j=0 G j ) − G i+1 is pre (i + 1)-isolated.Moreover, V supports G i+1 for the family S (the property 2).Further, Lemma 8.5 implies that (H − i j=0 G j ) − G i+1 is (i + 1)-isolated (the property 1).
This ends the inductive step.
Proof of Lemma 8.5.
Before we prove Lemma 8.5 we prove an easy lemma about the Λ X functions.
Proof.Let us recall definition of Λ X (H).It is the sum of all edges e in H such that X ⊆ e.
Let e be a hyperedge in H such that X ⊆ e.It suffices to prove that µ(e) appears the same number of times on both sides of the equation.On the right side e is added k−m l−m times, as µ(e) appears once in the expression Λ X (H).On the left side the number of times when µ(e) is added is equal to the number of l-element supersets of X that are included in e.
Proof of Lemma 8.5.Let X be a set of vertices as in Definition 8.4.We need to prove that for any X ⊆ X such that |X | = m it holds that Λ X (H) = 0.
Let Y 1 , Y 2 . . .Y n be all the (m − 1)-subsets of X and let X 1 , X 2 . . .X n be all the m-subsets of X .
Because of Lemma 8.7, we know that, for any Y i , the equation This system of equation may be rewritten in matrix form Cu = 0 where m−1 rows indexed with (m − 1)-element subsets of the set X , such that each individual entry represents inclusion between the index of the row and the index of the column.So up to permutation of rows and columns C is the 2m − 1, m, m − 1 matrix ( •, •, • is defined in Definition 6.1).
But according to Lemma 6.3 the rank of the matrix C is maximal, which implies u = 0 is the only solution of the system of equations.Thus, Λ X j (H) = 0 for any j ≤ n and consequently H is m-isolated.
Proof of Lemma 8.6.
The proof of Lemma 8.6 requires some preparation.
The above operation is called X -cut of F. For X ∩ W = ∅ we define the reverse operation called enriching F with X .It is denoted by F |+X and its effect is the minimal in the sense of inclusion 1 (k + |X |)-hypergraph F such that F |−X = F. 1 we say that a hypergraph includes a second hypergraph if the set of vertices of the first hypergraph includes the set of vertices of the second one and every hyperedge of the second hypergraph is also a hyperedge of the first hypergraph.Proof.In the equation below, e and e are indexes, while W, Y and X are fixed.
In the proof, instead of constructing G directly, we build a finite sequence of hypergraphs H 0 . . .H last , such that: of {G} such that S ⊆ Z-Sums(Eq(G)).Existence of such an algorithm is a consequence of the following lemmas.Lemma 9.2.Let G be a k-hypergraph and 0 ≤ m ≤ k.Suppose there are (m, a)-simple and (m, b)-simple hypergraphs that are elements of Z-Sums(Eq(G)).Then (m, a + b)-simple hypergraph is also a member of Z-Sums(Eq(G)).
Lemma 9.3.Let G be a k-hypergraph and V be its set of vertices.Suppose, X ∈ [V ] m where m ≤ k.Let a = Λ X (G).Then there is a (m, a)-simple k-hypergraph S m a such that S m a ∈ Z-Sums(Eq(G)).Indeed, using this two lemmas we can produce all elements in the simplification.Suppose, m ≤ k and a ∈ Z-Sums({Λ X (G) | X ∈ [V ] m }).We have to show that (m, a)-simple hypergraph is in Z-Sums(Eq(G)).Suppose, a = c 1 a 1 + c 2 a 2 . . .c l a l where a i ∈ {Λ X (G) | X ∈ [V ] m } and c i ∈ Z.Because of Lemma 9.2 it suffices to prove that (m, a i )-simple hypergraphs are in Z-Sums(Eq(G)).But this is exactly Lemma 9.3.So it remains to provide proofs of lemmas 9.2 and 9.3.The proof of Lemma 9.2 is easy so we start from it and then we concentrate on the more complicated proof of Lemma 9.3.
We may proceed to the proof of Lemma 9.3.We start with an operator that is used in the following proofs.Definition 9.4.Suppose G is a k-hypergraph and α ∈ V ert(G), α / ∈ V ert(G) are two vertices.Let σ α : V ert(G) ∪ {α } → V ert(G) ∪ {α } such that: The proof of Lemma 9.3 is by induction on k.We encapsulate the most important steps of the proof in four lemmas.Lemma 9.5 is an auxiliary lemma.Lemma 9.6 forms the induction base.Lemmas 9.7 and 9.8 cover the induction step.• Property 4. X contains exactly one vertex from every pair α i , β i for 0 < i ≤ m + 1.Here, there are two cases: α ∈ X or α ∈ X .We consider only one of them as the second one is similar.Suppose that α ∈ X .Then Λ |X ∩B| a, as required.• Property 5. X has m + 1 elements but it is not one of the sets considered in Property 4.
-The third case is almost the same as second. - π i is the identity on α, α , so.

Vol. 18:4
To prove Point 2 it suffices to prove that for each i, it holds that where G = τ {α,α } (G).We simplify further To prove Equation 9.2 we have to show that for every k for every potential hyperedge) the weight of that subset on both sides of Equation 9.2 is the same.
, so we formalise this as a following equation.For any We consider 4 cases depending on whether α, α ∈ X .
• If X does not contain α and α then Λ X (G , so the equality holds.• The case where X contains only α is similar.
• If X contains both α and α then 0 So both sides of Equation 9.3 are 0. Thus, Equality 9.2 holds for every X .
. Indeed, it is a hypergraph with only one nonisolated vertex α and Λ Proof of Lemma 9.7.We show how to construct the hypergraph . Now, because of Lemma 9.5, for some α / ∈ V ert(G), the 11:29 Lemma 9.8.Let k ∈ N. Suppose that Theorem 9.1 holds if restricted to k-hypergraphs.Let G be a (k + 1)-hypergraph and Λ ∅ (G) = a.Then there is a (0, a)-simple (k + 1)-hypergraph S 0 a such that S 0 a ∈ Z-Sums(Eq(G)).Proof of Lemma 9.8.First, observe that S 0 a has to satisfy only two properties: it has 2k + 1 vertices and the sum of weights of all hyperedges equals a.
Thus, the lemma is a consequence of a procedure (defined below) that takes a (k + 1)hypergraph F with l vertices and if l > 2k + 1 then it returns a (k + 1)-hypergraph F such that Λ ∅ (F) = Λ ∅ (F ), |V ert(F )| < l, and F ∈ Z-Sums(Eq(F)).We start from a hypergraph G.We apply the procedure until we produce a hypergraph with no more than than 2k + 1 vertices.It is the required simple hypergraph.We are guaranteed to finish, as with each application the number of vertices decreases.Moreover, each application of the procedure does not change Λ ∅ , so we know that Λ ∅ of the produced graph is equal to a, as required.
We define O as follows.Let S be an {F |−α }-simplified family of hypergraphs.By Ŝ we denote the set of all k-hypergraphs S such that V ert(S) = V ert(F) \ {α} and S ∈ Eq(S).Because of Theorem 8.2 we know that F |−α ∈ Z-Sums(Eq(S)) is supported by V ert(F) \ {α} i.e. ) is in Z-Sums(Eq(F)).
Before we prove the theorem, let us recall its statement.Theorem 2.2 For every fixed arity k ∈ N, the N-solvability problem is in NExp-Time.
We prove Theorem 2.2 by showing that in NExp-Time it is possible to reduce Nsolvability to Z-solvability.The produced instance of Z-solvability is of exponential size, and can be solved in Exp-Time because of Theorem 2.1.
Before we start, we need to recall some facts about solution of systems of linear equations.
Hybrid linear sets.
Definition 11.1.A set of vectors is called hybrid linear if it is the smallest set that includes a finite set B, called a base, and that is closed under the addition of elements from a finite set P, called periods.In the paper [Pot91], Pottier provides bounds on the norms of B and P. We present these bounds next.
For a vector v ∈ Z m we introduce two norms: the infinity norm v ∞ and the norm one v 1 , defined as follows: Lemma 11.5.Let U 1 and U 2 be two finite sets of vectors in Z d such that every vector in U 2 is nonreversible in U = U 1 ∪ U 2 .Suppose y ∈ N-Sums(U).For any solution: The main part of the proof of Theorem 2.2.
The idea of this reduction is as follows.We use I for a finite family of data vectors in [D] k − → Z d and v for a target vector.Similarly to Definition 11.4 reversibility can be defined for data vectors.Namely, a data vector y is reversible in a set of data vectors I if −y ∈ N-Sums(Perm(I)).
The schema of the proof may be followed on the diagram in Figure 3. Next, we define a homomorphism from data vectors [D] k → Z d to vectors in Z d .The homomorphism is

Figure 1 :
Figure 1: The graph representing the data vector v αβγ .

Figure 2 :
Figure 2: Idea behind the construction in Example 1.2.
Further, for a data permutation Vol.18:4 LINEAR EQUATIONS FOR UNORDERED DATA VECTORS IN [D] k → Z d .11:7 π : D − → D by v • π we mean the data vector defined as follows hypergraph Z-sum input: A finite set H of hypergraphs and a target hypergraph H, all of the same arity and dimension.output: Is H a Z-sum of H up to equivalence?Lemma 3.2.The Z-solvability problem reduces in logarithmic space to the hypergraph Z-sum problem.The reduction preserves the arity and dimension.

Definition 4. 3 .
For every vector a ∈ Z d we define an a-edge simple graph S •−• a (shown on the left) and an a-vertex simple graph S • a (shown on the right):

Definition 6. 1 .
Let a ∈ N and A be an a-set.For a ≥ b ≥ c, a, b, c ∈ N, we define a matrix a, b, c as follows: (1) it has a b columns and a c rows, (2) columns and rows are indexed with b-element subsets of A and c-element subsets of A, respectively, (3) a, b, c [C, B] = 1 if C ⊆ B and 0 otherwise, where C is an index of a row and B is an index of a column.The definition is up to reordering of rows and columns.Example 6.2.Suppose A = {α, β, γ, δ}.
Proof of Lemma 9.2.Let S m a ∈ Z-Sums(Eq(G)) be the (m, a)−simple hypergraph and S m b ∈ Z-Sums(Eq(G)) be the (m, b)−simple hypergraph.We can write S m a = i a i G i where a i ∈ Z and G i ∈ Eq(G) S m b = j b j G j where b j ∈ Z and G j ∈ Eq(G) We recall the notation used in Definition 7.2 of simple hypergraphs; the vertices of S m a can be split into A S m a , B S m a , C S m a and similarly vertices of S m b are in A S m b , B S m b , C S m b .According to Definition 7.2, the elements of sets A S m a , B S m a are paired, formally there is a bijection f : A S m a → B S m a , similarly elements of sets A S m b and B S m b are paired, formally there is a bijection f : A S m b → B S m b .Then there is a bijection π between vertices that transfers A S m a to A S m b , B S m a to B S m b , C S m a to C S m b , and that it preserves the pairings i.e. π(f Lemma 9.5.Let G be a k-hypergraph and S m a be an (m, a)-simple k-hypergraph.Suppose, α, α / ∈ V ert(S m a ) are two vertices and S m a ∈ Z-Sums(Eq(G |−α )).Then: (1) S m+1 a = S m a |+α − S m a |+α is an (m + 1, a)-simple (k + 1)-hypergraph, (2) S m+1 a ∈ Z-Sums(Eq(G)).Proof.We start by showing Point 1.Let V ert(S m a ) = A ∪ B ∪ C, where A, B, C are as in the definition of (m, a)-simple hypergraph (Definition 7.2).Let A = {α 1 , α 2 . . .α m } and B = {β 1 , β 2 . . .β m }.We define A def = {α 1 , α 2 . . .α m , α m+1 = α} and B def = {β 1 , β 2 . . .β m , β m+1 = α }.To show that S m+1 a is (m + 1, a)-simple we split V ert(S m a ) ∪ {α, α } to A , B , and C and we verify Properties 1 to 7. Properties 1 to 3 are trivial.Properties 4 to 6 speak about function Λ X (S m+1 a ) where: Suppose π i are bijections such that π i are identity on {α, α } and G i = G |−α • π i .The bijections π i are well defined as sets of vertices of G |−α and G i do not contain neither α nor α .Because of Point 1, we may write the following equation S m+1 a Lemma 9.6[HLT17, Th 15].Let G = (V , µ ) be a 1-hypergraph.Then for any X ⊆V and |X | ≤ 1 there is a (|X |, Λ X (G))-simple 1-hypergraph S |X | Λ X (G) , such that S |X | Λ X (G) ∈ Z-Sums(Eq(G)).Proof.We consider two cases (i) |X | = 1 and (ii) |X | = 0.The first case.Let X = {α}, a = Λ α (G), and α / ∈ V .We define S • a def = G − τ {α,α } (G).For any vertex β = α, α we have Λ β (S • a ) = 0.So S • a has two nonisolated vertices {α, α } and satisfies all properties of a (1, a)-simple 1-hypergraph.The second case.Let Λ ∅

F
|−α = i a i S *i where a i ∈ Z andS * i ∈ α i is any vertex in the set V ert(F)\(V ert(S * i )∪{α}) .Observe, that |V ert(S * i )∪ {α}| ≤ 2k + 1 < |V ert(F)| as |V ert(F)| > 2k + 1 , so α i can be always picked.O satisfies the required properties: (1) O ∈ Z-Sums(Eq(F)).It is sufficient to show that S * i|+α − S * i|+α i ∈ Z-Sums(Eq(F)).Because of the assumption that Theorem 9.1 holds for k-hypergraphs we know that S * i ∈ Z-Sums(Eq(F |−α )).If we combine this with Lemma 9.5 (point 2) then we get that each hypergraph (S * i|+α − S * i|+α i Theorem 11.2 [Pot91].Let M be a d × m-matrix with integer entries and y ∈ Z d .The set of nonnegative integer solutions of linear equations M • x = y is a hybrid linear set.The base B and periods P are as follows: • Base: it is the set of minimal, in the pointwise sense, solutions of M • x = y.• Periods: is the set of minimal nontrivial solutions of M • x = 0. (11.1)

Figure 3 :
Figure 3: The diagram of the proof of Theorem 2.2.
where a v , b w ∈ N, it holds thatw∈U 2 b w ≤ |U 2 | • ( U 1,∞ + y ∞ + 2) d+|U | i.e.w∈U 2 b w is bounded exponentially.Proof of Lemma 11.5.The set of solutions of Equation 11.2 is hybrid linear, and given by some B, P ⊂ N |U 1 ∪U 2 | .Every period is a solution of the equation0 = v∈U 1 a v • v + w∈U 2 b w • w. (11.3)From the definition of reversibility we get that in any solution of Equation11.3 all b w are equal to 0. Thus, in every solution of Equation 11.2, the sumw∈U 2 b w is bounded by |U 2 | • B ∞ ≤ |U 2 | • ( U 1,∞ + y ∞ + 2) d+|U |, where the last inequality is given by Lemma 11.3.
Proof of the claim.We construct G gradually as a sum of a sequence of graphs G 1 , G 2 . . .G last .The sequence G i is constructed in parallel with a sequence H i where H 0 = H and