Parameterized Complexity of Weighted Team Definability

In this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analogue of one of the most studied problems in parameterized complexity, the notion of weighted Fagin-definability, which is formulated in terms of satisfaction of first-order formulas with free relation variables. We focus on the parameterized complexity of weighted team definability for a fixed formula phi of central team-based logics. Given a first-order structure A and the parameter value k as input, the question is to determine whether A,T models phi for some team T of size k. We show several results on the complexity of this problem for dependence, independence, and inclusion logic formulas. Moreover, we also relate the complexity of weighted team definability to the complexity classes in the well-known W-hierarchy as well as paraNP.


Introduction
In this article, we study the parameterized complexity of weighted team definability for logics in team semantics.Team definability is a natural analogue of the notion of Fagin-definability whose weighted version can be used to characterize the W-hierarchy in parameterized complexity [2].We give several results on the complexity of this problem for dependence, independence, and inclusion logic formulas.The birth of the nowadays established logics of dependence and independence can be traced back to the introduction of dependence logic in 2007 [24].In team semantics, formulas are interpreted by sets of assignments (teams) instead of a single assignment as in Tarski's semantics of first-order logic.Syntactically dependence logic extends first-order logic by new dependence atomic formulas (dependence atoms) dep(x; y) expressing that the values of variables x functionally determine the value of the variable y in the team under consideration.Independence and inclusion logics are further extensions of first-order logic by independence atoms x⊥ z y and inclusion atoms x ⊆ y which essentially correspond to embedded multivalued dependences and inclusion dependences from database theory [9,7].
For the applications, it is important to understand the complexity theoretic aspects of team-based logics.During the past ten years, the expressivity and complexity theoretic aspects of logics in first-order (also propositional [26], modal [15,16], temporal [10] and probabilistic [3]) team semantics have been studies extensively (see, e.g., [14,19,13,4]).The baseline for these studies are the well-known results stating that the sentences of dependence logic and independence logic are equivalent to existential second-order logic while inclusion logic corresponds to positive greatest fixed point logic and thereby captures P over finite (ordered) structures [8].In team semantics results for sentences of the logic do not immediately extend to open formulas.In particular, the open formulas of dependence logic correspond in expressive power to sentences of ESO with an extra relation encoding the team that occurs only negatively in the sentence [18].For independence logic, the requirement of negativity can be lifted [7].For inclusion logic an analogous result shows that any first-order sentence ϕ(R) whose truth is preserved under R-unions can be expressed by an inclusion logic formula ϕ * (x).In other words, for all A and teams T = ∅: where rel(T ) is a relation encoding the team T [8].These result can be used to relate weighted team definability to weighted Fagin-definablity.However, it is instructive to note that, due to higher expressive power of the logics considered in this article, the syntactic complexity of a formula does not in general correlate with the complexity of the model-checking of the formula.In particular, any formula of dependence and independence logic is logically equivalent to a formula with ∀∃-quantifier prefix [24,Theorem 6.15] [18,Theorem 4.9].
A formalism to enhance the understanding of the inherent intractability of computational problems is brought by the framework of parameterized complexity [1].Here, one aims to find parameters relevant for practice allowing to solve the problem by algorithms running in time f (k) • n O (1) , for some computable function f , where k is the parameter value and n is the input length.Problems with such a running time are called fixed-parameter tractable (FPT) and correspond to efficient computation in the parameterized setting.The problems solvable within the runtimes of the form f (k) • n O (1) with respect to nondeterministic machines belong to the complexity class paraNP ⊇ FPT.Moreover, restricting the amount of nondeterminism allows to study a subclass W[P] ⊆ paraNP.The complexity class W[P] is defined via nondeterministic machines that have at most h(k) • log n many nondeterministic steps, where h is a computable function.In between FPT and W[P], a presumably infinite W-hierarchy is contained: It is unknown whether any of these inclusions is strict.Showing W[1]-hardness of a problem intuitively corresponds to being intractable in the parameterized world.

Our contributions
We define and study the parameterized complexity of weighted team definability with respect to formulas of several team-based logics.Moreover, we establish the relationship between our framework and the problem of weighted Fagin definability.In more details, we explore the complexity of weighted team definability in parameterized setting for dependence, independence and inclusion logic formulas as well as sentences.Thereby, we prove and obtain novel logical characterizations of, and new complete problems for, the aforementioned central parameterized complexity classes, i.e., the W-hierarchy, W[P], and paraNP.Table 1 gives a partial overview of our results concerning weighted team definability.

Related work
The complexity of counting/enumerating satisfying teams for a fixed firstorder formula of team-based logic has been studied before [11,12].Furthermore, there are also recent works on the parameterized complexity model-checking and satisfiability for propositional and first-order team-based logics [22,20,21,17].Regarding the descriptive complexity, Downey et al. [2] explored the logical characterization of the classes in the W-hierarchy.

Preliminaries
We require a basic knowledge of standard notions from classical complexity theory [23].The classical complexity classes we encounter mostly in this work are P and NP together with their respective completeness notions, employing polynomial time many-one reductions (≤ P m ).Moreover, we assume the reader is familiar with the basic first-order (predicate) logic [5].In the following, we define a few important classes of first-order formulas which are relevant to the results in this work.

FO-Formula Classes
The class of all first-order formulas is denoted by FO.Let τ be a relational vocabulary and R ∈ τ be a relation symbol of arity r.An atomic formula is a formula of the form x = y or R(x 1 , . . ., x r ).A literal is an atomic or a negated atomic formula.A quantifier-free formula is a formula that contains no quantifiers and a formula is in negation normal form (NNF) if the negation symbols occurs only front of atoms.A formula ϕ is in prenex normal form if ϕ has the form Q 1 x 1 . . .Q n x n ψ, where ψ is quantifier free and Q 1 , . . ., Q n ∈ {∃, ∀}.The classes Σ 0 and Π 0 both consist of quantifier free formulas.Then, for t ≥ 0, the class Σ t+1 includes all formulas of the form ∃x 1 . . .∃x ϕ, where ϕ ∈ Π t .Similarly, Π t+1 includes all formulas of the form ∀x 1 . . .∀x ϕ, where ϕ ∈ Σ t .

Fagin Definability
The first-order variables range over individual elements of the universe.In second-order logic, one also quantifies relation variables which range over relations on the universe.We now introduce first-order formulas where we also allow relation variables.Let τ be a vocabulary, X i for i ≤ n be free relation variables of arity s i and ϕ(X 1 , . . ., X n ) be a FO-formula in τ .Moreover, let A be a τ -structure and S i ⊆ A si be relations over A for i ≤ n.Then we say that the tuple S = (S 1 , . . ., S n ) is a solution for ϕ in A if A |= ϕ( S).We call the following decision problem, the problem Fagin-defined by ϕ.

Input:
A τ -structure A. Question: Is there a solution for ϕ in A?
Let Θ ⊆ F O be a class of formulas, then by FD-Θ we denote the class of all problems FD ϕ such that ϕ ∈ Θ.The following result regarding FO is known.
Next we introduce the following weighted version of Fagin definabilty, where we restrict our solution to have a specific size for a single free relation symbol S of arity s.

Input:
A τ -structure A and k ∈ N. Question: Is there a solution for ϕ of cardinality k?
As before, for a class Θ ⊆ F O of formulas, we denote by WD-Θ the class of all problems WD ϕ such that ϕ ∈ Θ.
Example 2. The problem Clique is defined as follows.Given a graph G := (V, E) and k ∈ N. Is there a set S ⊆ V such that |S| = k and (u, v) ∈ E for every x, y ∈ S? Then Clique is WD ϕc , where Moreover, Let DominatingSet be the problem to determine if a graph G contains a set S ⊆ V such that |S| = k and every vertex in V \ S is incident to some vertex in S? Then DominatingSet is in WD-Π 2 since the problem is WD ϕ d , where ϕ d (X) := ∀x∃y X(y) ∧ (E(x, y) ∨ x = y) .
Parameterized Complexity Theory A parameterized problem (PP) P ⊆ Σ * × N is a subset of the crossproduct of an alphabet and the natural numbers.For an instance (x, k) ∈ Σ * × N, k is called the (value of the) parameter.A parameterization is a polynomial-time computable function that maps a value from x ∈ Σ * to its corresponding k ∈ N. The problem P is said to be fixed-parameter tractable (or in the class FPT) if there exists a deterministic algorithm A and a computable function f such that for all (x, k) ∈ Σ * × N, algorithm A correctly decides the membership of (x, k) ∈ P and runs in time f (k) • |x| O (1) .The problem P belongs to the class XP if A runs in time |x| f (k) on a deterministic machine.Abusing a little bit of notation, we write C-machine for the type of machines that decide languages in the class C, and we will say a function f is C-computable if it can be computed by a machine on which the resource bounds of the class C are imposed.The class paraNP includes problems decidable by a nondeterministic algorithm A which runs in time f (k) • |x| O (1) for some computable function f .One can define a parameterized complexity class paraC corresponding to a complexity class C via a precomputation on the parameter.

Definition 3. Let C be any complexity class. Then paraC is the class of all PPs
Notice that paraP = FPT and the two definitions of paraNP are equivalent.
A problem P is in the complexity class W[P], if it can be decided by a NTM running in time f (k) • |x| O (1) steps, with at most g(k)-many non-deterministic steps, where f, g are computable functions.Moreover, W[P] is contained in the intersection of paraNP and XP (for details see the textbook of Flum and Grohe [6]).
Let c ∈ N and P ⊆ Σ * × N be a PP, then the c-slice of P , written as P c is defined as . Finally, in order to show that a problem P is paraC-hard (for some complexity class C) it is sufficient to prove that for some c ∈ N, the slice P c is C-hard in the classical setting.
To define the complexity classes in W-hierarchy, the parameterized version of the problem WD ϕ is now defined as follows.

Input:
A τ -structure A and k ∈ N. Parameter: k.

Question:
Is there a solution for ϕ of cardinality k?
The complexity classes of the W-hierarchy are characterized via the following definition.

The class W[t] forms the t-th level of the W-hierarchy.
Alternatively, the W-hierarchy can be defined via the weighted satisfiability problem for propositional formulas.Let I be a non-empty index set and d ∈ N. Consider the following special subclasses of propositional formulas: The classes of the W-hierarchy are defined equivalently in terms of these problems.

p-WSAT(Γ
Figure 1 draws the complexity landscape with complete problems in parameterized complexity that are relevant.

Team-based Logics
We assume basic familiarity with predicate logic [5].We consider first-order vocabularies τ that are sets of function symbols and relation symbols with an equality symbol =.Let VAR be a countably infinite set of first-order variables.Terms over τ are defined in the usual way, and the set of well-formed formulas of first-order logic (FO) is defined by the following EBNF: If ψ is a formula, then we use VAR(ψ) for its set of variables, and Fr(ψ) for its set of free variables.We evaluate FO-formulas in τ -structures, which are pairs of the form A = (A, τ A ), where A is the domain of A (when clear from the context, we write A instead of dom(A)), and τ A interprets the function and relational symbols in the usual way (e.g., ). Dependence logic FO(dep) extends FO by dependence atoms of the form dep(t; u) where t and u are tuples of terms.Inclusion logic FO(⊆) in obtained by adding to FO the inclusion atoms of the form t ⊆ u for tuples t and u of terms.Finally, independence logic FO(⊥) extends FO by independence atoms of the form t⊥ v u for tuples t, u and v of terms.We call expressions of the kind t 1 = t 2 , R(t), dep(t; u), t ⊆ u and t⊥ v u atomic formulas.
The semantics is defined through the concept of a team.Let A be a structure and X ⊆ VAR, then an assignment s is a mapping s : X → A.
If s : X → A is an assignment and x ∈ VAR is a variable, then s x a : X ∪ {x} → A is the assignment that maps x to a and y ∈ X \ {x} to s(y).Let T be a team in A with domain X.Then any function f : T → P(A) \ {∅} can be used as a supplementing function of T to extend or modify T to the supplemented team For the case f (s) = A is the constant function we simply write T x A for T x f .The semantics of formulas is defined as follows.
Definition 8. Let τ be a vocabulary, A be a τ -structure and T be a team over A with domain X VAR.Then, For a structure A and a team T over X in A, we let rel(T ) denote the relation defined by T .That is, rel(T ) := { a | s(x) = a, s ∈ T }.Moreover, we say that a formula ϕ is flat if for any team T over Fr(ϕ) we have that A, T |= ϕ if and only if A, {s} |= ϕ for every s ∈ T .The FO-formulas satisfy this flatness property.Notice that, for FO-formulas, by singleton equivalence, team semantics and classical Tarski semantics coincide, i.e., A, {s} |= ϕ if and only if A |= s ϕ.Furthermore, note that A, T |= ϕ for all ϕ when T = ∅ (this is also called the empty team property).Finally, C-formulas for every C ∈ {F O(dep), FO(⊆), FO(⊥)} are local, that is, for a team T in A over domain X and a FO(dep)-formula ϕ, we have that A, T |= ϕ if and only if A, T Fr(ϕ) |= ϕ.
Weighted Team Definability Now we introduce a novel version of the weighted definability problem for formulas in team-based logics.Let C ∈ {FO(dep), FO(⊆), FO(⊥)}, ϕ be a fixed C-formula over free variables Fr(ϕ) and k ∈ N. Then given a structure A, the weighted-team definable problem WT ϕ asks if there is a team of size k for ϕ over Fr(ϕ) in A.

Input:
A τ -structure A and k ∈ N. Question: Is there a team T over Fr(ϕ) such that |T | = k and A, T |= ϕ?
Then the analogous parameterized version of WT ϕ is defined as follows.

Input:
A τ -structure A and k ∈ N. Parameter: k.

Question:
Is there a team T over Fr(ϕ) such that |T | = k and A, T |= ϕ?
Note that the problem WT ϕ references the set of free variables Fr(ϕ) of the formula ϕ.As a consequence, our parameterization is trivial for sentences since there are only two teams ∅ and {∅} with the empty team domain.As before, for a set Θ ⊆ C of formulas, we denote by WT-Θ the class of problems WT ϕ such that ϕ ∈ Θ.

3
Complexity Results for Weighted Team Definability

First-Order Formulas
We begin our study of the complexity for p-WT ϕ in the case ϕ is a pure FO-formula under team semantics.Notice that the consequence of disallowing free relation variables in ϕ is that p-WT ϕ is different than the weighted Fagin definability p-WD ϕ .The following theorem establishes that the two problems are also different from the classical complexity theoretic point of view.Here, we assume basic familiarity about the circuit complexity classes TC 0 and AC 0 (for an introduction into this area, see the textbook of Vollmer [25]).
Theorem 9.For any FO-formula ϕ the problem WT ϕ is in DLOGTIME-uniform TC 0 .
Proof.The proof uses the flatness property of FO-formulas under team semantics: It is well know that A |= s ϕ can be decided by AC 0 -circuits, whence the original question reduces to counting the number t of satisfying assignments of ϕ and checking whether t ≥ k.This can be easily simulated by DLOGTIME-uniform TC 0 circuits as we can hardcode all possible assignments into the circuit.Here, notice that ϕ is fixed and thereby the number of free variables are fixed to some constant c ∈ N.Then, the input is the structure A of size n yielding O(n c ) many assignments.

Inclusion Logic
In this section, we relate the W-hierachy and W[P] to weighted team definability for inclusion logic formulas.First observe that if ϕ is an FO(⊆)-sentence, then the problem p-WT ϕ is in FPT.This is due to the reason that the data complexity of fixed FO(⊆)-sentences is in P [8].

Proof.
Recall that an FO(⊆)-sentence ϕ has a satisfying team T in A if and only if A, {∅} |= ϕ.Then ϕ is true in A if and only if there is a team T such that |T | = 1 and A, T |= ϕ.

Now we prove, that p-WT ϕ can already be W[1]-hard when ϕ is a quantifier-free FO(⊆)formula with free variables.
Theorem 11.There is a quantifier-free FO(⊆)-formula ϕ such that the problem p-WT ϕ is W[1]-hard and in W [2].
Proof.We present a reduction from the W[1]-complete problem p-Clique to p-WT ϕ such that ϕ is a quantifier free FO(⊆)-formula.Let G := (V, E) be a graph and k ∈ N.Then, we let ϕ := E(x, y) ∧ x = y ∧ y ⊆ x ∧ x ⊆ y.We claim that G has a clique of size k if and only if G, T |= ϕ for a team T of size (k 2 − k).It is straightforward to check that the existence of a k-clique is equivalent to ϕ having a satisfying team of cardinality k(k − 1) with exactly the same values for x and y.
For containment in W [2], it suffices to note that the formula ϕ can be expressed as an FO-sentence ψ(S) with a ∀∃-quantifier prefix where the auxiliary binary predicate S encodes the team T .This gives an FPT-reduction between p-WT ϕ and p-WD ψ .The result follows since This result can be strengthened to more general formulas as witnessed by the following corollary.

Corollary 12. For any quantifier-free FO(⊆)-formula ϕ without ∨, the problem p-WT ϕ is W[1]-hard and in W[2].
Proof.For containment in W [2], it suffices to note that the any quantifier-free formula without disjunction can be expressed as an FO-sentence ψ(S) with a ∀∃-quantifier prefix where the auxiliary binary predicate S encodes the team T .

Theorem 13. There is an FO(⊆)-formula ϕ with ∀∃-quantifier prefix for which the problem p-WT ϕ is W[2]-complete.
Proof.We present a reduction from the W[2]-complete problem p-DominatingSet to p-WT ϕ such that ϕ is a FO(⊆)-formula with ∀∃-quantifier prefix.Let G := (V, E) be a graph and k ∈ N. Then we let, ϕ := ∀x∃y(y ⊆ z ∧ (E(x, y) ∨ x = y)).It is straightforward to check that G has a dominating set of size k if and only if G, T |= ϕ for a team T with domain {z} of size k.
For W[2]-membership, notice that for all graphs G and teams T : where ϕ d (X) is the first-order sentence encoding the problem DominatingSet (see Example 2).A formal proof for the above equivalence is similar to the one given in Theorem 18.
The next lemma sets the stage for generalizing the two previous theorems to arbitrary levels of the W-hierarchy.To formulate the result, we assume an encoding of a formula ψ ∈ Γ + t,d (and a truth assignment) by its syntax circuit A ψ = (A, E, I, o), where A is the set of subformulas of ψ, E is the immediate subformula relation, I ⊆ A are the variables of ψ, o is a constant symbol interpreted by ψ.Finally a free relation variable S ⊆ I can be used to represent a truth assignment for the variables.Note that our encoding of ψ works for any t ∈ N but for the definability result below t has to be fixed.Lemma 14.Let t ∈ N. Then there exists a fixed formula ϕ t ∈ FO(⊆) with one free variable z such that for all ψ ∈ Γ + t,d and k ≥ 1: ψ has a satisfying assignment of weight k if and only if A ψ , T |= ϕ t , for some team T of cardinality k.
Proof.Without loss of generality, we assume d = 1.For higher d-values, the presented proof easily generalizes via a conjunction/disjunction of arity d.By the results of Galliani and Hella [8], it suffices to show that the required formula can be expressed by a first-order sentence θ(S) in which the relation symbol S occurs only postively.Then the existence of ϕ t (z) satisfying for all non-empty T and rel(T ) = S follows.Note that θ(S) is not true under the assignment setting all the variables to false, but on the other hand ϕ t is always satisfied for T = ∅ by the empty team property.It is easy to check that θ(S) can be expressed as follows: The relation symbol S has only one occurrence in the formula and it is positive.Now by Proposition 20 of [8], there exists a formula ϕ t such (1) holds for the sentence ∀ x(S( x) → θ(S)) for all A and all T .It is easy to see that θ(S) is equivalent with ∀ x(S( x) → θ(S)) modulo the cases when S = ∅.In fact, it is straightforward to show that ϕ t can be obtained from θ(S) simply by replacing S(x t ) by the inclusion atom x t ⊆ z.The proof then is analogous to the proof of Theorem 18.
Notice further that the translation of the formula θ to an FO(⊆)-formula only introduces inclusion atoms and, in particular, does not require any further quantification.Therefore, the following corollary follows immediately from the proof in Lemma 14.
Corollary 15.Let t ≥ 2 be even.Then there is an FO(⊆)-Π t -formula ϕ t for which the problem p-WT ϕt is W Proof.For the W[t]-membership of p-WT ϕt , notice that the translation between θ and the FO(⊆)-formula ϕ t in the proof of Lemma 14 preserves a one-to-one correspondence between the solutions S for θ and satisfying teams T for ϕ t .In other words, θ has a solution of size k if and only if ϕ t has a satisfying team of size k.This yields W[t]membership since θ ∈ Π t for each t ≥ 1 (see Def. 5).The W[t]-hardness and the containment W[t] ⊆ [p-WT-FO(⊆)-Π t ] FPT for all even t ≥ 1 follows from Proposition 6.
We conclude this section by presenting the upper bounds for WT ϕ when ϕ is an arbitrary FO(⊆)-formula.

Theorem 16. [p-WT-FO(⊆)] FPT ⊆ W[P].
Proof.We prove this via the machine characterization of the class W[P], analogous to the proof for FO-formulas [6, Prop.5.3].Let ϕ be a FO(⊆)-formula with s free variables.An algorithm for the problem p-WT ϕ proceeds as follows: Given a structure A and a k, nondeterministically guess k times an assignment (i.e., an s-tuple of elements of A), then deterministically verify that the team T has cardinality k and A, T |= ϕ.Guessing T requires s • k • log |A| nondeterministic bits, and the verification that A, T |= ϕ can be done in deterministic polynomial time in |A| [8].Thus p-WT ϕ is in W[P] because the formula ϕ is fixed and s is a constant.Moreover, the containment [p-WT-FO(⊆)] FPT ⊆ W[P] holds since p-WT ϕ ∈ W[P] for an arbitrary but fixed FO(⊆)-formula ϕ.

Dependence Logic
First observe that if ϕ is a FO(dep)-sentence, then the problem p-WT ϕ is paraNP-complete.This is due to the reason that the data complexity of fixed FO(dep)-sentences is already NP-complete [24].The correctness of our reduction is established via the following claim and also shows that the formula ϕ is, in fact, equivalent to the familiar definition of independent sets via a Π 1 -formula; hence, p-WT ϕ is W[1]-complete.
Claim 19.There is a team T over x in A such that |T | = k and A, T |= ϕ if and only if there is an independent set in G of size k.

It remains to prove the claim. Suppose that
denote the supplemented team, that is, s i,j (x) = a i and s i,j (y) = a j for every a j ∈ A. We prove that S = {a i | ∃s ∈ T, s(x) = a i } constitutes an independent set in G. Let a i , a j ∈ S, then there are s i , s j ∈ T such that s i (x) = a i , s j (x) = a j .Suppose further that (a i , a j ) = e ∈ E G .Then, T |= P (e) ∧ E(a i , e) and T |= E(a j , e) but T |= dep(y; x) since there are s i,j , s j,j ∈ T such that s i,j (xy) = a i a j and s j,j = a j a j .In other words, s i,j (y) = s j,j (y) but s i,j (x) = s j,j (x).Consequently, T |= (¬P (y) ∨ ¬I(x, y) ∨ dep(y; x)) and T |= ϕ, which is a contradiction.
Conversely, if there is an independent set S of size k in G then we prove that T |= ϕ(x) for T = {s i | i ≤ k, s i (x) ∈ S}.Clearly, the supplemented team T (x, y) has the following effect: for every y that corresponds to an edge e between elements a i , a j ∈ A, at most one of its endpoint a i or a j is in T (x), which is the case if and only if S is in independent set.
Once again, we prove the next lemma that generalizes the previous theorem to arbitrary levels of the W-hierarchy.Lemma 20.Let t ∈ N. Then there exists a fixed formula ϕ t ∈ FO(dep) with one free variable z such that for all ψ ∈ Γ − t,d and k ≥ 1: ψ has a satisfying assignment of weight k if and only if A ψ , T |= ϕ t , for some team T of cardinality k.
Proof.Without loss of generality, we assume that d = 1.Otherwise, the presented proof will easily generalize to larger values of d by a disjunction/conjunction of arity d.By the results of [18], it suffices to show that the required formula can be expressed by a first-order sentence θ(S) in which the relation symbol S occurs only negatively.Then the existence of ϕ t (z) satisfying for all non-empty T and rel(T ) = S follows.Now, it is easy to check that θ(S) can be expressed as follows: The relation symbol S appears only once in the formula and this appearance is negative.
Notice further that the translation of the formula θ to a FO(dep)-formula only introduces dependence atoms and, in particular, does not require any further quantification.Therefore, the following corollary (with proof analogous to Corollary 15) follows.Recall that every dependence logic formula can be put into the ∀∃-normal form.As a result, tracking the quantifier prefix in Lemma 20 is not useful and we get the much stronger statement that the whole W-hierarchy is already contained in FO(dep)-Π 2 .Proof.Hardness follows from Theorem 17.For membership, we present the following nondeterministic algorithm that runs in polynomial time in the size of A. Notice that since the formula is fixed, we have fixed many connectives including splits and existential quantifiers.The idea of the algorithm is that it guesses a team T of size k, as well as, a sequence T i for i ∈ N of teams which corresponds to the operations of duplication/supplementation and splits according to the formula ϕ.In other words, let ϕ and ψ is a quantifier free FO(dep)-formula.Then the algorithm has the following steps.
Guess a team T 0 of size k over Fr(ϕ).
For each i ≤ , guess a team T i over Fr(ϕ) ∪ {x 1 , . . ., x i } such that: if Once the team T has been guessed, it remains to determine whether T |= ψ.Since the data complexity of FO(dep) is still NP-complete for quantifier free formulas, this step is non-trivial.Nevertheless, we can list recursively all the subformulas of ψ in terms of its syntax tree.This helps in labelling a subteams of T according to the connectives of ψ.
Notice that for atomic formulas the truth evaluation T α |= α can be determined in polynomial time.Moreover, the intermediate steps including the verification of team duplication and supplementation can also be determined in polynomial time.This results in paraNP-membership of WT ϕ for a FO(dep)-formula ϕ.

Independence Logic
In this section, we turn to independence logic.The following theorem is obtained from the results in the previous sections and the fact that any ESO-sentence ψ(S) (with an extra relation encoding the team) can be represented by an independence logic formula [7].
Theorem 23. 1.For all t ∈ N there is an FO(⊥)-formula ϕ t such that p-WT ϕt is W[t]-complete.
For the second claim, we use the fact that p-WSAT(CIRC) is W[P]-complete, where CIRC is the class of all propositional formulas encoded as Boolean circuits.Note that the circuit value problem can be readily expressed by an ESO-sentence ψ(S), where S represents an input for the circuit.More precisely, assume we a given a DAG (A, E, D, K, I, o) encoding a Boolean circuit.Here A is the set of nodes/gates, E is the edge relation, I ⊆ A are the input gates of the circuit, o is the unique output, D ⊆ A is the set of OR-gates, and K ⊆ A the set of AND-gates.A Boolean input for the circuit is represented by a subset S ⊆ I, i.e., a gate g gets input 1 if and only if g ∈ S. Now in ESO we can existentially quantify a proof tree witnessing the circuit accepting the input S. In other words, we quantify a subset P ⊆ A such that o ∈ A, P ∩ I = S, for all g ∈ P ∩ D there exists at least one g ∈ P such that E(g , g), for all g ∈ P ∩ K and all g , if E(g , g) then g ∈ P .It is straightforward to check that the above conditions can be expressed in first-order logic.
Finally, the hardness part of the third claim follows again from the fact that FO(dep) is a sublogic of FO(⊥) and the containment proof is analogous to that of Theorem 22.

Conclusion
We have defined and studied the parameterized complexity of weighted team definability with respect to formulas of several team-based logics.Our results show that for plain first-order formulas weighted team definability differs greatly from weighted Fagin definability; the former being computationally much simpler.For dependence, independence and inclusion logic formulas, the complexity of weighted team definability ranges between the classes W[t] and paraNP.Now, these results provide a wide range of natural complete problems for the aforementioned complexity classes enriching the landscape in a nontrivial way.Interestingly,

FPTFigure 1
Figure 1 Landscape of relevant parameterized complexity classes with complete problems.The definition of several of these complete problems are mentioned in the relevant proofs.

Theorem 17 .
There is a FO(dep)-sentence ϕ, such that the problem p-WT ϕ is paraNPcomplete.Proof.Recall that a FO(dep)-sentence ϕ has a satisfying team T if and only if {∅} |= ϕ.For hardness, consider the data complexity of the model checking for FO(dep)-sentences.The problem asks whether an input structure A satisfies a fixed FO(dep)-sentence ϕ.Then ϕ is true in A if and only if A, {∅} |= ϕ if and only if there is a team T such that |T | = 1 and A, T |= ϕ.Now, we relate the W-hierarchy to the weighted definability for dependence logic.This also settles the complexity of p-WT for FO(dep)-formulas.In the following, we prove that already one universal quantifier is enough in FO(dep) to define W[1]-complete problems.Theorem 18.There is a FO(dep)-formula ϕ with only one universal quantifier such that the problem p-WT ϕ is W[1]-complete.Proof.We present a reduction from the W[1]-complete problem p-IndependentSet to p-WT ϕ such that ϕ is FO(dep)-formula with only one universal quantifier.An input to IndependentSet is a graph G := (V, E) and a number k ∈ N. The question is whether there is a set S of size k in G such that (a, b) ∈ E for every a, b ∈ S. We let τ := {N 1 , P 1 , I 2 } as our vocabulary where N, P are unary relations and I is a binary relation symbol.Moreover the τ -structure A is such that: dom(A) := V ∪ E, N A := V, P A := E and I A simulates the edge relation E G .That is, I := { (a, b), (c, b) | a, c ∈ V, and b ∈ P denotes the edge (a, c) ∈ E }. Finally we define a FO(dep)-formula ϕ over a single free variable x as in the following.ϕ(x) := ∀y N (x) ∧ (¬P (y) ∨ ¬I(x, y) ∨ dep(y; x))
the sentences in the considered logics depict different complexities: namely, membership in FPT for FO(⊆) and paraNP-completeness for FO(dep) and FO(⊥).The main open question is whether the converse directions of Corollary 15 or Theorem 16 can be proven, i.e., if one of the inclusions t∈N W[t] ⊆ [p-WT-FO(⊆)] FPT ⊆ W[P] is in fact an equality.

Table 1
Partial overview of our results concerning weighted team definability with pointers to the respective theorem or corollary.
d ) denote the class of all positive (negative) formulas in Γ t,d .The parameterized weighted satisfiability problem (WSAT) for propositional formulas is defined as below.