Model Checking and Validity in Propositional and Modal Inclusion Logics

Propositional and modal inclusion logic are formalisms that belong to the family of logics based on team semantics. This article investigates the model checking and validity problems of these logics. We identify complexity bounds for both problems, covering both lax and strict team semantics. By doing so, we come close to finalising the programme that ultimately aims to classify the complexities of the central reasoning problems for modal and propositional dependence, independence, and inclusion logics.


Introduction
Team semantics is the mathematical framework of modern logics of dependence and independence, which, unlike Tarski semantics, is not based on singletons as satisfying elements (e.g., first-order assignments or points of a Kripke structure) but on sets of such elements. More precisely, a first-order team is a set of first-order assignments that have the same domain of variables. As a result, a team can be interpreted as a database table, where variables correspond to attributes and assignments to records. Team semantics originates from the work of Hodges [19], where it was shown that Hintikka's IF-logic can be based on a compositional (as opposed to game-theoretic) semantics. In 2007, Väänänen [31] proposed a fresh approach to logics of dependence and independence. Väänänen adopted team semantics as a core notion for his dependence logic. Dependence logic extends first-order logic by atomic statements such as the value of variable x is determined by the value of y. Clearly such a statement is not meaningful under a single assignment, however, when evaluated over a team such a statement corresponds precisely to functional dependence of database theory when the team is interpreted as a database table.
Besides functional dependence, there are many other important dependency notions used in fields like statistics and database theory, which give rise to interesting logics based on team semantics. The two most widely studied of these new logics are independence logic of Grädel and Väänänen [12], and inclusion logic of Galliani [7]. Inclusion logic extends first-order logic by atomic statements of the form x ⊆ y, which is satisfied in a team X if any value that appears as a value for x in X also appears as a value of y in X. Dependence and independence logics are equal-expressive with existential second-order logic and accordingly capture the complexity class NP [31,12]. Surprisingly, inclusion logic has the same expressive power as positive greatest fixed point logic GFP + [9]. Since on finite structures, GFP + coincides with least fixed point logic LFP, it follows from the Immermann-Vardi-Theorem that inclusion logic captures the complexity class P on finite ordered structures. Interestingly under a semantical variant of inclusion logic called strict semantics the expressive power of inclusion logic rises to existential second-order logic [8]. Moreover, the fragment of inclusion logic (under strict semantics) in which only k universally quantified variables may occur captures the complexity class NTIME RAM (n k ) (i.e., structures that can be recognised by a nondeterministic random access machine in time O(n k )) [15]. That being so, indeed, inclusion logic and its fragments have very interesting descriptive complexity theoretic properties.
In this paper, we study propositional and modal inclusion logic under both the standard semantics (i.e., lax semantics) and strict semantics. The research around propositional and modal logics with team semantics has concentrated on classifying the complexity and definability of the related logics. Due to very active research efforts, the complexity and definability landscape of these logics is understood rather well; see the survey of Durand et al. [5] and the references therein for an overview of the current state of the research. In the context of propositional logic (modal logic, resp.) a team is a set of propositional assignments with a common domain of variables (a subset of the domain a Kripke structure, resp.). Extended propositional inclusion logic (extended modal inclusion logic, resp.) extends propositional logic (modal logic, resp.) with propositional inclusion atoms ϕ ⊆ ψ, where ϕ and ψ are formulae of propositional logic (modal logic, resp.). The following definability results hold for the standard lax semantics. A class of team pointed Kripke models is definable in extended modal inclusion logic iff M, ∅ is in the class for every model M, the class is closed under taking unions, and the class is closed under the so-called team kbisimulation, for some finite k [18]. From this, a corresponding characterization for extended propositional inclusion logic follows directly. In [26,27] (global) model definability and frame definability of team based modal logics are studied. It is shown that surprisingly, in both cases, (extended) modal inclusion logic collapses to modal logic. This paper investigates the complexity of the model checking and the validity problem for propositional and modal inclusion logic. The complexity of the satisfiability problem of modal inclusion logic was studied by Hella et al. [16]. The study on the validity problem of propositional inclusion logic was initiated by Hannula et al. [13], where the focus was on more expressive logics in the propositional setting. Consequently, the current paper directly extends the research effort initiated in these papers. It is important to note that since the logics studied in this paper, are closed under negation, the connection between the satisfiability problem and the validity problem fails. In [13] it was shown that, under lax semantics, the validity problem for propositional inclusion logic is coNP-complete. Here we obtain an identical result for the strict semantics. However, surprisingly, for model checking the picture looks quite different. We establish that whereas the model checking problem for propositional inclusion logic is P-complete under lax semantics, the problem becomes NPcomplete for the strict variant. Also surprisingly, for model checking, we obtain remarkable in the modal setting; modal inclusion logic is P-complete under lax semantics and NPcomplete under strict semantics. Nevertheless, for the validity problem, the modal variants are much more complex; we establish coNEXP-hardness for both strict and lax semantics. For an overview of the results of this paper together with the known complexity results from the literature, see tables 3-5 on page 17.

Propositional logics with team semantics
Let D be a finite, possibly empty set of proposition symbols. A function s : D → {0, 1} is called an assignment. A set X of assignments s : D → {0, 1} is called a team. The set D is the domain of X. We denote by 2 D the set of all assignments s : D → {0, 1}. If p = (p 1 , . . . , p n ) is a tuple of propositions and s is an assignment, we write s( p) for (s(p 1 ), . . . , s(p n )).
Let Φ be a set of proposition symbols. The syntax of propositional logic PL(Φ) is given by the following grammar: We denote by |= PL the ordinary satisfaction relation of propositional logic defined via assignments in the standard way. Next we give team semantics for propositional logic.
Definition 1 (Lax team semantics) Let Φ be a set of atomic propositions and let X be a team. The satisfaction relation X |= ϕ is defined as follows.
X |= p ⇔ ∀s ∈ X : s(p) = 1, The lax team semantics is considered the standard semantics for team-based logics. In this paper, we also consider a variant of team semantics called the strict team semantics. In strict team semantics, the above clause for disjunction is redefined as follows: When L denotes a team-based propositional logic, we let L s denote the variant of the logic with strict semantics. Moreover, in order to improve readability, for strict semantics we use |= s instead of |=. As a result lax semantics is used unless otherwise specified. The next proposition shows that the team semantics and the ordinary semantics for propositional logic defined via assignments (denoted by |= PL ) coincide.

Proposition 2 ([31])
Let ϕ be a formula of propositional logic and let X be a propositional team. Then X |= ϕ iff ∀s ∈ X : s |= PL ϕ.
The syntax of propositional inclusion logic PInc(Φ) is obtained by extending the syntax of PL(Φ) by the grammar rule ϕ ::= p ⊆ q, where p and q are finite tuples of proposition variables with the same length. The semantics for propositional inclusion atoms is defined as follows: X |= p ⊆ q iff ∀s ∈ X ∃t ∈ X : s( p) = t( q).
Remark 3 Extended propositional inclusion logic is the variant of PInc in which inclusion atoms of the form ϕ ⊆ ψ, where ϕ and ψ are tuples of PL-formulae, are allowed. It is easy to see that this extension does not increase complexity of the logic and on that account, in this paper, we only consider the non-extended variant.

Complexity of Propositional Inclusion Logic
We now define the model checking, satisfiability, and validity problems in the context of team semantics. Let L be a propositional logic with team semantics. A formula ϕ ∈ L is satisfiable, if there exists a non-empty team X such that X |= ϕ. A formula ϕ ∈ L is valid, if X |= ϕ holds for all teams X such that the proposition symbols in ϕ are in the domain of X. The satisfiability problem SAT(L) and the validity problem VAL(L) are defined in the obvious way: Given a formula ϕ ∈ L, decide whether the formula is satisfiable (valid, respectively). For the model checking problem MC(L) we consider combined complexity: Given a formula ϕ ∈ L and a team X, decide whether X |= ϕ. See Table 1 for known complexity results for PL and PInc, together with partial results of this paper.
It was shown in [13] that the validity problem of PInc is coNP-complete. Here we establish that the corresponding problem for PInc s is also coNP-complete. Our proof is similar to the one in [13]. However the proof of [13] uses the fact that PInc is union closed, while the same is not true for PInc s (cf. Example 5).
Lemma 6 Let X be a propositional team and ϕ ∈ PInc s . If {s} |= s ϕ for every s ∈ X then X |= s ϕ.
Proof The proof is by a simple induction on the structure of the formula. The cases for atomic formulae and conjunction are trivial. The case for disjunction is easy: Assume that {s} |= s ϕ ∨ ψ for every s ∈ X. Consequently for every s ∈ X either {s} |= s ϕ or {s} |= s ψ. As a result there exists Y and Z such that Y ∪ Z = X, Y ∩ Z = ∅, ∀s ∈ Y : {s} |= s ϕ, and ∀s ∈ Z : {s} |= s ψ. By the induction hypothesis Y |= s ϕ and Z |= s ψ. Consequently, X |= s ϕ ∨ ψ.

Theorem 7
The validity problem for PInc s is coNP-complete w.r.t. ≤ log m . Proof The coNP-hardness follows via Proposition 2 from the fact that the validity problem of PL is coNP-hard. Accordingly, it suffices to show VAL(PInc s ) ∈ coNP. It is easy to check that, by Lemma 6, a formula ϕ ∈ PInc s is valid iff it is satisfied by all singleton teams {s}. Note also that, over a singleton team {s}, an inclusion atom (p 1 , . . . , p n ) ⊆ (q 1 , . . . , q n ) is equivalent to the PL-formula 1 A logic L is downward closed if the implication X |= ϕ and Y ⊆ X ⇒ Y |= ϕ holds for every formula ϕ ∈ L and teams X and Y .
Denote by ϕ * the PL-formula obtained by replacing all inclusion atoms in ϕ by their PLtranslations. By the above, ϕ is valid iff ϕ * is valid. Since VAL(PL) is in coNP the claim follows.

Model checking in lax semantics is P-complete
In this section we construct a reduction from the monotone circuit value problem to the model checking problem of PInc. For a deep introduction to circuits see [33] by Vollmer.
Definition 8 A monotone Boolean circuit with n input gates and one output gate is a 3-tuple C = (V, E, α), where (V, E) is a finite, simple, directed, acyclic graph, and α : V → {∨, ∧, x 1 , . . . , x n } is a function such that the following conditions hold: 1. Every v ∈ V has in-degree 0 or 2.
2. There exists exactly one w ∈ V with out-degree 0. We call this node w the output gate of C and denote it by g out .

For each
Let C = (V, E, α) be a monotone Boolean circuit with n input gates and one output gate. Any sequence b 1 , . . . , b n ∈ {0, 1} of bits of length n is called an input to the circuit C. A function β : is called the valuation of the circuit C under the input b 1 , . . . , b n . The output of the circuit C is then defined to be β(g out ).
The monotone circuit value problem (MCVP) is the following decision problem: Given a monotone circuit C and an input b 1 , . . . , b n ∈ {0, 1}, is the output of the circuit 1?
Proof We will establish a LOGSPACE-reduction from MCVP to the model checking problem of PInc under lax semantics. Since MCVP is P-complete, the claim follows. More precisely, we will show how to construct, for each monotone Boolean circuit C with n input gates and for each input b for C, a team X C, b and a PInc-formula ϕ C such that We use teams to encode valuations of the circuit. For each gate v i of a given circuit, we identify an assignment s i . The crude idea is that if s i is in the team under consideration, the value of the gate v i with respect to the given input is 1. The formula ϕ C is used to quantify a truth value for each Boolean gate of the circuit, and then for checking that the truth values of the gates propagate correctly. We next define the construction formally and then discuss the background intuition in more detail.
Let C = (V, E, α) be a monotone Boolean circuit with n input gates and one output gate and let b = (b 1 . . . b n ) ∈ {0, 1} n be an input to the circuit C. We define that V = {v 0 , . . . , v m } and that v 0 is the output gate of C. Define For each i ≤ m, we define the assignment s i : τ C → {0, 1} as follows: Furthermore, we define s ⊥ (p) = 1 iff p = p ⊥ or p = p ⊤ . We note that the assignment s ⊥ will be the only assignment that maps p ⊥ to 1. We make use of the fact that for each gate v i of C, it holds that s ⊥ (p i ) = 0. We define that is, X C, b consists of assignments for each of the Boolean gates, assignments for those input gates that are given 1 as an input, and of the auxiliary assignment s ⊥ .
Let X be any nonempty subteam of X C, b such that s ⊥ ∈ X. We have Recall the intuition that s i ∈ X should hold iff the value of the gate v i is 1. Define The idea of the reduction is the following: The disjunction in φ C is used to guess a team Y for the right disjunct that encodes the valuation β of the circuit C. The right disjunct is then evaluated with respect to the team Y with the intended meaning that β(v i ) = 1 whenever s i ∈ Y . Note that Y is always as required in (1). The formula ψ out=1 is used to state that β(v 0 ) = 1, whereas the formulae ψ ∧ and ψ ∨ are used to propagate the truth value 1 down the circuit. The assignment s ⊥ and the proposition p ⊥ are used as an auxiliary to make sure that Y is nonempty and to deal with the propagation of the value 0 by the subformulae of the form p i ⊆ p j . Now observe that the team X C, b can be easily computed by a logspace Turing machine which scans the input for ∧-gates, ∨-gates, and true input gates, and then outputs the corresponding team members s i in a bitwise fashion. The formula ϕ C can be computed in logspace as well: 1. the left disjunct does not depend on the input, 2. for ψ ∧ we only need to scan for the ∧-gates and output the inclusion-formulae for the corresponding edges, 3. for ψ ∨ we need to maintain two binary counters for i and j, and use them for searching for those disjunction gates that satisfy i < j.
Consequently, the reduction can be computed in logspace.
For the proof of the above lemma it is not important that lax semantics is considered; the same proof works also for the strict semantics. However, as we will show next, we can show a stronger result for the model checking problem of PInc s ; namely that it is NP-hard. In Section 5.1 we will show that the model checking problem for modal inclusion logic with lax semantics is in P (Lemma 22). Since PInc is essentially a fragment of this logic, by combining Lemmas 10 and 22, we obtain the following theorem.

Model checking in strict semantics is NP-complete
In this section we reduce the set splitting problem, a well-known NP-complete problem, to the model checking problem of PInc s .

Definition 12
The set splitting problem is the following decision problem: Input: A family F of subsets of a finite set S.
Problem: Do there exist subsets S 1 and S 2 of S such that 1. S 1 and S 2 are a partition of S (i.e., S 1 ∩ S 2 = ∅ and S 1 ∪ S 2 = S), 2. for each A ∈ F , there exist a 1 , a 2 ∈ A such that a 1 ∈ S 1 and a 2 ∈ S 2 ?
The following proof relies on the fact that strict semantics is considered. It cannot hold for lax semantics unless P = NP.
Proof We give a reduction from the set splitting problem [10,SP4] to the model checking problem of PInc under strict semantics. Let F be an instance of the set splitting problem. We stipulate that F = {B 1 , . . . , B n } and that F = {a 1 , . . . , a k }, where n, k ∈ N. We will introduce fresh proposition symbols p i and q j for each point a i ∈ F and set B j ∈ F . We will then encode the family of sets F by assignments over these proposition symbols; each assignment s i will correspond to a unique point a i . Formally, let τ F denote the set {p 1 , . . . , p k , q 1 , . . . , q n , p ⊤ , p c , p d } of proposition symbols. For each i ∈ {1, . . . , k, c, d}, we define the assignment s i : τ F → {0, 1} as follows: Define X F := {s 1 , . . . , s k , s c , s d }, that is, X F consists of assignments s i corresponding to each of the points a i ∈ F and of two auxiliary assignments s c and s d . Note that the only assignment in X F that maps p c (p d , resp.) to 1 is s c (s d , resp.) and that every assignment maps p ⊤ to 1. Moreover, note that for 1 ≤ i ≤ k and 1 ≤ j ≤ n, s i (q j ) = 1 iff a i ∈ B j . Now define We claim that X F |= s ϕ F iff the output of the set splitting problem with input F is "yes". The proof is straightforward. Note that X F |= s ϕ F holds iff X F can be partitioned into two subteams Y 1 and Y 2 such that It is now evident that if X F |= s ϕ F holds then the related subteams Y 1 and Y 2 directly construct a positive answer to the set splitting problem. Likewise, any positive answer to the set splitting problem can be used to directly construct the related subteams Y 1 and Y 2 .
In order to compute the assignments s i and by this the team X F on a logspace machine we need to implement two binary counters to count through 1 ≤ i ≤ k for the propositions p i and 1 ≤ j ≤ n for the propositions q i . The formula ϕ F is constructed in logspace by simply outputting it step by step with the help of a binary counter for the interval 1 ≤ i ≤ n. As a result the whole reduction can be implemented on a logspace Turing machine.
In Section 5.1 we establish that the model checking problem of modal inclusion logic with strict semantics is in NP (Theorem 25). Since PInc is essentially a fragment of this logic, together with Lemma 14, we obtain the following theorem.

Modal logics with team semantics
Let Φ be a set of proposition symbols. The syntax of modal logic ML(Φ) is generated by the following grammar: By ϕ ⊥ we denote the formula that is obtained from ¬ϕ by pushing all negation symbols to the atomic level. Accordingly, T [R]S holds if and only if for every w ∈ T , there exists some v ∈ S such that wRv, and for every v ∈ S, there exists some w ∈ T such that wRv. We are now ready to define team semantics for ML. Analogously to the propositional case, we also consider the strict variant of team semantics for modal logic. In the strict team semantics, we have the following alternative semantic definitions for the disjunction and diamond (where W denotes the domain of M).
When L is a team-based modal logic, we let L s to denote its variant with strict semantics. As in the propositional case, for strict semantics we use |= s instead of |=. The formulae of ML have the following flatness property.   Table 2: Complexity of satisfiability, validity and model checking for modal logics under both systems of semantics. The given complexity classes refer to completeness results and "-h." denotes hardness. The complexities for Minc and EMinc coincide, see Theorems 24, 25, and 33.
The syntax of modal inclusion logic Minc(Φ) and extended modal inclusion logic EMinc(Φ) is obtained by extending the syntax of ML(Φ) by the following grammar rule for each n ∈ N: where ϕ 1 , ψ 1 , . . . , ϕ n , ψ n ∈ ML(Φ). Additionally, for Minc(Φ), we require that ϕ 1 , ψ 1 , . . . , ϕ n , ψ n are proposition symbols in Φ. The semantics for these inclusion atoms is defined as follows: The following proposition is proven in the same way as the analogous results for firstorder inclusion logic [7]. A modal logic L is union closed if M, T |= ϕ and M, S |= ϕ implies that M, T ∪ S |= ϕ, for every ϕ ∈ L.
Analogously to the propositional case, it is easy to see that, by Proposition 18, for ML the strict and the lax semantics coincide. Again, as in the propositional case, this does not hold for Minc or EMinc. (Note that since PInc s is not union closed (cf. Example 5) neither Minc s nor EMinc s is as well.) In contrary to the propositional case, Lemma 6 fails in the modal case as the following example illustrates.  Table 2 for known complexity results on ML and Minc, together with partial results of this paper.

Complexity of model checking
Let M be a Kripke model, T be a team of M, and ϕ be a formula of Minc. By maxsub(T, ϕ), we denote the maximum subteam T ′ of T such that M, T ′ |= ϕ. Since Minc is union closed (cf. Proposition 19), such a maximum subteam always exists.

Lemma 21
If ϕ is a proposition symbol, its negation, or an inclusion atom, then maxsub(T, ϕ) can be computed in polynomial time with respect to |T | + |ϕ|.
Proof If ϕ is a proposition symbol or its negation, the claim follows from flatness in a straightforward way. Assume then that T = {w 1 , . . . w n } and ϕ = p 1 , . . . , p k ⊆ q 1 , . . . , q k . Let G = (V, E) be a directed graph such that V = T and (u, v) ∈ E iff the value of p i in u is the same as the value of q i in v, for each 1 ≤ i ≤ k.
The graph G describes the inclusion dependencies between the points in the following sense: if w ∈ maxsub(T, ϕ), then there exists some v ∈ maxsub(T, ϕ) such that (w, v) ∈ E. Clearly G can be computed in time O(n 2 k). In order to construct maxsub(T, ϕ), we, round by round, delete all vertices from G with out-degree 0. Formally, we define a sequence G 0 , . . . , G n of graphs recursively. We define that G 0 := G and that G j+1 is the graph obtained from G j by deleting all of those vertices from G j that have out-degree 0 in G j . Let i be the smallest integer such that G i = (V i , E i ) has no vertices of out-degree 0. Clearly i ≤ n, and moreover, G i is computable from G in time O(n 3 ). It is easy to check that V i = maxsub(T, ϕ).
For the following Lemma it is crucial that lax semantics is considered. The lemma cannot hold for strict semantics unless P = NP.

Lemma 22 MC(Minc) under lax semantics is in P.
Proof We will present a labelling algorithm for model checking M, T |= ϕ. Let subOcc(ϕ) denote the set of all occurrences of subformulae of ϕ. Below we denote occurrences as if they were formulae, but we actually refer to some particular occurrence of the formula.
A function f : subOcc(ϕ) → P(W ) is called a labelling function of ϕ in M. We will next give an algorithm for computing a sequence f 0 , f 1 , f 2 , . . ., of such labelling functions.
• For odd i ∈ N, define f i (ψ) bottom up as follows: • For even i ∈ N larger than 0, define f i (ψ) top to bottom as follows: By a straightforward induction on i, we can prove that f i+1 (ψ) ⊆ f i (ψ) holds for every ψ ∈ subOcc(ϕ). The only nontrivial induction step is that for f i+1 (θ) and f i+1 (γ), when i + 1 is even and ψ = θ ∧ γ. To deal with this step, observe that, by the definition of f i+1 It follows that there is an integer j ≤ 2 · |W | · |ϕ| such that f j+2 = f j+1 = f j . We denote this fixed point f j of the sequence f 0 , f 1 , f 2 , . . . by f ∞ . By Lemma 21 the outcome of maxsub(·, ·) is computable in polynomial time with respect to its input. That being, clearly f i+1 can be computed from f i in polynomial time with respect to |W |+ |ϕ|. On that account f ∞ is also computable in polynomial time with respect to |W | + |ϕ|.
We will next prove by induction on ψ ∈ subOcc(ϕ) that M, f ∞ (ψ) |= ψ. Note first that there is an odd integer i and an even integer j such that f ∞ = f i = f j .
3. Assume that ψ = θ ∨γ. Using the fact that T θ ⊆ T ψ , and the two induction hypotheses on i and ψ, we see that 4. Assume that ψ = ✸θ. By the induction hypothesis on i, we have T θ ⊆ f i (θ), and by the induction hypothesis on ψ, we have 5. The case ψ = ✷θ is similar to the previous one; we omit the details.
It follows now that . . , ϕ n be exactly those subformulae of ϕ that occur as a parameter of some inclusion atom in ϕ and let p 1 , . . . , p n be distinct fresh proposition symbols. Let V ′ be a valuation defined as follows Let ϕ * denote the formula obtained from ϕ by simultaneously substituting each ϕ i by p i . It is easy to check that (W, R, V ), T |= ϕ if and only if (W, R, V ′ ), T |= ϕ * . Moreover, ϕ * can be clearly computed form ϕ in polynomial time. Likewise, V ′ can be computed in polynomial time; since each ϕ i is a modal formula the truth set of that formula in (W, R, V ) can be computed in polynomial time by the standard labelling algorithm used in modal logic (see e.g., [1]), and the numbers of such computations is bounded above by the size of ϕ. As a consequence the result follows form Lemma 22.
By combining Lemmas 10, 22, and 23, we obtain the following theorem. Proof The NP-hardness follows from the propositional case, i.e., by Lemma 14.
The obvious brute force algorithm for model checking for EMinc works in NP: For disjunctions and diamonds, we use nondeterminism to guess the correct partitions or successor teams, respectively. Conjunctions are dealt sequentially and for boxes the unique successor team can be computed by brute force in quadratic time. Checking whether a team satisfies an inclusion atom or a (negated) proposition symbol can be computed by brute force in polynomial time (this also follows directly from Lemma 21).

Dependency quantifier Boolean formulae
Deciding whether a given quantified Boolean formula (qBf) is valid is a canonical PSPACEcomplete problem. Dependency quantifier Boolean formulae introduced by Peterson et al. [25] are variants of qBfs for which the corresponding decision problem is NEXP-complete. In this section, we define the related coNEXP-complete complementary problem. For the definitions related to dependency quantifier Boolean formulae, we follow Virtema [32].
QBfs extend propositional logic by allowing a prenex quantification of proposition symbols. Formally, the set of qBfs is built from formulae of propositional logic by the following grammar: ϕ : where p is a propositional variable (i.e., a proposition symbol) and θ is formula of propositional logic. The semantics for qBfs is defined via assignments s : PROP → {0, 1} in the obvious way. When C is a set of propositional variables, we denote by c the canonically ordered tuple of the variables in the set C. When p is a propositional variable and b ∈ {0, 1} is a truth value, we denote by s(p → b) the modified assignment defined as follows: A formula that does not have any free variables is called closed. We denote by QBF the set of exactly all closed quantified Boolean formulae.

Proposition 26 ([30])
The validity problem of QBF is PSPACE-complete w.r.t. ≤ log m . A simple qBf is a closed qBf of the type ϕ := ∀p 1 · · · ∀p n ∃q 1 · · · ∃q k θ, where θ is a propositional formula and the propositional variables p i , q j are all distinct. Any tuple (C 1 , . . . , C k ) such that C 1 , . . . , C k ⊆ {p 1 , . . . , p n } is called a constraint for ϕ. Intuitively, a constraint C j = {p 1 , p 3 } can be seen as a dependence atom dep(p 1 , p 3 , q j ) 2 . A constraint C j = {p 1 , p 3 } can be also interpreted to indicate that the semantics of ∃g j is defined, if skolemised, via a Skolem function f j (p 1 , p 3 ).
A dependency quantifier Boolean formula is a pair (ϕ, C), where ϕ is a simple quantified Boolean formula and C is a constraint for ϕ. We say that (ϕ, C) is valid, if ϕ is valid under the constraint C. Let DQBF denote the set of all dependency quantifier Boolean formulae.
It is straightforward to see that non-validity problem of DQBF is the complement problem of the validity problem of DQBF. Accordingly, the following corollary follows.

Corollary 30
The non-validity problem of DQBF is coNEXP-complete w.r.t. ≤ log m .

Complexity of the validity problem is coNEXP-hard
In this section we give a reduction from the non-validity problem of DQBF to the validity problem of Minc.
Proof We provide a ≤ log m -reduction from the non-validity problem of DQBF to the validity problem of Minc.
Recall Definition 29. In our reduction we will encode all the possible modified assignments of Definition 29 by points in Kripke models. First we enforce binary (assignment) trees of depth n in our structures. Leafs of the binary tree will correspond to the set of assignments where ✷ i ϕ := i many ✷ · · · ✷ ϕ is the i-times concatenation of ✷. The formula tree p,n forces a complete binary assignment tree of depth n for proposition symbols p 1 , . . . , p n . Notice that tree p,n is an ML-formula and consequently flat (see Proposition 18). Let ℓ := max{|C 1 |, . . . , |C k |}. Then define The formula ϕ struc enforces the full binary assignment tree w.r.t. the p i s, enforces in its leaves trees of depth ℓ for variables t i , identifies the truth of θ by a proposition p θ at the depth n + ℓ as well as 1 by t ⊤ and 0 by t ⊥ , and then stores the values of the p j s and q r s consistently in their subtrees of relevant depth. The points at depth n are used to encode the modified assignments of Definition 29.
Recall again Definition 29 and consider the simple qBf ∀p 1 · · · ∀p n ∃q 1 · · · ∃q k θ with constraint (C 1 , . . . , C k ). Then consider some particular Kripke model with the structural properties described above. Shift your attention to those points in the enforced tree that are in depth n. Note first that if we restrict our attention to proposition symbols p 1 , . . . , p n all assignments are present. In fact the number points corresponding to some particular assignment can by anything ≥ 1. Values for the proposition symbols q j and consequently for the functions f j arise from the particular model; essentially, since we are considering validity, all possible values will be considered. In fact, in some particular models, the values of q j s are not functionally determined according to the related constraint C j . We will next define a formula that will deal with those models in which, for some j, the values for q j do not give rise to a function f j in Definition 29. These unwanted models have to be "filtered" out by the formula through satisfaction. This violation is expressed via ϕ cons defined as follows. Below we let n j = |C j |.
Assume that t ⊤ and t ⊥ correspond to the constant values 1 and 0, respectively. Moreover, for the time being, suppose that the values for the proposition symbols t i have been existentially quantified (we will later show how this is technically done). Now the formula ϕ cons essentially states that there exists a q j that does not respect the constraint C j . Finally define By ϕ ⊥ struc , we denote the negation normal form of the ML-formula ¬ϕ struc . An important observation is that since ϕ struc is an ML-formula, it is flat. Now the formula ϕ non-val is valid if and only if holds for every team pointed Kripke model M, T that satisfies the structural properties forced by ϕ struc . Let us now return to the formula (2). There, we assumed that the proposition symbols t i had been quantified and that the symbols p ⊤ and p ⊥ correspond to the logical constants. The latter part we already dealt with in the formula ϕ struc . Recall that ϕ struc forces full binary assignment trees for the t i s that start from depth n. The quantification of the t i s is done by selecting the corresponding successors by the diamonds ✸ ℓ in the formula (3). If M, T is such that, for some j, q j does not respect the constraint C j , we use ✸ ℓ to guess a witness for the violation. It is then easy to check that the whole team obtained by evaluating the diamond prefix satisfies the formula ϕ cons . On the other hand, if M, T is such that for each j the value of q j respects the constraint C j , then the subformula p ⊥ ⊆ p θ forces that there exists a point w in the team obtained from T by evaluating the modalities in (4) such that M, {w} |= p θ . In our reduction this means that w gives rise to a propositional assignment that falsifies θ as required in Definition 29.
It is now quite straightforward to show that a simple qBf ∀p 1 · · · ∀p n ∃q 1 · · · ∃q k θ is nonvalid under a constraint (C 1 , . . . , C k ) iff the Minc-formula ϕ non-val obtained as described above is valid.
In the following we show the correctness of the constructed reduction. By the observation made in (4) it suffices to show the following claim: Claim ∀p 1 · · · ∀p n ∃q 1 · · · ∃q k θ is non-valid under (C 1 , . . . , C k ) iff M, T |= ✷ n ✸ ℓ (ϕ cons ∨ p ⊥ ⊆ p θ ) holds for every team pointed Kripke model M, T that satisfies the structural properties forced by ϕ struc .
Proof (Proof of Claim) "⇒": Assume that the formula ϕ := ∀p 1 · · · ∀p n ∃q 1 · · · ∃q k θ is non-valid under the constraint (C 1 , . . . , C k ). As a consequence, for every sequence of functions f 1 , . . . , f k of appropriate arities there exists an assignment s : We will show that for each team pointed Kripke model M, T that satisfies the structural properties forced by ϕ struc . Let M, T be an arbitrary team pointed Kripke model that satisfies the required structural properties. Denote by S the team obtained from T after evaluating the first n ✷-symbols in (6). Note that each tuple of values assigned to p := (p 1 , . . . , p n ) is realised in S as the tree structure enforces all possible assignments over p. Due to the forced structural properties, S and of any team obtainable from S by evaluating the k ✸-symbols in (6) realise exactly the same assignments for {p 1 , . . . , p n , q 1 , . . . q k }. Let S k denote the set of exactly all points reachable from S by paths of length exactly k. For each point w denote by w( q) the value of q in the world w. Note that for every ℓ-tuple of bits b and every point w ∈ S there exists a point v ∈ S k such that v(p 1 , . . . , p n , q 1 , . . . q k ) = w(p 1 , . . . , p n , q 1 , . . . q k ) and v(t 1 , . . . , t ℓ ) = b. Moreover, for any fixed b, the team {w ∈ S k | w(t 1 , . . . , t ℓ ) = b} is obtainable from S by evaluating the k ✸-symbols in (6). We have two cases: 1. There exists a constraint C i , 1 ≤ i ≤ k, and points w, w ′ ∈ S with w( c i ) = w ′ ( c i ) but w(q i ) = w ′ (q i ). Now let S ′ be a team obtained from S by evaluating the k ✸-symbols in (6) such that, for every w ′ ∈ S ′ , w(t 1 , . . . , t ℓ ) is an expansion of w( c i ). Now clearly M, S ′ |= ϕ cons and as a consequence M, S |= ✸ ℓ (ϕ cons ∨ p ⊥ ⊆ p θ ). From this (6) follows.

For each
. . , f k be some functions that arise from the fact that the constraints (C 1 , . . . , C k ) are satisfied in S. Since, by assumption, ϕ is non-valid under the constraint (C 1 , . . . , C k ), it follows that there exists an assignment s : {p 1 , . . . , p n } → {0, 1} such that (5) holds. Now recall that each tuple of values assigned to p := (p 1 , . . . , p n ) is realised in S. Accordingly, in particular, s and s(q 1 → f 1 s( c 1 )), . . . , q k → f k (s( c k ))) are realised in S. For this reason M, S |= ✸ ℓ (p ⊥ ⊆ p θ ), from which (6) follows in a straightforward manner.
"⇐": Assume that M, T |= ✷ n ✸ ℓ (ϕ cons ∨ p ⊥ ⊆ p θ ) holds for every team pointed Kripke model M, T that satisfies the structural properties forced by ϕ struc . We need to show that ϕ is non-valid under the constraint (C 1 , . . . , C k ). In order to show this, let f 1 , . . . , f k be arbitrary functions with arities that correspond to the constraint (C 1 , . . . , C k ). Let M, T be a team pointed Kripke model and S a team of M such that a) M, T satisfies the structural properties forced by ϕ struc , b) S is obtained from T by evaluating the n ✷-symbols, c) f i w( c i ) = w(q i ), for each w ∈ S and 1 ≤ i ≤ k.
It is easy to check that such a model and teams always exist. From the assumption we then obtain that M, S |= ✸ ℓ (ϕ cons ∨ p ⊥ ⊆ p θ ).
But since the values of q i s, by construction, do not violate the constraint (C 1 , . . . , C k ), we obtain, with the help of the structural properties, that for (7) to hold is must be the case that M, S k |= p ⊥ ⊆ p θ , where S k is some team obtained from S by evaluating the k ✸-symbols in (7). But this means that there exists an assignment s : {p 1 , . . . , p n } → {0, 1} such that s(q 1 → f 1 s( c 1 )), . . . , q k → f k (s( c k ))) |= θ.
Consequently, the claim holds.
In order to compute ϕ non-val two binary counters bounded above by n + k + ℓ need to be maintained. Note that log(n + k + ℓ) is logarithmic with respect to the input length. That being the case, the reduction is computable in logspace and the lemma applies.
The construction in the previous proof works also for strict semantics. In the proof of the claim a small adjustment is needed to facilitate the strict semantics of diamond. As a result we obtain the following. Proof Let ϕ be a formula of EMinc and k the modal depth of ϕ. Let ϕ 1 , . . . , ϕ n be exactly those subformulae of ϕ that occur as a parameter of some inclusion atom in ϕ and let p 1 , . . . , p n be distinct fresh proposition symbols. Define where ϕ ⊥ subst denotes the negation normal form of ¬ϕ subst and ϕ + is the formula obtained from ϕ by simultaneously substituting each ϕ i by p i . It is easy to check that ϕ is valid if and only if the Minc formula ϕ * is. Clearly ϕ * is computable from ϕ in polynomial time.

Conclusion
In this paper we investigated the computational complexity of model checking and validity for propositional and modal inclusion logic in order to complete the complexity landscape of these problems in the mentioned logics. In particular we emphasise on the subtle influence of which semantics is considered: strict or lax. The model checking problem for these logics under strict semantics is NP-complete and under lax semantics P-complete. The validity problem is shown to be coNP-complete for the propositional strict semantics case. For the modal case we achieve a coNEXP lower bound under lax as well as strict semantics. The upper bound is left open for further research. It is however easy to establish that, if closed under polynomial reductions, the complexities of VAL(Minc) and VAL(EMinc), and VAL(Minc s ) and VAL(EMinc s ) coincide, respectively, see Proposition 33.   7 Related work and further research Tables 3-5 give an overview of the current state of research for satisfiability, model checking and validity in the propositional and modal team semantics setting for both strict and lax variants. In the tables AEXP[poly] refers to alternating exponential time with polynomially many alternations. We also identify the unclassified cases open for further research. As these tables also mention atoms which have not been considered elsewhere in this paper, we will introduce them shortly: Let p, q, and r be tuples of proposition symbols and q a proposition symbol. Then dep( p, r) is a propositional dependence atom and q ⊥ p r is a conditional independence atom with the following semantics: X |= dep( p, q) ⇔ ∀s, t ∈ X : s( p) = t( p) implies s(q) = t(q).
Intuitively, q ⊥ p r states that for any fixed value for p, q and r are informationally independent. We also consider the contradictory negation ∼ in our setting: Semantics for these atoms in the modal setting is defined analogously. When C is a set of atoms, we denote by PL(C) and ML(C) the extensions of PL and ML, in the team semantics setting, by the atoms in C, respectively.
A fruitful direction for future research is to study automatic reasoning in the team semantics setting. ? ? ? Table 5: Complexity of Validity, where all = {dep(·) , ⊆, ⊥, ∼}. Complexity classes refer to completeness results, "-h." denotes hardness and "∈" denotes containment. ⋆: Proof for lax semantics works also for strict semantics.
?: No nontrivial result is known.