Canonical Models and the
Complexity of Modal Team Logic

It is well-known that non-linear quantifier dependencies, such as $w$ depending only on $z$ in the sentence $\forall x\,\exists y\,\forall z\,\exists w\,\varphi$, cannot be expressed in first-order logic. To overcome this restriction, logics of incomplete information such as independence-friendly logic [HS89] have been studied. Later, Hodges [Hod97] introduced team semantics to provide these logics with a compositional interpretation. The fundamental idea is to not consider single assignments to free variables, but instead whole sets of assignments, called teams.

In this vein, Väänänen [Vää07] expressed non-linear quantifier dependencies by the dependence atom ${=\!\!(x_{1},\ldots,x_{n},y)}$, which intuitively states that the values of $y$ in the team functionally depend on those of $x_{1},\ldots,x_{n}$. Logics with numerous other non-classical atoms such as independence $\perp$ [GV13], inclusion $\subseteq$ and exclusion $\mid$ [Gal12] have been studied since, and manifold connections to scientific areas such as statistics, database theory, physics, cryptography and social choice theory have emerged (see also Abramsky et al. [AKVV16]).

Team semantics have also been adapted to a range of propositional [YV16, HKLV16], modal [Vää08], and temporal logics [KMV15, KMVZ18]. Besides propositional dependence logic $\mathsf{PDL}\xspace$ [YV16] and modal dependence logic $\mathsf{MDL}\xspace$ [Vää08], also propositional and modal logics of independence and inclusion have been studied [KMSV17, HKVV15, HS15, Han17]. Unlike in the first-order setting, the atoms such as the dependence atom range over flat formulas. For example, the instance ${=\!\!(p_{1},\ldots,p_{n},\Diamond\mathsf{unsafe})}$ of a modal dependence atom may specify that the reachability of an unsafe state is a function of $p_{1}\cdots p_{n}$, but instead of exhibiting the explicit function, the atom only stipulates its existence.

Most team logics lack the Boolean negation, and adding it as a connective ${\sim}$ usually increases both the expressive power and the complexity tremendously. The respective extensions of propositional and modal logic are called propositional team logic $\mathsf{PTL}\xspace$ [HKLV16, YV17, HKVV18] and modal team logic $\mathsf{MTL}\xspace$ [Mül14, KMSV15]. With ${\sim}$, these logics can express all the non-classical atoms mentioned above, and in fact are expressively complete for their respective class of models [KMSV15, YV17]. For these reasons, they are both interesting and natural logics.

The expressive power of $\mathsf{MTL}\xspace$ is well-understood [KMSV15], and a complete axiomatization was presented by the author [Lüc18a]. Yet the complexity of the satisfiability problem has been an open question [Mül14, KMSV15, DKV16, HKMV17]. Recently, certain fragments of $\mathsf{MTL}\xspace$ with restricted negation were shown $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp,\mathrm{poly})$-complete using the well-known filtration method [Lüc17]. In the same paper, however, it was shown that no elementary upper bound for full $\mathsf{MTL}\xspace$ can be established by the same approach, whereas the best known lower bound is $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp,\mathrm{poly})$-hardness, inherited from propositional team logic [HKVV18].

Contribution. We show that $\mathsf{MTL}\xspace$ is complete for a non-elementary class we call $\mathrm{TOWER}\xspace(\mathrm{poly})$, which contains the problems decidable in a runtime that is a tower of nested exponentials of polynomial height. Likewise, we show that the fragments $\mathsf{MTL}\xspace_{k}$ of bounded modal depth $k$ are complete for classes we call $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp_{k+1},\mathrm{poly})$ and which corresponds to $(k+1)$-fold exponential runtime and polynomially many alternations. These results fill a long-standing gap in the active field of propositional and modal team logics (see Table 1).

In our approach, we consider so-called canonical models. Loosely speaking, a canonical model satisfies every satisfiable formula in some of its submodels, and such models have been long known for, e.g.\xspace, many systems of modal logic [BRV01]. In Section 4, we adapt this notion for modal logics with team semantics, and prove that such models exist for $\mathsf{MTL}\xspace$. This enables us to reduce the satisfiability problem to simple model checking, albeit on models that are of non-elementary size with respect to ${|\nobreak\Phi\nobreak|}+k$, where $\Phi$ are the available propositional variables and $k$ is a bound on the modal depth.

Nonetheless, this approach is essentially optimal: In Section 5 and 6, we show that $\mathsf{MTL}\xspace$ can, in a certain sense, efficiently enforce canonical models, that is, with formulas that are of size polynomial in ${|\nobreak\Phi\nobreak|}+k$. In this vein, we then obtain the matching complexity lower bounds in Section 7 and 8, where we encode computations of non-elementary length in such large models.

Finally, in Section 9 we extend the preliminary version of this paper [Lüc18b] and consider restrictions of $\mathsf{MTL}\xspace$ to specific frame classes, and to so-called strict team-semantical connectives.

The length of (the encoding of) $x$ is denoted by ${|\nobreak x\nobreak|}$. We assume the reader to be familiar with alternating Turing machines [CKS81] and basic complexity theory. When a problem is hard or complete for a complexity class, in this paper we are always referring to logspace reductions.

The class $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp,\mathrm{poly})$ (also known as $\mathrm{AEXPTIME}\xspace(\mathrm{poly})$) contains the problems decidable by an alternating Turing machine in time $2^{p(n)}$ with $p(n)$ alternations, where $p$ is a polynomial. We generalize it to capture the elementary hierarchy as follows.

Let $\exp_{0}(n)\vcentcolon=n$ and $\exp_{k+1}(n)\vcentcolon=2^{\exp_{k}(n)}$. A function $f\colon\mathbb{N}\xspace\to\mathbb{N}\xspace$ is elementary if it is computable in time ${\mathcal{O}(\exp_{k}(n))}$ for some fixed $k$. In this paper, we consider the elementary hierarchy with polynomially many alternations:

For $k\geq 0$, $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp_{k},\mathrm{poly})$ is the class of problems decidable by an alternating Turing machine with at most $p(n)$ alternations and runtime at most $\exp_{k}(p(n))$, for a polynomial $p$. Note that setting $k=0$ or $k=1$ yields the classes $\mathrm{PSPACE}\xspace$ and $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp,\mathrm{poly})$, respectively [CKS81]. Schmitz [Sch16] proposed the following non-elementary class that contains $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp_{k},\mathrm{poly})$ for all $k$. {defiC}[[Sch16]] $\mathrm{TOWER}\xspace$ is the class of problems decidable by a deterministic Turing machine in time (or equivalently, space) $\exp_{f(n)}(1)$ for an elementary function $f$.

A suitable notion of reduction for this class is the following: An elementary reduction from $A$ to $B$ is an elementary function $f$ such that $x\in A\Leftrightarrow f(x)\in B$. $A\leq^{\mathrm{elem}}_{\mathrm{m}}B$ means that there exists an elementary reduction from $A$ to $B$.

[[Sch16]] $\mathrm{TOWER}\xspace$ is closed under $\leq^{\mathrm{elem}}_{\mathrm{m}}$.

The next class results from imposing a polynomial bound on the number of exponentials in the definition of $\mathrm{TOWER}\xspace$, which leads to a strict subclass. {defi} $\mathrm{TOWER}\xspace(\mathrm{poly})$ is the class of problems that are decided by a deterministic Turing machine in time (or equivalently, space) $\exp_{p(n)}(1)$ for some polynomial $p$.

The reader may verify that both $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp_{k},\mathrm{poly})$ and $\mathrm{TOWER}\xspace(\mathrm{poly})$ are closed under $\leq^{\mathrm{P}}_{\mathrm{m}}$ and $\leq^{\log}_{\mathrm{m}}$. Furthermore, by the time hierarchy theorem, $\mathrm{TOWER}\xspace(\mathrm{poly})\subsetneq\mathrm{TOWER}\xspace$.

To the author’s best knowledge, neither has been explicitly considered before. However, candidates for natural complete problems exist. Although not proved complete, several problems in $\mathrm{TOWER}\xspace(\mathrm{poly})$ are provably non-elementary, such as the satisfiability problem of separated first-order logic [Voi17], the equivalence problem for star-free expressions [SM73], or the first-order theory of finite trees [CH90], to only name a few. We refer the reader also to the survey of Meyer [Mey74].

Another example is the two-variable fragment of first-order team logic, $\mathsf{FO}\xspace^{2}({\sim})$. It is related to $\mathsf{MTL}\xspace$ in the same fashion as classical two-variable logic $\mathsf{FO}\xspace^{2}$ to $\mathsf{ML}\xspace$. By reduction from $\mathsf{MTL}\xspace$ to $\mathsf{FO}\xspace^{2}({\sim})$, the satisfiability problem of $\mathsf{FO}\xspace^{2}({\sim})$ is $\mathrm{TOWER}\xspace(\mathrm{poly})$-complete problems as a corollary of our main result, Theorem 31, while its fragments $\mathsf{FO}\xspace^{2}_{k}({\sim})$ of bounded quantifier rank $k$ are $\mathrm{ATIME}\xspace\text{-}\mathrm{ALT}\xspace(\exp_{k+1},\mathrm{poly})$-hard [Lüc18c].

Next, we justify why we use only $\leq^{\log}_{\mathrm{m}}$-reductions (or polynomial time reductions in general) in this paper instead of $\leq^{\mathrm{elem}}_{\mathrm{m}}$.

We fix a countably infinite set $\mathcal{PS}$ of propositional symbols. Modal team logic $\mathsf{MTL}\xspace$, introduced by Müller [Mül14], extends classical modal logic $\mathsf{ML}\xspace$. Formulas of classical $\mathsf{ML}\xspace$ are built following the grammar

where $p\in\mathcal{PS}$ and $\top$ is constant truth. $\mathsf{MTL}\xspace$ extends $\mathsf{ML}\xspace$ by the grammar

where $\alpha$ denotes an $\mathsf{ML}\xspace$-formula.

The set of propositional variables occurring in a formula $\varphi\in\mathsf{MTL}\xspace$ is $\mathsf{Prop}(\varphi)$. We use the common abbreviations $\bot\vcentcolon=\neg\top$, $\alpha\rightarrow\beta\vcentcolon=\neg\alpha\lor\beta$ and $\alpha\leftrightarrow\beta\vcentcolon=(\alpha\land\beta)\lor(\neg\alpha\land\neg\beta)$. For easier distinction, we have classical $\mathsf{ML}\xspace$-formulas denoted by $\alpha,\beta,\gamma,\ldots$ and reserve $\varphi,\psi,\vartheta,\ldots$ for general $\mathsf{MTL}\xspace$-formulas.

The modal depth $\mathsf{md}(\varphi)$ of a formula $\varphi$ is recursively defined:

$\mathsf{ML}\xspace_{k}$ and $\mathsf{MTL}\xspace_{k}$ are the fragments of $\mathsf{ML}\xspace$ and $\mathsf{MTL}\xspace$ with modal depth $\leq k$, respectively. If the propositions are restricted to a fixed set $\Phi\subseteq\mathcal{PS}$ as well, then the fragment is denoted by $\mathsf{ML}\xspace_{k}^{\Phi}$, or $\mathsf{MTL}\xspace^{\Phi}_{k}$, respectively.

Let $\Phi\subseteq\mathcal{PS}$ be finite. A Kripke structure (over $\Phi$) is a tuple $\mathcal{K}=(W,R,V)$, where $W$ is a set of worlds or points, $(W,R)$ is a directed graph called frame, and $V\colon\Phi\to\mathfrak{P}(W)$ is the valuation, with $\mathfrak{P}(X)$ being the power set of $X$.

Occasionally, by slight abuse of notation, we use the inverse mapping $V^{-1}\colon W\to\mathfrak{P}(\Phi)$ defined by $V^{-1}(w)\vcentcolon=\{p\in\Phi\mid w\in V(p)\}$ instead of $V$, i.e.\xspace, the set of propositions that are true in a given world. If $w\in W$, then $(\mathcal{K},w)$ is called pointed structure. $\mathsf{ML}\xspace$ is evaluated on pointed structures in the classical Kripke semantics.

By contrast, $\mathsf{MTL}\xspace$ is evaluated on pairs $(\mathcal{K},T)$ called structures with teams, where $\mathcal{K}=(W,R,V)$ is a Kripke structure and $T\subseteq W$ is called team (in $\mathcal{K}$). Every team $T$ has an image $RT\vcentcolon=\{v\mid w\in T,(w,v)\in R\}$, and for $w\in W$, we simply write $Rw$ instead of $R\{w\}$. $R^{i}T$ is inductively defined as $R^{0}T\vcentcolon=T$ and $R^{i+1}T\vcentcolon=RR^{i}T$. An $R$-successor team (or simply successor team) of $T$ is a team $S$ such that $S\subseteq RT$ and $T\subseteq R^{-1}S$, where $R^{-1}\vcentcolon=\{(v,w)\mid(w,v)\in R\}$. Intuitively, $S$ is formed by picking at least one $R$-successor of every world in $T$. The semantics of $\mathsf{MTL}\xspace$ can now be defined as follows. ¹¹1Often, the “atoms” of $\mathsf{MTL}\xspace$ are restricted to literals $p,\neg p$ instead of $\mathsf{ML}\xspace$-formulas $\alpha$. However, this implies a restriction to formulas in negation normal form, and both definitions are equivalent due to the flatness property of $\mathsf{ML}\xspace$ (cf. [KMSV15, Proposition 2.2]).

We often omit $\mathcal{K}$ and write only $T\vDash\varphi$ (for team semantics) or $w\vDash\alpha$ (for Kripke semantics).

An $\mathsf{MTL}\xspace$-formula $\varphi$ is satisfiable if it is true in some structure with team over $\mathsf{Prop}(\varphi)$, which is then called a model of $\varphi$. Analogously, $\varphi$ is valid if it is true in every structure with team (over $\mathsf{Prop}(\varphi)$). For a logic $\mathcal{L}$, the sets of all satisfiable resp. valid formulas of $\mathcal{L}$ are $\mathsf{SAT}(\mathcal{L})$ and $\mathsf{VAL}(\mathcal{L})$, respectively.

In the literature on team semantics, the empty team is usually excluded in the above definition, since most ${\sim}$-free logics with team semantics have the empty team property, i.e.\xspace, the empty team satisfies every formula [Vää08, KMSV17, HS15]. However, this distinction is unnecessary for $\mathsf{MTL}\xspace$: $\varphi$ is satisfiable iff $\top\lor\varphi$ is satisfied by some non-empty team²²2Note that $\top\lor\varphi$ is not a tautology in general, since $\lor$ is not the Boolean disjunction. Rather, $\top\lor\varphi$ existentially quantifies a subteam where $\varphi$ holds. In fact, $\top\lor\varphi$ is a tautology if and only if $\varphi$ holds in the empty team., and $\varphi$ is satisfied by some non-empty team iff ${\sim}\bot\land\varphi$ is satisfiable.

The modality-free fragment $\mathsf{MTL}\xspace_{0}$ syntactically coincides with propositional team logic $\mathsf{PTL}\xspace$ [HKLV16, HKVV18, YV17]. The usual interpretations of the latter, i.e.\xspace, sets of Boolean assignments, can easily be represented as teams in Kripke structures. For this reason, we treat $\mathsf{PTL}\xspace$ and $\mathsf{MTL}\xspace_{0}$ as identical in this article.

Note that the connectives $\lor$, $\rightarrow$ and $\neg$ are not the Boolean disjunction, implication and negation, except on singleton teams, which correspond to Kripke semantics. Using $\land$ and ${\sim}$ however, we can define team-wide Boolean disjunction $\varphi_{1}\ovee\varphi_{2}\vcentcolon={\sim}({\sim}\varphi_{1}\land{\sim}\varphi_{2})$ and material implication $\varphi_{1}\rightarrowtriangle\varphi_{2}\vcentcolon={\sim}\varphi_{1}\ovee\varphi_{2}$.

The notation $\Box^{i}\varphi$ is defined via $\Box^{0}\varphi\vcentcolon=\varphi$ and $\Box^{i+1}\varphi\vcentcolon=\Box\Box^{i}\varphi$, and analogously for $\Diamond^{i}\varphi$. To express that at least one element of a team satisfies $\alpha\in\mathsf{ML}\xspace$, we use $\mathsf{E}\alpha\vcentcolon={\sim}\neg\alpha$.

$\mathsf{MTL}\xspace$ can express the (extended) dependence atom ${=\!\!(\alpha_{1},\ldots,\alpha_{n-1},\alpha_{n})}$ of (extended) modal dependence logic [Vää08, EHM⁺13], which states that the truth value of $\alpha_{n}$ is a function of the truth values of $\alpha_{1},\ldots,\alpha_{n-1}$, where $\alpha_{1},\ldots,\alpha_{n}\in\mathsf{ML}\xspace$. It is definable in $\mathsf{MTL}\xspace$ as ${\sim}\left[\top\lor{\sim}\left(\bigwedge_{i=1}^{n-1}{=\!\!(\alpha_{i})}\rightarrowtriangle{=\!\!(\alpha_{n})}\right)\right]$, where ${=\!\!(\alpha)}\vcentcolon=\alpha\ovee\neg\alpha$ is the constancy atom, stating that the truth value of $\alpha\in\mathsf{ML}\xspace$ is constant throughout the team.

The well-known bisimulation relation $\rightleftharpoons^{\Phi}_{k}$ fundamentally characterizes the expressive power of modal logic [BRV01] and plays a key role in our results.

Let $\Phi\subseteq\mathcal{PS}$ and $k\geq 0$. For $i\in\{1,2\}$, let $(\mathcal{K}_{i},w_{i})$ be a pointed structure, where $\mathcal{K}_{i}=(W_{i},R_{i},V_{i})$. Then $(\mathcal{K}_{1},w_{1})$ and $(\mathcal{K}_{2},w_{2})$ are $(\Phi,k)$-bisimilar, in symbols $(\mathcal{K}_{1},w_{1})\rightleftharpoons^{\Phi}_{k}(\mathcal{K}_{2},w_{2})$, if

• •

  $\forall p\in\Phi\colon w_{1}\in V_{1}(p)\Leftrightarrow w_{2}\in V_{2}(p)$,

• •

  and if $k>0$,

  • –

    $\forall v_{1}\in R_{1}w_{1}\colon\exists v_{2}\in R_{2}w_{2}\colon(\mathcal{K}_{1},v_{1})\rightleftharpoons^{\Phi}_{k-1}(\mathcal{K}_{2},v_{2})$ (forward condition),

  • –

    $\forall v_{2}\in R_{2}w_{2}\colon\exists v_{1}\in R_{1}w_{1}\colon(\mathcal{K}_{1},v_{1})\rightleftharpoons^{\Phi}_{k-1}(\mathcal{K}_{2},v_{2})$ (backward condition).

[[GO07, Theorem 32]] Let $\Phi\subseteq\mathcal{PS}$ be finite, $k\geq 0$, and let $(\mathcal{K},w)$ be a pointed structure. Then there is a formula $\zeta\in\mathsf{ML}\xspace^{\Phi}_{k}$ such that for all pointed structures $(\mathcal{K}^{\prime},w^{\prime})$ we have $(\mathcal{K}^{\prime},w^{\prime})\vDash\zeta$ if and only if $(\mathcal{K},w)\rightleftharpoons^{\Phi}_{k}(\mathcal{K}^{\prime},w^{\prime})$.

The notion of bisimulation was lifted to team semantics by Hella et al. [HLSV14, KMSV17, KMSV15]:

Let $\Phi\subseteq\mathcal{PS}$ and $k\geq 0$. For $i\in\{1,2\}$, let $(\mathcal{K}_{i},T_{i})$ be a structure with team. Then $(\mathcal{K}_{1},T_{1})$ and $(\mathcal{K}_{2},T_{2})$ are $(\Phi,k)$-team-bisimilar, written $(\mathcal{K}_{1},T_{1})\rightleftharpoons^{\Phi}_{k}(\mathcal{K}_{2},T_{2})$, if

• •

  $\forall w_{1}\in T_{1}\colon\exists w_{2}\in T_{2}\colon(\mathcal{K}_{1},w_{1})\rightleftharpoons^{\Phi}_{k}(\mathcal{K}_{2},w_{2})$,

• •

  $\forall w_{2}\in T_{2}\colon\exists w_{1}\in T_{1}\colon(\mathcal{K}_{1},w_{1})\rightleftharpoons^{\Phi}_{k}(\mathcal{K}_{2},w_{2})$.

As a consequence, the forward and backward condition from Definition 3 can be equivalently stated in terms of team-bisimilarity of the respective image teams. A similar characterization exists for team-bisimilarity:

Note that the analog of condition 4 in Proposition 3 for team bisimulation is not equivalent: It is possible that $(\mathcal{K}_{1},T_{1})\rightleftharpoons^{\Phi}_{0}(\mathcal{K}_{2},T_{2})$ and $(\mathcal{K}_{1},R_{1}T_{1})\rightleftharpoons^{\Phi}_{k-1}(\mathcal{K}_{2},R_{2}T_{2})$, but $(\mathcal{K}_{1},T_{1})\not\rightleftharpoons^{\Phi}_{k}(\mathcal{K}_{2},T_{2})$.

Many modal logics admit a “universal” model, also called canonical model. The defining property of a canonical model is that it simultaneously witnesses all satisfiable (sets of) formulas in some of its points. These models are a popular tool for proving the completeness of manifold systems of modal logics; for the explicit construction of such a model for $\mathsf{ML}\xspace$, consult, e.g.\xspace, Blackburn et al. [BRV01, Section 4.2].

Unfortunately, any canonical model for $\mathsf{ML}\xspace$ is necessarily infinite, and consequently impractical for complexity theoretic considerations. Instead, we use so-called $(\Phi,k)$-canonical models for finite $\Phi\subseteq\mathcal{PS}$ and $k\in\mathbb{N}\xspace$; as the name suggests they are canonical for the fragment $\mathsf{ML}\xspace^{\Phi}_{k}$. While these models are finite, by Proposition 3 their size is at least the number of equivalence classes of $\rightleftharpoons^{\Phi}_{k}$. We call the equivalence classes of $\rightleftharpoons^{\Phi}_{k}$ types.

A first issue arises since types are then proper classes, and in team semantics, we need to speak about sets of types. For this reason, we begin this section by defining types on proper set-theoretic grounds, by indentifying the type of a point with the set of formulas that are true in it, which is a standard approach in first-order model theory.