CoNP Complexity for Combinations of Non-normal Modal Logics

Abstract

We study the complexity of the validity/derivability problem for combinations of non-normal modal logics in the form of logic fusions, possibly extended with simple interaction axioms. We first present cut-free sequent calculi for these logic combinations. Then, we introduce hypersequent calculi with invertible rules, and show that they allow for a coNP proof search procedure. In the last part of the paper, we consider the case of combinations of logics sharing a universal modality. Using the hypersequent calculi, we show that these logics remain coNP-complete, and also provide an equivalent axiomatisation for them.

1 Introduction

Modal logics that combine different modalities have widespread diffusion. On the one hand, modal logics designed for applications usually contain multiple operators, possibly with interactions among them. On the other hand, non-standard modal logics, such as intuitionistic or description modal logics, have been connected with classical logics with combined modalities [18, 19, 46, 47], an observation that allowed for a fruitful transfer of results among the different formalisms.

Concerning logics designed for applications, several systems contain modalities that display a non-normal behaviour, as they do not satisfy some principles that are validated by any normal operator. Significant examples are epistemic logics without omniscience [4], deontic logics [1], agency and ability logics [6, 14, 26], coalition logics [37, 43]. At the same time, the recent introduction of non-normal systems based on intuitionistic or description logic [9, 10, 12, 40, 41] naturally raises the question of their connections with classical systems with combined non-normal modalities.

Multimodal logics obtained as combinations of normal systems have been extensively studied, with a specific focus on fusions and products [19, 20, 45], and the transfer of properties from the single systems to their combinations. Concerning fusions of normal logics, it is known for instance that decidability, interpolation [45] and semantic completeness [17, 30] are always preserved, whereas the complexity of the satisfiability/validity problem is not: while fusions of PSpace logics generally remain PSpace, the same does not hold for fusions of systems with coNP validity (respectively, NP satisfiability) problem, as witnessed by the PSpace bimodal logics $\mathsf {S5_2}$, $\mathsf {KD45_2}$, $\mathsf {K4.3_2}$ and $\mathsf {S4.3_2}$ [25, 42], in contrast with their coNP monomodal counterparts¹ (see [19] for an overview on transfer results).

Although most studies focus on combinations of normal modal logics, similar questions have been also addressed for fusions of non-normal systems. In particular, decidability [3, 23] and superamalgamation [21, 22] (an algebraic property corresponding to a form of interpolation) are known to be preserved, while completeness is not [15, 16]. By contrast, less is understood about the transfer of complexity results, which is the topic of the present work.

Non-normal modal logics (NNMLs in the following) are good examples of coNP modal logics. These logics are defined by extending classical propositional logic with the congruence rule $A \leftrightarrow B / \Box A \leftrightarrow \Box B$ and combinations of standard modal axioms (cf. Sec. 2). As shown by Vardi [44], in this family of logics, the complexity of the validity problem strictly depends on the presence of the agglomeration axiom $\Box A \wedge \Box B \rightarrow \Box (A \wedge B)$: the logics with this axiom are in PSpace, whereas the logics without it are coNP-complete.² Differently from the coNP normal systems mentioned above, the same complexity bounds hold for the multi-modal formulations of these logics where all modalities are of the same kind [44]. For this reason, combinations of NNMLs are promising in terms of preservation of coNP complexity.

In this paper, we investigate the complexity of the validity problem for some kinds of combinations of coNP NNMLs. In particular, we consider all coNP NNMLs of the classical cube [7, 34] as well as their coNP extensions with non-iterative modal axioms. We first consider the fusions of NNMLs, roughly corresponding to the disjoint union of the modal axiomatisations of the combined systems, as well as their extensions with interaction axioms of the form $\Box _i A \rightarrow \Box _j A$ (that correspond, for instance, to the well-known principles of ‘ought implies can’ and ‘does implies can’ of deontic and agency logics (see e.g. [1, 6, 14])). In the last part of the paper we also consider the case of combinations of NNMLs sharing a universal modality. While most studies on property transfers are based on algebraic or model-theoretical techniques, we adopt here a proof-theoretical approach. We first present cut-free sequent calculi for these logic combinations. Then we present a reformulation of the calculi in terms of hypersequents where all the rules are invertible, and show that they provide a coNP decision procedure for validity in the logics. In the last part of the paper, we consider the case of combinations of logics sharing a universal modality. Using the hypersequent calculi, we show that these logics remain coNP-complete.

2 Non-normal Modal Logics and Their Combinations

Given a set of unary modalities $\{\Box _1, ..., \Box _n\}$, we denote $\mathcal {L}[\Box _1,...,\Box _n]$ the propositional modal language based on a set $Atm= \{ p_1, p_2, p_3, ...\}$ of countably many propositional variables, containing the Boolean operators $\bot$, $\rightarrow$, and the modalities $\Box _1, ..., \Box _n$. We consider $\top , \lnot , \wedge , \vee , \Diamond _i$ to be defined as usual.

Non-normal monomodal logics are defined in a language $\mathcal {L}[\Box _i]$, for some $i\in \mathbb {N}$, by extending any axiomatisation of classical propositional logic (containing modus ponens), formulated in $\mathcal {L}[\Box _i]$, with the rule ${RE}_i$ below, and a combination of the following axioms:

The minimal non-normal monomodal logic defined in $\mathcal {L}[\Box _i]$, denoted by $\textsf{E}_i$, only contains ${RE}_i$ (that is, it does not contain any additional modal axiom). Given a list of modal axioms $\varSigma _i$ in $\mathcal {L}[\Box _i]$ (without repetitions), the other non-normal monomodal systems are denoted by $\mathsf {E\Sigma }_i$. We call monotonic any system $\mathsf {E\Sigma }_i$ such that ${M}_i\in \varSigma _i$. Moreover, we use $\textsf{L}_i$ to denote any logic defined in $\mathcal {L}[\Box _i]$.

Fig. 1.

Diagram of non-normal monomodal logics.

We consider the standard notion of derivability in axiomatic modal systems: a rule $B_1, ..., B_n / A$ is derivable in a logic $\textsf{L}_i$ if there is a finite sequence of formulas ending with A in which every formula is an (instance of an) axiom of $\textsf{L}_i$, or it belongs to $\{B_1, ..., B_n\}$, or it is obtained from previous formulas by the application of a rule of $\textsf{L}_i$. A formula A is derivable in $\textsf{L}_i$, written $\vdash _{\textsf{L}_i} A$, if the rule $\emptyset / A$ is derivable in $\textsf{L}_i$. Finally, a formula A is (locally) derivable from a set of formulas $\varPhi$ in $\textsf{L}_i$, written $\varPhi \vdash _{\textsf{L}_i} A$, if there is a finite set $\{B_1, ..., B_n\}\subseteq \varPhi$ such that $\vdash _{\textsf{L}_i} B_1\wedge ... \wedge B_n \rightarrow A$. We recall that the axioms ${M}_i$ and ${N}_i$ are respectively equivalent to the monotonicity rule $A \rightarrow B / \Box _i A \rightarrow \Box _i B$ and to the necessitation rule $A / \Box _i A$. Note also that the axioms ${P}_i$ and ${D}_i$ are equivalent in normal modal logics (i.e., modal logics extending $\textsf{K}_i$), but are not equivalent in non-normal ones. In particular, the following derivability relations hold: $\vdash _{\textsf{ET}_i} {P}_i$, $\vdash _{\textsf{ET}_i} {D}_i$, $\vdash _{\textsf{EMD}_i} {P}_i$, $\vdash _{\textsf{END}_i} {P}_i$. By virtue of these relations, the considered family contains 17 distinct monomodal logics, displayed in Fig. 1.

In this paper, we study multimodal logics obtained by combining non-normal monomodal logics in the following way. First, let $\textsf{L}_1, ..., \textsf{L}_n$ be n non-normal monomodal logics respectively formulated in the languages $\mathcal {L}[\Box _1]$, ..., $\mathcal {L}[\Box _n]$ sharing the same propositional variables and Boolean operators, but with distinct modalities $\Box _1$, ..., $\Box _n$. Moreover, let $\mathcal {I}$ be an acyclic set of pairs (i, j) with $1 \le i, j \le n$ (that is, there is no chain $(i, j_1)$, $(j_1,j_2)$, ..., $(j_k, i)$).

Definition 1

The combination $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ is the smallest multimodal logic in the language $\mathcal {L}[\Box _1,...,\Box _n]$ that contains $\textsf{L}_1 \cup ... \cup \textsf{L}_n$ as well as the interaction axioms $\Box _i A \rightarrow \Box _j A$, for all $(i,j)\in \mathcal {I}$, and is closed under the rules of $\textsf{L}_1$, ..., $\textsf{L}_n$ (that is, modus ponens and ${RE}_1$, ..., ${RE}_n$).

Note that $\langle \textsf{L}_1...\textsf{L}_n\emptyset \rangle$ corresponds to the fusion of $\textsf{L}_1, ..., \textsf{L}_n$ [45]. The reason for restricting to acyclic sets $\mathcal {I}$ is that in presence of cycles $(i, j_1)$, $(j_1,j_2)$, ..., $(j_k, i)$, the modalities $\Box _i$, $\Box _{j_1}$, ..., $\Box _{j_k}$ become all indistinguishable. In the following, for every logic $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, we denote $\mathcal {I^*}$ the transitive closure of $\mathcal {I}$.

The standard semantics of non-normal monomodal logics is given in terms of so-called neighbourhood models. Dealing with multimodal logics, we consider here models endowed with n neighbourhood functions, one for each modality.

Definition 2

A n-neighbourhood model is a tuple $\mathcal M= (\mathcal W,\mathcal N_1, ..., \mathcal N_n, \mathcal V)$, where $\mathcal W$ is a non-empty set of worlds, $\mathcal V:Atm\longrightarrow \mathcal P(\mathcal W)$ is a valuation function, and each $\mathcal N_i$ is a neighbourhood function $\mathcal W\longrightarrow \mathcal P(\mathcal P(\mathcal W))$ possibly satisfying the following conditions for all $w \in \mathcal W$, where $\alpha , \beta \subseteq \mathcal W$:

$$\begin{array}{l l l l l l l l} ({M}_i\hbox {-c}) &{} \text {if }\alpha \in \mathcal N_i(w) \text { and }\alpha \subseteq \beta , \text { then }\beta \in \mathcal N_i(w); &{}&{} ({N}_i\hbox {-c}) &{} \mathcal W\in \mathcal N_i(w); \\ ({T}_i\hbox {-c}) &{} \text {if }\alpha \in \mathcal N_i(w), \text { then }w\in \alpha ; &{}&{} ({P}_i\hbox {-c}) &{} \emptyset \notin \mathcal N_i(w); \\ ({D}_i\hbox {-c}) &{} \text {if }\alpha \in \mathcal N_i(w), \text { then }\mathcal W\setminus {\alpha }\notin \mathcal N_i(w); &{}&{} ({Int}_{ij}\hbox {-c}) \ {} &{} \mathcal N_i(w) \subseteq \mathcal N_j(w). \\ \end{array}$$

Given a monomodal logic $\mathsf {E\Sigma }_i$ and a neighbourhood function $\mathcal N_i$, we say that $\mathcal N_i$ is a $\mathsf {E\Sigma }_i$-function if it satisfies Condition ($\sigma _i$-c), for every ${\sigma _i} \in {\varSigma _i}$. Moreover, we say that a model $\mathcal M= (\mathcal W,\mathcal N_1, ..., \mathcal N_n, \mathcal V)$ is a model for a multimodal logic $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, or it is a $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-model, if $\mathcal N_i$ is a $\textsf{L}_i$-function for all $1 \le i \le n$, and $\mathcal M$ satisfies (${Int}_{ij}$-c) for all $(i,j)\in \mathcal {I}$.

The relation $\mathcal M, w \Vdash A$ is defined as usual for propositional variables and Boolean connectives, while for $\Box _i$ it is as follows, where $\llbracket A\rrbracket _{\mathcal M} = \{v \mid \mathcal M, v \Vdash A\}$:

$$\mathcal M, w \Vdash \Box _i A \ \ \text { iff }\ \ \llbracket A\rrbracket _{\mathcal M}\in \mathcal N_i(w).$$

We consider the usual notions of validity in a model $\mathcal M$ and validity in a class of models $\mathcal {C}$: $\mathcal {M}\,\models \,A$ iff $\mathcal {M}, w \Vdash A$, for all w of $\mathcal M$; and $\mathcal {C}\,\models \,A$ iff $\mathcal {M}\,\models \,A$, for all $\mathcal {M}\in \mathcal {C}$, respectively. In the following, we omit to specify $\mathcal M$, and simply write $w \Vdash A$ or $\llbracket A\rrbracket$, when it is clear from the context.

In this paper, we study the complexity of the validity problem for the logics $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, that is, the problem of deciding, given a formula A of $\mathcal {L}[\Box _1,...,\Box _n]$, whether A is valid in the class of all $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-models. Due to the following completeness result, the validity problem for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ is equivalent to the derivability problem for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, that is, the problem of deciding whether A is derivable in the axiomatic system $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ (Definition 1).

Theorem 1

A formula A of $\mathcal {L}[\Box _1,...,\Box _n]$ is derivable in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ if and only if it is valid in the class of all $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-models.

Proof

Soundness is routine by showing that all axioms and rules are, respectively, valid and validity preserving in the corresponding models. For completeness, we adapt the standard proof for non-normal monomodal logics (cf. [7]). As usual, we call $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-maximal consistent (or maxcons) any set $\varPhi$ of formulas of $\mathcal {L}[\Box _1,...,\Box _n]$ such that $\varPhi \not \vdash _{\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle }\bot$, and for all $A \in \mathcal {L}[\Box _1,...,\Box _n]$, $A \notin \varPhi$ implies $\varPhi \cup \{A\}\vdash _{\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle } \bot$. Moreover, we denote $[A]$ the class of $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-maxcons sets s.t. $A \in \varPhi$. The usual properties of maxcons sets hold, in particular: if $\varPhi \not \vdash _{\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle } \bot$, then there is $\varPsi$ $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-maxcons s.t. $\varPhi \subseteq \varPsi$. We define the canonical model for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ as $\mathcal M= ( \mathcal W,\mathcal N_1, ..., \mathcal N_n, \mathcal V)$, where $\mathcal W$ is the class of all $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-maxcons sets, and for all $p\in Atm$, $\mathcal V(p) = [p]$. Moreover, for all $1 \le i \le n$ and all $\varPhi \in \mathcal W$, we define $\alpha \in \mathcal N_i(\varPhi )$ iff $\alpha = [A]$ for some $\Box _j A \in \varPhi$ s.t. $j = i$ or $(j,i)\in \mathcal {I^*}$, or $\alpha \supseteq [B]$ for some $\Box _k B\in \varPhi$ s.t. $k = i$ or $(k,i)\in \mathcal {I^*}$, and ${M}_i\in \textsf{L}_i$, or ${M}_k\in \textsf{L}_k$, or ${M}_u\in \textsf{L}_u$ for some u s.t. $(k,u),(u,i)\in \mathcal {I^*}$. We can show that $\mathcal M$ is a $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-model, and that for all $A\in \mathcal {L}[\Box _1,...,\Box _n]$, $\llbracket A\rrbracket = [A]$. Then the completeness of $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ follows in the usual way.    $\square$

3 Sequent Calculi

In this section, we present sequent calculi for all the considered combinations of NNMLs. We show that the calculi are sound and cut-free complete with respect to the corresponding axiomatic systems.

In the following, we use capital Greek letters $\varGamma , \varDelta , \varPi , \varTheta$ to denote possibly empty multisets of formulas. As usual, we call sequent any pair $\varGamma \Rightarrow \varDelta$ of finite multisets of formulas. Sequents are interpreted in the language of the logic by the formula interpretation $\iota (\varGamma \Rightarrow \varDelta ) = \bigwedge \varGamma \rightarrow \bigvee \varDelta$, if $\varGamma \not = \emptyset$, and $\iota (\varGamma \Rightarrow \varDelta ) = \bigvee \varDelta$, if $\varGamma = \emptyset$, where $\bigvee \emptyset = \bot$.

Fig. 2.

Sequent rules.

Sequent calculi for non-normal monomodal logics are studied in [27, 28, 31, 34, 36].³ For each logic $\textsf{L}_i$, the corresponding sequent calculus $\mathbb {S}.\textsf{L}_i$ contains the propositional rules $\textsf{init}$, ${\bot }_\textsf{L}$, ${\rightarrow }_\textsf{L}$, ${\rightarrow }_\textsf{R}$ and suitable modal rules from Fig. 2, as summarised in Fig. 3.

Fig. 3.

Modal rules of sequent calculi for non-normal monomodal logics.

Concerning the other rules in Fig. 2, note that the order of the indexes i, j is relevant for $\textsf{e}_{ij}$ and $\textsf{m}_{ij}$ ($\Box _i A$ is in $\varGamma$ while $\Box _j B$ is in $\varDelta$), while it is not relevant for $\textsf{d}_{ij}$ and $\textsf{md}_{ij}$ (both $\Box _i A$ and $\Box _j B$ are in $\varGamma$). Accordingly, we assume $\textsf{d}_{ij}= \textsf{d}_{ji}$ and $\textsf{md}_{ij}= \textsf{md}_{ji}$, whereas $\textsf{e}_{ij}\not = \textsf{e}_{ji}$ and $\textsf{m}_{ij}\not = \textsf{m}_{ji}$. The sequent calculi for the combinations of NNMLs are defined as follows.

Definition 3

The sequent calculus $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ contains, for all $1\le i\le n$, all the rules of $\mathbb {S}.\textsf{L}_i$ different from $\textsf{d}_i'$, as well as the following rules:

$\textsf{e}_{ij}$,:

if $(i,j)\in \mathcal {I^*}$, and $\textsf{m}_i\notin \mathbb {S}.\textsf{L}_i$, and $\textsf{m}_j\notin \mathbb {S}.\textsf{L}_j$, and there is no k such that $(i, k), (k,j)\in \mathcal {I^*}$ and $\textsf{m}_{k}\in \mathbb {S}.\textsf{L}_{k}$;

$\textsf{m}_{ij}$,:

if $(i,j)\in \mathcal {I^*}$, and $\textsf{m}_i\in \mathbb {S}.\textsf{L}_i$ or $\textsf{m}_j\in \mathbb {S}.\textsf{L}_j$ or there is k s.t. $\textsf{m}_{k}\in \mathbb {S}.\textsf{L}_{k}$ and $(i, k), (k,j)\in \mathcal {I^*}$;

$\textsf{n}_i$,:

if there is j such that $(j,i)\in \mathcal {I^*}$ and $\textsf{n}_j\in \mathbb {S}.\textsf{L}_j$;

$\textsf{d}_i$,:

if there is j such that $(i,j)\in \mathcal {I^*}$, and $\textsf{e}_{ij}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and $\textsf{d}_j\in \mathbb {S}.\textsf{L}_{j}$;

$\textsf{md}_i$,:

if there is j such that $(i,j)\in \mathcal {I^*}$, and $\textsf{m}_{ij}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and $\textsf{d}_j\in \mathbb {S}.\textsf{L}_{j}$ or $\textsf{md}_j\in \mathbb {S}.\textsf{L}_{j}$;

$\textsf{d}_{ij}$,:

if there is k such that (1) $(i,k)\in \mathcal {I^*}$, and (2) $(j,k)\in \mathcal {I^*}$ or $k = j$, and (3) $\textsf{d}_k\in \mathbb {S}.\textsf{L}_k$, and (4) $\textsf{e}_{ik},\textsf{e}_{jk}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$,

$\textsf{md}_{ij}$,:

if there is k s.t. (1) $(i,k)\in \mathcal {I^*}$, (2) $(j,k)\in \mathcal {I^*}$ or $k = j$, (3) $\textsf{d}_k\in \mathbb {S}.\textsf{L}_k$ or $\textsf{md}_k\in \mathbb {S}.\textsf{L}_k$, and (4) $\textsf{m}_{ik}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{m}_{jk}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$;

$\textsf{p}_i$,:

if there is j such that $j = i$ or $(i,j)\in \mathcal {I^*}$, and $\textsf{p}_j\in \mathbb {S}.\textsf{L}_{j}$ or there is k such that $\textsf{n}_k\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and $\textsf{d}_{jk}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{md}_{jk}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$;

$\textsf{d}_i'$,:

if $\textsf{p}_i\notin \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and there is j s.t. $j = i$ or $(i,j)\in \mathcal {I^*}$, and $\textsf{d}_j'\in \mathbb {S}.\textsf{L}_j$.

$\textsf{t}_i$,:

if there is j such that $(i,j)\in \mathcal {I^*}$ and $\textsf{t}_j\in \mathbb {S}.\textsf{L}_j$.

The rules listed in Definition 3 are necessary in order to ensure cut-free completeness of the sequent calculi in presence of interactions. Two examples of calculi resulting from the definition are as follows:

$$\begin{array}{rcl} \mathbb {S}\langle \textsf{EN}_1,\textsf{ET}_2,\textsf{EM}_3\{(1,2),(2,3)\}\rangle &{} = &{} \{\textsf{e}_1, \textsf{n}_1, \textsf{e}_2, \textsf{t}_2, \textsf{m}_3, \textsf{m}_{1,2}, \textsf{m}_{1,3}, \textsf{m}_{2,3}, \textsf{t}_1, \\ {} &{}&{} \ \textsf{n}_2, \textsf{n}_3\}; \\ \mathbb {S}\langle \textsf{EN}_1,\textsf{EM}_2,\textsf{ED}_3\{(1,3),(2,3)\}\rangle &{} = &{} \{\textsf{e}_1, \textsf{n}_1, \textsf{m}_2, \textsf{e}_3, \textsf{d}_3, \textsf{e}_{1,3}, \textsf{m}_{2,3}, \textsf{n}_3, \textsf{d}_{1,3}, \\ {} &{}&{} \ \textsf{d}_1, \textsf{md}_{2,3}, \textsf{p}_3, \textsf{p}_1, \textsf{p}_2\}.\\ \end{array} $$

As usual, initial sequents are formulated only for propositional variables but can be extended to arbitrary formulas. We say that a rule is admissible in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ if whenever the premisses are derivable in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, the conclusion is also derivable, and that a single-premiss rule is height-preserving admissible in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ (hp-admissible for short) if whenever the premiss is derivable, the conclusion is derivable with a derivation of at most the same height. Moreover, we say that a rule $\mathcal S_1, ..., \mathcal S_n/ \mathcal S'$ is height-preserving invertible in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ (hp-invertible) if the rule $\mathcal S'/\mathcal S_i$ is hp-admissible for all premisses $\mathcal S_i$. One can show that the propositional rules of $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ are hp-invertible, by contrast the modal rules are not (with the exception of $\textsf{t}_i$). As an easy example, consider the sequents $p \Rightarrow q$ and $\Box _i p \Rightarrow \Box _i q, \Box _i (p \vee r)$, respectively premiss and conclusion of an instance of $\textsf{m}_i$, where the conclusion is derivable and the premiss is not.

Proposition 1

In every calculus $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, the following structural rules $\textsf{Lwk}$, $\textsf{Rwk}$, $\textsf{Lctr}$ and $\textsf{Rctr}$ are hp-admissible, and the following rule $\textsf{cut}$ is admissible:

Proof

Hp-admissibility of $\textsf{Lwk}$, $\textsf{Rwk}$, $\textsf{Lctr}$ and $\textsf{Rctr}$ is proved as usual by mutual induction on the height of the derivation of their premisses (with $\textsf{d}_i'$ ensuring that contraction is admissible also in the calculi with $\textsf{d}_i$). Admissibility of $\textsf{cut}$ is proved by induction on the lexicographically ordered pairs (c, h), where c is the weight of the cut formula, and $h = h_1 + h_2$ is the cut height, where $h_1$ and $h_2$ are the heights of the derivations of the premisses of $\textsf{cut}$. The proof is standard and distinguishes some cases according to whether the cut formula is or not principal in the last rules applied in the derivation of the premisses of cut. Here we only show two representative cases, where the cut formula is principal in the last rule applied in the derivation of both premisses of $\textsf{cut}$.

($\textsf{e}_{iu} - \textsf{md}_{uj}$) The derivation on the left is converted into the one on the right:

where the application of $\textsf{cut}$ has a lower height, and $\textsf{md}_{ij}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ by Definition 3. Indeed, $\textsf{e}_{iu}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ implies $(i,u)\in \mathcal {I^*}$. Moreover, since $\textsf{md}_{uj}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, following Definition 3 there are three possibilities: (1) $(u,j)\in \mathcal {I^*}$, and $\textsf{m}_{uj}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and $\textsf{d}_j\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{md}_j\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$; or (2) $(j,u)\in \mathcal {I^*}$, and $\textsf{m}_{ju}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and $\textsf{d}_u\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{md}_u\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$; or (3) there is k such that $(u,k),(j,k)\in \mathcal {I^*}$, and $\textsf{m}_{uk}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{m}_{jk}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and $\textsf{d}_k\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{md}_k\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. If (1), then $(i,j)\in \mathcal {I^*}$ and $\textsf{m}_{ij}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. If (2), then $(i,u),(j,u)\in \mathcal {I^*}$. If (3), then $(i,k),(j,k)\in \mathcal {I^*}$. In all these cases, by Definition 3, $\textsf{md}_{ij}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$.

($\textsf{m}_{ij} - \textsf{p}_j$) The derivation on the left is converted into the one on the right:

where the application of $\textsf{cut}$ has a lower height, and $\textsf{p}_i\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ by Definition 3. Indeed, $\textsf{m}_{ij}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ implies $(i,j)\in \mathcal {I^*}$. Moreover, since $\textsf{p}_{j}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ we have three possibilities: (1) $\textsf{p}_{j}\in \mathbb {S}.\textsf{L}_j$; or (2) there is k such that $(j,k)\in \mathcal {I^*}$ and $\textsf{p}_{k}\in \mathbb {S}.\textsf{L}_k$; or (3) there are k, u such that $(j,k),(k.u)\in \mathcal {I^*}$, $\textsf{n}_{u}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and $\textsf{d}_{ku}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{md}_{ku}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. If (1), then by Definition 3, $\textsf{p}_i\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. If (2) or (3), then $(i,k)\in \mathcal {I^*}$, and in both cases by Definition 3, $\textsf{p}_i\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$.    $\square$

Theorem 2

$\varGamma \Rightarrow \varDelta$ is derivable in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ if and only if $\bigwedge \varGamma \rightarrow \bigvee \varDelta$ is derivable in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$

Proof

($\Rightarrow$) For each rule $\mathcal S/ \mathcal S'$ or $\mathcal S_1,\mathcal S_2 / \mathcal S'$ of $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, we need to show that the corresponding rule $\iota (\mathcal S) / \iota (\mathcal S')$ or $\iota (\mathcal S_1),\iota (\mathcal S_2) / \iota (\mathcal S')$ is derivable in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. We consider as an example the rule $\textsf{md}_{ij}$, and write $\vdash$ for $\vdash _{\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle }$. First, it is easy to see that $\vdash \Box _i A \rightarrow \Box _j A$ for all $(i,j)\in \mathcal {I^*}$. Now suppose that $\vdash A \wedge B \rightarrow \bot$, hence $\vdash A \rightarrow \lnot B$. By Definition 3, there is k such that $(i,k)\in \mathcal {I^*}$ or $k = i$, $(j,k)\in \mathcal {I^*}$ or $k = j$, $\textsf{d}_k\in \mathbb {S}.\textsf{L}_k$ or $\textsf{md}_k\in \mathbb {S}.\textsf{L}_k$, and $\textsf{m}_{ik}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{m}_{jk}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. Then, by def. of monomodal calculi, ${D}_k\in \textsf{L}_k$. Suppose that $\textsf{m}_{ik}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. One can show that the rule $C\rightarrow D/ \Box _i C \rightarrow \Box _k D$ is derivable in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ for any C, D. Then since $\vdash A \rightarrow \lnot B$, we have $\vdash \Box _i A \rightarrow \Box _k\lnot B$. Moreover, we have $\vdash \Box _j B \rightarrow \Box _k B$. Then by ${D}_k$, $\vdash \Box _i A \wedge \Box _j B \rightarrow \bot$, thus $\vdash \bigwedge \varGamma \wedge \Box _i A \wedge \Box _j B \rightarrow \bigvee \varDelta$ for all $\varGamma$, $\varDelta$. If $\textsf{m}_{jk}\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ the proof is analogous. ($\Leftarrow$) By showing that all axioms and rules of $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ are derivable, respectively admissible, in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, with modus ponens simulated by $\textsf{cut}$ in the usual way.    $\square$

In this paper, we provide a proof of coNP-complexity for the validity problem for the logics $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ following a strategy based on a reformulation of the calculi $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ in terms of hypersequents, as explained in the next section. Alternatively, it could be possible to devise a strategy directly based on the calculi $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ only.⁴ To this goal, two key observations are in order. First, it is easy to see that in any proof tree $\mathcal {T}$ for $\varGamma \Rightarrow \varDelta$ in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, every branch of $\mathcal {T}$ has polynomial length with respect to the length n of $\varGamma \Rightarrow \varDelta$. Second, for every non-invertible modal rule, at most quadratically many premisses (w.r.t. n) are possible. This would allow one to obtain certificates for non-derivability in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ verifiable in polynomial time by a deterministic Turing machine. We leave as future work further investigation in this direction.

4 Invertible Calculi and CoNP Complexity

In this section, we present a proof of coNP complexity for the logics $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ based on a reformulation of the sequent calculi $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ where all the rules are invertible. In particular, in order to make the modal rules invertible, we rewrite all the rules using hypersequents, following the strategy of [11]. We show that the hypersequent calculi $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ provide a coNP decision procedure for the validity problem in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. Specifically, we present a coNP proof search algorithm in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ that explicitly constructs a derivation for every valid hypersequent/formula. Moreover, we show that from every failed derivation one can extract a countermodel of the input hypersequent: this means that we can construct a countermodel of every non-valid formula.

A hypersequent $\mathcal H$ [2] is a finite multiset of sequents, and is written $\varGamma _1 \Rightarrow \varDelta _1 \mid ... \mid \varGamma _k \Rightarrow \varDelta _k$, where $\varGamma _1 \Rightarrow \varDelta _1$, ..., $\varGamma _k \Rightarrow \varDelta _k$ are called the components of $\mathcal H$. The hypersequent rules for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ are direct reformulation of the sequent rules, and are displayed in Fig. 4. Essentially, backward applications of the hypersequent modal rules introduce a new component which coincides with the premiss of the corresponding sequent rule. In this way, all information contained in the conclusion is preserved into the premisses, thus making alternative rule applications still possible in bottom-up proof search. Concerning the propositional rules, we consider a cumulative formulation of them where the principal formulas are kept into the premisses. As we will see, this allows us to easily extract countermodels from failed proofs.

Differently from sequents, hypersequents cannot be interpreted as formulas of $\mathcal {L}[\Box _1,...,\Box _n]$ (we will come back to this problem in the next section). Hypersequents are evaluated on n-neighbourhood models as: $\mathcal M, w \Vdash \varGamma \Rightarrow \varDelta$ if and only if $\mathcal M, w \Vdash \iota (\varGamma \Rightarrow \varDelta )$; $\mathcal M\,\models \,\varGamma \Rightarrow \varDelta$ if and only if $\mathcal M, w \Vdash \varGamma \Rightarrow \varDelta$, for all w of $\mathcal M$; and $\mathcal M\,\models \,\varGamma _1 \Rightarrow \varDelta _1 \mid ... \mid \varGamma _k \Rightarrow \varDelta _k$ if and only if $\mathcal M\,\models \,\varGamma _\ell \Rightarrow \varDelta _\ell$, for some $1 \le \ell \le k$.

Fig. 4.

Hypersequent rules.

Definition 4

The hypersequent calculus $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ is defined as $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ (Definition 3), with the difference that the rules are formulated in their hypersequent version (Fig. 4).

We first show that the calculi are sound and complete with respect to the corresponding logics. Since hypersequents do not have a formula interpretation, we consider a semantic proof of soundness.

Proposition 2

If $\mathcal H$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, then $\mathcal H$ is valid in every n-neighbourhood model for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$.

Proof

It is immediate to see that the initial hypersequents $\textsf{init}$ and ${\bot }_\textsf{L}$ are valid in every model. We need to show that all rules of $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ are validity preserving in every model for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. We consider as an example the rule $\textsf{md}_{ij}$: Suppose that $\mathcal M\models \mathcal H\mid \varGamma , \Box _i A, \Box _j B \Rightarrow \varDelta \mid A, B \Rightarrow$. If $\mathcal M\models \mathcal H\mid \varGamma , \Box _i A, \Box _j B \Rightarrow \varDelta$ we are done. Otherwise $\mathcal M\models A, B \Rightarrow$, that is, $\llbracket A\rrbracket \subseteq \llbracket \lnot B\rrbracket$. As a consequence of Definition 3, $\textsf{md}_{ij}$ belongs to $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ in two cases: (1) $(i,j)\in \mathcal {I^*}$ and $\mathcal M$ satisfies (${D}_{j}$-c) and (${M}_{i}$-c) or (${M}_{j}$-c), or (2) there is k such that $(i,k),(j,k)\in \mathcal {I^*}$ and $\mathcal M$ satisfies (${D}_{k}$-c) and (${M}_{i}$-c) or (${M}_{j}$-c) or (${M}_{k}$-c). If (1), then suppose $w\Vdash \Box _i A$, that is $\llbracket A\rrbracket \in \mathcal N_i(w)$. If (${M}_{i}$-c), then $\llbracket \lnot B\rrbracket \in \mathcal N_i(w)$, and by (${Int}_{ij}$-c), $\llbracket \lnot B\rrbracket \in \mathcal N_j(w)$. Otherwise by (${Int}_{ij}$-c), $\llbracket A\rrbracket \in \mathcal N_j(w)$, and by (${M}_{j}$-c), $\llbracket \lnot B\rrbracket \in \mathcal N_j(w)$. Thus by (${D}_{j}$-c), $\llbracket B\rrbracket \notin \mathcal N_j(w)$. If (2), let us assume (${M}_{k}$-c), the other cases being similar. Suppose $w\Vdash \Box _i A \wedge \Box _j B$. Then $\llbracket A\rrbracket \in \mathcal N_i(w)$ and $\llbracket B\rrbracket \in \mathcal N_j(w)$. By (${Int}_{ik}$-c) and (${Int}_{jk}$-c), $\llbracket A\rrbracket ,\llbracket B\rrbracket \in \mathcal N_k(w)$, thus $\llbracket B\rrbracket ,\llbracket \lnot B\rrbracket \in \mathcal N_k(w)$, against (${D}_{k}$-c). Thus in both cases $w\not \Vdash \Box _i A \wedge \Box _j B$. Since this holds for every w, we have $\mathcal M\models \Box _i A, \Box _j B \Rightarrow$, hence $\mathcal M\models \mathcal H\mid \varGamma , \Box _i A, \Box _j B \Rightarrow \varDelta$.    $\square$

To prove completeness, we consider here a simple proof that relies on the cut-free completeness of the sequent calculi, although a direct proof of cut elimination analogous to the one in the previous section could be given. The proof is based on the following observation, which can be easily proved by induction on the height of the derivation of the premiss of the rules.

Lemma 1

The rules of external weakening and external contraction are height-preserving admissible in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$:

Proposition 3

If $\varGamma \Rightarrow \varDelta$ is derivable in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, then $\varGamma \Rightarrow \varDelta$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$.

Proof

By induction on the height of the derivation of $\varGamma \Rightarrow \varDelta$ in $\mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, considering the last rule applied in the derivation. For initial sequents and propositional rules the proof is immediate. For modal rules, suppose that $\varGamma \Rightarrow \varDelta$ is obtained from $\mathcal S_1$ and (possibly) $\mathcal S_2$ by the application of the sequent rule $R$. Then by i.h., $\mathcal S_1$ and $\mathcal S_2$ are derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and by $\textsf{Ewk}$, $\varGamma \Rightarrow \varDelta \mid \mathcal S_1$ and $\varGamma \Rightarrow \varDelta \mid \mathcal S_2$ are derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. Then by the hypersequent version of the rule $R$, $\varGamma \Rightarrow \varDelta$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$.    $\square$

Another immediate consequence of the height-preserving admissibility of external weakening is that all the rules of $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ are height-preserving invertible in the calculi. It follows that one single proof search is sufficient to establish whether a hypersequent is derivable or not. However, as a difference with sequent rules, backward applications of the hypersequent rules increase the complexity of the hypersequents, thus proof search in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ does not terminate per se. In order to retrieve termination but also obtain an optimal proof search, following [11] (cf. also [32]), we consider a proof search strategy based on the following loop checking condition and on a fixed order of rule applications.

Definition 5

An application of a hypersequent rule with premisses $\mathcal G_1$, ..., $\mathcal G_n$ and conclusion $\mathcal H$ satisfies the local loop checking condition (LLCC) if for each premiss $\mathcal G_i$, there exists a component $\varGamma \Rightarrow \varDelta$ in $\mathcal G_i$ such that for no component $\varPi \Rightarrow \varTheta$ of the conclusion $\mathcal H$ we have $\textsf{set}(\varGamma ) \subseteq \textsf{set}(\varPi )$ and $\textsf{set}(\varDelta ) \subseteq \textsf{set}(\varTheta )$. Moreover, having fixed an enumeration $R_1, ..., R_m$ of the rules of $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, we say that the backward application of a rule $R_i$ with conclusion $\mathcal H$ satisfies the priority order (PO) if there is no $R_j$ backward applicable to $\mathcal H$ with $j < i$.

Bottom-up proof search with LLCC and PO is described by Algorithm 1. We now show that bottom-up proof search with LLCC and PO is complete, and that it provides a coNP procedure for deciding derivability in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$.

Proposition 4

If $\mathcal H$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, then it is derivable with a derivation in which all rule applications satisfy the LLCC and the PO.

Proof

First, we show by induction on the height n of the derivation $\mathcal {D}$ of $\mathcal H$ in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ that if $\mathcal H$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, then it is derivable respecting the LLCC: If $n = 0$, then $\mathcal H$ is an initial hypersequent and $\mathcal {D}$ trivially satisfies LLCC. For $n + 1$, let R be the last rule applied in $\mathcal {D}$. If R satisfies the LLCC, then we apply the i.h. to its premisses and are done. Otherwise, there is a premiss $\mathcal G_i$ of R such that for all components $\varGamma \Rightarrow \varDelta$ in $\mathcal G_i$, there is $\varPi \Rightarrow \varTheta$ in $\mathcal H$ s.t. $\textsf{set}(\varGamma ) \subseteq \textsf{set}(\varPi )$ and $\textsf{set}(\varDelta ) \subseteq \textsf{set}(\varTheta )$. Then $\mathcal H$ can be obtained from $\mathcal G_i$ by means of height-preserving applications of the structural rules. Again, by applying the i.h. we obtain a derivation of $\mathcal H$ where every rule application satisfies the LLCC. Moreover, given the invertibility of the rules, any derivation can be transformed into one satisfying PO by rearranging the order of the rule applications.    $\square$

Proposition 5

For every logic $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, Algorithm 1 runs in coNP.

Proof

The algorithm is presented in the form of a non-deterministic Turing machine with only universal states (that is, states that are accepting if every transition leads to some accepting state), thus in order to prove that it runs in coNP, we need to show that every computation takes polynomial time. Let $\mathcal H$ be the input hypersequent and n be the size of $\mathcal H$ defined as the sum of the lengths of the formulas occurring in it. Since every backward application of a rule introduces a formula or a component, the number of possible rule applications, whence the number of computation steps, is bounded by the maximal length of the hypersequents that can be generated by the procedure. Given that all formulas occurring in a hypersequent are subformulas of some formulas occurring in $\mathcal H$, and that the LLCC avoids multiple occurrences of the same formulas in the same components, every component has length at most $\mathcal O(n)$. Moreover, new components are generated by a modal formula or a pair of modal formulas. Because of the LLCC, no matter in which component their occur, the same formula or pair of formulas cannot generate more than one component. Then the number of components is bounded by $\mathcal O(n) + \mathcal O(n) + \mathcal O(n^2)$. It follows that every hypersequent has a maximal length of $\mathcal O(n^3)$. Finally, checking that a premiss does not violate the LLCC takes polynomial time in the length of the conclusion. Thus the whole execution takes polynomial time.    $\square$

In order for the procedure to succeed, it is necessary that all executions terminate on an initial hypersequent, hence a single failed execution is sufficient to ensure the non-derivability of the input hypersequent. In this latter case, the procedure constructs a hypersequent which is not initial and it is such that no rule is backward applicable to it without violating the LLCC. We call such a hypersequent saturated. We now show that from a saturated hypersequent we can extract a countermodel of the input hypersequent.

Definition 6

Let $\mathcal H= \varGamma _1 \Rightarrow \varDelta _1 \mid \ldots \mid \varGamma _k \Rightarrow \varDelta _k$ be a saturated hypersequent returned by Algorithm 1 on input $\mathcal G$ and $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. For all formulas B occurring in $\mathcal H$ and all $1 \le i \le n$, we define

$$\begin{array}{rcl} \lfloor B \rfloor _i &{} = &{} \{\ell \mid B \in \varGamma _\ell \} ; \\ \lceil B \rceil _i &{} = &{} {\left\{ \begin{array}{ll} \mathcal W\setminus \{\ell \mid B \in \varDelta _\ell \}, &{} \text {if }\textsf{L}_i \text { is not monotonic;} \\ \mathcal W, &{} \text {if }\textsf{L}_i\text { is monotonic;} \end{array}\right. } \\ \eta _i &{} = &{} {\left\{ \begin{array}{ll} \{\mathcal W\}, &{} \text {if there is }j\text { such that }j = i \text { or }(i,j)\in \mathcal {I^*}, \text { and }{N}_j\in \textsf{L}_j;\\ \emptyset , &{} \text {otherwise.} \end{array}\right. }\\ \end{array}$$

Then the model $\mathcal M= ( \mathcal W, \mathcal N_1, ..., \mathcal N_n, \mathcal V)$ is defined with $\mathcal W= \{\ell \mid \varGamma _\ell \Rightarrow \varDelta _\ell \in \mathcal H\}$; for all $p\in Atm$, $\mathcal V(p) = \{\ell \mid p \in \varGamma _\ell \}$; and for all $1 \le i \le n$ and all $1 \le \ell \le k$,

$$\begin{array}{cc} \mathcal N_i(\ell ) = \eta _i \cup \{\alpha \subseteq \mathcal W\mid \text {there is } \Box _j B\in \varGamma _\ell \text { such that } j = i \text { or } (j,i)\in \mathcal {I^*}, \\ \text { and } \lfloor B \rfloor _j \subseteq \alpha \subseteq \lceil B \rceil _j\}. \end{array}$$

Proposition 6

Let $\mathcal H$ be a saturated hypersequent returned by Algorithm 1 on input $\mathcal G$ and $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, and $\mathcal M$ be the model defined on the basis of $\mathcal H$ as in Definition 6. Then for all formulas B and all worlds $\ell$ of $\mathcal M$, it holds:

• if $B\in \varGamma _\ell$, then $\mathcal M, \ell \Vdash B$;

• if $B\in \varDelta _\ell$, then $\mathcal M, \ell \not \Vdash B$.

Moreover, $\mathcal M$ is a $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-model.

Proof

The first claim is proved by induction on the construction of B. For $B = p$, $B = \bot$ and $B = C \wedge D$ the proof is standard. Suppose $B = \Box _i C \in \varGamma _\ell$. By i.h., $\lfloor C \rfloor _i\subseteq \llbracket C\rrbracket \subseteq \lceil C \rceil _i$. Then by definition, $\llbracket C\rrbracket \in \mathcal N_i(\ell )$, thus $\mathcal M, \ell \Vdash \Box _i C$. Now suppose $B = \Box _i C \in \varDelta _\ell$. If there is no $\Box _i D \in \varGamma _\ell$ or $\Box _j D \in \varGamma _\ell$ with $(j,i)\in \mathcal {I^*}$, then if $\eta _i = \emptyset$, then $\mathcal N_i(\ell ) = \emptyset$, hence $\mathcal M, \ell \not \Vdash \Box _i C$. If instead $\eta _i = \{\mathcal W\}$, then $\mathcal N_i(\ell ) = \{\mathcal W\}$, moreover by Definition 3, $\textsf{n}_i\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, hence by Definition 4, $\textsf{n}_i\in \mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. Thus, since $\mathcal H$ is saturated, there is $\varGamma _m \Rightarrow \varDelta _m$ in $\mathcal H$ where $C \in \varDelta _m$, then by i.h., $\mathcal M, m \not \Vdash C$, hence $\llbracket C\rrbracket \not =\mathcal W$, thus $\llbracket C\rrbracket \notin \mathcal N_i(\ell )$, hence $\mathcal M, \ell \not \Vdash \Box _i C$. Otherwise let $\Box _j D \in \varGamma _\ell$ with $j = i$ or $(j,i)\in \mathcal {I^*}$. If $\textsf{L}_i$ is monotonic, then by the rule $\textsf{m}_{ji}$ there is $\varGamma _m \Rightarrow \varDelta _m$ in $\mathcal H$ such that $D \in \varGamma _m$ and $C\in \varDelta _m$, while if $\textsf{L}_i$ is not monotonic, then by the rule $\textsf{e}_{ji}$ there is $\varGamma _m \Rightarrow \varDelta _m$ in $\mathcal H$ such that $D \in \varGamma _m$ and $C\in \varDelta _m$, or $C \in \varGamma _m$ and $D\in \varDelta _m$. In the first case, by i.h., $\lfloor D \rfloor _j\not \subseteq \llbracket C\rrbracket$, and in the second case, $\lfloor D \rfloor _j\not \subseteq \llbracket C\rrbracket$ or $\llbracket C\rrbracket \not \subseteq \lceil D \rceil _j$. Since this holds for all $\Box _j D \in \varGamma _\ell$ with $j = i$ or $(j,i)\in \mathcal {I^*}$, $\llbracket C\rrbracket \notin \mathcal N_i(\ell )$, thus $\mathcal M, \ell \not \Vdash \Box _i C$.

We now prove that $\mathcal M$ is a $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-model. From the definition of $\mathcal N_i$ it follows immediately that (${Int}_{ij}$-c) is satisfied for all $(i,j)\in \mathcal {I^*}$, that (${M}_i$-c) is satisfied if ${M}_i \in \textsf{L}_i$, and that (${N}_i$-c) is satisfied if ${N}_i \in \textsf{L}_i$. We show (${D}_i$-c) as an example for the other conditions: Suppose that ${D}_i\in \textsf{L}_i$ and, by contradiction, $\alpha \in \mathcal N_i(\ell )$ and $\mathcal W\setminus \alpha \in \mathcal N_i(\ell )$. By def. of the monomodal calculi, $\textsf{d}_i\in \mathbb {S}.\textsf{L}_i$ or $\textsf{md}_i\in \mathbb {S}.\textsf{L}_i$. Moreover, by def. of $\mathcal N_i$, there is $\Box _j B\in \varGamma _\ell$ s.t. $j = i$ or $(j,i)\in \mathcal {I^*}$, and $\lfloor B \rfloor _j\subseteq \alpha \subseteq \lceil B \rceil _j$, and either there is $\Box _u C\in \varGamma _\ell$ s.t. $u = i$ or $(u,i)\in \mathcal {I^*}$, and $\lfloor C \rfloor _u \subseteq \mathcal W\setminus \alpha \subseteq \lceil C \rceil _u$, which implies $\lfloor B \rfloor _j\cap \lfloor C \rfloor _u=\emptyset$ and $\mathcal W\setminus \lceil B \rceil _j\cap \mathcal W\setminus \lceil C \rceil _u=\emptyset$, or $\mathcal W\setminus \alpha = \mathcal W$ and $\eta _i = \{\mathcal W\}$. There are four possible cases. (1) If $j = u$ and $B = C$, then by Definition 3, $\textsf{d}_j'\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{p}_j\in \mathbb {S}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, hence by Definition 4, $\textsf{d}_j'\in \mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{p}_j\in \mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. Thus by saturation of $\mathcal H$, there is $\varGamma _m\Rightarrow \varDelta _m$ in $\mathcal H$ s.t. $B\in \varGamma _m$ or $B\in \varDelta _m$. Then $m\in \lfloor B \rfloor _j$ or $m\in \mathcal W\setminus \lceil B \rceil _j$. Since $\lfloor B \rfloor _j=\lfloor C \rfloor _u$ and $\lceil B \rceil _j=\lceil C \rceil _u$, this gives a contradiction. (2) If $j = u$ and $B \not = C$, by Definition 3 and 4 we have $\textsf{d}_j\in \mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{md}_j\in \mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. (3) If $j \not = u$, by Definition 3 and 4, $\textsf{d}_{ju}\in \mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ or $\textsf{md}_{ju}\in \mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. In both cases, by saturation there is $\varGamma _m\Rightarrow \varDelta _m$ in $\mathcal H$ s.t. $B,C\in \varGamma _m$ or $B,C\in \varDelta _m$, which implies $m\in \lfloor B \rfloor _j\cap \lfloor C \rfloor _j$ or $m\in \mathcal W\setminus \lceil B \rceil _j\cap \mathcal W\setminus \lceil C \rceil _j$, giving a contradiction. (4) $\mathcal W\setminus \alpha = \mathcal W$ and $\eta _i = \{\mathcal W\}$, that is $\alpha = \emptyset$. By Definition 3 and 4, $\textsf{p}_j\in \mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$. Thus there is $\varGamma _m\Rightarrow \varDelta _m$ in $\mathcal H$ s.t. $B\in \varGamma _m$, then $\lfloor B \rfloor _j\not =\emptyset$, then $\alpha \not =\emptyset$, giving a contradiction. It follows that $\alpha \notin \mathcal N_i(\ell )$ or $\mathcal W\setminus \alpha \notin \mathcal N_i(\ell )$.    $\square$

Note that the model $\mathcal M$ of Proposition 6 is also a countermodel for the input hypersequent $\mathcal G$. Indeed, since backward rule applications never delete formulas or components, for all components $\varGamma \Rightarrow \varDelta$ in $\mathcal G$, there is $\varPi \Rightarrow \varTheta$ in $\mathcal H$ such that $\textsf{set}(\varGamma ) \subseteq \textsf{set}(\varPi )$ and $\textsf{set}(\varDelta ) \subseteq \textsf{set}(\varTheta )$. Thus the world corresponding to $\varPi \Rightarrow \varTheta$ in $\mathcal M$ falsifies also $\varGamma \Rightarrow \varDelta$. In the light of this model extraction, Algorithm 1 can be easily reformulated in order to provide a NP decision procedure for the satisfiability problem in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, with the algorithm taking as input hypersequents of the form $A \Rightarrow$. On the basis of the above results, we can conclude the following.

Theorem 3

The validity problem for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ is coNP-complete.

Fig. 5.

Hypersequent rules for universal modality.

5 Adding the Universal Modality

As we have seen, hypersequents cannot be interpreted in the language of NNMLs. The reason is that the hypersequent construct “$\mid$” semantically corresponds to a disjunction of validities of sequents. In order to make the hypersequent calculi fully internal, we now extend the language with a universal modality $\mathcal {U}$, and add to the calculi suitable hypersequent rules for it. This operation allows us to treat another kind of logic combinations, namely the combination of NNMLs whose common language also contains $\mathcal {U}$ (together with the propositional variables and the Boolean connectives). Differently from the combinations introduced in Sect. 2, we define these logic combinations not based on the axiomatic systems, but based on the hypersequent calculi. We show that this extension of the calculi still provides a coNP proof search procedure, and also allows one to extract suitable countermodels. Based on the hypersequent calculi and the formula interpretation of the hypersequents, we also provide an axiomatisation for the resulting logics.

Let $\mathcal {L}[\Box _1,...,\Box _n]^\mathcal {U}$ be the language containing the modalities $\Box _1$, ..., $\Box _n$ as well as $\mathcal {U}$. Hypersequents are now interpreted in $\mathcal {L}[\Box _1,...,\Box _n]^\mathcal {U}$ by considering the standard formula interpretation of hypersequent calculi for $\textsf{S5}$ [2, 38]:

$$\iota (\varGamma _1 \Rightarrow \varDelta _1 \mid ... \mid \varGamma _n \Rightarrow \varDelta _n) = \mathcal {U}(\bigwedge \varGamma _1 \rightarrow \bigvee \varDelta _1) \vee ... \vee \mathcal {U}(\bigwedge \varGamma _n \rightarrow \bigvee \varDelta _n).$$

Moreover, let $\textsf{L}_1, ..., \textsf{L}_n$ be n non-normal monomodal logics respectively formulated in the languages $\mathcal {L}[\Box _1]$, ..., $\mathcal {L}[\Box _n]$, with $\Box _1$, ..., $\Box _n$ all distinct but sharing the same propositional variables, Boolean operators, and universal modality $\mathcal {U}$.

Definition 7

For every calculus $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ from Sect. 4, the corresponding calculus $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ in $\mathcal {L}[\Box _1,...,\Box _n]^\mathcal {U}$ contains the rules of $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$, plus the rules ${\mathcal {U}}_\textsf{L}$, ${\mathcal {U}}_\textsf{R}$ and $\mathcal {U}_{\textsf{t}}$ in Fig. 5. Moreover, we call $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$-model any $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$-model (Definition 2), where $\mathcal {U}$ is interpreted as $\mathcal M, w \Vdash \mathcal {U} A$ if and only if $\mathcal M, v \Vdash A$ for all worlds v of $\mathcal M$.

The rules for $\mathcal {U}$ are taken from [38] (see also [39] for similar rules, while different hypersequent rules for $\textsf{S5}$ can be found in [29] and references therein). We start by showing that some of the results proved for $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ immediately extend to $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$.

Proposition 7

If $\mathcal H$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$, then $\mathcal H$ is valid in every $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$-model.

Proof

By extending the proof of Proposition 2. We consider as an example the rule ${\mathcal {U}}_\textsf{L}$: Suppose that $\mathcal M\,\models \,\mathcal H\mid \varGamma , \mathcal {U} A \Rightarrow \varDelta \mid \varSigma , A \Rightarrow \varPi$. If $\mathcal M\,\models \,\mathcal H\mid \varGamma , \mathcal {U} A \Rightarrow \varDelta$ we are done. Otherwise $\mathcal M\,\models \,\varSigma , A \Rightarrow \varPi$, and since $\mathcal M\models \mathcal {U} A$ or $\mathcal M\,\models \,\lnot \mathcal {U} A$, from $\mathcal M\not \models \varGamma , \mathcal {U} A \Rightarrow \varDelta$ we get $\mathcal M\models \mathcal {U} A$. Then $\mathcal M\,\models \,\varSigma \Rightarrow \varPi$.    $\square$

Proposition 8

Algorithm 1 on inputs $\mathcal H$ in $\mathcal {L}[\Box _1,...,\Box _n]^\mathcal {U}$ and $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ runs in coNP.

Proof

The proof is exactly as the one of Proposition 5, observing that every formula $\mathcal {U} A$ can generate at most one component (cf. [32]). Note that LLCC and Algorithm 1 remain well-defined on the new inputs.    $\square$

Proposition 9

Let $\mathcal H= \varGamma _1 \Rightarrow \varDelta _1 \mid \ldots \mid \varGamma _k \Rightarrow \varDelta _k$ be a saturated hypersequent returned by Algorithm 1 on input $\mathcal G$ and $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$, and $\mathcal M= ( \mathcal W, \mathcal N_1, ..., \mathcal N_n, \mathcal V)$ be the model defined on the basis of $\mathcal G$ as in Definition 6. Then for all formulas B of $\mathcal {L}[\Box _1,...,\Box _n]^\mathcal {U}$ and all $\ell \in \mathcal W$, it holds: if $B\in \varGamma _\ell$, then $\mathcal M, \ell \Vdash B$, and if $B\in \varDelta _\ell$, then $\mathcal M, \ell \not \Vdash B$. Moreover, $\mathcal M$ is a $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$-model.

Proof

The proof extends the one of Proposition 6 with the case $B = \mathcal {U} C$, which is standard: If $\mathcal {U} C\in \varGamma _\ell$, then by ${\mathcal {U}}_\textsf{L}$ and $\mathcal {U}_{\textsf{t}}$, $C\in \varGamma _m$ for all $m\in \mathcal W$, then by i.h., $\mathcal M, m \Vdash C$ for all $m\in \mathcal W$, that is $\mathcal M, \ell \Vdash \mathcal {U} C$. If $\mathcal {U} C\in \varDelta _\ell$, then by ${\mathcal {U}}_\textsf{R}$ there is $\varGamma _m \Rightarrow \varDelta _m$ in $\mathcal H$ with $C\in \varDelta _m$. By i.h., $\mathcal M, m \not \Vdash C$, thus $\mathcal M, \ell \not \Vdash \mathcal {U} C$.    $\square$

As before, on the basis of Proposition 9, we can obtain from the algorithm a NP decision procedure for satisfiability of $\mathcal {L}[\Box _1,...,\Box _n]^\mathcal {U}$ formulas in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$-models. As a further consequence, Proposition 9 entails that the calculi $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ are complete with respect to the corresponding models. Indeed, if the proof search procedure fails on input $\mathcal H$, then it constructs a saturated hypersequent $\mathcal G$ that extends $\mathcal H$. From Proposition 9 we get a $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$-countermodel of $\mathcal G$, whence of $\mathcal H$, which means that $\mathcal H$ is not $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$-valid.

Theorem 4

$\mathcal H$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ with LLCC and PO if and only if $\mathcal H$ is valid in every $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$-model.

We now take advantage of the completeness of the calculi $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ and of the formula interpretation of hypersequents to provide an axiomatisation for the corresponding logics.

Definition 8

A logic $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ is axiomatically defined as the corresponding logic $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ (Definition 1), but, for each $1 \le i \le n$, replacing ${RE}_i$, ${M}_i$, ${N}_i$, ${D}_i$ and ${P}_i$ with the corresponding axiom ${E}^{\mathcal {U}}_{i}$, ${M}^{\mathcal {U}}_{i}$, ${N}^{\mathcal {U}}_{i}$, ${D}^{\mathcal {U}}_{i}$ and ${P}^{\mathcal {U}}_{i}$ below, and adding ${K}_{\mathcal {U}}$, ${T}_{\mathcal {U}}$, ${5}_{\mathcal {U}}$ and ${RN}_{\mathcal {U}}$ ($\textsf{S5}$ axioms for $\mathcal {U}$):

${T}_i$ is the only axiom that does not change. $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ is an extension of $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ as ${RE}_i$ is derivable in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ for all $1 \le i \le n$, and ${M}_i$, ${N}_i$, ${D}_i$ or ${P}_i$ is derivable if, respectively, ${M}^{\mathcal {U}}_{i}$, ${N}^{\mathcal {U}}_{i}$, ${D}^{\mathcal {U}}_{i}$ or ${P}^{\mathcal {U}}_{i}$ belongs to $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$. Consider as an example ${M}_i$: From $A \wedge B \rightarrow A$, by ${RN}_\mathcal {U}$, $\mathcal {U}(A \wedge B \rightarrow A)$, then by ${M}^{\mathcal {U}}_{i}$, $\mathcal {U}(\Box _i (A \wedge B) \rightarrow \Box _i A)$, thus by ${T}_\mathcal {U}$, $\Box _i (A \wedge B) \rightarrow \Box _i A$. We now show that each logic $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ is equivalent to the corresponding calculus $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$.

Proposition 10

If A is derivable in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$, then $\Rightarrow A$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$, and if $\mathcal H$ is derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$, then $\iota (\mathcal H)$ is derivable in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$.

Proof

For the first claim, one can show that the axioms of $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ are derivable in $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$. For the second claim, we prove that for every rule $\mathcal H/ \mathcal H'$ or $\mathcal H_1,\mathcal H_2 / \mathcal H'$ of $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$, the corresponding rule $\iota (\mathcal H) / \iota (\mathcal H')$ or $\iota (\mathcal H_1),\iota (\mathcal H_2) / \iota (\mathcal H')$ is derivable in $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$. The proof follows the lines of the proof of Theorem 2 ($\Rightarrow$), considering that depending on the logics, additional axioms such as $\mathcal {U}(A \rightarrow B) \wedge \mathcal {U}(B \rightarrow A) \rightarrow \mathcal {U}(\Box _i A \rightarrow \lnot \Box _j\lnot B)$ can be derivable.

Finally, considering the properties of the calculi $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ and their equivalence with the systems $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$, we can conclude the following.

Theorem 5

$\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ is sound and complete with respect to the class of all $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$-models. Moreover, the validity problem for $\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ is coNP-complete.

6 Conclusion

We have proved that the validity/derivability problem for fusions of standard coNP NNMLs, as well as for their extensions with interaction axioms of the form $\Box _i A \rightarrow \Box _j A$, remains coNP-complete, and that the same result holds for combinations of logics sharing also a universal modality. In this respect, combinations of NNMLs display a different behaviour than combinations of standard coNP normal logics such as $\textsf{S5}$, $\textsf{KD45}$, $\mathsf {K4.3}$ and $\mathsf {S4.3}$, whose fusions are instead PSpace.

As we have seen, fully invertible hypersequent calculi offer a good point of view on the problem, as they allow one to decompose its global complexity into the one of the single rule applications. As a further advantage, the hypersequent calculi $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle$ allow one to explicitly construct derivations of valid hypersequents/formulas, as well as to construct countermodels of non-valid hypersequents/formulas. Furthermore, after the integration of the rules for $\mathcal {U}$ from [38], the calculi $\mathbb {H}\langle \textsf{L}_1...\textsf{L}_n\mathcal {I}\rangle ^{\mathcal {U}}$ directly construct countermodels where both $\mathcal {U}$ and the neighbourhood functions behave correctly. This can be compared with alternative techniques such as the submodel generation [5] that might be non-trivial to apply in presence of the neighbourhood functions.

On the other hand, the definition of cut-free calculi for the logics with interaction axioms requires an intricate combinatorial analysis, in future work we would like to study calculi that allow for a modular definition of the logic combinations. We would also like to study logics with iterative axioms such as ${4}$, ${5}$, ${B}$, as well as product-like combinations for NNMLs.