# Residues of connections and the Chevalley-Weil formula for curves

Donu Arapura  Department of Mathematics
Purdue University
West Lafayette, IN 47907
U.S.A.

###### Abstract.

Given a finite group of automorphisms of a compact Riemann surface, the
Chevalley-Weil formula computes the character valued Euler characteristic of
an equivariant line bundle. The goal of this article is to give a proof by
computing using residues of a Gauss-Manin connection.

###### Key words and phrases:

Riemann surface, character

###### 2020 Mathematics Subject Classification:

14H55, 14H37

Partially supported by a grant from the Simons foundation

Let $G$ be a finite group of automorphisms of a compact Riemann surface $X$.
Then $G$ will act on the space of holomorphic $1$-forms, more generally on the
space of sections of powers of the canonical bundle $K$, or even more
generally on the cohomology of an equivariant holomorphic line bundle $L$. The
virtual representation $H^{0}(X,L)-H^{1}(X,L)$ is determined by its character
$\chi_{G}(L)$, and there are at least two ways in which one might try to
compute it. First of all, one could try to find a formula for the character as
a complex valued function on the group. The holomorphic Lefschetz fixed point
theorem of Atiyah and Bott [AB] gives a formula for $\chi_{G}(L)$ in this
form:

$\chi_{G}(L)(g)=\sum_{p}\frac{\nu_{p}(g)}{1-\tau_{p}(g)},\>g\not=1$

where the sum runs over fixed points of $g$, and $\tau_{p}$ and $\nu_{p}$ are
the characters of the action of the isotropy groups $G_{p}$ on $T_{p}^{*}$ and
$L_{p}$. The second, and arguably more useful type of formula, is an
expression for $\chi_{G}(L)$ as a linear combination of characters of
irreducible representations, the set of which we denote by $\widehat{G}$.
Chevalley and Weil [CW] gave such a formula when $L$ is a positive power of
the canonical bundle. Formulas of this type for more general $L$ were
considered by Ellingsrud and Lønsted [EL]. When $G$ acts freely on $X$,
$\chi_{G}(L)$ is a multiple of the character of the regular representation
$\chi_{reg}$. More specifically,

$\chi_{G}(L)=\left(\frac{1}{|G|}\deg L+1-h\right)\chi_{reg}$

where $h$ is the genus of $X/G$. If the action is not free, then

$\chi_{G}(L)=\left(\frac{1}{|G|}\deg
L+1-h\right)\chi_{reg}-\sum_{\xi\in\widehat{G}}m_{\xi}(L)\xi$

where the coefficients $m_{\xi}(L)$ are given by explicit formulas depending
on $\xi,\tau_{p}$ and $\nu_{p}$. This can be made much more explicit when $G$
is cyclic, and we will see that the holomorphic Lefschetz fixed point theorem
for finite order automorphisms of Riemann surfaces is a straightforward
consequence of this.

Our main goal is give a new proof of the (generalized) Chevalley-Weil formula.
The argument is neither very efficient nor particularly general, but we hope
that it is instructive. We start by proving a general residue theorem, which
expresses the degree of a vector bundle on a compact Riemann surface as minus
the sum of traces of residues of a singular holomorphic connection. Next given
an equivariant line bundle $L$, we decompose the direct image
$\pi_{*}L=\bigoplus V_{\xi}$ into isotypic components, where $\pi:X\to X/G$ is
the projection. The heart of the proof is the computation of the degrees of
the bundles $V_{\xi}$ by using the residue theorem applied to certain Gauss-
Manin connections. With these in hand, the formula follows easily from
Riemann-Roch on $X/G$.

##  1\. Characters

We start by recalling a few facts from representation theory needed below [S].
Let $G$ be a finite group of order $N$. Let $F(G)$ denote the ring of class
functions on $G$. The vector space $F(G)$ has a basis given by the set
$\widehat{G}$ of characters of irreducible representations. In fact,
$\widehat{G}$ forms an orthonormal basis with respect to the inner product

$\langle\chi,\eta\rangle=\langle\chi,\eta\rangle_{G}=\frac{1}{N}\sum_{g\in
G}\chi(g)\overline{\eta(g)}$

Thus any element $\eta\in F(G)$ can be expanded as a Fourier series

$\eta=\sum_{\chi\in\widehat{G}}\langle\eta,\chi\rangle\chi$

We write

$\chi_{reg}(g)=\sum_{\chi\in\widehat{G}}\chi(1)\chi(g)=\begin{cases}N&\text{if
$g=1$}\\\ 0&\text{otherwise}\end{cases}$

for the character of the regular representation ${\mathbb{C}}[G]$. If
$\chi\in\widehat{G}$, and $M$ is a $G$-module, then the $\chi$-isotypic
component $M_{\chi}\subset M$ is given by $e_{\chi}M$, where

$e_{\chi}=\frac{\chi}{N}\sum_{g\in G}\chi(g^{-1})g\in{\mathbb{C}}[G]$

is a central idempotent.

If $H\subset G$ is a subgroup, let $\operatorname*{Ind}:F(H)\to F(G)$ be
induction. This takes the character of an $H$-module $M$ to the character of
the induced $G$-module ${\mathbb{C}}[G]\otimes_{{\mathbb{C}}[H]}M$. In the
opposite direction we have restriction. Frobenius reciprocity says that these
operations are adjoint, i.e. if $\chi\in F(H),\eta\in F(G)$,

$\langle\chi,\eta|_{H}\rangle_{H}=\langle\operatorname*{Ind}\chi,\eta\rangle_{G}$

##  2\. Statement of the Chevalley-Weil formula

Let $X$ be a compact Riemann surface. Fix a finite subgroup $G\subseteq
Aut(X)$ of order $N$, and let $\pi:X\to Y=X/G$ be the quotient. For each $p\in
X$, let $G_{p}=\\{g\in G\mid gp=p\\}$ be the isotropy group. It acts
faithfully on the cotangent space $T_{p}^{*}=m_{p}/m_{p}^{2}$. In particular,
it is cyclic of finite order, say $N_{p}$. Let $\tau_{p}$ be the character of
the representation of $G_{p}$ on $T_{p}^{*}$. We can choose an isomorphism
$G_{p}\cong{\mathbb{Z}}/N$ such that $\tau_{p}(1)=e^{2\pi i/N}=\zeta$, and a
local coordinate $x$ at $p$ so that $\tau_{p}(1)$ acts by $x\mapsto\zeta x$.

Let $\psi:L\to X$ be a line bundle in the geometric sense, i.e. $L$ is a
complex manifold, with holomorphic projection $\psi$ satisfying the usual
conditions. There is an action of ${\mathbb{C}}^{*}$ on $L$ preserving the
fibres. Let $Aut^{{\mathbb{C}}^{*}}(L)$ denote the group of holomorphic
automorphisms of the manifold $L$ which commute with the action of
${\mathbb{C}}^{*}$. It follows easily that an element of
$Aut^{{\mathbb{C}}^{*}}(L)$ preserves the zero section. Thus we have a
homomorphism

$p:Aut^{{\mathbb{C}}^{*}}(L)\to Aut(X)$

which sends $\gamma$ to its restriction to the zero section. Let

$Aut(X)_{L}=\\{g\in Aut(X)\mid g^{*}L\cong L\\}$

denote the isotropy group of $L\in Pic(X)$. There is an exact sequence

$1\to{\mathbb{C}}^{*}\to Aut^{{\mathbb{C}}^{*}}(L)\xrightarrow{p}Aut(X)_{L}\to
1$

[B, lemma 3.4.1]. Let us suppose that $G\subseteq Aut(X)_{L}$, or equivalently
that $L\in Pic(X)^{G}$ is invariant under $G$. A $G$-linearization of $L$ is a
splitting of

(1) $1\to{\mathbb{C}}^{*}\to p^{-1}G\to G\to 1,$

or in other words a lifting of $G$ to a subgroup of
$Aut^{{\mathbb{C}}^{*}}(L)$. A $G$-equivariant line bundle on $X$ is a line
bundle with a $G$-linearization. Equivariant bundles and invariant bundles are
not quite the same thing. If $Pic_{G}(X)$ denotes the group of equivariant
line bundles, then there is forgetful homomorphism

$Pic_{G}(X)\to Pic(X)^{G}$

In general, it is neither injective nor surjective, but it will have finite
kernel and cokernel. To see this, observe that by basic facts about group
cohomology [Br, chap IV], the obstruction to splitting (1) lies in
$H^{2}(G,{\mathbb{C}}^{*})$. If the obstruction vanishes, then the set of
splittings is parametrized by $H^{1}(G,{\mathbb{C}}^{*})\cong
Hom(G,{\mathbb{C}}^{*})$. Both cohomology groups are finite.

Given a $G$-equivariant line bundle $L$, $G$ will act on $L$ by bundle
automorphisms. Consequently, it will act on the space of holomorphic sections
$\mathcal{L}(U)={\mathcal{O}}_{X}(L)(U)$ for any $G$-invariant set $U$.
Therefore $G$ will act on the Čech complex
$\check{C}(\mathcal{U},\mathcal{L})$ for any covering by $G$-invariant open
Stein sets. (Note that such a cover exists, because we may use the preimage of
a Stein cover of the Riemann surface $X/G$.) Thus $G$ acts on
$H^{i}(X,\mathcal{L})$. Let $\chi_{G}(L)$ denote the character of the virtual
representation

$H^{0}(X,\mathcal{L})-H^{1}(X,\mathcal{L})$

In other words, $\chi_{G}(L)$ is the character of $H^{0}(X,\mathcal{L})$ minus
the character of $H^{1}(X,\mathcal{L})$.

Here are a few examples of equivariant line bundles.

######  Example 2.1.

For any homomorphism $h:G\to{\mathbb{C}}^{*}$, the trivial bundle
${\mathbb{C}}_{X}={\mathbb{C}}\times X$ has a $G$-linearization, where
$g(v,x)=(h(g)v,gx)$. We usually take $h=1$. Then the action on sections
${\mathcal{O}}_{X}(U)$ is the usual action on functions $gf(x)=f(g^{-1}x)$

######  Example 2.2.

The canonical bundle $K$ has a natural $G$-linearization such that the
$G$-action on sections is the action $\omega\mapsto g^{*}\omega$ on $1$-forms.
There is an induced linearization on any power $K^{\otimes n}$.

######  Example 2.3.

If $D=\sum n_{p}p$ is a divisor, then recall that we have a line bundle whose
sheaf of sections is

${\mathcal{O}}_{X}(D)(U)=\\{f\text{ meromorphic on }U\mid\forall p\in
U,ord_{p}(f)\geq-n_{p}\\}$

If $D$ is $G$-invariant, ${\mathcal{O}}_{X}(D)$ has a $G$-linearlization such
that the action is $gf(x)=f(g^{-1}x)$. All $G$-equivariant line bundles arise
this way; more precisely

$Pic_{G}(X)\cong\frac{Div(X)^{G}}{\pi^{*}Princ(Y)}$

[Bo, thm 2.3].

If $L$ is equivariant, then $G_{p}$ will act on the fibre $L_{p}$. Let
$\nu_{p}$ denote the character of this representation.

######  Theorem 2.4.

Let $X,G$ be as above, and let $L$ be an equivariant line bundle on $X$. Let
$Y=X/G$ have genus $h$. Then

(2) $\chi_{G}(L)=\left(\frac{1}{N}\deg
L+1-h\right)\chi_{reg}-\sum_{\xi\in\widehat{G}}m_{\xi}(L)\xi$

where

$m_{\xi,p}(L)=\frac{\xi(1)}{N}\sum_{i=0}^{N_{p}-1}i\langle\xi|_{G_{p}},\nu_{p}\cdot\tau_{p}^{i}\rangle_{G_{p}}$
$m_{\xi}(L)=\sum_{p\in X}m_{\xi,p}(L)$

######  Remark 2.5.

  1. (1)

This is a special case of a more general result due to Ellingsrud and Lønsted
[EL]. When $L=K^{\otimes n}$, the formula goes back to Chevalley and Weil
[CW].

  2. (2)

The sums $m_{\xi}(L)$ are clearly finite since $m_{\xi,p}(L)=0$ unless
$G_{p}\not=\\{1\\}$. We also note that $m_{\xi,p}(L)$ only depends on the
orbit by the arguments of section 4. So we can rewrite

$m_{\xi}(L)=\sum_{q\in Y}|G_{p}|m_{\xi,p}(L)$

where we choose one $p\in X$ over each $q$.

  3. (3)

The formula can be made more explicit in certain cases, see section 5.

######  Corollary 2.6.

If $G$ acts freely, then

$\chi_{G}(L)=\left(\frac{1}{N}\deg L+1-h\right)\chi_{reg}$

In particular, $\chi_{G}(L)$ is a multiple of the regular representation.

##  3\. The Residue theorem

Let $V$ be a holomorphic vector bundle on compact Riemann surface $Y$, and let
$\mathcal{V}={\mathcal{O}}_{Y}(V)$ be its sheaf of holomorphic sections. Let
$\Omega_{Y}^{1}$ be the sheaf of holomorphic $1$-forms. A holomorphic
connection on $V$ (or $\mathcal{V}$) is a ${\mathbb{C}}$-linear map

$\nabla:\mathcal{V}\to\Omega_{Y}^{1}\otimes\mathcal{V}$

satisfying the Leibnitz rule. Note that the curvature necessarily vanishes,
because it locally a matrix of holomorphic $2$-forms. Therefore by Chern-Weil
theory [GH, chap 3, §3], the degree $\deg V:=\int_{Y}c_{1}(V)=0$. This
provides an obstruction to the existence of such a connection. We can avoid it
by allowing singularities. Given a finite set $S\subset Y$, let
$\Omega_{Y}^{1}(\infty S)$ ($\Omega_{X}^{1}(*S)$, resp. $\Omega_{Y}^{1}(\log
S)$) be the sheaf of holomorphic forms on $Y-S$ (with poles of finite order,
resp. order at most $1$, on points of $S$). Given a vector bundle $V$ on $Y$,
a ${\mathbb{C}}$-linear map

$\nabla:\mathcal{V}\to\Omega_{Y}^{1}(\infty S)\otimes\mathcal{V}$

satisfying the Leibnitz rule will be referred to as a singular holomorphic
connection with singularities along $S$. We call $\nabla$ meromorphic (resp.
logarithmic) if it takes values in $\Omega_{Y}^{1}(*S)\otimes\mathcal{V}$
(resp. $\Omega_{Y}^{1}(\log S)\otimes\mathcal{V}$). Any vector bundle $V$
carries a meromorphic connection, namely $\nabla=d$ where
$V|_{X-S}\cong{\mathcal{O}}_{X-S}^{n}$ is a local trivialization of the
underlying algebraic vector bundle (which exists by GAGA). When $V$ is a line
bundle, we will give a more precise construction in lemma 4.3.

The standard reference for logarithmic connections is Deligne [D], although we
will not really need much beyond the definition of residue, which we now
explain. Given $s\in S$, and a basis $\\{v_{i}\\}$ of the stalk
$\mathcal{V}_{s}$ as an ${\mathcal{O}}_{s}$-module, a singular holomorphic
connection is determined by a matrix of $1$-forms $A=(\alpha_{ij})$ given by

$\nabla v_{i}=\sum_{j}\alpha_{ij}\otimes v_{j}$

The forms $\alpha_{ij}$ are holomorphic in a punctured disk about $s$.
Therefore given a local parameter $y$ at $s$, we can form a Laurent expansion

$A=\sum_{n=-\infty}^{\infty}A_{n}y^{n}dy$

where the coefficients are constant matrices. In the logarithmic case,
$A_{-1}$ gives a well defined endomorphism of the fibre $End(V_{s})$ [D, p
78]. For a general singular holomorphic connection, we claim something weaker:

######  Lemma 3.1.

Given a singular holomorphic connection, the trace
$\operatorname*{tr}A_{-1}\in{\mathbb{C}}$ is well defined, i.e. it is
independent of the choice of basis of $\mathcal{V}_{s}$ and local parameter.

###### Proof.

Suppose that $v_{i}^{\prime}=\sum f_{ij}v_{j}$ is a new basis. If
$F=(f_{ij})$, then the new connection matrix is given by
$A^{\prime}=FAF^{-1}+dF\cdot F^{-1}$. Therefore
$\operatorname*{tr}A^{\prime}=\operatorname*{tr}A+\operatorname*{tr}(dF\cdot
F^{-1})$, and so $\operatorname*{tr}A_{-1}^{\prime}=\operatorname*{tr}A_{-1}$.
Cauchy’s integral formula shows that $\operatorname*{tr}A_{-1}$ is independent
of the local parameter. ∎

We denote $\operatorname*{tr}A_{-1}$ by
$\operatorname*{tr}{\operatorname*{Res}}_{s}(\nabla)$.

######  Theorem 3.2.

If $(V,\nabla)$ is vector bundle equipped with a singular holomorphic
connection as above, then

$\deg V=-\sum_{s\in S}\operatorname*{tr}{\operatorname*{Res}}_{s}(\nabla)$

######  Remark 3.3.

  1. (1)

The theorem generalizes the well known fact that the sum of residues of
meromorphic $1$-form $\omega$ is zero [GH, p 222]. This follows by applying
the theorem when $V$ is trivial with connection $d+\omega$.

  2. (2)

When the connection is logarithmic, this is a special case of a theorem of
Esnault and Verdier [EV, appendix B].

###### Proof.

Choose $\epsilon>0$, so that the closed disks $D_{s}(\epsilon)$ of radius
$\epsilon$ (with respect to a metric) around $s$ are disjoint and contained in
coordinate neighbourhoods. By a standard partition of unity argument, we can
construct a $C^{\infty}$ connection $\nabla^{\prime}$ on $V$ which agrees with
$\nabla$ on $\bigcup_{s}D_{s}(\epsilon/2)$. The curvature $\Theta$ of
$\nabla^{\prime}$ must vanish on $Y-\bigcup_{s}D_{s}(\epsilon)$, because it
coincides with the curvature of $\nabla$ on this set. By the Chern-Weil
formula [GH, chap 3, §3]

$\deg V=\frac{i}{2\pi}\int_{Y}\operatorname*{tr}(\Theta)=\sum_{s\in
S}\frac{i}{2\pi}\int_{D_{s}(\epsilon)}\operatorname*{tr}(\Theta)$

Let $B_{s}=(\beta_{ij})$ denote the connection matrix for $\nabla^{\prime}$
with respect to a trivialization of $V$ on $D_{s}(\epsilon)$. Then

$\Theta=dB_{s}+B_{s}\wedge B_{s}$

by the usual formula for curvature (see [GH, p 401] although the sign there
differs from ours). Therefore

$\operatorname*{tr}\Theta=\operatorname*{tr}(dB_{s})+\sum_{ij}\beta_{ij}\wedge\beta_{ij}=d\operatorname*{tr}B_{s}$

Since $B_{s}$ is the connection matrix of $\nabla$ on the annulus
$D_{s}(\epsilon)-D_{s}(\epsilon/2)$,

$\operatorname*{tr}B_{s}=(y^{-1}\operatorname*{tr}{\operatorname*{Res}}_{s}\nabla+f(y))dy$

on the annulus, where $f$ is holomorphic on $D_{s}(\epsilon)-\\{s\\}$ with
zero residue at $s$. Therefore by Stokes’ and Cauchy’s theorems

$\frac{i}{2\pi}\int_{D_{s}(\epsilon)}\operatorname*{tr}(\Theta)=-\frac{1}{2\pi
i}\int_{\partial
D_{s}(\epsilon)}\operatorname*{tr}(B_{s})=-\operatorname*{tr}{\operatorname*{Res}}_{s}(\nabla)$

∎

######  Corollary 3.4.

$\sum_{s\in
S}\operatorname*{tr}{\operatorname*{Res}}_{s}(\nabla)\in{\mathbb{Z}}$

##  4\. Proof of Chevalley-Weil

Let $\pi:X\to Y$ be a nonconstant holomorphic map between compact Riemann
surfaces. Suppose that $L$ is a holomorphic line bundle on $X$ with a
logarithmic connection

$\nabla:L\to\Omega_{X}^{1}(\log S)\otimes L$

After increasing $S$ if necessary, we can assume that it contains all the
ramification points of $\pi$. Then $T=\pi(S)$ will contain the branch points.
Let $\mathcal{L}={\mathcal{O}}_{X}(L)$, and $\mathcal{V}=\pi_{*}\mathcal{L}$.
The direct image

$\mathcal{V}=\pi_{*}\mathcal{L}\xrightarrow{\pi_{*}\nabla}\pi_{*}(\Omega_{X}^{1}(\log
S)\otimes\mathcal{L})\cong\Omega_{Y}^{1}(\log T)\otimes\mathcal{V}$

defines a logarithmic connection$\tilde{\nabla}$, which we refer to as the
Gauss-Manin connection.

We give a local description. Choose $q\in Y$, and let
$\\{p_{1},\ldots,p_{m}\\}=\pi^{-1}(q)$. Let $n_{i}$ denote the ramification
index of $p_{i}$. Choose local coordinates about $p_{i}$ and $q$ so that
$y=x^{n_{i}}$. Let us also choose a local generator $\lambda$ for $L$ at
$p_{i}$. Let $\mathcal{W}_{i}$ be the free ${\mathcal{O}}_{p_{i}}$-module with
basis $\lambda,x\lambda,\ldots,x^{n_{i}-1}\lambda$. Then

$\mathcal{V}_{q}=\bigoplus_{i=1}^{m}\mathcal{W}_{i}$

The Gauss-Manin connection respects this decomposition. We will compute it on
$\mathcal{W}_{i}$. Let us suppose that $\nabla$ has residue $r_{i}$ at
$p_{i}$. We can write

(3) $\begin{split}\tilde{\nabla}(x^{j}\lambda)&=\frac{dx}{x}\otimes
r_{i}x^{j}\lambda+\frac{dx}{x}\otimes jx^{j}\lambda+\ldots\\\
&=\frac{dy}{y}\otimes\left(\frac{r_{i}+j}{n_{i}}\right)x^{j}\lambda+\ldots\end{split}$

where the omitted terms are holomorphic.

######  Lemma 4.1.

$\operatorname*{tr}{\operatorname*{Res}}_{q}\tilde{\nabla}=\sum_{i=1}^{m}{\operatorname*{Res}}_{p_{i}}\nabla+\frac{1}{2}\sum_{i=1}^{m}(n_{i}-1)$

###### Proof.

Using (3), we obtain

$\operatorname*{tr}{\operatorname*{Res}}_{q}\tilde{\nabla}=\sum_{i=1}^{m}\sum_{j=0}^{n_{i}-1}\frac{r_{i}+j}{n_{i}}=\sum_{i=1}^{m}\left(r_{i}+\frac{n_{i}-1}{2}\right)$

∎

######  Corollary 4.2.

Let $R$ be the ramification divisor of $\pi$ then

$\deg\mathcal{V}=\deg L-\frac{1}{2}\deg R$

###### Proof.

This follows from the lemma and theorem 3.2. ∎

Let us now assume that $G\subseteq Aut(X)$ is a subgroup of order $N$, and
that $Y=X/G$. The ramification index of $p\in X$ is exactly $N_{p}=|G_{p}|$.
In particular, $p$ is ramified if and only if it is a fixed point for some
$g\not=1$. The fibre $\pi^{-1}(\pi(p))$ is precisely the orbit $G\cdot p$.

######  Lemma 4.3.

If $L$ is a $G$-equivariant line bundle on $X$, then there exists a
$G$-invariant logarithmic connection

$\nabla:L\to\Omega_{X}^{1}(\log S)\otimes L$

for some invariant $S\subset X$.

###### Proof.

First suppose that $\deg L<0$, then for $m\gg 0$, $L^{\otimes-m}$ will have a
nonzero section $\sigma^{\prime}$. Then
$\sigma^{\vee}=\sigma^{\prime}\otimes\ldots\otimes\sigma^{\prime}$ will give
an invariant section of $L^{\otimes-n}$, where $n=mN$. The dual gives an
invariant injective map $\sigma:{\mathcal{O}}_{X}(L^{\otimes
n})\to{\mathcal{O}}_{X}$. Then $\nabla(\lambda)=d\sigma(\lambda^{\otimes
n})/\sigma(\lambda^{\otimes n})\otimes\lambda$ will have the desired
properties. We can do the general case, by writing $L=L_{1}\otimes
L_{2}^{-1}$, with $\deg L_{i}<0$, and using the tensor product and dual
connections. ∎

The direct image $\mathcal{V}=\pi_{*}\mathcal{L}$ is locally free with
$G$-action. We can decompose

(4)
$\mathcal{V}=\bigoplus_{\xi\in\widehat{G}}\mathcal{V}_{\xi},\>\mathcal{V}_{\xi}=e_{\xi}\mathcal{V}$

We have

$e_{\xi}H^{i}(X,\mathcal{L})=H^{i}(Y,\mathcal{V}_{\xi})$

Therefore by Riemann-Roch

(5)
$\langle\chi_{G}(L),\xi\rangle=\chi(Y,\mathcal{V}_{\xi})=\deg\mathcal{V}_{\xi}+\xi(1)(1-h)$

where $\chi(Y,\mathcal{V}_{\xi})$ is the Euler characteristic of
$\mathcal{V}_{\xi}$. The proof of theorem 2.4 will follow from the computation
of the degree $\deg\mathcal{V}_{\xi}$, which we now explain.

Let us choose an invariant connection

$\nabla:L\to\Omega_{X}^{1}(\log S)\otimes L$

as in the above lemma. Then the Gauss-Manin connection $\tilde{\nabla}$ is
$G$-invariant, so we can decompose $\tilde{\nabla}$ as a sum
$\bigoplus\tilde{\nabla}_{\xi}$ with respect to (4). Choose $p=p_{1}\in T$,
and let $q=\pi(p)$. Let $n=N_{p}$, $m=N/n$,
and$\\{p_{1},\ldots,p_{m}\\}=G\cdot p_{1}=\pi^{-1}(q)$. The modules with
connection $(\mathcal{W}_{i},\tilde{\nabla}|_{\mathcal{W}_{i}})$ are now
isomorphic, so it suffices to look at $\mathcal{W}=\mathcal{W}_{1}$. Let
$W={\mathbb{C}}\otimes_{{\mathcal{O}}_{p}}\mathcal{W}$, and
$V={\mathbb{C}}\otimes_{{\mathcal{O}}_{q}}\mathcal{V}_{q}$ denote the fibres.
The first vector space $W$ is a ${\mathbb{C}}[G_{p}]$-module. The basis
vectors $x^{i}\lambda$ span $G_{p}$-submodules ${\mathbb{C}}_{\nu\tau^{i}}$
with character $\nu\tau^{i}$. Therefore

$W=\bigoplus_{i=0}^{n-1}{\mathbb{C}}_{\nu\tau^{i}}$

We note that $\\{\nu,\nu\tau,\ldots,\nu\tau^{n-1}\\}=\widehat{G}_{p}$. Since
$G_{p}$ is abelian, the $G_{p}$-module

$W\cong{\mathbb{C}}[G_{p}]$

The vector space $V$ is the ${\mathbb{C}}[G]$-module induced from the
$G_{p}$-module $W$. Consequently $V\cong{\mathbb{C}}[G]$. It will useful to
weight the summands, by considering the class function

(6)
$\rho_{q}=\sum_{i=0}^{n-1}\left(\frac{r+i}{n}\right)\operatorname*{Ind}(\nu\tau^{i})$

We can see from (3) and the above remarks that

(7)
$\operatorname*{tr}{\operatorname*{Res}}_{q}\tilde{\nabla}_{\xi}=\langle\rho_{q},\xi\rangle_{G}$

###### Proof of theorem 2.4.

It suffices to prove that

$\langle\chi_{G}(L),\xi\rangle=\left(\frac{1}{N}\deg
L+1-h\right)\xi(1)-m_{\xi}(L)$

for every $\xi\in\widehat{G}$. By comparing with (5), we see that if suffices
to prove that

(8) $\deg\mathcal{V}_{\xi}=\frac{1}{N}(\deg L)\xi(1)-m_{\xi}(L)$

By theorem 3.2,

$\deg\mathcal{V}_{\xi}=-\sum_{q\in
T}\operatorname*{tr}{Res}_{q}\tilde{\nabla}_{\xi}=-\sum_{q\in
T}\langle\rho_{q},\xi\rangle_{G}$

Let us write $\rho_{q}=\rho_{q}^{\prime}+\rho_{q}^{\prime\prime}$, where

$\rho_{q}^{\prime}=\sum_{i=0}^{n-1}\left(\frac{r}{n}\right)\operatorname*{Ind}(\nu\tau^{i})$
$\rho_{q}^{\prime\prime}=\sum_{i=0}^{n-1}\left(\frac{i}{n}\right)\operatorname*{Ind}(\nu\tau^{i})$

As noted above,

$\rho_{q}^{\prime}=\left(\frac{r}{n}\right)\operatorname*{Ind}W=\left(\frac{r}{n}\right)\chi_{reg}$

Therefore

$\langle-\rho_{q}^{\prime},\xi\rangle=\left(\frac{r}{n}\right)\xi(1)=rm\frac{\xi(1)}{N}$

The quantities $r$ and $m$ depend on $q$, even though the notation does not
reflect this. Summing over $q$, and using theorem 3.2 yields

$\sum_{q}rm=-\deg L$

Therefore

$\sum_{q}\langle-\rho_{q}^{\prime},\xi\rangle=\frac{1}{N}(\deg L)\xi(1)$

From Frobenius reciprocity, we obtain

$\sum_{q}\langle\rho_{q}^{\prime\prime},\xi\rangle=m_{\xi}(L)$

This proves (8). ∎

##  5\. Cyclic and dihedral groups

We make the Chevalley-Weil formula more explicit in a couple of cases. With
the previous notation, we calculate the local coefficients $m_{\xi,p}(L)$ when
$G$ is cyclic or dihedral.

Let $G=\langle g|g^{N}=1\rangle$. Then $\widehat{G}$ is also a cyclic group of
order $N$ generated by

$\xi(g^{j})=\exp(2\pi ij/N)$

The dual of the inclusion $G_{p}\subseteq G$ gives a surjective homomorphism
$\widehat{G}\to\widehat{G}_{p}$. Since the character $\tau_{p}$ generates
$\widehat{G}_{p}$, we can assume without loss of generality that
$\xi|_{G_{p}}=\tau_{p}$. The character $\nu_{p}$ is given by
$\xi^{m}|_{G_{p}}$ for nonnegative integer $m<N_{p}$. Given an integer $x$,
let $\overline{x}\in\\{0,1,\ldots N_{p}-1\\}$ be its residue modulo $N_{p}$.

######  Proposition 5.1.

With the notation above

$m_{\xi^{k},p}(L)=\frac{\overline{k-m}}{N}$

###### Proof.

The orthonormality of $\widehat{G}$ implies that

(9)
$\begin{split}\langle\xi^{k}|_{G_{p}},\nu_{p}\cdot\tau_{p}^{i}\rangle_{G_{p}}=\begin{cases}1&\text{if
$k\equiv i+m\mod N_{p}$}\\\ 0&\text{otherwise}\end{cases}\end{split}$

Substituting into

$m_{\xi^{k},p}(L)=\frac{\xi^{k}(1)}{N}\sum_{i=0}^{N_{p}-1}i\langle\xi^{k}|_{G_{p}},\nu_{p}\cdot\tau_{p}^{i}\rangle_{G_{p}}$

proves the proposition. ∎

Next suppose that $G=D_{n}$ is the dihedral group of order $N=2n$. This group
has a presentation

$D_{n}=\langle r,s\mid r^{n}=1,s^{2}=1,srs=r^{-1}\rangle$

Let $R\subset D_{n}$ be the subgroup generated by $r$. The characters in
$\widehat{G}$ are easy to write down [S, §5.3]. When $n$ is even,

$\widehat{G}=\\{\chi_{h}\mid 0<h<n/2\\}\cup\\{\psi_{i}\mid i=1,2,3,4\\}$

where the characters are given by the table

| $r^{k}$ | $sr^{k}$
---|---|---
$\chi_{h}$ | $2\cos\frac{2\pi hk}{n}$ | $0$
$\psi_{1}$ | $1$ | $1$
$\psi_{2}$ | $1$ | $-1$
$\psi_{3}$ | $(-1)^{k}$ | $(-1)^{k}$
$\psi_{4}$ | $(-1)^{k}$ | $(-1)^{k+1}$

In more conceptual terms, $\chi_{h}$ is induced from $\xi^{h}$, where $\xi$ is
a generator of $\widehat{R}$. We also have $\chi_{h}|_{R}=\xi^{h}+\xi^{n-h}$.
When $n$ is odd $\psi_{3}$ and $\psi_{4}$ are omitted, otherwise $\widehat{G}$
is the same.

A nontrivial cyclic subgroup of $D_{n}$ is easily seen to be either a subgroup
$R$, or the subgroup $\\{1,sr^{k}\\}$ for some $k$. For simplicity, we just
treat the case where $G_{p}\subseteq R$.

######  Proposition 5.2.

Suppose that $G_{p}\subseteq R$. Lift $\tau_{p}$ to a generator
$\xi\in\widehat{R}$, and $\nu_{p}$ to $\xi^{m}$. Then

$m_{\chi_{h},p}(L)=2\left(\frac{\overline{h-m}+\overline{n-h-m}}{N}\right)$

and

$m_{\psi_{i},p}(L)=\frac{\overline{-m}}{N},\quad i=1,2$

When $n$ is even

$m_{\psi_{i},p}(L)=\frac{\overline{n/2-m}}{N},\,\quad i=3,4$

###### Proof.

From the properties above

$\langle\chi_{h}|_{G_{p}},\nu_{p}\cdot\tau_{p}^{i}\rangle_{G_{p}}=\begin{cases}1&\text{if
$h\equiv i+m\mod N_{p}$ or $n-h\equiv i+m\mod N_{p}$}\\\
0&\text{otherwise}\end{cases}$

Substituting into the formula for $m_{\chi_{h},p}(L)$ yields the first
equation. The proofs of the remaining equations are similar. ∎

##  6\. The Lefschetz formula

In this section, we want to explain the relation to the holomorphic Lefschetz
fixed point formula of Atiyah and Bott [AB, thm 4.12]. We refer to their paper
for the general statement, as well as for some historical information about
it. First let us we briefly digress from Riemann surfaces to explain that the
conclusion of corollary 2.6 is true in much greater generality.

######  Proposition 6.1.

  1. (1)

If a finite group $G$ acts freely on a finite simplicial complex $X$, then the
character $\chi_{G}(X)$ of the virtual representation
$\sum(-1)^{i}H^{i}(X,{\mathbb{C}})$ is an integer multiple of a regular
representation.

  2. (2)

If $G$ acts freely on a compact complex manifold $X$, and $V$ is an
equivariant vector bundle, then the character $\chi_{G}(V)$ of the virtual
representation $\sum(-1)^{i}H^{i}(X,V)$ is an integer multiple of a regular
representation.

###### Proof.

By the Lefschetz fixed point theorem [H, p 179] $\chi_{G}(X)(g)=0$ when
$g\not=1$. This implies that $\chi_{G}(X)$ is an integer multiple of the
regular representation by [S, ex 2.7]. The proof of the second statement is
identical except that we use the holomorphic Lefschetz fixed point theorem
[AB, thm 4.12].

∎

######  Corollary 6.2.

In case (1) (resp. 2), if $\chi(X)$ (resp. $\chi(V)$) denotes the topological
(resp. holomorphic) Euler characteristic of $X$ (resp. $V$), then

$\chi_{G}(X)=\frac{\chi(X)}{|G|}\chi_{reg}$
$\chi_{G}(V)=\frac{\chi(V)}{|G|}\chi_{reg}$

The coefficients on the right hand side of the two equations are integers.

###### Proof.

We know that $\chi_{G}(X)=c\chi_{reg}$ (resp. $\chi_{G}(V)=c\chi_{reg}$) for
some $c\in{\mathbb{Z}}$. Evaluation at $g=1$ gives the value of $c$. ∎

We will deduce the holomorphic Lefschetz formula for finite order
automorphisms of a Riemann surface from the Chevalley-Weil theorem by a fairly
elementary reduction.

######  Theorem 6.3.

Let $g$ be a nontrivial finite order automorphism of a compact Riemann surface
$X$. Let $L$ be a line bundle, which is equivariant for the group generated by
$g$, and let $X^{g}=\\{p\mid g(p)=p\\}$, and let $\tau_{p},\nu_{p}$ be as in
section 2. Then

$\sum_{i=0}^{1}(-1)^{i}\operatorname*{tr}\left[g:H^{i}(X,\mathcal{L})\to
H^{i}(X,\mathcal{L})\right]=\sum_{p\in X^{g}}\frac{\nu_{p}(g)}{1-\tau_{p}(g)}$

###### Proof.

Let $G\subseteq Aut(X)$ be the group generated by $g$, and let $N$ denote its
order. Since $g\not=1$, we have $\chi_{reg}(g)=0$. Therefore theorem 2.4
reduces to

$\chi_{G}(L)(g)=\sum_{p\in X}M_{p}$

where

$M_{p}=-\sum_{\xi\in\widehat{G}}m_{\xi,p}(L)\xi(g)$

Of course $M_{p}=0$ unless $N_{p}\not=1$. Let us assume this. Let $n=N/N_{p}$
denote the index of $G_{p}$ in $G$. Employing the statement and notation of
proposition 5.1, we find

$\begin{split}M_{p}&=-\frac{1}{N}\sum_{k=m}^{N-1+m}\,\overline{k-m}\,\zeta^{k}\\\
&=-\frac{1}{N}\sum_{i=0}^{N_{p}-1}\sum_{\ell=0}^{n-1}i\zeta^{i+m+\ell
N_{p}}\\\ \end{split}$

where the last expression results from the substitutions $i=\overline{k-m}$
and $\ell=(k-m-i)/N_{p}$. Therefore

(10)
$M_{p}=-\frac{\zeta^{m}}{N}\left(\sum_{i=0}^{N_{p}-1}i\zeta^{i}\right)\left(\sum_{\ell=0}^{n-1}(\zeta^{N_{p}})^{\ell}\right)$

We now separate the argument into two cases depending on $n$. First suppose
that $n\not=1$. Then we claim that

$M_{p}=0$

which would be predicted by the theorem because $X^{g}=\\{p\mid g\in
G_{p}\\}=\\{p\mid N=N_{p}\\}$. To see this, observe that $\zeta^{N_{p}}$ is a
primitive $n$th root of unity, therefore

$\sum_{\ell=0}^{n-1}(\zeta^{N_{p}})^{\ell}=0$

Substituting this into (10) shows that $M_{p}=0$ as claimed.

Let us now assume that $n=1$, which means that $N_{p}=N\not=1$ so that $p\in
X^{g}$. One checks the identity

$\sum_{i=0}^{N-1}i\zeta^{i}=\frac{N}{\zeta-1}$

by cross multiplying and observing

$\sum_{i=0}^{N-1}i\zeta^{i}(\zeta-1)=N\zeta^{N}-\sum_{i=1}^{N}\zeta^{i}=N$

Therefore by (10)

$\begin{split}M_{p}&=-\frac{\zeta^{m}}{N}\left(\sum_{i=0}^{N-1}i\zeta^{i}\right)\\\
&=\frac{\zeta^{m}}{1-\zeta}\\\ &=\frac{\nu_{p}(g)}{1-\tau_{p}(g)}\end{split}$

∎

## References

  * [AB] M. Atiyah, R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (1968)
  * [Bo] N. Borne, Une formule de Riemann-Roch équivariante pour les courbes, Canad. J. Math. 55 (2003)
  * [B] M. Brion, Linearization of algebraic group actions, Handbook of group actions. Volume IV, International Press (2018)
  * [Br] K. Brown, Cohomology of groups, Springer (1982)
  * [CW] C. Chevalley, A. Weil, Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers, Abh. Math. Semin. Univ. Hamb. (1934)
  * [D] P. Deligne, Equations différentielles à points singuliers réguliers, Springer (1970)
  * [EL] Ellingsrud, Lønsted, An equivariant Lefschetz formula for finite reductive groups, Math, Ann. (1980)
  * [EV] H. Esnault, E. Viehweg, Logarithmic de Rham complexes and vanishing theorems, Invent. Math. (1986)
  * [GH] P. Griffiths, J. Harris, Principles of algebraic geometry, John Wiley & Sons. (1978)
  * [H] A. Hatcher, Algebraic topology, Cambridge University Press (2002).
  * [S] J.P. Serre, Linear representations of finite groups. Springer (1977)


$\displaystyle=\gamma^{\boxminus}\ominus\beta^{\boxminus}$

######  Definition 5.11 (Embedding of
$\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}$
into
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$).

Let
$\alpha\in\mathcal{L}_{\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}}$
be outer-$\neg$-free, we define $\alpha^{\Box,\lozenge}$ as follows.

$\displaystyle(\mathsf{B}\phi)^{\Box,\lozenge}$
$\displaystyle=\mathsf{Pr}_{1}(\Box_{1}\phi)$
$\displaystyle(\mathsf{Pl}\phi)^{\Box,\lozenge}$
$\displaystyle=\mathsf{Pr}_{2}(\lozenge_{2}\phi)$
$\displaystyle({\sim_{\mathsf{N}}}\beta)^{\Box,\lozenge}$
$\displaystyle={\sim_{\mathsf{N}}}(\beta^{\Box,\lozenge})$
$\displaystyle(\beta\circ\gamma)^{\Box,\lozenge}$
$\displaystyle=\beta^{\Box,\lozenge}\circ\gamma^{\Box,\lozenge}$
($\circ\in\\{\wedge,\rightarrowtriangle\\}$)

We can now prove the following statement.

######  Theorem 5.12.

  1. 1.

$\alpha\in\mathcal{L}_{\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}}$
is
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$-valid
iff (a) $e_{1}(\alpha^{\boxplus})=1$ in every
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ model and (b)
$e_{1}(\alpha^{\boxminus})=0$ in every
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ model.

  2. 2.

$\alpha\in\mathcal{L}_{\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}}$
is
$\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}$-valid
iff $\alpha^{\Box,\lozenge}$ is
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$-valid.

######  Proof 5.13.

We begin with (1). Consider the ‘only if’ direction: we assume that either (i)
$e_{1}(\alpha^{\boxplus})=x<1$ in some
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ model or (ii)
$e_{1}(\alpha^{\boxminus})=y>0$ in some
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ model. We prove by
induction that in this case, there are some
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$
models where $e^{\prime}_{1}(\alpha)=x$ or $e^{\prime}_{2}(\alpha)=y$,
respectively.

Let $\alpha^{\boxplus}=\mathsf{Pr}(\Box\phi)$ and
$e_{1}(\mathsf{Pr}(\Box\phi))=x<1$. This means that
$\pi(|\mathsf{Pr}(\Box\phi)|^{+})=x$. Using Remark 4.9, we obtain that there
is a belief function $\mathtt{bel}$ s.t. $\mathtt{bel}(|\phi|^{+})=x<1$. Thus,
$e^{\mathtt{bel}}_{1}(\mathsf{B}\phi)=x<1$ for the evaluation
$e^{\mathtt{bel}}_{1}$ induced by $\mathtt{bel}$, as required. The induction
steps can be obtained by a simple application of the induction hypothesis
since $\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ and
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$
use
${\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}_{(\triangle,\rightarrow)}$
as their outer-layer logic. Thus, (i) is tackled.

For (ii), we proceed similarly. Let
$\alpha^{\boxminus}=\mathsf{Pr}(\Box\neg\phi)=y>0$. Again, we obtain that
there is a belief function $\mathtt{bel}$ s.t.
$\mathtt{bel}(|\neg\phi|^{+})=x>0$ (i.e., $\mathtt{bel}(|\phi|^{-})=y>0$).
Hence, $e^{\mathtt{bel}}_{2}(\mathsf{B}\phi)=x>0$ for the evaluation induced
by $\mathtt{bel}$. The cases of propositional connectives can be tackled
similarly, so, we only consider
$\alpha^{\boxminus}=\gamma^{\boxminus}\ominus\beta^{\boxminus}$. Let
$e_{1}(\gamma^{\boxminus}\ominus\beta^{\boxminus})=y>0$. Thus,
$z\\!=\\!e_{1}(\gamma^{\boxminus})\\!>\\!e_{1}(\beta^{\boxminus})\\!=\\!z^{\prime}$
with $z\\!-\\!z^{\prime}\\!=\\!y$. Applying the induction hypothesis, we have
that $e^{\prime}_{2}(\gamma)\\!=\\!z$,
$e^{\prime}_{2}(\beta)\\!=\\!z^{\prime}$, and $z\\!-\\!z^{\prime}\\!=\\!y$.
Hence, $e^{\prime}_{2}(\beta\rightarrow\gamma)=y>0$.

Let us now deal with the ‘if’ direction. Assume that $\alpha$ is not
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$-valid,
i.e., either (i) there is a
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$
model where $e_{1}(\alpha)=x<1$ or (ii) there is a model where
$e_{2}(\alpha)=y>0$. We prove by induction that there is a
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ model s.t.
$e^{\prime}_{1}(\alpha^{\boxplus})=x<1$ or
$e^{\prime}_{1}(\alpha^{\boxminus})=y>0$.

First, if $\alpha=\mathsf{B}\phi$ and $e_{1}(\mathsf{B}\phi)=x<1$ in some
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$
model, then we have that there is a $\mathsf{BD}$ $\mathtt{bel}$-model s.t.
$\mathtt{bel}(|\phi|^{+})=x$. Hence, by Theorem 4.10, there is a $\mathsf{BD}$
probabilistic Kripke model s.t. $\pi(|\Box\phi|^{+})=x$, and thus,
$e^{\pi}_{1}(\mathsf{Pr}(\Box\phi))=x<1$ for the induced valuation
$e^{\pi}_{1}$. In the second case, we have $e_{2}(\mathsf{B}\phi)=y>0$ in some
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$
model. Thus, $\mathtt{bel}(|\phi|^{-})=y$ which is equivalent to
$\mathtt{bel}(|\neg\phi|^{+})=x$. Again, applying Theorem 4.10, we obtain that
there is a probabilistic Kripke model with $\pi(|\Box\neg\phi|^{+})=y>0$.
Hence, $e^{\pi}_{1}(\mathsf{Pr}(\Box\neg\phi))=y>0$, as required. The cases of
propositional connectives can be obtained by simple applications of the
induction hypothesis as in the ‘only if’ direction.

(2) can be shown similarly. Assume that $e_{1}(\alpha^{\Box,\lozenge})=x<1$ in
some
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$
model. Applying Remark 4.9, one can show by induction that there is a
$\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}$
model with $e^{\prime}_{1}(\alpha)=x$. Conversely, using Theorem 4.10, one
obtains that if $\alpha$ _is not
$\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}$-valid_,
then $\alpha^{\Box,\lozenge}$ is not
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$-valid
either.

###  5.3 Complexity

Theorem 5.12 gives us the reduction from validity (and satisfiability) in
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$
and
$\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}$
to $\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ and
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$.
It is, thus, sufficient to prove the $\mathsf{NP}$-completeness of
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ and
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$.
Recall that in the proof of Theorem 3.35, we used canonical models of
$\mathsf{BD}$ built over the powerset of all literals occurring in a formula
which we encoded via $u_{v}$ variables. We define canonical models for
$\mathsf{BD}$ with $\mathbf{S5}$ modalities.

######  Definition 5.14 (Canonical model for $\mathsf{BD}$ with $\mathbf{S5}$
$\Box$).

Let $\Gamma\subseteq\mathcal{L}_{\mathsf{BD}^{1}_{\Box}}$ be finite with
$\mathsf{Lit}[\Gamma]=\\{l_{1},\ldots,l_{n}\\}$. The _canonical model of
$\Gamma$_ is the following structure:

$\displaystyle\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}}$
$\displaystyle=\biguplus\limits^{\mathbb{B}_{n}}_{i=1}\left\langle
2^{\mathsf{Lit}[\Gamma]},R_{i},v^{+},v^{-}\right\rangle$

with $\mathbb{B}_{n}$ being the $n$’th Bell’s number (i.e., the number of all
equivalence relations on the set with $n$ elements), $R_{i}$’s corresponding
to all possible equivalence relations on $2^{\mathsf{Lit}[\Gamma]}$, and
valuations defined as follows:

$\displaystyle\forall w\in 2^{\mathsf{Lit}[\Gamma]}:w\in
v^{+}(p)\Leftrightarrow p\in w\text{ and }w\in v^{-}(p)\Leftrightarrow\neg
p\in w$

It is clear that
$\left|\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}}\right|=\mathbb{B}_{n}\cdot
2^{n}$ which is significantly below $2^{n^{2}+n}$ for $|\Gamma|=n$. Likewise,
if we need two equivalence relations (as in
$\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}$),
the size of $\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}}$ will be bounded
from above by $2^{2n^{2}+n}$. Since $\mathsf{BD}$ with $\mathbf{S5}$
modalities is locally tabular, it is also easy to check that if there is a
$\mathsf{BD}$ probabilistic Kripke model $\mathfrak{M}$ with

$\displaystyle\pi(\sigma_{1})=x_{1},\ldots,\pi(\sigma_{m})=x_{m}$ (6)

then there is a probability measure $\pi^{\mathsf{C}}$ on
$\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}}$ s.t.

$\displaystyle\pi^{\mathsf{C}}(\sigma_{1})=x_{1},\ldots,\pi^{\mathsf{C}}(\sigma_{m})=x_{m}$
(7)

Indeed, a probabilistic model $\mathfrak{M}$ satisfying (6) can be w.l.o.g.
assumed to be finite and, in addition, to be a submodel of
$\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}}$ (since $\mathsf{BD}$ with
$\mathbf{S5}$ modalities is locally tabular). But then, the measure on
$\mathfrak{M}$ can be extended to a measure on
$\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}}$.

The proof of $\mathsf{NP}$-completeness of
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ and
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$
mostly follows that of Theorem 3.35. Thus, we need a tableaux calculus for
$\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}$.
It was given by Bílková et al., (2021) and we recall it here. All notions —
open and closed branches, interpretations of entries, and satisfying
valuations of branches — are the same as in
$\mathcal{T}\\!\\!\left({\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}_{(\triangle,\rightarrow)}\right)$
(cf. Definitions 3.25 and 3.27), so we only list the rules in the next
definition.

######  Definition 5.15
($\mathcal{T}\\!\\!\left(\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}\right)$
— the tableau calculus for
$\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}$).

The rules are as follows.

$\displaystyle\neg\\!\leqslant_{1}\\!\dfrac{\neg\phi\leqslant_{1}i}{\phi\leqslant_{2}i}$
$\displaystyle\neg\\!\leqslant_{2}\\!\dfrac{\neg\phi\leqslant_{2}i}{\phi\leqslant_{1}i}$
$\displaystyle\neg\\!\geqslant_{1}\\!\dfrac{\neg\phi\geqslant_{1}i}{\phi\geqslant_{2}i}$
$\displaystyle\neg\\!\geqslant_{2}\\!\dfrac{\neg\phi\geqslant_{2}i}{\phi\geqslant_{1}i}$
$\displaystyle{\sim_{\mathsf{N}}}\\!\leqslant_{1}\\!\dfrac{{\sim_{\mathsf{N}}}\phi\leqslant_{1}i}{\phi\geqslant_{1}1-i}$
$\displaystyle{\sim_{\mathsf{N}}}\\!\leqslant_{2}\\!\dfrac{{\sim_{\mathsf{N}}}\phi\leqslant_{2}i}{\phi\leqslant_{1}i}$
$\displaystyle{\sim_{\mathsf{N}}}\\!\geqslant_{1}\\!\dfrac{{\sim_{\mathsf{N}}}\phi\geqslant_{1}i}{\phi\leqslant_{1}1-i}$
$\displaystyle{\sim_{\mathsf{N}}}\\!\geqslant_{2}\\!\dfrac{{\sim_{\mathsf{N}}}\phi\geqslant_{2}i}{\phi\geqslant_{2}i}$
$\displaystyle\rightarrowtriangle\leqslant_{1}\dfrac{\phi_{1}\rightarrowtriangle\phi_{2}\leqslant_{1}i}{i\geq
1\left|\begin{matrix}\phi_{1}\geqslant_{1}1\\!-\\!i\\!+\\!j\\\
\phi_{2}\\!\leqslant_{1}\\!j\\\ j\leq i\end{matrix}\right.}$
$\displaystyle\rightarrowtriangle\leqslant_{2}\dfrac{\phi_{1}\rightarrowtriangle\phi_{2}\leqslant_{2}i}{\begin{matrix}\phi_{1}\\!\leqslant_{2}\\!i\\!+\\!j\\\
\phi_{2}\\!\leqslant_{1}\\!1\\!-\\!j\end{matrix}}$
$\displaystyle\rightarrowtriangle\geqslant_{1}\dfrac{\phi_{1}\rightarrowtriangle\phi_{2}\geqslant_{1}i}{\begin{matrix}\phi_{1}\\!\leqslant_{1}\\!1\\!-\\!i\\!+\\!j\\\
\phi_{2}\\!\geqslant_{1}\\!j\end{matrix}}$
$\displaystyle\rightarrowtriangle\geqslant_{2}\dfrac{\phi_{1}\rightarrowtriangle\phi_{2}\geqslant_{2}i}{i\leq
0\left|\begin{matrix}\phi_{1}\\!\geqslant_{2}\\!i\\!+\\!j\\\
\phi_{2}\\!\geqslant_{1}\\!1\\!-\\!j\\\ j\leq 1-i\end{matrix}\right.}$
$\displaystyle\wedge\leqslant_{1}\dfrac{\phi_{1}\wedge\phi_{2}\leqslant_{1}i}{\phi_{1}\leqslant_{1}i\mid\phi_{2}\leqslant_{1}i}$
$\displaystyle\wedge\leqslant_{2}\dfrac{\phi_{1}\wedge\phi_{2}\leqslant_{2}i}{\begin{matrix}\phi_{1}\leqslant_{2}i\\\
\phi_{2}\leqslant_{2}i\end{matrix}}$
$\displaystyle\wedge\geqslant_{1}\dfrac{\phi_{1}\wedge\phi_{2}\geqslant_{1}i}{\begin{matrix}\phi_{1}\geqslant_{1}i\\\
\phi_{2}\geqslant_{1}i\end{matrix}}$
$\displaystyle\wedge\geqslant_{2}\dfrac{\phi_{1}\wedge\phi_{2}\geqslant_{2}i}{\phi_{1}\geqslant_{2}i\mid\phi_{2}\geqslant_{2}i}$

######  Remark 5.16.

Note that all connectives in
$\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}$
are continuous and thus all rules of
$\mathcal{T}\\!\\!\left(\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}\right)$
can be linearised. This will require the introduction of another sort of
variables that range over $\\{0,1\\}$. To make the presentation of both
tableaux calculi uniform, however, we decided against the linear version of
the tableau rules.

######  Theorem 5.17.

Satisfiability in $\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$ and
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$
is $\mathsf{NP}$-complete.

######  Proof 5.18.

Since
${\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}_{(\triangle,\rightarrow)}$
and
$\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}$
are $\mathsf{NP}$-hard, we focus on the $\mathsf{NP}$-membership proof. Let us
begin with $\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$.

We show how to check the satisfiability of
$\alpha\in\mathcal{L}_{\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}}$
in nondeterministic polynomial time. Assume that $\alpha$ is built over
$\mathsf{Pr}\sigma_{1}$, …, $\mathsf{Pr}\sigma_{n}$ and assume w.l.o.g. that
$|\mathsf{Lit}(\alpha)|=n$ (just as in Theorem 3.35, we can add superfluous
modal atoms to make their number equal to the number of literals in $\alpha$)
and that $\alpha$ is outer-$\neg$-free (this holds because
$e(\neg\mathsf{Pr}\sigma)=e(\mathsf{Pr}\neg\sigma)$ in every model).

We begin with substituting all modal atoms of $\alpha$ with fresh variables
$q_{\sigma_{i}}$ and construct a
$\mathcal{T}\\!\\!\left({\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}_{(\triangle,\rightarrow)}\right)$
constraint tableau beginning with $\\{\alpha^{-}\leqslant_{1}c,c<1\\}$. We can
check in nondeterministic polynomial time whether this tableau is closed. If
it is, $\alpha^{-}$ is not satisfiable (cf. Lemma 5.7) in
${\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}_{(\triangle,\rightarrow)}$,
whence $\alpha$ is not
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$-satisfiable. Otherwise,
we consider the set
$\mathfrak{Br}=\\{\mathcal{B}_{1},\ldots,\mathcal{B}_{w}\\}$ of all open
branches of the tableau, guess some $\mathcal{B}\in\mathfrak{Br}$, and
consider its corresponding instance of the bMIP.

$\displaystyle z_{1}\triangledown t_{1},\ldots,z_{n}\triangledown
t_{n},k_{1}\leq k^{\prime}_{1},\ldots,k_{r}\leq k^{\prime}_{r},m_{1}\geq
1,\ldots,m_{s}\geq 1,m^{\prime}_{1}\leq 0,\ldots,m^{\prime}_{t}\leq 0$
($[\mathrm{MIP}(1)_{\Box}]^{\mathcal{B}}$)

Here, $z_{i}$’s correspond to the values of $q_{\sigma_{i}}$’s in
$\alpha^{-}$, and $t_{i}$’s are linear polynomials that label
$q_{\sigma_{i}}$’s. Numerical constraints give us $k$’s, $k^{\prime}$’s, $m$’s
and $m^{\prime}$’s. Denote the number of inequalities and the number of
variables in $[\mathrm{MIP}(1)_{\Box}]^{\mathcal{B}}$ with $l_{1}$ and
$l_{2}$, respectively. It is clear that $l_{1}=O(\mathcal{l}(\alpha^{-}))$ and
$l_{2}=O(\mathcal{l}(\alpha^{-}))$.

We need to check whether $z_{i}$’s are coherent as probabilities of
$\sigma_{i}$’s. To do this, we introduce $\mathbb{B}_{n}\cdot 2^{n}$ new
variables of the form $u^{j}_{v}$ with $j\in\\{1,\ldots,\mathbb{B}_{n}\\}$ and
$v\in\\{0,1\\}^{n}$. Here, $j$ indexes _copies of $2^{\mathsf{Lit}(\alpha)}$_
with different equivalence relations thereon and $v$ encodes the valuation of
literals of $\alpha$ in the corresponding state of the canonical model (cf.
Definition 5.14). Additionally, we put $a^{j}_{i,v}=1$ if
$\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}},(v,j)\vDash^{+}\sigma_{i}$
and $a^{j}_{i,v}=0$, otherwise. Here, $(v,i)$ stands for the state in the
$j$’th copy of $2^{\mathsf{Lit}(\alpha)}$ that makes true the literals denoted
by $v$: e.g., for $\mathsf{Lit}(\alpha)=\\{p_{1},\neg p_{1},p_{2},\neg
p_{2}\\}$, $u^{2}_{1100}$ encodes the state $\\{p_{1},\neg p_{1}\\}$ in the
second copy of $2^{\mathsf{Lit}(\alpha)}$. Note that since modalities do not
nest and since $\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}}$ is a
disjoint union of models, it takes only deterministic polynomial time to check
whether
$\mathfrak{M}^{\mathsf{\mathsf{BD}\mathbf{{S5}}}},(v,j)\vDash^{+}\sigma_{i}$.

We then add the following equalities.

$\displaystyle\sum\limits^{\mathbb{B}_{n}}_{j=1}\sum\limits_{v}u^{j}_{v}$
$\displaystyle=1$
$\displaystyle\sum\limits^{\mathbb{B}_{n}}_{j=1}\sum\limits_{v}(a^{j}_{i,v}\cdot
u^{j}_{v})$ $\displaystyle=z_{i}$
($[\mathrm{MIP}(2\exp)_{\Box}]^{\mathcal{B}}$)

Again, just as in Theorem 3.35,
$[\mathrm{MIP}(1)_{\Box}]^{\mathcal{B}}$$\cup$$[\mathrm{MIP}(2\exp)_{\Box}]^{\mathcal{B}}$
has a solution over $[0,1]$ iff $\alpha$ is
$\mathsf{Pr}^{(\triangle,\rightarrow)}_{\mathbf{S5}}$-satisfiable. Moreover,
the number of (in)equalities in this system is
$l_{1}+n+1=O(\mathcal{l}(\alpha))$. Thus we can apply the result of (Fagin et
al.,, 1990, Lemma 2.5) and obtain that
$[\mathrm{MIP}(1)_{\Box}]^{\mathcal{B}}$$\cup$$[\mathrm{MIP}(2\exp)_{\Box}]^{\mathcal{B}}$
has a solution with at most $l_{1}+n+1$ non-zero entries. We then guess a list
$L$ of at most $l_{1}+n+1$ pairs $(v,j)$. The size of $L$ is $n\cdot\lceil\log
j\rceil\cdot(l_{1}+n+1)=O(n^{4})$ since $j\leq\mathbb{B}_{n}<2^{n^{2}}$. We
compute the values of $a^{j}_{i,v}$’s for $i\leq n$ and $(v,j)\in L$ which
gives us the following bMIP.

$\displaystyle\sum\limits_{(v,j)\in L}u^{j}_{v}$ $\displaystyle=1$
$\displaystyle\sum\limits_{(v,j)\in L}(a^{j}_{i,v}\cdot u^{j}_{v})$
$\displaystyle=z_{i}$ ($[\mathrm{MIP}(2\mathrm{poly})_{\Box}]^{\mathcal{B}}$)

Since the size of
$[\mathrm{MIP}(1)_{\Box}]^{\mathcal{B}}$$\cup$$[\mathrm{MIP}(2\mathrm{poly})_{\Box}]^{\mathcal{B}}$
is polynomial in $\mathcal{l}(\alpha^{-})$, the rest of the proof is identical
to that of Theorem 3.35.

The $\mathsf{NP}$-membership proof for
$\mathsf{Pr}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}_{\mathbf{S5}}$
is mostly the same. The main differences are as follows. First of all, there
are two modalities — $\mathsf{Pr}_{1}$ and $\mathsf{Pr}_{2}$; this is why,
when transforming $\alpha$ into $\alpha^{-}$, we replace
$\mathsf{Pr}_{1}\sigma_{i}$’s with $q_{\sigma_{1_{i}}}$’s and
$\mathsf{Pr}_{2}\sigma_{i}$’s with $q_{\sigma_{2_{i}}}$’s. Thus, we introduce
two sorts of $z$’s ($z_{1_{i}}$’s and $z_{2_{i}}$’s) in the bMIP corresponding
to an open branch in the
$\mathcal{T}\\!\\!\left(\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}\right)$
tableau for $\alpha^{-}$. Second, when introducing $u^{j}_{v}$’s we let $j$
range over $\\{1,\ldots,\mathbb{B}^{2}_{n}\\}$ (i.e., up to _the square of the
$n$’th Bell’s number_. Third, when we guess the list $L$ of pairs $(v,j)$, we
have that $j\leq\mathbb{B}^{2}_{n}<2^{2n^{2}}$. This, however, still produces
the list of the size $n\cdot\lceil\log j\rceil\cdot(l_{1}+n+1)=O(n^{4})$.

######  Corollary 5.19.

Satisfiability in
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$
and
$\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}$
is $\mathsf{NP}$-complete.

######  Proof 5.20.

Since
${\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}_{(\triangle,\rightarrow)}$
and
$\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}$
are $\mathsf{NP}$-hard, the statement follows mmediately from Lemma 5.7 and
Theorems 5.12 and 5.17.

##  6 Conclusion

In this paper, we considered several two-layered logics for reasoning with
probability measures and belief functions defined in the
$\mathsf{BD}$-framework. In particular, we established that the logics of
$\mathbf{4}$\- and $\pm$-probabilities can be faithfully embedded into one
another (Theorems 3.14 and 3.16), constructed a complete Hilbert
axiomatisation for the logic of $\mathbf{4}$-probabilities (Theorem 3.23). We
also established $\mathsf{NP}$-completeness for logics of $\pm$\- and
$\mathbf{4}$-probabilities as well as for logics of belief and plausibility
functions in $\mathsf{BD}$ (Theorem 3.35 and Corollary 5.19). Still, several
important questions remain open.

First of all, our belief and plausibility functions are close to
$\pm$-probabilities because they assign each statement $\phi$ two measures:
the measure of its positive extension and the measure of its negative
extension. It is thus, instructive to define $\mathbf{4}$-valued belief and
plausibility functions that will assign measures to $\phi$ according to the
pure belief, pure disbelief, conflict and uncertainty extensions (cf.
Convention 2). This is not a trivial task since belief and plausibility
functions are not additive, whence, the measures of these extensions cannot be
directly obtained from $\mathtt{bel}(|\phi|^{+})$ and
$\mathtt{bel}(|\phi|^{-})$.

Second, observe that axiomatisations of
$\mathsf{Bel}^{{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}^{2}}_{\triangle}$
and
$\mathsf{Bel}^{\mathsf{N}{\mathchoice{\mbox{\sf\L}}{\mbox{\sf\L}}{\mbox{\sf\scriptsize\L}}{\mbox{\sf\tiny\L}}}}$
by Bílková et al., (2023) are _infinite_. In the case of belief functions over
the classical logic, Godo et al., (2001, 2003) show how to produce a  _finite_
axiomatisation for the two-layered logics of belief functions in the classical
case using the representation of $\mathtt{bel}(\|\phi\|)$ as
$\pi(\|\Box\phi\|)$. Recall that a similar property (Theorem 4.10) holds for
the $\mathsf{BD}$ as well. Can we use it to obtain a finite axiomatisation of
belief and plausibility functions over $\mathsf{BD}$?

Third and finally, in Bílková et al., 2023a , we proposed two-layered logics
for _qualitative reasoning_ about different uncertainty measures. In
particular, we constructed logics for qualitative reasoning with belief
functions and probabilities. These logics utilised the bi-Gödel logic
($\mathsf{biG}$) in the outer layer. Since $\mathsf{biG}$ is also
$\mathsf{NP}$-complete the question arises whether we can apply the technique
of Hájek and Tulipani, (2001) to prove the $\mathsf{NP}$-completeness of two-
layered logics for qualitative reasoning about uncertainty.

###### Acknowledgements.

The first author was supported by grant №22-01137S of Grantová Agentura České
Republiky. The second and third authors were supported by grant
ANR-19-CE48-0006 of Agence Nationale de la Recherche. The third author was
also supported by grant ANR-19-CHIA-0014 of Agence Nationale de la Recherche.
The paper is a part of the MOSAIC project financed by the European Union’s
Marie Skłodowska-Curie grant №101007627.

No conflict of interest to declare.

## References

  * Arieli and Avron, (2017) Arieli, O. and Avron, A. 2017\.  Four-valued paradefinite logics.  Studia Logica, 105(6):1087–1122.
  * Baaz, (1996) Baaz, M. 1996\.  Infinite-valued Gödel logics with $0$-$1$-projections and relativizations.  In Gödel’96: Logical foundations of mathematics, computer science and physics—Kurt Gödel’s legacy, Brno, Czech Republic, August 1996, proceedings, volume 6, pp. 23–34. Association for Symbolic Logic.
  * Baldi et al., (2020) Baldi, P., Cintula, P., and Noguera, C. 2020\.  Classical and fuzzy two-layered modal logics for uncertainty: Translations and proof-theory.  International Journal of Computational Intelligence Systems, 13(1):988–1001.
  * Belnap, (1977) Belnap, N. 1977\.  A Useful Four-Valued Logic.  In Dunn, J. M. and Epstein, G., editors, Modern Uses of Multiple-Valued Logic, pp. 5–37, Dordrecht. Springer Netherlands.
  * Belnap, (2019) Belnap, N. 2019\.  How a computer should think.  In Omori, H. and Wansing, H., editors, New Essays on Belnap-Dunn Logic, volume 418 of Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science). Springer, Cham.
  * Bílková et al., (2021) Bílková, M., Frittella, S., and Kozhemiachenko, D. 2021\.  Constraint tableaux for two-dimensional fuzzy logics.  In Das, A. and Negri, S., editors, Automated Reasoning with Analytic Tableaux and Related Methods, volume 12842 of Lecture Notes in Computer Science, pp. 20–37. Springer International Publishing.
  * (7) Bílková, M., Frittella, S., Kozhemiachenko, D., and Majer, O. 2023a.  Qualitative reasoning in a two-layered framework.  International Journal of Approximate Reasoning, 154:84–108.
  * (8) Bílková, M., Frittella, S., Kozhemiachenko, D., and Majer, O. 2023b.  Two-layered logics for paraconsistent probabilities.  In Hansen, H., Scedrov, A., and de Queiroz, R., editors, Logic, Language, Information, and Computation. WoLLIC 2023, volume 13923 of Lecture notes in computer science, pp. 101–117. Springer Nature Switzerland, Cham.
  * Bílková et al., (2023) Bílková, M., Frittella, S., Kozhemiachenko, D., Majer, O., and Manoorkar, K. 2023\.  Describing and quantifying contradiction between pieces of evidence via Belnap Dunn logic and Dempster-Shafer theory.  In Miranda, E., Montes, I., Quaeghebeur, E., and Vantaggi, B., editors, Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, volume 215 of Proceedings of Machine Learning Research, pp. 37–47. PMLR.
  * Bílková et al., (2023) Bílková, M., Frittella, S., Kozhemiachenko, D., Majer, O., and Nazari, S. 2023\.  Reasoning with belief functions over Belnap–Dunn logic.  Annals of Pure and Applied Logic, 103338.
  * Bílková et al., (2020) Bílková, M., Frittella, S., Majer, O., and Nazari, S. 2020\.  Belief based on inconsistent information.  In International Workshop on Dynamic Logic, pp. 68–86. Springer.
  * Bueno-Soler and Carnielli, (2016) Bueno-Soler, J. and Carnielli, W. 2016\.  Paraconsistent probabilities: Consistency, contradictions and Bayes’ theorem.  Entropy (Basel), 18(9):325.
  * Běhounek et al., (2011) Běhounek, L., Cintula, P., and Hájek, P. 2011\.  Introduction to Mathematical Fuzzy Logic.  In Cintula, P., Hájek, P., and Noguera, C., editors, Handbook of Mathematical Fuzzy Logic, volume 37 of Studies in logic, pp. 1–102. College Publications.
  * Dautović et al., (2021) Dautović, Š., Doder, D., and Ognjanović, Z. 2021\.  An epistemic probabilistic logic with conditional probabilities.  In Logics in Artificial Intelligence. JELIA 2021, volume 12678 of Lecture notes in computer science, pp. 279–293. Springer International Publishing, Cham.
  * Delgrande and Renne, (2015) Delgrande, J. and Renne, B. 2015\.  The logic of qualitative probability.  In Twenty-Fourth International Joint Conference on Artificial Intelligence.
  * Delgrande et al., (2019) Delgrande, J., Renne, B., and Sack, J. 2019\.  The logic of qualitative probability.  Artificial Intelligence, 275:457–486.
  * Dubois, (2008) Dubois, D. 2008\.  On ignorance and contradiction considered as truth-values.  Logic Journal of the IGPL, 16(2):195–216.
  * Dunn, (1976) Dunn, J. 1976\.  Intuitive semantics for first-degree entailments and ‘coupled trees’.  Philosophical Studies, 29(3):149–168.
  * Dunn, (2010) Dunn, J. 2010\.  Contradictory information: Too much of a good thing.  Journal of Philosophical Logic, 39:425––452.
  * Fagin and Halpern, (1991) Fagin, R. and Halpern, J. 1991\.  Uncertainty, belief, and probability.  Computational Intelligence, 7(3):160–173.
  * Fagin et al., (1990) Fagin, R., Halpern, J. Y., and Megiddo, N. 1990\.  A logic for reasoning about probabilities.  Information and Computation, 87:78–128.
  * Gärdenfors, (1975) Gärdenfors, P. 1975\.  Qualitative probability as an intensional logic.  Journal of Philosophical Logic, pp. 171–185.
  * Godo et al., (2001) Godo, L., Hájek, P., and Esteva, F. 2001\.  A Fuzzy Modal Logic for Belief Functions.  In Nebel, N., editor, Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence, IJCAI 2001, Seattle, Washington, USA, August 4-10, 2001, pp. 723–732. Morgan Kaufmann.
  * Godo et al., (2003) Godo, L., Hájek, P., and Esteva, F. 2003\.  A fuzzy modal logic for belief functions.  Fundamenta Informaticae, 57(2–4):127–146.
  * Hähnle, (1992) Hähnle, R. 1992\.  A new translation from deduction into integer programming.  In International Conference on Artificial Intelligence and Symbolic Mathematical Computing, pp. 262–275. Springer.
  * Hähnle, (1994) Hähnle, R. 1994\.  Many-valued logic and mixed integer programming.  Annals of mathematics and Artificial Intelligence, 12(3-4):231–263.
  * Hájek, (1996) Hájek, P. 1996\.  Getting belief functions from Kripke models.  International Journal of General Systems, 24(3):325–327.
  * Hájek, (1998) Hájek, P. 1998\.  Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic.  Kluwer.
  * Hájek et al., (1995) Hájek, P., Godo, L., and Esteva, F. 1995\.  Fuzzy logic and probability.  In Besnard, P. and Hanks, S., editors, UAI ’95: Proceedings of the Eleventh Annual Conference on Uncertainty in Artificial Intelligence, Montreal, Quebec, Canada, August 18-20, 1995, pp. 237–244. Morgan Kaufmann.
  * Hájek and Tulipani, (2001) Hájek, P. and Tulipani, S. 2001\.  Complexity of fuzzy probability logics.  Fundamenta Informaticae, 45(3):207–213.
  * Klein et al., (2021) Klein, D., Majer, O., and Rafiee Rad, S. 2021\.  Probabilities with gaps and gluts.  Journal of Philosophical Logic, 50(5):1107–1141.
  * Mares, (1997) Mares, E. 1997\.  Paraconsistent probability theory and paraconsistent bayesianism.  Logique et analyse, pp. 375–384.
  * Metcalfe et al., (2008) Metcalfe, G., Olivetti, N., and Gabbay, D. 2008\.  Proof Theory for Fuzzy Logics.  Applied Logic Series 36. Springer.
  * Nelson, (1949) Nelson, D. 1949\.  Constructible falsity.  The Journal of Symbolic Logic, 14(1):16–26.
  * Odintsov and Wansing, (2017) Odintsov, S. and Wansing, H. 2017\.  Disentangling FDE-Based Paraconsistent Modal Logics.  Studia Logica, 105(6):1221–1254.
  * Omori and Wansing, (2017) Omori, H. and Wansing, H. 2017\.  40 years of FDE: An introductory overview.  Studia Logica, 105(6):1021–1049.
  * Rodrigues et al., (2021) Rodrigues, A., Bueno-Soler, J., and Carnielli, W. 2021\.  Measuring evidence: a probabilistic approach to an extension of Belnap–Dunn logic.  Synthese, 198(S22):5451–5480.
  * Rodriguez et al., (2022) Rodriguez, R., Tuyt, O., Esteva, F., and Godo, L. 2022\.  Simplified Kripke semantics for K45-like Gödel modal logics and its axiomatic extensions.  Studia Logica, 110(4):1081–1114.
  * Shafer, (1976) Shafer, G. 1976\.  A mathematical theory of evidence.  Princeton university press.
  * Zhou, (2013) Zhou, C. 2013\.  Belief functions on distributive lattices.  Artificial Intelligence, 201:1–31.