* Mott and Littleton (1938) N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34, 485 (1938).
  * Frenkel (1931) J. Frenkel, Phys. Rev. 37, 17 (1931).
  * Qiu  _et al._ (2015) D. Y. Qiu, T. Cao, and S. G. Louie, Phys. Rev. Lett. 115, 176801 (2015).
  * Qiu  _et al._ (2021a) D. Y. Qiu, G. Cohen, D. Novichkova, and S. Refaely-Abramson, Nano Lett. 21, 7644 (2021a).
  * Toyozawa (1958) Y. Toyozawa, Progr. Theoret. Phys. 20, 53 (1958).
  * Antonius and Louie (2022) G. Antonius and S. G. Louie, Phys. Rev. B Condens. Matter 105, 085111 (2022).
  * Chen  _et al._ (2020) H.-Y. Chen, D. Sangalli, and M. Bernardi, Phys. Rev. Lett. 125, 107401 (2020).
  * Kozin  _et al._ (2021) V. K. Kozin, V. A. Shabashov, A. V. Kavokin, and I. A. Shelykh, Phys. Rev. Lett. 126, 036801 (2021).
  * Onga  _et al._ (2017) M. Onga, Y. Zhang, T. Ideue, and Y. Iwasa, Nat. Mater. 16, 1193 (2017).
  * Kuga  _et al._ (2008) S.-I. Kuga, S. Murakami, and N. Nagaosa, Phys. Rev. B Condens. Matter 78, 205201 (2008).
  * Yao and Niu (2008) W. Yao and Q. Niu, Phys. Rev. Lett. 101, 106401 (2008).
  * Heller and Marcus (1951) W. R. Heller and A. Marcus, Phys. Rev. 84, 809 (1951).
  * Hopfield and Thomas (1960) J. J. Hopfield and D. G. Thomas, J. Phys. Chem. Solids 12, 276 (1960).
  * Mostofi  _et al._ (2014) A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 185, 2309 (2014).
  * Pizzi  _et al._ (2020) G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Poncé, T. Ponweiser, J. Qiao, F. Thöle, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. Mostofi, and J. R. Yates, J. Phys. Condens. Matter 32, 165902 (2020).
  * Born  _et al._ (1956) Born, M., Huang, and K.,  _Dynamical theory of crystal lattices_ (Clarendon Press, Oxford, 1956).
  * Knox (1963) R. S. Knox,  _Theory of excitons_ (Academic Press, New York, 1963).
  * Gonze and Lee (1997) X. Gonze and C. Lee, Phys. Rev. B Condens. Matter 55, 10355 (1997).
  * Onida  _et al._ (2002) G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).
  * Löwdin (1951) P. Löwdin, J. Chem. Phys. 19, 1396 (1951).
  * Benedict (2002) L. X. Benedict, Phys. Rev. B Condens. Matter 66, 193105 (2002).
  * Qiu  _et al._ (2021b) D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Phys. Rev. B Condens. Matter 103, 045117 (2021b).
  * Tang  _et al._ (2022) F. Tang, Z. Li, C. Zhang, S. G. Louie, R. Car, D. Y. Qiu, and X. Wu, Proc. Natl. Acad. Sci. U. S. A. 119, e2201258119 (2022).
  * Ismail-Beigi (2006) S. Ismail-Beigi, Phys. Rev. B Condens. Matter 73, 233103 (2006).
  * Gonze  _et al._ (1994) X. Gonze, J.-C. Charlier, D. C. Allan, and M. P. Teter, Phys. Rev. B Condens. Matter 50, 13035 (1994).
  * Wang and Berkelbach (2020) X. Wang and T. C. Berkelbach, J. Chem. Theory Comput. 16, 3095 (2020).
  * Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  * Kohn and Sham (1965) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  * Hamann (2013) D. R. Hamann, Phys. Rev. B Condens. Matter 88, 085117 (2013).
  * van Setten  _et al._ (2018) M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D. R. Hamann, X. Gonze, and G.-M. Rignanese, Comput. Phys. Commun. 226, 39 (2018).
  * Giannozzi  _et al._ (2017) P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. Otero-de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, J. Phys. Condens. Matter 29, 465901 (2017).
  * Adler (1962) S. L. Adler, Phys. Rev. 126, 413 (1962).
  * Wiser (1963) N. Wiser, Phys. Rev. 129, 62 (1963).
  * Shirley (1998) E. L. Shirley, Phys. Rev. B Condens. Matter 58, 9579 (1998).
  * Shirley  _et al._ (1996) E. L. Shirley, L. J. Terminello, J. E. Klepeis, and F. J. Himpsel, Phys. Rev. B Condens. Matter 53, 10296 (1996).
  * Wang  _et al._ (2003) N.-P. Wang, M. Rohlfing, P. Krüger, and J. Pollmann, Phys. Rev. B Condens. Matter 67, 115111 (2003).
  * Deilmann (2018) T. Deilmann, Phys. Rev. B Condens. Matter 98, 165105 (2018).
  * Benedict  _et al._ (1998) L. X. Benedict, E. L. Shirley, and R. B. Bohn, Phys. Rev. Lett. 80, 4514 (1998).
  * Gatti and Sottile (2013) M. Gatti and F. Sottile, Phys. Rev. B Condens. Matter 88, 155113 (2013).
  * Sun  _et al._ (2020) J. Sun, J. Yang, and C. A. Ullrich, Phys. Rev. Res. 2, 013091 (2020).
  * Onodera and Toyozawa (1967) Y. Onodera and Y. Toyozawa, J. Phys. Soc. Jpn. 22, 833 (1967).
  * Verdi and Giustino (2015) C. Verdi and F. Giustino, Phys. Rev. Lett. 115, 176401 (2015).
  * Jhalani  _et al._ (2020) V. A. Jhalani, J.-J. Zhou, J. Park, C. E. Dreyer, and M. Bernardi, Phys. Rev. Lett. 125, 136602 (2020).
  * Dexter (1953) D. L. Dexter, J. Chem. Phys. 21, 836 (1953).
  * Förster (1948) T. Förster, Ann. Phys. 437, 55 (1948).
  * Főrster (1959) T. Főrster, Discuss. Faraday Soc. 27, 7 (1959).


# Field Knowledge as a Dual to Distributed Knowledge

A characterization by weighted modal logic

Xiaolong Liang  School of Philosophy, Shanxi University
92 Wucheng Road, Taiyuan, 030006, Shanxi, P.R. China  Yì N. Wáng  Department
of Philosophy (Zhuhai), Sun Yat-sen University
2 Daxue Road, Zhuhai, 519082, Guangdong, P.R. China

###### Abstract

The study of group knowledge concepts such as mutual, common, and distributed
knowledge is well established within the discipline of epistemic logic. In
this work, we incorporate epistemic abilities of agents to refine the formal
definition of distributed knowledge and introduce a formal characterization of
field knowledge. We propose that field knowledge serves as a dual to
distributed knowledge. Our approach utilizes epistemic logics with various
group knowledge constructs, interpreted through weighted models. We delve into
the eight logics that stem from these considerations, explore their relative
expressivity and develop sound and complete axiomatic systems.

111This is a preprint of the paper published in Liao et al. (eds.) Fourth
International Workshop on Logics for New-Generation Artificial Intelligence
(LNGAI 2024), pp. 9–31, College Publications, 24 June 2024.
https://www.collegepublications.co.uk/LNGAI/?00004

##  1 Introduction

The introduction is segmented into two sections. In Section 1.1, we elucidate
our interpretation of the concepts _distributed knowledge_ and _field
knowledge_. Section 1.2 is dedicated to detailing our methodology for modeling
these concepts within the context of weighted (or labeled) modal logic.

###  1.1 Group notions of knowledge

Alice and Bob, both instrumentalists with additional expertise in philosophy
and mathematics respectively, engage in a conversation that shapes and
reflects their knowledge. Classical epistemic logic [12, 8, 16] offers a
framework for dissecting their individual and collective knowledge, utilizing
tools like Kripke semantics among others.

Their mutual knowledge (a.k.a. everyone’s knowledge or general knowledge)
consists of statements known by both, essentially an intersection of their
individual knowledge. In Kripke semantics, this is interpreted by the _union_
of their respective epistemic uncertainty relations. Common knowledge is
recursive mutual knowledge: they know $\varphi$, know that they know
$\varphi$, and so on, ad inf. It is modeled by the transitive closure of the
union of their uncertainty relations.

Distributed knowledge signifies the sum of knowledge Alice and Bob would have
after full communication, but it is not merely the union of what each knows.
Though an interpretation based on the _intersection_ of their individual
uncertainty relations does not fully align with its intended meaning either
[18, 3], as it stands, this prevalent definition treats _mutual knowledge as
the semantic dual to distributed knowledge_.

In our scenario, we conceptualize distributed knowledge in light of the
professional competencies of Alice and Bob. Their distributed knowledge of a
statement $\varphi$ is not understood as just a matter of aggregate knowledge,
but rather the outcome of their combined expertise in musical instruments,
philosophy, and mathematics. That is, $\varphi$ is their distributed knowledge
if their collaborative proficiency across these domains enables them to deduce
$\varphi$. Thus, we redefine _distributed knowledge_ as the _union_ of Alice
and Bob’s epistemic abilities, diverging from its classical interpretation.

Upon reevaluating distributed knowledge, we introduce the allied concept of
_field knowledge_. This notion encapsulates knowledge that stems from their
shared discipline – musical instruments, in this case. A statement $\varphi$
falls under Alice and Bob’s field knowledge if it is derivable exclusively
from their musical background. The formal interpretation of this concept will
be presented in Section 2.1, where we propose that _field knowledge
semantically functions as the dual to distributed knowledge_.

Developing a coherent characterization for the emergent concepts of
distributed and field knowledge presents its challenges within classical
epistemic logics. We aim to craft a comprehensive framework that encompasses
these new ideas while preserving the established interpretations of mutual and
common knowledge.

###  1.2 Modeling knowledge in weighted modal logic

Even though the concept of _similarity_ is intrinsically linked to _knowledge_
, it has not been traditionally emphasized or explicitly incorporated in the
classical representation of knowledge within the field of epistemic logic [12,
8, 16]. Over recent years, researchers have started to probe this relationship
more deeply, marking a fresh direction in the field [17, 7]. The technical
framework for exploring this relationship has its roots in weighted modal
logics [14, 13, 11]. This approach offers a quantitative way of considering
similarity, allowing for a more nuanced understanding of knowledge.

In this paper, we adapt the concept of similarity from the field of data
mining, where it is primarily used to quantify the likeness between two data
objects. In data mining, distance and similarity measures are generally
specific algorithms tailored to particular scenarios, such as computing the
distance and similarity between matrices, texts, graphs, etc. (see, e.g., [1,
Chapter 3]). There is also a body of literature that outlines general
properties of distance and similarity measures. For instance, in [20], it is
suggested that typically, the properties of _positivity_ (i.e., $\forall
x\forall y:s(x,y)=1\Rightarrow x=y$) and _symmetry_ (i.e., $\forall x\forall
y:s(x,y)=s(y,x)$) hold for $s(x,y)$ – a binary numerical function that maps
the similarity between points $x$ and $y$ to the range $[0,1]$.

Our primary interest here does not lie in the measures of similarity
themselves, but rather in modeling similarity and deriving from it the
concepts of knowledge. Our work distinguishes itself from recent advancements
in epistemic logic interpreted through the concepts of similarity or distance.
One key difference is that we employ the standard language of epistemic logic.
We do not explicitly factor in the degree of similarity into the language,
maintaining the traditional structure of epistemic logic while reinterpreting
its concepts in the light of similarity.

In our setting, the phrase “$a$ knows $\varphi$” ($K_{a}\varphi$) can be
interpreted as “$\varphi$ holds true in all states that, in $a$’s perception
by its expertise, resemble the actual state.” A “state” in this context can be
seen as a data object – the focus of data mining. But it could also be treated
as an epistemic object, a possible situation, and so forth. We generalize the
similarity function by replacing its range [0, 1] with an arbitrary set of
epistemic abilities. The degrees of similarity may not have a comparable or
ordered relationship.

The primary focus of this paper is on _group knowledge_ , as elaborated in
Section 1.1. We explore epistemic logics across all combinations of these
group knowledge notions. As mutual knowledge is definable by individual
knowledge (with only finitely many agents), we have formulated eight logics
(with or without common, distributed, and field knowledge). The syntax and
semantics of them are introduced in Sections 2.1–2.2, and in Section 2.3, we
compare the expressive power of these languages.

For the axiomatization of the logics, we introduce sound and strongly complete
axiomatic systems for the logics excluding common knowledge. For those
incorporating common knowledge, we present sound and weakly complete axiomatic
systems (owing to the lack of compactness for the common knowledge operators).
These systems are then categorized based on whether their completeness results
are obtainable via translation of models (Section 3.2.1) or the canonical
model method (Section 3.2.2), shown via a path-based canonical model (Sections
3.2.3 and 3.2.4), or require a finitary method leading to a weak completeness
result (Section 3.2.5).

##  2 Logics

In this section, we present a comprehensive framework composed of eight
distinctive logics. We supplement our discussion with illustrative examples,
offering a visual representation of the models and their accompanying
semantics.

###  2.1 Syntax

Our study utilizes formal languages rooted in the standard language of multi-
agent epistemic logic [8, 16], with the addition of modalities that represent
group knowledge constructs. We particularly concentrate on the constructs of
_common knowledge_ , _distributed knowledge_ and _field knowledge_.

In terms of our assumptions, we consider Prop as a countably infinite set of
propositional variables, and Ag as a finite, nonempty set of agents.

######  Definition 2.1.

(formal languages) The languages utilized in our study are defined by the
following rules, where the name of each language is indicated in parentheses
on the left-hand side:

$\displaystyle(\mathcal{EL})$
$\displaystyle\varphi::=p\mid\neg\varphi\mid(\varphi\rightarrow\varphi)\mid
K_{a}\varphi$ $\displaystyle(\mathcal{ELC})$
$\displaystyle\varphi::=p\mid\neg\varphi\mid(\varphi\rightarrow\varphi)\mid
K_{a}\varphi\mid C_{G}\varphi$ $\displaystyle(\mathcal{ELD})$
$\displaystyle\varphi::=p\mid\neg\varphi\mid(\varphi\rightarrow\varphi)\mid
K_{a}\varphi\mid D_{G}\varphi$ $\displaystyle(\mathcal{ELF})$
$\displaystyle\varphi::=p\mid\neg\varphi\mid(\varphi\rightarrow\varphi)\mid
K_{a}\varphi\mid F_{G}\varphi$ $\displaystyle(\mathcal{ELCD})$
$\displaystyle\varphi::=p\mid\neg\varphi\mid(\varphi\rightarrow\varphi)\mid
K_{a}\varphi\mid C_{G}\varphi\mid D_{G}\varphi$
$\displaystyle(\mathcal{ELCF})$
$\displaystyle\varphi::=p\mid\neg\varphi\mid(\varphi\rightarrow\varphi)\mid
K_{a}\varphi\mid C_{G}\varphi\mid F_{G}\varphi$
$\displaystyle(\mathcal{ELDF})$
$\displaystyle\varphi::=p\mid\neg\varphi\mid(\varphi\rightarrow\varphi)\mid
K_{a}\varphi\mid D_{G}\varphi\mid F_{G}\varphi$
$\displaystyle(\mathcal{ELCDF})$
$\displaystyle\varphi::=p\mid\neg\varphi\mid(\varphi\rightarrow\varphi)\mid
K_{a}\varphi\mid C_{G}\varphi\mid D_{G}\varphi\mid F_{G}\varphi$

where $p\in\text{{Prop}}$, $a\in\text{{Ag}}$, and $G$ represents a nonempty
subset of Ag, signifying a group. We also employ other boolean connectives,
including conjunction ($\wedge$), disjunction ($\vee$), and equivalence
($\leftrightarrow$). $E_{G}\varphi$ is a shorthand for $\bigwedge_{a\in
G}K_{a}\varphi$ (note that $G$ is finite).

We employ formulas such as $K_{a}\varphi$ to depict: “Agent $a$ knows
$\varphi$.” This is often referred to as _individual knowledge_. Similarly,
formulas like $C_{G}\varphi$, $D_{G}\varphi$, $E_{G}\varphi$ and
$F_{G}\varphi$ are used to convey that $\varphi$ is _common knowledge_ ,
_distributed knowledge_ , _mutual knowledge_ (or _everyone’s knowledge_) and
field knowledge of group $G$, respectively. When the group $G$ is a simple
set, e.g., $\\{a,b\\}$, we write $C_{ab}\varphi$ as a shorthand for
$C_{\\{a,b\\}}\varphi$, and likewise for the operators $D_{ab}$, $E_{ab}$ and
$F_{ab}$.

Before delving into the formal semantics of these formulas, it is important to
first establish the semantic models that will be used for the intended logics.

###  2.2 Semantics

We introduce a type of _similarity models_ for the interpretation of the
languages.

######  Definition 2.2.

(similarity models) A _similarity model_ (_model_ for short) is a quintuple
$(W,A,E,C,\nu)$ where:

  * •

$W$ is a nonempty set of states or nodes, referred to as the _domain_ ;

  * •

$A$ is an arbitrary set of abstract epistemic _abilities_ (e.g., one’s
expertise or profession), which could be empty, finite or infinite;222We have
opted not to fix the set $A$ of abilities as a given parameter of the logic,
in contrast to the set Ag of agents. The primary reason for this decision is
our intention to examine models that may extend the set $A$ (see Section
3.2.1). It is important to note that the validities and subsequent
axiomatization of our logic remain unaffected when $A$ is considered to be an
infinite set of abilities defined as a parameter of the logic.

  * •

$E:W\times W\to\wp(A)$, known as an _edge function_ , assigns each pair of
states a set of epistemic abilities, meaning that the two states are
indistinguishable for individuals possessing only these epistemic abilities;

  * •

$C:\text{{Ag}}\to\wp(A)$ is a _capability function_ that assigns each agent a
set of epistemic abilities;

  * •

$\nu:W\to\wp(\text{{Prop}})$ is a valuation.

and conforms to the following conditions (for all $s,t\in W$):

  * •

Positivity: if $E(s,t)=A$, then $s=t$;

  * •

Symmetry: $E(s,t)=E(t,s)$.

The above definition warrants further elucidation. Firstly, our approach
adopts a broad interpretation of epistemic abilities that may not necessarily
be arranged in a linear order, although such an arrangement is plausible [7,
15]. Secondly, we perceive the edge function $E$ as a representation of the
relation of similarity between states. The conditions of positivity and
symmetry serve as generalized forms of common conditions employed to
characterize similarity between data objects, as demonstrated in [20].333An
implicit condition often assumed, the converse of positivity, posits
$E(s,t)=A$ if $s=t$. This condition entails the reflexivity of graphs,
depicted by the characterization axiom T (i.e.,
$K_{a}\varphi\rightarrow\varphi$). In the realm of data mining, this condition
implies that if two data objects are identical (i.e., they refer to the same
data object), they would receive the maximum value from any similarity
measure. This, however, is not always guaranteed. Transitivity is usually not
a characteristic of this framework (that $x$ and $y$, $y$ and $z$ are similar,
does not necessarily mean that $x$ and $z$ are similar), resulting in the
failure to uphold the principle of positive introspection
($K_{a}\varphi\rightarrow K_{a}K_{a}\varphi$). Although it is easy to impose
transitivity, we choose not to enforce it here. Our framework allows a more
discerning evaluation of the tenability of positive introspection, and for an
examination of other significant constraints, see [15]. Thirdly, in this
context, similarities are deemed objective, signifying their constancy across
diverse agents.

######  Example 2.3.

Alice and Bob are denoted as $a$ and $b$, respectively. Consider the fields
mentioned in the beginning of the paper: musical instruments ($\alpha$),
philosophy ($\beta$) and mathematics ($\gamma$), which are regarded as
epistemic abilities in this example. As set up in the beginning of the paper,
Alice is a philosopher, Bob a mathematician, and both of them are also
instrumentalists. Four possible states are named $s_{1},\dots,s_{4}$, in which
$s_{1}$ is the factual state. From the viewpoint of an instrumentalist, all
the states look no difference. From the perspective of a philosopher, no
difference is between $s_{1}$ and $s_{3}$, and between $s_{2}$ and $s_{4}$. As
for a mathematician, $s_{1}$ is indistinguishable to $s_{2}$, and $s_{3}$ to
$s_{4}$. Consider the following propositions:

$p$: A standard modern piano has 88 keys in total.

$q$: Knowledge is defined by “justified true belief.”

$r$: Fermat’s Last Theorem has been proved.

The above scenario can be abstracted to a pointed model $(M,s_{1})$, such that
$M=(W,A,E,C,\nu)$ and:

  * •

$W=\\{s_{1},s_{2},s_{3},s_{4}\\}$; $A=\\{\alpha,\beta,\gamma\\}$;

  * •

$E(s_{1},s_{2})=E(s_{3},s_{4})=\\{\alpha,\gamma\\}$,
$E(s_{1},s_{3})=E(s_{2},s_{4})=\\{\alpha,\beta\\}$,
$E(s_{1},s_{4})=E(s_{2},s_{3})=\\{\alpha\\}$, and for all $x\in W$,
$E(x,x)=\\{\alpha,\beta,\gamma\\}$;

  * •

$C(a)=\\{\alpha,\beta\\}$ and $C(b)=\\{\alpha,\gamma\\}$;

  * •

$\nu(s_{1})=\\{p,r\\}$, $\nu(s_{2})=\\{p,q,r\\}$, $\nu(s_{3})=\\{p\\}$ and
$\nu(s_{4})=\\{p,q\\}$.

Figure 1 illustrates the model $M$ introduced above, where the factual state
$s_{1}$ is framed by a rectangle and other states by an eclipse.

$\textstyle{\frac{s_{1}}{p,r}}$$\scriptstyle{\alpha,\beta,\gamma}$$\scriptstyle{\alpha,\gamma}$$\scriptstyle{\alpha,\beta}$$\scriptstyle{\alpha}$$\textstyle{\frac{s_{2}}{p,q,r}}$$\scriptstyle{\alpha,\beta,\gamma}$$\scriptstyle{\alpha,\beta}$$\scriptstyle{\alpha}$$\textstyle{\frac{s_{3}}{p}}$$\scriptstyle{\alpha,\beta,\gamma}$$\scriptstyle{\alpha,\gamma}$$\textstyle{\frac{s_{4}}{p,q}}$$\scriptstyle{\alpha,\beta,\gamma}$

$\begin{array}[]{l}C(a)=\\{\alpha,\beta\\}\\\ C(b)=\\{\alpha,\gamma\\}\\\
\end{array}$

Figure 1: Illustration of the model in Example 2.3.

In the real world ($s_{1}$), one may come up with the following true
sentences:

  * •

$K_{a}(p{\wedge}\neg q){\wedge}\neg(K_{a}r{\vee}K_{a}\neg r)$ (Alice knows $p$
and $\neg q$, but doesn’t know whether $r$.)

  * •

$K_{b}(p{\wedge}r)\land\neg(K_{a}q{\vee}K_{a}\neg q)$ (Bob knows that $p$ and
$r$, but doesn’t know whether $q$.)

  * •

$D_{ab}(p\wedge\neg q\wedge r)$ (It is Alice and Bob’s distributed knowledge
that $p$, not $q$, and $r$.)

  * •

$F_{ab}p\land\neg(F_{ab}q\vee F_{ab}\neg q)\land\neg(F_{ab}r\vee F_{ab}\neg
r)$ (While $p$ is Alice and Bob’s field knowledge, $q$ and $r$ are not.)

We now introduce a formal semantics that makes the model in Example 2.3 indeed
yields the true sentences listed above.

######  Definition 2.4.

Given a formula $\varphi$, a model $M=(W,A,E,C,\nu)$ and a state $s\in W$, we
say $\varphi$ is _true_ or _satisfied_ at $s$ in $M$, denoted
$M,s\models\varphi$, if the following hold (the case for $E_{G}\psi$ is
redundant, but included for clarification):

$\begin{array}[]{lll}M,s\models p&\iff&p\in\nu(s)\\\
M,s\models\neg\psi&\iff&\text{not }M,s\models\psi\\\
M,s\models(\psi\rightarrow\chi)&\iff&\text{if }M,s\models\psi\text{ then
}M,s\models\chi\\\ M,s\models K_{a}\psi&\iff&\text{for all $t\in W$, if
$C(a)\subseteq E(s,t)$ then $M,t\models\psi$}\\\ M,s\models
E_{G}\psi&\iff&\text{$M,s\models K_{a}\psi$ for all $a\in G$}\\\ M,s\models
C_{G}\psi&\iff&\text{for all positive integers $n$, $M,s\models
E_{G}^{n}\psi$}\\\ M,s\models D_{G}\psi&\iff&\text{for all $t\in W$, if
$\bigcup_{a\in G}C(a)\subseteq E(s,t)$ then $M,t\models\psi$}\\\ M,s\models
F_{G}\psi&\iff&\text{for all $t\in W$, if $\bigcap_{a\in G}C(a)\subseteq
E(s,t)$ then $M,t\models\psi$,}\\\ \end{array}$

where $E_{G}^{n}\psi$ is defined recursively as $E^{1}_{G}E_{G}^{n-1}\psi$,
with $E_{G}^{1}\psi$ to mean $E_{G}\psi$. The concepts of _validity_ and
_satisfiability_ have their classical meaning.

In the definition above, the interpretation of $K_{a}\psi$ includes a
condition “$C(a)\subseteq E(s,t)$,” which intuitively means that, “Agent $a$,
with its abilities, cannot discern between states $s$ and $t$.” Thus, the
formula $K_{a}\psi$ expresses that $\psi$ is true in all states $t$ that $a$
cannot differentiate from the current state $s$.

$E_{G}\psi$ stands for the conventional notion of _everyone’s knowledge_ , or
_mutual knowledge_ as we call, stating that “Everyone in group $G$ knows
$\psi$” (see [10] for details).

_Common knowledge_ ($C_{G}\psi$) follows the classical fixed-point
interpretation as $E_{G}C_{G}\psi$. In other words, $C_{G}\psi$ implies that,
“Everyone in group $G$ knows that $\psi$ is true, and everyone in $G$ knows
about this first-order knowledge, and also knows about this second-order
knowledge, and so on.”

The concept of _distributed knowledge_ ($D_{G}\psi$) in this paper diverges
from the traditional definitions found in literature. We redefine distributed
knowledge as being attainable by pooling together individual abilities. In
practice, we swap the intersection of individual uncertainty relations with
the union of individual epistemic abilities. Thus, $\psi$ is deemed
distributed knowledge among group $G$ if and only if $\psi$ holds true in all
states $t$ that, when utilizing all the epistemic abilities of agents in group
$G$, cannot be differentiated from the present state.

An additional type of group knowledge, termed _field knowledge_ ($F_{G}\psi$),
states that $\psi$ is field knowledge if and only if $\psi$ is true in all
states $t$ that, using the shared abilities of group $G$, cannot be
differentiated from the current state. We will examine its logical properties
in greater detail later on.

Upon defining the semantics, we derive eight logics, each associated with one
of the languages. We denote these logics using upright Roman capital letters.
E.g., the logic corresponding to the interpretation of the language
$\mathcal{ELF}$ is represented as ELF.

######  Example 2.5.

Consider the model illustrated in Figure 2, we have the following:

  1. (1)

$M,s_{2}\models C_{ab}p\land E_{ab}p\land D_{ab}p\land\neg F_{ab}p$ (note that
for $C_{ab}p$ we check whether $p$ is true in all states along an “$ab$-path”
– connected via $\\{\lambda,\pi\\}$, $\\{\lambda,\mu\\}$ or
$\\{\lambda,\pi,\mu\\}$ edges – that is, whether $p$ is true at $s_{2}$
through $s_{4}$, regardless of $s_{1}$.)

  2. (2)

$M,s_{2}\models F_{ab}q\land E_{ab}q\land D_{ab}q\land\neg C_{ab}q$

  3. (3)

$M,s_{3}\models D_{ab}r\land\neg K_{a}r\land\neg K_{b}r\land\neg E_{ab}r$

$\textstyle{\frac{s_{1}}{q}}$$\scriptstyle{\lambda}$$\scriptstyle{\lambda}$$\textstyle{\frac{s_{2}}{p,q}}$$\scriptstyle{\lambda,\pi,\mu}$$\scriptstyle{\lambda,\pi}$$\textstyle{\frac{s_{3}}{p,q,r}}$$\scriptstyle{\lambda,\pi,\mu}$$\scriptstyle{\lambda,\mu}$$\textstyle{\frac{s_{4}}{p}}$$\scriptstyle{\lambda,\pi}$

$\begin{array}[]{l}C(a)=\\{\lambda,\pi\\}\\\ C(b)=\\{\lambda,\mu\\}\\\
\end{array}$

Figure 2: Illustration of a model for Example 2.5. We do not draw a line
between two nodes when the edge between them is with no label (i.e., labeled
by an empty set, e.g., between $s_{1}$ and $s_{3}$).

From the above, it is clear that none of the following formula schemes are
valid: $C_{G}\varphi\rightarrow F_{G}\varphi$, $E_{G}\varphi\rightarrow
F_{G}\varphi$, $D_{G}\varphi\rightarrow F_{G}\varphi$,
$F_{G}\varphi\rightarrow C_{G}\varphi$, $E_{G}\varphi\rightarrow
C_{G}\varphi$, $D_{G}\varphi\rightarrow C_{G}\varphi$,
$D_{G}\varphi\rightarrow E_{G}\varphi$, $D_{G}\varphi\rightarrow K_{a}\varphi$
(where $a\in G$). In particular, that $\varphi$ is common knowledge implies
that $\varphi$ is distributed knowledge ($\models C_{G}\varphi\rightarrow
D_{G}\varphi$), but does not imply that it is field knowledge ($\not\models
C_{G}\varphi\rightarrow F_{G}\varphi$). The underlying reasoning for this is
that when professions intersect, the range of uncertain states can potentially
expand significantly – sometimes even more so than the increase that occurs
when taking the transitive closure in the case of common knowledge.
Consequently, this expansion of uncertainty can lead to a substantial
contraction of field knowledge. Nonetheless, the standard principles
pertaining to individual, common, and distributed knowledge from classical
logic remain applicable, as indicated by the following proposition.

######  Proposition 2.6.

We have the following validities for any given formula $\varphi$, any agent
$a$ and any groups $G$ and $H$ (proofs omitted):

  1. (1)

$K_{a}(\varphi\rightarrow\psi)\rightarrow(K_{a}\varphi\rightarrow K_{a}\psi)$

  2. (2)

$\varphi\rightarrow K_{a}\neg K_{a}\neg\varphi$

  3. (3)

$C_{G}\varphi\rightarrow\bigwedge_{a\in G}K_{a}(\varphi\wedge C_{G}\varphi)$

  4. (4)

$D_{\\{a\\}}\varphi\leftrightarrow K_{a}\varphi$

  5. (5)

$D_{G}\varphi\rightarrow D_{H}\varphi$ (with $G\subseteq H$)

  6. (6)

$\varphi\rightarrow D_{G}\neg D_{G}\neg\varphi$

  7. (7)

$F_{\\{a\\}}\varphi\leftrightarrow K_{a}\varphi$

  8. (8)

$F_{G}\varphi\rightarrow F_{H}\varphi$ (with $H\subseteq G$)

  9. (9)

$\varphi\rightarrow F_{G}\neg F_{G}\neg\varphi$

###  2.3 Expressivity

We adopt the conventional method for assessing a language’s expressive power,
which entails benchmarking it against the expressive capabilities of other
languages. For an exact articulation of the relations in expressive power
between two compatibly interpreted languages, we direct the reader to [21,
Def. 8.2].

It is evident that the expressive power of all eight languages under
consideration can be evaluated against one another. The comparative outcomes
are encapsulated in Figure 3, and the proofs are left in Appendix A.

---
$\textstyle{\mathcal{ELCDF}}$$\textstyle{\mathcal{ELCD}}$$\textstyle{\mathcal{ELCF}}$$\textstyle{\mathcal{ELDF}}$$\textstyle{\mathcal{ELC}}$$\textstyle{\mathcal{ELD}}$$\textstyle{\mathcal{ELF}}$$\textstyle{\mathcal{EL}}$

(a) when $|\text{{Ag}}|=1$

---
$\textstyle{\mathcal{ELCDF}}$$\textstyle{\mathcal{ELCD}}$$\textstyle{\mathcal{ELCF}}$$\textstyle{\mathcal{ELDF}}$$\textstyle{\mathcal{ELC}}$$\textstyle{\mathcal{ELD}}$$\textstyle{\mathcal{ELF}}$$\textstyle{\mathcal{EL}}$

(b) when $|\text{{Ag}}|\geq 2$

Figure 3: The above two diagrams illustrate the relative expressive power of
the languages. An arrow pointing from one language to another implies that the
second language is at least as expressive as the first. The “at least as
expressive as” relationship is presumed to be _reflexive_ and _transitive_ ,
meaning that a language is considered at least as expressive as another if a
path of arrows exists leading from the second to the first (self-loops exist
for all, but omitted). A lack of a path of arrows from one language to another
indicates that the first language is _not_ at least as expressive as the
second. This implies that either the two languages are incomparable or that
the first language is more expressive than the second.

##  3 Axiomatization

We will present sound and complete axiomatic systems for the logics introduced
in the preceding section. The names of these systems will be designated with
bold capital letters. For example, the axiomatic system for the logic ELF is
denoted as $\mathbf{ELF}$.

###  3.1 Axiomatic systems

The $\mathbf{K}$ is a widely recognized axiomatic system for modal logic (here
it refers to the multi-modal version with each $K_{a}$ functioning as a box
operator). For simplicity, the axiom schemes are referred to as axioms in this
context. The axiom system KB is obtained by augmenting the system $\mathbf{K}$
with an additional axiom B (i.e., $\varphi\rightarrow K_{a}\neg
K_{a}\neg\varphi$). In this context, we represent KB as $\mathbf{K}$ $\oplus$
B, where the symbol $\oplus$ acts like a union operation for sets of axioms
and/or rules. For a comprehensive understanding of these axiomatic systems for
modal logic, please refer to, say, [5]. Now, the system $\mathbf{EL}$ that we
introduce for our base logic EL is in fact KB.

Common knowledge is characterized by a set $\mathbf{C}$ consisting of the
following two inductive principles, represented by an axiom and a rule, which
can be found in [8]:

  * (C1)

$C_{G}\varphi\rightarrow\bigwedge_{a\in G}K_{a}(\varphi\wedge C_{G}\varphi)$

  * (C2)

from $\varphi\rightarrow\bigwedge_{a\in G}K_{a}(\varphi\wedge\psi)$ infer
$\varphi\rightarrow C_{G}\psi$

Our the system $\mathbf{ELC}$ is then represented as
$\mathbf{EL}\oplus\mathbf{C}$.

Distributed knowledge is characterized by a set $\mathbf{D}$ of additional
axioms:

  * (KD)

$D_{G}(\varphi\rightarrow\psi)\rightarrow(D_{G}\varphi\rightarrow D_{G}\psi)$

  * (D1)

$D_{\\{a\\}}\varphi\leftrightarrow K_{a}\varphi$

  * (D2)

$D_{G}\varphi\rightarrow D_{H}\varphi$ with $G\subseteq H$

  * (BD)

$\varphi\rightarrow D_{G}\neg D_{G}\neg\varphi$

The resulting system for the logic ELD is then denoted as
$\mathbf{ELD}=\mathbf{EL}\oplus\mathbf{D}$.

Field knowledge is characterized by the following set $\mathbf{F}$:

  * (KF)

$F_{G}(\varphi\rightarrow\psi)\rightarrow(F_{G}\varphi\rightarrow F_{G}\psi)$

  * (F1)

$F_{\\{a\\}}\varphi\leftrightarrow K_{a}\varphi$

  * (F2)

$F_{G}\varphi\rightarrow F_{H}\varphi$ with $H\subseteq G$

  * (BF)

$\varphi\rightarrow F_{G}\neg F_{G}\neg\varphi$

  * (NF)

from $\varphi$ infer $F_{G}\varphi$

While the set $\mathbf{F}$ might at first glance seem analogous to
$\mathbf{D}$, there exists a subtle yet crucial difference between the axioms
M2 and D2 – specifically, the positions of the groups $G$ and $H$ are swapped.
This distinction necessitates the introduction of the necessitation rule NF,
while within the system $\mathbf{ELD}$, the rule “from $\varphi$ infer
$D_{G}\varphi$” is derivable. The validity of these axioms can be verified
with relative ease, and it is particularly noteworthy how the altered order of
$G$ and $H$ precisely mirrors the union/intersection of epistemic abilities as
observed in the semantic interpretations. Similarly, $\mathbf{ELF}$ is
represented as $\mathbf{EL}\oplus\mathbf{F}$.

Moving towards more complex axiomatic systems, they are constructed in a
similar manner. For any given string $\mathbf{\Xi}$ comprising elements from
the set $\mathbf{\\{C,D,M\\}}$:

> The axiomatic system $\mathbf{EL\Xi}$ consists of all axioms and rules of
> $\mathbf{EL}$, along with those of the sets denoted by each character in
> string $\mathbf{\Xi}$.

To illustrate, when $\mathbf{\Xi}$ is the string “$\mathbf{CF}$,”
$\mathbf{ELCF}$ stands for $\mathbf{EL}\oplus\mathbf{C}\oplus\mathbf{F}$, and
when $\mathbf{\Xi}$ is the string “$\mathbf{CDF}$,” then $\mathbf{ELCDF}$
equates to $\mathbf{EL}\oplus\mathbf{C}\oplus\mathbf{D}\oplus\mathbf{F}$. For
an extreme case, when $\mathbf{\Xi}$ is an empty string, $\mathbf{EL\Xi}$
simply stands for $\mathbf{EL}$.

We now turn our attention to validating these axiomatic systems to be sound
and complete for the corresponding logics. Soundness signifies that all the
theorems of an axiomatic system are valid sentences of the corresponding
logic. This can be simplified to the task of verifying that all the axioms of
the system are valid, and that all the rules preserve this validity. The
soundness of the proposed axiomatic systems can be confirmed without much
difficulty. Though we omit the proof, we state it as the following theorem. We
will follow this up with the completeness results in the subsequent section.

######  Theorem 3.1 (soundness).

Every axiomatic system introduced in this section is sound for its
corresponding logic. ∎

###  3.2 Completeness

In this section, we aim to demonstrate the completeness of all eight axiomatic
systems that were introduced earlier.

It is a widely accepted fact in classical epistemic logic that the inclusion
of common knowledge can cause a logic to lose its _compactness_. This leads to
the situation where its axiomatic system is not strongly complete, but only
weakly complete (see, e.g., [5, 21]). This is also the case in our context. As
a consequence, we will demonstrate that the four systems that do not include
common knowledge are strongly complete axiomatic systems for their
corresponding logics, while the other four systems that do incorporate common
knowledge are only weakly complete.

The structure of this section is predicated on the various proof techniques we
employ. We start with a method that reduces the satisfiability from classical
epistemic logics to the logics we have proposed (Section 3.2.1). However, this
technique is only applicable to the system $\mathbf{EL}$; the canonical model
method also works, and we provide a definition in Section 3.2.2 for the
reference of the reader. When dealing with systems that incorporate either
distributed or field knowledge, but not both (i.e., $\mathbf{ELD}$ and
$\mathbf{ELF}$), we utilize a path-based canonical model method (Section
3.2.3). For the system that includes both distributed and field knowledge,
namely $\mathbf{ELDF}$, while the path-based canonical model method still
applies, a slightly more nuanced approach is required (Section 3.2.4). Lastly,
for the remaining systems incorporating common knowledge, we merge the
finitary method (which involves constructing a closure) with the methods
mentioned above (Section 3.2.5).

####  3.2.1 Proof by translation of satisfiability

######  Definition 3.2.

(translation $\rho$) The mapping $\cdot^{\rho}$ from symmetric Kripke models
to similarity models is such that, for any symmetric Kripke model $N=(W,R,V)$,
$N^{\rho}$ is the similarity model $(W,\text{{Ag}}\cup\\{b\\},E,C,\nu)$ with
the same domain and:

  * •

$E$ is such that for all $s,t\in W$, $E(s,t)=\\{a\in\text{{Ag}}\mid(s,t)\in
R(a)\\}$,

  * •

$b$ is a new agent which does not appear in Ag,

  * •

$C$ is such that for all $a\in\text{{Ag}}$, $C(a)=\\{a\\}$, and

  * •

$\nu$ is such that for all $s\in W$, $\nu(s)=\\{p\in\text{{Prop}}\mid s\in
V(p)\\}$.

In the translated model $N^{\rho}$ of the above definition, the set of
epistemic abilities is appointed as $\text{{Ag}}\cup\\{b\\}$. We use agents as
labels of edges, which can intuitively be understood as an agent’s inability
to distinguish the ongoing state from current state when considering their
epistemic abilities as a whole. In the subsequent lemma, we demonstrate that
this translation preserves truth.

######  Lemma 3.3.

The following hold (proof in Appendix B):

  1. (1)

Given a symmetric Kripke model $N$, its translation $N^{\rho}$ is a similarity
model;

  2. (2)

For any $\mathcal{ELCD}$-formula $\varphi$, any symmetric Kripke model $N$ and
any state $s$ of $N$, $N,s\Vdash\varphi$ iff $N^{\rho},s\models\varphi$.444The
operator $F_{G}$ is undefined in classical epistemic logic, so there is no
point to consider a language incorporated with this operator.

The system $\mathbf{EL}$ is known to be complete for symmetric Kripke models.
By Lemma 3.3, such a model can be translated to a similarity model in a way
that preserves truth. So we get the following.

######  Theorem 3.4.

$\mathbf{EL}$ is strongly complete for EL.∎

We note that it is possible to use the same method to achieve completeness
results for the systems $\mathbf{ELC}$, $\mathbf{ELD}$ and $\mathbf{ELCD}$, as
long as their completeness results in classical epistemic logic exist.
However, to the best of our knowledge, the completeness of these systems in
Kripke semantics, while expected, has never been explicitly established.

####  3.2.2 Proof by the canonical model method

We have demonstrated the completeness of $\mathbf{EL}$ using the method of
translation. This method is efficient and relies on the completeness result
for its classical counterpart interpreted via Kripke semantics. In other
words, the translation method cannot be employed for logics whose classical
counterparts have not been introduced or studied (for example, logics with
field knowledge), or when a completeness result does not exist for their
classical counterparts (for instance, ELC, ELD and ELCD). In this section, we
introduce a direct proof of the completeness of $\mathbf{EL}$ using the
canonical model method and extend this to completeness proofs for other logics
in later sections.

######  Definition 3.5.

The _canonical model for EL_ is the tuple $\textnormal{\sf M}=(\textnormal{\sf
W},\textnormal{\sf A},\textnormal{\sf E},\textnormal{\sf C},\textnormal{\sf
v})$ such that:

  * •

W is the set of all maximal $\mathbf{EL}$-consistent sets of
$\mathcal{EL}$-formulas;

  * •

$\textnormal{\sf A}=\wp(\text{{Ag}})$, the power set of all agents;

  * •

$\textnormal{\sf E}:\textnormal{\sf W}\times\textnormal{\sf
W}\to\wp(\textnormal{\sf A})$ is defined such that for any
$\Phi,\Psi\in\textnormal{\sf W}$,
$E(\Phi,\Psi)=\bigcup_{a\in\text{{Ag}}}E_{a}(\Phi,\Psi)$, where

$E_{a}(\Phi,\Psi)=\left\\{\begin{array}[]{ll}\textnormal{\sf C}(a),&\text{if
$\\{\chi\mid K_{a}\chi\in\Phi\\}\subseteq\Psi$ and $\\{\chi\mid
K_{a}\chi\in\Psi\\}\subseteq\Phi$},\\\
\emptyset,&\text{otherwise;}\end{array}\right.$

  * •

$\textnormal{\sf C}:\text{{Ag}}\to\wp(\textnormal{\sf A})$ is such that for
any agent $a$, $\textnormal{\sf C}(a)=\\{G\subseteq\text{{Ag}}\mid a\in G\\}$;

  * •

$\textnormal{\sf v}:\textnormal{\sf W}\to\wp(\text{{Prop}})$ is such that for
any $\Phi\in\textnormal{\sf W}$, $\textnormal{\sf
v}(\Phi)=\\{p\in\text{{Prop}}\mid p\in\Phi\\}$.

The discerning reader may first ascertain that the canonical model for EL is
indeed a model, and then proceed to demonstrate the completeness of
$\mathbf{EL}$ by employing conventional techniques. The specifics of these
procedures are left in Appendix C.

####  3.2.3 Proof using a path-based canonical model

When addressing logics that incorporate concepts of distributed and field
knowledge, the conventional canonical model technique is not suitable. To
overcome this challenge, there exists a methodology for logics with
distributed knowledge [9]. This technique, which traces its origins to the
unraveling methods of [19], starts by analogously treating distributed
knowledge as forms of individual knowledge. The process involves creating a
pseudo model that embodies these aspects. This pseudo model is then unraveled
into a tree-like structure with paths. The resulting structure is further
processed through an identification/folding step to yield the target model.

An simplified approach has been proposed in [22], where the construction of a
path-based, tree-like model, referred to as a _standard model_ , is advocated.
This approach eliminates the intermediate steps of unraveling and
identification/folding. We embrace this latter approach here, setting out to
construct a standard model directly.

##### Completeness of $\mathbf{ELD}$

######  Definition 3.6.

$\langle\Phi_{0},G_{1},\Phi_{1},\dots,G_{n},\Phi_{n}\rangle$ is called a
_canonical path_ for ELD, if:

  * •

$\Phi_{0},\Phi_{1},\dots,\Phi_{n}$ represent maximal $\mathbf{ELD}$-consistent
sets of $\mathcal{ELD}$-formulas,

  * •

$G_{1},\dots,G_{n}$ denote groups of agents, i.e., nonempty subsets of Ag.

In the context of a canonical path
$s=\langle\Phi_{0},G_{1},\Phi_{1},\dots,G_{n},\Phi_{n}\rangle$, we denote
$\Phi_{n}$ as $tail(s)$. This also applies to canonical paths defined later.

######  Definition 3.7.

The _standard model for ELD_ is the tuple $\textnormal{\sf M}=(\textnormal{\sf
W},\textnormal{\sf A},\textnormal{\sf E},\textnormal{\sf C},\textnormal{\sf
v})$ such that:

  * •

W is the set of all canonical paths for ELD;

  * •

$\textnormal{\sf A}=\wp(\text{{Ag}})$;

  * •

$\textnormal{\sf E}:\textnormal{\sf W}\times\textnormal{\sf
W}\to\wp(\textnormal{\sf A})$ is defined such that for any
$s,t\in\textnormal{\sf W}$, $\textnormal{\sf E}(s,t)=$

$\left\\{\begin{array}[]{ll}\bigcup_{a\in G}\textnormal{\sf C}(a),&\text{if
$t$ is $s$ extended with $\langle G,\Psi\rangle$},\\\\[-3.0pt]
&\text{$\\{\chi\mid D_{G}\chi\in tail(s)\\}\subseteq\Psi$ and $\\{\chi\mid
D_{G}\chi\in\Psi\\}\subseteq tail(s)$};\\\ \bigcup_{a\in G}\textnormal{\sf
C}(a),&\text{if $s$ is $t$ extended with $\langle G,\Psi\rangle$},\\\\[-3.0pt]
&\text{$\\{\chi\mid D_{G}\chi\in tail(t)\\}\subseteq\Psi$ and $\\{\chi\mid
D_{G}\chi\in\Psi\\}\subseteq tail(t)$};\\\
\emptyset,&\text{otherwise;}\end{array}\right.$

  * •

$\textnormal{\sf C}:\text{{Ag}}\to\wp(\textnormal{\sf A})$ is such that for
any agent $a$, $\textnormal{\sf C}(a)=\\{G\in\text{{Ag}}\mid a\in G\\}$;

  * •

$\textnormal{\sf v}:\textnormal{\sf W}\to\wp(\text{{Prop}})$ is such that for
all $s\in\textnormal{\sf W}$, $\textnormal{\sf v}(s)=\\{p\in\text{{Prop}}\mid
p\in tail(s)\\}$.

######  Lemma 3.8 (standardness).

The standard model for ELD is a model.

######  Proof 3.9.

Note that $\emptyset\notin\textnormal{\sf C}(a)$ for any agent $a$. This
implies that for any $s,t\in\textnormal{\sf W}$, $\textnormal{\sf
E}(s,t)\neq\textnormal{\sf A}$, thereby meeting the criterion of positivity.
Additionally, the condition of symmetry is fulfilled as E is a commutative
function.

######  Lemma 3.10 (Truth Lemma).

Let $\textnormal{\sf M}=(\textnormal{\sf W},\textnormal{\sf A},\textnormal{\sf
E},\textnormal{\sf C},\textnormal{\sf v})$ be the standard model for ELD. For
any $s\in\textnormal{\sf W}$ and $\mathcal{ELD}$-formula $\varphi$,
$\varphi\in tail(s)$ iff $\textnormal{\sf M},s\models_{\text{ELD}}\varphi$.

######  Proof 3.11.

We prove it by induction of $\varphi$. The boolean cases are easy by the
definition of $\nu$ and the induction hypothesis. The only interested cases
are $\varphi=K_{a}\psi$ and $\varphi=D_{G}\psi$.

Case $\varphi=K_{a}\psi$: very similar to the case for $D_{G}\psi$ given
below.

Case $\varphi=D_{G}\psi$: Suppose $D_{G}\psi\notin tail(s)$, but
$\textnormal{\sf M},s\models_{\text{ELD}}D_{G}\psi$. By definition,
$\bigcup_{a\in G}\textnormal{\sf C}(a)\subseteq\textnormal{\sf E}(s,t)$
implies $\textnormal{\sf M},t\models_{\text{ELD}}\psi$ for any
$t\in\textnormal{\sf W}$. We can extend $\\{\neg\psi\\}\cup\\{\chi\mid
D_{G}\chi\in tail(s)\\}\cup\\{\neg D_{G}\neg\chi\mid\chi\in tail(s)\\}$ to
some maximal $\mathbf{ELD}$-consistent set $\Delta^{+}$. Similar to the proof
of Lemma C.3 we get $\bigcup_{a\in G}\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(s,t)$ where $t$ extends $s$ with $\langle
G,\Delta^{+}\rangle$. By the induction hypothesis we have $\textnormal{\sf
M},t\models_{\text{ELD}}\neg\psi$. A contradiction!

For the opposite direction, suppose $D_{G}\psi\in tail(s)$, but
$\textnormal{\sf M},s\not\models_{\text{ELD}}D_{G}\psi$. Then there exists
$t\in W$ such that $\textnormal{\sf M},t\not\models_{\text{ELD}}\psi$ and
$\bigcup_{a\in G}C(a)\subseteq E(s,t)$. This implies that there exists
$H\supseteq G$, such that $\\{\chi\mid D_{H}\chi\in tail(s)\\}\subseteq
tail(t)$ and $\\{\chi\mid D_{H}\chi\in tail(t)\\}\subseteq tail(s)$. Since
$D_{G}\psi\in tail(s)$ implies $D_{H}\psi\in tail(s)$, we have $\psi\in
tail(t)$. By the induction hypothesis we have $\textnormal{\sf
M},t\models_{\text{ELD}}\psi$, also leading to a contradiction.

######  Theorem 3.12.

$\mathbf{ELD}$ is strongly complete for ELD. ∎

##### Completeness of $\mathbf{ELF}$

The completeness of $\mathbf{ELF}$ can be demonstrated in a manner that
parallels the completeness of $\mathbf{ELD}$. While we will not delve into the
intricate details of the proofs, we will outline the necessary adaptations to
the definitions of the standard model.

A canonical path for ELF mirrors that for ELD. The only modification required
is the adjustment of the maximal consistent sets to align with the axiomatic
system being considered. When defining the standard model for ELF, we replace
W with the set of all canonical paths for ELF, and let

$\textnormal{\sf E}(s,t)=\left\\{\begin{array}[]{ll}\bigcap_{a\in
G}\textnormal{\sf C}(a),&\text{if $t$ is $s$ extended with $\langle
G,\Psi\rangle$,}\\\\[-3.0pt] &\text{$\\{\chi\mid F_{G}\chi\in
tail(s)\\}\subseteq\Psi$ and $\\{\chi\mid F_{G}\chi\in\Psi\\}\subseteq
tail(s)$},\\\ \bigcap_{a\in G}\textnormal{\sf C}(a),&\text{if $s$ is $t$
extended with $\langle G,\Psi\rangle$,}\\\\[-3.0pt] &\text{$\\{\chi\mid
F_{G}\chi\in tail(t)\\}\subseteq\Psi$ and $\\{\chi\mid
F_{G}\chi\in\Psi\\}\subseteq tail(t)$},\\\
\emptyset,&\text{otherwise.}\end{array}\right.$

Note that $\bigcup_{a\in G}\textnormal{\sf C}(a)=\\{H\mid H\cap
G\neq\emptyset\\}$, which includes all the shared epistemic abilities of those
groups $H$ that intersects with $G$. Additionally, $\bigcap_{a\in
G}\textnormal{\sf C}(a)=\\{H\mid G\subseteq H\\}$, which represents all the
supersets of $G$.

By using analogous proof structures, we can demonstrate the standardness of
these models, show a Truth Lemma, and establish the completeness.

######  Theorem 3.13.

$\mathbf{ELF}$ is strongly complete for ELF. ∎

####  3.2.4 Incorporation of both distributed and field knowledge

We now discuss the logic and its axiomatic system that incorporate both
distributed and field knowledge but exclude common knowledge, namely the logic
ELDF and the axiomatic system $\mathbf{ELDF}$. The construction process
requires careful consideration of the intricate interaction between the two
types of knowledge modalities.

######  Definition 3.14.

$\langle\Phi_{0},I_{1},\Phi_{1},\dots,I_{n},\Phi_{n}\rangle$ is a _canonical
path for ELDF_, if:

  * •

$\Phi_{0},\Phi_{1},\dots,\Phi_{n}$ are maximal $\mathbf{ELDF}$-consistent sets
of $\mathcal{ELDF}$-formulas;

  * •

$I_{1},\dots,I_{n}$ are of the form $(G,d)$ or $(G,m)$, with $G$ denoting a
group, and “$d$” and “$m$” being just two distinct characters.

######  Definition 3.15.

The _standard model for ELDF_ is a tuple $\textnormal{\sf M}=(\textnormal{\sf
W},\textnormal{\sf A},\textnormal{\sf E},\textnormal{\sf C},\textnormal{\sf
v})$ where A, C and v are defined just as in the standard model for ELD
(Definition 3.7), and:

  * •

W is the set of all canonical paths for ELDF;

  * •

$\textnormal{\sf E}:\textnormal{\sf W}\times\textnormal{\sf
W}\to\wp(\textnormal{\sf A})$ is such that for any $s,t\in\textnormal{\sf W}$,
$\textnormal{\sf E}(s,t)=$

$\left\\{\begin{array}[]{ll}\bigcup_{a\in G}\textnormal{\sf C}(a),&\text{if
$t$ extends $s$ with $\langle(G,d),\Psi\rangle$,}\\\ &\text{$\\{\chi\mid
D_{G}\chi\in tail(s)\\}\subseteq\Psi$ and $\\{\chi\mid
D_{G}\chi\in\Psi\\}\subseteq tail(s)$},\\\ \bigcup_{a\in G}\textnormal{\sf
C}(a),&\text{if $s$ extends $t$ with $\langle(G,d),\Psi\rangle$,}\\\
&\text{$\\{\chi\mid D_{G}\chi\in tail(t)\\}\subseteq\Psi$ and $\\{\chi\mid
D_{G}\chi\in\Psi\\}\subseteq tail(t)$},\\\ \bigcap_{a\in G}\textnormal{\sf
C}(a),&\text{if $t$ extends $s$ with $\langle(G,m),\Psi\rangle$,}\\\
&\text{$\\{\chi\mid F_{G}\chi\in tail(s)\\}\subseteq\Psi$ and $\\{\chi\mid
F_{G}\chi\in\Psi\\}\subseteq tail(s)$},\\\ \bigcap_{a\in G}\textnormal{\sf
C}(a),&\text{if $s$ extends $t$ with $\langle(G,m),\Psi\rangle$,}\\\
&\text{$\\{\chi\mid F_{G}\chi\in tail(t)\\}\subseteq\Psi$ and $\\{\chi\mid
F_{G}\chi\in\Psi\\}\subseteq tail(t)$},\\\
\emptyset,&\text{otherwise.}\end{array}\right.$

The standardness and the Truth Lemma can be achieved in a similar way, and we
leave a proof of the Truth Lemma in Appendix D for the careful reader.

######  Theorem 3.16.

$\mathbf{ELDF}$ is strongly complete for ELDF. ∎

####  3.2.5 Proof by a finitary standard model

We now delineate the extension of the completeness results to the rest of the
logics with common knowledge, deploying a finitary method for this purpose. We
can only achieve weak completeness due to the non-compact nature of the common
knowledge modality. A difficulty, except for ELC, is that we also need to
address the modality for distributed or field knowledge.

In this section, we focus on providing the completeness proofs for
$\mathbf{ELCDF}$. By making simple adaptations, one can obtain the
completeness of the axiomatic systems for their sublogics with common
knowledge. We adapt the definition of the _closure_ of a formula presented in
[22], to cater to formulas with modalities $D_{G}$ and/or $F_{G}$.

######  Definition 3.17.

For an $\mathcal{ELCDF}$-formula $\varphi$, we define $cl(\varphi)$ as the
minimal set satisfying the subsequent conditions:

  1. (1)

$\varphi\in cl(\varphi)$;

  2. (2)

if $\psi$ is in $cl(\varphi)$, so are all subformulas of $\psi$;

  3. (3)

$\psi\in cl(\varphi)$ implies ${\sim}\psi\in cl(\varphi)$, where
$\mathord{\sim}\psi=\neg\psi$ if $\psi$ is not a negation and
$\mathord{\sim}\psi=\chi$ if $\psi=\neg\chi$;

  4. (4)

$K_{a}\psi\in cl(\varphi)$ implies $D_{\\{a\\}}\psi,F_{\\{a\\}}\psi\in
cl(\varphi)$;

  5. (5)

$D_{\\{a\\}}\psi\in cl(\varphi)$ implies $K_{a}\psi\in cl(\varphi)$;

  6. (6)

For groups $G$ and $H$, if $H$ appears in $\varphi$, then $D_{G}\psi\in
cl(\varphi)$ implies $D_{H}\psi\in cl(\varphi)$;

  7. (7)

$C_{G}\psi\in cl(\varphi)$ implies $\\{K_{a}\psi,K_{a}C_{G}\psi\mid a\in
G\\}\subseteq cl(\varphi)$;

  8. (8)

$F_{G}\psi\in cl(\varphi)$ implies $\\{K_{a}\psi\mid a\in G\\}\subseteq
cl(\varphi)$;

  9. (9)

For groups $G$ and $H$, if $H$ appears in $\varphi$, then $F_{G}\psi\in
cl(\varphi)$ implies $F_{H}\psi\in cl(\varphi)$.

Considering that the initial two clauses exclusively introduce subformulas of
$\varphi$, the subsequent three clauses incorporate formulas in a constrained
manner, and given that there are a finite number of groups mentioned in
$\varphi$, with each group containing only a finite number of agents, we can
readily confirm that $cl(\varphi)$ for any given formula $\varphi$.

Subsequently, we introduce the concept of a _maximal consistent set of
formulas within a closure_. For a comprehensive definition, which is naturally
contingent on the specific axiomatic system under consideration, we refer to
established literature, for example, [21].

A _canonical path for ELCDF in $cl(\varphi)$_ is defined similarly to that for
ELDF (Def. 3.14). Given an $\mathcal{ELCDF}$-formula $\varphi$, we can
construct the _standard model for ELCDF with respect to $cl(\varphi)$_ in a
manner that closely mirrors the construction of the standard model for ELDF
(as per Def. 3.15). The primary differences lie in bounding the canonical
paths by the closure and adjusting the logics accordingly. More specifically,
we need to: (1) replace all occurrences of “$\mathcal{ELDF}$” with
“$\mathcal{ELCDF}$,” and “ELDF” with “ELCDF”; (2) within the definition of W,
replace “canonical paths for ELDF” with “canonical paths for ELCDF in
$cl(\varphi)$.” Moreover, it is easy to confirm that the standard model for
ELCDF (in any closure of a given formula) is a model.

######  Lemma 3.18 (Truth Lemma).

Given an $\mathcal{ELCDF}$-formula $\theta$, and let $\textnormal{\sf
M}=(\textnormal{\sf W},\textnormal{\sf A},\textnormal{\sf E},\textnormal{\sf
C},\textnormal{\sf v})$ be the standard model for ELCDF with respect to
$cl(\theta)$, for any $s\in\textnormal{\sf W}$ and $\varphi\in cl(\theta)$, we
have $\varphi\in tail(s)$ iff $\textnormal{\sf
M},s\models_{\text{ELCDF}}\varphi$.

######  Proof 3.19.

We show the lemma by induction on $\varphi$. We omit the straightforward cases
(partially found in Appendix E) and focus on the cases concerning common
knowledge.

Suppose $C_{G}\psi\in tail(s)$, but $\textnormal{\sf
M},s\not\models_{\text{ELCDF}}C_{G}\psi$, then there are
$s_{i}\in\textnormal{\sf W}$, $a_{i}\in G$, $0\leq i\leq n$ for some
$n\in\mathbb{N}$ such that: $s_{0}=s$, $\textnormal{\sf
M},s_{n}\not\models\psi$ and $\textnormal{\sf
C}(a_{i})\subseteq\textnormal{\sf E}(s_{i-1},s_{i})$ for $1\leq i\leq n$.
Since $\textnormal{\sf C}(a_{i})\subseteq\textnormal{\sf E}(s_{i-1},s_{i})$,
we have either $\\{\chi\mid D_{H}\chi\in tail(s_{i-1})\\}\subseteq
tail(s_{i})$ for some $H$ containing $a_{i}$ or $\\{\chi\mid
F_{\\{a_{i}\\}}\chi\in tail(s_{i-1})\\}\subseteq tail(s_{i})$. In both cases,
$\\{\chi\mid K_{a_{i}}\chi\in tail(s_{i-1})\\}\subseteq tail(s_{i})$. Since
$C_{G}\psi\in tail(s_{i})$ implies $K_{a_{i}}C_{G}\psi,K_{a_{i}}\psi\in
tail(s_{i})$, we can infer that $C_{G}\psi,\psi\in tail(s_{n})$. By the
induction hypothesis, $\textnormal{\sf M},s_{n}\models_{\text{ELCDF}}\psi$,
leading to a contradiction.

Suppose $C_{G}\psi\not\in tail(s)$, but $\textnormal{\sf
M},s\models_{\text{ELCDF}}C_{G}\psi$. Thus for any $s_{i}\in\textnormal{\sf
W}$ and $a_{i}\in G$ where $0\leq i\leq n$, such that: $s_{0}=s$ and
$\textnormal{\sf C}(a_{i})\subseteq\textnormal{\sf E}(s_{i-1},s_{i})$, we have
$M,s_{n}\models_{\text{ELCDF}}\psi$ and
$M,s_{n}\models_{\text{ELCDF}}C_{G}\psi$. Collect all such possible $s_{n}$
above and $s$ into the set $\mathcal{S}$; similarly collect all the
$tail(s_{n})$ and $tail(s)$ into the set $\Theta$. We define
$\delta=\bigvee_{t\in\mathcal{S}}\widehat{tail(t)}$, where for any
$t\in\textnormal{\sf W}$, $\widehat{tail(t)}=\bigwedge tail(t)$. (In general,
for any finite set $\Psi$ of formulas, we write $\widehat{\Psi}$ for
$\bigwedge\Psi$.) We claim that $\vdash_{\text{ELCDF}}\delta\rightarrow
K_{a}\delta$ and $\vdash_{\text{ELCDF}}\delta\rightarrow K_{a}\psi$ for any
$a\in G$. By this claim and (C2) we have
$\vdash_{\text{ELCDF}}\delta\rightarrow C_{G}\psi$, and then by
$\widehat{tail(s)}\rightarrow\delta$ we have $\widehat{tail(s)}\rightarrow
C_{G}\psi$. In this way $C_{G}\psi\in tail(s)$, which leads to a
contradiction. As for the proof of the claim:

(1) Suppose $\nvdash_{\text{ELCDF}}\delta\rightarrow K_{a}\delta$, then
$\delta\wedge\neg K_{a}\delta$ is consistent. Then there exists
$t_{0}\in\mathcal{S}$ such that $\widehat{tail(t_{0})}\wedge\neg K_{a}\delta$
is consistent. Notice that $\vdash_{\text{ELCDF}}\bigvee_{t\in\textnormal{\sf
W}}\widehat{tail(t)}$, hence we have a consistent set
$\widehat{tail(t_{0})}\wedge\neg K_{a}\neg\widehat{tail(t_{1})}$ for some
$t_{1}\in\textnormal{\sf W}\setminus\mathcal{S}$; for otherwise we have
$\textnormal{\sf W}\setminus\mathcal{S}=\emptyset$, hence
$\vdash_{\text{ELCDF}}\delta$, which leads to
$\vdash_{\text{ELCDF}}K_{a}\delta$, contradicting with
$\nvdash_{\text{ELCDF}}\delta\rightarrow K_{a}\delta$. Thus we have
$\\{\chi\mid K_{a}\chi\in tail(t_{0})\\}\subseteq tail(t_{1})$, which implies
$\\{\chi\mid D_{\\{a\\}}\chi\in tail(t_{0})\\}\subseteq tail(t_{1})$.
Moreover, for any $\chi$, if $D_{\\{a\\}}\chi\in tail(t_{1})$, then
$\widehat{tail(t_{0})}\wedge\neg K_{a}\neg D_{\\{a\\}}\chi$ is consistent,
hence $\widehat{tail(t_{0})}\wedge\chi$ is also consistent, thus $\chi\in
tail(t_{0})$. So we have $\\{\chi\mid D_{\\{a\\}}\chi\in
tail(t_{1})\\}\subseteq tail(t_{0})$. Now we let $t_{2}$ be $t_{0}$ extended
with $\langle(\\{a\\},d),tail(t_{1})\rangle$, we have $\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(t_{0},t_{2})$. Hence $t_{2}\in\mathcal{S}$
but $tail(t_{2})=tail(t_{1})\notin\Theta$. A contradiction!

(2) Suppose $\nvdash_{\text{ELCDF}}\delta\rightarrow K_{a}\psi$, then
$\delta\wedge\neg K_{a}\psi$ is consistent. So there exists
$t_{0}\in\mathcal{S}$ such that $\widehat{tail(t_{0})}\wedge\neg K_{a}\psi$ is
consistent. Thus $\\{\mathord{\sim}\psi\\}\cup\\{\chi\mid D_{\\{a\\}}\chi\in
tail(t_{0})\\}\cup\\{\neg D_{\\{a\\}}\mathord{\sim}\chi\in
cl(\theta)\mid\chi\in tail(t_{0})\\}$ is consistent. Hence it can be extended
to some max consistent subset $\Delta^{+}$ in $cl(\theta)$. Let $t_{1}$ be
$t_{0}$ extended with $\langle(\\{a\\},d),\Delta^{+}\rangle$, we have
$\\{\chi\mid D_{\\{a\\}}\chi\in tail(t_{0})\\}\subseteq tail(t_{1})$.
Moreover, if $D_{\\{a\\}}\chi\in tail(t_{1})=\Delta^{+}$, then $\neg
D_{\\{a\\}}\chi\notin\Delta^{+}$, thus $\neg
D_{\\{a\\}}\mathord{\sim}\neg\chi=\neg D_{\\{a\\}}\chi\in cl(\theta)$ and
$\neg\chi\notin tail(t_{0})$, which implies $\chi\in tail(t_{0})$. So we also
have $\\{\chi\mid D_{\\{a\\}}\chi\in tail(t_{1})\\}\subseteq tail(t_{0})$.
Thus we have $\textnormal{\sf C}(a)\subseteq\textnormal{\sf E}(t_{0},t_{1})$.
Hence $t_{1}\in\mathcal{S}$ and then $\textnormal{\sf
M},t_{1}\models_{\text{ELCDF}}\psi$, which contradicts with
$\mathord{\sim}\psi\in tail(t_{1})$ by the induction hypothesis.

######  Theorem 3.20.

$\mathbf{ELCDF}$ is weakly complete for ELCDF. ∎

Building on the discussion earlier in this section, the completeness proofs
for $\mathbf{ELC}$, $\mathbf{ELCD}$ and $\mathbf{ELCF}$ can be derived from
the proofs for $\mathbf{ELCDF}$. For brevity, however, the intricate details
of these adaptations are not included in this paper.

##  4 Conclusion

We examined epistemic logics with all combinations of common, distributed and
field knowledge, interpreted in scenarios that consider agents’ epistemic
abilities, such as professions. We adopted a type of similarity model that
extends from a Kripke model by adding weights to edges, and studied the
axiomatization of the resulting logics.

The framework of our logics presents diverse possibilities for characterizing
the concept of _knowability_. Apart from interpreting knowability as known
after a single announcement [4], a group announcement [2], or after a group
resolves their knowledge [3], it is now conceivable to perceive knowability as
known after an agent acquires certain skills (epistemic abilities) from some
source or from a given group. Our framework also enables us to easily
characterize _forgetability_ or _degeneration_ through changes in epistemic
abilities, a process that is not as straightforward in classical epistemic
logic.

Looking ahead, we aim to explore more sophisticated conditions on the
similarity relation, such as those introduced in [6]. It would also be of
interest to compare our framework with existing ones that use the same style
of models, as presented in [17, 7].

## References

  * [1] Aggarwal, C. C., “Data Mining: The Textbook,” Springer, 2015.
  * [2] Ågotnes, T., P. Balbiani, H. van Ditmarsch and P. Seban, _Group announcement logic_ , Journal of Applied Logic 8 (2010), pp. 62–81.
  * [3] Ågotnes, T. and Y. N. Wáng, _Resolving distributed knowledge_ , Artificial Intelligence 252 (2017), pp. 1–21.
  * [4] Balbiani, P., A. Baltag, H. van Ditmarsch, A. Herzig, T. Hoshi and T. de Lima, _‘knowable’ as ‘known after an announcement’_ , The Review of Symbolic Logic 1 (2008), pp. 305–334.
  * [5] Blackburn, P., M. de Rijke and Y. Venema, “Modal logic,” Cambridge University Press, 2001.
  * [6] Chen, S., B. Ma and K. Zhang, _On the similarity metric and the distance metric_ , Theoretical Computer Science 410 (2009), pp. 2365–2376.
  * [7] Dong, H., X. Li and Y. N. Wáng, _Weighted modal logic in epistemic and deontic contexts_ , in: S. Ghosh and T. Icard, editors, _Proceedings of the Eighth International Conference on Logic, Rationality and Interaction (LORI 2021)_ , Lecture Notes of Theoretical Computer Science 13039 (2021), pp. 73–87.
  * [8] Fagin, R., J. Y. Halpern, Y. Moses and M. Y. Vardi, “Reasoning about knowledge,” The MIT Press, 1995.
  * [9] Fagin, R., J. Y. Halpern and M. Y. Vardi, _What can machines know? on the properties of knowledge in distributed systems_ , Journal of the ACM 39 (1992), pp. 328–376.
  * [10] Halpern, J. Y. and Y. Moses, _A guide to completeness and complexity for modal logics of knowledge and belief_ , Artificial Intelligence 54 (1992), pp. 319–379.
  * [11] Hansen, M., K. G. Larsen, R. Mardare and M. R. Pedersen, _Reasoning about bounds in weighted transition systems_ , Logical Methods in Computer Science 14 (2018), pp. 1–32.
  * [12] Hintikka, J., “Knowledge and Belief: An Introduction to the Logic of Two Notions,” Cornell University Press, Ithaca, New York, 1962.
  * [13] Larsen, K. G. and R. Mardare, _Complete proof systems for weighted modal logic_ , Theoretical Computer Science 546 (2014), pp. 164–175.
  * [14] Larsen, K. G. and A. Skou, _Bisimulation through probabilistic testing_ , Information and Computation 94 (1991), pp. 1–28.
  * [15] Liang, X. and Y. N. Wáng, _Epistemic logic via distance and similarity_ , in: _Proceedings of PRICAI 2022: Trends in Artificial Intelligence_ (2022), pp. 32–45.
  * [16] Meyer, J.-J. C. and W. van der Hoek, “Epistemic Logic for AI and Computer Science,” Cambridge University Press, 1995.
  * [17] Naumov, P. and J. Tao, _Logic of confidence_ , Synthese 192 (2015), pp. 1821–1838.
  * [18] Roelofsen, F., _Distributed knowledge_ , Journal of Applied Non-Classical Logics 16 (2007), pp. 255–273.
  * [19] Sahlqvist, H., _Completeness and correspondence in the first and second order semantics for modal logic_ , in: S. Kanger, editor, _Proceedings of the Third Scandinavian Logic Symposium_ , Studies in Logic and the Foundations of Mathematics 82, Elsevier, 1975 pp. 110 – 143.
  * [20] Tan, P.-N., M. Steinbach and V. Kumar, “Introduction to data mining,” Pearson, 2005.
  * [21] van Ditmarsch, H., W. van der Hoek and B. Kooi, “Dynamic Epistemic Logic,” Synthese Library 337, Springer Netherlands, 2008.
  * [22] Wáng, Y. N. and T. Ågotnes, _Simpler completeness proofs for modal logics with intersection_ , in: M. A. Martins and I. Sedlár, editors, _Dynamic Logic: New Trends and Applications_ , Lecture Notes in Computer Science 12569 (2020), pp. 259–276.

##  Appendix A Proofs regarding Experssivity

In Figure 3, every language, with the exception of $\mathcal{ELCDF}$, has an
arrow pointing to its immediate superlanguages. This is clearly true, as by
definition, every language is at most as expressive as its superlanguages. In
the case when $|\text{{Ag}}|=1$, a reverse arrow also exists between languages
that either both contain common knowledge or neither contain common knowledge.

######  Lemma A.1.

When $|\text{{Ag}}|=1$ (i.e., when there is only one agent available in the
language),

  1. (1)

$\mathcal{EL}\equiv\mathcal{ELD}\equiv\mathcal{ELF}\equiv\mathcal{ELDF}$

  2. (2)

$\mathcal{ELC}\equiv\mathcal{ELCD}\equiv\mathcal{ELCF}\equiv\mathcal{ELCDF}$

  3. (3)

$\mathcal{ELDF}\prec\mathcal{ELC}$, and hence any language in the first clause
are less expressive than any language in the second clause.

######  Proof A.2.

1 & 2\. In the case when $|\text{{Ag}}|=1$ there is only one agent, and since
$D_{\\{a\\}}\varphi$ and $F_{\\{a\\}}\varphi$ are equivalent to
$K_{a}\varphi$, the operators for distributed and mutual knowledge are
redundant in this case. Hence the lemma.

3\. We show that $\mathcal{ELC}\not\preceq\mathcal{ELDF}$, and so
$\mathcal{ELDF}\prec\mathcal{ELC}$ since
$\mathcal{ELDF}\equiv\mathcal{EL}\preceq\mathcal{ELC}$ by the first clause.
Suppose towards a contradiction that there exists a formula $\varphi$ of
$\mathcal{ELDF}$ equivalent to $C_{\\{a\\}}p$. Consider the set
$\Phi=\\{E_{\\{a\\}}^{n}p\mid n\in\mathbb{N}\\}\cup\\{\neg C_{\\{a\\}}p\\}$.
It is not hard to see that any finite subset of $\Phi$ is satisfiable, but not
$\Phi$ itself. Let $k$ be the length of $\varphi$ (refer to a modal logic
textbook for its definition), and suppose $\\{E_{\\{a\\}}^{n}p\mid
n\in\mathbb{N},n\leq k\\}\cup\\{\neg\varphi\\}$ is satisfied at a state $w$ in
a model $M=(W,A,E,C,\nu)$. For any $s,t\in W$, we say that $s$ reaches $t$ in
one step if $C(a)\subseteq E(s,t)$. Consider the model
$M_{k}=(W_{k},A,E,C,\nu)$, where $W_{k}$ is set of states reachable from $w$
in at most $k$ steps. We can verify that
$M_{k},w\models\\{E_{\\{a\\}}^{n}p\mid
n\in\mathbb{N}\\}\cup\\{\neg\varphi\\}$, which implies that $\Phi$ is
satisfiable, leading to a contradiction.

We now proceed to elucidate the absence of arrows in the figure for the case
when $|\text{{Ag}}|\geq 2$.

######  Lemma A.3.

When $|\text{{Ag}}|\geq 2$ (i.e., when there are at least two agents available
in the language),

  1. (1)

For any superlanguage $\mathcal{L}$ of $\mathcal{ELC}$, and any sublanguage
$\mathcal{L}^{\prime}$ of $\mathcal{ELDF}$, it is not the case that
$\mathcal{L}\preceq\mathcal{L}^{\prime}$;

  2. (2)

For any superlanguage $\mathcal{L}$ of $\mathcal{ELD}$, and any sublanguage
$\mathcal{L}^{\prime}$ of $\mathcal{ELCF}$, it is not the case that
$\mathcal{L}\preceq\mathcal{L}^{\prime}$;

  3. (3)

For any superlanguage $\mathcal{L}$ of $\mathcal{ELF}$, and any sublanguage
$\mathcal{L}^{\prime}$ of $\mathcal{ELCD}$, it is not the case that
$\mathcal{L}\preceq\mathcal{L}^{\prime}$.

######  Proof A.4.

1\. The Proof of Lemma A.1(3) can be used here to show that
$\mathcal{ELC}\not\preceq\mathcal{ELDF}$ also when $|\text{{Ag}}|\geq 2$.

2\. Consider models $M=(W,A,E,C,\nu)$ and
$M^{\prime}=(W^{\prime},A,E^{\prime},C,\nu^{\prime})$, where $A=\\{1,2,3\\}$,
$C(a)=\\{1,2\\}$, $C(b)=\\{1,3\\}$ (if there are more agents in the language,
they are irrelevant here), and are illustrated below.

$M$
$\textstyle{\frac{u_{1}}{p}}$$\scriptstyle{1,2}$$\scriptstyle{1,3}$$\textstyle{\frac{u_{2}}{p}}$$\scriptstyle{1,2}$$\textstyle{\frac{u_{4}}{p}}$$\scriptstyle{1,3}$$\textstyle{\frac{u_{3}}{p}}$
$M^{\prime}$ $\textstyle{\frac{u^{\prime}}{p}}$$\scriptstyle{1,2,3}$

We can show by induction that for any formula $\varphi$ of $\mathcal{ELCF}$,
$M,u_{1}\models\varphi$ iff $M^{\prime},u^{\prime}\models\varphi$. On the
other hand, $M,u_{1}\models D_{ab}\bot$ but $M^{\prime},u^{\prime}\not\models
D_{ab}\bot$. It means that no $\mathcal{ELCF}$-formula can discern between
$M,u_{1}$ and $M^{\prime},u^{\prime}$, while languages with distributed
knowledge can. Thus the lemma holds.

3\. Consider similarity models $M=(W,A,E,C,\nu)$ and
$M^{\prime}=(W^{\prime},A,E^{\prime},C,\nu^{\prime})$, where $A=\\{1,2,3\\}$,
$C(a)=\\{1,2\\}$, $C(b)=\\{1,3\\}$, and are illustrated below.

$M$
$\textstyle{\frac{u_{1}}{p}}$$\scriptstyle{1,2,3}$$\scriptstyle{1}$$\textstyle{\frac{u_{2}}{}}$$\scriptstyle{1,2,3}$
$M^{\prime}$ $\textstyle{\frac{u^{\prime}}{p}}$$\scriptstyle{1,2,3}$

We can show by induction that for any formula $\varphi$ of $\mathcal{ELCD}$,
$M,u_{1}\models\varphi$ iff $M^{\prime},u^{\prime}\models\varphi$. Meanwhile,
we have $M,u_{1}\not\models F_{ab}p$ and $M^{\prime},u^{\prime}\models
F_{ab}p$. It follows that no $\mathcal{ELCD}$-formula can discern between
$M,u_{1}$ and $M^{\prime},u^{\prime}$, while languages with field knowledge
can. Thus the lemma holds.

As per Figure 3, Lemma A.3 suggests that there is not an arrow or a path of
arrows leading from $\mathcal{ELC}$ (or any language having an arrow or a path
of arrows originating from $\mathcal{ELC}$) to $\mathcal{ELDF}$ (or any
language with an arrow or a path of arrows pointing to $\mathcal{ELDF}$).
Similar relationships exist between $\mathcal{ELD}$ and $\mathcal{ELCF}$, and
between $\mathcal{ELF}$ and $\mathcal{ELCD}$. Furthermore, in Figure 3, if
there is an arrow or a path of arrows leading from one language to another,
and not the other way round, this signifies that the first language is less
expressive than the second. If there is no arrow or path of arrows in either
direction between two languages, they are deemed incomparable. These
observations lead us directly to the following corollary.

######  Corollary A.5.

When $|\text{{Ag}}|\geq 2$,

  1. (1)

$\mathcal{EL}\prec\mathcal{ELC}$, $\mathcal{ELD}\prec\mathcal{ELCD}$,
$\mathcal{ELF}\prec\mathcal{ELCF}$ and $\mathcal{ELDF}\prec\mathcal{ELCDF}$;

  2. (2)

$\mathcal{EL}\prec\mathcal{ELD}$, $\mathcal{ELC}\prec\mathcal{ELCD}$,
$\mathcal{ELF}\prec\mathcal{ELDF}$ and $\mathcal{ELCF}\prec\mathcal{ELCDF}$;

  3. (3)

$\mathcal{EL}\prec\mathcal{ELF}$, $\mathcal{ELC}\prec\mathcal{ELCF}$,
$\mathcal{ELD}\prec\mathcal{ELDF}$ and $\mathcal{ELCD}\prec\mathcal{ELCDF}$;

  4. (4)

$\mathcal{ELC}$, $\mathcal{ELD}$ and $\mathcal{ELF}$ are pairwise
incomparable;

  5. (5)

$\mathcal{ELCD}$, $\mathcal{ELCF}$ and $\mathcal{ELDF}$ are pairwise
incomparable;

  6. (6)

$\mathcal{ELC}$ is incomparable with $\mathcal{ELDF}$;

  7. (7)

$\mathcal{ELD}$ is incomparable with $\mathcal{ELCF}$;

  8. (8)

$\mathcal{ELF}$ is incomparable with $\mathcal{ELCD}$.

##  Appendix B Proof of Lemma 3.3

We provide a proof for Lemma 3.3 while first repeating it:

######  Lemma B.1.

The following hold:

  1. (1)

Given a symmetric Kripke model $N$, its translation $N^{\rho}$ is a similarity
model;

  2. (2)

For any $\mathcal{ELCD}$-formula $\varphi$, any symmetric Kripke model $N$ and
any state $s$ of $N$, $N,s\Vdash\varphi$ iff $N^{\rho},s\models\varphi$.

Proof. (i) Let $N=(W,R,V)$ be a symmetric Kripke model, and its translation
$N^{\rho}=(W,\text{{Ag}}\cup\\{b\\},E,C,\nu)$. For any
$a\in\text{{Ag}}\cup\\{b\\}$ and $s,t\in W$, we have:

$\begin{array}[]{llll}a\in E(s,t)&\iff&(s,t)\in
R(a)&\text{(Def.~{}\ref{def:trans-w})}\\\ &\iff&(t,s)\in R(a)&\text{(since
$R(a)$ is symmetric)}\\\ &\iff&a\in
E(t,s).&\text{(Def.~{}\ref{def:trans-w})}\\\ \end{array}$

Hence $N^{\rho}$ satisfies symmetry. Furthermore, $N^{\rho}$ satisfies
positivity, as there cannot be any $s,t\in W$ such that $E(s,t)=A\cup\\{b\\}$
and $s\neq t$. Hence $N^{\rho}$ is a similarity model.

(ii) Let $N=(W,R,V)$ and its translation
$N^{\rho}=(W,\text{{Ag}}\cup\\{b\\},E,C,\nu)$. We show the lemma by induction
on $\varphi$. The cases involving atomic propositions, Boolean connectives,
and common knowledge are straightforward to verify since their semantic
definitions follow a consistent pattern that facilitates smooth inductive
reasoning. In this proof, we focus specifically on the cases for individual
and distributed knowledge. It should be noted that the case for individual
knowledge can be regarded as a particular instance of distributed knowledge;
however, we include the details here for readers who seek a thorough clarity:

$\begin{array}[]{rcll}N,s\Vdash K_{a}\psi&\iff&\text{for all $t\in W$, if
$(s,t)\in R(a)$ then $N,t\Vdash\psi$}&\\\ &\iff&\text{for all $t\in W$, if
$a\in E(s,t)$ then $N,t\Vdash\psi$}&\\\ &\iff&\text{for all $t\in W$, if
$C(a)\subseteq E(s,t)$ then $N,t\Vdash\psi$}&\\\ &\iff&\text{for all $t\in W$,
if $C(a)\subseteq E(s,t)$ then $N^{\rho},t\models\psi$}&\\\
&\iff&N^{\rho},s\models K_{a}\psi.&\\\\[10.00002pt] N,s\Vdash
D_{G}\psi&\iff&\text{for all $t\in W$, if $(s,t)\in\bigcap_{a\in G}R(a)$, then
$N,t\Vdash\psi$}&\\\ &\iff&\text{for all $t\in W$, if $(s,t)\in R(a)$ for all
$a\in G$, then $N,t\Vdash\psi$}&\\\ &\iff&\text{for all $t\in W$, if
$C(a)\subseteq E(s,t)$ for all $a\in G$, then $N,t\Vdash\psi$}&\\\
&\iff&\text{for all $t\in W$, if $\bigcup_{a\in G}C(a)\subseteq E(s,t)$, then
$N,t\models\psi$}&\\\ &\iff&\text{for all $t\in W$, if $\bigcup_{a\in
G}C(a)\subseteq E(s,t)$, then $N^{\rho},t\models\psi$}&\\\
&\iff&N^{\rho},s\models D_{G}\psi.&\\\ \end{array}$

##  Appendix C Completeness of $\mathbf{EL}$ by the Canonical Model Method

######  Lemma C.1 (canonicity).

The canonical model for EL is a similarity model.

######  Proof C.2.

Let $\textnormal{\sf M}=(\textnormal{\sf W},\textnormal{\sf A},\textnormal{\sf
E},\textnormal{\sf C},\textnormal{\sf v})$ be the canonical model for EL.
Notice that $\emptyset\notin\textnormal{\sf C}(a)$ for any agent $a$, so
$\textnormal{\sf E}(s,t)\neq\textnormal{\sf A}$ for any $s,t\in\textnormal{\sf
W}$, ensuring positivity. The symmetry of the model is evident as
$\textnormal{\sf E}(s,t)=\textnormal{\sf E}(t,s)$ for any
$s,t\in\textnormal{\sf W}$. Therefore, M is a similarity model.

######  Lemma C.3 (Truth Lemma).

Let $\textnormal{\sf M}=(\textnormal{\sf W},\textnormal{\sf A},\textnormal{\sf
E},\textnormal{\sf C},\textnormal{\sf v})$ be the canonical model for EL. For
any $\Gamma\in\textnormal{\sf W}$ and any $\mathcal{EL}$-formula $\varphi$, we
have $\varphi\in\Gamma$ iff $\textnormal{\sf
M},\Gamma\models_{\text{EL}}\varphi$.

######  Proof C.4.

We will only demonstrate the case when $\varphi$ is of the form $K_{a}\psi$
here.

Assuming $K_{a}\psi\in\Gamma$, but $\textnormal{\sf
M},\Gamma\not\models_{\text{EL}}K_{a}\psi$, there would exist a
$\Delta\in\textnormal{\sf W}$ such that $\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(\Gamma,\Delta)$ and $\textnormal{\sf
M},\Delta\not\models_{\text{EL}}\psi$. Consequently, $\\{\chi\mid
K_{a}\chi\in\Gamma\\}\subseteq\Delta$ (otherwise $\\{a\\}\notin\textnormal{\sf
E}(\Gamma,\Delta)$, contradicting $\\{a\\}\in\textnormal{\sf C}(a)$). Thus,
$\psi\in\Delta$. It follows from the induction hypothesis that
$\textnormal{\sf M},\Delta\models_{\text{EL}}\psi$, which results in a
contradiction.

For the opposite direction, suppose $K_{a}\psi\notin\Gamma$, but
$\textnormal{\sf M},\Gamma\models_{\text{EL}}K_{a}\psi$, then for any
$\Delta\in\textnormal{\sf W}$, $\textnormal{\sf C}(a)\subseteq\textnormal{\sf
E}(\Gamma,\Delta)$ implies $\textnormal{\sf M},\Delta\models_{\text{EL}}\psi$.
First, we assert that $\\{\neg\psi\\}\cup\\{\chi\mid
K_{a}\chi\in\Gamma\\}\cup\\{\neg K_{a}\neg\chi\mid\chi\in\Gamma\\}$ is
$\mathbf{EL}$ consistent. If not, note that for any $\eta\in\\{\chi\mid
K_{a}\chi\in\Gamma\\}$, we have $\neg K_{a}\neg K_{a}\eta\in\\{\neg
K_{a}\neg\chi\mid\chi\in\Gamma\\}$. As $\vdash_{\mathbf{EL}}\neg K_{a}\neg
K_{a}\eta\rightarrow\eta$, it follows that $\\{\neg\psi\\}\cup\\{\neg
K_{a}\neg\chi\mid\chi\in\Gamma\\}$ is not $\mathbf{EL}$ consistent. Therefore,
we have $\vdash_{\mathbf{EL}}\big{(}\bigwedge_{\chi\in\Gamma_{0}}\neg
K_{a}\neg\chi\big{)}\rightarrow\psi$ for some finite subset $\Gamma_{0}$ of
$\Gamma$. This leads to
$\vdash_{\mathbf{EL}}K_{a}\big{(}(\bigwedge_{\chi\in\Gamma_{0}}\neg
K_{a}\neg\chi)\rightarrow\psi\big{)}$, and hence
$\vdash_{\mathbf{EL}}\bigwedge_{\chi\in\Gamma_{0}}K_{a}\neg
K_{a}\neg\chi\rightarrow K_{a}\psi$. Since we have
$\vdash_{\mathbf{EL}}\chi\rightarrow K_{a}\neg K_{a}\neg\chi$ for any
$\chi\in\Gamma_{0}$, it follows that we have
$\vdash_{\mathbf{EL}}\big{(}\bigwedge_{\chi\in\Gamma_{0}}\chi\big{)}\rightarrow
K_{a}\psi$. This deduction implies that $K_{a}\psi\in\Gamma$, which
contradicts our previous assumption. Now, let us extend the set
$\\{\neg\psi\\}\cup\\{\chi\mid K_{a}\chi\in\Gamma\\}\cup\\{\neg
K_{a}\neg\chi\mid\chi\in\Gamma\\}$ to some maximal $\mathbf{EL}$-consistent
set $\Delta^{+}$ of $\mathcal{EL}$-formulas. Notice that $K_{a}\chi\in\Gamma$
implies $\chi\in\Delta^{+}$ for any $\chi$. Furthermore, if we suppose
$\chi\notin\Gamma$, then $\neg\chi\in\Gamma$, which leads to $\neg
K_{a}\neg\neg\chi\in\Delta^{+}$, implying $\neg K_{a}\chi\in\Delta^{+}$.
Therefore, $K_{a}\chi\in\Delta^{+}$ implies $\chi\in\Gamma$ for any $\chi$.
Given these stipulations, we find that $\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(\Gamma,\Delta)$. However, by using the
induction hypothesis, we see that $M,\Delta\not\models_{\text{EL}}\psi$. As a
result, $M,\Gamma\not\models_{\text{EL}}K_{a}\psi$. This conclusion
contradicts our previous assumptions, confirming this direction of the lemma.

With the Truth Lemma, we can state the following theorem:

######  Theorem C.5 (completeness of $\mathbf{EL}$, with a direct proof).

For any $\mathcal{EL}$-formula $\varphi$ and any set $\Phi$ of
$\mathcal{EL}$-formulas, if $\Phi\models_{\text{EL}}\varphi$, then
$\Phi\vdash_{\mathbf{EL}}\varphi$.

######  Proof C.6.

To prove this, suppose the contrary: $\Phi\not\vdash_{\mathbf{EL}}\varphi$. In
this case, the set $\Phi\cup\\{\neg\varphi\\}$ can be extended to a maximal
$\mathbf{EL}$-consistent set $\Delta^{+}$. In the canonical model for EL,
denoted M, we have $\textnormal{\sf M},\Delta^{+}\models\chi$ for any formula
$\chi\in\Phi\cup\\{\neg\varphi\\}$. This conclusion leads to
$\Phi\not\models_{\text{EL}}\varphi$.

##  Appendix D Truth Lemma for Theorem 3.16

######  Lemma D.1.

Let $\textnormal{\sf M}=(\textnormal{\sf W},\textnormal{\sf A},\textnormal{\sf
E},\textnormal{\sf C},\textnormal{\sf v})$ be the standard model for ELDF. For
any $s\in\textnormal{\sf W}$ and any $\mathcal{ELDF}$-formula $\varphi$,
$\varphi\in tail(s)$ if and only if $\textnormal{\sf
M},s\models_{\text{ELDF}}\varphi$.

######  Proof D.2.

The proof is by induction on $\varphi$, and we only display the cases for
modalities.

Case $\varphi=K_{a}\psi$. Suppose $K_{a}\psi\in tail(s)$, but $\textnormal{\sf
M},s\not\models_{\text{ELDF}}K_{a}\psi$, then there exists
$t\in\textnormal{\sf W}$ such that $\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(s,t)$ and $\textnormal{\sf
M},t\not\models_{\text{ELDF}}\psi$. Therefore, $\\{\chi\mid D_{G}\chi\in
tail(s)\\}\subseteq tail(t)$ for some group $G$ containing $a$ or $\\{\chi\mid
X_{\\{a\\}}\chi\in tail(s)\\}\subseteq tail(t)$. In both scenarios, $\psi\in
tail(t)$ since $K_{a}\psi\in tail(s)$ implies $D_{G}\psi,F_{\\{a\\}}\psi\in
tail(s)$. By the induction hypothesis, we have $\textnormal{\sf
M},t\models_{\text{ELDF}}\psi$, which leads to a contradiction. Suppose
$K_{a}\psi\notin tail(s)$, but $\textnormal{\sf
M},s\models_{\text{ELDF}}K_{a}\psi$, then $\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(s,t)$ implies $\textnormal{\sf
M},t\models_{\text{ELDF}}\psi$ for any $t\in\textnormal{\sf W}$. Extend
$\\{\neg\psi\\}\cup\\{\chi\mid K_{a}\chi\in tail(s)\\}\cup\\{\neg
K_{a}\neg\chi\mid\chi\in tail(s)\\}$ to some maximal
$\mathbf{ELDF}$-consistent set $\Delta^{+}$, thus $\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(s,t)$ where $t$ extends $s$ with
$\langle(\\{a\\},d),\Delta^{+}\rangle$. By the induction hypothesis, we have
$\textnormal{\sf M},t\models_{\text{ELDF}}\neg\psi$, which is contradictary.

Case $\varphi=D_{G}\psi$. Suppose $D_{G}\psi\in tail(s)$, but $\textnormal{\sf
M},s\not\models_{\text{ELDF}}D_{G}\psi$, then there exists some
$t\in\textnormal{\sf W}$ such that $\bigcup_{a\in G}\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(s,t)$ and $\textnormal{\sf
M},t\not\models_{\text{ELDF}}\psi$. Therefore, $\\{\chi\mid D_{H}\chi\in
tail(s)\\}\subseteq tail(t)$ for some group $H$ such that $G\subseteq H$. We
have $\psi\in tail(t)$ since $D_{G}\psi\in tail(s)$ implies $D_{H}\psi\in
tail(s)$. By the induction hypothesis, we have $\textnormal{\sf
M},t\models_{\text{ELDF}}\psi$, which leads to a contradiction. Suppose
$D_{G}\psi\notin tail(s)$, but $\textnormal{\sf
M},s\models_{\text{ELDF}}D_{G}\psi$, then $\bigcup_{a\in G}\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(s,t)$ implies $\textnormal{\sf
M},t\models_{\text{ELDF}}\psi$ for any $t\in\textnormal{\sf W}$. Extend
$\\{\neg\psi\\}\cup\\{\chi\mid D_{G}\chi\in tail(s)\cup\\{\neg
D_{G}\neg\chi\mid\chi\in tail(s)\\}\\}$ to some maximal
$\mathbf{ELDF}$-consistent set $\Delta^{+}$, thus $\bigcup_{a\in
G}\textnormal{\sf C}(a)\subseteq\textnormal{\sf E}(s,t)$ where $t$ extends $s$
with $\langle(G,d),\Delta^{+}\rangle$. We have $\textnormal{\sf
M},t\models_{\text{ELDF}}\neg\psi$ by the induction hypothesis. A
contradiction!

Case $\varphi=F_{G}\psi$. Suppose $F_{G}\psi\in tail(s)$, but $\textnormal{\sf
M},s\not\models_{\text{ELDF}}F_{G}\psi$, then there exists
$t\in\textnormal{\sf W}$ such that $\bigcap_{a\in G}\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(s,t)$ and $\textnormal{\sf
M},t\not\models_{\text{ELDF}}\psi$. Therefore, $\\{\chi\mid F_{H}\chi\in
tail(s)\\}\subseteq tail(t)$ for some group $H$ such that $H\subseteq G$ or
$\\{\chi\mid D_{J}\chi\in tail(s)\\}\subseteq tail(t)$ for some group $J$ such
that $G\cap J\neq\emptyset$. In both scenarios, we have $\psi\in tail(t)$
since $F_{G}\psi\in tail(s)$ implies $F_{H}\psi,D_{J}\psi\in tail(s)$. By the
induction hypothesis, we have $\textnormal{\sf M},t\models_{\text{ELDF}}\psi$,
which leads to a contradiction. Suppose $F_{G}\psi\notin tail(s)$, but
$\textnormal{\sf M},s\models_{\text{ELDF}}F_{G}\psi$, then $\bigcap_{a\in
G}\textnormal{\sf C}(a)\subseteq\textnormal{\sf E}(s,t)$ implies
$\textnormal{\sf M},t\models_{\text{ELDF}}\psi$ for any $t\in\textnormal{\sf
W}$. Extend $\\{\neg\psi\\}\cup\\{\chi\mid F_{G}\chi\in tail(s)\\}\cup\\{\neg
F_{G}\neg\chi\mid\chi\in tail(s)\\}$ to some maximal
$\mathbf{ELDF}$-consistent set $\Delta^{+}$, thus $\bigcap_{a\in
G}\textnormal{\sf C}(a)\subseteq\textnormal{\sf E}(s,t)$ where $t$ extends $s$
with $\langle(G,m),\Delta^{+}\rangle$. By the induction hypothesis,
$\textnormal{\sf M},t\models_{\text{ELDF}}\neg\psi$, which leads to a
contradiction.

##  Appendix E Extra Cases for the Proof of Lemma 3.18

The proof of Lemma 3.18 in the main text (p. 3.18) only contains the case for
common knowledge. Here we supplement the cases for distributed and field
knowledge for the careful reader (individual knowledge can be treated as a
special case of the two).

Case $\varphi=D_{G}\psi$. The direction from $D_{G}\psi\in tail(s)$ to
$\textnormal{\sf M},s\models_{\text{ELCDF}}D_{G}\psi$ is similarly to the
proof of Lemma D.1. For the other direction, suppose $D_{G}\psi\not\in
tail(s)$, but $\textnormal{\sf M},s\models_{\text{ELCDF}}D_{G}\psi$, then
$\bigcup_{a\in G}\textnormal{\sf C}(a)\subseteq\textnormal{\sf E}(s,t)$
implies $\textnormal{\sf M},t\models_{\text{ELCDF}}\psi$ for any
$t\in\textnormal{\sf W}$. Notice that $\\{\mathord{\sim}\psi\\}\cup\\{\chi\mid
D_{G}\chi\in tail(t_{0})\\}\cup\\{\neg D_{G}\mathord{\sim}\chi\in
cl(\theta)\mid\chi\in tail(t_{0})\\}$ is a consistent subset of $cl(\theta)$.
Extend it to a maximal $\mathbf{ELCDF}$-consistent set $\Delta^{+}$ in
$cl(\theta)$. Thus, by a similar method to the proof of
$\vdash_{\mathbf{ELCDF}}\delta\to K_{a}\psi$ in Lemma 3.18, we have
$\\{\chi\mid D_{G}\chi\in tail(s)\\}\subseteq\Delta^{+}$ and $\\{\chi\mid
D_{G}\chi\in\Delta^{+}\\}\subseteq tail(s)$. Let $t$ be $s$ extended with
$\langle(G,d),\Delta^{+}\rangle$, we have $\bigcup_{a\in G}\textnormal{\sf
C}(a)\subseteq\textnormal{\sf E}(s,t)$. By the induction hypothesis we have
$\textnormal{\sf M},t\not\models_{\text{ELCDF}}\psi$, contradicting with
$\textnormal{\sf M},s\models_{\text{ELCDF}}D_{G}\psi$.

The case when $\varphi=F_{G}\psi$ is similar to the case for distributed
knowledge except that we extend the consistent set
$\\{\mathord{\sim}\psi\\}\cup\\{\chi\mid F_{G}\chi\in
tail(t_{0})\\}\cup\\{\neg F_{G}\mathord{\sim}\chi\in cl(\theta)\mid\chi\in
tail(t_{0})\\}$ to get a maximal $\Delta^{+}$ in the closure, and let $t$ be
$s$ extended with $\langle(G,m),\Delta^{+}\rangle$.