# Gödel logics: Prenex fragments ††thanks: Research supported by FWF grant P
36571

Matthias Baaz
Technische Universität Wien
Institut für Algebra und Diskrete Mathematik
A–1040 Vienna, Austria
<EMAIL_ADDRESS>
Mariami Gamsakhurdia
Technische Universität Wien
Institut für Algebra und Diskrete Mathematik
A–1040 Vienna, Austria
<EMAIL_ADDRESS>

###### Abstract

In this paper, we provide a complete classification for the first-order Gödel
logics concerning the property that the formulas admit logically equivalent
prenex normal forms. We show that the only first-order Gödel logics that admit
such prenex forms are those with finite truth value sets since they allow all
quantifier-shift rules and the logic $G_{\uparrow}$ with only one accumulation
point at $1$ in the infinite truth values set. In all the other cases, there
are generally no logically equivalent prenex normal forms. We will also see
that $G_{\uparrow}$ is the intersection of all finite first-order Gödel
logics.
The second part of this paper investigates the existence of effective
equivalence between the validity of a formula and the validity of some prenex
normal form. The existence of such a normal form is obvious for finite valued
Gödel logic and $G_{\uparrow}$. Gödel logics with an uncountable truth value
set admit the prenex normal forms if and only if every surrounding of $0$ is
uncountable or $0$ is an isolated point. Otherwise, uncountable Gödel logics
are not recursively enumerable, however, the prenex fragment is always
recursively enumerable. Therefore, there is no effective translation between
the valid formula and the valid prenex normal form. However, the existence of
effectively constructible validity equivalent prenex forms for the countable
case is still up for debate.

_Keywords_ Proof Theory $\cdot$ Gödel Logics $\cdot$ Prenex Normal Forms
$\cdot$ Skolemization $\cdot$ Satisfiability $\cdot$ Validity $\cdot$ Delta
Operator

##  1 Introduction

###  1.1 Prenex Fragments

The existence of an equivalent prenex normal form is one of the oldest
recognized characteristics of classical predicate logic: to every formula
corresponds an equivalent prenex formula. The prenex formulas have many
features that usual formulas do not have; One can skolemize, in certain cases,
it admits Herbrand’s theorem or some infinite form of it. This concept of
prenexation is not as simple as one might think, because the prenex forms can
be linguistically different. There might exist prenex formulas, although not
all quantifier shift rules are valid. A natural example is to have some
fragment of non-classic logic, where every sentence is either true or false;
Then we have prenex forms and prenexation has nothing to do with quantifier
shifts.
The expressive power of prenex fragments is easy to see in classical logic
because it coincides with the whole logic; It is also easy to see for
intuitionistic logic since the validity of prenex formulas is decidable.
Prenex fragments in non-classical logics normally do not coincide with the
logic, but they are easier to handle syntactically and semantically.
Typically, the prenex fragment has a structure that can be analyzed by
algebraic tools. As a result, even though the logic might not interpolate
(such as first-order S5), the prenex fragment does [23],[30]. Since Gödel
logics are intermediary logics, the expressibility of prenex fragments is in
doubt.
Of course, it is easy to see that one can not hope to obtain prenex normal
forms in the usual sense in Gödel logics. Note that the absence of the
quantifier-shift rules does not guarantee that no prenex normal form exists.
However, proving that such prenex forms do not exist is more difficult because
the formula may have a completely different nature.
Usually, quantifier shifting rules are applied to construct these prenex
normal forms. Since the skolem functions relate to quantifier shifting, the
prenex fragments of Gödel logics admit skolemization while the logics
themselves usually do not. Furthermore, the prenex fragments admit Herbrand’s
theorems or approximative versions (for example, $G_{\uparrow}$). In addition,
they provide general normal form theorems that demonstrate the lower bound of
the complexity of particular Gödel logics. The length of direct, i.e. cut-free
proofs and Herbrand disjunctions, however, is greatly influenced by the
shifting of quantifiers 111This is where Herbrand’s mistake in the original
proof of Herbrand’s theorem came from: he believed that the quantifier
shifting did not alter the Herbrand disjunction, and his reasoning was limited
to prenex forms [19]..
Interestingly, the 1-satisfiability of prenex formulas in Gödel logics is a
classical satisfiability (unlike validity). This is important for database-
like structures. For ontologies, it is important to prove that they are
consistent (1-satisfiable). The fact that rules subsume another one is of
minor importance. The many-valued structure being non-contradictory can be,
therefore, reduced to a classical satisfiability problem.
As a first example note that prenexation does not work for $G_{[0,1]}$ when 0
is not isolated, since the formula $(\neg\forall xA(x)\wedge\forall x\neg\neg
A(x))$ does not admit a prenex normal form. This result can be extended to all
Gödel logics where there is at least one accumulation point from above, even
if it is not 0.
The logic might not admit prenex forms in the original language, but it might
admit prenex normal forms in an extended language:

Example (First-order Gödel on $[0,1]$ logic with propositional quantifiers):
There are only three quantifier-shift rules that are not intuitionistically
valid in first-order Gödel logics

$\displaystyle(S1)\quad\forall x(A(x)\vee B)\supset(\forall xA(x)\vee B)$
$\displaystyle(S2)\quad(B\supset\exists xA(x))\supset(\exists xB\supset\vee
A(x))$ $\displaystyle(S3)\quad(\forall xA(x)\supset B)\supset(\exists
x(A(x)\supset B)$

($x$ is not free in $B$).
Using propositional quantifiers prenexation is possible in $G_{[0,1]}$ as the
following quantifier shifts are valid

$\displaystyle(Q1)\quad\forall xA(x)\supset B\leftrightarrow\forall Z\exists
x(A(x)\supset Z\vee Z\supset B)$ $\displaystyle(Q2)\quad(A\supset\exists
xB(x))\leftrightarrow\forall Z\exists x(A\supset Z\vee Z\supset B(x)).$

For the right direction use the validity of $A\supset B\supset((A\supset
C)\vee(C\supset B))$ for all formulas $C$, for the left direction use the
validity of the quantified propositional expression of the Takeuti-Titani Rule
$\forall Z(A\supset Z\vee Z\supset A)\supset(A\supset B)$. All other classical
quantifier shift rules are valid in the extended language.

In this paper, we study two different problems:

  1. 1.

Logical equivalence

  2. 2.

Effective validity equivalence

We give a comprehensive classification of first-order Gödel logics after the
existence of the equivalent prenex normal forms.

###  1.2 First-order Gödel Logics

We begin with a preliminary discussion of the syntax and semantics of first-
order Gödel logics, followed by some of the more interesting special cases and
their relationships.
First-order Gödel logics are a family of many-valued logics where the truth
values set (known also as _Gödel set_) $V$ is closed222Gödel logics are
usually defined using the single truth value set $[0,1]$. For propositional
logic the choice of any infinite subset of $V\subseteq[0,1]$ leads to the same
propositional logic (set of tautologies). In the first order case, where
quantifiers will be interpreted as infima and suprema, a closed subset of
$[0,1]$ is necessary. subset of the full $[0,1]$ interval that includes both
$0$ and $1$. We fix a standard first-order language $\mathcal{L}$ with
finitely or countably many predicate symbols $P$ and finitely or countably
many function symbols $f$ for every finite arity $k$.

######  Definition 1.1.

First-order Gödel logic $G_{V}$ is a logic defined over the _Gödel set_ $V$
(any closed set of real numbers, $V\subseteq[0,1]$ containing $0$ and $1$)
given by the following evaluation function $\mathcal{I}$ on $V$

$\displaystyle(1)$ $\displaystyle\quad\mathcal{I}(\bot)=0$ $\displaystyle(2)$
$\displaystyle\quad\mathcal{I}(A\wedge
B)=min\\{\mathcal{I}(A),\mathcal{I}(B)\\}$ $\displaystyle(3)$
$\displaystyle\quad\mathcal{I}(A\vee
B)=max\\{\mathcal{I}(A),\mathcal{I}(B)\\}$ $\displaystyle(4)$
$\displaystyle\quad\mathcal{I}(A\supset
B)=\begin{cases}\mathcal{I}(B)&\text{if\/ $\mathcal{I}(A)>\mathcal{I}(B)$,}\\\
1&\text{if\/ $\mathcal{I}(A)\leq\mathcal{I}(B)$.}\end{cases}$
$\displaystyle(5)$ $\displaystyle\quad\mathcal{I}(\forall
xA(x))=\text{inf}\\{\mathcal{I}(A(u))\>u\in U\\}$ $\displaystyle(6)$
$\displaystyle\quad\mathcal{I}(\exists
xA(x))=\text{sup}\\{\mathcal{I}(A(u))\>u\in U\\}$

######  Remark 1.1.

Note that the fourth operator is characteristic operator of Gödel logics, so-
called Gödel conditional.

######  Remark 1.2.

We use the abbreviation $A<B$ for $(A\supset B)\supset A$, which means that
the value of $A$ is bigger than the value of $B$ with the exception that both
take the value $1$ and $A\leftrightarrow B$ for $(A\supset B)\land(B\supset
A)$.

Properties of first-order Gödel logics are based on the order-theoretic
structure of their set of truth values rather than the truth values
themselves. In particular, different logics are distinguished by the number
and type of accumulation points and their respective Cantor–Bendixon ranks
[3],[25],[40]. This raises the question of an intentional333 We use
intentional in the sense that logical properties (e.g., set of validities,
entailment relation, set of satisfiable formulas) change when the
characterising object changes. characterization of Gödel logics. While it is
technically convenient to represent Gödel logics via their set of truth
values, it has the disadvantage that uncountably many truth value sets have
indistinguishable logical properties. For example, consider

$V_{1}=\\{1/n:n\in N^{+}\\}\cup\\{0\\}$

$V_{2}=\\{1/2n:n\in N^{+}\\}\cup\\{0\\}$.

The logics $G_{V_{1}}$ and $G_{V_{2}}$ have exactly the same properties. This
leads to the representation of Gödel logics as a set of valid sentences.

######  Definition 1.2.

We define the set of valid formulas $G_{V}$ (referred to as Gödel logic) as
the set of formulas $A$ of $\mathcal{L}$ such that the evaluation of $A$ is 1
for all $V$-interpretations $\mathcal{I}$.

While focusing on validities, we must address insufficiencies with respect to
satisfiability and entailment. It is unclear how to describe the sets of
1-satisfiable sentences in these logics. Furthermore, whereas there are
uncountable many entailment relations, there are only countable many sets of
validities [[20],[22]]. Even the 3-valued Gödel logic $G_{3}$ corresponds to
uncountably many Gödel sets $\\{0,c,1\\}$ for any $c\in(0,1)$. However, this
definition would be problematic if different sets of satisfiable formulas
correspond to identical Gödel logics depending on the topological structure.
This is prevented by a functional dependency between the set of sets of valid
formulas and the set of sets of $1$-satisfiable formulas [15]. As a
generalisation of classical satisfiability, we introduce the following
concepts:

######  Definition 1.3 (satisfiability).

The formula in Gödel logic is _1-satisfiable ( $>0$ satisfiable)_ if there
exists at least one interpretation that assigns 1 ($>0$) to the formula.

######  Definition 1.4 (Validity).

The formula in Gödel logic is _valid_ (_$ >0$-valid_) if the formula evaluates
to 1 ($>0$) under every interpretation.

An interesting aspect to note is that in most investigations of non-classical
logics, validity is prioritised over $1$-satisfiability. These two are, of
course, dual in classical logic, in the sense that $A$ is valid iff $\neg A$
is not satisfiable. This duality does not hold in many-valued logics:
Consider, for example, the first-order Gödel logic based on $\\{1-1/n:n\in
N\\}\cup\\{1\\}$. There, validity is not recursively enumerable, but both the
complement of $1$-satisfiability and the complement of $>0$ -satisfiability
are recursively enumerable (see [15] Lemma 37 and 38).
Equivalence between formulas in Gödel logic are defined in the usual way,i.e.,
two formulas are equivalent if they are equal under all interpretations.

######  Definition 1.5.

For Gödel logics there is an asymmetry between $0$ and $1$, this is because
$0$ can be distinguished from other values (by using the negation), while $1$
cannot be distinguished. The reason is that all connectives and quantifiers
are continuous at $1$. To overcome this asymmetry the absoluteness operator
${\mathop{\triangle}}$ has been introduced in [3].

$\displaystyle\mathcal{I}({\mathop{\triangle}}A)=\begin{cases}1&\text{if
$\mathcal{I}(A)=1$,}\\\ 0&\text{otherwise}\end{cases}$

We consider Gödel logics with or without an absoluteness operator
${\mathop{\triangle}}$ and denote by $G_{V}^{\mathop{\triangle}}$ or $G_{V}$,
respectively.

We recall several definitions from topology and descriptive set theory.

######  Definition 1.6.

(Limit point, perfect space, perfect set). A _non-isolated point_ of a
topological space is a point $x$ such that for every open neighbourhood $U$ of
$x$ there exists a point $y\in U$ with $y\neq x$. A _limit point_ of a
topological space is a point $x$ that is not isolated. A space is _perfect_ if
all its points are limit points. A set $P\subseteq R$ is _perfect_ if it is
closed and together with the topology induced from $R$ is a perfect space.

It is obvious that all (non-trivial) closed intervals are perfect sets, as
well as all countable unions of (non-trivial) intervals. But all these sets
generated from closed intervals have the property that they are ‘everywhere
dense,’ i.e., contained in the closure of their inner component.

Let us list all the occurring axioms and rules:

(I1) $\displaystyle\quad(A\to B)\to((B\to C)\to(A\to C))$ (I2)
$\displaystyle\quad A\lor A\to A,A\to A\land A$ (I3) $\displaystyle\quad A\to
A\lor B,A\land B\to A$ (I4) $\displaystyle\quad A\lor B\to B\lor A,A\land B\to
B\land A$ (I5) $\displaystyle\quad(A\to B)\to(C\lor A\to C\lor B)$ (I6)
$\displaystyle\quad(A\land B\to C)\to(A\to(B\to C))$ (I7)
$\displaystyle\quad(A\to(B\to C))\to(A\land B\to C)$ (I8)
$\displaystyle\quad\bot\to A$ (I9) $\displaystyle\quad\forall xA(x)\to A(t)$
(I10) $\displaystyle\quad A(t)\to\exists xA(x)$ Additional axioms for
${\mathop{\triangle}}$: $\displaystyle{\mathnormal{\mathrm{\Delta}}1}$
$\displaystyle\quad{\mathop{\triangle}}A\lor\neg{\mathop{\triangle}}A$
$\displaystyle{\mathnormal{\mathrm{\Delta}}2}$
$\displaystyle\quad{\mathop{\triangle}}(A\lor
B)\to({\mathop{\triangle}}A\lor{\mathop{\triangle}}B)$
$\displaystyle{\mathnormal{\mathrm{\Delta}}3}$
$\displaystyle\quad{\mathop{\triangle}}A\to A$
$\displaystyle{\mathnormal{\mathrm{\Delta}}4}$
$\displaystyle\quad{\mathop{\triangle}}A\to{\mathop{\triangle}}{\mathop{\triangle}}A$
$\displaystyle{\mathnormal{\mathrm{\Delta}}5}$
$\displaystyle\quad{\mathop{\triangle}}(A\to
B)\to({\mathop{\triangle}}A\to{\mathop{\triangle}}B)$ Additional axioms for
Gödel logics: qs $\displaystyle\quad\forall x(B^{(x)}\lor
A(x))\to(B^{(x)}\lor\forall xA(x))$ lin $\displaystyle\quad(A\to B)\lor(B\to
A)$ $\displaystyle{\hbox{iso}_{0}}$ $\displaystyle\quad\forall x\neg\neg
A(x)\to\neg\neg\forall xA(x)$ $\displaystyle{\hbox{iso}_{1}}$
$\displaystyle\quad{\mathop{\triangle}}\exists xA(x)\to\exists
x{\mathop{\triangle}}A(x)$ $\displaystyle{\hbox{fin}(n)}$
$\displaystyle\quad(\top\to p_{1})\lor(p_{1}\to
p_{2})\lor\ldots\lor(p_{n-2}\to p_{n-1})\lor(p_{n-1}\to\bot)$

and the set of rules is

(MP) $B\lx@proof@logical@and AA\to B$ (I${\mathop{\triangle}}$)
${\mathop{\triangle}}AA$ (Q$\forall$) $B^{(x)}\to\forall xA(x)B^{(x)}\to A(x)$
(Q$\exists$) $\exists xA(x)\to B^{(x)}A(x)\to B^{(x)}$ (where $B^{(x)}$ means
that $x$ is not free in $B$).

The names of the axioms can be explained as follows: qs stands for ‘quantifier
shift’, lin for ‘linearity’, ${\hbox{iso}_{0}}$ for ‘isolation axiom of $0$’,
${\hbox{iso}_{1}}$ for ‘isolation axiom of $1$’, and ${\hbox{fin}(n)}$ for
‘finite with $n$ elements’.

The axioms for isolation have been chosen to be purely universally (case $0$)
and purely existentially (case $1$). ${\hbox{iso}_{0}}$ expresses that if all
instances are larger than $0$, then also the infimum. ${\hbox{iso}_{1}}$
expresses that if the supremum is $1$, then there is also an instance that is
$1$. There are alternative axioms that are equivalent:

$\displaystyle{\hbox{iso}_{0}}^{\prime}$ $\displaystyle\quad\neg\forall
xA(x)\to\exists x\neg A(x)$ $\displaystyle{\hbox{iso}_{1}}^{\prime}$
$\displaystyle\quad\forall
x\neg{\mathop{\triangle}}A(x)\to\neg{\mathop{\triangle}}\exists xA(x)$

###  1.3 Relationships between first-order Gödel logics

The relationships between finite and infinite valued propositional Gödel
logics are well understood. Any choice of an infinite set of truth values
results in the same propositional Gödel logic. In the first-order case, the
situation is more interesting. We establish several important results
regarding the relationships between various first-order Gödel logics. We first
consider several ‘prototypical’444This is because logic is defined
extensionally as the set of formulas valid in this truth value set, so the
Gödel logics on different truth value sets may coincide. Gödel sets:

$V_{[0,1]}=[0,1]$

$V_{\downarrow}=\\{0\\}\cup\\{1/k:k\geq 1\\}$

$V_{\uparrow}=\\{1\\}\cup\\{1-1/k:k\geq 1\\}$

$V_{m}=\\{1\\}\cup\\{1-1/k:1\geq k\geq m-1\\}$

The corresponding Gödel logics are $G_{[0,1]}$, $G_{\downarrow}$,
$G_{\uparrow}$, $G_{m}$. $G_{[0,1]}$ is _standard_ Gödel logic. The logic
$G_{\downarrow}$ also turns out to be closely related to some temporal logics.
Below we present relationships between some first-order Gödel logics
[11],[12].

######  Proposition 1.1.

Whenever $V\subseteq V^{\prime}$ then $G_{V^{\prime}}\subseteq G_{V}$.

###### Proof.

If $V\subseteq V^{\prime}$ are both Gödel sets, and $\mathcal{I}$ is an
interpretation into $V^{\prime}$, then $\mathcal{I}$ can also be seen as an
interpretation into $V$, and the values $\mathcal{I}(A)$, computed recursively
using (1) – (6) in Definition 1.2, do not depend on whether we consider
$\mathcal{I}$ as a $V^{\prime}$-interpretation or a $V$-interpretation.
Consequently, if $V^{\prime}\subseteq V$, there are more interpretations into
$V$ than into $V^{\prime}$, hence if $\Gamma\models_{V^{\prime}}A$ then also
$\Gamma\models_{V}A$ and $G_{V^{\prime}}\subseteq G_{V}$. ∎

The result can be generalized to the embeddings between Gödel sets.

######  Definition 1.7 (g-embedding).

A _g-embedding_ $h$ from a Gödel set $V_{1}$ to a Gödel set $V_{2}$ is an
embedding $h:V_{1}\to V_{2}$ which is continuous (in the usual topology of
closed sets on $\mathbb{R}$), strictly monotone on $V_{1}$, and preserves $0$
and $1$.

######  Proposition 1.2.

[12] The following containment relationships hold:
1) $G_{m}\supsetneq G_{m+1}$
2) $G_{m}\supsetneq G_{\uparrow}\supsetneq G_{[0,1]}$
3) $G_{m}\supsetneq G_{\downarrow}\supsetneq G_{[0,1]}$

###### Proof.

The only challenging part is demonstrating strict containment. Note that the
$(Fin)$ $(\top\supset A_{1})\vee(A_{1}\supset A_{2})\vee(A_{2}\supset
A_{3})\vee\dots\vee(A_{m-2}\supset A_{m-1})\vee(A_{m-1}\supset\bot)$ is valid
in $G_{m}$ but not in $G_{m+1}$. Consider the formulas

(1) $C_{\uparrow}=\exists x(A(x)\supset\forall yA(y))$

(2) $C_{\downarrow}=\exists x(\exists yA(y)\supset A(x))$.

$C_{\downarrow}$is valid in all $G_{m}$, $G_{\uparrow}$. On the other hand,
$C_{\uparrow}$ is valid in all $G_{m}$ and $G_{\uparrow}$ but not in
$G_{\downarrow}$. However, none of them is valid in $G_{[0,1]}$ [[12],
Corollary 2.9]. ∎

######  Remark 1.3.

The formulas $C_{\uparrow}$ and $C_{\downarrow}$ are important in the study of
first-order infinite valued Gödel logics. $C_{\uparrow}$ expresses the fact
that every infimum is a minimum, while $C_{\downarrow}$ states that every
supremum is a maximum.

######  Proposition 1.3.

$G_{[0,1]}=\bigcap_{V}G_{V}$ where $V$ ranges over all Gödel sets. $G_{[0,1]}$
logic is contained in every Gödel logic. So it is an intersection of all Gödel
logics.

###### Proof.

If $\Gamma\mathrel{\models}_{V}A$ for every Gödel set $V$, then it does so in
particular for $V=[0,1]$. Conversely, if
$\Gamma\mathrel{\mathrel{\nvDash}}_{V}A$ for a Gödel set $V$, there is a
$V$-interpretation $\mathcal{I}$ with $\mathcal{I}(\Gamma)>\mathcal{I}(A)$.
Since $\mathcal{I}$ is also a $[0,1]$-interpretation,
$\Gamma\mathrel{\mathrel{\nvDash}}_{[0,1]}A$. ∎

##  2 Gluing Argument

In this section, we present a crucial tool to show that $G_{[0,1]}$ does not
admit logically equivalent prenex normal form whenever $0$ is not an isolated
point, which can be extended to all cases when 0 is isolated from the truth
value set. So we literally cut the truth set at the point $\omega$, we keep
the lower part and we glue the upper part to 1. In this way, we simply change
interpretations on atoms; The ones below $\omega$ remain and those above
$\omega$ will become $1$. In principle, it does not matter what value $\omega$
we fix, but if $\omega$ is not isolated from the truth values set the
construction does not work for the universal quantifier.

######  Lemma 2.1 (Gluing lemma for formulas with $\exists$).

Let $\mathcal{I}$ be an interpretation into $V\subseteq[0,1]$. Let us fix a
value $\omega\in[0,1]$ and define

$\mathcal{I}_{\omega}(\mathcal{P})=\begin{cases}\mathcal{I}(\mathcal{P})&\text{if\/
$\mathcal{I}(\mathcal{P})\leq\omega$,}\\\ 1&\text{otherwise}\end{cases}$

for atomic formula $\mathcal{P}$ in $\mathcal{L}^{\mathcal{I}}$. Then
$\mathcal{I}_{\omega}$is an interpretation into $V$ such that

$\mathcal{I}_{\omega}(\mathcal{B})=\begin{cases}\mathcal{I}(\mathcal{B})&\text{if\/
$\mathcal{I}(\mathcal{B})\leq\omega$,}\\\ 1&\text{otherwise}\end{cases}$

###### Proof.

The proof is by induction principle after the complexity of the formulas that
any subformula $\mathcal{B}$ of $\mathcal{A}$ with
$\mathcal{I}(\mathcal{B})\leq\omega$ has
$\mathcal{I}^{\prime}(\mathcal{B})=\mathcal{I}(\mathcal{B})$. For
$\mathcal{B}$ being an atomic formula it is obvious by definition. We
distinguish cases according to the logical form of $\mathcal{B}$:
$\mathcal{B}\equiv D\land E$. If $\mathcal{I}(\mathcal{B})\leq\omega$, then,
without loss of generality, assume
$\mathcal{I}(\mathcal{B})=\mathcal{I}(D)\leq\mathcal{I}(E)$. By induction
hypothesis, $\mathcal{I}^{\prime}(D)=\mathcal{I}(D)$ and
$\mathcal{I}^{\prime}(E)\geq\mathcal{I}(E)$, so
$\mathcal{I}^{\prime}(\mathcal{B})=\mathcal{I}(\mathcal{B})$. If
$\mathcal{I}(\mathcal{B})>\omega$, then $\mathcal{I}(D)>\omega$ and
$\mathcal{I}(E)>\omega$, by induction hypothesis
$\mathcal{I}^{\prime}(D)=\mathcal{I}^{\prime}(E)=1$, thus,
$\mathcal{I}^{\prime}(\mathcal{B})=1$.
$\mathcal{B}\equiv D\lor E$. If $\mathcal{I}(\mathcal{B})\leq v$, then,
without loss of generality, assume
$\mathcal{I}(\mathcal{B})=\mathcal{I}(D)\geq\mathcal{I}(E)$. By induction
hypothesis, $\mathcal{I}^{\prime}(D)=\mathcal{I}(D)$ and
$\mathcal{I}^{\prime}(E)=\mathcal{I}(E)$, so
$\mathcal{I}^{\prime}(\mathcal{B})=\mathcal{I}(\mathcal{B})$. If
$\mathcal{I}(\mathcal{B})>\omega$, then, again without loss of generality,
$\mathcal{I}(\mathcal{B})=\mathcal{I}(D)>\omega$, by induction hypothesis
$\mathcal{I}^{\prime}(D)=1$, thus, $\mathcal{I}^{\prime}(\mathcal{B})=1$.
$\mathcal{B}\equiv D\to E$. Since $\omega<1$, we must have
$\mathcal{I}(D)>\mathcal{I}(E)=\mathcal{I}(\mathcal{B})$. By induction
hypothesis, $\mathcal{I}^{\prime}(D)\geq\mathcal{I}(D)$ and
$\mathcal{I}^{\prime}(E)=\mathcal{I}(E)$, so
$\mathcal{I}^{\prime}(\mathcal{B})=\mathcal{I}(\mathcal{B})$. If
$\mathcal{I}(\mathcal{B})>\omega$, then
$\mathcal{I}(D)\geq\mathcal{I}(E)=\mathcal{I}(\mathcal{B})>\omega$, by
induction hypothesis
$\mathcal{I}^{\prime}(D)=\mathcal{I}^{\prime}(E)=\mathcal{I}^{\prime}(\mathcal{B})=1$.
$\mathcal{B}\equiv\exists xD(x)$. First assume that
$\mathcal{I}(\mathcal{B})\leq\omega$. Since $D(c)$ evaluates to a value less
or equal to $\omega$ in $\mathcal{I}$ and, by induction hypothesis, in
$\mathcal{I}^{\prime}$ also the supremum of these values is less or equal to
$\omega$ in $\mathcal{I}^{\prime}$, thus
$\mathcal{I}^{\prime}(\mathcal{B})=\mathcal{I}(\mathcal{B})$. If
$\mathcal{I}(\mathcal{B})>\omega$, then there is a $c$ such that
$\mathcal{I}(D(c))>\omega$, by induction hypothesis
$\mathcal{I}^{\prime}(D(c))=1$, thus, $\mathcal{I}^{\prime}(\mathcal{B})=1$. ∎

The gluing lemma holds in general for all formulas iff the fixed point
$\omega$ is isolated from the truth set from above.

######  Corollary 2.0.1 (Gluing lemma for for formulas with $\forall$).

If $\omega\notin Val(\mathcal{I},\mathcal{B})$, thus $\omega$ is isolated
point from above, then the gluing lemma can be extended to the universal
quantifier.

###### Proof.

Let $\mathcal{B}\equiv\forall xD(x)$. This is the crucial part, since the
above gluing argument has one proviso, it does not work for $\forall$ if
$\omega$ is not isolated from above. First, assume that
$\mathcal{I}(\mathcal{B})<\omega$. Then there is a witness $c$ such that
$\mathcal{I}(\mathcal{B})\leq\mathcal{I}(D(c))<\omega$ and, by induction
hypothesis, also $\mathcal{I}^{\prime}(D(c))<\omega$ and therefore,
$\mathcal{I}^{\prime}(\mathcal{B})=\mathcal{I}(\mathcal{B})$. For
$\mathcal{I}(\mathcal{B})>\omega$ it is obvious that
$\mathcal{I}^{\prime}(\mathcal{B})=\mathcal{I}(\mathcal{B})=1$. Finally assume
that $\mathcal{I}(\mathcal{B})=\omega$. If this infimum would be proper, i.e.
no minimum, then the value of all witnesses under $\mathcal{I}^{\prime}$ would
be $1$, but the value of $\mathcal{B}$ under $\mathcal{I}^{\prime}$ would be
$\omega$, which would contradict the definition of the semantic of the
$\forall$ quantifier. Since all infima are minima, there is a witness $c$ such
that $\mathcal{I}(D(c))=\omega$ and therefore, also
$\mathcal{I}^{\prime}(D(c))=\omega$ and thus
$\mathcal{I}^{\prime}(\mathcal{B})=\mathcal{I}(\mathcal{B})$. ∎

As a consequence of the gluing lemma we have the following results:

######  Proposition 2.1.

In the following cases, 1-satisfiability in Gödel logics is classical
satisfiability:
1) In the propositional case
2) In the first-order case, where the truth value set is arbitrary but 0 is
isolated
3) In the case of the prenex fragment for any truth value set
4) In the case of the existential fragments for any truth value set

###### Proof.

The first two cases follow by induction on the complexity of the formulas. 3)
The change of values of the modified interpretations in the case of $\omega$
being non-isolated does not diminish the truth of prenex formulas. 4) It
follows from the fact that purely existential formulas are always classically
satisfiable.∎

We know that the gluing lemma works whenever there is no universal quantifier
or the gluing point is isolated. Since in finite-valued Gödel logics the
points are isolated, and so is in $G_{\uparrow}$, therefore we have the
following result:

######  Proposition 2.2.

[12] $G_{\uparrow}=\bigcap_{m\geq 2}G_{m}$ is an intersection of all finite-
valued Gödel logics.

###### Proof.

The idea of the proof is the following: formula valid in $G_{\uparrow}$ is
valid in all finite-valued Gödel logics. On the other hand, the formula not
valid in $G_{\uparrow}$ is not valid in at least one finite-valued Gödel
logic. Because the counterexample has a certain finite value, we glue
everything from above this value to $1$, then we get a counterexample in
finite-valued Gödel logic.
The right inclusion follows from proposition 1.2 It is sufficient to show that
if $G_{\uparrow}\nvDash A$ for $A$ being existential, then $G_{m}\nvDash A$
for some $m$. Given any interpretation $\mathcal{I}$ and a fixed value
$\omega<1$, we define modified interpretation $\mathcal{I}_{\omega}$ and apply
gluing lemma to the formula $A$. Now suppose that $\mathcal{I}\nvDash A$ then,
clearly, if $A$ is an existential formula of the type
$\exists\bar{x}B(\bar{x})$ then $\mathcal{I}(A)=\omega<1$ and there are only
finitely many truth values below $\omega$ in $V$; let us say $\omega=1-1/k$.
Since $\mathcal{I}(B)\leq\omega<1$, by gluing lemma
$\mathcal{I}_{\omega}(B)\leq\omega$ and so $\mathcal{I}_{\omega}(A)\leq\omega$
but $\mathcal{I}_{\omega}$ is a $G_{k+1}$ interpretation and
$\mathcal{I}_{\omega}\nvDash A$. ∎

So here is our first result on the level of classification, specifically for
$G_{\downarrow}$:

######  Proposition 2.3.

For all first-order Gödel logics, when $0$ is not an isolated point, formulas
do not in generally allow the existence of an equivalent prenex normal form.

###### Proof.

Consider the formula $F:=\neg\forall xA(x)\wedge\forall x\neg\neg A(x)$ which
states that ’for a suitable choice of $A(x)$, $0$ is a true infimum’. Suppose
there exists an equivalent prenex normal form for some quantifier-free $P$ and
some string of quantifiers $Qx^{\prime}$

$\displaystyle F\Leftrightarrow Qx^{\prime}Px^{\prime}.$

We can find a suitable interpretation of $A(x)$ (descending to $0$ but never
reaching it) that makes the formula $F$ true. As a consequence, we have that
the formula $F$ is $1$-satisfiable. Since the equivalence is valid, the prenex
normal formula is also $1$-satisfiable. Moreover prenex formula is classically
satisfiable so the gluing argument always works but the original formula $F$
can not be classically satisfiable, which leads to a contradiction. ∎

######  Remark 2.1.

As a consequence, we have that, firstly, $1$-unsatisfiability is the same for
all Gödel logics where $0$ is an isolated point. So the refutation of the
$1$-satisfiability is recursive, because it is recursive in classic logic. The
same is true for the existential fragments.

##  3  $>0$ Valid Equivalent Prenex Normal Forms

The gluing argument can be used to solve another problem, namely if every
double negated formula is equivalent to a double negated prenex normal form,
in this case, classification is simple.
Recall that

######  Proposition 3.1.

1-validity of the double negated formula $\neg\neg A$ is equivalent to
$>0$-validity of $A$.

###### Proof.

If we take the formula $A$ which has a value $>0$, then $\neg A$ has truth
value 0, and therefore $\neg\neg A$ evaluates to 1. On the other hand, if we
take $\neg\neg A$ which is evaluated to 1, then $\neg A$ has truth value 0,
and therefore $A$ has value $>0$. ∎

######  Proposition 3.2.

If $0$ is not isolated in $V$, the first-order Gödel logic $G_{V}$ does not
allow in general the existence of the $>0$ valid equivalent prenex normal
form.

###### Proof.

Suppose that every formula admits its $>0$ valid equivalent prenex formula.
Recall that $>0$ valid is expressed by double negation. Let us define a
formula and its prenex normal form such that the equivalence is $>0$ valid:

$\neg\neg(\neg\forall xA(x)\wedge\forall x\neg\neg A(x)\Leftrightarrow
Qx^{\prime}Px^{\prime})$.

The proof is similar to the proof of proposition 2.2, the only difference is
that instead of $1$-satisfiable we consider $>0$ satisfiable. ∎

######  Proposition 3.3.

For all first-order Gödel logics when $0$ is an isolated point, every formula
allows equivalent prenex normal forms such that the equivalence is $>0$-valid.

###### Proof.

Before we start the proof, recall that intuitionistic predicate logic
$\mathbf{IL}$ is contained in all first-order Gödel logics, and all the
Intuitionistically sound axioms and rules hold in $G_{V}$. We will show that
for every formula, its double negation is valid in Gödel logic when 0 is an
isolated point, using Glivenko translation + Kuroda shift. One straightforward
way is to consider Hilbert-style calculus with Modus-Ponens for the Gödel
logic.
It is an easy observation that all axioms and rules of $\mathbf{IL}$ are
intuitionistically sound with double negation. For any formula $A$, we have
$A\rightarrow\neg\neg A$ holds in $\mathbf{IL}$, it means that if we have
assumption $A$ true, then we can derive $\neg\neg A$. Also, the axiom of the
excluded middle with double negation $\neg\neg(\neg A\vee A)$ holds in
$\mathbf{IL}$. By induction, we show that every classically valid $A$ not
including $\forall$ can be derived $\neg\neg A$ in $\mathbf{IL}$. This is
because Modus Ponens with double negation also holds

$\displaystyle(\mathrm{\neg\neg MP})$ $\neg\neg
B\lx@proof@logical@and\neg\neg(A\rightarrow B)\neg\neg A$

and

($\neg\neg$Q$\exists$) $(\neg\neg\exists xA(x))\to B^{(x)}\neg\neg(A(x))\to
B^{(x)}$ (where $B^{(x)}$ means that $x$ is not free in $B$).

Now for the $\forall$

($\neg\neg$Q$\forall$) $B^{(x)}\to(\neg\neg\forall
xA(x))B^{(x)}\to\neg\neg(A(x))$

this is obtained by Glivenko translation

$\displaystyle\vbox{$\forall x\neg\neg A(x)\neg\neg(t)$}.$

adding Kuroda shift (by simply applying double negation before the whole
formula and after $\forall$)

$\displaystyle\vbox{$\neg\neg\forall xA(x)\forall x\neg\neg A(x)$}.$

Note that $\forall x\neg\neg A(x)\rightarrow\neg\neg\forall xA(x)$ is
${\hbox{iso}_{0}}$ and is satisfied by assumption. Thus, we have shown that
for every classically valid formula, its double negation is valid in Gödel
logic when 0 is an isolated point, this is obtained by Glivenko translation +
Kuroda shift. We also know that classically, every formula is equivalent to a
prenex normal form. Consequently, the double negation of the equivalence is
also valid in Gödel logic, since it corresponds to $>0$ validity of the
equivalence, which ends our proof. ∎

So by gluing argument, we have shown that no ($>0$ valid) equivalent prenex
normal form exists, if $0$ is not an isolated point in the truth value set. It
is natural to ask whether it is true in general that whenever 0 is isolated,
the equivalent prenex normal form exists. From now on, we investigate the
existence of the equivalent prenex normal form for the first-order Gödel
logics according to the number and type of accumulation points in their truth
set.

##  4 At Least One Accumulation Point From Above

In this section, we consider the first-order Gödel logics with at least one
accumulation point from above in their truth value set. The idea is to reflect
the truth values above the accumulation point $X$ and associate it with the
logic $G_{[X,1]}$. We look at it as a subinterval, where we fix the value for
this lower (accumulation) point. Then, of course, we can find an isomorphism
to the full interval from this subinterval, which contains only this stretched
upper part, under the condition that everything is interpreted above this
accumulation point and nothing is interpreted below.

######  Theorem 4.1.

In first-order Gödel logics $G_{V}$ such that at least one accumulation point
from above exists, there is no valid equivalent prenex normal form.

###### Proof.

The idea of the proof is that we introduce for the lower accumulation point
from above a new propositional variable $X$ that behaves like ’falsum’ in the
usual sense. We reflect the truth values above $X$ and associate it with the
logic $G_{[X,1]}$. Then we may extend the interval $[X,1]$ to $[0,1]$ interval
and apply the result from the previous section. Let us consider the old
formula

(1) $\neg\forall xA(x)\wedge\forall x\neg\neg A(x)$.

and modify it in such a way that nothing has to be interpreted below $X$. Now
we write the following literal interpretation of (1) by adding atomic formulas
and adding $X$ to every atomic formulas

(2) $F\equiv(\forall x(A(x)\vee X)\supset X)\wedge(\forall x((A(x)\vee
X)\supset X)\supset(A(x)\vee X))$.

Let $Qx^{\prime}Px^{\prime}$ be the equivalent prenex normal form

$F\Leftrightarrow Qx^{\prime}Px^{\prime}$.

Then we know that this prenex normal form has to be evaluated above $X$, since
everything has value above $X$. However, the matrix of the prenex formula
might contain the $\bot$, therefore we replace them with a new ’falsum’ $X$
and apply the gluing argument (this does not change the evaluation, since the
prenex formula is interpreted above $X$). There is a suitable interpretation
of $A(x)$ such that the formula $F$ is true. Therefore, it is 1-satisfiable,
and so is the prenex formula. We can glue everything over $X$ to $1$ in the
prenex formula. Then $X$ behaves as usual ’Falsum’, so the prenex formula has
a 2-valued model, where the values are the evaluation of $X$ and 1. However
the original formula $F$ can not have such 2-valued model. ∎

##  5  No Accumulation Point From Above

Now we consider first-order Gödel logics with truth set where all accumulation
points are from below and there is no accumulation point from above. We will
distinguish the following different subcases:

  1. 1.

No accumulation points exist at all (the finite case)

  2. 2.

Only one accumulation point from below at $1$ exists ( $G_{\uparrow}$ )

  3. 3.

Only accumulation points from below exist but at least one is an inner
accumulation point, which is less than $1$.

Prenexation is, of course, feasible in cases where all classical quantifier
shift rules are available. This holds for all finitely valued first-order
Gödel logics and their intersection $G_{\uparrow}$ (the only infinite-valued
Gödel logic where every formula is equivalent to a prenex normal form).

######  Proposition 5.1.

All finite first-order Goedel logics $G_{m}$ admit equivalent prenex normal
form.

###### Proof.

All finite-valued first-order Gödel logics $G_{m}$ admit the classical
quantifier-shift rules, including the three quantifier-shift rules which are
not valid in intuitionistic logic

$\displaystyle(S1)\quad\forall x(A(x)\vee B)\supset(\forall xA(x)\vee B)$
$\displaystyle(S2)\quad(B\supset\exists xA(x))\supset(\exists xB\supset\vee
A(x))$ $\displaystyle(S3)\quad(\forall xA(x)\supset B)\supset(\exists
x(A(x)\supset B)$

($x$ is not free in $B$) This is easily seen by considering the validity of
the following formulas

(1) $C_{\uparrow}=\exists x(A(x)\supset\forall yA(y))$

(2) $C_{\downarrow}=\exists x(\exists yA(y)\supset A(x))$.

The formulas $C_{\uparrow}$ and $C_{\downarrow}$ are important in the study of
first-order infinite valued Gödel logics. $C_{\uparrow}$ expresses the fact
that every infimum is a minimum, while $C_{\downarrow}$ expresses that every
supremum is a maximum, consequently they are valid $G_{m}$. (S1) is valid in
all Gödel logic, $(S2)$ is equivalent to $C_{\uparrow}$ and $(S3)$ is
equivalent to $C_{\downarrow}$. Consequently, all finite-valued first-order
Gödel logics $G_{m}$ admit logically equivalent prenex normal forms. ∎

$G_{\uparrow}$ is thus the only infinite-valued Gödel logic where every
formula is equivalent to a prenex formula.

######  Proposition 5.2.

$G_{\uparrow}$ admits logically equivalent prenex normal forms.

###### Proof.

We have shown that $G_{\uparrow}$ is the intersection of all finite first-
order Goedel logics. Consequently, all classical quantifier shifting rules
hold in $G_{\uparrow}$. ∎

For the third subcase, we use a completely different approach. In the previous
discussion, we have defined a formula that describes the fact that the objects
descend to $0$ but never reach. Now we do the opposite. We define a formula
that describes a value is not an accumulation point in the interpretation and
we consider, unlike the previous cases, models of various sizes. Note that to
argue in that way we use the property that there is no accumulation point from
above, because of this we are allowed usual model-theoretic skolemization for
universal quantifiers. $\forall xA(x)\to A(f)$ is trivial, for the other
direction we need to introduce skolem axioms$A(f)\to\forall xA(x)$ which is
allowed since the infimum is always a minimum.

######  Proposition 5.3.

For all first-order Gödel logics that have only accumulation points from below
such that at least one accumulation point is inside this interval, no
equivalent prenex normal form exists.

###### Proof.

Step 1: Let us describe what it means "$X$ not being a true accumulation
point" for a suitable interpretation of atoms $X$ and $A(x)$ assuming that
$X<1$:

$F\equiv\quad$$\forall x(A(x)<X)\supset\exists xA(x)<X$

Step 2: Assume there is a valid equivalent prenex normal form
$Qx^{\prime}Px^{\prime}$. Then this prenex normal form, for all
interpretations of $X$ and $A(x)$ should hold.
Step 3: Concentrate on interpretations under which all $A(x)<X<1$. Show that
among these interpretations, there are some under which the equivalence is
false.
Step 4: Since, by assumption, there is no accumulation point from above we add
skolem axioms to555It is like in classic logic one can simply pick an element
from above as there is no true infimum, ’every infimum is minimum’., skolemize
$\forall$ on the side of the prenex form and reduce it to a purely existential
formula.
Step 5: Consider an $1$-element model. For any choice of $X$ and $A(x)$ the
formula $F$ is always true.
Step 6: Extend the $1$-element model by adding an ascending sequence of points
never reaching $X$ in such a way that the formula $F$ becomes false, the
existential formula remains true. We add points, and at the same time, we
evaluate the skolem axiom. We always take the minimum among the points that
make the skolem axioms true. This is possible since there is no accumulation
point from above.
But by assumption, this equivalence has to hold for all interpretations of $X$
and $A(x)$, so it should be $1$-satisfiable for the interpretation when all
$A(x)$ are below $X$. But the formula $F$ might not be true, since for a
suitable choice of ascending sequence of $A(x)$ there might be an inner
accumulation point. Thus, the equivalence does not hold. $F$ will not be true
at accumulation point because accumulation point always take the value among
the $A(x)$’s and $A(x)<1$, because all $A(x)<X$, but the supremum of $\exists
xA(x)$ is not less then X, but X.∎

##  6  Gödel logics with Delta operator $G_{V}^{\Delta}$

In this chapter, we investigate first-order Gödel logics with the absoluteness
operator ${\mathop{\triangle}}$ after the existence of the equivalent prenex
normal form. Recall that

$\displaystyle\mathcal{I}({\mathop{\triangle}}A)=\begin{cases}1&\text{if
$\mathcal{I}(A)=1$,}\\\ 0&\text{otherwise}\end{cases}$

This absoluteness operator allows to represent the complement of
$1$-satisfiability by $\lnot{\mathop{\triangle}}A$, and to represent the
complement of ${>}0$-satisfiability by $\lnot A$ (note that $\lnot\lnot A$
means that $A$ has value $>0$, and therefore $\lnot\lnot\lnot A$ represents
the complement of ${>}0$-satisfiability, and $\lnot\lnot\lnot A$ reduces to
$\lnot A$ already intuitionistically.) It is shown in [15] that – contrary to
classical logic – there is no faithful and uniform (across all truth value
sets) recursive translation which reduces the validity of first-order Gödel
sentences to the complement of $1$-satisfiability.

######  Remark 6.1.

1-satisfiability of formulas with $\Delta$ does not imply classical
satisfiability. For example the formula $\neg\neg
A\land\neg{\mathop{\triangle}}A$ which says that $A$ is between 0 and 1, is
1-satisfiable with ${\mathop{\triangle}}$ but not classically 1-satisfiable.

Now we will answer the main question of this section: ’Can there be a prenex
normal form with ${\mathop{\triangle}}$ when there is none without
${\mathop{\triangle}}$? ’
We formulate our argument in the restricted semantics, where all atoms besides
$\top$ are interpreted below 1. The point of restricted semantics is that we
can evaluate ${\mathop{\triangle}}$ (because the value of
${\mathop{\triangle}}X$ for some atom $X$ which admits a value below 1 is 0).
Therefore any formula with ${\mathop{\triangle}}$ coincides with the
disjunction of chains of a formula without ${\mathop{\triangle}}$. It is clear
that all counterexamples work for restricted semantics. We show that
counterexamples work also if the prenex formulas contain
${\mathop{\triangle}}$.
Let us first recall some important notions we will use in this section:

######  Definition 6.1.

A _chain_ $C$ over the set of propositional variables $X=[X_{1},...X_{n}]$ is
an expression

$\displaystyle(\bot\vartriangleright_{1}X_{\Pi(1)})\land(X_{\Pi(1)}\vartriangleright_{2}X_{\Pi(2)})\land\dots\land(X_{\Pi(n)}\vartriangleright_{n+1}\top)$

abbreviated as

$\displaystyle\bot\vartriangleright_{1}X_{\Pi(1)}\vartriangleright_{2}\top\vartriangleright_{1}X_{\Pi(2)}\vartriangleright_{3}\dots\vartriangleright_{n}X_{\Pi(n)}\vartriangleright_{n+1}\top$

where $\vartriangleright_{i}\in\\{<,\leftrightarrow\\}$ and $\Pi$ is
permutation on $[1\dots n]$.

Every chain describes an order type of the variables $X$. We define the two
types of chain normal forms which we will use later.

######  Definition 6.2.

($CNF_{\mathop{\triangle}}-1$). The first _chain normal form with
${\mathop{\triangle}}$_ is the following expression

$\displaystyle\lor C\land\psi_{c}(A)$

where $\lor C$ are all chains and $\psi_{c}(A)$ is an evaluation666The
evaluation changes subformula step by step according to the information
contained in the chain. over the chain $C$ of $A$ with variables among the
variables of the chain.

######  Definition 6.3.

($CNF_{\mathop{\triangle}}-2$) The second _chain normal form with
${\mathop{\triangle}}$_ in the restricted semantics is obtained from the first
chain normal form with ${\mathop{\triangle}}$ after the following steps:
1) If the evaluation of $A$ is false we delete the chain,
2) if the evaluation of $A$ is true we leave the chain as it is,
3) if the evaluation of $A$ is an atom and is not 1 we delete the whole chain.

######  Lemma 6.1.

The second chain normal form of a formula with ${\mathop{\triangle}}$ in the
restricted semantics coincides with a disjunction of chains without
${\mathop{\triangle}}$.

###### Proof.

Let us take a formula $F$ without ${\mathop{\triangle}}$ and develop the
second chain normal form. According to Definition 6.3 the disjuncts might be
among the following three cases:
1) $C\land\top$,
2) $C\land\bot$ or
3) $C\land X$ for some atom $X$ and chain $C$.
Since we are in the restricted semantics, all atoms are interpreted below 1.
Now we put ${\mathop{\triangle}}$ in front of the disjuncts and we distribute.
3) becomes false because atoms can not be interpreted at 1 so the evaluation
of ${\mathop{\triangle}}X$ is 0 and we remove it. 2) will be removed by
definition and therefore remains only 1) which is equivalent to
${\mathop{\triangle}}C$. In the restricted semantics, ${\mathop{\triangle}}C$
coincides with $C$, thus the second chain normal form will coincide with the
disjunction of chains without ${\mathop{\triangle}}$. ∎

As a consequence of this lemma the prenex formulas with ${\mathop{\triangle}}$
coincides with the prenex formulas without ${\mathop{\triangle}}$ in the
restricted semantics.

######  Theorem 6.1.

There is no equivalent prenex normal form with ${\mathop{\triangle}}$ for the
formulas in the first-order Gödel logics with ${\mathop{\triangle}}$ when
there is none for the formulas in the first-order Gödel logics without
${\mathop{\triangle}}$.

###### Proof.

Suppose that the formula without prenex normal form without
${\mathop{\triangle}}$ has prenex normal form with ${\mathop{\triangle}}$.
Then the equivalence holds also in the restricted semantics. The matrix of the
prenex formula corresponds to the disjunction of chains. From the above lemma
follows that any formula with ${\mathop{\triangle}}$ coincides with the
disjunction of chains of the formula without ${\mathop{\triangle}}$. In
restricted semantics adding ${\mathop{\triangle}}$ to the chains does not
change anything. this contradicts to our assumption. ∎

  * •

Thus, all the negative results from the first-order Gödel logics without
${\mathop{\triangle}}$ can be transferred to first-order Gödel logics with
${\mathop{\triangle}}$.

  * •

There are only two positive cases to investigate.

It is easy to see that the gluing argument is not valid in all first-order
Gödel logics with ${\mathop{\triangle}}$.

######  Proposition 6.1.

Finite-valued Gödel logics with ${\mathop{\triangle}}$ still admit equivalent
prenex normal form.

###### Proof.

By the following two equivalences,

$\displaystyle{\mathop{\triangle}}\forall xA(x)\leftrightarrow\forall
x{\mathop{\triangle}}A(x)$ $\displaystyle{\mathop{\triangle}}\exists
xA(x)\leftrightarrow{\mathop{\triangle}}A(x)$

one can shift quantifiers, so in the finite case with ${\mathop{\triangle}}$,
all quantifier shifting rules are valid, and therefore equivalent prenex
normal form exists. ∎

######  Proposition 6.2.

$G_{\uparrow}^{\mathop{\triangle}}$ is not an intersection of finite valued
Gödel logics.

###### Proof.

Since in finite-valued logics also with ${\mathop{\triangle}}$ all quantifier-
shifts rules are valid, but not in $G_{\uparrow}^{\mathop{\triangle}}$ (e.g.,
${\mathop{\triangle}}\exists xA(x)\leftrightarrow{\mathop{\triangle}}$ A(x) is
not valid), therefore $G_{\uparrow}^{\mathop{\triangle}}$ is not an
intersection of finite Gödel logics anymore. ∎

######  Proposition 6.3.

$G_{\uparrow}^{\Delta}$ admits no equivalent prenex normal form.

###### Proof.

The proof is similar to Proposition 5.3. In
$G_{\uparrow}^{\mathop{\triangle}}$, we formulate an expression that takes
value either true or false

$F\equiv\forall
x\neg{\mathop{\triangle}}A(x)\rightarrow\neg{\mathop{\triangle}}\exists
xA(x)$.

The formula says that ’1 is not an accumulation point’; More precisely, the
formula is false at 1 when 1 is the accumulation point. Assuming this formula
has an equivalent prenex normal form, it can also be evaluated as true or
false. we can use the skolem axioms

$A(f)\rightarrow\forall xA(x)$

to reduce universal quantifiers to skolem functions, we are allowed to do so
since there are no accumulation points from above and it can not happen that
the infimum is empty. With the skolem axioms, we reduce the prenex formula to
the purely existential formula. We know the formula is true and thus the
existential formula is also true in a $1$-element model. It is true in the
bigger model too whereas $F$ is not true for a suitable choices of
interpretation. ∎

######  Lemma 6.2.

In any Gödel logic, given an interval $(0,c]$ such that no atoms are
interpreted in this interval, they can be interpreted only at 0 or above $c$,
then it follows that every formula is either interpreted at 0 or $\geq c$.

###### Proof.

The proof is simply by induction. For atoms, it is stated; For propositional
connectives, it is clear due to Gödel conditional. For $\forall$ there might
exist a proper infimum which is $c$. For $\exists$, since all elements are
interpreted bigger then $c$, then the supremum is also bigger then $c$. ∎

######  Theorem 6.2.

In first-order infinite-valued Gödel logics with ${\mathop{\triangle}}$ when 0
is an isolated point, in general there exists no $>0$-valid equivalent prenex
normal form.

###### Proof.

Suppose there is an accumulation point from above.
Step 1: Consider the expression ’not being an accumulation point from above’

$\displaystyle F\equiv(\forall
x{\mathop{\triangle}}(X<A(x))\to{\mathop{\triangle}}(X<\forall xA(x))))$

This formula has the property that it evaluates either to true or false.
Step 2: Since the surrounding of 0 is isolated, we fix some lower point $c$ of
the surrounding of 0. Consider the interpretation where the surrounding of 0
is empty and all $A(x)$ are interpreted above $c$ or is 0. Thus, the infimum
might be empty if $A(x)$ is not interpreted 0. Recall that we are in
restricted semantics, where nothing is interpreted between 0 and $c$.
Step 3: Suppose $F$ admits $>0$-valid equivalent prenex form

$\displaystyle\neg\neg(F\leftrightarrow Qx^{\prime}Px^{\prime}).$

As everything besides 0 is interpreted above or equal $c$, we can find an
interpretation of the skolem functions such that the skolem axioms are
interpreted above $c$. Consequently, there is a $>0$ valid equivalence between
the formula $F$ and existential prenex formula.
Step 4: Repeat the same argument as in Proposition 5.3. just in this case the
equivalence is ($>c>0$)-valid. Consider a 1-element model, the equivalence is
$>0$ in the mentioned interpretation. Now we extend the 1-element model by
adding a descending chain of points that converges to $X$ but never reaches
$X$, in such a way that the existential formula remains $>0$-valid, and we
make sure that the skolem axioms remain $>0$ valid by simply adding the lower
element for the infimum which is always $\geq c$ if it is not 0.
Step 5: However, at the accumulation point within the interval $(0,c]$, the
formula $F$ becomes 0 for a suitable interpretation in the restricted
semantics, but the prenex formula remains $>0$-valid. So the equivalence is 0
and not $>0$.
In case there are no accumulation points from above we consider the following
formula:

$\displaystyle F\equiv\forall
x{\mathop{\triangle}}(A(x)<X)\to{\mathop{\triangle}}(\exists xA(x)<X).$

We repeat the same argument as above. ∎

######  Corollary 6.2.1.

There is no $>0$-valid equivalent prenex normal form for formulas in general
in $G_{\uparrow}^{\mathop{\triangle}}$.

##  7 Validity Equivalence

In this section, we consider a weaker form of prenexation in Gödel logics
namely that the validity of a formula is effectively777Can be constructed one
to one recursive procedure that assigns equivalent valid prenex formula to a
given original formula. equivalent to the validity for some prenex normal
form. (This question is senseless if the equivalence is admitted to be non-
effective: take $\bot$ and $\top$, respectively).
In this case e.g. $\mathbf{G}_{[0,1]}$ admits prenex forms as derivability in
$\mathbf{G}_{[0,1]}$ can be expressed by Kleene’s T888It is a universal Turing
machine for classical logic that expresses the fact that the formula can be
derived iff it is in the recursively enumerable set. within
$\mathbf{G}_{[0,1]}$ putting double negation in front of all atoms and
shifting the quantifiers in the classical way.
This normal form does not always exist as is demonstrated by the fact that
$G_{V}$ with $V$ uncountable but $0$ not in the perfect subset is not
recursively enumerable but its prenex fragment is.
The problem of the effective existence of validity (1-satisfiability)
equivalent prenex forms can be sharpened to the effective existence of
validity (1- satisfiability) equivalent purely existential999Note that in the
case of validity, universal quantifiers are replaced. In the case of
1-satisfiability existential quantifiers are replaced. (universal) forms.

######  Proposition 7.1.

The following holds in first-order Gödel logic:
1\. Prenex formulas admit validity equivalent standard skolemizations,
2\. Prenex formulas admit 1-satisfiability equivalent standard skolemizations.

###### Proof.

1) It is sufficient to prove for an arbitrary formula $F$ and a new function
$f$ the following:

$\displaystyle\models\exists\bar{x}\forall
yF(\bar{x},y)\Leftrightarrow\models\exists\bar{x}F(\bar{x},f(\bar{x})).$

The proof is by induction. The direction from left to right is obvious.
For the other direction, suppose that $\nvDash\exists\bar{x}\forall
yF(\bar{x},y)$ then for some interpretation $\mathcal{I}\omega$

$\displaystyle sup\\{\mathcal{I}\omega(\forall
yF(\bar{c},y))=\omega_{\bar{c}}\\}\leq\omega<1.$

Using the axiom of choice we can assign a value for every $f(\bar{c})$ such
that $\mathcal{I}_{\omega}(F(\bar{c},f(\bar{c})))$ is in between
$\omega_{\bar{c}}$ and $\omega_{\bar{c}}+(1-\omega)/2$. As a consequence

$\displaystyle
sup\\{\mathcal{I}\omega(F(\bar{c},f(\bar{c})))<(1-\omega)/2\\}\\}\leq\omega+(1-\omega)/2<1.$

Thus the right is not valid.
2) It is obvious since 1-satisfiability coincide with classical
satisfiability. ∎

######  Remark 7.1.

As a consequence a prenex formula is valid if the corresponding existential
formula is valid.

Validity in first-order Gödel logic is characterized by the following theorem.

######  Theorem 7.1.

[15] A first-order Gödel logic $G_{V}$ is recursively enumerable iff one of
the following conditions is satisfied:
1\. $V$ is finite,
2\. $V$ is uncountable101010A Gödel set is uncountable iff it contains a non-
trivial dense linear subordering., and $0$ is an isolated point,
3\. $V$ is uncountable, and every neighbourhood of $0$ is uncountable.

In all the remaining cases, i.e., logics with countable truth value set and
those with uncountable truth value set where there is a countable
neighbourhood of $0$ and $0$ is not isolated; the respective logics are not
r.e.

######  Remark 7.2.

Note that finitely valued logics of different cardinalities differ and
finitely valued logics of the same cardinality coincide. 2) and 3) in the
above theorem characterize single logic each. 3) is axiomatized by the axioms
and rules of intuitionistic predicate calculus extended by (Lin) and (QS). 2)
is axiomatized by the addition of $Iso_{0}$ axiom to 3). In case $V$ has a
finite cardinality $n$, then 1) is axiomatized by adding the respective axiom
Fin(n) to 3).

The following propositions characterize the Gödel logics with uncountable
Gödel set:

######  Proposition 7.2.

The only Gödel logics with an uncountable set that admit validity equivalent
prenex normal forms are in case when $0$ is isolated or $0$ is in perfect set.

###### Proof.

The reason is that the prenex fragment in the uncountable case is always
recursively enumerable. It was shown by [15] that when $0$ is isolated or $0$
is in perfect set it is recursively enumerable. On the other hand, when every
surrounding of $0$ is countable ( $0$ is not in the perfect set), the set is
not recursively enumerable but the prenex fragment is. Therefore there is no
equivalent valid prenex formula since there is no effective recursive
construction. Note that if the logic is r.e. Kleene’s T can be used to express
the formulas and Kleene’s T can be represented in Goedel logics by adding n
double negation to every atom. This enables classical quantifier shifting
rules. ∎

######  Remark 7.3.

Note that all non-recursively enumerable Gödel logics allow the expression of
validity in all finite domains.

######  Proposition 7.3.

Finite valued Gödel logics and $G_{\uparrow}$ admits validity equivalent
prenex normal forms.

###### Proof.

In all cases where there exists an equivalent prenex form by quantifier
shifts, there is also an effective validity equivalent prenex normal forms,
therefore finite valued Gödel logics and $G_{\uparrow}$ also have effective
validity equivalent prenex forms. ∎

##  8 Full Classification Tables

The results of this paper can be summarized in the following tables:

Logically Equivalent Prenex Normal Forns
---
Gödel set $V$ | without ${\mathop{\triangle}}$ | with ${\mathop{\triangle}}$ | validity $>0$ | Validity $>0$ with ${\mathop{\triangle}}$
finite | ✓ | ✓ | ✓ | ✓
$G_{\uparrow}$ | ✓ | × | ✓ | ×
Countable without $G_{\uparrow}$ | × | × | × | ×
0 isolated | × | × | ✓ | ×
0 not isolated | × | × | × | ×
Validity Equivalent Prenex Normal Forns
---
Goedel set $V$ | with ${\mathop{\triangle}}$ | without ${\mathop{\triangle}}$
Finite | ✓ | ✓
Uncountable | 0 isolated | ✓ | ✓
0 is in perfect set | ✓ | ✓
0 not in perfect set | × | ×
Countable | Open

##  9 Further Research

For the countable case, one can ask whether there is an effectively
constructible validity equivalence between formulas and existential formulas.
The existence of such prenex forms depends on reflective properties of the
non-recursively enumerable Gödel logics. Hájek [38] has provided a non-
arithmetic lower bound and Aguilera, Bydžovský, Fernández-Duque [2] have
provided a corresponding upper bound for $G_{\downarrow}$ ($G_{\downarrow}$ is
the countable Gödel logic with sole accumulation point at 0). An idea would be
the use of partial truth definitions. On the other hand the countability of
the logic might provide isolated points which provide a partition of the
truth-value set.
We conjecture the following:

######  Conjecture 1.

A first-order Gödel logic admits effective validity equivalent prenex normal
form iff it is arithmetical.

1) We can investigate whether purely existential prenex formulas form an
arithmetical set. In this case we will obtain a full characterization.
2) We can study other Gödel sets after the property of the logic being
arithmetical.

######  Remark 9.1.

A logic being non-arithmetical does not mean that a single formulas are also
non-arithmetical.

## References

  * [1] Juan P Aguilera and Matthias Baaz.  Unsound inferences make proofs shorter. arXiv preprint arXiv:1608.07703, 2016.
  * [2] Juan Pablo Aguilera, Jan Bydžovský, and David Fernández-Duque.  A nonhyperarithmetical Gödel logic.  In International Symposium on Logical Foundations of Computer Science, pages 1–8. Springer, 2022.
  * [3] Matthias Baaz.  Infinite-valued Gödel logic with 0-1-projections and relativisations.  In Petr Hájek, editor, Gödel’96: Logical Foundations of Mathematics, Computer Science, and Physics, volume 6 of Lecture Notes in Logic, pages 23–33. Springer-Verlag, Brno, 1996.
  * [4] Matthias Baaz and Agata Ciabattoni.  A Schütte-Tait style cut-elimination proof for first-order Gödel logic.  In Uwe Egly and Christian G. Fermüller, editors, Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX 2002, volume 2381 of LNCS, pages 24–38, Berlin, 2002\. Springer.
  * [5] Matthias Baaz, Agata Ciabattoni, and Christian G. Fermüller.  Hypersequent calculi for Gödel logics: A survey.  Journal of Logic and Computation, 13(6):835–861, 2003.
  * [6] Matthias Baaz, Agata Ciabattoni, and Christian G. Fermüller.  Monadic fragments of Gödel logics: Decidability and undecidability results.  In Nachum Dershowitz and Andrei Voronkov, editors, Logic for Programming, Artificial Intelligence, and Reasoning, volume 4790/2007 of Lecture Notes in Computer Science, pages 77–91, 2007.
  * [7] Matthias Baaz, Agata Ciabattoni, and Norbert Preining.  SAT in monadic Gödel logics: A borderline between decidability and undecidability.  In Hiroakira Ono, Makoto Kanazawa, and Ruy de Queiroz, editors, Logic, Language, Information and Computation, 16th Workshop, WoLLIC 2009, number 5514 in LNAI, pages 113–123, 2009.
  * [8] Matthias Baaz, Agata Ciabattoni, and Norbert Preining.  First-order satisfiability in Gödel logics: An NP-complete fragment.  Theoretical Computer Science, 412:6612–6623, 2011.
  * [9] Matthias Baaz, Agata Ciabattoni, and Richard Zach.  Quantified propositional Gödel logic.  In Andrei Voronkov and Michel Parigot, editors, Logic for Programming and Automated Reasoning. 7th International Conference, LPAR 2000, volume 1995 of LNAI, pages 240–256, Berlin, 2000. Springer.
  * [10] Matthias Baaz and Christian G. Fermüller.  A resolution mechanism for prenex Gödel logic.  In Anuj Dawar and Helmut Veith, editors, Computer Science Logic, 24th International Workshop, CSL 2010, 19th Annual Conference of the EACSL, volume 6247 of LNCS, pages 67–79, Berlin, 2010. Springer.
  * [11] Matthias Baaz, Alexander Leitsch, and Richard Zach.  Completeness of a first-order temporal logic with time-gaps.  Theoretical Computer Science, 160(1–2):241–270, 1996.
  * [12] Matthias Baaz, Alexander Leitsch, and Richard Zach.  Incompleteness of a first-order Gödel logic and some temporal logics of programs.  In Computer Science Logic, volume 1092/1996 of Lecture Notes in Computer Science, pages 1–15, 1996.
  * [13] Matthias Baaz and Norbert Preining.  Quantifier elimination for quantified propositional logics on Kripke frames of type omega.  Journal of Logic and Computation, 18:649–668, 2008.
  * [14] Matthias Baaz, Norbert Preining,  Gödel–Dummett logics,  in: Petr Cintula, Petr Hájek, Carles Noguera (Eds.) Handbook of Mathematical Fuzzy Logic vol.2, College Publications, 2011, pp.585–626, chapterVII.
  * [15] Matthias Baaz and Norbert Preining.  On the classification of first order Goedel logics.  Ann. Pure Appl. Log 170:36–57, 2019.
  * [16] Matthias Baaz, Norbert Preining, and Richard Zach.  Characterization of the axiomatizable prenex fragments of first-order Gödel logics.  In 33rd IEEE International Symposium on Multiple-Valued Logic (ISMVL 2003), pages 175–180, Los Alamitos, 2003. IEEE Computer Society.
  * [17] Matthias Baaz, Norbert Preining, and Richard Zach.  Completeness of a hypersequent calculus for some first-order Göde logics with delta.  In 36th International Symposium on Multiple-valued Logic (ISMVL 2006). IEEE Computer Society, 2006.
  * [18] Matthias Baaz, Norbert Preining, and Richard Zach.  First-order Gödel logics.  Annals of Pure and Applied Logic, 147(1–2):23–47, 2007.
  * [19] Matthias Baaz and Helmut Veith.  An axiomatization of quantified propositional Gödel logic using the Takeuti-Titani rule.  In Logic Colloquium 1999, volume 13 of Lecture Notes in Logic, pages 91–104, 2000.
  * [20] Matthias and Helmut Veith.  Interpolation in Fuzzy Logic  Archive for Mathematical Logic, accepted for publication.
  * [21] Matthias and Helmut Veith.  Quantifier Elimination in Fuzzy Logic  CSL 1998.
  * [22] Matthias Baaz and Richard Zach.  Compact propositional Gödel logics.  In Proceedings of 28th International Symposium on Multiple-Valued Logic, pages 108–113, Los Alamitos, CA, 1998. IEEE Computer Society Press.
  * [23] Matthias Baaz and Richard Zach.  Hypersequents and the proof theory of intuitionistic fuzzy logic.  In Peter G. Clote and Helmut Schwichtenberg, editors, Proceedings of 14th CSL Workshop, volume 1862 of Lecture Notes in Computer Science, pages 187–201, Berlin, 2000. Springer-Verlag.
  * [24] J.R Buechi.  On a decision method in restricted second order arithmetic.  In Logic, Methodology, and Philosophy of Science  Proc. of the 1960 Congress, Stanford University Press, pp. 1-11.
  * [25] Arnold Beckmann, Martin Goldstern, and Norbert Preining.  Continuous Fraïssé conjecture.  Order, 25(4):281–298, 2008.
  * [26] Arnold Beckmann and Norbert Preining.  Linear Kripke frames and Gödel logics.  Journal of Symbolic Logic, 72(1):26–44, 2007.
  * [27] Arnold Beckmann, Norbert Preining,  Deciding logics of linear Kripke frames with scattered end pieces,  Soft Comput. 21(1) (2017) 191–197.
  * [28] Arnold Beckmann, Norbert Preining,  Separating intermediate predicate logics of well-founded and dually well-founded structures by monadic sentences,  J. Logic Comput. 25(3) (2015) 527–547.
  * [29] Agata Ciabattoni.  A proof-theoretical investigation of global intuitionistic (fuzzy) logic.  Archive for Mathematical Logic, 44(4):435–457, 2005.
  * [30] Brian A. Davey.  On the lattice of subvarieties.  Houston Journal of Mathematics, 5(2):183–192, 1979.
  * [31] Michael Dummett.  A propositional calculus with denumerable matrix.  Journal of Symbolic Logic, 24(2): 97–106, 1959.
  * [32] Melvin Chris Fitting.  Intuitionistic Logic, Model Theory and Forcing.  Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam/London, 1969.
  * [33] Melvin Chris Fitting.  Proof Methods for Modal and Intuitionistic Logics.  Kluwer, 1983.
  * [34] Roland Fraïssé.  Sur la comparaison des types d’ordres.  Comptes Rendus Hebdomadaire des Séances de l’Académie des Sciences, Paris, 226:1330–1331, 1948.
  * [35] Kurt Gödel.  Zum intuitionistischen Aussagenkalkül.  Anzeiger Akademie der Wissenschaften Wien, 69: 65–66, 1932.
  * [36] S. Gottwald.  Mehrwertige Logik.  Berlin 1989.
  * [37] Petr Hájek.  Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic.  Kluwer, Dordrecht, 1998.
  * [38] Petr Hájek. A non-arithmetical Gödel logic.  Logic Journal of the IGPL, 13(4):435–441, 2005.
  * [39] Alfred Horn.  Logic with truth values in a linearly ordered Heyting algebras.  Journal of Symbolic Logic, 34(3):395–408, 1969.
  * [40] Alexander S. Kechris.  Classical Descriptive Set Theory, volume 159 of Graduate Texts in Mathematics.  Springer, 1995.
  * [41] L. Kreiser, S. Gottwald, W. Stelzner  Nichtklassische Logik.  Akademie-Verlag, Berlin 1988.
  * [42] L. Maksimova.  Craig’s interpolation theorem and amalgamable varieties.  Doklady Akademii Nauk SSSR, 237/6, 1281-1284, 1977.
  * [43] Richard Mansfield.  Perfect subsets of definable sets of real numbers.  Pacific Journal of Mathematics, 35:451–457, 1970.
  * [44] Yiannis N. Moschovakis.  Descriptive Set Theory, volume 100 of Studies in Logic and the Foundations of Mathematics.  North-Holland, Amsterdam, 1980.
  * [45] Hiroakira Ono.  Kripke models and intermediate logics.  Publications of the Research Institute for Mathematical Sciences Kyoto University, 6:461–476, 1971.
  * [46] Hiroakira Ono.  A study of intermediate predicate logics.  Publications of the Research Institute for Mathematical Sciences Kyoto University, 8:619–649, 1972/73.
  * [47] Norbert Preining.  Complete Recursive Axiomatizability of Gödel Logics.  PhD thesis, Vienna University of Technology, Austria, 2003.
  * [48] Norbert Preining,  Gödel logics and Cantor–Bendixon analysis,  in: M. Baaz, A. Voronkov (Eds.), Logic for Programming and Automated Reasoning, LPAR 2002. Proceedings, in: Lecture Notes in Artificial Intelligence, vol.2514, Springer, 2002, pp.327–336.
  * [49] George N. Raney.  Completely distributive complete lattices.  Proceedings of the American Mathematical Society, 3:677–680, 1952\.
  * [50] Joseph G. Rosenstein.  Linear Orderings.  Academic Press, New York, 1982.
  * [51] Mitio Takano.  Another proof of the strong completeness of the intuitionistic fuzzy logic.  Tsukuba Journal of Mathematics, 11(1):101–105, 1987.
  * [52] Mitio Takano.  Ordered sets R and Q as bases of Kripke models.  Studia Logica, 46:137–148, 1987.
  * [53] Gaisi Takeuti  Proof Theory,  2nd ed., North-Holland , 1987.
  * [54] Gaisi Takeuti and Satoko Titani.  Intuitionistic fuzzy logic and intuitionistic fuzzy set theory.  Journal of Symbolic Logic, 49(3):851–866, 1984.
  * [55] Anne S. Troelstra.  Aspects of constructive mathematics.  In Jon Barwise, editor, Handbook of Mathematical Logic, pages 973–1052. North-Holland, 1977.
  * [56] Reinhard Winkler.  How much must an order theorist forget to become a topologist?  In Contributions of General Algebra 12, Proc. of the Vienna Conference, pages 420–433, Klagenfurt, Austria, 1999. Verlag Johannes Heyn.


# New $X_{0,1}(2900)$-like exotic states in $b$-baryon decays

Yu-Kuo Hsiao  Corresponding author<EMAIL_ADDRESS>School of Physics and
Information Engineering, Shanxi Normal University, Linfen 041004, China  Yao
Yu  Corresponding author<EMAIL_ADDRESS>Chongqing University of Posts &
Telecommunications, Chongqing 400065, China

###### Abstract

In the $B^{+}\to D^{+}D^{-}K^{+}$ decay, LHCb has reported the observation of
the open-charm exotic states $X_{0,1}^{0}\equiv X_{0,1}(2900)^{0}$ with four
different quark flavors $(ud\bar{s}\bar{c})$, where the subscripts (0,1)
denote the spins. To confirm the discovery, we propose
$\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)}$ in the final state
interaction, where $X_{0,1}^{\prime\,0(--)}$ with $su\bar{d}\bar{c}$
($ds\bar{u}\bar{c}$) are the new $X_{0,1}$-like exotic states. More
specifically, $\Lambda_{c}^{+}D_{s}^{-}$ in
$\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-}$ are transformed as
$\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)}$, by exchanging $\pi^{+(-)}$. As
the order of magnitude estimates, we calculate ${\cal
B}(\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)})=(2.3\pm 0.6,4.3\pm
0.8^{+3.3}_{-2.5})\times 10^{-4}$. In addition, we estimate other $b$-baryon
decays with the $X_{0,1}$-like states, such as ${\cal
B}(\Xi_{b}^{0(-)}\to\Xi_{c}^{0(+)}(2645)X_{0}^{\prime\,0(--)},\Lambda_{b}\to\Xi_{c}^{\prime\,0}X_{0,1}^{0})\sim
10^{-5}$. While one needs $\Lambda_{b}\to\Sigma_{c}^{0(++)}M_{c}M$ to observe
$X_{0,1}^{\prime\,0(--)}\to M_{c}M$ with $M_{c}M=D_{s}^{-}\pi^{+}$,
$\bar{D}^{0}\bar{K}^{0}$ $(D_{s}^{-}\pi^{-}$, $D^{-}K^{-})$, ${\cal
B}(\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)},X_{0,1}^{\prime\,0(--)}\to
M_{c}M)\sim 10^{-4}$ are accessible to the LHCb experiment.

##  I introduction

The LHCb Collaboration has recently observed $X_{0,1}^{0}\equiv
X_{0,1}(2900)^{0}$ from the $B^{+}\to D^{+}D^{-}K^{+}$ decay Aaij:2020hon ;
Aaij:2020ypa , where the subscript 0(1) denotes the spin. With the resonant
strong decays $X_{0,1}^{0}\to D^{-}K^{+}$ in the $D^{-}K^{+}$ invariant mass
spectrum, one concludes that $X_{0,1}^{0}$ are the first open-charm exotic
states, which consist of four quarks with different flavors
$\bar{c}d\bar{s}u$. Explicitly, the masses and decay widths of $X_{0,1}^{0}$
are given by Aaij:2020hon ; Aaij:2020ypa

$\displaystyle(m_{X_{0}^{0}},m_{X_{1}^{0}})$ $\displaystyle=$
$\displaystyle(2.866\pm 0.007\pm 0.002,2.904\pm 0.005\pm
0.001)~{}\text{GeV}\,,$ $\displaystyle(\Gamma_{X_{0}^{0}},\Gamma_{X_{1}^{0}})$
$\displaystyle=$ $\displaystyle(57\pm 12\pm 4,110\pm 11\pm 4)~{}\text{MeV}\,,$
(1)

where $m_{X_{0}^{0}}\simeq m_{X_{1}^{0}}$ and
$\Gamma_{X_{0}^{0}}\simeq\Gamma_{X_{1}^{0}}/2$. From the strong decays
$X(5568)^{\pm}\to B_{s}^{0}\pi^{\pm}$, the “open-beauty” exotic state
$X(5568)$ with four different quark flavors $\bar{b}s\bar{d}u$
$(\bar{b}s\bar{u}d)$ was once reported to be observed by D0 Collaboration
D0:2016mwd . Unfortunately, the discovery was not confirmed by LHCb and other
experiments Aaij:2016iev ; Sirunyan:2017ofq ; Aaltonen:2017voc ;
Aaboud:2018hgx . Since it is possible that $X_{0,1}^{0}$ can be further
examined as the tetraquarks, which definitely improves our knowledge of QCD
and the quark model, one needs to provide the different decays to confirm the
discovery.

Theoretical attempts have been given to understand the open-charm exotic
states Molina:2010tx ; Karliner:2020vsi ; He:2020jna . Considering the light
quark $q=(u,d,s)$ as a triplet (3) under the $SU(3)$ flavor symmetry
($SU(3)_{f}$), the open-charm exotic states $q_{1}q_{2}\bar{q}_{3}\bar{c}$ are
in the irreducible forms of $(3\times
3\times\bar{3})\bar{c}=(3+3+\bar{6}+15)\bar{c}$, where $\bar{6}$ and 15 are
able to include three different quark flavors. Consequently, one classifies
$X_{0}^{0}$ and $X_{1}^{0}$ into $\bar{6}$ and $15$, with $X_{0}^{0}=(ud-
du)\bar{s}\bar{c}$ and $X_{1}^{0}=(ud+du)\bar{s}\bar{c}$, respectively
He:2020jna . Moreover, $X_{0(1)}^{0}$ is assigned the quantum numbers
$J^{P}=0^{+}(1^{-})$, which is based on the first radially (orbitally) excited
state $2S$ ($1P$) of the two-body Coulomb and chromomagnetic interaction
models. In this classification, there can be $X_{0,1}$-like exotic states,

$\displaystyle X^{\prime\,--}_{0,1}=(ds\mp
sd)\bar{u}\bar{c}\,,\;X^{\prime\,0}_{0,1}=(su\mp
us)\bar{d}\bar{c}\,,\;Y^{\prime\,-}_{0,1}=(sn-ns)\bar{n}\bar{c}\,,$ (2)

with $J^{P}=(0^{+},1^{-})$, $n=u$, $d$, and $m_{X_{0(1)}^{\prime}}\simeq
m_{X_{0(1)}}$. In addition, $X_{0(1)},X_{0(1)}^{\prime}\to DM$ are related
with the same strong coupling constant. Therefore, the observation of
$X^{\prime}_{0,1}$ can be regarded to confirm the discovery of $X_{0,1}$.

As the discovery channel, $B^{+}\to D^{+}X_{0(1)}^{0},X_{0(1)}^{0}\to
D^{-}K^{+}$ has been interpreted to proceed through the final state
interaction Burns:2020epm ; Burns:2020xne ; Chen:2020eyu ; Liu:2020orv .
Following the $B^{+}\to D_{s}^{*+}\bar{D}^{0}$ weak decay,
$D_{s}^{*+}\bar{D}^{0}$ in the rescattering effect are transformed as
$D^{+}X_{0,1}^{0}$, which is by exchanging $K^{0}$. Moreover, the same
mechanism has been applied to the production of the hidden-charm (strange)
pentaquark ${\cal P}_{c(s)}$ with $c\bar{c}uud$ ($c\bar{c}uds$) in
$\Lambda_{b}\to J/\Psi{\cal P}_{c(s)}$ Wu:2021dmq ; Liu:2015fea ; Aaij:2020gdg
; Aaij:2015tga . Therefore, it is reasonable to consider the $b$-baryon decays
with the new $X_{0,1}^{0}$-like exotic states in the final state interaction.
Specifically, we propose
$\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)}$ as depicted in Fig.
1, with $X_{0,1}^{\prime\,0}$ and $X_{0,1}^{\prime\,--}$ in Eq. (2).

To estimate $\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)}$, the
couplings in the triangle loop should be crucial. One has observed ${\cal
B}(\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-})\simeq 10^{-2}$ and ${\cal
B}(\Sigma_{c}^{0(++)}\to\Lambda_{c}^{+}\pi^{-(+)})\simeq 100\%$ pdg ,
indicating the sizeable weak and strong couplings of the baryon decays. In
addition, the coupling of $X_{0(1)}^{\prime}\to D_{s}^{-}\pi$ is not small,
due to the $SU(3)$ flavor ($SU(3)_{f}$) symmetry that has been enabled to
relate $X_{0,1}^{\prime}$ and $X_{0,1}^{0}$ decays He:2020jna . Hence, ${\cal
B}(\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)})$ are anticipated
to be as accessible as ${\cal B}(B^{+}\to D^{+}X_{0,1}^{0})$ to the LHCb
experiment. In this paper,
$\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)}$ will be demonstrated
as the promising decay channels to confirm the existence of the
$X_{0,1}^{0}$-like exotic states.

##  II Formalism

Figure 1: Rescattering $\Lambda_{b}\to\Sigma_{c}X_{0,1}^{\prime}$ decays: the
left panel presents the quark lines, and the right panel the momentum flows.

The open-charm exotic states $X_{0,1}^{0}$ observed in $B^{+}\to
D^{+}D^{-}K^{+}$ need to be confirmed by different decays. We think
$\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0,1}^{\prime\,0(--)}$ may be promising.
Like $X_{0,1}^{0}(\bar{c}d\bar{s}u)$, $X_{0,1}^{\prime\,0}(\bar{c}s\bar{d}u)$
and $X_{0,1}^{\prime\,--}(\bar{c}s\bar{u}d)$ also consist of four different
quark flavors. See Fig. 1, $\Lambda_{b}\to\Sigma_{c}X_{0,1}^{\prime}$ are the
triangle-rescattering decays, separated into two parts. The first part is the
weak decay $\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-}$, which proceeds through
the $\Lambda_{b}\to\Lambda_{c}^{+}$ transition and $D_{s}^{-}$ meson
production. By following Refs. Hsiao:2018zqd ; Hsiao:2017umx ;
Leibovich:2003tw , we derive the amplitude of
$\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-}$ as

$\displaystyle{\cal M}_{b}(\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-})$
$\displaystyle=$
$\displaystyle\bar{u}_{\Lambda_{c}}(F_{b}^{+}-F_{b}^{-}\gamma_{5})u_{\Lambda_{b}}\,,$
(3)

where

$\displaystyle
F_{b}^{+}=C_{\text{w}}m_{-}\bigg{[}f_{1}+\left(\frac{m_{D_{s}}^{2}}{m_{+}m_{-}}\right)f_{3}\bigg{]}\,,\,\,\,\,\,\,\,\,\,F_{b}^{-}=C_{\text{w}}m_{+}\bigg{[}g_{1}+\left(\frac{m_{D_{s}}^{2}}{m_{+}^{2}}\right)g_{3}\bigg{]}\,.$
(4)

In the above, we define $m_{\pm}=m_{\Lambda_{b}}\pm m_{\Lambda_{c}}$ and
$C_{\text{w}}=i({G_{F}}/{\sqrt{2}})\,a_{1}V_{cb}V^{*}_{cs}f_{D_{s}}$, where
$G_{F}$, $V_{ij}$, $f_{D_{s}}$ and $f_{1,3}(g_{1,3})$ are the Fermi constant,
CKM matrix element, decay constant, and $\Lambda_{b}\to\Lambda_{c}^{+}$
transition form factors Feldmann:2011xf , respectively, while $a_{1}$ results
from the factorization Hsiao:2020gtc ; Hsiao:2014mua . The second part as the
rescattering effect proceeds through the strong decays, such that
$\Lambda_{c}^{+}D_{s}^{-}$ are turned into $\Sigma_{c}X_{0,1}^{\prime}$ by
emitting or receiving a pion. Accordingly, the amplitudes are given by
He:2020jna ; Albertus:2005zy ; Can:2016ksz

$\displaystyle{\cal M}_{0}(X_{0}^{\prime}\to D^{-}_{s}\pi)$ $\displaystyle=$
$\displaystyle g_{0}\,,\;\hat{\cal M}_{1}(X_{1}^{\prime}\to
D^{-}_{s}\pi)=g_{1}\epsilon\cdot(p_{D_{s}}-p_{\pi})\,,\;$ $\displaystyle{\cal
M}_{c}(\Sigma_{c}\to\Lambda^{+}_{c}\pi)$ $\displaystyle=$ $\displaystyle
g_{c}\bar{u}_{\Lambda_{c}}\gamma_{5}u_{\Sigma_{c}}\,,$ (5)

with $g_{c,0,1}$ the strong coupling constants and $\epsilon_{\mu}$ the
polarization four vector, where ${\cal
M}_{c}(\Sigma_{c}\to\Lambda^{+}_{c}\pi)$ is presented as that used in the
quark model and lattice QCD.

To proceed, we assemble the two parts as the rescattering amplitudes, given by
Yu:2020vlt ; Hsiao:2019ait

$\displaystyle{\cal M}(\Lambda_{b}\to\Sigma_{c}X_{0,1}^{\prime})$
$\displaystyle=$ $\displaystyle\int\frac{d^{4}{q}_{3}}{(2\pi)^{4}}\frac{{\cal
M}_{b}{\cal M}^{{\dagger}}_{c}{\cal
M}_{0,1}}{(q_{1}^{2}-m_{D_{s}}^{2})(q_{2}^{2}-m_{\pi}^{2})(q_{3}^{2}-m_{\Lambda_{c}}^{2})}\,,$
(6)

where ${\cal M}_{1}\equiv\hat{\cal M}_{1}F_{\Lambda}(q_{2}^{2})$, and
$F_{\Lambda}(q_{2}^{2})\equiv(\Lambda^{2}-m^{2}_{\pi})/(\Lambda^{2}-q^{2}_{2})$
is the form factor that takes care of the divergence in
$\Lambda_{b}\to\Sigma_{c}X_{1}^{\prime}$. Besides, $q_{1}=p_{1}-q_{3}$ and
$q_{2}=p_{3}-q_{3}$ correspond to the momentum flows in Fig. 1. In the general
form, one expresses the amplitudes of
$\Lambda_{b}\to\Sigma_{c}X_{0,1}^{\prime}$ as

$\displaystyle{\cal M}(\Lambda_{b}\to\Sigma_{c}X_{0}^{\prime})$
$\displaystyle=$
$\displaystyle\bar{u}_{\Sigma_{c}}(F_{0}^{+}-F_{0}^{-}\gamma_{5})u_{\Lambda_{b}}\,,$
$\displaystyle{\cal M}(\Lambda_{b}\to\Sigma_{c}X_{1}^{\prime})$
$\displaystyle=$
$\displaystyle\bar{u}_{\Sigma_{c}}[(F_{1}^{+}\gamma^{\mu}-F_{1}^{-}\gamma^{\mu}\gamma^{5})+(G_{1}^{+}p_{3}^{\mu}-G_{1}^{-}p_{3}^{\mu}\gamma^{5})]u_{\Lambda_{b}}\epsilon_{\mu}\,.$
(7)

To obtain $F_{0,1}^{\pm}$ and $G_{1}^{\pm}$, we need to integrate over the
variables of the triangle loop in Eq. (6), for which the equations are given
by tHooft:1978jhc ; Hahn:1998yk ; Denner:2005nn

$\displaystyle\\{C_{0};C^{\mu};C^{\mu\nu}\\}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q_{3}}{i\pi^{2}}\frac{\\{1;q_{3}^{\mu};q_{3}^{\mu}q_{3}^{\nu}\\}}{(q_{3}^{2}-m_{\Lambda_{c}}^{2})[(q_{3}-p_{1})^{2}-m_{D_{s}}^{2}][(q_{3}-p_{3})^{2}-m_{\pi}^{2}]}\,,$
(8)

such that we obtain

$\displaystyle C^{\mu}$ $\displaystyle=$ $\displaystyle-
p_{1}^{\mu}C_{1}-p_{3}^{\mu}C_{2}\,,$ $\displaystyle C^{\mu\nu}$
$\displaystyle=$ $\displaystyle
g^{\mu\nu}C_{00}+p_{1}^{\mu}p_{1}^{\nu}C_{11}+p_{3}^{\mu}p_{3}^{\nu}C_{22}+(p_{1}^{\mu}p_{3}^{\nu}+p_{1}^{\nu}p_{3}^{\mu})C_{12}\,,$
(9)

with the parameters $C_{0},C_{1},C_{2},C_{00},C_{11},C_{12},C_{22}$ to be
given in the numerical analysis. By replacing $m_{\pi}$ in Eq. (8) with
$\Lambda$, we obtain $\\{C^{\prime}_{0};C^{\prime\mu};C^{\prime\mu\nu}\\}$ and
then define another set of parameters:
$\tilde{C}_{i(ij)}=C_{i(ij)}-C^{\prime}_{i(ij)}$, where $i=(0,1,2)$ and
$ij=(00,11,22,12)$. We hence derive that

$\displaystyle F_{0}^{\pm}$ $\displaystyle=$ $\displaystyle\frac{\mp
ig_{c}g_{0}F_{b}^{\mp}}{16\pi^{2}}\bigg{(}m_{\Lambda_{c}}C_{0}\pm
m_{\Lambda_{b}}C_{1}+m_{\Sigma_{c}}C_{2}\bigg{)}\,,\;F_{1}^{\pm}=\frac{ig_{c}g_{1}F_{b}^{\mp}}{8\pi^{2}}\tilde{C}_{00}\,,\;$
$\displaystyle G_{1}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{-ig_{c}g_{1}F_{b}^{\mp}}{8\pi^{2}}\bigg{[}m_{\Lambda_{c}}(\tilde{C}_{0}+\tilde{C}_{1}+\tilde{C}_{2})$
(10) $\displaystyle\pm
m_{\Lambda_{b}}(\tilde{C}_{1}+\tilde{C}_{11}+\tilde{C}_{12})+m_{\Sigma_{c}}(\tilde{C}_{2}+\tilde{C}_{12}+\tilde{C}_{22})\bigg{]}\,.$

##  III Numerical results and Discussions

In the numerical analysis, we adopt
$(V_{cb},V_{cs})=(A\lambda^{2},1-\lambda^{2}/2)$ and $f_{D_{s}}=(249.9\pm
0.5)$ MeV, with $A=0.790\pm 0.017$ and $\lambda=0.22650\pm 0.00048$ pdg . We
get $(f_{1},g_{1})=(0.59,0.53)$ and $(f_{3},g_{3})=(-0.02,-0.03)$ from the
lattice QCD calculation Detmold:2015aaa . Using Eqs. (3) and (4), and ${\cal
B}(\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-})=(1.10\pm 0.10)\%$ pdg , we extract
$a_{1}=0.93\pm 0.04$, where $a_{1}$ of ${\cal O}(1.0)$ demonstrates the
feasibility of the generalized factorization Hsiao:2018zqd ; Hsiao:2017umx ;
Leibovich:2003tw . From $B(\Sigma_{c}\to\Lambda_{c}\pi)\simeq 100\%$ pdg , we
determine $g_{c}=19.1$ GeV. The $SU(3)_{f}$ symmetry has been enabled to
relate $X_{0,1}^{\prime\,0(--)}\to D_{s}^{-}\pi^{+(-)}$ and $X_{0,1}^{0}\to
D^{-}K^{+}$, such that we obtain $g_{0}=(2.85\pm 0.32)$ GeV and $g_{1}=4.63\pm
0.25$ from $\Gamma_{X_{0,1}^{0}}$ in Eq. (I); besides, we take
$m_{X_{0,1}^{\prime}}\simeq m_{X_{0,1}^{0}}$.

The integration of the triangle loop in
$\Lambda_{b}\to\Sigma_{c}X_{0}^{\prime}$ gives

$\displaystyle(C_{0},C_{1},C_{2})=(0.37-0.32i,-1.50-2.30i,-0.91-0.55i)~{}\text{GeV}^{-2}\,.$
(11)

On the other hand, $\Lambda_{b}\to\Sigma_{c}X_{1}^{\prime}$ encounters the
logarithmic divergence, for which the cutoff $\Lambda$ needs to be introduced.
Since $\Lambda$ of ${\cal O}(1.0~{}\text{GeV})$ has been commonly used as a
phenomenological parameter Tornqvist:1993ng ; Li:1996yn ; Wu:2019vbk , we take
$\Lambda=(1.00,1.25,1.50)$ GeV for the demonstration, which results in

$\displaystyle\tilde{C}_{0}$ $\displaystyle=$
$\displaystyle(0.26+0.17i,0.36+0.10i,0.41+0.05i)~{}\text{GeV}^{-2}\,,$
$\displaystyle\tilde{C}_{1}$ $\displaystyle=$
$\displaystyle(0.95-2.15i,0.35-2.75i,-0.11-2.94i)~{}\text{GeV}^{-2}\,,$
$\displaystyle\tilde{C}_{2}$ $\displaystyle=$
$\displaystyle(0.06-0.24i,0.08-0.35i,0.05-0.45i)~{}\text{GeV}^{-2}\,,$
$\displaystyle\tilde{C}_{00}$ $\displaystyle=$
$\displaystyle(-1.02-0.51i,-1.62-0.45i,-2.17-0.25i)\,,$
$\displaystyle\tilde{C}_{11}$ $\displaystyle=$
$\displaystyle(-0.52+1.85i,-0.01+2.19i,0.33+2.23i)~{}\text{GeV}^{-2}\,,$
$\displaystyle\tilde{C}_{12}$ $\displaystyle=$
$\displaystyle(-0.02+0.19i,0.00+0.26i,0.03+0.31i)~{}\text{GeV}^{-2}\,,$
$\displaystyle\tilde{C}_{22}$ $\displaystyle=$
$\displaystyle(0.01+0.06i,0.02+0.08i,0.02+0.11i)~{}\text{GeV}^{-2}\,.$ (12)

As a consequence, we obtain

$\displaystyle{\cal
B}_{0}(\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0}^{\prime\,0(--)})$
$\displaystyle=$ $\displaystyle(2.3\pm 0.6)\times 10^{-4}\,,$
$\displaystyle{\cal
B}_{1}(\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{1}^{\prime\,0(--)})$
$\displaystyle=$ $\displaystyle(4.3\pm 0.8^{+3.3}_{-2.5})\times 10^{-4}\,,$
(13)

where ${\cal B}_{0,1}$ as large as $10^{-4}$ are not caused by the triangle
singularity Liu:2020orv . Indeed, the sizeable weak and strong coupling
constants play the key role. However, the calculations are at most as accurate
as the order of magnitude estimations due to the large uncertainties in Eq.
(III). Explicitly, the first uncertainties combine the errors from
$V_{cb(cs)}$, $f_{D_{s}}$, $g_{0,1}$, $a_{1}$, and the second ones from
$\Lambda$, where we simply take $\Lambda=(1.25\pm 0.25)$ GeV for a naive
estimate of ${\cal B}(\Lambda_{b}\to\Sigma_{c}X_{1}^{\prime})$, in accordance
with $\tilde{C}_{i}$ and $\tilde{C}_{ij}$ in Eq. (III).

In the heavy-baryon chiral perturbation theory Cheng:2015naa ; Yan:1992gz ,
the amplitude of $\Sigma_{c}\to\Lambda^{+}_{c}\pi$ in Eq. (II) has another
form: ${\cal
M}^{\prime}_{c}(\Sigma_{c}\to\Lambda^{+}_{c}\pi)=g^{\prime}_{c}\bar{u}_{\Lambda_{c}}(\gamma_{\mu}\gamma_{5}p_{\pi}^{\mu})u_{\Sigma_{c}}$,
where $g^{\prime}_{c}=4.0$. Subsequently, $p_{\pi}^{\mu}$ added to the
triangle loop causes the logarithmic (linear) divergence for
$\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0(1)}^{\prime\,0(--)}$, which corresponds
to the integration with $C^{\mu\nu}$ ($C^{\mu\nu\rho}$). Note that
$C^{\mu\nu\rho}$ has an extra $q_{3}^{\rho}$ compared to $C^{\mu\nu}$ in Eq.
(8), whose detailed form can be found in tHooft:1978jhc ; Hahn:1998yk ;
Denner:2005nn . We hence obtain different results, given by

$\displaystyle{\cal
B}^{\prime}_{0}(\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0}^{\prime\,0(--)})$
$\displaystyle=$ $\displaystyle(3.1\pm 0.9^{+0.5}_{-0.9})\times 10^{-4}\,,$
$\displaystyle{\cal
B}^{\prime}_{1}(\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{1}^{\prime\,0(--)})$
$\displaystyle=$ $\displaystyle(4.5\pm 0.8^{+3.1}_{-2.5})\times 10^{-3}\,,$
(14)

where ${\cal B}^{\prime}_{0}(\Lambda_{b}\to\Sigma_{c}X_{0})$ is slightly
deviated from ${\cal B}_{0}(\Lambda_{b}\to\Sigma_{c}X_{0})$; however, due to
the cutoff, it is more uncertain. On the other hand, ${\cal
B}^{\prime}_{1}(\Lambda_{b}\to\Sigma_{c}X_{1})$ is 10 times larger than ${\cal
B}_{1}(\Lambda_{b}\to\Sigma_{c}X_{1})$, presenting a sensitivity to
$p_{\pi}^{\mu}$ from the linear divergence. To distinguish between the two
different strong couplings, ${\cal B}_{0}/{\cal B}_{1}\simeq 0.5$ and ${\cal
B}^{\prime}_{0}/{\cal B}^{\prime}_{1}\simeq 0.07$ can be used for the future
experimental examination.

The exotic $X_{0,1}^{0}$ are observed in $B^{+}\to D^{+}D^{-}K^{+}$. Likewise,
we expect $X_{0}^{\prime\,0(--)}$ to be observed in
$\Lambda_{b}\to\Sigma_{c}^{0(++)}M_{c}M$, which receives the resonant
contributions from
$\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{0}^{\prime\,0(--)},X_{0}^{\prime\,0(--)}\to
M_{c}M$ with $M_{c}M=D_{s}^{-}\pi^{+}$, $\bar{D}^{0}\bar{K}^{0}$
$(D_{s}^{-}\pi^{-}$, $D^{-}K^{-})$. Approximately, we present the resonant
branching fractions as

$\displaystyle{\cal
B}(\Lambda_{b}\to\Sigma_{c}X_{0,1}^{\prime},X_{0,1}^{\prime}\to
M_{c}M)\simeq{\cal B}(\Lambda_{b}\to\Sigma_{c}X_{0,1}^{\prime}){\cal
B}(X_{0,1}^{\prime}\to M_{c}M)\,,$ (15)

with ${\cal B}(\Lambda_{b}\to\Sigma_{c}X_{0,1}^{\prime})$ in Eq. (III). In the
$SU(3)_{f}$ symmetry, ${\cal B}(X_{0,1}^{\prime}\to M_{c}M)$ are given by
He:2020jna

$\displaystyle{\cal B}(X_{0,1}^{\prime\,0(--)}\to
D_{s}^{-}\pi^{\pm})=(51,53)\%\,,$ $\displaystyle{\cal
B}(X_{0,1}^{\prime\,0(--)}\to\bar{D}^{0}\bar{K}^{0}(D^{-}K^{-}))=(49,47)\%\,.$
(16)

We can hence estimate ${\cal
B}(\Lambda_{b}\to\Sigma_{c}^{++}X_{0}^{\prime\,--}X_{0,1}^{\prime\,--}\to
D^{-}K^{-})$ as large as $10^{-4}$, which seems to be very accessible to the
LHCb experiment. However, there might exist other resonant decays to interfere
with $\Lambda_{b}\to\Sigma_{c}^{++}X_{0}^{\prime\,--},X_{0,1}^{\prime\,--}\to
D^{-}K^{-}$, which would cause a complicated amplitude analysis. For example,
the amplitude analysis of $B^{+}\to D^{+}X_{0,1}^{0},X_{0,1}^{0}\to
D^{-}K^{+}$ is complicated, which is due to the resonant decays $B^{+}\to
M_{c\bar{c}}K^{+},M_{c\bar{c}}\to D^{+}D^{-}$ with $M_{c\bar{c}}=\psi(3770)$,
$\chi_{c0(c2)}(3770)$, and $\psi(4040,4160,4415)$ to interfere with $B^{+}\to
D^{+}X_{0,1}^{0},X_{0,1}^{0}\to D^{-}K^{+}$ Aaij:2020hon ; Aaij:2020ypa .

Apart from
$\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-}\to\Sigma_{c}X_{0,1}^{\prime}$ with
$\pi$ exchange, other $b$-baryon decays can also produce the $X_{0,1}$-like
states of $Y_{1}^{\prime\,-}(\bar{c}s\bar{n}n)$,
$Y_{1}^{-}(\bar{c}n\bar{s}n)$, and
$\bar{X}_{0,1}^{\prime\,0}(c\bar{s}d\bar{u})$ in Eq. (2). Taking the quark
level $b\to c\bar{c}s$, $c\bar{c}d$ and $c\bar{u}d$ weak decays as the
examples, the possible rescattering decays are given by

$\displaystyle b\to c\bar{c}s:\;$
$\displaystyle\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-}\to\Sigma^{+}_{c}Y_{1}^{\prime\,-}\,,\;\Xi_{b}^{0(-)}\to\Xi_{c}^{+(0)}D_{s}^{-}\to\Xi_{c}^{+(0)}(2645)Y_{1}^{\prime\,-}\;\;\;\text{(with
$\pi^{0}$ exchange)}\,,$
$\displaystyle\Xi_{b}^{0}\to\Xi_{c}^{+}D_{s}^{-}\to\Xi_{c}^{0}(2645)X_{0,1}^{\prime\,0}\,,\;\Xi_{b}^{-}\to\Xi_{c}^{0}D_{s}^{-}\to\Xi_{c}^{+}(2645)X_{0,1}^{\prime\,--}\;\;\text{(with
$\pi^{\pm}$ exchange)}\,,$ $\displaystyle b\to c\bar{c}d:\;$
$\displaystyle\Lambda_{b}\to\Lambda_{c}^{+}D^{-}\to\Xi_{c}^{\prime\,0}X_{0,1}^{0}\;\;\;\;\text{(with
$K^{\pm}$ exchange)}\,,$ $\displaystyle b\to c\bar{u}d:\;$
$\displaystyle\Lambda_{b}\to\Lambda_{c}^{+}\pi^{-}\to\Lambda^{(*)}\bar{X}_{0,1}^{\prime\,0}\,,\;\Lambda_{b}\to\Lambda_{c}^{+}\pi^{-}\to\Sigma^{(*)0}\bar{X}_{0,1}^{\prime\,0}\,,$
$\displaystyle\Xi_{b}^{0(-)}\to\Xi_{c}^{+(0)}\pi^{-}\to\Xi^{(*)0(-)}\bar{X}_{0,1}^{\prime\,0}\;\;\;\;\;\;\text{(with
$D_{s}^{\pm}$ exchange)}\,,$ (17)

where ${\bf B}^{*}$ stands for the higher-wave baryon state, and
$\Xi_{c}(2645)$ is able to decay into $\Xi_{c}\pi$. These decays are worthy of
the future explorations; particularly, we estimate that ${\cal
B}(\Lambda_{b}\to\Sigma^{+}_{c}Y_{1}^{\prime\,-})\sim 10^{-4}$, ${\cal
B}(\Xi_{b}^{0(-)}\to\Xi_{c}^{0(+)}(2645)X_{0}^{\prime\,0(--)})\sim 10^{-5}$
and ${\cal B}(\Lambda_{b}\to\Xi_{c}^{\prime\,0}X_{0,1}^{0})\sim 10^{-5}$,
which suggest interesting measurements. As the final remark, we emphasize that
our estimations rely on the observed weak decays in the triangle loop, such as
the color-allowed processes $\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-}$,
$\Xi_{b}^{0(-)}\to\Xi_{c}^{+(0)}D_{s}^{-}$ and
$\Lambda_{b}\to\Lambda_{c}^{+}D^{-}(\pi^{-})$. However, there should be other
triangle rescatterings to be discovered in the future work. For example, those
proceed through the similar color-allowed $\Lambda_{b}$ decay processes but
with $\Lambda_{c}^{+}D_{s}^{-}$ or $\Lambda_{c}^{+}D^{-}(\pi^{-})$ replaced by
other particles of the same quark contents, or those through the color-
suppressed ones, which are not necessarily suppressed in the final state
interaction. Moreover, when the open-charm exotic particles are interpreted as
the bound states Molina:2010tx , it allows for the rescatterings with the
exchanges of the vector mesons, which also deserve future investigation.

##  IV Conclusions

In summary, we have proposed the rescattering decays
$\Lambda_{b}\to\Lambda_{c}^{+}D_{s}^{-}\to\Sigma_{c}X_{0,1}^{\prime}$ with the
$\pi$ exchange, in order to provide the different decays to confirm the
$X_{0,1}$-like exotic states observed in $B^{+}\to D^{+}D^{-}K^{+}$. As the
order of magnitude estimates, we have calculated that ${\cal
B}(\Lambda_{b}\to\Sigma_{c}^{(++)}X_{0}^{\prime\,0(--)})=(2.3\pm 0.6)\times
10^{-4}$ and ${\cal
B}(\Lambda_{b}\to\Sigma_{c}^{0(++)}X_{1}^{\prime\,0(--)})=(4.3\pm
0.8^{+3.3}_{-2.5})\times 10^{-4}$. Other possible $b$-baryon decays with the
$X_{0,1}$-like exotic states have also been estimated, such as ${\cal
B}(\Lambda_{b}\to\Sigma^{+}_{c}Y_{1}^{\prime\,-})\sim 10^{-4}$, ${\cal
B}(\Xi_{b}^{0(-)}\to\Xi_{c}^{0(+)}(2645)X_{0}^{\prime\,0(--)})\sim 10^{-5}$
and ${\cal B}(\Lambda_{b}\to\Xi_{c}^{\prime\,0}X_{0,1}^{0})\sim 10^{-5}$. To
measure $X_{0,1}^{\prime\,0(--)}\to M_{c}M$ in the $M_{c}M$ invariant mass
spectrum, where $M_{c}M$ can be $D_{s}^{-}\pi^{+}$, $\bar{D}^{0}\bar{K}^{0}$
$(D_{s}^{-}\pi^{-}$, $D^{-}K^{-})$, we have introduced the three-body decay
channel $\Lambda_{b}\to\Sigma_{c}^{0(++)}M_{c}M$. The branching fraction
estimated at the level of $10^{-4}$ can be accessible to the near future
measurements.

## ACKNOWLEDGMENTS

YY would like to thank Dr. Xu-Chang Zheng for useful discussions. YKH was
supported in part by NSFC (Grant No. 11675030). YY was supported in part by
NSFC (Grant No. 11905023) and CQCSTC (Grants No. cstc2020jcyj-msxmX0555, No.
cstc2020jcyj-msxmX0810).

## References

  * (1) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 125, 242001 (2020).
  * (2) R. Aaij et al. [LHCb Collaboration], Phys. Rev. D 102, 112003 (2020).
  * (3) V.M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. 117, 022003 (2016).
  * (4) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 117, 152003 (2016); 118, 109904(A) (2017).
  * (5) A.M. Sirunyan et al. [CMS Collaboration], Phys. Rev. Lett. 120, 202005 (2018).
  * (6) T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 120, 202006 (2018).
  * (7) M. Aaboud et al. [ATLAS Collaboration], Phys. Rev. Lett. 120, 202007 (2018).
  * (8) R. Molina, T. Branz, and E. Oset, Phys. Rev. D 82, 014010 (2010).
  * (9) M. Karliner and J.L. Rosner, Phys. Rev. D 102, 094016 (2020).
  * (10) X.G. He, W. Wang and R. Zhu, Eur. Phys. J. C 80, 1026 (2020).
  * (11) T.J. Burns and E.S. Swanson, Phys. Rev. D 103, 014004 (2021).
  * (12) T.J. Burns and E.S. Swanson, Phys. Lett. B 813, 136057 (2021).
  * (13) Y.K. Chen, J.J. Han, Q.F. Lu, J.P. Wang and F.S. Yu, Eur. Phys. J. C 81, 71 (2021).
  * (14) X.H. Liu, M.J. Yan, H.W. Ke, G. Li and J.J. Xie, Eur. Phys. J. C 80, 1178 (2020).
  * (15) R. Aaij et al. [LHCb Collaboration], Sci. Bull. 66, 1278 (2021).
  * (16) R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 115, 072001 (2015).
  * (17) Q. Wu, D.Y. Chen and R. Ji, Chin. Phys. Lett. 38, 071301 (2021).
  * (18) X.H. Liu, Q. Wang and Q. Zhao, Phys. Lett. B 757, 231 (2016).
  * (19) P.A. Zyla et al. [Particle Data Group], PTEP 2020, 083C01 (2020).
  * (20) Y.K. Hsiao and C.Q. Geng, Eur. Phys. J. C 77, 714 (2017).
  * (21) Y.K. Hsiao and C.Q. Geng, Phys. Lett. B 782, 728 (2018).
  * (22) A.K. Leibovich, Z. Ligeti, I.W. Stewart and M.B. Wise, Phys. Lett. B 586, 337 (2004).
  * (23) T. Feldmann and M.W.Y. Yip, Phys. Rev. D 85, 014035 (2012); 86, 079901(E) (2012).
  * (24) Y.K. Hsiao, L. Yang, C.C. Lih and S.Y. Tsai, Eur. Phys. J. C 80, 1066 (2020).
  * (25) Y.K. Hsiao and C.Q. Geng, Phys. Rev. D 91, 116007 (2015).
  * (26) C. Albertus, E. Hernandez, J. Nieves and J. M. Verde-Velasco, Phys. Rev. D 72, 094022 (2005).
  * (27) K.U. Can, G. Erkol, M. Oka and T.T. Takahashi, Phys. Lett. B 768, 309 (2017).
  * (28) Y. Yu and Y.K. Hsiao, arXiv:2012.14575 [hep-ph].
  * (29) Y.K. Hsiao, Y. Yu and B.C. Ke, Eur. Phys. J. C 80, 895 (2020).
  * (30) G. t́ Hooft and M.J.G. Veltman, Nucl. Phys. B 153, 365 (1979).
  * (31) T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999).
  * (32) A. Denner and S. Dittmaier, Nucl. Phys. B 734, 62 (2006).
  * (33) W. Detmold, C. Lehner and S. Meinel, Phys. Rev. D 92, 034503 (2015).
  * (34) N.A. Tornqvist, Z. Phys. C 61, 525 (1994).
  * (35) X.Q. Li, D.V. Bugg and B.S. Zou, Phys. Rev. D 55, 1421 (1997).
  * (36) Q. Wu, D.Y. Chen, X.J. Fan and G. Li, Eur. Phys. J. C 79, 265 (2019).
  * (37) T.M. Yan, H.Y. Cheng, C.Y. Cheung, G.L. Lin, Y.C. Lin, and H.L. Yu, Phys. Rev. D 46, 1148 (1992); 55, 5851(E) (1997).
  * (38) H.Y. Cheng and C.K. Chua, Phys. Rev. D 92, 074014 (2015).