Dynamical System Modeling and Stability Investigation�DSMSI-2025

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May 08-10, 2025, Kyiv, Ukraine

Modal Systems with Separate Dominance of Experts

in Logics of Partial Predicates

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Oksana S. Shkilniak, Stepan S. Shkilniak

Taras Shevchenko National University of Kyiv, Ukraine�

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Introduction

Modal logics are used with great success to describe and model various subject areas and aspects of human activity.

Temporal logics: dynamic systems modeling, specification and verification of programs.

Epistemic logics: description of artificial intelligence systems, information and expert systems, databases and knowledge bases.

We propose special modal logics of the epistemic type, based on first-order logics of partial quasiary predicates. At the core of the studied logics lies the concept of a multiple-expert modal system (MEMS). To describe such systems, we use semantic models of the relational type.

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We consider systems with a finite set of intelligent experts or agents. Each of these experts can be in a certain situation (i.e., a certain state of the world).

For each expert, a binary relation is defined on the set of such states. In relational models, such relations are traditionally referred to as transition relations, or accessibility relations.

The presence of multiple experts implies that each of them may have their own opinion about a given situation. In the most general case, the opinion of one expert does not depend on the opinions of others, which complicates the process of decision making. A crucial step in coordinating such decisions is the introduction of a dominance relation on a set of experts.

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Each expert has their own accessibility relation for possible situations. In this setting, the notion of truth in a possible situation depends on the dominance relation between experts and may vary for different experts.

• Multiple-expert modal models of propositional level with dominance relations between experts were introduced in the work [M. Fitting. Many-Valued Modal Logics II (1992)]. Such systems were studied basing on relational models of propositional intuitionistic logic.
• Interactions between multiple intelligent agents were investigated, in particular, in [H. Rasiowa, W. Marek. On Reaching Consensus by Groups of Intelligent Agents (1989)]. The logical system described in the work uses an ordering of agents based on the sharpness of their perception. This ordering is essentially opposite to those considered in the work above and in the present one.

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Preliminary Concepts

The focus is on the semantic aspects of multiple-expert systems.

We remain within the framework of traditional {true, false}-valued logic and consider pure first-order logics of partial single-valued quasiary predicates.

We assume that truth values should be preserved under dominance: both true and false must be preserved, i.e.

• if expert f considers a statement Q to be true and dominates expert e, then e must also consider Q to be true;
• if f considers Q to be false, then e must also consider Q to be false.

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Such a dominance relation is naturally assumed to be transitive: if expert a dominates b, and b dominates c, then a dominates c.

But what if a single expert is dominated by two or more experts?

Inconsistencies may arise: one dominant expert may require Q to be true, while another may require Q to be false!

To avoid such issues, we will consider the concept of separate dominance, meaning that no more than one expert can directly dominate another.

Dynamical System Modeling and Stability Investigation, DSMSI-2025

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Multiple-Expert Modal Systems (MEMS)

MEMS is the object M = (Ex, , St, , Pr, Cm), Fr, I), where

1. Ex ≠ ∅ is a finite set (group) of experts.

2.  ⊆ Ex × Ex is a relation of immediate dominance on experts; e  g abbreviates  (e, g).

3. St ≠ ∅ is a set of possible states of the world, or situations.

4.  ⊆ Ex × St × St is an accessibility relation between possible states (situations), which depends on the expert being considered;

α _e β abbreviates (e, α, β)∈ .

5. St is specified as a set of algebraic structures α = (A_α, Pr_α), where A_α is a set of basic data of the state α, Pr_α is a set of quasiary predicates ^VA_α → {T, F} (called predicates of the state α).

is a set of basic data of the system M.

The predicates ^VA → {T, F} are called global.

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6. Cm is a set of compositions on Pr: ¬, ∨, ∃x; ⬜.

7. Fr is a set of formulas of the MEMS language.

8. I : Ps × Ex × St → {T, F} is an interpretation mapping for atomic formulas.

MEMS can be shorter denoted by M = (Ex, , St, , A, I).

We will consider the case of injective relation  ⊆ Ex × Ex, i.e., for each e∈Ex, there is at most one g∈Ex such that g  e: no more than one expert may immediately dominate a given expert – two or more experts cannot simultaneously immediately dominate a single expert.

The reflexive-transitive closure of  will be called the separate dominance relation and denoted by .

e  g means that the decision of the dominant expert e must be accepted by every subdominant to e expert g.

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The injective relation  ⊆ Ex × Ex can be represented as a forest of trees with vertices from Ex.

Example. Let Ex = {e, g, a, b, c, d, h, p, q, s, m, n};

e  a, e  b, e  c, g  d, g  h, a  p, a  q, d  m, d  n.

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The accessibility relation on states of the world  ⊆ Ex × St × St is connected to the dominance relation in the following way: α _e β аnd e  g ⇒ α _g β.

This means: if expert e considers that state β is accessible from state α (i.e. situation β is possible with respect to situation α), then any expert g over whom e dominates must also accept this.

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Dynamical System Modeling and Stability Investigation, DSMSI-2025

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Languages of pure first-order MEMS

The alphabet:

V – a set of variables;

Ps – a set of predicate symbols;

– a set of basic logical composions’ symbols;

Ms = {⬜} – a set of basic modal compositions.

The definition of the set Fm of formulas of the language:

Fa) Ps ⊆ Fm;

Fр) Φ∈Fm ⇒ ¬Φ∈Fm; Φ, Ψ∈Fm ⇒ ∨ΦΨ∈Fm;

F_R) Φ∈Fm ⇒

F_∃) Φ∈Fm ⇒ ∃xΦ∈Fm;

F_◻) Φ∈Fm ⇒ ⬜Φ∈Fm.

Formulas of the form р∈Ps will be called atomic.

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Formulas that contain symbols ⬜ of modal compositions are called modalized.

Formulas that do not contain modal composition symbols are called non-modalized.

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A language L of pure first-order MEMS is based on a language L_B of pure first-order logic of quasiary predicates.

In defining the set of formulas of this basic logical language L_B, we omit the clause F_◻.

The sets of non-modalized formulas of the MEMS language L and the set of formulas of its basic logical language L_B coincide.

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The interpretation mapping for atomic formulas on states of the world and experts is defined as:

I : Ps × Ex × St → PrP^V_A.

Here, PrP^V_A is the class of partial single-valued quasiary predicates V-A-quasiary predicates (P-predicates).

The interpretation mapping is extended for formulas as follows.

I¬) I(¬Φ, e, α) = ¬(I(Φ, e,  α).

I∨) I(∨ΦΨ, e, α) = ∨(I(Φ, e,  α), I(Φ, e,  α)).

IR)

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I∃) I(∃xΦ, e, α)(d) =

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I⬜) I(⬜Φ, e, α)(d)=

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I(Φ, e, α) is a predicate which is the value of a formula Φ for expert e in state α; it is denoted by

If for α∈St and e∈Ex there doesn’t exist such β that

then for every d∈^VA, we have I(⬜Φ, e, α)(d)⭡.

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The compositions ∀x and ◊ are treated as derived.

The formulas ¬∃x¬Φ and ¬⬜¬Φ are abbreviated as ∀xΦ and ◊Φ.

For ∀xΦ and ◊Φ, the interpretation mapping is defined as follows.

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I(∀xΦ, e, α)(d) =

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I(◊Φ, e, α)(d) =

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Dynamical System Modeling and Stability Investigation, DSMSI-2025

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Preservation of Truth Values under Dominance

The preservation of truth values under dominance is the defining property of MEMS of quasiary predicates: a decision made by an expert must also be accepted by every expert whom they dominate.

For formula predicates, the following condition must hold:

if e  g, then the value of must

induce the corresponding value (D)

For I(⬜Φ, e, α) = T, this means: expert e believes that Φ(d) is necessarily true in situation α, if every expert g whom e dominates, believes that Φ(d) is true in every situation δ that is possible with respect to α (i.e., in every state δ accessible from α).

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For I(⬜Φ, e, α) = F, this means: expert e believes that necessary truth of Φ(d) in situation α is refuted if they can refute Φ(d) in some situation δ that is possible with respect to α.

Therefore, property (D) can be formulated more precisely:

if e  g then we have (

and ). (D1)

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Now, it remains to specify the value of under the condition

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For the truth and falsity domains of predicates, (D1) can be presented as follows:

if e  g then we have (

and ) (D2)

Hence,

Therefore,

This means: moving from expert e to expert g such that e  g, reduces the uncertainty of the statement.

Under the condition e  g and in addition to the possibility of there are also possibilities of and

This means: subdominant expert g can freely choose the value of Φ(d) in situation α, if dominant expert e didn’t provide it.

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The preservation of truth values under dominance can be specified as the following Dom-P property :

1) if e  g and

2) if e  g and

3) if e  g and

the possible values are

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The “P” in the name of the property indicates that we are referring to the logic of partial single-valued quasiary predicates, or P-predicates.

Dynamical System Modeling and Stability Investigation, DSMSI-2025

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Initial Semantic Properties of MEMS

The Dom-P property can be formulated for atomic formulas first, and then extended to all formulas of the language:

Theorem 1. Let the Dom-P property hold for all Φ∈Ps, α∈St, d∈^VA_α and e, g∈Ex, under the condition e  g. Then the Dom-P property holds for all Φ∈Fm.

(The proof is by induction on the construction of the formula.)

Comparing the definitions of I(⬜Φ, e, α) = T and I(⬜Φ, e, α) = F, one may notice a certain difference in style.

A question arises: could it be possible to remove a reference to experts in and make it similar in form to the simpler definition of

The answer is no: this may violate the Dom-P property.

Dynamical System Modeling and Stability Investigation, DSMSI-2025

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On a set of formulas of MEMS, we introduce the following definitions.

Formula Φ is true (irrefutable) for expert e on state α (denoted α^e |= Φ), if is an irrefutable predicate.

Formula Φ is true (irrefutable) on state α (denoted α |= Φ), if α^e |= Φ for all e∈Ex.

Formula Φ is true (irrefutable) for expert e (denoted ^e |= Φ), if α^e |= Φ for all α∈St.

Formula Φ is true (irrefutable) in MEMS M (denoted M |= Φ), if α^e |= Φ for all α∈St and e∈Ex.

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M |= Φ ⇔ α |= Φ for all α∈St ⇔ ^e |= Φ for all e∈Ex.

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Let  be a class of MEMS of a certain type.

Formula Φ is -irrefutable (denoted ^ |= Φ), if M |= Φ for all MEMS M∈.

If  is given by default, we will simply say that Φ is MEMS-irrefutable and denote the fact by |= Φ.

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Theorem 2. Let L be a language of MEMS, L_B be a basic logical language for L, Φ be a non-modalized formula of the language L. Then

• is MEMS-irrefutable ⇔

Φ is an irrefutable formula of the basic language L_B.

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For MEMS, the general condition of definedness on states will be used: if Φ is a non-modalized formula, then under the condition d∉^VA_δ, we assume

Here, d_δ denotes a nominative set [va∈d | a∈A_δ ].

Informally, this means that predicates of state δ “perceive” only those components va which have basic data a∈A_δ.

For MEMS, the properties related to the interaction of modal compositions with renominations and quantifiers hold and coincide with those for TMS.

Logical consequence relations in MEMS will be defined on a set of formulas specified with both experts’ and states’ names (or, simply, with experts and states). This can be done similarly to the definitions of logical consequence relations on a set of formulas specified with states in TMS languages.

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Conclusion

Special modal logics of the epistemic type, which are based on pure first-order logics of partial quasiary predicates have been studied. At the foundation of the investigated logics lies the concept of a multiple-expert modal system (MEMS).

We propose modal systems with a relation of separate dominance defined on a set of experts, meaning no expert can be directly dominated by more than one other expert. Unlike known similar systems, our approach requires that both true and false values are preserved under dominance.

The language of pure first-order MEMS has been described. Special attention has been given to the features of the separate dominance relation. Conditions for truth value preservation under dominance have been formulated for atomic formulas and proven to extend to all formulas of the language.

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Thank you for your attention

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