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Context Localization for Generalized Level-Based Evaluation in Knowledge-Based Systems
Ondrej Hutník, Natália Puškárová
Intelligence
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Summary
This paper investigates the mathematical conditions under which filtering a score function by a specific context B is equivalent to localizing admissible knowledge contexts by intersecting them with B in generalized level-based evaluation systems. The authors identify two necessary and sufficient structural conditions for this consistency: monotonicity with respect to contexts (enlarging the conditioning set does not increase the local evaluation value) and a reduction property (sets not contained in B cannot produce positive localized support after masking). The study extends these findings to parameterized systems and interprets them as stability criteria for evidence selection and non-additive support evaluation.
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Ondrej Hutník → affiliatedwith → Pavol Jozef Šafárik University in Košice
confidence 99% · Ondrej Hutník Address: Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice
Natália Puškárová → affiliatedwith → Pavol Jozef Šafárik University in Košice
confidence 99% · Natália Puškárová Address: Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice
Context Localization → requires → Monotonicity with respect to contexts
confidence 95% · The main theorem shows that this consistency holds... if and only if two structural conditions are satisfied: monotonicity with respect to contexts...
Context Localization → requires → Reduction Property
confidence 95% · The main theorem shows that this consistency holds... if and only if two structural conditions are satisfied: ... and a reduction property excluding positive localized support outside B.
Generalized Level Measure → utilizes → Conditional Aggregation Operator
confidence 90% · The framework models situations where a structured nonnegative score... is evaluated through conditional aggregation tests... The generalized level measure maximizes a monotone set function...
Generalized Level Measure → utilizes → Monotone Measure
confidence 90% · The generalized level measure maximizes a monotone set function over all contexts whose aggregated support reaches a prescribed level.
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Abstract
Abstract:We study context localization for generalized level-based evaluation in knowledge-based systems. The framework models situations where a structured nonnegative score, defined on facts, rules, cases, criteria or evidence units, is evaluated through conditional aggregation tests on admissible knowledge contexts. The generalized level measure maximizes a monotone set function over all contexts whose aggregated support reaches a prescribed level. We characterize when filtering the score by a context $B$ is equivalent to localizing the admissible contexts by intersection with $B$. The main theorem shows that this consistency holds for all monotone set functions if and only if two structural conditions are satisfied: monotonicity with respect to contexts and a reduction property excluding positive localized support outside $B$. We analyze pointwise and block-generated mechanisms producing the reduction property, extend the result to parameterized systems, and interpret it as a stability criterion for context-dependent evidence selection, non-additive support evaluation and level-based knowledge aggregation.
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- Source: https://arxiv.org/abs/2608.27482v1
- Canonical: https://arxiv.org/abs/2608.27482v1
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Context Localization for Generalized Level-Based Evaluation in Knowledge-Based SystemsThanks: Author preprint. This manuscript is not the publisher’s version of record. The peer-reviewed version has been published in Knowledge-Based Systems; see the published article on ScienceDirect. Ondrej Hutník Address: Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01 Košice, Slovakia Email address: ondrej.hutnik@upjs.sk and Natália Puškárová Address: Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01 Košice, Slovakia Email address: natalia.puskarova@student.upjs.sk Abstract. We study context localization for generalized level-based evaluation in knowledge-based systems. The framework models situations where a structured nonnegative score, defined on facts, rules, cases, criteria or evidence units, is evaluated through conditional aggregation tests on admissible knowledge contexts. The generalized level measure maximizes a monotone set function over all contexts whose aggregated support reaches a prescribed level. We characterize when filtering the score by a context B is equivalent to localizing the admissible contexts by intersection with B. The main theorem shows that this consistency holds for all monotone set functions if and only if two structural conditions are satisfied: monotonicity with respect to contexts and a reduction property excluding positive localized support outside B. We analyze pointwise and block-generated mechanisms producing the reduction property, extend the result to parameterized systems, and interpret it as a stability criterion for context-dependent evidence selection, non-additive support evaluation and level-based knowledge aggregation. Key words and phrases: knowledge-based systems, context localization, generalized level measure, conditional aggregation operator, monotone measure, evidence selection, structured uncertainty 1. Motivation: context localization in knowledge-based evaluation In many knowledge-based systems, an output score is not obtained from a single scalar value but from a structured collection of facts, rules, cases, criteria, expert statements, documents or evidence units. Such units are rarely evaluated in isolation. Their relevance depends on the active query, decision context, user profile, diagnostic regime, information source, semantic constraint or observation window. The resulting system-level evaluation should therefore depend not only on the local scores assigned to the units, but also on which subsets of the knowledge base are admissible as coherent contexts for evaluation. This viewpoint is compatible with fuzzy, non-additive and approximate-reasoning models, where uncertain or imprecise requirements are evaluated through nonlinear satisfaction and aggregation mechanisms; see, for instance, [1, 12]. The localization problem considered in this paper can be stated informally as follows. If only a part B of the knowledge base is relevant for the current task, one may either first filter the score function outside B or, alternatively, evaluate admissible contexts in the original system and then localize their informational weight by intersection with B. These two procedures need not be equivalent for nonlinear conditional evaluation rules. Our goal is to characterize exactly when this equivalence holds. The classical level-set construction already contains the basic localization principle in its simplest form. Let f:X→[0,∞)f X→[0,∞) be a nonnegative measurable score function and let B∈ΣB∈ be the part of the domain that is active for the current task. For every u>0u>0, x∈X:f(x)B(x)⩾u=x∈X:f(x)⩾u∩B.\x∈ X:f(x) 1_B(x) u\=\x∈ X:f(x) u\∩ B. Consequently, for every monotone measure μ on the ambient measurable space, μ(x∈X:f(x)B(x)⩾u)=μ(x∈X:f(x)⩾u∩B),u>0.μ (\x∈ X:f(x) 1_B(x) u\ )=μ (\x∈ X:f(x) u\∩ B ), u>0. Thus, in the ordinary level-based setting, filtering the score outside B before evaluation is equivalent to localizing the resulting level set by intersection with B. The aim of the present paper is to identify when this elementary localization principle remains valid for generalized level measures generated by conditional aggregation operators. Let =(⋅|E):E∈ℰ A=\ A(·|E):E∈E\ be a family of conditional aggregation operators in the sense of [2], and let μ be a monotone measure on ℰE. In a knowledge-based interpretation, the sets E∈ℰE∈E are admissible knowledge contexts, (f|E) A(f|E) is the aggregated degree of support of the score function f inside E, and μ(E)μ(E) is a non-additive informational weight of the context. Following the generalized level-measure construction from [3], for u⩾0u 0, define Λμ,(f,u):=supμ(E):(f|E)⩾u,E∈ℰ. _μ, A(f,u):= \μ(E): A(f|E) u,E∈E\. This quantity may be interpreted as a level-constrained evidence selection value: at level u, one searches for admissible contexts E on which the score passes the conditional aggregation test, and then takes the largest support, confidence or relevance mass μ(E)μ(E). The localized counterpart used in the paper is not obtained by simply replacing ℰE with the subcollection of sets contained in B. Instead, each admissible set E is localized by intersection with B, i.e., Λμ,(f,u;B):=supμ(E∩B):(f|E)⩾u,E∈ℰ. _μ, A(f,u;B):= \μ(E∩ B): A(f|E) u,E∈E\. This is closer to the classical level-set identity, where the level set f⩾u\f u\ is replaced by its intersection with B. The central localization problem is therefore to determine when Λμ,(fB,u)=Λμ,(f,u,B),u>0. _μ, A(f 1_B,u)= _μ, A(f,u;B), u>0. In words, we ask when filtering the score outside B is equivalent to localizing the admissible knowledge contexts appearing in the generalized level measure. This equality is automatic for the ordinary level measure, but it is not automatic for arbitrary conditional aggregation rules. Averaging-type tests, for instance, generally do not have this property, since zeros introduced outside B may still affect an average computed on a set not contained in B. In contrast, infimum-type, essential-infimum-type, conjunctive, or suitable block-generated tests may possess a reduction mechanism: after multiplication by B 1_B, a set not contained in B cannot produce a positive level value. This is the role of the reduction property studied below. A second structural requirement is also needed. Even if the masked signal eliminates sets outside B, one must still compare the value of the conditional aggregation test on a set E with its value on the localized part E∩BE∩ B. This is controlled by monotonicity with respect to sets: enlarging the conditioning set should not increase the local evaluation value. The paper therefore first separates these two mechanisms. We discuss monotonicity with respect to sets in Section 3.1, then the reduction property is investigated in Section 3.2, and only afterwards combine them in the localization theorem Section 4. A parameterized version is natural in knowledge-based systems because the active evaluation structure may depend on a query, decision context, user profile, resolution level or admissibility regime. Changing this external parameter may change the admissible collection, the monotone measure, or the conditional aggregation rule. More generally, one may consider a parameterized system t=(ℰt,μt,t),t∈T, S_t=(E_t, _t, A_t), t∈ T, and the corresponding two-variable generalized level surface Λ(f;t,u)=supμt(E):t(f|E)⩾u,E∈ℰt, _ S(f;t,u)= \ _t(E): A_t(f|E) u,E∈E_t\, cf. [3]. Here t is the external parameter selecting the information structure and the local evaluation mechanism, while u is the level threshold used to test the aggregated support. Keeping these two variables separate is essential: diagonal expressions such as u=tu=t may be useful later, but they do not contain enough information to recover the full structural conditions behind localization. Thus, the motivating question can be summarized as follows. Given a collection ℰE of admissible sets and a family A of conditional aggregation rules, which structural conditions ensure that, for every f∈f∈F, every localizing set B∈ℰ0B∈E^0, every positive level u>0u>0, and every monotone measure μ on ℰE, the identity Λμ,(fB,u)=Λμ,(f,u,B) _μ, A(f 1_B,u)= _μ, A(f,u;B) holds? The answer given in the paper is that this identity is governed precisely by the combination of monotonicity with respect to sets and the reduction property. The parameterized version presented in Section 5 then applies the same principle pointwise in the external parameter and provides the basis for later interpretations in terms of level-dependent evaluation curves, stochastic processes, and level-dependent integral functionals. 2. Preliminaries We start with a nonempty set X. Whenever measurability is relevant, we fix an ambient σ-algebra Σ⊆2X 2^X. The role of Σ is only to specify which functions and sets are measurable. It is assumed to be sufficiently rich for the collections considered below. Throughout the paper, the only standing assumption on ℰ⊆ΣE is that it is a paving, that is, ∅∈ℰ. ∈E. We write ℰ0:=ℰ∖∅E^0:=E \ \. Additional structural assumptions on ℰE, such as closure under intersections, will be imposed explicitly when needed. The distinction between Σ and ℰE is intentional. The σ-algebra Σ is the ambient measurable structure, whereas ℰE is the working collection of admissible sets used for conditioning and for the suprema defining generalized level measures. Thus, Σ may be much larger than ℰE. For instance, ℰE may consist only of unions of blocks of a partition, events observable at a fixed time, scenario classes, or level-dependent feasible sets. In purely order-theoretic or finite discrete situations, we shall usually take Σ=2X =2^X. Hence, statements involving ℰ=2XE=2^X are understood in the discrete measurable setting, where all subsets of X are measurable. The class of all nonnegative bounded Σ -measurable functions on X is denoted by F. For f,g∈f,g∈F, we write f⩽gf g on X if f(x)⩽g(x)f(x) g(x) for every x∈Xx∈ X. The indicator function of a set B∈ΣB∈ is denoted by B 1_B, where B(x)=1,if x∈B,0,otherwise. 1_B(x)= cases1,&if x∈ B,\\ 0,&otherwise. cases Since ℰ⊆ΣE , for every f∈f∈F and every B∈ℰB∈E the masked function fBf 1_B again belongs to F. By a monotone measure on a collection ℰE we mean a nondecreasing set function μ:ℰ→[0,∞]μ E→[0,∞] such that μ(∅)=0μ( )=0 and μ(E)>0μ(E)>0 for at least one E∈ℰE∈E. Thus, whenever C,D∈ℰC,D∈E and C⊆DC D, one has μ(C)⩽μ(D)μ(C) μ(D). If X∈ℰX∈E, one may replace the last nontriviality condition by μ(X)>0μ(X)>0. For background on monotone measures, capacities and non-additive set functions, see [5, 9, 11]. The conditional aggregation tests used in this paper are represented by conditional aggregation operators. Such operators can be viewed as local nonlinear evaluation rules; the specific notion used below was introduced in [2]. For general background on nonlinear aggregation of several inputs, see [6]. Definition 2.1. Let ℰ⊆ΣE be a collection. A family =(⋅|E):E∈ℰ A=\ A(·|E):E∈E\ is called a family of conditional aggregation operators, briefly an FCA, if for every E∈ℰ0E∈E^0 the mapping (⋅|E):→[0,∞] A(·|E) F→[0,∞] satisfies: (C1) (f|E)⩽(g|E) A(f|E) A(g|E) for all f,g∈f,g∈F such that f⩽gf g on E; (C2) (Ec|E)=0 A( 1_E^c|E)=0, where Ec=X∖E^c=X E. For the empty set, we use the convention (⋅|∅)=+∞ A(·| )=+∞. With this convention, ∅ may be included in level suprema without causing exceptional cases: it always satisfies the level inequality, but its monotone measure is μ(∅)=0μ( )=0. Although the family A depends on ℰ,E, we will not indicate it in our notation when there is no doubt of confusion. The monotonicity axiom (C1) immediately implies the following locality property, which will be used repeatedly below. Lemma 2.2. If f=gf=g on E∈ℰE∈E, then (f|E)=(g|E) A(f|E)= A(g|E). Proof. For E≠∅E≠ , both f⩽gf g and g⩽fg f hold on E, and the assertion follows from (C1). For E=∅E= it follows from the convention above. ∎ 3. Structural conditions for localization The purpose of this section is to isolate two structural properties of the conditional aggregation rules which will be used later in the study of localization. The first property is monotonicity with respect to sets. It requires that enlarging the conditioning set cannot increase the corresponding local level value. This condition is natural for infimum-type, conjunctive, and worst-case evaluation mechanisms, but it is not automatic for arbitrary nonconstant families. The second property is a reduction property. It describes what happens when a signal is localized by multiplication with B 1_B. Informally, it requires that after such a localization, no set which is not contained in B can still produce a positive level value. This property may follow from a pointwise annihilator, but it may also be forced by the structure of the underlying collection itself. We discuss these two properties separately. This makes it possible to distinguish the order-theoretic mechanism, encoded by monotonicity with respect to sets, from the localization mechanism, encoded by the reduction property. 3.1. Monotonicity with respect to sets Definition 3.1. Let ℰ⊆ΣE be a collection. We say that an FCA =(⋅|E):E∈ℰ A=\ A(·|E):E∈E\ is nonincreasing with respect to sets if C,D∈ℰ0,C⊆D⟹(f|D)⩽(f|C)C,D∈E^0,\,\,C D A(f|D) A(f|C) (3.1) for every f∈f∈F. Remark 3.2. Although the empty set does not cause a formal difficulty in the definition above, due to the convention (⋅|∅)=∞ A(·| )=∞, we shall not regard it as relevant when discussing monotonicity with respect to sets. Thus, whenever structural properties of an FCA are considered, the substantive comparisons are those between nonempty sets. This convention avoids degenerate or pathological interpretations involving ∅ and keeps the order-theoretic conditions tied to genuine conditioning sets. A trivial way to obtain such a family is to use the same conditional aggregation rule on all sets. For instance, if inf(f|E):=infx∈Ef(x),E∈ℰ0, A (f|E):= _x∈ Ef(x), E∈E^0, then inf=inf(⋅|E):E∈ℰ A =\ A (·|E):E∈E\ is nonincreasing with respect to sets. However, the conditional aggregation rules assigned to different sets from the collection need not be the same. What is required is their compatibility with the inclusion ordering of the underlying sets. The following example shows that it is not sufficient to know that the individual conditional aggregation rules are, in isolation, nonincreasing under enlargement of their domains. The way in which different rules are assigned to different sets also matters. Example 3.3. Let X=1,2,3X=\1,2,3\, let ℰ0=1,2,1,2,3E^0=\\1,2\,\1,2,3\\, and consider functions f:X→[0,1]f X→[0,1]. Define a family A by (f|1,2,3) A(f|\1,2,3\) =minf(1),f(2),f(3), = \f(1),f(2),f(3)\, (f|1,2) A(f|\1,2\) =W(f(1),f(2)), =W(f(1),f(2)), where WW is the Lukasiewicz t-norm, W(a,b)=maxa+b−1,0.W(a,b)= \a+b-1,0\. Although both the minimum t-norm and the Lukasiewicz t-norm are nonincreasing with respect to the addition of further arguments, the mixed family A is not nonincreasing with respect to sets. Indeed, for f=(0.9,0.9,0.85)f=(0.9,0.9,0.85) we have 1,2⊆1,2,3\1,2\ \1,2,3\, but (f|1,2,3)=0.85>0.8=(f|1,2). A(f|\1,2,3\)=0.85>0.8= A(f|\1,2\). Thus, the implication (3.1) fails. The example indicates the correct principle. If larger sets are assigned stronger, that is, pointwise smaller, local evaluation mechanisms, then the family becomes nonincreasing with respect to sets. This can be made precise for the following ordered class of conjunctive rules, see [8]. Let T be a class of triangular norms on [0,1][0,1]. We write T1⪯T2T_1 T_2 if T1(a,b)⩽T2(a,b)T_1(a,b) T_2(a,b) for all a,b∈[0,1]a,b∈[0,1]. For a t-norm T, let T(n)T^(n) denote its standard n-ary extension, defined recursively by T(1)(x1)=x1,T(n+1)(x1,…,xn+1)=T(T(n)(x1,…,xn),xn+1).T^(1)(x_1)=x_1, T^(n+1)(x_1,…,x_n+1)=T (T^(n)(x_1,…,x_n),x_n+1 ). By associativity and commutativity, the value of T(n)T^(n) is independent of the order of the arguments. Proposition 3.4. Let X be finite, let ℰ⊆2XE 2^X be a paving, and let τ:ℰ0→τ E^0→ T be an assignment of t-norms to nonempty sets such that C⊆D⟹τ(D)⪯τ(C)C D τ(D) τ(C) for all C,D∈ℰ0C,D∈E^0. For E∈ℰ0E∈E^0, put τ(f|E):=τ(E)(|E|)((f(x))x∈E),f:X→[0,1]. A_τ(f|E):=τ(E)^(|E|) ((f(x))_x∈ E ), f X→[0,1]. Then τ:=τ(⋅|E):E∈ℰ A_τ:=\ A_τ(·|E):E∈E\ is nonincreasing with respect to sets. Proof. Let C,D∈ℰ0C,D∈E^0 with C⊆DC D. Put TD=τ(D)T_D=τ(D) and TC=τ(C)T_C=τ(C). By assumption, TD⪯TCT_D T_C. This order is inherited by the finite extensions of the same arity: for every m∈ℕm and every (x1,…,xm)∈[0,1]m(x_1,…,x_m)∈[0,1]^m, TD(m)(x1,…,xm)⩽TC(m)(x1,…,xm).T_D^(m)(x_1,…,x_m) T_C^(m)(x_1,…,x_m). Indeed, this follows by induction on m from the recursive definition of the finite extensions and the monotonicity of TCT_C. Applying this with m=|D|m=|D| gives TD(|D|)((f(x))x∈D)⩽TC(|D|)((f(x))x∈D).T_D^(|D|) ((f(x))_x∈ D ) T_C^(|D|) ((f(x))_x∈ D ). Since C⊆DC D and every t-norm satisfies T(a,b)⩽aT(a,b) a and T(a,b)⩽bT(a,b) b, adding further arguments cannot increase the value of the finite extension. Hence TC(|D|)((f(x))x∈D)⩽TC(|C|)((f(x))x∈C).T_C^(|D|) ((f(x))_x∈ D ) T_C^(|C|) ((f(x))_x∈ C ). Combining the two inequalities yields τ(f|D)⩽τ(f|C). A_τ(f|D) A_τ(f|C). Since f was arbitrary, τ A_τ is nonincreasing with respect to sets. ∎ Example 3.5. On the standard scale of t-norms, one has the pointwise ordering D⪯W⪯Π⪯M,D , where DD, WW, Π , and MM denote, respectively, the drastic, Lukasiewicz, product, and minimum t-norms, cf. [8]. Thus, the assignment E↦τ(E)E τ(E) has to reverse the inclusion order of the admissible sets: whenever C⊆DC D, the rule assigned to D must be pointwise smaller than the rule assigned to C. For instance, if C⊆DC D, then τ(D)=W⪯M=τ(C)τ(D)=W =τ(C), and consequently τ(f|D)⩽τ(f|C) A_τ(f|D) A_τ(f|C) for every f:X→[0,1]f X→[0,1]. The preceding construction can be stated more generally. Suppose that we have a collection ℬ=α:α∈Iwithα=α(⋅|E):E∈ℰ0, B=\ A^α:α∈ I\\,\,\, with\,\,\, A^α=\ A^α(·|E):E∈E^0\, where (I,⩽I)(I, _I) is a partially ordered set. Assume that each α A^α is nonincreasing with respect to sets and that the collection ℬ B is ordered pointwise in the sense: for all α,β∈Iα,β∈ I, α⩽Iβ⟹α(f|E)⩽β(f|E)α _Iβ A^α(f|E) A^β(f|E) for every E∈ℰ0E∈E^0 and every f∈f∈F. Proposition 3.6. Let (I,⩽I)(I, _I) be a partially ordered set and let ℬ=α:α∈I B=\ A^α:α∈ I\ be a pointwise ordered collection of local evaluation families, each nonincreasing with respect to sets. Let γ:ℰ0→Iγ E^0→ I be a nonincreasing selector, that is, C⊆D⟹γ(D)⩽Iγ(C).C D γ(D) _Iγ(C). Define γ(f|E):=γ(E)(f|E),E∈ℰ0. A_γ(f|E):= A^γ(E)(f|E), E∈E^0. Then γ=γ(⋅|E):E∈ℰ A_γ=\ A_γ(·|E):E∈E\ is nonincreasing with respect to sets. Proof. Let C,D∈ℰ0C,D∈E^0 with C⊆DC D. Since γ is nonincreasing, γ(D)⩽Iγ(C)γ(D) _Iγ(C). By the pointwise ordering, γ(D)(f|D)⩽γ(C)(f|D). A^γ(D)(f|D) A^γ(C)(f|D). Since γ(C) A^γ(C) is nonincreasing with respect to sets, γ(C)(f|D)⩽γ(C)(f|C). A^γ(C)(f|D) A^γ(C)(f|C). Therefore, γ(f|D)⩽γ(f|C) A_γ(f|D) A_γ(f|C), which proves the claim. ∎ Remark 3.7. Proposition 3.6 gives a general selection principle for constructing nonconstant families of conditional aggregation rules. One may start with a collection of families which are ordered pointwise by the parameter order ⩽I _I and which are already known to be nonincreasing with respect to sets. Then, for each set E, one selects one family from this collection. The resulting family remains nonincreasing with respect to sets provided that the selector is compatible with inclusion in the order-reversing sense: larger sets must be assigned pointwise smaller local mechanisms. 3.2. Reduction property and its structural mechanisms We now discuss the reduction property used later in the localization theorem. Its role is to ensure that, after a signal is masked by B 1_B, no set outside B can still produce a positive level value. We also record several structural mechanisms which imply this property. As in the case of monotonicity with respect to sets, the empty set does not play a substantive role. Indeed, ∅⊆B B for every B∈ℰB∈E, so the implication defining the reduction property is vacuous for E=∅E= . We therefore formulate the condition only for nonempty conditioning sets, where the elimination mechanism is nontrivial. Definition 3.8. Let ℰ⊆ΣE be a collection. We say that an FCA =(⋅|E):E∈ℰ A=\ A(·|E):E∈E\ has the reduction property, or simply property (R), if, for every f∈f∈F and every B,E∈ℰ0B,E∈E^0, E⊈B⟹(fB|E)=0.E B A(f 1_B|E)=0. Equivalently, (fB|E)>0⟹E⊆B. A(f 1_B|E)>0 E B. Thus, property (R) says that, after the function has been masked by B 1_B, a strictly positive aggregated value can occur only on sets contained in B. In the language of localized level functions, masking the signal outside B automatically removes all sets which are not contained in B. Lemma 3.9. If A has property (R), then, for all f∈f∈F, B,E∈ℰ0B,E∈E^0, and u>0u>0, (fB|E)⩾u⟹E⊆B. A(f 1_B|E) u E B. Proof. If E⊈BE B, then property (R) gives (fB|E)=0 A(f 1_B|E)=0, which is impossible when (fB|E)⩾u>0 A(f 1_B|E) u>0. ∎ The pointwise annihilator mechanism The simplest way to guarantee property (R) is to require that a single zero value inside the conditioning set annihilates the local level value. Definition 3.10. We say that 00 is a pointwise annihilator of an FCA A if, for every E∈ℰ0E∈E^0 and every f∈f∈F, (∃x∈E:f(x)=0)⟹(f|E)=0. (∃ x∈ E:\ f(x)=0 ) A(f|E)=0. Proposition 3.11. If 00 is a pointwise annihilator of A, then A has property (R). Proof. Let f∈f∈F and B,E∈ℰ0B,E∈E^0. Suppose that E⊈BE B. Then there exists x∈E∖Bx∈ E B, and hence (fB)(x)=0(f 1_B)(x)=0. By the pointwise annihilator property, (fB|E)=0 A(f 1_B|E)=0. Thus, A has property (R). ∎ Example 3.12. Assume that X is finite. The conditional aggregation rules min,p(f|E)=(minx∈Ef(x))p,p>0, A ,p(f|E)= ( _x∈ Ef(x) )^p, p>0, have 00 as a pointwise annihilator. Consequently, the corresponding FCA has property (R). On the scale [0,1][0,1], the same observation applies to conditional aggregation rules induced by triangular norms. If T is a t-norm and T(|E|)T^(|E|) denotes its standard finite extension, then T(f|E)=T(|E|)((f(x))x∈E),f:X→[0,1], A^T(f|E)=T^(|E|) ((f(x))_x∈ E ), f X→[0,1], has 00 as a pointwise annihilator, since every t-norm satisfies T(a,0)=0T(a,0)=0. In particular, for E=i1,…,imE=\i_1,…,i_m\, Π(f|E)=∏ℓ=1mf(iℓ),W(f|E)=max∑ℓ=1mf(iℓ)−(m−1),0 A (f|E)= _ =1^mf(i_ ), A^W(f|E)= \ _ =1^mf(i_ )-(m-1),0 \ both satisfy property (R). The pointwise annihilator can also be imposed on an arbitrary FCA by a simple filtering construction. Proposition 3.13. Let X be finite and let 0=0(⋅|E):E∈ℰ A^0=\ A^0(·|E):E∈E\ be an FCA. For E∈ℰ0E∈E^0, define (f|E)=0(f|E),if f(x)>0 for every x∈E,0,otherwise. A(f|E)= cases A^0(f|E),& if f(x)>0 for every x∈ E,\\ 0,&otherwise. cases Then =(⋅|E):E∈ℰ A=\ A(·|E):E∈E\ is an FCA, 00 is a pointwise annihilator of A, and hence A has property (R). Proof. We verify (C1). Let f,g∈f,g∈F, E∈ℰ0E∈E^0 and assume that f⩽gf g on E. If f(x)=0f(x)=0 for some x∈Ex∈ E, then (f|E)=0⩽(g|E) A(f|E)=0 A(g|E). If f(x)>0f(x)>0 for every x∈Ex∈ E, then also g(x)>0g(x)>0 for every x∈Ex∈ E, and therefore (f|E)=0(f|E)⩽0(g|E)=(g|E), A(f|E)= A^0(f|E) A^0(g|E)= A(g|E), by (C1) for 0 A^0. For (C2), note that Ec(x)=0 1_E^c(x)=0 for every x∈Ex∈ E, and hence (Ec|E)=0 A( 1_E^c|E)=0. Thus, A is an FCA. The pointwise annihilator property is immediate from the definition of A, and property (R) follows from Proposition 3.11. ∎ A concrete instance of Proposition 3.13 is (f|E)=(maxx∈Ef(x))∏x∈Ef(x)>0, A(f|E)= ( _x∈ Ef(x) ) _x∈ E 1_\f(x)>0\, where E is finite. The additional factor enforces the pointwise annihilator. Point-separating collections The pointwise annihilator is sufficient for property (R), but it is not necessary in general. It becomes necessary whenever the collection is rich enough to separate single points inside the admissible sets. Definition 3.14. We say that the collection ℰ⊆ΣE is point-separating if, for every E∈ℰ0E∈E^0 and every x∈Ex∈ E, there exists B∈ℰ0B∈E^0 such that x∉Bx∉ B and E∩B=E∖x.E∩ B=E \x\. Thus, a collection ℰE is point-separating if every point of every nonempty set can be removed by intersecting it with another nonempty set. This property itself does not require ℰE to be closed under intersections. However, in the finite discrete setting, point-separation becomes very restrictive once closure under finite intersections and the condition X∈ℰX∈E are imposed. A collection ℰ⊆ΣE is said to be intersection-stable if it satisfies C,D∈ℰ⟹C∩D∈ℰ.C,D∈E C∩ D∈E. (E) Proposition 3.15. Let X=[n]X=[n], n⩾2n 2, and let ℰ⊆2XE 2^X be an intersection-stable and point-separating collection. If X∈ℰX∈E, then ℰ=2XE=2^X. Proof. We first show that X∖x∈ℰX \x\∈E for every x∈Xx∈ X. Fix x∈Xx∈ X. Applying point-separation to E=XE=X and x∈Xx∈ X, we obtain Bx∈ℰ0B_x∈E^0 such that x∉Bxx∉ B_x and X∩Bx=X∖xX∩ B_x=X \x\. Since X∩Bx=BxX∩ B_x=B_x, it follows that Bx=X∖x∈ℰB_x=X \x\∈E. Now let E⊆XE X. If E=XE=X, then E∈ℰE∈E. If E≠XE≠ X, then E=⋂x∈X∖E(X∖x).E= _x∈ X E (X \x\ ). The intersection is finite, and each set X∖xX \x\ belongs to ℰE. By (E), we obtain E∈ℰE∈E. Hence ℰ=2XE=2^X. ∎ Remark 3.16. The assumption X∈ℰX∈E cannot be omitted. For instance, if X=1,2,3X=\1,2,3\, then ℰ=∅,1,2,3,1,2,1,3,2,3E=\ ,\1\,\2\,\3\,\1,2\,\1,3\,\2,3\\ is intersection-stable and point-separating, but X∉ℰX∉E. Thus, ℰ≠2XE≠ 2^X. Proposition 3.17. Assume that the collection ℰ⊆ΣE is point-separating. Then an FCA =(⋅|E):E∈ℰ A=\ A(·|E):E∈E\ has property (R) if and only if 00 is a pointwise annihilator of A. Proof. The implication from the pointwise annihilator to (R) follows from Proposition 3.11. Conversely, assume that A has property (R). Let E∈ℰ0E∈E^0, let x0∈Ex_0∈ E, and let f∈f∈F be such that f(x0)=0f(x_0)=0. By point-separation, there exists B∈ℰ0B∈E^0 such that x0∉Bx_0∉ B and E∩B=E∖x0E∩ B=E \x_0\. In particular, E⊈BE B. Moreover, fBf 1_B and f coincide on E: outside B, the only point of E is x0x_0, and f(x0)=0f(x_0)=0. Hence, by Lemma 2.2, (f|E)=(fB|E). A(f|E)= A(f 1_B|E). Since E⊈BE B, property (R) gives (fB|E)=0 A(f 1_B|E)=0. Therefore (f|E)=0 A(f|E)=0, and so 00 is a pointwise annihilator of A. ∎ The role of the underlying collection If ℰE is not point-separating, property (R) may still hold without a pointwise annihilator. In such cases the structure of ℰE itself may force the required reduction. Proposition 3.18. Assume that the nonempty members of ℰE are pairwise disjoint, that is, C∩D=∅C∩ D= for all C,D∈ℰ0C,D∈E^0 with C≠DC≠ D. Then every FCA =(⋅|E):E∈ℰ A=\ A(·|E):E∈E\ has property (R). Proof. Let B,E∈ℰ0B,E∈E^0 and suppose that E⊈BE B. Since the nonempty members of ℰE are pairwise disjoint, this implies E∩B=∅E∩ B= . Therefore fB=0f 1_B=0 on E. By Lemma 2.2 and (C2), (fB|E)=(Ec|E)=0 A(f 1_B|E)= A( 1_E^c|E)=0. Thus, A has property (R). ∎ More interesting examples occur when the admissible sets are generated by blocks. In the following construction the blocks are indexed, and this indexing determines the order in which the block values are inserted into the outer evaluation. Thus no symmetry of the outer evaluation is required. If one wants the construction to be independent of the labelling of the blocks, then symmetry of the outer evaluation should be imposed. Proposition 3.19. Let X=[n]X=[n] and let =D1,…,DkP=\D_1,…,D_k\ be a partition of X. Put ℰ=⋃j∈JDj:J⊆[k].E= \ _j∈ JD_j:J [k] \. Let ~=~(⋅|Dj):j∈[k] A=\ A(·|D_j):j∈[k]\ be a family of conditional aggregation rules on the blocks. Let Γm:[0,∞]m→[0,∞] _m [0,∞]^m→[0,∞], m=1,…,km=1,…,k, be nondecreasing in every coordinate and have 00 as an annihilator, that is, if zℓ=0z_ =0 for some ℓ , then Γm(z1,…,zm)=0. _m(z_1,…,z_m)=0. For E∈ℰ0E∈E^0, write J(E)=j∈[k]:Dj⊆E=j1<⋯<jm,J(E)=\j∈[k]:D_j E\=\j_1<·s<j_m\, and define (f|E)=Γm(~(f|Dj1),…,~(f|Djm)). A(f|E)= _m ( A(f|D_j_1),…, A(f|D_j_m) ). Then =(⋅|E):E∈ℰ A=\ A(·|E):E∈E\ is an FCA and has property (R). Proof. Let E∈ℰ0E∈E^0 and write J(E)=j1<⋯<jmJ(E)=\j_1<·s<j_m\. Let f,g∈f,g∈F and suppose that f⩽gf g on E. Then f⩽gf g on every block Djr⊆ED_j_r E. Hence ~(f|Djr)⩽~(g|Djr),r=1,…,m, A(f|D_j_r) A(g|D_j_r), r=1,…,m, by (C1) for ~ A. Since Γm _m is nondecreasing in every coordinate, we obtain (f|E)⩽(g|E). A(f|E) A(g|E). Thus, (C1) holds. For (C2), take f=Ecf= 1_E^c. If Djr⊆ED_j_r E, then Ec=0 1_E^c=0 on DjrD_j_r. By Lemma 2.2 and (C2) for ~ A, we have ~(Ec|Djr)=0,r=1,…,m. A( 1_E^c|D_j_r)=0, r=1,…,m. Thus, all inputs of Γm _m are equal to 00, and the annihilator property gives (Ec|E)=0. A( 1_E^c|E)=0. Hence, A is an FCA. It remains to prove (R). Let B,E∈ℰ0B,E∈E^0 and assume that E⊈BE B. Since both B and E are unions of blocks, there is a block Dj0D_j_0 such that Dj0⊆ED_j_0 E and Dj0∩B=∅D_j_0∩ B= . Consequently, fB=0f 1_B=0 on Dj0D_j_0. By Lemma 2.2 and (C2) for ~ A, we get ~(fB|Dj0)=0. A(f 1_B|D_j_0)=0. Thus, one of the inputs entering the outer evaluation is equal to 00. Since 00 is an annihilator of Γm _m, we obtain (fB|E)=0. A(f 1_B|E)=0. Therefore, A has property (R). ∎ Example 3.20. Let X=[n]X=[n] and let =D1,…,DkP=\D_1,…,D_k\ be a partition of X. Put ℰ=⋃j∈JDj:J⊆[k],E= \ _j∈ JD_j:J [k] \, and for E∈ℰ0E∈E^0 define block(f|E)=min1|Dj|∑i∈Djf(i):Dj⊆E. A block(f|E)= \ 1|D_j| _i∈ D_jf(i):D_j E \. Then block A block is an FCA and has property (R). Moreover, it is nonincreasing with respect to sets: if C⊆DC D, then the minimum defining block(f|D) A block(f|D) is taken over at least as many block averages as the minimum defining block(f|C) A block(f|C). If at least one block DjD_j contains more than one element, then 00 is not a pointwise annihilator of block A block. Indeed, take E=DjE=D_j, choose x0∈Djx_0∈ D_j, and define f(x0)=0f(x_0)=0 and f(x)=1f(x)=1 for every x∈Dj∖x0x∈ D_j \x_0\. Then block(f|E)=|Dj|−1|Dj|>0. A block(f|E)= |D_j|-1|D_j|>0. Thus, property (R) may hold even when the pointwise annihilator fails. The annihilator assumption on the outer evaluation is essential in this mechanism. Example 3.21. Let X=1,2,3,4X=\1,2,3,4\, D1=1,2D_1=\1,2\, D2=3,4D_2=\3,4\, and let ℰ=∅,D1,D2,XE=\ ,D_1,D_2,X\. Define max-block(f|E)=max1|Dj|∑i∈Djf(i):Dj⊆E. A -block(f|E)= \ 1|D_j| _i∈ D_jf(i):D_j E \. This is an FCA, but it does not have property (R). Indeed, take B=D1,E=X,f=(2,3,4,5).B=D_1, E=X, f=(2,3,4,5). Then fB=(2,3,0,0)f 1_B=(2,3,0,0), and hence 1|D1|∑i∈D1(fB)(i)=52,1|D2|∑i∈D2(fB)(i)=0. 1|D_1| _i∈ D_1(f 1_B)(i)= 52, 1|D_2| _i∈ D_2(f 1_B)(i)=0. Therefore, max-block(fB|X)=52>0, A -block(f 1_B|X)= 52>0, although X⊈BX B. Thus, property (R) fails. Almost-everywhere versions In measure-theoretic applications, exact pointwise annihilation is often too strong. A zero on a null set should usually be ignored. This leads to an almost-everywhere version of property (R). Remark 3.22. Let (X,Σ,λ)(X, ,λ) be a Lebesgue measure space and let essinf(f|E)=essinfx∈Ef(x). A essinf(f|E)= *ess\,inf_x∈ Ef(x). Then essinf A essinf is nonincreasing with respect to sets. It does not have a pointwise annihilator, because values on λ-null sets are ignored. However, for every u>0u>0, essinf(fB|E)⩾u⟹λ(E∖B)=0. A essinf(f 1_B|E) u λ(E B)=0. Indeed, if λ(E∖B)>0λ(E B)>0, then fB=0f 1_B=0 on a subset of E of positive λ-measure, and therefore essinf(fB|E)=0. A essinf(f 1_B|E)=0. Thus, the exact reduction property is replaced by the almost-everywhere reduction essinf(fB|E)⩾u>0⟹E⊆Bmodulo λ-null sets. A essinf(f 1_B|E) u>0 E B λ-null sets. Consequently, the order-theoretic localization theorem has a natural measure-theoretic analogue whenever the underlying sets and monotone measures are considered modulo null sets, or whenever the relevant set functions are insensitive to null modifications. This is in line with the treatment of null-additive set functions in non-additive measure theory [9]. Remark 3.23. The reduction property may arise from different mechanisms. It is implied by the pointwise annihilator, it is equivalent to the pointwise annihilator for point-separating collections, it is automatic for pairwise disjoint collections, and it can be produced by block-generated structures through an outer evaluation with 00 as an annihilator. In measure-theoretic settings, exact reduction is naturally replaced by reduction modulo null sets, as in the case of the essential infimum. 4. The localization theorem We now return to generalized level measures and show how the two structural conditions introduced above interact. The aim is to compare two ways of localizing the level analysis to a fixed set B: one may either first mask the signal, replacing f by fBf 1_B, or one may keep the original signal f and localize the admissible sets in the supremum by intersection with B. The localization theorem below identifies conditions under which these two procedures are equivalent: monotonicity with respect to sets controls the passage from larger admissible sets to their localized parts, while the reduction property ensures that the masked signal cannot produce positive level values on sets outside B. For an FCA A and a monotone measure μ on ℰE, define the generalized level measure Λμ,(f,u):=supμ(E):(f|E)⩾u,E∈ℰ,u⩾0, _μ, A(f,u):= \μ(E): A(f|E) u,\ E∈E\, u 0, (4.1) introduced in [2]. For B∈ℰ0B∈E^0 define its localized version by Λμ,(f,u;B):=supμ(E∩B):(f|E)⩾u,E∈ℰ,u⩾0. _μ, A(f,u;B):= \μ(E∩ B): A(f|E) u,\ E∈E\, u 0. (4.2) Theorem 4.1. Let ℰE be an intersection-stable collection and let A be an FCA on ℰE. The following conditions are equivalent: (i) for every f∈f∈F, every B∈ℰ0B∈E^0, every u>0u>0 and every monotone measure μ on ℰE, one has Λμ,(fB,u)=Λμ,(f,u,B); _μ, A(f 1_B,u)= _μ, A(f,u;B); (4.3) (i) the FCA A is nonincreasing with respect to sets and has property (R). Proof. (i)⇒ (i) Fix f∈f∈F, B∈ℰ0B∈E^0, u>0u>0 and a monotone measure μ on ℰE. We first reduce the left-hand side. If E∈ℰE∈E satisfies (fB|E)⩾u A(f 1_B|E) u, then either E=∅E= , in which case E⊆BE B, or E∈ℰ0E∈E^0, in which case Lemma 3.9 gives E⊆BE B. By Lemma 2.2, (fB|E)=(f|E) A(f 1_B|E)= A(f|E) whenever E⊆BE B. Hence, Λμ,(fB,u)=supμ(E):E⊆B,(f|E)⩾u,E∈ℰ. _μ, A(f 1_B,u)= \μ(E):E B,\ A(f|E) u,\ E∈E\. (4.4) Let E∈ℰE∈E satisfy (f|E)⩾u A(f|E) u and put F:=E∩BF:=E∩ B. Since ℰE is intersection-stable, F∈ℰF∈E. Since F⊆EF E, monotonicity with respect to sets gives (f|E)⩽(f|F), A(f|E) A(f|F), where the case F=∅F= is covered by the convention (⋅|∅)=∞ A(·| )=∞. Hence (f|F)⩾u A(f|F) u. Since F⊆BF B, the set F contributes to the reduced supremum (4.4), and therefore μ(E∩B)=μ(F)⩽Λμ,(fB,u).μ(E∩ B)=μ(F) _μ, A(f 1_B,u). Taking the supremum over all E∈ℰE∈E satisfying (f|E)⩾u A(f|E) u gives Λμ,(f,u,B)⩽Λμ,(fB,u). _μ, A(f,u;B) _μ, A(f 1_B,u). Conversely, let F∈ℰF∈E contribute to the reduced supremum (4.4). Choose E:=FE:=F. Then E∩B=FE∩ B=F and (f|E)⩾u A(f|E) u, so μ(F)⩽Λμ,(f,u,B).μ(F) _μ, A(f,u;B). Taking the supremum in (4.4) gives Λμ,(fB,u)⩽Λμ,(f,u,B). _μ, A(f 1_B,u) _μ, A(f,u;B). Thus, Λμ,(fB,u)=Λμ,(f,u,B) _μ, A(f 1_B,u)= _μ, A(f,u;B). (i)⇒ (i) First we prove property (R). Suppose that, for some f∈f∈F, B∈ℰ0B∈E^0 and E0∈ℰ0E_0∈E^0, we have (fB|E0)>0andE0⊈B. A(f 1_B|E_0)>0 E_0 B. Choose u>0u>0 such that (fB|E0)⩾u A(f 1_B|E_0) u; if the value is infinite, take any u>0u>0. Define μB(F)=0,F⊆B,1,F⊈B,F∈ℰ. _B(F)= cases0,&F B,\\ 1,&F B, cases F∈E. This is a monotone measure on ℰE. Since E0E_0 contributes to the left-hand side of (4.3) and μB(E0)=1 _B(E_0)=1, the left-hand supremum is equal to 11. On the right-hand side, every set has the form F∩B⊆BF∩ B B, hence μB(F∩B)=0 _B(F∩ B)=0. The right-hand supremum is therefore 00, a contradiction. Thus, no such E0E_0 exists, and property (R) holds. It remains to prove monotonicity. Let C,D∈ℰ0C,D∈E^0 with C⊆DC D, and fix f∈f∈F. We show that (f|D)⩽(f|C) A(f|D) A(f|C). If (f|D)=0 A(f|D)=0, there is nothing to prove. If 0<(f|D)<∞0< A(f|D)<∞, put u:=(f|D)u:= A(f|D). If (f|D)=∞ A(f|D)=∞, let u>0u>0 be arbitrary; the following argument then gives (f|C)⩾u A(f|C) u for every u>0u>0, which implies (f|C)=∞ A(f|C)=∞. For such a finite positive u, or for an arbitrary u>0u>0 in the infinite case, define μC(F):=C⊆F,F∈ℰ. _C(F):= 1_\C F\, F∈E. This is a monotone measure on ℰE because C≠∅C≠ . Apply (4.3) with B=CB=C. Since D contributes to the right-hand side, (f|D)⩾u A(f|D) u, and D∩C=CD∩ C=C, the right-hand supremum is equal to 11. Hence the left-hand supremum is equal to 11. As μC _C is 0,1\0,1\-valued, there exists E∈ℰ0E∈E^0 such that (fC|E)⩾u A(f 1_C|E) u and C⊆E.C E. By Lemma 3.9, applied with B=CB=C, we get E⊆CE C. Therefore E=CE=C. By Lemma 2.2, (f|C)=(fC|C)⩾u. A(f|C)= A(f 1_C|C) u. In the finite positive case this yields (f|C)⩾(f|D) A(f|C) A(f|D). In the infinite case it holds for every u>0u>0, hence (f|C)=∞ A(f|C)=∞. Thus, A is nonincreasing with respect to sets. ∎ In point-separating intersection-stable collections, the reduction property admits a simpler interpretation. In this case, property (R) is equivalent to the existence of a pointwise annihilator. Consequently, once monotonicity with respect to sets is assumed, the localization identity can be characterized directly in terms of the pointwise behaviour of the conditional aggregation rules. Corollary 4.2. Assume that ℰE is intersection-stable and point-separating, and that A is nonincreasing with respect to sets. Then the following assertions are equivalent: (i) 00 is a pointwise annihilator of A; (i) A has property (R); (i) the localization identity (4.3) holds for every f∈f∈F, every B∈ℰ0B∈E^0, every u>0u>0, and every monotone measure μ on ℰE. Proof. The equivalence of (i) and (i) follows from Proposition 3.17. Since A is assumed to be nonincreasing with respect to sets, the equivalence of (i) and (i) follows from Theorem 4.1. ∎ Remark 4.3. The restriction u>0u>0 in the localization identity is essential. At level u=0u=0, all admissible sets satisfy the level condition because the local evaluation values are nonnegative. Thus, masking by B 1_B does not exclude sets outside B at the zero level, and the reduction property cannot force the desired localization. 5. Localization in parameterized systems We now pass from a single collection and a single local evaluation system to a parameterized family of such objects, cf. [3]. The parameter may represent time, scale, a clinical regime, a confidence level, or any other external variable which changes the observable sets, the underlying monotone measure, or the local evaluation mechanism. The important point is that this external parameter must be kept separate from the level variable used in the generalized level measure. Let T be a non-empty parameter set. A parameterized system is a family of triples t=(ℰt,μt,t),t∈T, S_t=(E_t, _t, A_t), t∈ T, where each ℰtE_t is an intersection-stable collection, each μt _t is a monotone measure on ℰtE_t, and t=t(⋅|E):E∈ℰt A_t=\ A_t(·|E):E∈E_t\ is an FCA on ℰtE_t. We write ℰ=(ℰt)t∈T,=(μt)t∈T,∙|∙=(t)t∈T. E=(E_t)_t∈ T, μ=( _t)_t∈ T, A_ | =( A_t)_t∈ T. As before, the convention t(⋅|∅)=∞ A_t(·| )=∞ is used when ∅∈ℰt ∈E_t. For f∈f∈F, t∈Tt∈ T and u⩾0u 0, define the parameterized generalized level surface by Λ(f;t,u):=supμt(E):t(f|E)⩾u,E∈ℰt. _ S(f;t,u):= \ _t(E): A_t(f|E) u,\ E∈E_t\. (5.1) For B∈ℰt0B∈E_t^0 define the corresponding localized surface by Λ(f;t,u;B):=supμt(E∩B):t(f|E)⩾u,E∈ℰt. _ S(f;t,u;B):= \ _t(E∩ B): A_t(f|E) u,\ E∈E_t\. (5.2) The level surface may be defined also at u=0u=0. However, the localization identity below is stated for u>0u>0, since the reduction property eliminates sets outside B only at positive levels. Definition 5.1. Fix t∈Tt∈ T. We say that the parameterized FCA ∙|∙ A_ | (i) is nonincreasing with respect to sets at t if, for all C,D∈ℰtC,D∈E_t with C⊆DC D and f∈f∈F, t(f|D)⩽t(f|C); A_t(f|D) A_t(f|C); (i) has the reduction property at t, denoted (Rt), if, for all f∈f∈F and B,E∈ℰt0B,E∈E_t^0, E⊈B⟹t(fB|E)=0.E B A_t(f 1_B|E)=0. Thus, the two structural conditions from the non-parameterized setting are required only after the parameter has been fixed. No compatibility between different parameter values is needed for the pointwise localization statement. Theorem 5.2. Let t∈Tt∈ T be fixed. The following conditions are equivalent: (i) for every f∈f∈F, every B∈ℰt0B∈E_t^0, every u>0u>0 and every monotone measure νt _t on ℰtE_t, supνt(E):t(fB|E)⩾u,E∈ℰt \ _t(E): A_t(f 1_B|E) u,E∈E_t\ = = (5.3) supνt(E∩B):t(f|E)⩾u,E∈ℰt. \ _t(E∩ B): A_t(f|E) u,E∈E_t\. (i) ∙|∙ A_ | is nonincreasing with respect to sets at t and has property (Rt). Consequently, if condition (i) holds for every t∈Tt∈ T, then, for every choice of localizing sets Bt∈ℰt0B_t∈E_t^0, Λ(fBt,t,u)=Λ(f,t,u,Bt) _ S(f 1_B_t;t,u)= _ S(f;t,u;B_t) (5.4) holds for every t∈Tt∈ T, every u>0u>0 and every f∈f∈F. Proof. For fixed t∈Tt∈ T, apply Theorem 4.1 to the intersection-stable collection ℰtE_t and the FCA t A_t. This gives the equivalence of (i) and (i). The final assertion follows by taking νt=μt _t= _t and B=BtB=B_t. ∎ Remark 5.3. The parameter t and the level u play different roles. The parameter t selects the underlying collection, the monotone measure and the local evaluation system, while u is the threshold at which the level analysis is performed. If the theory were formulated only on the diagonal u=tu=t, then equality of the two suprema would test localization only at that particular level. It would not recover the full monotonicity condition t(f|D)⩽t(f|C) A_t(f|D) A_t(f|C) for arbitrary values of t(f|D) A_t(f|D), because the proof of monotonicity in Theorem 4.1 tests the localization identity at the level u=t(f|D)u= A_t(f|D) whenever this value is finite and positive. This level need not coincide with the parameter t. The two-variable formulation (t,u)(t,u) is therefore the natural one; the diagonal case is only a specialization. Context-dependent knowledge evaluation The parameterized framework can be interpreted as a formal model for context-dependent evaluation in knowledge-based systems, in the broad sense of structured symbolic or evidence-based information processing; see, e.g., [4, 10]. We use this interpretation in a structural sense. The aim is not to model the full inference mechanism of an arbitrary knowledge-based system, but rather the common situation in which a system evaluates a query, hypothesis, decision alternative or diagnostic state by selecting admissible collections of information units and aggregating their local degrees of support. Let X denote the universe of elementary information units. Depending on the system, elements of X may represent facts, rules, symptoms, cases, documents, expert statements, criteria or elementary pieces of evidence. A nonnegative function f∈f∈F assigns to each unit a score, such as relevance to a query, reliability, compatibility with a hypothesis, degree of activation, confidence, or local support value. A parameter t∈Tt∈ T may represent a query, decision context, resolution level, user profile, time instant or regime of admissibility. The collection ℰtE_t represents the admissible knowledge contexts available at parameter value t. These contexts are not arbitrary subsets of X. They may be constrained by logical consistency, semantic compatibility, observability, modular structure, expert-source restrictions, scenario constraints, or the architecture of a rule base. Thus, a set E∈ℰtE∈E_t should be read as an information context which is admissible for the current evaluation. The monotone measure μt _t assigns an informational weight to such a context. This weight may represent support, coverage, importance, confidence, relevance, or a non-additive capacity of the information carried by E. The conditional aggregation operator t(⋅|E) A_t(·|E) evaluates the quality of the score function inside the context E. Hence the level condition t(f|E)⩾u A_t(f|E) u means that the context E passes the local acceptability test at level u. The generalized level measure Λ(f;t,u)=supμt(E):t(f|E)⩾u,E∈ℰt _ S(f;t,u)= \ _t(E): A_t(f|E) u,\ E∈E_t\ therefore has a direct selection meaning: it is the maximal informational weight of an admissible context whose aggregated support reaches the required level. In this sense, the generalized level measure is the value of a level-constrained evidence selection problem over admissible knowledge contexts. Localization enters when the current task makes only a part B of the knowledge base relevant. The set B may be determined by a query filter, a clinical regime, a user profile, a semantic restriction, an active rule module, a trusted source set, or a context window. There are then two natural ways of performing the evaluation. One may first restrict the score function to the relevant part of the knowledge base, replacing f by fBf 1_B, and then solve the level-constrained selection problem. Alternatively, one may evaluate contexts in the original system and localize only their informational weights by replacing E with E∩BE∩ B. The localization identity Λ(fB,t,u)=Λ(f,t,u,B) _ S(f 1_B;t,u)= _ S(f;t,u;B) is therefore not merely a formal equality. It expresses a consistency requirement for context filtering: pre-filtering the knowledge base and post-filtering the selected evidence contexts lead to the same level-based evaluation. In a knowledge-based system, this means that the final score is invariant under the order in which contextual restriction and level-based evidence selection are performed. Theorem 5.2 shows that this consistency is governed by two structural mechanisms. Monotonicity with respect to sets says that enlarging an admissible context cannot improve the local lower-level evaluation. This is natural for conjunctive, infimum-type, worst-case and robust support rules, where adding further pieces of information may introduce additional constraints. The reduction property says that after the score has been masked outside B, no context which is not contained in B can still pass a positive level test. In knowledge-based terminology, information units outside the active context cannot contribute to a positive localized support value. This distinction is important. The reduction property need not be a pointwise property of individual information units. In modular knowledge bases, it may be produced by the structure of admissible contexts. Suppose, for example, that the knowledge base is decomposed into modules t=Dt,1,…,Dt,kt,P_t=\D_t,1,…,D_t,k_t\, where the modules represent rule groups, expert sources, symptom clusters, document classes, scenario blocks, or other semantically coherent information units. If ℰtE_t consists of unions of such modules, one may first evaluate the score inside each module and then combine the module scores by an outer aggregation with 00 as an annihilator. If a module lies entirely outside the active context B, masking by B 1_B gives a zero module score, and the outer aggregation propagates this zero to the whole context. Thus, localization consistency may be enforced at the modular level even when a single zero value inside a module does not annihilate the local module evaluation. The theorem can therefore be read as a structural criterion for reliable context-dependent knowledge evaluation. It identifies when a generalized level-based score is stable under query restriction, removal of irrelevant evidence, or activation of a context window. Such stability is essential in systems where admissible information states are constrained by semantic, logical, modular, or observational structure, and where the evaluation of a hypothesis or decision alternative is based on non-additive aggregation of local support. A localization-constrained selection problem The preceding interpretation leads to a simple selection problem. Let U be a nonempty set of hypotheses, configurations or decision alternatives, and suppose that each a∈a is represented by a structured support signal fa∈f_a∈F. For a fixed parameter t, the value Vt(a,u):=Λ(fa,t,u)V_t(a,u):= _ S(f_a;t,u) can be read as the largest informational weight of an admissible context on which alternative a reaches the aggregated support level u. If only a localized part Bt∈ℰt0B_t∈E_t^0 of the knowledge structure is relevant, the corresponding localized score is VtB(a,u):=Λ(fa,t,u,Bt).V_t^B(a,u):= _ S(f_a;t,u;B_t). The localization theorem gives structural conditions under which the two natural procedures, masking the support signal first or localizing the admissible contexts in the evaluation, lead to the same value. This matters when irrelevant, unreliable, delayed or contextually excluded information units should not affect the final comparison of hypotheses or alternatives. Fixing a positive level u and a class ℬt⊆ℰt0B_t E_t^0 of admissible localization contexts, one obtains maxa∈,B∈ℬtΛ(fa,t,u,B). _a ,\,B _t _ S(f_a;t,u;B). Under the assumptions of Theorem 5.2, this problem can equivalently be written as maxa∈,B∈ℬtΛ(faB,t,u), _a ,\,B _t _ S(f_a 1_B;t,u), so that the selection can be performed either on localized admissible contexts or on masked support signals. The equality is not merely formal: without monotonicity with respect to sets or without the reduction property, the two formulations may rank alternatives differently. Thus, the localization theorem provides a consistency principle for level-based knowledge evaluation with context-dependent feasible information. Local and global localization sets The parameterized localization theorem is local in the parameter. At each parameter value, one may choose a different localizing set Bt∈ℰt0B_t∈E_t^0. If ∙|∙ A_ | is nonincreasing with respect to sets at t and has property (Rt) for every t∈Tt∈ T, then Λ(fBt,t,u)=Λ(f,t,u,Bt),Bt∈ℰt0,u>0. _ S(f 1_B_t;t,u)= _ S(f;t,u;B_t), B_t∈E_t^0, u>0. (5.5) If one wants to localize by one fixed set B for all parameter values, then B must belong to all the corresponding collections. Hence the natural domain of global localization is ℰT0:=⋂t∈Tℰt0.E_T^0:= _t∈ TE_t^0. If B∈ℰT0B∈E_T^0, then (5.4) gives Λ(fB,t,u)=Λ(f,t,u,B),t∈T,u>0. _ S(f 1_B;t,u)= _ S(f;t,u;B), t∈ T, u>0. (5.6) If ℰT0=∅E_T^0= , then no non-trivial global localization set exists. This is not a defect of the theorem but a structural feature of the parameterized information structures. Interpretation in stochastic processes The framework also admits a natural stochastic interpretation. Let (Ω,ℱ,(ℱt)t⩾0,)( ,F,(F_t)_t 0, P) be a filtered probability space [7] and let Y=(Yt)t⩾0Y=(Y_t)_t 0 be a non-negative adapted process. For each time t, the sets in ℱtF_t represent events observable at time t. Thus one may take, for instance, X=Ω,ℰt⊆ℱt,μt=|ℰt.X= , E_t F_t, _t= P|_E_t. The choice ℰt=ℱtE_t=F_t corresponds to the full information available at time t, while smaller collections ℰtE_t may represent restricted observable regimes, scenario classes, or coarser information structures. A first canonical choice is the essential-infimum conditional aggregation rule tessinf(Z|E)=essinfω∈EZ(ω),E∈ℱt. A essinf_t(Z|E)= *ess\,inf_ω∈ EZ(ω), E∈F_t. With the usual convention modulo null sets, the generalized level measure reduces to the ordinary level measure whenever Z is ℱtF_t-measurable: sup(E):essinfEZ⩾u,E∈ℱt=(Z⩾u). \ P(E): *ess\,inf_EZ u,E∈F_t\= P(Z u). (5.7) Indeed, the condition essinfEZ⩾u *ess\,inf_EZ u is equivalent to E⊆Z⩾uE \Z u\ up to a P-null set; since Z⩾u∈ℱt\Z u\∈F_t, the largest observable choice is the event Z⩾u\Z u\ itself. Taking Z=YtZ=Y_t gives sup(E):essinfEYt⩾u,E∈ℱt=(Yt⩾u). \ P(E): *ess\,inf_EY_t u,E∈F_t\= P(Y_t u). If B∈ℱtB∈F_t, then for u>0u>0, sup(E):essinfE(YtB)⩾u,E∈ℱt=(Yt⩾u∩B). \ P(E): *ess\,inf_E(Y_t 1_B) u,E∈F_t\= P(\Y_t u\∩ B). (5.8) Thus, the localization theorem generalizes the elementary fact that the probability of a level event after restriction to an event B is obtained by intersecting the level set with B. A less extreme interpretation is obtained from conditional means over information atoms. Suppose that, for a fixed t, the collection ℰtE_t is generated by a finite ℱtF_t-measurable partition t=Dt,1,…,Dt,ktP_t=\D_t,1,…,D_t,k_t\ with (Dt,j)>0 P(D_t,j)>0, namely ℰt=⋃j∈JDt,j:J⊆[kt].E_t= \ _j∈ JD_t,j:J [k_t] \. For a non-negative integrable random variable Z, define tatom(Z|E):=min1(Dt,j)∫Dt,jZ:Dt,j⊆E,E∈ℰt0. A atom_t(Z|E):= \ 1 P(D_t,j) _D_t,jZ\,d P:D_t,j E \, E∈E_t^0. The value on the empty set is understood according to the standing convention. This operator evaluates an event E by the worst conditional mean over the information atoms contained in E. Hence, the condition tatom(Z|E)⩾u A atom_t(Z|E) u means that every information atom included in E has conditional mean at least u. The operator tatom A atom_t is nonincreasing with respect to sets, since enlarging E adds further atoms to the minimum. It also has the reduction property. Indeed, if B,E∈ℰt0B,E∈E_t^0 and E⊈BE B, then, because both B and E are unions of atoms, there exists an atom Dt,j⊆ED_t,j E such that Dt,j∩B=∅D_t,j∩ B= . On this atom ZB=0Z 1_B=0, and therefore the corresponding conditional mean is zero. Consequently, tatom(ZB|E)=0. A atom_t(Z 1_B|E)=0. Thus, tatom A atom_t satisfies the structural assumptions of the localization theorem. In this case the generalized level measure has a transparent form. Put mt,j(Z):=1(Dt,j)∫Dt,jZ.m_t,j(Z):= 1 P(D_t,j) _D_t,jZ\,d P. Then tatom(Z|E)⩾u A atom_t(Z|E) u holds precisely when all atoms contained in E satisfy mt,j(Z)⩾um_t,j(Z) u. Hence sup(E):tatom(Z|E)⩾u,E∈ℰt=(⋃j:mt,j(Z)⩾uDt,j). \ P(E): A atom_t(Z|E) u,E∈E_t\= P ( _\j:m_t,j(Z) u\D_t,j ). (5.9) Equivalently, this is the probability of a level event for the conditional mean of Z with respect to the finite information structure generated by tP_t. If Z=YtZ=Y_t is measurable with respect to this finite information structure, then the atomwise conditional mean coincides with YtY_t on each atom, and (5.9) reduces to the ordinary level-set probability of YtY_t. If Z=YsZ=Y_s for s>ts>t, then the same expression gives the level-set probability of the forecast of the future value YsY_s based on the finite information structure tP_t. This example shows that the abstract conditional aggregation rule need not be interpreted only as an essential infimum. It may also represent a robust evaluation of conditional averages over information states. The resulting generalized level measure then describes the largest observable event on which the process satisfies a prescribed lower bound in this information-dependent sense. Conclusion We have studied a localization principle for generalized level measures generated by conditional aggregation operators. The motivation comes from knowledge-based, fuzzy and non-additive systems in which a structured score is evaluated through admissible knowledge contexts and local aggregation rules. The main result shows that the equality between the generalized level measure of the masked signal fBf 1_B and the localized generalized level measure of f is governed by two independent structural requirements. The first one is monotonicity with respect to sets, which controls the passage from a set to its localized part. The second one is the reduction property, which ensures that after multiplication by B 1_B no nonempty set outside B can still produce a positive level value. Several mechanisms leading to the reduction property were identified. The pointwise annihilator is the simplest sufficient condition, but it is not necessary in general. In particular, block-generated collections show that localization may hold even when zeros do not annihilate the conditional aggregation rule at the point level. In such cases the reduction is produced by the structure of the collection together with an outer aggregation mechanism which propagates zero values from whole blocks. Point-separating collections represent the opposite situation: there the reduction property is equivalent to the existence of a pointwise annihilator. The parameterized version of the theory shows that the external parameter and the level threshold must be kept separate. A parameter t may determine the admissible collection, the monotone measure, and the conditional aggregation rule, whereas the level u determines the threshold at which the generalized level measure is evaluated. This leads naturally to a two-variable generalized level surface Λ(f;t,u)=supμt(E):t(f|E)⩾u,E∈ℰt. _ S(f;t,u)= \ _t(E): A_t(f|E) u,\ E∈E_t\. The diagonal choice u=tu=t may be useful in applications, but it is only a specialization of the two-variable formulation. This distinction is essential, since the proof of monotonicity requires testing the localization identity at levels determined by the values of the conditional aggregation rules themselves, not only at prescribed diagonal levels. When T⊆(0,∞)T (0,∞), the diagonal Sf(t):=Λ(f;t,t)=supμt(E):t(f|E)⩾t,E∈ℰtS_f(t):= _ S(f;t,t)= \ _t(E): A_t(f|E) t,\ E∈E_t\ may be interpreted as a level-dependent evaluation curve. Under the structural assumptions of Theorem 5.2, one obtains SfB(t)=SfB(t),S_f 1_B(t)=S_f^B(t), where SfB(t):=Λ(f;t,t;B)=supμt(E∩B):t(f|E)⩾t,E∈ℰt,S_f^B(t):= _ S(f;t,t;B)= \ _t(E∩ B): A_t(f|E) t,\ E∈E_t\, for all positive levels for which B belongs to the corresponding collection. This is the level-dependent analogue of the classical identity μ(fB⩾t)=μ(f⩾t∩B).μ(\f 1_B t\)=μ(\f t\∩ B). The knowledge-based reading shows that the same theorem supplies a consistency principle for localized evaluation under structured uncertainty. When hypotheses, configurations or alternatives are represented by structured support signals, the theorem identifies when comparing masked signals is equivalent to comparing localized admissible evidence contexts. In particular, localization-constrained selection problems can be formulated either through localized contexts or through masked input signals, and the two formulations coincide precisely under the structural assumptions isolated in the paper. The diagonal curves also provide a natural input for Choquet–Stieltjes-type integral functionals. If φ:[0,∞)→[0,∞) [0,∞)→[0,∞) is nondecreasing and mφm_ is the associated Stieltjes measure, one may formally consider ,φ(f):=∫(0,∞)Sf(t)dmφ(t),ChS_ S, (f):= _(0,∞)S_f(t)\,\,dm_ (t), whenever the integral is well-defined. The localization identity then yields the representation ,φ(fB)=∫(0,∞)SfB(t)dmφ(t),ChS_ S, (f 1_B)= _(0,∞)S_f^B(t)\,\,dm_ (t), provided that B is available at the relevant levels and the integrability assumptions are satisfied. Thus, the localization theorem supplies a structural basis not only for generalized level analysis, but also for level-dependent non-additive integral scores built from localized evaluation information. The framework also admits a stochastic interpretation. For a filtered probability space and an adapted nonnegative process, sets in the corresponding collections may be interpreted as events observable at a given time, while the conditional aggregation rule may encode an essential lower bound, a robust conditional mean over information blocks, or another local performance criterion. These observations suggest several directions for further work. One may study additional classes of conditional aggregation rules satisfying the two structural conditions, develop almost-sure versions of the localization theorem for measure-theoretic settings, or impose regularity assumptions on the parameterized system in order to obtain measurable and integrable generalized level curves. Such developments would connect the present order-theoretic localization principle with knowledge-based evaluation, fuzzy and non-additive systems, non-additive integration, stochastic-process models, and applications in structured time-dependent data. Acknowledgments This work was supported by the Slovak Research and Development Agency under contract No. APVV-21-0468. The second author also acknowledges the support of the internal scientific grant vvgs-2026-3897. Disclosure statement No potential conflict of interest was reported by the authors. Data availability statement No datasets were generated or analyzed during the current study. References [1] Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Management Science 17(4) (1970), B141–B164. [2] Boczek, M., Halčinová, L., Hutník, O., Kaluszka, M.: Novel survival functions based on conditional aggregation operators. Inf. Sci. 580 (2021), 705–719. [3] Boczek, M., Hutník, O., Kaluszka, M., Kleinová, M.: Generalized level measure based on a family of conditional aggregation operators. Fuzzy Sets and Systems 457 (2023), 180–196. [4] Brachman, R.J., Levesque, H.J.: Knowledge Representation and Reasoning. Morgan Kaufmann, 2004. [5] Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Springer, 2016. [6] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, 2009. [7] Kallenberg, O.: Foundations of Modern Probability, Third Edition. Springer, 2021. [8] Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Springer, 2000. [9] Pap, E.: Null-Additive Set Functions. Kluwer-Ister Science, 1995. [10] van Harmelen, F., Lifschitz, V., Porter, B. (eds.): Handbook of Knowledge Representation. Elsevier, 2008. [11] Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, 2009. [12] Zimmermann, H.-J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1(1) (1978), 45–55.